x Global Lorentzian Geometry Beem

bout the first edition . . . ... a definitive source-book. "

-General Relativity and Gravitation

bout the second edition . . .

his fully revised and updated Second Edition of an incomparable referenceltext 5dges the gap between modem differential geometry and the mathematical ~ysics of general relativity by providing an invariant treatment of Lorentzian 2ometry-reflecting the more complete understanding of Lorentzian geometry zhieved since the publication of the previous edition.

arefully comparing and contrasting Lorentzian geometry with Riemannian geo- ietry throughout, Global Lorentzian Geometry, Second Edition offers a compre- znsive treatment of the space-time distance function not available in other books ... :cent results on the general instability in the space of Lorentzian metrics for a wen manifold of both geodesic completeness and geodesic incompleteness ... new laterial on geodesic connectibili ty... a more in-depth explication of the behavior I the sectional curvature function in a neighborhood of a degenerate two-plane .and more.

bout the authors . . . 3HN K. BEEM is a Professor of Mathematics at the University of Missouri, olumbia. The coauthor or coeditor of three books and the author or coauthor of ver 60 professional papers, he is a member of the American Mathematical ociety, the Mathematical Association of America, and the International Society )r General Relativity and Gravitation. Dr. Beem received the Ph.D. degree !968) in mathematics from the University of Southern California, Los Angeles.

AUL E. EHRLICH is a Professor of Mathematics at the University of Florida, iainesville. A member of the American Mathematical Society, the Mathematical .ssociation of America, and the International Society for General Relativity and iravitation, he is the author or coauthor of numerous professional papers that :flect his research interests in differential geometry and general relativity. Dr. .hrlich received the Ph.D. degree (1974) in mathematics from the State University f New York at Stony Brook.

~ V I N L. EASLEY is an Associate Professor of Mathematics at Truman State Jniversity, Kirksville, Missouri. He is a member of the American Mathematical ociety, the Mathematical Association of America, and the American Association 3r the Advancement of Science. Dr. Easley received the Ph.D. degree (1991) in lathematics from the University of Missouri, Columbia, where he studied ~rentzian geometry under Professors Beem and Ehrlich.

' R E A N D A P P L I E D M A T H E M A T I C S

A Series of Monographs and Textbooks

GLOBAL LORENTZ GEOMETRY Second Edition

John K. Beem Paul E. Ehrlich Kevin L. Faslev

GLOBAL LORENTZIAN GEOMETRY

PURE AND APPLIED MATHEMATICS

A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS

Earl J . Taft Rutgers University

New Brunswick, New Jersey

EDITORIAL BOARD

Zuhair Nashed University of Delaware

Newark, Delaware

M. S. Baouendi Anil Nerode University of California, Cornell University

Sun Diego Donald Passman

Jane Cronin University of Wisconsin, Rugers University Madison

Jack K. Hale Fred S. Roberts Georgia Institute of Technology Rugers University

S. Kobayashi Gian-Carlo Rota University of California, Massachusetts Institute of

Berkeley Technology

Marvin Marcus David L. Russell University of California, Virginia Polytechnic Institute

Santa Barbara and State University

W. S. Massey Walter Schempp Yale University Universitiit Siegen

Mark Teply University of Wisconsin,

Milwaukee

MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS

1. K. Yano, Integral Formulas in Riemannian Geometry (1 970) 2. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970) 3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; A. Littlewood,

trans.) (1 970) 4. B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation

ed.; K. Makowski, trans.) (1971) 5. L. Narici et a/., Functional Analysis and Valuation Theory (1 971 ) 6. S. S. Passman, Infinite Group Rings (1 971 ) 7. L. Dornhoff, Group Representation Theory. Part A: Ordinary Representation Theory.

Part B: Modular Representation Theory (1 971, 1972) 8. W. Boothby and G. L. Weiss, eds., Symmetric Spaces (1 972) 9. Y. Matsushima, Differentiable Manifolds (E. T. Kobayashi, trans.) (1 972)

10. L. E. Ward, Jr., Topology (1972) 1 1. A. Babakhanian, Cohomological Methods in Group Theory (1 972) 12. R. Gilmer, Multiplicative Ideal Theory (1 972) 13. J. Yeh, Stochastic Processes and the Wiener lntegral (1 973) 14. J. Barros-Nero, lntroduction to the Theory of Distributions (1 973) 15. R. Larsen, Functional Analysis (1 973) 16. K. Yano and S. lshihara, Tangent and Cotangent Bundles (1 973) 17. C. Procesi, Rings with Polynomial Identities (1 973) 18. R. Hermann, Geometry, Physics, and Systems (1 973) 19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1 973) 20. J. Dieudonn&, lntroduction to the Theory of Formal Groups (1 973) 21. 1. Vaisman, Cohomology and Differential Forms (1 973) 22. B.-Y. Chen, Geometry of Submanifolds (1973) 23. M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973, 1975) 24. R. Larsen, Banach Algebras (1 973) 25. R. 0. Kujala andA. 1. Vitrer, eds., Value Distribution Theory: Part A; Part 8: Deficit

and Bezout Estimates by Wilhelm St011 (1 973) 26. K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974) 27. A. R. Magid, The Separable Galois Theory of Commutative Rings (1 974) 28. B. R. McDonald, Finite Rings with Identity (1 974) 29. J. Satake, Linear Algebra (S. Koh et al., trans.) (1975) 30. J. S. Golan, Localization of Noncommutative Rings (1 975) 31. G. Klambauer, Mathematical Analysis (1 975) 32. M . K. Agoston, Algebraic Topology (1976) 33. K. R. Goodearl, Ring Theory (1 976) 34. L. E. Mansfield, Linear Algebra with Geometric Applications (1 976) 35. N. J. Pullman, Matrix Theorv and Its ADDlications (1 976) 36. B. R. ~ c ~ o n a l d , ~eometr ic Algebra over Local Rings (1 976) 37. C. W. Groetsch, Generalized Inverses of Linear Operators (1 977)

J. E. Kuczkowskiand J. L. Gersting, Abstract Algebra (1977) C. 0. Christenson and W. L. Voxman, Aspects of Topology (1 977) M. Nagata, Field Theory (1 977) R. L. Long, Algebraic Number Theory (1 977) W. F. Pfeffer, lntegrals and Measures (1 977) R. L. Wheeden and A. Zygmund, Measure and Integral (1 977) J. H. Curtiss, lntroduction to Functions of a Complex Variable (1 978) K. Hrbacek and T. Jech, lntroduction to Set Theory (1978) W. S. Massey, Homology and Cohomology Theory (1 978) M. Marcus, lntroduction to Modern Algebra (1 978) E. C. Young, Vector and Tensor Analysis (1978) S. B. Nadler, Jr., Hyperspaces of Sets ( 1 978) S. K. Segal, Topics in Group Kings (1 978) A. C. M. van Rooij, Non-Archimedean Functional Analysis (1 978) L. Corwin and R. Szczarba, Calculus in Vector Spaces (1 979) C. Sadosky, interpolation of Operators and Singular Integrals (1 979)

54. J. Cronin, Differential Equations (1 980) 55. C. W. Groetsch, Elements of Applicable Functional Analysis (1980) 56. 1. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1 980) 57. H. I. Freedan, Deterministic Mathematical Models in Population Ecology (1 980) 58. S. B. Chae, Lebesgue Integration (1980) 59. C. S. Rees etal., Theory and Applications of Fourier Analysis (1 981) 60. L. Nachbin, lntroduction to Functional Analysis (R. M. Aron, trans.) (1981) 61. G. Onech and M. Orzech, Plane Algebraic Curves (1 981) 62. R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysis

(1 981) 63. W. L. Voxman and R. H. Goetschel, Advanced Calculus (1 981) 64. L. J. Corwin and R. H. Szczarba, Multivariable Calculus (1982) 65. V. I. Ist@tescu, lntroduction t o Linear Operator Theory (1981) 66. R. D. Jsirvinen, Finite and Infinite Dimensional Linear Spaces (1981) 67. J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry (1981) 68. D. L. Armacost, The Structure of Locally Compact Abelian Groups (1 981) 69. J. W. Brewer and M. K. Smith, eds., Emmy Noether: A Tribute (1 981) 70. K. H. Kim, Boolean Matrix Theory and Applications (1 982) 71 . T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1 982) 72. D. B.Gauld, Differential Topology (1 982) 73. R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983) 74. M. Carmeli, Statistical Theory and Random Matrices (1 983) 75. J. H. Carruth et a/., The Theory of Topological Semigroups (1 983) 76. R. L. Faber, Differential Geometry and Relativity Theory (1983) 77. S. Bamett, Polynomials and Linear Control Systems (1983) 78. G. Karpilovsky, Commutative Group Algebras (1983) 79. F. Van Oystaeyen and A. Verschoren, Relative lnvariants of Rings (1 983) 80. 1. Vaisman, A First Course in Differential Geometry (1984) 81. G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1 984) 82. T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1 984) 83. K. Goebeland S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive

Mappings (1 984) 84. T. Albu and C. Ngstasescu, Relative Finiteness in Module Theory (1 984) 85. K. Hrbacek and T. Jech, lntroduction t o Set Theory: Second Edition (1 984) 86. F. Van Oystaeyen and A. Verschoren, Relative lnvariants of Rings (1 984) 87. B. R. McDonald, Linear Algebra Over Commutative Rings (1984) 88. M. Namba, Geometry of Projective Algebraic Curves 11 984) 89. G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1985) 90. M. R. Bremner et a/., Tables of Dominant Weight Multiplicities for Representations of

Simple Lie Algebras (1 985) 91. A. E. Fekete, Real Linear Algebra (1 985) 92. S. B. Chae, Holomorphy and Calculus in Normed Spaces (1 985) 93. A. J. Jerri, lntroduction t o Integral Equations w i th Applications (1 985) 94. G. Karpilovsky, Projective Representations of Finite Groups (1985) 95. L. Nariciand E. Beckenstein, Topological Vector Spaces (1 985) 96. J. Weeks, The Shape of Space (1985) 97. P. R. Gribik and K. 0. Kortanek, Extremal Methods of Operations Research (1 985) 98. J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic Analysis

(1986) 99. G. D. Crown etal., Abstract Algebra (1986)

100. J. H. Carruth etal., The Theory of Topological Semigroups, Volume 2 (1 986) 101. R. S. Doran and V. A. Belfi, Characterizations of C*-Algebras (1986) 102. M. W, Jeter, Mathematical Programming (1 986) 103. M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations wi th

Applications (1 986) 104. A. Verschoren, Relative lnvariants of Sheaves (1 987) 105. R. A. Usmani, Applied Linear Algebra (1 987) 106. P. Blass and J. Lang, Zariski Surfaces and Differential Equations in Characteristic p

> 0 (1 987) 107. J. A. Reneke eta/., Structured Hereditary Systems (1 987) 108. H. Busemann and B. B. Phadke, Spaces wi th Distinguished Geodesics (1987) 109. R. Harre, lnvertibility and Singularity for Bounded Linear Operators (1 988)

110. G. S. Ladde e t a/., Oscillation Theory of Differential Equations w i th Deviating Arguments (1 987)

1 1 1. L. Dudkin et al., Iterative Aggregation Theory (1 987) 1 12. T. Okubo, Differential Geometry (1 987) 113. D. L. Stancland M. L. Stancl, Real Analysis wi th Point-Set Topology (1 987) 114. T. C. Gard, lntroduction t o Stochastic Differential Equations (1988) 115. S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux (1 988) 1 16. H. Strade and R. Farnsteiner, Modular Lie Algebras and Their Representations (1 988) 1 17. J. A. Huckaba, Commutative Rings wi th Zero Divisors (1 988) 1 18. W. D. Wallis, Combinatorial Designs (1 988) 1 19. W. Wipslaw, Topological Fields (1 988) 1 20. G. Karpilovsky, Field Theory (1 988) 121. S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded

Rings (1 989) 122. W. Kozlowski, Modular Function Spaces (1 988) 123. E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps (1 989) 124. M. Pavel, Fundamentals of Pattern Recognition (1989) 125. V. Lakshmikantham eta/., Stability Analysis of Nonlinear Systems (1 989) 126. R. Sivaramakrishnan, The Classical Theory of Arithmetic Functions (1 989) 127. N. A. Watson, Parabolic Equations on an lnfinite Strip (1 989) 128. K. J. Hastings, lntroduction t o the Mathematics of Operations Research (1 989) 129. B. fine, Algebraic Theory of the Bianchi Groups (1 989) 130. D. N. Dikranjan et a/., Topological Groups (1 989) 131. J. C. Morgan 11, Point Set Theory (1 990) 132. P. Biler and A. Witkowski, Problems in Mathematical Analysis (1 990) 133. H. J. Sussmann, Nonlinear Controllability and Optimal Control (1 990) 134. J.-P. Florens et a/., Elements of Bayesian Statistics (1 990) 135. N. Shell, Topological Fields and Near Valuations (1 990) 136. B. F. Doolin and C. F. Martin, lntroduction t o Differential Geometry for Engineers

(1 990) 137. S. S. Holland, Jr., Applied Analysis by the Hilbert Space Method (1 990) 138. J. Oknirfski, Semigroup Algebras (1 990) 139. K. Zhu, Operator Theory in Function Spaces (1 990) 140. G. B. Price, An lntroduction t o Multicomplex Spaces and Functions (1 991 ) 141 . R. B. Darst, lntroduction t o Linear Programming (1 99 1 ) 142. P. L. Sachdev, Nonlinear Ordinary Differential Equations and Their Applications

(1991) 143. T. Husain, Orthogonal Schauder Bases (1 991 ) 144. J. Foran, Fundamentals of Real Analysis (1 991 ) 145. W. C. Brown, Matrices and Vector Spaces (1991) 146. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces (1 991 ) 147. J. S. Golan and T. Head, Modules and the Structures of Rings (1 991) 148. C. Small, Arithmetic of Finite Fields (1 991) 149. K. Yang, Complex Algebraic Geometry (1 991) 150. D. G. Hoffman etal., Coding Theory (1991) 151. M. 0. Gonzdlez, Classical Complex Analysis ( 1 992) 152. M. 0. Gonzdlez, Complex Analysis (1 992) 153. L. W. Baggett, Functional Analysis (1 992) 154. M. Sniedovich, Dynamic Programming (1 992) 155. R. P. Agarwal, Difference Equations and Inequalities (1 992) 156. C. Brezinski, Biorthogonality and Its Applications t o Numerical Analysis (1 992) 157. C. Swarrz, An lntroduction to Functional Analysis (1 992) 158. S. B. Nadler, Jr., Continuum Theory (1 992) 159, M. A. Al-Gwaiz, Theory of Distributions (1 992) 160. E. Perry, Geometry: Axiomatic Developments wi th Problem Solving (1 992) 161. E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and

Engineering (1 992) 162. A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analysis

(1 992) 163. A. Charlier e t a/., Tensors and the Clifford Algebra (1 992) 164. P. Biler and T. Nadzieja, Problems and Examples in Differential Equations (1 992) 165. E. Hansen, Global Optimization Using Interval Analysis (1 992)

S. Guerre-Delabridre, Classical Sequences in Banach Spaces (1 992) Y. C. Wong, Introductory Theory of Topological Vector Spaces (1 992) S. H. Kulkarniand B. V. Limaye, Real Function Algebras (1 992) W. C. Brown, Matrices Over Commutative Rings (1 993) J. Loustau and M. Dillon, Linear Geometry wi th Computer Graphics (1 993) W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations (1 993) E. C. Young, Vector and Tensor Analysis: Second Edition (1 993) T. A. Bick, Elementary Boundary Value Problems (1 993) M. Pavel, Fundamentals of Pattern Recognition: Second Edition (1 993) S. A. Albeverio eta/., Noncommutative Distributions (1 993) W. Fulks, Complex Variables (1 993) M. M. Rao, Conditional Measures and Applications (1 993) A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes (1 994) P. Neittaanmakiand D. Tiba, Optimal Control of Nonlinear Parabolic Systems (1 994) J. Cronin, Differential Equations: lntroduction and Qualitative Theory, Second Edition (1 994) S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinu- ous Nonlinear Differential Equations (1 994) X. Mao. Exoonential Stabilitv of Stochastic Differential Eauations 11 994)

183. B. S. ~homion, Symmetric Properties of Real Functions ( i 994) 184. J. E. Rubio. Ootimization and Nonstandard Analvsis (1 994) 185. J. L. Bueso eral., Compatibility, Stability, and sheaves (1 995) 186. A. N. Micheland K. Wang, Qualitative Theory of Dynamical Systems (1 995) 187. M. R. Darnel, Theory of Lattice-Ordered Groups (1 995) 188. Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational

Inequalities and Applications (1 995) 189. L. J. Convin and R. H. Szczarba, Calculus in Vector Spaces: Second Edition (1995) 190. L. H. Erbe eta/., Oscillation Theory for Functional Differential Equations (1995) 191. S. Aaaian et a/.. Binaw Polvnomial Transforms and Nonlinear Diaital Filters 11 995) 192. M. l .v~i/ ' , ~orm '~s t ima t i ons for Operation-Valued Functions and-~pplications (1 995) 193. P. A. Grillet, Seminroups: An lntroduction t o the Structure Theorv (1995) 194. S. ~ichenassam~,- onl linear Wave Equations (1 996) 195. V. F. Krotov, Global Methods in Optimal Control Theory (1 996) 196. K. I. Beidar eta/., Rings with Generalized Identities (1 996) 197. V. I. Arnautov et a/., lntroduction t o the Theory of Topological Rings and Modules

(1 996) 198. G. Sierksma, Linear and Integer Programming (1 996) 199. R. Lasser, lntroduction to Fourier Series (1 996) 200. V. Sima, Algorithms for Linear-Quadratic Optimization (1 996) 201. D. Redmond, Number Theory (1 996) 202. J. K. Beem eta/. Global Lorentzian Geometry: Second Edition (1 996)

Additional Volumes in Preparation

GLOBAL LORENTZIAN GEOMETRY Second Edition

John K. Beem Department of Mathematics University of Missouri- Columbia Columbia, Missouri

Paul E. Ehrlich Department of Mathematics University of Florida- Gainesville Gainesville, Florida

Kevin L. Easlev d

Department of Mathematics Truman State University Kirks ville, Missouri

Marcel Dekker, Inc. New YorkeBasel Hong Kong

PREFACE TO THE SECOND EDITION Library of Congress Cataloging-in-Publication Data

Beem, John K. Global Lorentzian geometry. - 2nd ed. /John K. Beern, Paul E. Ehrlich,

Kevin L. Easley. p. cm. - (Monographs and textbooks in pure and applied mathematics ;

202) Includes bibliographical references and index. ISBN 0-8247-9324-2 (pbk. : alk. paper) 1. Geometry, Differential. 2. General relativity (Physics).

I. Ehrlich, Paul E. 11. Easley, Kevin L. 111. Title. IV. Series. QA649.B42 1996 5 1 6 . 3 ' 7 6 ~ 2 0 96-957

CIP

The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the address below.

This book is printed on acid-free paper.

Copyright O 1996 by MARCEL DEKKER, INC. All Rights Reserved.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.

MARCEL DEKKER, INC. 270 Madison Avenue, New York, New York 10016

Current printing (last digit): 1 0 9 8 7 6 5 4 3 2 1

PRINTED IN THE UNITED STATES OF AMERICA

The second edition of this book continues the study of Lorentzian geometry,

the mathematical theory used in general relativity. Chapters 3 through 12 con-

tain material, slightly revised in some cases, which was discussed in Chapters 2

through 11 of the first edition. Much new material has been added to Chapters

7 and 11, and new Chapters 13 and 14 have been written reflecting the more

complete and detailed understanding that has been gained in the intervening

years on many of the topics treated in the first edition. Inspired by an example

of P. Williams (1984), additional material on the instability of both geodesic

completeness and geodesic incompleteness for general space-times has been

provided in Section 7.1. Section 7.4 has been added giving sufficient condi-

tions on a spacetime to guarantee the stability of geodesic completeness for

metrics in a neighborhood of a given complete metric. New material has also

been added to Section 11.3 on the topic of geodesic connectibility. Appendixes

A, B, and C of the first edition have now been expanded into Chapter 2,

which also contains new material on the generic condition as well as Section

2.3, which gives a proof that the null cone determines the metric up to a con-

formal factor in any semi-Riemannian manifold which is neither positive nor

negative definite. Also, a deeper treatment of the behavior of the sectional

curvature function in a neighborhood of a degenerate two-plane is given in

Chapter 2.

In the concluding Chapter 11 of the first edition, which is now Chapter 12,

we showed how the Lorentzian distance function and causally disconnecting

sets could be used to obtain the Hawking-Penrose Singularity Theorem con-

cerning geodesic incompleteness of the space-time manifold. Around 1980,

S. T. Yau suggested that the concept of "curvature rigidity," well known for

iv PREFACE TO THE SECOND EDITION

the differential geometry of complete Riemannian manifolds since the publica-

tion of Cheeger and Ebin (1975), might be applied to the seemingly unrelated

topic of singularity theorems in space-time differential geometry. (Earlier. Ge-

roch (1970b) had suggested that spatially closed space-times should fail to be

timelike geodesically incomplete only under special circumstances.) As a step

toward this conjectured rigidity of geodesic incompleteness, the Lorentzian

analogue of the Cheeger-Gromoll Splitting Theorem for complete Riemann-

ian manifolds of nonnegative Ricci curvature needed to be obtained. This

was accomplished in a series of research papers published between 1984 and

1990. Aspects of the proof of the Lorentzian Splitting Theorem are discussed

in the new Chapter 14. Another new chapter in the second edition, Chapter

13, draws upon investigations of Ehrlich and Emch (1992a,b,c, 1993) and is

devoted to a study of the geodesic behavior and causal structure of a class

of geodesically complete Ricci flat space-times, the gravitational plane waves,

which were introduced into general relativity as astrophysical models. These

space-times provide interesting and nontrivial examples of astigmatic conju-

gacy [cf. Penrose (1965a)l and of the nonspacelike cut locus, a concept dis-

cussed in Chapter 8 of the first edition.

As for the first edition, this book is written using the notations and meth-

ods of modern differential geometry. Thus the basic prerequisites remain a

working knowledge of general topology and differential geometry. This book

should be accessible to advanced graduate students in either mathematics or

mathematical physics.

The list of works to which we are indebted for the two editions is quite

extensive. In particular, we would like to single out the following five im-

portant sources: The Large Scale Stmcture of Space-time by S. W. Hawk-

ing and G. I?. R. Ellis; Techniques of Diflerential Topology in Relativity by

R. Penrose; Riemannsche Geometric im Grossen by D. Gromoll, W. Klin-

genberg, and W. Meyer; the 1977 Diplomarbeit a t Bonn University, Exzstenz

und Bedeutung von konjugierten Werten in der Raum-Zeit, by G. Bolts; and

Semi-Riemannian Geometry by B. O'Neill.

In the time from the late 1970's when we wrote the first edition to the

PREFACE TO THE SECOND EDlTION v

present, it has been interesting to observe the enormous expansion in the

journal literature on space-time differential geometry. This is reflected in the

substantial growth of the list of references for the second edition. However,

this wealth of new material has precluded our treating many interesting new

developments in space-time geometry since 1980 which are less closely tied in

with the overall viewpoint and selection of topics originally discussed in the

first edition.

The authors would like to thank all those who have provided encouraging

comments about the first edition and urged us to issue a second edition after

the first edition had gone out of print, especially Gregory Galloway, Steven

Harris, Andrzej Kr6lak, Philip Parker, and Susan Scott. We thank Gerard

Emch for insisting that the second edition be undertaken, and the first two

authors thank Stephen Summers and Maria Allegra for independently sug-

gesting that a third author be added to the team to share the duties of the

completion of this revision. It is also a pleasure to thank Maria Allegra and

Christine McCafferty at Marcel Dekker, Inc., for working with us to see the

second edition into print. We are also indebted to Lia Petracovici for much

helpful proofreading and to Todd Hammond for valuable technical advice con-

cerning A M S W .

John K. Beem

Paul E. Ehrlich

Kevin L. Easley

PREFACE TO THE FIRST EDITION

This book is about Lorentzian geometry, the mathematical theory used in

general relativity, treated from the viewpoint of global differential geometry.

Our goal is to help bridge the gap between modern differential geometry and

the mathematical physics of general relativity by giving an invariant treat-

ment of global Lorentzian geometry. The growing importance in physics of

this approach is clearly illustrated by the recent Hawking-Penrose singularity

theorems described in the text of Hawking and Ellis (1973).

The Lorentzian distance function is used as a unifying concept in our book.

Furthermore, we frequently compare and contrast the results and techniques

of Lorentzian geometry to those of Riemannian geometry to alert the reader

to the basic differences between these two geometries.

This book has been written especially for the mathematician who has a

basic acquaintance with Riemannian geometry and wishes to learn Lorentzian

geometry. Accordingly, this book is written using the notation and methods

of modern differential geometry. For readers less familiar with this notation,

we have included Appendix A which gives the local coordinate representations

for the symbols used.

The basic prerequisites for this book are a working knowledge of general

topology and differential geometry. Thus this book should be accessible to

advanced graduate students in either mathematics or mathematical physics.

In writing this monograph, both authors profited greatly from the oppor-

tunity to lecture on part of this material during the spring semester, 1978, a t

the University of Missouri-Columbia. The second author also gave a series of

lectures on this material in Ernst Ruh's seminar in differential geometry a t

Bonn University during the summer semester, 1978, and would like to thank

... vlll PREFACE TO THE FIRST EDITION

Professor Ruh for giving him the opportunity to speak on this material. We

would like to thank C. Ahlbrandt, D. Carlson, and M. Jacobs for several help

ful conversations on Section 2.4 and the calculus of variations. We would like

to thank M. Engman, S. Harris, K. Nomizu, T. Powell, D. Retzloff, and H. Wu

for helpful comments on our preliminary version of this monograph. We also

thank S. Harris for contributing Appendix D to this monograph and J.-H. Es-

chenburg for calling our attention to the Diplomarbeit of Bolts (1977). To

anyone who has read either of the excellent books of Gromoll, Klingenberg,

and Meyer (1975) on Riemannian manifolds or of Hawking and Ellis (1973)

on general relativity, our debt to these authors in writing this work will be

obvious. It is also a pleasure for both authors to thank the Research Council

of the University of Missouri-Columbia and for the second author to thank

the Sonderforschungsbereich Theoretische Mathematik 40 of the Mathematics

Department, Bonn University, and to acknowledge an NSF Grant MCS77-

18723(02) held at the Institute for Advanced Study, Princeton, New Jersey,

for partial financial support while we were working on this monograph. Fi-

nally it is a pleasure to thank Diane Coffman, DeAnna Williamson, and Debra

Retzloff for the patient and cheerful typing of the manuscript.

John K. Beem

Paul E. Ehrlich

CONTENTS

Preface to the Second Edition iii

Preface to the First Edition vii

List of Figures xiii

1. Introduction: Riemannian Themes in Lorentzian Geometry 1

2. Connections and Curvature 15

2.1 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Semi-Riemannian Manifolds 20

. . . . . . . . . . . . . . . . . . . . 2.3 Null Cones and Semi-Riemannian Metrics 25

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Sectional Curvature 29

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Generic Condition 33

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Einstein Equations 44

3. Lorentzian Manifolds and Causality 49

3.1 Lorentzian Manifolds and Convex Normal Neighborhoods . . . . . . 50

3.2 Causality Theory of Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

. . . . . . . . . . . . . . . . . 3.3 Limit Curves and the C" Topology on Curves 72

3.4 Two-Dimensional Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.5 The Second Fundamental Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.6 WarpedProducts .......................................... 94

3.7 Semi-Riemannian Local Warped Product Splittings . . . . . . . . . . . 117

x CONTENTS

4 . Lorentzian Distance 135

............................. 4.1 Basic Concepts and Definitions 135

4.2 Distance Preserving and Homothetic Maps .................. 151

. . . . . . . . . . . . . 4.3 The Lorentzian Distance Function and Causality 157

. . . . . . . . . . . . . 4.4 Maximal Geodesic Segments and Local Causality 166

5 . Examples of Space-times 173

5.1 Minkowski Space-time .................................... 174

5.2 Schwarzschild and Kerr Space-times ........................ 179

5.3 Spaces of Constant Curvature .............................. 181

5.4 Robertson-Walker Space-times ............................ 185 I 5.5 Bi-Invariant Lorentzian Metrics on Lie Groups ............... 190

I 6 . Completeness a n d Extendibility 197

6.1 Existence of Maximal Geodesic Segments . . . . . . . . . . . . . . . . . . . . 198

6.2 Geodesic Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.3 Metric Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

6.4 IdealBoundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

6.5 Local Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

6.6 Singularities ............................................. 225

7 . Stability of Completeness and Incompleteness 239

7.1 Stable Properties of Lor(M) and Con(M) ................... 241

7.2 The C1 Topology and Geodesic Systems . . . . . . . . . . . . . . . . . . . . . 247

7.3 Stability of Geodesic Incompleteness for Robertson-Walker Space-times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

7.4 Sufficient Conditions for Stability ........................... 263

8 . Maximal Geodesics and Causally Disconnected Space-times 271

8.1 Almost Maximal Curves and Maximal Geodesics ............. 273

8.2 Nonspacelike Geodesic Rays in Strongly Causal Space-times . . . 279

8.3 Causally Disconnected Space-times and Nonspacelike GwdesicLines ........................................... 282

CONTENTS xi

9 . T h e Lorentzian C u t Locus 295

9.1 The Timelike Cut Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

9.2 The Null Cut Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

9.3 The Nonspacelike Cut Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

9.4 The Nonspacelike Cut Locus Revisited . . . . . . . . . . . . . . . . . . . . 318

10 . Morse Index Theory o n Lorentzian Manifolds 323

........................ 10.1 The Timelike Morse Index Theory 327

10.2 The Timelike Path Space of a Globally Hyperbolic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Space-time 354

10.3 The Null Morse Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

11 . Some Results in Global Lorentzian Geometry 399

11.1 The Timelike Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

11.2 Lorentzian Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . 406

11.3 Lorentzian Hadamard-Cartan Theorems . . . . . . . . . . . . . . . . . . 411

12 . Singularities 425

12.1 JacobiTensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

12.2 The Generic and Timelike Convergence Conditions . . . . . . . . . 433

12.3 Focal Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

12.4 The Existence of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

12.5 SmoothBoundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

13 . Gravitational P lane Wave Space-times 479

13.1 The Metric. Geodesics. and Curvature . . . . . . . . . . . . . . . . . . . . 480

13.2 Astigmatic Conjugacy and the Nonspacelike Cut Locus . . . . . 486

13.3 Astigmatic Conjugacy and the Achronal Boundary . . . . . . . . . 493

14 . T h e Splitting Problem in Global Lorentzian Geometry 501

14.1 The Busemann Function of a Timelike Geodesic Ray . . . . . . . . 507

14.2 Co-rays and the Busemann Function . . . . . . . . . . . . . . . . . . . . . . 519

14.3 The Level Sets of the Busemann Function . . . . . . . . . . . . . . . . . 538

14.4 The Proof of the Lorentzian Splitting Theorem ............. 549

14.5 Rigidity of Geodesic Incompleteness ...................... 563

xii CONTENTS !

Appendixes 567 A. Jacobi Fields and Toponogov's Theorem for Lorentzian

. . . . . . . . . . . . . . . . . . . . . . . . . . . Manifolds by Steven G. Harris 567

B. From the Jacobi, to a Riccati, t o the Raychaudhuri Equation: Jacobi Tensor Fields and the Exponential Map 1 Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

LIST OF FIGURES

References

List of Symbols

Index

Figure Page Brief Description of Figure

chronological and causal future

basis of Alexandrov topology, I f ( p ) n I - ( q )

reverse triangle inequality for p << T << q

failure of causal continuity

nonspacelike curves may be imprisoned in compact sets

hierarchy of causality conditions

limit curve convergence but not C0 convergence

C0 convergence but not limit curve convergence

for the proof of Proposition 3.34


for the proof of Theorem 3.43

warped product structure of M x f H


yn -+ y , but lim L (7,) = 0 < L ( y )

d ( p , q ) = cc in Reissner-Nordstrijm space-time

pn + p, but d(p, q ) < lim inf d ( ~ n , q )

inner ball B+(p, E )

outer balls O+(P, E ) and 0- (p, E )

causally continuous with d ( p , q) not continuous

horismos E + ( p ) in Minkowski space-time

unit sphere K ( p , 1 )

horismos in W: = L~ with one point removed

conformal Wy = Ln with null, spacelike, timelike infinity

xiv I

LIST OF FIGURES !

Figure Page Brief Description of Figure

Penrose diagram for Minkowski space-time

Kruskal diagram for Schwarzschild space-time

de Sitter space-time

2-dimensional universal anti-de Sitter space-time

spacelike and null complete, but timelike incomplete

d(p , x,) -+ 0, but no accumulation point for {x,)

TIP'S and TIF's

local b-boundary extension

compact closure in a local extension

local extension of Minkowski space-time


Reissner-Nordstrom space-time with e2 = m2


example with s(v) = m, v, -t v, and s(v,) 5 4

simply connected but not future 1-connected


focal points in the Euclidean plane

focal points t o a spacelike submanifold

a focal point example in Wi = L3

Cauchy development D+(S)

a trapped set example in S1 x S1

a causal completion example

a causally disconnecting set example

causal disconnection for a Robertson-Walker space-time

the conjugate locus and astigmatic conjugacy

the null tail and astigmatic conjugacy

CHAPTER 1

INTRODUCTION: RIEMANNIAN THEMES IN LORENTZIAN GEOMETRY

In the 1970's, progress on causality theory, singularity theory, and black

holes in general relativity, described in the influential text of Hawking and

Ellis (1973), resulted in a resurgence of interest in global Lorentzian geometry.

Indeed, a better understanding of global Lorentzian geometry was required for

the development of singularity theory. For example, i t was necessary to know

that causally related points in globally hyperbolic subsets of spacetimes could

be joined by a nonspacelike geodesic segment maximizing the Lorentzian arc

length among all nonspacelike curves joining the two given points. In addi-

tion, much work done in the 1970's on foliating asymptotically flat Lorentzian

manifolds by families of maximal hypersurfaces has been motivated by general

relativity [cf. Choquet-Bruhat, Fischer, and Marsden (1979) for a partial list

of references].

All of these results naturally suggested that a systematic study of global

Lorentzian geometry should be made. The development of "modern" global

Riemannian geometry as described in any of the standard texts [cf. Bishop

and Crittenden (1964), Gromoll, Klingenberg, and Meyer (1975), Helgason

(1978), Hicks (1965)] supported the idea that a comprehensive treatment of

global Lorentzian geometry should be grounded in three fundamental topics:

geodesic and metric completeness, the Lorentzian distance function, and a

Morse index theory valid for nonspacelike geodesic segments in an arbitrary

Lorentzian manifold.

Geodesic completeness, or more accurately geodesic incompleteness, played

a crucial role in the development of singularity theory in general relativity and

has been thoroughly explored within this framework. However, the Lorentzian

distance function had not been as well investigated, although it had been of

2 1 INTRODUCTION

some use in the study of singularities [cf. Hawking (1967), Hawking and Ellis

(1973), Tipler (1977a), Beern and Ehrlich (1979a)l. Some of the properties of

the Lorentzian distance function needed in general relativity had been briefly

described in Hawking and Ellis (1973, pp. 215-217). Further results relat-

ing Lorentzian distance to causality and the global behavior of nonspacelike

geodesics had been given in Beem and Ehrlich (1979b). Hence, as part of the

first edition, a systematic study of the Lorentzian distance function was made.

Uhlenbeck (1975), Everson and Talbot (1976), and Woodhouse (1976) had

studied Morse index theory for globally hyperbolic space-times, and we had

sketched [cf. Beem and Ehrlich (1979c,d)] a Morse index theory for nonspace-

like geodesics in arbitrary space-times. But no complete treatment of this

theory for arbitrary space-times had been published prior to the first edition

of the current book.

The Lorentzian distance function has many similarities with the Riemann-

ian distance function but also many differences. Since the Lorentzian distance

function is still less well known, we now review the main properties of the Rie-

mannian distance function and then compare and contrast the corresponding

results for the Lorentzian distance function.

For the rest of this portion of the introduction, we will let (N, go) denote a

Riemannian manifold and (M, g) denote a Lorentzian manifold, respectively.

Thus N is a smooth paracompact manifold equipped with a positive definite

inner product golp : T,N x TpN -+ R on each tangent space T,N. In addition,

if X and Y are arbitrary smooth vector fields on N , the function N -+ R given

by p 4 go(X(p), Y(p)) is required to be a smooth function. The Riemannian

structure go : T N x T N 4 R then defines the Riemannian distance function

as follows. Let Q , , denote the set of piecewise smooth curves in N from p

to q. Given a curve c E flp,, with c : [O,1] + N, there is a finite partition

0 = t l < t z < . . . < tk = 1 such that c 1 It,, ti+l] is smooth for each i. The

Riemannian arc length of c with respect to go is defined as

RIEMANNIAN THEMES IN LORENTZIAN GEOMETRY 3

The Riemannian distance do(p, q) between p and q is then defined to be

For any Riemannian metric go for N , the function do : N x N + [0, co) has

the following properties:

(1) ~ o ( P , q) = do(q,p) for all P, q E N. (2) do(p, q) I do (P, r ) + do(r, q) for all P, q, r E N . (3) do(p, q) = 0 if and only if p = q.

More surprisingly,

(4) do : N x N -+ 10, co) is continuous, and the family of metric balls

for all p E N and E > 0 forms a basis for the given manifold topology.

Thus the metric topology and the given ~narlifold topology coincide. Further-

more, by a result of Whitehead (1932), given any p E N, there exists an R > 0

such that for any E with 0 < E < R, the metric ball B ( p , E) is geodesically

convex. Thus for any E with 0 < E < R, the set B(p, E) is diffeomorphic to the

n-disk, n = dim(N), and the set {q E N : do(p,q) = E ) is diffeomorphic to sn-1

Removing the origin from R2 equipped with the usual Euclidean metric and

setting p = (-1, O), q = (1, O), one calculates that do(p, q) = 2 but finds no

curve c E R,,, with Lo(c) = &(p,q) and also no smooth geodesic from p to

q. Thus the following questions arise naturally. Given a manifold N, find

conditions on a Riemannian metric go for N such that

(1) All geodesics in N may be extended to be defined on all of R.

(2) The pair (N, do) is a complete metric space in the sense that all Cauchy

sequences converge.

(3) Given any two points p,q E N, there is a smooth geodesic segment

c E Q,,, with Lo(c) = do(p, q).

A distance realizing geodesic segment as in (3) is called a minimal geodesic

segment. The word minimal is used here since the definition of Riemannian

4 1 INTRODUCTION

distance implies that Lo(y) 2 do(p, q) for all y E R,,,. More generally, one

may define an arbitrary piecewise smooth curve y E R,,, to be minimal if

Lo(y) = &(p, q). Using the variation theory of the arc length functional, it

may be shown that if y f a,,, is minimal, then y may be reparametrized to a

smooth geodesic segment.

The question of finding criteria on go such that (I), (2), or (3) holds was

resolved by H. Hopf and W. Rinow in their famous paper (1931). In modern

terminology the Hopf-Rinow Theorem asserts the following.

Theorem (Hopf-Rinow). For any Riemannian manifold (N, go) the fol-

lowing are equivalent:

(1) Metric completeness: (N, &) is a complete metric space.

(2) Geodesic completeness: For any v E T N , the geodesic c(t) in N with

c'(0) = v is defined for all positive and negative real numbers t E R.

(3) For some p E N , the exponential map exp, is defined on the entire

tangent space TpN to N at p.

(4) Finite compactness: Every subset K of N that is do bounded (i.e.,

sup{&(p, q) : p, q E K) < m) has compact closure.

Furthermore, if any one of (1) through (4) holds, then

(5) Minimal geodesic connectibility: Given any p, q E N , there exists a

smooth geodesic segment c from p to q with Lo(c) = do(p, q).

A Riemannian manifold (N, go) is said to be complete provided any one (and

hence all) of conditions (1) through (4) is satisfied. It should be stressed that

the Hopf-Rinow Theorem guarantees the equivalence of metric and geodesic

completeness and also that all fiemannian metrics for a compact smooth

manifold are complete. Unfortunately, none of these statements is valid for

arbitrary Lorentzian manifolds.

A remaining question for noncompact but paracompact manifolds is the

existence of complete Riemannian metrics. This was settled by Nomizu and

Ozeki's (1961) proof that given any Riemannian metric go for N, there is a

complete Riemannian metric for N globally conformal to go. Since any para-

compact connected smooth manifold N admits a Riemannian metric by a par-


tition of unity argument, it follows that N also admits a complete Riemannian

metric.

We now turn our attention to the Lorentzian manifold (M, g). A Lorentzian

metric g for the smooth paracompact manifold M is the assignment of a nonde-

generate bilinear form glp : T,M x TpM -+ R with diagonal form (-, +, . . . , +)

to each tangent space. It is well known that if M is compact and x ( M ) # 0,

then M admits no Lorentzian metric. On the other hand, any noncompact

manifold admits a Lorentzian metric. Geroch (1968a) and Marathe (1972)

have also shown that a smooth Hausdorff manifold which admits a Lorentzian

metric is paracompact.

Nonzero tangent vectors are classified as timelike, spacelike, nonspacelike,

or null according to whether g(v,v) < 0, > 0, 5 0, or = 0, respectively.

[Some authors use the convention (+, -, . . . , -) for the Lorentzian metric,

and hence all of the inequality signs in the above definition are reversed for

them.] A Lorentzian manifold (M, g) is said to be time oriented if M admits

a continuous, nowhere vanishing, timelike vector field X. This vector field is

used to separate the nonspacelike vectors a t each point into two classes called

future directed and past directed. A space-time is then a Lorentzian manifold

(M,g) together with a choice of time orientation. We will usually work with

space-times below.

In order to define the Lorentzian distance function and discuss its properties,

we need to introduce some concepts from elementary causality theory. It is

standard to write p << q if there is a future directed piecewise smooth timelike

curve in M from p to q, and p < q if p = q or if there is a future directed

piecewise smooth nonspacelike curve in M from p to q. The chronological past

and future of p are then given respectively by I-(p) = {q E M : q << p) and

I+(p) = {q E M : p << q). The causal past and future of p are defined as

J-(p) = {q E M : q < p) and J+(p) = {q E M : p < q). The sets I-(p) and

I+(p) are always open in any space-time, but the sets J-(p) and J+(p) are

neither open nor closed in general (cf. Figure 1.1).

The causal stmcture of the space-time (M, g) may be defined as the collec-

tion of past and future sets at all points of M together with their properties.

6 1 INTRODUCTION

:..:..:...... ....;, / . . . . . . . . . . . . . . . . :.. :_. :_. :.. , . . . . . . . . . . .

- - +i, . . . . . . . . . . . . . . . . . . . . .?. . . . . . . . . .'.. :.. :.. :./ . . . . . . . . . . .

t remove this closed set

r

FIGURE 1.1. The chronological (respectively, causal) future of a

point consists of all points which can be reached from that point

by future directed timelike (respectively, nonspacelike) curves. In

this example, the causal future J + ( r ) of r is the closure of the

chronological future I + ( r ) of r. However, the set J+(q) is not the

closure of I+(q). In particular, the point w is in the closure of I+(q)

but is not in J+(q).

I t may be shown that two strongly causal Lorentzian metrics gl and g2 for

M determine the same past and future sets at all points if and only if the

two metrics are globally conformal [i.e., gl = Rg2 for some smooth function

0 : M --+ (0, a)]. Letting C(M,g) denote the set of Lorentzian metrics glob-

ally conformal to g, it follows that properties suitably defined using the past

and future sets hold simultaneously either for all metrics in C(M, g) or for no

metric in C(M,g). Thus all of the basic properties of elementary causality


theory depend only on the conformal class C(M,g) and not on the choice of

Lorentzian metric representing C(M, g).

Perhaps the two most elementary properties to require of the conformal

structure C(M, g) are either that (M, g) be chronological or that (M, g) be

causal. A space-time (M,g) is said to be chronological if p $! I+(p) for all

p E M. This means that (M,g) contains no closed timelike curves. The

space-time (M, g) is said to be causal if there is no pair of distinct points

p, q E M with p < q < p. This is equivalent to requiring that (M, g) contain

no closed nonspacelike curves.

Already at this stage, a basic difference emerges between Lorentzian and

Riemannian geometry. On physical grounds, the space-times of general rela-

tivity are usually assumed to be chronological. But it is easy to show that if

M is compact, (M, g) contains a closed timelike curve. Thus the space-times

usually considered in general relativity are assumed to be noncompact.

In general relativity each point of a Lorentzian manifold corresponds to an

event. Thus the existence of a closed timelike curve raises the possibility that

a person might traverse some path and meet himself at an earlier age. Wlore

generally, closed nonspacelike curves generate paradoxes involving causality

and are thus said to "violate causality." Even if a space-time has no closed

nonspacelike curves, it may contain a point p such that there are future di-

rected nonspacelike curves leaving arbitrarily small neighborhoods of p and

then returning. This behavior is said to be a violation of strong causality at

p. Space-times with no such violation are strongly causal.

The strongly causal space-times form an important subclass of the causal

space-times. For this class of space-times the Alexandrov topology with basis

{I+(p) n I-(q) : p, q E M) for M and the given manifold topology are related

as follows [cf. Kronheimer and Penrose (1967), Penrose (1972)l.

Theorem. The following are equivalent:

(1) (MI g) is strongly causal.

(2) The Alexandrov topology induced on M agrees with the given manifold

topology.

(3) The Alexandrov topology is Hausdorff

1 INTRODUCTION

q

FIGURE 1.2. Sets of the form I+(p)nI-(q) with arbitrary p, q E M

form a basis of the Alexandrov topology. This topology is always

at least as coarse as the original topology on M. The Alexandrov

topology agrees with the original topology if and only if (M,g) is

strongly causal.

Ure are ready a t last to define the Lorentzian distance function

d = d(g) : M x M -+ [0, m]

of an arbitrary space-time. If c : [O, 11 -+ M is a piecewise smooth nonspacelike

curve differentiable except a t 0 = t l < t2 < ... < t k = 1, then the length

L(c) = L,(c) of c is given by the formula

If p (< q, there are timelike curves from p to q (very close to piecewise null

curves) of arbitrarily small length. Hence the infimum of Lorentzian arc length

of all piecewise smooth curves joining any two chronologically related points


p (< q is zero. On the other hand, if p (< q and p and q lie in a geodesically

convex neighborhood U, the future directed timelike geodesic segment in U

from p to q has the largest Lorentzian arc length among all nonspacelike curves

in U from p to q. Thus the following definition for d(p, q) is natural: fixing

a point p E M, set d(p, q) = 0 if q J+(p), and otherwise calculate d(p, q)

for q E J+(p) as the supremum of Lorentzian arc length of all future directed

nonspacelike curves from p to q. Thus if q E J+(p) and y is any future

directed nonspacelike curve from p to q, we have L(y ) 5 d(p, q). Hence unlike

the Riemannian distance function, the Lorentzian distance function is not

a priori finite-valued. Indeed, a secalled totally vicious space-time may be

characterized in terms of its Lorentzian distance function by the property that

d ( p , q) = m for all p, q E M. Also, if (M, g) is nonchronological and p E I+(p),

it follows that d(p,p) = w.

The Reissner-Nordstrijm space-times, physically important examples of ex-

act solutions to the Einstein equations in general relativity, also contain pairs

of chronologically related distinct points p << q with d(p, q) = co.

By definition of Lorentzian distance, d(p, q) = 0 whenever q f M - J'(p),

We have even seen that d(p, p) = rn is possible. Thus for arbitrary Lorentzian

manifolds there is no analogue for property (3) of the Riemannian distance

function. Also, the Lorentzian distance function tends from its definition to

be a nonsymmetric distance. In particular, for any space-time it may be shown

that if 0 < d(p, q) < m, then d(q,p) = 0. But the Lorentzian distance function

does possess a useful analogue for property (2) of the Riemannian distance

function. Namely, d(p, q) 2 d(p, r) + d(r, q) for all p, q, T f IvI with p < r < q.

The reversal of inequality sign is to be expected since nonspacelike geodesics

in a Lorentzian manifold locally maximize rather than minimize arc length.

Since d(p,q) > 0 if and only if q E I+(p), and d(q,p) > 0 if and only if

q E I-(p), the distance function determines the chronology of (M, g). On

the other hand, conformally changing the metric changes distance but not the

chronology, so that the chronology does not determine the distance function.

Clearly, the distance function does not determine the sets J+(p) or J-(p) since

d(p, q) = 0 not only for q E J+(p) - I+(p) but also for q E M - J+(p).

1 INTRODUCTION

FIGURE 1.3. If r is in the causal future of p and q is in the causal

future of r, then the distance function satisfies the reverse triangle

inequality d(p, q) > d(p, r ) + d(r, q). The reverse triangle inequality

will not be valid in general for some point r' which is not causally

between p and q.

Property (4) of the Riemannian distance function is the continuity of this

function for all Riemannian metrics. For space-times, on the other hand, the

Lorentzian distance function may fail to be upper semicontinuous. Indeed, the

continuity of d : M x M + 10, m ] has the following consequence for the causal

structure of (M,g) [cf. Theorem 4.241. If (M,g) is a distinguishing space-

time and d is continuous, then (M, g) is causally continuous (cf. Chapter 3 for

definitions of these concepts). Hence it is necessary to accept the lack of con-

tinuity and lack of finiteness of the Lorentzian distance function for arbitrary

space-times. The Lorentzian distance function is a t least lower semicontinu-

ous where it is finite. This may be combined with the upper semicontinuity

in the C0 topology of the Lorentzian arc length functional for strongly causal

space-times to construct distance realizing nonspacelike geodesics in certain

classes of space-times (cf. Sections 8.1 and 8.2).


With these remarks in mind, it is natural to ask if some class of spacetimes

for which the Lorentzian distance function is finite-valued and/or continuous

may be found. The globally hyperbolic spacetimes turn out to be such a

class. Here a space-time (M, g) is said to be globally hyperbolic if it is strongly

causal and satisfies the condition that J+(p)nJ-(q) is compact for allp,q E M.

It has been most useful in proving singularity theorems in general relativity

to know that if (M,g) is globally hyperbolic, then its Lorentzian distance

function is finite-valued and continuous. Oddly enough, the finiteness of the

distance function, rather than its continuity, characterizes globally hyperbolic

space-times in the following sense (cf. Theorem 4.30). We say that a space-

time (M, g) satisfies the finite distance condition provided that d(g)(p, q) < oo for all p,q E M. It may then be shown that the strongly causal Lorentzian

manifold (M, g) is globally hyperbolic if and only if (M, g') satisfies the finite

distance condition for all g' C(M, g).

Motivated by the definition of minimal curve in Riemannian geometry, we

make the following definition for space-times.

Definition. (Mazimal Curve) A future directed nonspacelike curve y

from p to q is said to be maximal if L(y) = d(p, q).

It may be shown (cf. Theorem 4.13), just as for minimal curves in Rie-

mannian spaces, that if y is a maximal curve from p to q, then -f may be

reparametrized to a nonspacelike geodesic segment. This result may be used

to construct geodesics in strongly causal spacetimes as limit curves of appro-

priate sequences of "almost maximal" nonspacelike curves (cf. Sections 8.1,

8.2).

In view of (5) of the Hopf-Rinow Theorem for Riemannian manifolds, it

is reasonable to look for a class of space-times satisfying the property that if

p < q, there is a maximal geodesic segment from p to q. Using the compactness

of J+(p) n J-(q), one can show that globally hyperbolic space-times always

contain maximal geodesics joining any two causally related points.

We are finally led to consider what can be said about Lorentzian analogues

for the rest of the Hopf-Rinow Theorem. Here, however, every conceivable

thing goes wrong. Thus much of the difficulty in Lorentzian geometry from

12 1 INTRODUCTION

the viewpoint of global Riemannian geometry or its richness from the viewpoint

of singularity theory in general relativity stems from the lack of a sufficiently

strong analogue of the Hopf-Rinow Theorem.

We now give a basic definition which corresponds to (2) of the Hopf-Rinow

Theorem.

Definition. (Timelike, Null, and Spacelike Geodesic Completeness) A

space-time (M,g) is said to be timelike (respectively, null, spacelike, non-

spacelike) complete if all timelike (respectively, null, spacelike, nonspacelike)

geodesics may be defined for all values -co < t < cm of an aRne parameter t.

A space-time which is nonspacelike incomplete thus has a timelike or null

geodesic which cannot be defined for all values of an affine parameter. Such

space-times are said to be singular in the theory of general relativity.

It is first important to note that global hyperbolicity does not imply any

of these forms of geodesic completeness. This may be seen by fixing points

p and q in Minkowski space with p << q and equipping M = I+(p) n I-(q)

with the Lorentzian metric it inherits as an open subset of Minkowski space.

This space-time M is globally hyperbolic. Since geodesics in M are just the

restriction of geodesics in Minkowski space to M , it follows that every geodesic

in M is incomplete.

It was once hoped that timelike completeness might imply null complete-

ness, etc. However, a series of examples has been given by Kundt, Geroch,

and Beem of globally hyperbolic space-times for which timelike geodesic com-

pleteness, null geodesic completeness, and spacelike geodesic completeness are

all logically inequivalent. Thus, there are globalIy hyperbolic spacetimes that

are s~acelike and timelike complete but null incomplete.

Metric completeness and geodesic completeness [(I) iff (2) of the Hopf-

Rinow Theorem] are unrelated for arbitrary Lorentzian manifolds. There are

also Lorentzian metrics which are timelike geodesically complete but also con-

tain points p and q with p << q such that no timelike geodesic from p to q

exists (cf. Figure 6.1).

On the brighter side, there is some relationship between (1) and (4) of the

Hopf-Rinow Theorem for globally hyperbolic space-times. Since d(p, q) = 0


if q $ J+(p), convergence of arbitrary sequences in (M,g) with respect to

the Lorentzian distance function does not make sense. But timelike Cauchy

completeness may be defined (cf. Section 6.3). It can be shown for globally

hyperbolic space-times that timelike Cauchy completeness and a type of finite

compactness are equivalent.

In addition, results analogous to the Nomizu-Ozeki Theorem mentioned

above for Riemannian metrics have been obtained. For instance, given any

strongly causal space-time (M, g), there is a conformal factor fl : M -+ (0, m)

such that the space-time (M, fig) is timelike and null geodesically conlplete

(cf. Theorem 6.5). It is still unknown, however, whether this result can be

strengthened to include spacelike geodesic completeness as well.

It should now be clear that while there are certain similarities between

the Lorentzian and the Riemannian distance functions, especially for globally

hyperbolic space-times, there are also striking differences. In spite of these

differences, the Lorentzian distance function has many uses similar to those of

the Riemannian distance function.

In Chapter 8 the Lorentzian distance function is used in constructing max-

imal nonspacelike geodesics. These maximal geodesics play a key role in the

proof of singularity theorems (cf. Chapter 12). In Chapter 9 the Lorentzian

distance function is used to define and study the Lorentzian cut locus.

In Chapter 10 a Morse index theory is developed for both timelike and null

geodesics. A number of global results for Lorentzian manifolds are obtained

in Chapter 11 using the index theory and the Lorentzian distance function.

Also, results are presented concerning the relationship of geodesic connectibil-

ity to geodesic pseudoconvexity and geodesic disprisonrnent. In Chapter 13 a

nontrivial example is given of the cut locus structure and certain associated

achronal boundaries for the class of gravitational plane wave space-times from

general relativity. Finally, Chapter 14 treats the concept of rigidity of geo-

desic incompleteness and the Lorentzian Splitting Theorem for space-times

satisfying the timelike convergence condition which also contain a complete

Lorentzian distance-realizing timelike geodesic line. In this setting, the almost

maximal curves of Chapter 8 again play a role [cf. Galloway and Horta (1995)l.

CHAPTER 2

CONNECTIONS AND CURVATURE

Let (M, g) be an n-dimensional manifold M with a semi-Riemannian met-

ric g of arbitrary signature (-, . . . , -, +, . . . , +). Then there exists a unique

connection V on M which is both compatible with the metric tensor g and

torsion free. This connection, which is called the Levi-Czvita connection of

(M, g ) , satisfies the same formal relations whether or not (M, g) is positive

definite. Thus geodesics, curvature, Ricci curvature, scalar curvature, and sec-

tional curvature may all be defined for semi-Riemannian manifolds using the

same formulas as for positive definite Riemannian manifolds. The only diffi-

culty which arises is that the sectional curvature is not defined for degenerate

sections of the tangent space when (M,g) is not of constant curvature. In fact,

the sectional curvature is only bounded near all degenerate sections a t a point

in the case of constant sectional curvature [see Kulkarni (1979)l. This generic

"blow up" of the sectional curvature at degenerate sections corresponds to a

generic unboundedness of tidal accelerations [cf. Beem and Parker (1990)] for

objects moving arbitrarily close to the speed of light.

In the first section of this chapter there is an introduction to connections,

and in the second section semi-Riemannian manifolds are discussed. Riemann-

ian manifolds are positive definite semi-Riemannian manifolds and thus have

metrics of signature (+, +, . - . , +). Consequently, the metric induced on the

tangent space of a Riemannian manifold is Euclidean, The types of semi-

Riemannian manifolds of primary interest in this book are Lorentzian mani-

folds. These manifolds have metric tensors of signature (-, +, . . . , +). Thus,

the tangent spaces of a Lorentzian manifold have induced Minkowskian met-

rics. The tangent vectors a t p E M of length zero form the null cone at p. For

semi-Riemannian manifolds which are neither positive nor negative definite, a

16 2 CONNECTIONS AND CURVATURE

null cone is an (n - 1)-dimensional surface in the tangent space. In the third

section of this chapter we show that null cones determine the metric up to a

conformal factor for metrics which are not definite. In the fourth section we

consider sectional curvature. This curvature is related to tidal accelerations

using the Jacobi equation. The Jacobi equation analyzes the relative behavior

of "nearby" geodesics and will be of fundamental importance in later c h a p

ters. We introduce the generic condition in the fifth section. This condition

corresponds to a tidal acceleration assumption. The Einstein equations are

introduced in the last section. These are partial differential equations relating

the metric tensor and its &st two derivatives to the energy-momentum ten-

sor T. The Einstein equations thus link geometry in terms of the metric and

curvature to physics in terms of the distribution of mass and energy.

The manifold M will always be smooth, paracompact, and Hausdorff. The

tangent bundle will be denoted by TM, and the tangent space at p E M will

be denoted by TpM. If (U, x) is an arbitrary chart for M, then (x', x2,. . . , xn)

will denote local coordinates on M and 6/6x1, 6/6x2,. . . , 6/6xn will denote

the natural basis for the tangent space.

2.1 Connections

Let X(M) denote the set of all smooth vector fields defined on M , and let

$(M) denote the ring of all smooth real-valued functions on M. A connection

is a function

V : X(M) x X(M) + X(M)

with the properties that

(2.1) Vv(X + Y) = v v x + VvY,

(2.2) Vfv+hw(X) = f V v X + hVwX, and

(2.3) Vv(fX) = f v v x + Vtf ) X

for all f , h E z (M) and all X, Y, V, W E X(M).

The vector V X ( ~ ) Y = VxYIp at the point p E M depends only on the

connection V, the value X(p) = Xp of X at p, and the values of Y along any

2.1 CONNECTIONS 17

smooth curve which passes through p and has tangent X(p) at p [cf. Hicks

(1965, p. 57)]. To see this, let El, Ez,. . . , E n be smooth vector fields defined

near p which form a basis of the tangent space at each point in a neighborhood

of p. Then X(p) = C Xi(p) Ei (p) and Y = C Yi Ei. Hence

It follows that Xi(p), Yi(p), and X(p)(Yi) determine VxYI, completely if

the VE,(,)Ei's are known.

Given the connection V on M and a curve y : [a, b ] -+ M , we may define

parallel translation of vector fields along y. Here a vector field Y along y is a

smooth mapping Y : [a, b ] + TM such that Y(t) E T,(t)M for each t E [a, b ] .

For to E [a, b ] we may locally extend Y to a smooth vector field defined on a

neighborhood of $to). Then we may consider the vector field V,,(,]Y along

y. The preceding arguments show that this vector field along y is independent

of the local extension, and consequently V,,Y (= Y') is well defined. A vector

field Y along y which satisfies V,lY(t) = 0 for all t E [a, b ] is said to move by

parallel translation along y. A geodesic c : (a, b) -+ M is a smooth curve of M

such that the tangent vector c' moves by parallel translation along c. In other

words, c is a geodesic if

(2.4) Vc~c1 = 0. (geodesic equation)

A pregeodesic is a smooth curve c which may be reparametrized to be a geo-

desic. Any parameter for which c is a geodesic is called an afine parameter.

If s and t are two affine parameters for the same pregeodesic, then s = at + b

for some constants a , b f R. A pregeodesic is said to be complete if for some

affine parametrization (hence for all affine parametrizations) the domain of the

parametrization is all of R. The equation VC,c1 = 0 may be expressed as a


system of linear differential equations. To this end, we let (U, ( x l , x2 , . . . , x n ) )

be local coordinates on M and let d / d x l , d /dx2 , . . . , d/dxn denote the natural

basis with respect to these coordinates. The connection coefficients rfj of V

with respect to ( x l , x2, . . . , xn) are defined by

" d

(2.5) v (5) = f - (connection coefficients) k= I

dxk

Using these coefficients we may write equation (2.4) as the system

d2xk " dxi dxj (2.6) + x- = 0. (geodesic equatzons in coordinates) dt i,j=l dt

The connection coefficients (Christoffel symbols of the second kind) may

also be used to give a local representation of the action of V. If the vector

fields X and Y have local representations as

n d n

d x = x ~ ' ( x ) ~ and Y = ~ Y ' ( ~ ) - ,

i=l i=l axz

then VxY has a local representation

The vector field V x Y is said to be the covariant derivative of Y with respect

to X . A semicolon is used to denote covariant differentiation with respect to a

natural basis vector. If X = d / d x J , then the components of VxY = Va/az ,Y

are denoted by Y k i j where

The Lie bracket of the ordered pair of vector fields X and Y is a vector field

[X, Y ] which acts on a smooth function f by [ X , Y ] ( f ) = X ( Y ( f)) - Y ( X ( f ) ) .

If X = C ~ " x ) ) d / d x ~ and Y = C Y i ( x ) d / d x i , then

(Lie bracket)

2.1 CONNECTIONS 19

The torsion tensor T of V is the function T : X ( M ) x X(M) + X(M) given

by

(2.8) T ( X , Y ) = VxY - V y X - [ X , Y] . (torsion tensor)

The mapping T is said to be f -bilinear since it is linear in both arguments

and also satisfies T ( f X , Y ) = T ( X , f Y ) = f T ( X , Y ) for smooth functions f .

The value T ( X , Y)I, depends only on the connection V and the values X ( p )

and Y ( p ) . Consequently, T determines a bilinear map T,M x T,M + T,M

a t each point p E M . Using the skew symmetry ( [ X , Y ] = -[Y, XI ) of the Lie

bracket, it is easily seen that T ( X , Y) = -T (Y , X ) , and hence T is also skew.

Since [d /dx i , d / d x j ] = 0 for all 1 5 i , j < n, it follows that

Consequently,

where

Tik j = rtj - I'$ (torsion components)

Clearly, the torsion tensor provides a measure of the nonsymmetry of the

connection coefficients. Hence, T = 0 if and only if these coefficients are

symmetric in their subscripts. A connection V with T = 0 is said to be

torsion free or symmetric.

The curvature R of V is a function which assigns to each pair X , Y E X ( M )

the f-linear map R ( X , Y ) : X ( M ) -+ X(M) given by

Curvature provides a measure of the nuncommutativity of Vx and Vy. I t

should be noted that some authors define the curvature as the negative of the

above definition. Consequently, they differ in sign for some of the definitions

of curvature quantities given below. The curvature R represents a tensor field.


Hence, the map (X, Y, Z) -+ R(X, Y)Z from X(M) x X(M) x T(M) to X(M) is

f-trilinear, and the vector R(X, Y)Zlp depends only on X(p), Y(p), Z(p), and

V. If x, y, z E TpM, one may extend these vectors to corresponding smooth

vector fields X, Y, Z and define R(x, y)z = R(X, Y)Zlp.

If w E T,*M is a cotangent vector at p and x, y, z E TpM are tangent vectors

at p, then one defines

for X , Y, and Z smooth vector fields extending x, y, and z, respectively. The

curvature tensor R is a (1,3) tensor field which is given in local coordinates

by

where the curvature components R z j k m are given by

Notice that R(X, Y)Z = -R(Y, X)Z, R(w, X, Y, 2) = -R(w, Y, X, Z), and

Rijkm = - ~ i ,mk. F'urthermore, if X = CXid/dxi , Y = Yi6'/dx" Z =

C Zia/dxi, and w = C widxi, then

and

Consequently, one has R(dxi, d/dxk, a/dxm, a/dxj) = Rijkm.

2.2 Semi-Riemannian Manifolds

A semi-Riemannian metric g for a manifold M is a smooth symmetric tensor

field of type (0,2) on M which assigns to each point p E M a nondegenerate

2.2 SEMI-RIEMANNIAN MANIFOLDS 21

inner product glp : TpM x TpM -+ R of signature (-, . . . , -, +, . . . , +). Here

nondegenerate means that for each nontrivial vector v E T,M there is some

w E TpM such that gp(v, w) # 0. If g has components gij in local coordinates,

then the nondegeneracy assumption is equivalent to the condition that the

determinant of the matrix (gij) be nonzero.

Although we consider only smooth metrics, some authors have studied dis-

tributional semi-Riemannian metrics [cf. Parker (1979), Taub (1980)l. Also,

a number of authors have studied semi-Riemannian metrics for which degen-

eracy is allowed [cf. Bejancu and Duggal (1991), Katsuno (1980), Kossowski

(1987, 1989)l.

In local coordinates (U, (xl, x2,. . . , xn)) on M, the metric g is represented

by n

g 1 U = gij(r) dxi 8 dxi (metric tensor) i,j=l

with

,, - ,, (symmetric) and det(gij) # 0. (nondegenerate) 9 ' . - 9 . .

If g has s negative eigenvalues and r = n - s positive eigenvalues, then the

signature of g will be denoted by (s, r). For each fixed p E M , there exist lo-

cal coordinates (U, (xl, x2,. . . , xn)) such that gp = g I TpM can be represented

as the diagonal matrix diag{-, . - . , -, +, . . . , +). For each semi-FLiemannian

manifold (M, g) there is an associated semi-Riemannian manifold (M, -9) ob-

tained by replacing g with -g. Aside from some minor changes in sign, there is

no essential difference between (M, g) and (M, -g). Thus, results for spaces of

signature (s, r ) may always be translated into corresponding results for spaces

of signature (r, s) by appropriate sign changes and inequality reversals.

Two vectors in TpM are orthogonal if their inner product with respect to

gp is zero. Note that when g has eigenvalues with different signs, there will

be some nontrivial vectors which are orthogonal to themselves (i.e., satisfy

g(v,v) = 0). These are known as null vectors. A given vector is said to be a

unit vector if it has inner product with itself equal to either +1 or -1. Thus,

an orthonormal basis {el, ez, . . . ,en) of T,M satisfies Jg(e,, e,)] = 6',.


Given a semi-Riemannian manifold (M,g), there is a unique connection V

on M such that

(2.16) Z(g(X, Y)) = g(VzX, Y) + g(X, V z Y ) (metric compatible)

and

(2.17) [X, Y] = VxY - V y X (torsion free)

for all X, Y, Z E X(M). This connection V is called the Levi-Czvita connection

of (M,g). As indicated above, (2.16) is the condition that the connection V

be compatible with the metric g, and (2.17) is the condition that V be torsion

free. Setting 2 = c' in (2.16), one finds that parallel translation of vector fields

along any smooth curve c of M preserves inner products. For semi-Riemannian

manifolds, the connection coefficients are given by

1 " dgia 89. . (2.18) rk . = - Igak (- - -' + %) '3 2

(connection coeficzents) a=l dxj dxa dx2

where gij represents the (2,O) tensor defined by

n

(2.19) C g i a 9 a j = bij for 1 5 i , j 5 n. a=l

The local representations gij and gij may be used to raise and lower indices.

For example, if the upper index of the curvature tensor is lowered, one obtains

the components of the Riemann-Christoffel tensor which is also known as the

couariant curvature tensor.

n

(2.20) &jkm = ):gat Rajkrn (couariant curvature components) a=l

Alternatively, one may define the Riemann-Christoffel tensor as the (0,4)

tensor such that

2.2 SEMI-RIEMANNIAN MANIFOLDS 23

Some standard curvature identities satisfied by the components of this tensor

are &jkm = Rkrnrj = -Rjikrn = -Rijmk and Rilkm + Rikmj + Rimjk = 0.

The trace of the curvature tensor is the Ricci curvature, a symmetric (0,2)

tensor. For each p E M , the Ricci curvature may be interpreted as a symmetric

bilinear map Ric, : T p M x T p M + R. To evaluate Ric(u, w), let e l , ea, . . . , en

be an orthonormal basis for T p M . Then

or equivalently,

n

(2.23) Ric(v,w) = x g ( e i , ei) x(ei , v: ei, w). r=l

One may express v and w in the natural basis as v = C v a d / d z b n d w =

C wid/dxi and then write

n

where n

(2.25) Ri, = C Raiaj. (Ricci curvature components) a=l

If one uses the Einstein summation convention of summing over repeated

indices, then equations (2.24) and (2.25) become Ric(v, w) = Rijviw3 and

Rij = Raiaj, respectively. The Ricci tensor is the ( 1 , l ) tensor field which

corresponds to the Ricci curvature. The components of the Ricci tensor may

be obtained by raising one index of the Ricci curvature. Either index may be

raised since the Ricci curvature is symmetric. Thus,

n n

(2.26) R> = ) g a ' ~ a 3 = ) g a a ~ j a . (Rzccz tensor components) a=l a=l

The trace of the Ricci curvature is the scalar curvature T. Historically, this

function has been denoted by the much used symbol R. Accordingly, n

(2.27) 7 = R = Raa. (scalar curuature) a=l


Thus i f e l , e2, . . . ,en is an orthonormal basis of T p M , one has

n

(2.28) T = R = g(ei, ei) Ric(ei, ei). i=l

The gradient and Hessian are defined for semi-Riemannian manifolds just

as for Riemannian manifolds. I f f : M --+ R is a smooth function, then df is

a ( 0 , l ) tensor field (i.e., one-form) on M , and grad f is the (1,O) tensor field

(i.e., vector field) which corresponds to df. Thus,

(2.29) Y ( f ) = df ( Y ) = g(grad f , Y ) (gradient)

for an arbitrary vector field Y . In local coordinates (U, ( x l , x2 , . . . , xn)), the

vector field grad f is represented by

* . . a f a (2.30) grad f = gZ3 - -. (gradient using coordinates)

i,i=l axi 8x3

The Hessian Hf is defined to be the second covariant differential o f f:

~f = V ( V f ). (Hessian)

For a given f E 5 ( M ) , the Hessian Hf is a symmetric (0,2) tensor field which

is related to the gradient o f f through the formula

for arbitrary vector fields X and Y . The Laplacian Af = div (grad f ) is now

defined to be the divergence o f the gradient o f f.

A tangent vector v f T p M is classified as timelike, nonspacelike, null, or

spacelike i f g(v, v ) is negative, nonpositive, zero, or positive, respectively:

g(v, v ) < 0, (timelike)

g(v, v ) 1 0, (nonspacelike or causal)

g(v, v ) = 0, (null or lightlike)

g(v, v ) > 0. (spacelike)

2.3 NULL CONES AND SEMI-RIEMANNIAN METRICS 25

A Riemannian manifold ( M , g) is a semi-Riemannian manifold o f signature

(0, n ) [i.e., (+, . . . , + ) I . Thus, on each tangent space T p M o f a Riemannian

manifold the metric gp is positive definite. Consequently, the metric induced

on each tangent space is Euclidean, and all (nontrivial) vectors for Riemannian

manifolds are spacelike.

A Lorentzian manifold is a semi-Riemannian manifold ( M , g) o f signature

( 1 , n - 1) [i.e., (- ,+,. . . ,+)I. At each point p f M the induced metric on

the tangent space is Minkowskian. Each point o f a Lorentzian manifold has

timelike, null, and spacelike tangent vectors. A smooth curve is said to be

timelike, null, or spacelike i f its tangent vectors are always timelike, null, or

spacelike, respectively.

A timelike curve in a Lorentzian manifold corresponds to the path of an

observer moving at less than the speed of light. Null curves correspond to

moving at the speed of light, and spacelike curves correspond to the geometric

equivalent of moving faster than light. Although relativity predicts that physical

particles cannot move faster than light, spacelike curves are of clear geometric

interest.

A vector field X on M is timelike i f g (X , X ) < 0 at all points o f M . A

Lorentzian manifold with a given timelike vector field X is said to be time

oriented b y X . A space-time is a time oriented Lorentzian manifold. Not all

Lorentzian manifolds may be time oriented, but a Lorentzian manifold which

is not time orientable always admits a two-fold cover which is time orientable

(cf . Chapter 3).

2.3 Null Cones and Semi-Riemannian Metrics

One o f the folk theorems o f general relativity asserts that the space-time

metric is determined up to a conformal factor by the set o f null vectors. In

this section we examine more generally to what extent the null vectors o f a

nondefinite nondegenerate inner product on a vector space determine the given

inner product and obtain as a consequence in Theorem 2.3 an elementary proof

o f this folk theorem for spacetimes [cf. Ehrlich (1991)l.


Lemma 2.1. Let V be a real n-dimensional vector space, n 2 2, and let

g and h be two nondefinite nondegenerate inner products on V of arbitrary

signature. Suppose that g and h satisfy the condition that for any v f V,

(2.31 ) g(v,v) = 0 iff h(v, v) = 0.

Th cn

(1) either g and h or g and -h have the same signature, and

(2) there exists X # 0 such that

(2.32 ) h(v, w) = Xg(v, w)

for all v,w E V.

Furthermore, i f g has signature ( s , ~ ) with T # s, then X > 0 if g and h have

the same signature and X < 0 if g and -h have the same signature.

Proof. Flrst conslder the case that r = s = 1. Let {el, e2) be an arbitrary

g-orthonormal basis for V such that g(el ,el) = -1 and g(e2, e2) = +1. Then

771 = el + e2 and 172 = el - e2 are both g-null vectors. Hence, setting h,, =

h(e,, e,), we obtain the system of equations

0 = h(711,771) = h l l + h22 + 2h12,

0 = h(~2,772) = h l l + h22 - 2h12,

from which we conclude hll = -hz2 and h12 = 0. Since g(e,,e,) # 0, we

also have hil, hz2 # 0. Thus if we set X = -hll, then h = Xg with X > 0 if

h(el ,el) < 0 and X < 0 if h(el,el) > 0.

Multiplying g by -1 if necessary, we consider the case that g has signature

(r,s) with r 2 1 and s 2 2. Let {el,. . . ,e,, e,+l,. . . , en ) be any g-orthonormal

basis for V with {el,. . . , e,} g-unit timelike and {e,+l,. . . ,en) g-unit space-

like. Again by (2.31), we have hJ3 = h(e,, e,) # 0 for all j. Fix any j , k > r + 1

with j # k. Then we have a one-parameter family of g-null vectors

(2.33) v(8) = el + cos(8) . e, + sin(8) . ek

2 3 NULL CONES AND SEMI-RIEMANNIAN METRICS 27

for any 8 f R. In vlew of (2.31), we obtain

(2.34) 0 = h(v(@), ~ ( 0 ) )

= hll + cos2(8) . h,, + sin2(6) . hkk

+ 2 cos(8) . hl, + 2sin(8) . hlk + sin(28) . hJk.

Taking 8 = 0 and 8 = .rr respectively in (2.34) yields

0 = hi1 + h,, + 2h1,,

and 0 = hll + h,, - 2hl, ,

from which we conclude hl, = 0 and h,, = -hll for all J 2 r + 1. With this

information, equation (2.34) reduces to the equation

0 = hl l - (cos2 8 + sin2 6) . hll + sln(26) hJk

or simply

0 = sin(28) - hJk.

Hence, hJk = 0 for all j, k 2 r + 1, j # k .

We now have for r = 1 that hpq = 0 if p # q , and h l l = -hz2 = -hS3 =

... - - -hnn. Hence if X = -hll > 0, then g and h both have signature

(-, +, . . . , +), whereas if X = -hli < 0, then h has signature (+, -, . . . , -). In either case, h = Xg with X = -hll as required.

If r 2 2, we have a little more work left. First, consideration of the one-

parameter family

with j, k 5 r and j # k yields hJk = h,, = 0 and hll = hz2 = .. . = h,, =

-h,,. It remains to show that hl, = 0 and h,, = 0 if p < r < q. But for this

we need simply consider the one-parameter null family

x(0) = cos(8) . el + sin(8) . ep + eq,

since the equation 0 = h(x(8), x(8)) reduces to


in view o f our above results. Thus we have obtained hll = hz2 = . . . =

h,, = -h,+l,,+l = ... = -hnn and hJk = 0 i f j # k , from which the desired

conclusion follows as above.

As a corollary, we obtain

Corollary 2.2. Let V be a real n-dimensional vector space, n 2 3, and

assume V is equipped with inner products g and h , both o f Lorentzian signa-

ture (-, +, . . . , +). Suppose g and h determine the same null vectors. Then

h = Xg for some constant A > 0.

W i t h this last result applied to each tangent space, we obtain the desired

result.

Theorem 2.3. Let M be a smooth manifold of dimension n 2 3 with

metrics g and h, both o f Lorentzjan signature (-, +, . . . , +). Suppose for any

v E T M that g(v ,v) = 0 i f f h(v ,v) = 0. Then there exists a smooth function

R : M -+ (0, m) such that h = Rg.

A somewhat more general class o f objects than globally conformally r e

lated metrics has been considered in differential geometry and general rel-

ativity, namely, conformal transformations. Here a global diffeomorphism

F : ( M , g ) -+ ( N , h ) between two semi-Riemannian manifolds ( M I g) and

( N , h) is said to be a global conformal transformation i f there exists a smooth

function R : M -t (0 , m) such that

(2.35) h(F*v, F,w) = R(p) g(v, w)

for all v , w f T p M and p E M . In the Riemannian case, such maps would be

angle preserving. Evidently, condition (2.35) implies that ( M , g ) and ( N , h )

have the same signature.

Conversely, given two semi-Ftiemannian manifolds ( M , g) and ( N , h ) , we

may apply Lemma 2.1 t o g and F'h in order t o obtain that the condition

(2.36) g(v, v ) = 0 i f f h(F.v, F,v) = 0 for all v f T M

serves to ensure that F is a global conformal transformation o f ( M , g ) onto

either ( N , h ) or ( N , -h). Especially for space-times, one has the following

well-known extension of Theorem 2.3.

2.4 SECTIONAL CURVATURE 29

Corollary 2.4. Let ( M I g) and ( N , h ) be smooth manifolds of dimension

n 2 3 having Lorentzian signature (-, +, . . . , +). Suppose that f : M -+ N is

a diffeomorphism which satisfies condition (2.36). Then f is a global conformal

transformation o f ( M , g) onto ( N , h).

One could paraphrase this last result as follows: a diffeomorphism o f space-

times which preserves null vectors is a conformal transformation. Hence Corol-

lary 2.4 is a differentiable version o f a much deeper a priori topological result

o f Hawking, King, and McCarthy (1976).

2.4 Sectional Curvature

Let ( M , g) be a semi-Riemannian manifold. A two-dimensional linear sub-

space E of TpM is called a plane section. The plane section E is said to be

nondegenerate if for each nontrivial vector v 6 E there is some vector u E E

with g (u ,v ) # 0. This is equivalent to the requirement that gp I E be a nonde

generate inner product on E. I f v and w form a basis o f the plane section E ,

then g(v, v)g(w, w ) - [g(v, w)j2 is a nonzero quantity i f and only i f E is non-

degenerate. This quantity represents the square o f the semi-Euclidean area o f

the parallelogram determined by v and w. The plane E is said to be timelike

i f the signature o f gp I E is (1 , I ) , i.e., (-, +). It is spacelike i f the signature is

(0,2), i.e., (+, +). For Lorentzian manifolds, degenerate planes are called either null or lightlike

and always have signature (0, +). A null plane in T p M is a plane tangent to

the null cone in T p M . Thus, in the Lorentzian case a degenerate plane contains

exactly one generator o f the null cone.

Using the square o f the semi-Euclidean area o f the parallelogram determined

by the basis vectors {v , w ) , one has the following classification for plane sec-

tions o f Lorentzian manifolds:

g(v, v)g(w, w ) - Ig(v, w)I2 < 0, (tzmelike plane)

g(v, v)g(w, w ) - [g(v, w)I2 = 0 , (degenerate plane)

g(v, v)g(w, W ) - lg(u, w)I2 > 0. (spacelike plane)

T h e sectional curvature o f the nondegenerate plane section E with basis


{v, w) where v = Cvid /dx i and w = C wid/dxi is then given by

- - R w , v, w, V) (sectional curvature) g(v, v)g(w1 w) - [g(v, w)I2

For positive definite manifolds the Ricci curvature evaluated at a unit vector

is sometimes thought of as more or less an average sectional curvature weighted

by a factor of (n- I). More precisely, let w be a unit vector at p in the Riemann-

ian manifold (M, g), extend to an orthonormal basis {el, ez, . . . , en-l, en = w),

and let Ei = span{ei, w) for 1 5 i 5 n - 1. Equations (2.22) and (2.37) yield

. -

Ric(w, w) = C g ( ~ ( e i , W)W, ei) = C K(p, E,).

It is instructive to contrast the above with the Ricci curvature evaluated a t

a unit timelike vector in a Lorentzian manifold. In this case the interpretation

is in terms of the negative of the timelike sectional curvature. For let u be a

unit timelike vector at a point p of the Lorentzian manifold (M,g). Extend to

an orthonormal basis {el, e2,. . . , en-1, en = u) and let Ei = span{ei, u) for

I 5 i < n - I . Equations (2.22) and (2.37) yield

Thus, if u is a timelike vector with Ric(u, u) > 0, then in some sense the

"average" sectional curvature for planes in the pencil of u is negative.

For Riemannian manifolds one has a number important "pinching" theo-

rems [cf. Cheeger and Ebin (1975)l. However, similar results fail for Lorentzian

manifolds. In particular, if the sectional curvatures of timelike planes are

bounded both above and below, and if dim(M) > 3, then (M, g) has constant

curvature [cf. Harris (1982a), Dajczer and Nomizu (l980a)l. Nevertheless, fam-

ilies of Lorentzian manifolds conformal to ones of constant curvature may be

2 4 SECTIONAL CURVATURE 31

constructed which have all timelike sectional curvatures bounded in one direc-

tion [cf. Harris (1979)j. However, if dim(M) > 3 and if the sectional curvatures

of all nondegenerate planes are bounded either from above or from below, then

the sectional curvature is constant [cf. Kulkarni (1979)l. Sectional curvature

has been further investigated by Dajczer and Nomizu (1980b), Nomizu (1983),

Beem and Parker (1984), and Cordero and Parker (1995a,c).

A semi-Riemannian manifold (M, g) which has the same sectional curvature

on all (nondegenerate) sections is said to have constant curvature. If (M, g) has

constant curvature c, then R(X, Y)Z = c [g(Y, Z ) X - g(X, Z)Y] [cf. Graves

and Nomizu (1978)l. Thorpe (1969) showed that the sectional curvature can

only be continuously extended to degenerate planes in the case of constant

curvature. Spaces of constant sectional curvature have been investigated in

connection with the space form problem [cf. Wolf (1961, 1974)l.

The curvature tensor, Ricci curvature, scalar curvature, and sectional cur-

vature may all be calculated in local coordinates using the metric tensor com-

ponents and the first two partial derivatives of these con~por~ents. Thus, the

metric tensor determines the curvatures. In contrast, curvature does not nec-

essarily determine the metric. Nevertheless, for most Lorentzian manifolds

the metric will be determined either completely or up to a constant conformal

factor given sufficient information concerning curvature in local coordinates

[cf. Hall (1983, 1984), Hall and Kay (1988), Ihrig (1975), Quevedo (1992)j.

Conjugate points along geodesics may be defined using Jacobi fields. These

objects are studied in Chapter 10 in connection with the development of the

Morse index theory for timelike and null geodesics. If c : (a, b) -+ M is a

geodesic, then the smooth vector field J : (a, b) -+ T M along c is a Jacobi field

if it satisfies the Jacobi equation,

(2.38) J" + R(J, cJ)cJ = 0, (Jacobz equatzon)

where J" = V,,(V,lJ). In an intuitive sense, one thinks of a Jacobi field

as representing the relative displacement of "nearby" geodesics [cf. Hawking

and Ellis (1973), Hicks (1965), or Misner, Thorne and Wheeler (1973)l. In

particular, let c be a unit speed timelike geodesic. Then c represents the path

of a "freely falling" particle moving a t less than the speed of light. Taking J


as a Jacobi field which is orthogonal to the tangent vector c', one interprets J

as a vector from the original particle to another particle moving on a nearby

timelike geodesic, and one interprets J" as the relative (or tidal) acceleration of

the second particle as measured by the first. If g(J, J ) # 0, then the definition

of sectional curvature together with g(cl, c') = -1 and g(J, c') = 0 yield

At points where J does not vanish, the vector J / d m is a unit vector in

the direction of J. Using J" = -R(J, cl)c', one finds that the radial component

of the tidal acceleration is given by

This equation shows that for "close" particles the radial component of the tidal

acceleration varies directly with the separation distance IJI and with the sec-

tional curvature K(cl, J ) of the plane containing both c' and J. Thus, using

our sign conventions, a timelike plane with positive sectional curvature cor-

responds to freely falling particles accelerating away from each other, and

negative sectional curvature of a timelike plane corresponds to particles accel-

erating toward each other. Since Ric(cl, c') > 0 corresponds to the timelike

planes containing c' having negative average sectional curvature, it follows that

Ric(cl, c') > 0 corresponds to average attractive (i.e., focusing) tidal forces. I t

should be kept in mind that some authors use different sign conventions and

may have sectional curvature equal to the negative of ours.

For a constant value of I JI the maximum tidal acceleration will be radial

[cf. Beem and Parker (1990, p. 820)j. Thus at any fied value to, an observer

traversing the timelike geodesic c will have zero tidal accelerations if and only

if all planes E containing cl(to) have zero sectional curvature.

2.5 THE GENERIC CONDITION

2.5 The Generic Condition

The sectional curvature can be used to study the generic condition, which

will be of importance in the singularity theorems to be considered in C h a p

ter 12. If W = Wad/dxa is a tangent vector, then the values W a

are the contravariant components. Using the metric g one has values Wb =

Cr=l gabWa which are the covariant components. Thus, xr=l Wb dxb is the

cotangent vector corresponding to the original W . The generic condition is

said to be satisfied for a vector W at p E M if

n

(2.40) wC wdwfa R ~ ~ ~ ~ [ ~ wfl # 0. (generic condition) c.d=l

~f condition (2.40) fails to hold (i.e., if C:,,=, WCWdWI,Rblcdl,Wfl = o), then

W will be called nongeneric.

The generic condition is said to hold for a geodesic c : (a, b) -+ M if at some

point c(t0) the tangent vector to the geodesic is generic, i.e., one has (2.40)

satisfied with W = cl(to). Notice from continuity that if (2.40) holds for some

W = cl(to), then it will hold for W = cl(t) whenever t is sufficiently close to

to.

We will show in Proposition 2.6 that for a vector W which is nonnull (i.e.,

g(W, W ) # O), the nongeneric condition is equivalent to requiring that all

plane sections containing W have zero sectional curvature. For null vectors,

the nongeneric condition is slightly more complicated.

Lemma 2.5. Let Rabcd represent the components of the curvature tensor

with respect to an oi.thonormal basis {vl , vz, . . . , v,) of T,M. The vector

W = vn satisfies the generic condition (2.40) iff there exist b and e with

1 < b, e < n - 1 such that Rbnne # 0.

Proof. The components of W are given by W 1 = . . . = Wn-' = Wl =

. . . = Wn-1 = 0, W" = 1, and W, = +l or -1. Consequently,


- wa Rbnn j We + Wb Rann j We)

- bna Rbnn j bne + bnb Rann j S n e ) .

It is easily seen that this expression is nonzero i f and only i f Rbnne # 0 for

s o m e b , e w i t h l ~ b , e ~ n - 1 .

Proposition 2.6. If W E T p M is a nonnull vector, then W fails to be

generic (i.e., is nongeneric) i f f for each nondegenerate plane section E contain-

ing W the sectional curvature K ( p , E ) vanishes.

Proof. W e may assume without loss o f generality that W is a unit vector.

Set W = vn and extend to an orthonormal basis { v l , v2, . . . , v,) o f T p M . Let

Rabcd represent the components o f the curvature tensor with respect t o this

basis.

From Lemma 2.5, i f W fails to satisfy the generic condition, then Rbnne = 0

for all 1 5 b, e 5 n - 1. This and the skew symmetry of Rabcd in both the

first pair o f indices and the second pair o f indices yield Rbnne = 0 for all

1 5 b, e 5 n. Hence Rnbne = -Rbnne = 0 for all such b and e. I t follows that

i f U is another tangent vector at p, one has z:,b,c,e=l RabceWaUbWCUe =

~ ~ , e = l R,~,,U~U" = 0. Fkom (2.37), we conclude that i f E is the plane

spanned by { W , U), and i f E is nondegenerate, then K(p, E ) = 0.

Conversely, assume all nondegenerate planes containing W have zero sec-

tional curvature. Then (2.37) easily shows that all terms o f the form Rbnnb

must be zero. Assume b # e. Notice that E = span{vb,vb + 2ve) cannot - - be degenerate. Using (2.37) we obtain R(vn, vb, v,, vb) = R(vn, v,, v,, v,) = - - R(vn, vb + 2ve, vn, vb + 2ve) = 0. The multilinearity of R and standard curva- - ture identities then yield 0 = R(vn, vb + 2ve, v,, vb + 2ve) = 4 @v,, vb, v,, we) .

This shows f i b n e = 0 and hence Rbnne = 0. Using Lemma 2.5, it follows that

W fails to satisfy the generic condition as desired. U

2.5 THE GENERIC CONDITION 35

Proposition 2.6 implies that the only way for a timelike geodesic c to fail to

satisfy the generic condition is for the corresponding observer to fail to ever

experience any tidal accelerations.

In Chapter 12 we will make use o f an alternative formulation o f the generic

condition using the curvature tensor. Let c be a unit speed timelike geo-

desic, p = c(to), and W = cl(to). The set of vectors orthogonal to W is an

( n - 1)-dimensional linear subspace W' = V L ( c ( t o ) ) lying in T p M , and the

metric induced on this linear subspace is positive definite. Thus vL(c ( to ) ) is

a spacelike hyperplane in T p M . I f y E T p M , then g (W, R(y, W ) W ) = 0 which

implies that R ( y , W ) W lies in VL(c( to)) . It follows that the curvature tensor

R induces a linear map from VL(c ( to ) ) to VL(c( to)) :

Since g I VL(c( to ) ) is positive definite, this curvature map is nontrivial i f and

only i f there arc vectors yl, y2 f V L ( c ( t o ) ) with g ( y n , R ( y l , W ) W ) # 0. The

next result shows this map is nontrivial i f and only i f W is generic.

Proposition 2.7. If W = cl(to) is a timelike vector in the Lorentzian

manifold ( M , g) , then the following three conditions are equivalent:

(1 ) The timelike vector W is generic.

( 2 ) At least one plane containing W has nonzero sectional curvature.

(3) R( . , W ) W is not the trivial map.

Proof. Clearly, Proposition 2.6 shows the first two conditions are equiva-

lent. Let Rabcd represent the components o f the covariant curvature tensor with

respect to an orthonormal basis {v l ,v2 , . . .vn-l,vn = W ) of T p M . I f W sat-

isfies the generic condition, then Lemma 2.5 implies that -Rbnen = Rbnne # 0

for some 1 5 b, e 5 n - 1. Consequently, g(vb, R(ve , vn)vn) # 0 which shows

R(v,, vn)vn is a nontrivial vector. Thus, when W is generic the map R( . , W ) W

is not the trivial map.

Conversely, assume that R ( . , W ) W is not trivial. Let yl , yz be vectors in

W L = spanjvl, v2, . . . , vn-l) with g(y2, R(y l , vn)vn) # 0. This last inequality

together with the multilinearity o f 2(~, Z , X , Y ) = g(W, R ( X , Y ) Z ) yield b


and e with 1 5 b, e < n - 1 such that Rbnne # 0. Thus, Lemma 2.5 shows that

W is generic as desired.

The next proposition shows that a sufficient condition for the nonnull vector

W to be generic is that Ric(W, W) # 0.

Proposition 2.8. If W E TpM is a nonnull vector with Ric(W, W) # 0,

then W is generic.

Proof. We may assume without loss of generality that W is a unit vector

and extend to an orthonormal basis {vl, v2,. . . , vn-1, vn = W) of TpM. Then,

which shows that Ric(W, W) # 0 implies Rnin # 0 for some 1 5 i 5 n - 1.

The result now follows from Lemma 2.5. 0

In particular, this last proposition shows that if c : (a, b) --+ M is a timelike

geodesic with Ric(c1, c') > 0 for some t = t o , then c is generic.

In order to investigate the generic condition for (nontrivial) null vectors,

we first define a pseudo-orthonoma1 basis Icf. Hawking and Ellis (1973)l. Let

{vl, 7.12,. . . , vn-2) be n - 2 unit spacelike vectors in TpM that are mutually

orthogonal, and let W, N be two null vectors which satisfy g(W, N ) = -1

and which are both orthogonal to the first n - 2 vectors. Then a pseudo-

orthonormal basis of TpM is formed by {vl, va, . . . , vn-2, N, W). In this basis

the metric tensor at p has the form

g. . V - - 6.. ,3 and g in- l=g in=O f o r l < i , j S n - 2 ,

with g = -1 and gnn = gn-ln-l = 0.

It is not difficult to show that any nontrivial null vector W can be extended

to a pseudo-orthonormal basis. One first takes any timelike plane E containing


W and finds an orthonormal basis {en-', en) of E with W = (en-l +en)/&,

en-1 spacelike, and en timelike. Then one extends to an orthonormal basis

{el, e2,. . . ,en) and sets vi = ei for 1 < i 5 n - 2. Finally, one assigns

vn-1 = N = (-en-l +en)/&, and vn = W.

The next lemma provides the null version of Lemma 2.5 using a pseudo-

orthonormal basis. In the lemma below, the values of b and e only go to n - 2

as opposed to n - 1 in Lemma 2.5.

Lemma 2.9. Let Rabcd represent the components of the curvature tensor

with respect to a pseudo-orthonormal basis { q , . . . , vn-2, vn-1 = N, vn = W)

ofTpM. The null vector W = vn satisfies thegeneric condition (2.40) iff there

exist b and e with 1 5 b, e 5 n - 2 such that Rbnne # 0.

Proof. The components of W are given by W' = . .. = Wn-' = Wl =

. . = Wn-2 = Wn = 0, Wn = 1, and Wn-l = -1. Consequently,

- wa Rbnnf We + Wb Rannf We)

It is easily seen that this expression is zero for all a , b, e, f whenever one or

more of a , b, e, f equal n. If it is nonzero for some a, b, e, f , then exactly one of

a , b must be n - 1, and also exactly one of e, f must be n - 1. It follows that

x:d=l WCWdWIaRblcdIeWfl # 0 iff Rbnne # 0 for some 1 5 b, e 5 n - 2.

If n = 2, then Lemma 2.9 shows all null vectors fail to be generic since there

are no values of b and e with 1 5 b, e I n - 2.

Corollary 2.10. If dim(M) = 2, then all null vectors are nongeneric

If W is a nontrivial null vector in TpM, then the orthogonal space W-' =

N(W) = {v E TpM / g(W, v) = 0) is an (n - 1)-dimensional linear subspace

of the tangent space and contains the null vector W. The set N(W) is a

hyperplane which is tangent to the null cone at p along one generator. The i


signature of g I N(W) is degenerate of order one and positive of order n - 2.

Notice that g(W, R(v, W)W) = 0 implies that R(v, W)W lies in N(W).

The one-dimensional linear subspace determined by W will be denoted by

[W]. Let G(W) = N(W)/[W] be the (n - 2)-dimensional quotient space and

T : N(W) -+ G(W) be the natural projection map. Since R(W, W) = 0 and

the multilinearity of the curvature tensor yield R(v + aW, W)W = R(v, W)W

for all v, each element of a given coset of G(W) is mapped to the same element

of N(W) by R( . , W)W. It follows that one has a linear map

defined by (G, W) W = x (R(v, W) W) where v is any vector in 7r-l(8). The

degenerate metric g 1 N(W) projects to a positive definite metric 3 on G(W)

since vl,vz E N(W) must satisfy g(v1 + aW,v2 + bW) = g(vl,v2) for all real

a, b. Thus,

(2.44) - g(G1, G2) = g(v1, v2) (quotient metric on G(W))

where V1 = x(v1) and 5 2 = x(v2).

The next result is the null version of Proposition 2.7. We do not give a

statement corresponding to condition (2) of Proposition 2.7. In general, a null

vector may lie in planes with nonzero sectional curvature and yet fail to be

generic.

Proposition 2.11. Let (M, g) be a Lorentzian manifold, and let W E TpM

be a nontrivial null vector. Then the following two conditions are equivalent:

(1) The null vector W is generic.

(2) R ( . , W)W is not the trivial map.

Proof. Let Rabcd represent the components of the covariant curvature tensor

with respect to a pseudo-orthonormal basis {vl,. . . ,v,-2,vn-l,vn = W) of

T,M, N(W) = WL, and G(W) = N(W)/[W] as above. Notice that N(W) =

span(vl, ~ 2 , . . . , vn-2, W).

Assuming W is generic, Lemma 2.9 yields b and e with Rbnne # 0 for 1 5 b, e 5 n - 2. Consequently, g ( ~ ( v b ) , Z(T(V,), W)W) = g(vb, R(ve, W)W) =

Rbnen = -Rbnne # 0 which shows that R ( . , W)W is not trivial.


Conversely, assume R ( . , W)W is not the trivial map. Since 3 is positive

definite, there must exist E and 8 in G(W) with 3 ( i i , R(v, W)W ) # 0.

Choose u f n-'(E) and v E a-'(5). Then multilinearity, the fact that

3 ( E , R(u, W)W ) = g(u, R(v, W)W) # 0, and R(W, W)W = 0 together im-

ply there must be b and e, 1 5 b,e < n - 2, such that g(vb, R(ve, W)W) # 0.

Hence, Rbnen = -Rbnne # 0, and W is generic by Lemma 2.9. 0

The next proposition shows that Ric(W, W) # 0 is a sufficient condition for

a vector W to be generic. This was already proven for timelike and spacelike

vectors in Proposition 2.8.

Proposition 2.12. Let (MI g) be a Lorentzian manifold, and let W E TpM

be a tangent vector. If Ric(W, W) # 0, then W is generic.

Proof. We may assume without loss of generality that W is a null vector

because of Proposition 2.8. It is always possible to construct both an orthonor-

ma1 basis {el, e2, . . . ,en) and a pseudo-orthonormal basis

with v, = e, for 15 i 5 n - 2,

and en timelike. Let Rabcd be the components of the Riemann-Christoffel

curvature tensor with respect to the pseudo-orthonormal basis and Rabcd be

the components with respect to the orthonormal basis. Then,

40

Hence,

2 CONNECTIONS AND CURVATURE

which shows that Ric(W, W ) # 0 implies %,in # 0 for some 1 5 i 5 n - 2.

The result now follows from Lemma 2.9. 0

The next corollary follows from Corollary 2.10 and Proposition 2.12

Corollary 2.13. Let ( M , g ) be a Lorentzian manifold with d im(M) = 2.

I f W is null, then Ric(W, W ) = 0.

The generic condition holds generically in each tangent space o f any point

where there is some component o f the curvature tensor not equal to zero. More

precisely, one can establish the following result [cf. Beem and Harris (1993a,

p. 950)].

Proposition 2.14. Let ( M , g) be a Lorentzian manifold of dimension four

and let p E M . If T p M has five nonnull and nongeneric vectors with four

o f them linearly independent and with the fifth not in any plane determined

by any two o f the original four, then the curvature tensor vanishes at p. In

particular, i f any component o f the curvature tensor fails to be zero at p, then

one cannot find five nonnull nongeneric vectors at p in general position.

The situation for nongeneric null vectors is somewhat different. Having all

null vectors nongeneric does not necessarily imply zero curvature at a point.

For dimension three and higher, all null vectors at a point will be nongeneric

i f and only i f there is constant sectional curvature at the point. In dimension

four, one can have a "cubic" o f nongeneric null vectors at a point and yet fail


t o have constant sectional curvature at that point. Using a cubic definition

for "generically situated" [cf. Beem and Harris (1993b, pp. 969-972)], one may

establish the following result.

Proposition 2.15. Let ( M , g) be a Lorentzian manifold o f dimension four,

and let p E M. I f T,M has 11 null nongeneric vectors generically situated,

then all sectional curvatures at p are equal. In particular, i f ( M , g) does not

have constant sectional curvature at p, then the generic null directions at p

form an open dense subset o f the two-sphere o f all null directions at p.

This last proposition yields that "generically" a given null direction at a

point p satisfies the generic condition.

Assume that ( M , g) is a four-dimensional Lorentzian manifold, and recall

that the Jacobi equation is J" + R(J, cl)c' = 0. Let c be a unit speed timelike

vector, and assume { E l , E2, E3, E4) is an orthonormal basis along c which

moves by parallel translation along c and satisfies Eq = cl(t) . I f J = c : ~ ~ JzEl

is a Jacobi field, the Jacobi equation can then be written as

3

= - C ~ ' 4 k 4 J~ (Jacobi equation) k=l

using ~ ~ 4 . 4 4 = 0. Since { E l , E2, Esl E4) is an orthonormal basis with c l ( t ) =

E4, the vectors { E l , E2, E3) are unit spacelike vectors, and the Jacobi equation

becomes

W e now illustrate the unboundedness of tidal accelerations for observers

with velocity vectors close to the direction o f a null vector W which sat-

isfies the generic condition [cf. Beem and Parker (1990)l. To this end, as-

sume that (U, Xl,X2,X3,~4) are local coordinates near c(t0) = p. Assume

also that the natural basis for the x,-coordinates at p is the orthonormal

set { E l , E2, E3, Ed). Denote the components o f the curvature tensor for this


basis by Rabcd. Regard 8 as temporarily fixed, and define new coordinates

(YI, ~ 2 , ~ 3 , ~ 4 ) near P by YI = 21, y2 = 22, y3 = 5 3 cosh(8) - 2 4 sinh(B), and

y4 = -23 sinh(8) + 2 4 cosh(8). Let the natural basis a t p for the y-coordinates - - - -

be denoted by {El , E p , E3, E4), and let the curvature tensor components at p - - - -

with respect to this basis be denoted by Rabcd. The new basis {E l , E2, E3, E 4 )

is also orthonormal, and if one lets 8 --+ +m, the direction of & converges to

the null direction determined by W = E3 + E4. This change of coordinates

corresponds in T,M t o what is called a "pure boost." The new curvature

components fZabcd are related to the original components Rabcd by

Using XI = yl, x2 = y2, 2 3 = y3 cosh(0) + y4sinh(8), and x 4 = y~sinh(8) + y4 cosh(8), one finds

An observer traversing the timelike geodesic which has tangent vector - - -

a t p has a rest space at p given by F: = span{El, E2, E3). TO investigate

tidal accelerations for this observer, one may consider the Jacobi equation -

d2J'/dt2 = - ~ i , ~ Rt4k4Jk using vectors in this rest space of the form J =

c:=, J"% with IJI = 1. Notice that the tidal accelerations will be bounded

as 8 4 +m if and only if the components fZz4k4 are bounded for large 8, and


this will hold if and only if the following equations hold:

The vector W is nongeneric if and only if (2.53), (2.54), and (2.55) all hold,

and it is nondestructive if (2.51)-(2.55) all hold [cf. Beem and Parker (1990),

Beem and Harris (1993a,b)]. Tidal forces and radiation for a falling body in

Schwarzschild space-time have been studied by Mashhoon (1977). Also, tidal

impulses have been investigated by Mashhoon and McClune (1993). Further

results on nondestructive directions have been obtained by Hall and Hossack

(1993).

Remark 2.16. In order to get a physical interpretation of (2.51)-(2.55),

consider a "freely falling" steel ball with center having velocity vector E4 at

p. Assume the ball has rest mass mo and radius = a. Tidal accelerations for

points on the surface of the ball will correspond to IJI = a. Recall that the

formula for the special relativistic increase in mass (or energy) is given by

where y = 1 / d m = cosh(8), v is the speed measured with respect to

the original rest frame, and c is the speed of light. Of course, v -+ c corresponds

to 8 -+ +m. In other words, the classical special relativistic magnification

factor of mass is y = cosh(8). If one of (2.53), (2.54), or (2.55) fails to hold,

then W = E3 + E4 is generic, and the increase in tidal accelerations for large 8

is approximately proportional to -y2 which is the square of the increase in the

mass. For particles approaching the speed of light in a direction corresponding

to a generic null direction, eventually the increase in tidal accelerations will

become a bigger difficulty than the increase in mass. If (2.53)-(2.55) all hold,


but at least one of (2.51) or (2.52) fails to hold, then the increase in tidal

accelerations for large B is approximately proportional to y. In this case, the

null direction corresponding to W is (tidally) destructive but is not generic.

2.6 T h e Einstein Equations

In this section we give a brief description of the Einstein equations. A

heuristic derivation of these equations may be found in Frankel (1979, Chap-

ter 3). Since these equations apply to manifolds of dimension four, we restrict

our attention in this section to this dimension. The Einstein equations re-

late purely geometric quantities to the energy-momentum tensor T, which is a

physical quantity. They may thus be used to state energy conditions in terms

of T. In the case of a perfect fluid, the energy-momentum tensor also takes

a simple form. This is important in general relativity because the matter of

the universe is assumed to behave like a perfect fluid in the standard cosmo-

logical models. The physical motivation for studying Lorentzian manifolds is

the assumption that a gravitational field may be effectively modeled by some

Lorentzian metric g defined on a suitable four-dimensional manifold M. Since

every manifold which admits a Lorentzian metric clearly admits uncountably

many such metrics, it is necessary to decide both which manifold and which

Lorentzian metric on that manifold should be used to model a given grav-

itational problem. The Einstein equations relate the metric tensor g, Ricci

curvature Ric, and scalar curvature T (= R) to the energy-momentum tensor

T. The tensor T is to be determined from physical considerations dealing with

the distribution of matter and energy [cf. Hawking and Ellis (1973, Chapter

3), Misner, Thorne, and Wheeler (1973, Chapter 5)]. The Einstein equations

may be written invariantly as

1 (2.57) Ric - - R g + A g = 87rT (Einstein equations)

2

where A is a constant known as the cosmological constant. The constant factor

of 87r is present for scaling purposes. In local coordinates, one has

1 (2.58) Rij - -Rgij + hgi j = 87rTij (Einstein equations in coordinates)

2

2.6 THE EINSTEIN EQUATIONS 45

where 1 < i, j 5 4. The Ricci curvature and scalar curvature involve the first

and second partial derivatives of the components gij of the metric g but do not

involve any higher derivatives. Hence, the Einstein equations represent (non-

linear) partial differential equations in the metric and its first two derivatives.

These sixteen equations reduce to ten because all of the tensors in equation

(2.58) are symmetric. There is a further reduction to six equations [cf. Misner,

Thorne, and Wheeler (1973, p. 409)] using the curvature identity

which yields four conservation laws given by

4

C ~ ' j ; j = O. (conservation laws) j=1

Here, "; j" denotes covariant differentiation in the xj direction (i.e., 'i7a/dzl).

The Einstein equations do not determine the metric on M without sufficient

boundary conditions. For example, let M = { ( t , r ) It E R and r > 2m) x S2.

Then M is topologically lR2 x S2. Let A = 0 and T = 0, and set dR2 =

do2 + ~ i n ~ ( e ) d $ ~ . Then M with this A and T admits both the flat metric

ds2 = -dt2 + dr2 + r2dR2 as well as the Schwarzschild metric

(2.59) ds2 = - dr2 + r2dR2 (Schwarzschild)

as solutions to the Einstein equations. Each of these rnetrics is asymptotically

flat, and each is Ricci JEat (i.e., Ric = 0). However, the Schwarzschild metric

has a nonzero curvature tensor, and hence the two metrics cannot be isometric.

Nevertheless, a counting argument shows that, in general, one expects the Ein-

stein equations to determine the metric up to diffeomorphism [cf. Hawking and

Ellis (1973, p. 74)j. First, notice that the metric tensor g has 16 components

which, by symmetry, reduce to ten independent components. Furthermore,

four of these ten components can be accounted for by the dimension of M

which allows four degrees of freedom. Thus the metric tensor is thought of


as having six independent components after symmetry and diffeomorphism

freedom are taken into account. Consequently, the Einstein equations yield

six independent equations to determine six essential components of the metric

tensor.

More rigorous approaches to the problem of existence and uniqueness of

solutions to the Einstein equations using Cauchy surfaces with initial data may

be found in a number of articles and books such as Chrusciel (1991), Hawking

and Ellis (1973, Chapter 7), Marsden, Ebin, and Fischer (1972, pp. 233-264),

and Choquet-Bruhat and Geroch (1969).

The Einstein equations may be used to relate the timelike convergence con-

dition (Ric(v, v) 2 0 for all timelike, and hence also all null vectors v) to the

energy-momentum tensor. In order to evaluate the scalar curvature R in terms

of T at p E M , let {el, e 2 , . . . , en ) be an orthonormal basis of T,M, and use

equation (2.57) to obtain

Using the fact that the scalar curvature R is the trace of the Ricci curvature,

this last equation becomes

Hence,

(2.60) R = - 8 ~ tr(T) + 411.

The Einstein equations become

1 Ric - - 2 ( - 8 ~ tr(T) + 4A)g + Ag = 8x2".

Thus,

2.6 THE EINSTEIN EQUATIONS 47

This last equation shows that the condition Ric(v,v) 2 0 is equivalent to

the inequality T(v, v) 2 [tr(T)/2 - A/8n] g(v, v). If follows that when A = 0

and dim(M) = 4, the condition

is equivalent to the condition

[cf. Hawking and Ellis (1973, p. 95)]. Note that (2.60) and (2.61) show that if

A = 0, then T = 0 (i.e., vacuum) is equivalent to Ric = 0 (i.e., Ricci flat).

The Einstein equations are fundamental in the construction of cosmological

models. Consider a fluid which moves through space. This motion generates

timelike flow lines in space-time. Let v be the unit speed timelike vector field

which is everywhere tangent to the flow lines of the fluid. The fluid is said to be

a perfect fluid if it has an energy density p , pressure p, and energy-momentum

tensor T such that

(2.62) T = (p + p) w 8 u + p g, (perfect fiuid)

which is

Tij = (p+p)vivj +pg i j

in local coordinates. Here w = z v i d x i is the one-form corresponding to the

vector field v = via/dxi. It follows from the above form of T that a perfect

fluid is an isotropic fluid which is free of shear and viscosity. Let (M,g)

be a manifold for which T has the above perfect fluid form. If the vectors

{el, e:!, e3, e4) form an orthonormal basis for T,M, then the trace of T may be

calculated as follows:

4

(2.63) tr(T) = g(ei, el) T(e,, e,) i=l

= -(P+P) $ 4 ~

= 3p - p.


Using equation (2.61), it follows that the timelike convergence condition for a

perfect fluid is equivalent to

for all timelike (and null) w. For Lorentzian manifolds, it is easy to verify

that the inner product of a timelike vector and another timelike (or nontrivial

null) vector is nonzero. Thus, we may assume without loss of generality that

g(v, w) # 0. Using equation (2.62) we obtain

which simplifies to

Since g(w, w) 5 0 and g(v, w) # 0, equation (2.64) shows that a negative

cosmological constant has the effect of making the timelike convergence condi-

tion more plausible and that a positive cosmological constant has the opposite

effect. Einstein originally introduced the cosmological constant because the

Einstein equations with A = 0 predict a universe which is either expanding

or contracting, and in the early part of this century it was believed that the

universe was essentially static. After the discovery that the universe was ex-

panding, the original motivation for the cosmological constant was removed;

however, removing A from the theory has been more difficult. While astro-

nomical experiments have failed to detect a A different from zero, one may

always argue that A is so small that the experiments have not been sufficiently

sensitive.

Discussions of the experimental evidence for general relativity may be found

in a number of books such as Misner, Thorne, and Wheeler (1973) and Will

(1981).

CHAPTER 3

LORENTZIAN MANIFOLDS AND CAUSALITY

Sections 3.1 and 3.2 give a brief review of elementary causality theory basic

to this monograph as well as to general relativity. Then Section 3.3 describes an

important relationship between the limit curve topology and the C0 topology

for sequences of nonspacelike curves in strongly causal space-times. Namely,

if y : [a, b ] --+ M is a future directed nonspacelike limit curve of a sequence

{y,) of future directed nonspacelike curves, then a subsequence converges to y

in the C0 topology. This result is useful for constructing maximal geodesics in

strongly causal spacetimes using the Lorentzian distance function (cf. Chapter

8 and Chapter 12, Section 4).

In Section 3.4 we study the causal structure of two-dimensional Lorentzian

manifolds. In particular, we show that if (M, g) is a space-time homeomorphic

to R2, then (M,g) is stably causal.

Section 3.5 gives a brief discussion of the theory of Lorentzian submani-

folds and the second fundamental form needed for our discussion of singularity

theory in Chapter 12.

An important splitting theorem of Geroch (1970a) guarantees that a glob-

ally hyperbolic space-time may be written as a topological (although not nec-

essarily metric) product R x S where S is a Cauchy hypersurface. This result

suggests that product space-times of the form (R x M, -dt2 @ g) with (M, g)

a Riemannian manifold should be studied. While this class of space-times in-

cludes Minkowski space and the Einstein static universe, it fails to include the

physically important exterior Schwarzschild and Robertson-Walker solutions

to Einstein's equations.

In Sections 3.6 and 3.7 we study a more general class of product space-times,

the so-called warped products, which are spacetimes MI x f M2 with metrics

50 3 LORENTZIAN MANIFOLDS AND CAUSALITY

of the form gl @I fg2. This class of metrics, studied for Riemannian manifolds

by Bishop and O'Neill (1969) and later for semi-Riemannian manifolds by

O'Neill (1983), includes products, the exterior Schwarzschild space-times, and

Robertson-Walker space-times. The following result, which may be regarded

as a "metric converse" to Geroch's splitting theorem, is typical of the results

of Section 3.6. Let (R x M, -dt2 @ g) be a Lorentzian product manifold with

(M, g) an arbitrary Riemannian manifold. Then the following are equivalent:

(1) (M, g) is a complete Riemannian manifold.

(2) (R x M, -dt2 @ g) is globally hyperbolic.

(3) (R x M, -dt2 @I g) is geodesically complete.

3.1 Lorentzian Manifolds a n d Convex Normal Neighborhoods

Let M be a smooth connected paracompact Hausdorff manifold, and let

TM denote the tangent bundle of M with n : T M -+ M the usual bundle map

taking each tangent vector to its base point. Recall that a Lorentzzan metric g

for M is a smooth symmetric tensor field of type (0,2) on M such that for each

p f M, the tensor glp : TpM x TpM -+ R is a nondegenerate inner product

of signature (1, n - 1) [i.e., (-, +, . . . , +)I. All noncompact manifolds admit

Lorentzian metrics. However, a compact manifold admits a Lorentzian metric

if and only if its Euler characteristic vanishes [cf. Steenrod (1951, p. 207)].

The space of all Lorentzian metrics for M will be denoted by Lor(M).

A continuous vector field X on M is timelike if g(X(p),X(p)) < 0 for all

points p E M . In general, a Lorentzian manifold does not necessarily have

globally defined timelike vector fields. If (M,g) does admit a timelike vector

field X E X(M), then (M,g) is said to be time oriented by X . The timelike

vector field X divides all nonspacelike tangent vectors into two separate classes,

called future and past directed. A nonspacelike tangent vector v E T,M is

said to be future [respectively, past] directed if g(X(p), v) < 0 [respectively,

g(X(p),v) > 01. A Lorentzian manifold (M,g) is said to be time onentable

if (M,g) admits a time orientation by some timelike vector field X. In this

case, (M,g) admits two distinct time orientations defined by X and -X,

respectively. A time oriented Lorentzian manifold is called a space-time.

3.1 LORENTZIAN MANIFOLDS, CONVEX NORMAL NEIGHBORHOODS 51

More precisely, we have the following definition.

Definition 3.1. (Space-time) A space-time (M,g) is a connected Cm Hausdorff manifold of dimension two or greater which has a countable basis,

a Lorentzian metric g of signature (-, +?. . . , +), and a time orientation.

We now show how to construct a time oriented two-sheeted Lorentzian

covering manifold n : ( E , i j ) -+ (M,g) for any Lorentzian manifold (M, g)

which is not time orientable.

To this end, first let (M,g) be an arbitrary Lorentzian manifold. Fix a base point PO E M . Give a time orientation to TpoM by choosing a timelike

tangent vector vo E TpoM and defining a nonspacelike w E TpoM to be future

[respectively, past] directed if g(v0, w) < 0 [respectively, g(v0, w) > 01. Now let

q be any point of M. Piecewise smooth curves y : [O, 11 -+ M with y(0) = po

and $1) = q may be divided into two equivalence classes as follows. Given

~ 1 ~ 7 2 : [0,11 -+ M with ~ l ( 0 ) = ~ ( 0 ) = PO and yl(1) = y2(1) = 91 let Vl

(respectively, T/z) be the unique parallel field along yl (respectively, yz) with

Vl(0) = V2(0) = vo. Wesay that yl and 7 2 areequivalent ifg(Vl(l), V2(1)) < 0.

If yl and 7 2 are homotopic curves from po to q, then yl and 7 2 are equivalent.

But equivalent curves are not necessarily homotopic.

Given y : [O, 11 -+ M with y(0) = po, let [y] denote the equivalence class

of y. Let 2 consist of all such equivalence classes of piecewise smooth curves

y : [O, 11 -+ M with $0) = PO. Define n : z -+ M by ~ ( [ y ] ) = y(1). If (M, g) - is time orientable, then M = M . Otherwise, T : % -+ M is a two-sheeted

covering [cf. Markus (1955, p. 412)].

Suppose now that the Lorentzian manifold (M, g) is not time orientable. I t

is standard from covering space theory to give the set ii;j a topology and dif-

ferentiable structure such that n : M + M is a two-sheeted covering manifold. - Define a Lorentzian metric g for M by ij = n*g, i.e., ;(v, w) = g ( ~ , v , n,w).

Then the map 7r : z -+ M is a local isometry.

In order to show that (%,g) is time orientable, it is useful to establish a

preliminary lemma. Fix a base point Fo E T - ' ( ~ ~ ) for M. Let Go E T~,M be - the unique timelike tangent vector in TFo M with r,Go = VO.


Lemma 3.2. Let q E be arbitrary and let yl,;j;2 : [0, I] -+ % be two

piecewise smooth curves with Yl(0) = 72(O) = Po and ;j;l(l) = 72(1) = g. If

fi, 1/2 are the parallel vector fields along y1 and y2, respectively, with (0) = - V2(0) = 6, then g(v1(1), %(I)) < 0.

- Proof. Let y l = .n o yl and 72 = 7r o 72. Since .n : ( M , $ ) ---t (M, g) is

a local isometry, the vector fields Vl = .n,(vl) and 6 = 7riT,(v2) are parallel - fields along 71 and 7 2 , respectively, with Vl(0) = h(O) = n,vo = vo. Also,

g (K( l ) , Wl)) = g(x*G(l) ,x*%(l)) = g(G(l )> %(I)).

Suppose now that $(Ql(l), V2(1)) y! 0. Since ?1(1) and ?2(1) are timelike

tangent vectors, it follows that 5(?1(1), V2(1)) > 0. Thus g(Vl(l), h ( 1 ) ) > 0

at q = .n(@). By definition of the equivalence relation on piecewise smooth

curves from po to q, we have [yl] # [y2]. F'rom the construction of M , we know

that T1(l) = [yl] and T2(l) = Im]. Thus %,(I) # 72(1), in contradiction.

Theorem 3.3. Suppose that (M,g) is not time orientable. Then the two-

sheeted Lorentzian covering manifold (%, $) of (M, g) constructed above is

time orientable and hence is a space-time.

Proof. Let & and Go be as above. Given any T E %, let o : [O, 11 -+ % be - a smooth curve with a(0) = Po, a(1) = $ Let V be the unique parallel vector

field along a with ?(o) = Go. Set F + ( q ) = (timelike w E T& : ~ ( v ( l ) , w) < 0). By Lemma 3.2, the definition of F+(a ) is independent of the choice of

u. Hence if+ F+(T) consistently assigns a future cone to each tangent space

T~%, q E G. Now let h be an auxiliary positive definite Riemannian metric for %. We

may define a continuous nowhere zero timelike vector field X on by choosing

X(5) to be the vector in F+(q) which is the unique h-unit vector in F + ( q )

having a negative eigenvalue of g with respect to h. That is, we may find a

continuous function X : M + (-a, 0) and a continuous timelike vector field

X on % satisfying X ( q ) E F+(q"), h(X(T), X(c)) = 1, and $(X(g), v) =

X(y) h(X(q), v) for all v f T& and 5 E %. U

Implicit in the proof of Theorem 3.3 is an alternative definition for the time

orientability of a Lorentzian manifold (M, g). Namely, ( M g) is time orientable

3.1 LORENTZIAN MANIFOLDS, CONVEX NORhlAL NEIGHBORHOODS 53

if, fixing any base point po E M and timelike tangent vector vo E Tp,M, the

following condition is satisfied for all q f M. Let yl,yz : [0,1] --+ M be any

two smooth curves from p to q. If V, is the unique parallel vector field along

yi with V,(O) = vo for i = 1,2, then g(V1(l), V2(1)) < 0. This condition means

that parallel translation of the future cone determined by vo at po to any

other point q of M is independent of the choice of path from p to q. Hence

a consistent choice of future timelike vectors for each tangent space may be

made by parallel translation from po.

Recall that a smooth curve in (M,g) is said to be timelike (respectively,

nonspacelike, null, spacelike) if its tangent vector is always timelike (respec-

tively, nonspacelike, null, spacelike). As in the Riemannian case, a geodesic

c : (a , b) -+ M is a smooth curve whose tangent vector moves by parallel dis-

placement, i.e., V,jcl(t) = 0 for all t E (a, b). The tangent vector field cl(t) of

a geodesic c satisfies g(cl(t), cl(t)) = constant for all t 6 (a , b) since

Consequently, a geodesic which is timelike (respectively, null, spacelike) for

some value of its parameter is timelike (respectively, null, spacelike) for all

values of its parameter.

The exponential map expp : TpM + M is defined for Lorentzian manifolds

just as for Riemannian manifolds. Given v E TpM, let cu(t) denote the unique

geodesic in M with cv(0) = p and cul(0) = v. Then the exponential exp,(v)

of v is given by expp(v) = cv( l ) provided c,(l) is defined.

Let vl, v2, . . . , vn be any basis for the tangent space TpM. For sufficiently

small (21, 22,. . . , x,) E Rn, the map

xlvl + 22212 + . . . + X n V n --' ~ X P ~ ( X ~ U ~ + 2 2 ~ 2 + . . . + xnvn)

is a diffeomorphism of a neighborhood of the origin of T,M onto a neigh-

borhood U(p) of p in M . Thus, assigning coordinates (xl,x2,. . . ,xn) to the

point expp(xlvl + 22712 + . . . + xnvn) in U(p) defines a coordinate chart for M

called normal coordinates based at p for U(p). The set U(p) is said to be a

(simple) convex neighborhood of p if any two points in U(p) can be joined by


a unique geodesic segment of (M,g) lying entirely in U(p). Whitehead (1932)

has shown that any semi-Riemannian (hence Lorentzian) manifold has convex

neighborhoods about each point [cf. Hicks (1965: pp. 133-136)]. In fact, it may

even be assumed that for each q E U(p), there are normal coordinates based

a t q containing U(p). We call such a neighborhood U(p) a convex normal

neighborhood [cf. Hawking and Ellis (1973, p. 34)].

The next proposition is essential to the study of the local behavior of causal-

ity. A proof is given in Hawking and Ellis (1973, pp. 103-105). It is important

to carefully note the phrase "contained in U" in the statement of this result.

Proposit ion 3.4. Let U be a convex normal neighborhood of q. Then

the points of U which can be reached by timelike {respectively, nonspacelike)

curves contained in U are those of the form expq(u), u E TqM, such that

g(v, v) < 0 {respectively, g(v, v) 5 0).

3.2 Causality Theory of Space-times

In a space-time (M, g) a (nowhere vanishing) nonspacelike vector field along

a curve cannot continuously change from being future directed to being past

directed. It follows that a smooth timelike, null, or nonspacelike curve in

(M, g) is either always future directed or always past directed.

We will use the standard notation p << q if there is a smooth future directed

timelike curve from p to q, and p < q if either p = q or there is a smooth future

directed nonspacelike curve from p to q. Furthermore, p < q will mean p 6 q

and P # q-

A continuous curve y : (a, b) --+ M is said to be a future directed non-

spacelike curve if for each to E (a, b) there is an E > 0 and a convex normal

neighborhood U(y(t0)) of y(t0) with y(t0-E, to+€) C U(y(t0)) such that given

any tl , tz with to - E < t l < t2 < to + E, there is a smooth future directed non-

spacelike curve in (U(y(to)), g I U(y(t0))) from y(t1) to y(t2). It is necessary

to use the convex normal neighborhood U(y(to)) in this definition for the fol-

lowing reason. There exist space-times for which p << q for all (p, q) E M x M.

But in these space-times, any continuous curve y satisfies both y(tl) << y(ta)

and y(t2) << y(tl) for all tl and t2 in the domain of y.

3.2 CAUSALITY THEORY OF SPACE-TIMES 55

For a given p E M, the chronological future Ii(p), chronological past I-(p),

causal future Ji(p), and causal past J-(p) of p are defined as follows:

I f (p) = { q f M : p << q) , (chronological future)

I-(p) = { q E M : q << p), (chronological past)

~ + ( p ) = { q E M : p 6 q), (causal future)

J-(p) = { q E M : q < p). (causal past)

For general subsets S 2 M, the sets I i (S) , I - (S) , J+(S) , and J - (S) are

defined analogously: for example, I+(S) = { q E A4 / s << q for some s E S ) .

The relations << and Q are clearly transitive. Moreover,

p << q and q < r implies p << r,

and

p 6 q and q << r implies p << r

[cf. Penrose (1972, p. 14)]. If there is a future directed timelike curve from p

to q, there is a neighborhood U of q such that any point of U can be reached

by a future directed timelike curve. Consequently, it follows that

Lemma 3.5. If p is any point of the space-time (M,g), then I+(p) and

I-(p) are open sets of M.

An example has been given in Chapter 1, Figure 1.1, to show that the sets

J+(p) and J-(p) are neither open nor closed in general.

Two especially important classes of subsets of a space-time are future sets

and past sets. In this section we will restrict our attention to open future and

past sets, which may be defined as follows.

Definition 3.6. (Future and Past Sets) The (open) subset F (respec-

tively, P) of the space-time (M,g) is said to be a .future (respectively, past)

set if F = I+(F) (respectively, P = I-(P)).

These sets will be used in studying the causality of the gravitational plane

wave space-times in Chapter 13. Future and past sets have often proven useful


in causality proofs [cf. Hawking and Sachs (1974), Dieckmann (1987)l. Rather

complete discussions of this topic may be found in Penrose (1972, Section 3)

and in Hawking and Ellis (1973).

The simplest examples are formed by taking F = I+(S) or P = IP(S) ,

where S is an arbitrary subset of M. (Indeed, these sets may also be defined

in this manner.) They also share the following property with the chronological

past and future sets I-(p) and I+(p) of an arbitrary point p in M:

x in F and x << y implies y in F ; (future set)

x in P and y << x implies y in P. (past set).

The next two results give simple characterizations of the closures and bound-

aries of sets which are either future or past.

Proposition 3.7.

(1) If F is a future set, then F = {x E M : I+(x) c F) .

(2) If P is a past set, then P = {x E M : I-(2) C P}.

Proof. As usual, it suffices to establish (1). First, suppose I+(x) 2 F. We

may take {q,) 5 Z+(x) with lim,,,q, = x. Since {q,) 2 F, this yields

x E 7. Conversely, let x E 7 be given. Select any z E I+(x). Then x E I- (z)

which is open, and since x E 7, there exists y E I - (z )nF . But then y E F and

y << z implies z f F since F is a future set. Thus I+(x) C_ F as required. 0

Corollary 3.8. Let F (respectively, P) be a future (respectively, past) set.

Then

(1) B(F) = {x 6 M : x @ F and Z+(x) E F) , and

(2) d(P) = {x E M : x 4 P and I-(2) E P).

In the particular cases where F = I+(p) or P = I-(p), the following useful

characterizations of

-- -- I- (p) = J-(p) and I+(p) = J+ (p)


are obtained for arbitrary space-times:

- I+(p) = {x E M : I+(%) C I+(p)), (closure)

= {x E M : I-(2) I-(p)),

d(I+(p)) = {x E M : x 4f I+(p) and I+(x) C I+(p)), (boundary)

and d(I-(p)) = {x E M : x 4 I-(p) and I-(x) C I-(p)).

Past and future sets have played an important role in singularity theory

in general relativity as treated in Penrose (1972) or Hawking and Ellis (1973)

via the allied concept of achronal boundaries. A subset B of M is said to be

an achronal boundary if B = d(F) for some future set F . (Of course, it is

necessary to prove that a ( F ) is achronal, i.e., that no two points of d(F) may

be joined by a future timelike curve, for this definition to make sense.) In

particular, for any p in M the set

is a simple example of an achronal boundary. Achronal boundaries have the

following important regularity properties.

Theorem 3.9. The (nonempty) boundary d ( F ) of a future set is a closed

achronal (Lipschitz) topological hypersurface.

Discussions of this result are given in Penrose (1972, pp. 21-23) and in

Hawking and Ellis (1973, p. 187) from the viewpoint of applications in sin-

gularity theory. A rather complete treatment of a number of mathematical

aspects may be found in O'Neill (1983, pp. 413-415).

I t may happen that p E I+(p). If so, there is a closed timelike curve through

p, and the space-time is said to have a causality violation. For example, on

the cylinder M = S1 x W with the Lorentzian metric ds2 = -do2 + dt2, the

circles t = constant are closed timelike curves. In this space-time, I+(p) = M

for all p E M. A number of causality conditions have been defined in general

relativity in recent years because of the problems associated with examples of

causality violations.


Space-times which do not contain any closed timelike curves [i.e., p # I+(p)

for all p f M ] are said to be chronological. A space-time with no closed

nonspacelike curves is said to be causal. Equivalently, a causal space-time

contains no pair of distinct points p and q with p < q < p. The cylinder M =

S 1 x R with the Lorentzian metric ds2 = do d t is an example of a chronological

space-time that fails to be causal. The only closed nonspacelike curves in this

example are the circles t = constant, which with proper parametrization are

null geodesics.

The chronological condition is the weakest causality condition which will be

introduced. The next proposition guarantees that no compact space-time is

either causal or chronological.

Proposition 3.10. Any compact space-time (M, g ) contains a closed time-

like curve and thus fails to be chronological.

Proof. Since the sets of the form I + ( p ) are open, it may be seen that ( I f ( p ) :

p f M ) forms an open cover of M . By compactness, we may extract a finite

subcover ( I f ( P I ) , I f (p2) , . . . , I f ( p k ) ) . Now pl E I+(p ip) ) for some i ( 1 ) with

1 5 i ( 1 ) 5 k. Similarly, p,(l) E I+(pq2)) for some index i ( 2 ) with 1 5 i ( 2 ) 5 k .

Continuing inductively, we obtain an infinite sequence . . . << p , ( ~ ) << pi(2) << pi(l) << pl. Since k is finite, there are only a finite number of distinct P ~ ( ~ ) ' s .

Thus there are repetitions on the list, and from the transitivity of <<, it follows

that pi(,) f I'-(pi(,)) for some index pi(,). Thus (M,g) contains a closed

timelike curve through pi(n).

Tipler (1979) has proved that certain classes of compact space-times contain

closed timelike geodesics, not just closed timelike curves. Since the proof

uses the Lorentzian distance function as a tool, discussion of Tipler's result is

postponed until Section 4.1, Theorem 4.15.

A spacetime is said to be dzstinguishing if for all points p and q in M,

either I+(p) = I+(q) or I - ( p ) = I - ( q ) implies p = q. In a distinguishing

space-time, distinct points have distinct chronological futures and chronolog-

ical pasts. Thus, points are distinguished both by their chronological futures

and pasts.


A distinguishing space-time is said to be causally continuous if the set-

valued functions I + and I - are outer continuous. Since I+ and I - are always

inner continuous [cf. Hawking and Sachs (1974, p. 291)] , the causally con-

tinuous spacetimes are those distinguishing space-times for which both the

chronological future and past of a point vary continuously with the point.

Here I+ is said to be inner continuous a t p E M if for each compact set

K I+(p) , there exists a neighborhood U ( p ) of p such that K c I+(q) for

each q f U ( p ) . The set-valued function I+ is outer continuous a t p E M if -

for each compact set K E M - I+(p) , there exists some neighborhood U ( p ) -

of p such that K c M - I+(q) for each q E U ( p ) . Inner and outer continuity

of I - may be defined dually. An example of a space-time for which I - fails

t o be outer continuous is given in Figure 3.1. The concept of causal continu-

ity was introduced by Hawking and Sachs (1974). For these space-times the

causal structure may be extended to the causal boundary [cf. Budic and Sachs

(1974)l. Fbrthermore, a metrizable topology may be defined on the causal

completion of a causally continuous space-time [cf. Beem (1977)l.

An open set U in a space-time is said to be causally convex if no nonspace-

like curve intersects U in a disconnected set. Given p E M , the space-time

( M , g ) is said to be strongly causal at p if p has arbitrarily small causally con-

vex neighborhoods. Thus, p has arbitrarily small neighborhoods such that no

nonspacelike curve that leaves one of these neighborhoods ever returns. The

space-time (M, g) is strongly causal if it is strongly causal a t every point. It

may be shown that the set of points of an arbitrary space-time (M, g ) a t which

(M, g) is strongly causal is an open subset of M fcf. Penrose (1972, p. 30)]. I t

is not hard to show that strongly causal spacetimes are distinguishing.

Strongly causal space-times may be characterized in terms of the Alexan-

drov topology for M. The Alexandmv topology on an arbitrary space-time

(M,g) is the topology given M by taking as a basis all sets of the form

I+(p) n I - ( q ) with p, q E M (cf. Figure 1.2). The given manifold topology on

M is always at least as fine as the Alexandrov topology since I+(p) n I - ( q )

is an open set in the given topology by Lemma 3.5. The following result has

been obtained by Kronheimer and Penrose [cf. Penrose (1972, p. 34)] .


delete 1 /

FIGURE 3.1. A space-time which is not causally continuous is

shown. The map p -+ I-(p) fails to be outer continuous at the -

point q. The compact set K is contained in M - I - ( q ) , yet each

neighborhood U(q) of q contains some point T such that K is not -

contained in M - I-(r).

Proposition 3.11. The Alexandrov topology for (M, g) agrees with the

given manifold topology iff (M, g) is strongly causal.

Proof. Assume first that (M,g) is strongly causal. Then each p E M has

some convex normal neighborhood U(p) such that no nonspacelike curve in-

tersects U(p) more than once. The set U(p) is a convex normal neighborhood

of each of its points, and hence Proposition 3.4 implies that for each q E U(p),

the chronological future (respectively, past) of q in (U(p), g IU(,)) consists of


all points which can be reached by geodesic segments in U of the form exp,(tu)

for 0 5 t 5 1, where v is a future (respectively, past) directed timelike vector at

q. This demonstrates that the Alexandrov topology on (U(p), g l U ( p ) ) agrees

with the given manifold topology on U(p). Using the fact that no nonspacelike

curve of (M, g) intersects U(p) more than once, it follows that the Alexandrov

topology agrees with the given manifold topology.

Now assume that strong causality fails to hold at p E M. Then there is a

convex normal neighborhood V(p) of p such that if W(p) is any neighborhood

of p with W(p) V(p), a nonspacelike curve starts in W(p), leaves V(p),

and returns to W(p). It follows that all neighborhoods of p in the Alexandrov

topology contain points outside of V(p). Thus, the Alexandrov topology differs

from the given manifold topology. O

In order to study causality breakdowns and geodesic incompleteness in gen-

eral relativity, it is helpful to formulate the concept of inextendibility for non-

spacelike curves. This may be done as follows. Let y : [a, b) -+ M be a curve

in M. The point p E M is said to be the endpoint of y corresponding to t = b

if

lim y (t) = p. t - b -

If y : [a, b) -4 M is a future (respectively, past) directed nonspacelike curve

with endpoint p corresponding to t = b, the point p is called a future (re-

spectively, past) endpoint of y. A nonspacelike curve is said to be future

znextendible (or future endless) if it has no future endpoint. Dually, a past

inextendible nonspacelike curve is one that has no past endpoint.

Convention 3.12. A nonspacelike curve y : (a, b) -+ M is said to be

znextendible (or endless) if it is both future and past inextendible.

Causal space-times exist that contain inextendible nonspacelike curves hav-

ing compact closure. An example given by Carter is displayed in Figure 3.2

Icf. Hawking and Ellis (1973, p. 195)]. An inextendible nonspacelike curve

which has compact closure and hence is contained in a compact set is said to

be imprisoned. Thus, Carter's example shows that imprisonment can occur in

causal space-times.


identify

___f

z identify-

FIGURE 3.2. A causal space-time (M, g) is shown which has im-

prisoned nonspacelike curves that are inextendible. Let a be an

irrational number, and let M = W x S' x S1 = {(t, y, z) E IR3 :

(t, y, z) -- (t, y, z + l ) and (t, y, z) -- (t, y+1, z+a)). The Lorentzian

metric is given by ds2 = (cosht - 1)'(dy2 - dt2) - dt dy + dz2.

Let y : [a, b) -+ M be a future directed nonspacelike curve. Then y is said

to be future imprisoned in the compact set K if there is some to < b such

that y(t) E K for all to < t < b. The curve y is said to be partially future

imprisoned in the compact set K if there exists an infinite sequence t, f b

with y(t,) E K for each n.

If (M,g) is strongly causal and K is a compact subset of M , then K may

be covered with a finite number of convex normal neighborhoods {Ui) such

that no nonspacelike curve which leaves some U, ever returns to that Ui. This

observation leads to the following result.


Proposition 3.13. If (M, g) is strongly causal, then no inextendible non-

spacelike curve can be partially future (or past) imprisoned in any compact

set.

We now discuss another important class of space-times in general relativity,

stably causal space-times. For this purpose as well as for later use, it is helpful

to define the fine CT topologies on Lor(M).

Recall that Lor(M) denotes the space of all Lorentzian metrics on M. The

fine CT topologies on Lor(M) may be defined by using a fixed countable cov-

ering B = {Bi) of M by coordinate neighborhoods with the property that the

closure of each Bi lies in a coordinate chart of M and such that each compact

subset of M intersects only finitely many of the Bi's. This last requirement

is the condition that the covering be locally finite. Let 6 : M -+ (0 ,m) be

a continuous function. Then gl, g2 E Lor(M) are said to be 6 : M -+ (0, m )

close in the CT topology, written Igl - g21r < 6, if for each p E M all of the

corresponding coefficients and derivatives up to order r of the two metric ten-

sors gl and g2 are 6(p) close at p when calculated in the fixed coordinates of

all Bi 6 I3 which contain p. The sets {gl E Lor(M) : Igl - gzlr < 6 ) , with

g2 E Lor(M) arbitrary and 6 : M 4 (0 ,m) an arbitrary continuous function,

form a basis for the fine CT topology on Lor(M). This topology may be shown

to be independent of the choice of coordinate cover B.

The C' topologies for r = 0,1,2 may be given the following interpretations.

Remark 3.14. (1) Two Lorentzian metrics for M which are close in the

fine C0 topology have light cones which are close.

(2) Two Lorentzian metrics for M which are close in the fine C' topology have

geodesic systems which are close (cf. Section 7.2).

(3) Two Lorentzian metrics for M which are close in the fine C 2 topology have

curvature tensors which are close.

A space-time (M, g) is said to be stably causal if there is a fine CO neighbor-

hood U(g) of g in Lor(M) such that each gl E U(g) is causal. Thus a stably

causal space-time remains causal under small Co perturbations.


Stably causal space-times may be characterized in terms of a partial or-

dering < for Lor(M) defined using light cones to compare Lorentzian metrics.

Explicitly, if A is a subset of M , one defines gl < A g2 if for each p E A and

v E TpM with v # 0, gl(v,v) 5 0 implies g2(v,v) 5 0. One also defines

gl < A 92 if for each p E A and v E T,M with v # 0, gl(v,v) 5 0 implies

g2(v,v) < 0. We will write gl < g2 (respectively, gl < gz) for gl <M g2 (re-

spectively, gl <M g ~ ) . Thus gl < g2 means that a t every point of M the light

cone of gl is smaller than the light cone of g2, or g2 has wider light cones than

gl. I t may be shown that (M,g) is stably causal if and only if there exists

some causal gl E Lor(M) with g < gl.

A C0 function f : M -+ W is a global time function if f is strictly increasing

along each future directed nonspacelike curve. A space-time (MI g) admits

a global time function if and only if it is stably causal [cf. Hawking (1968),

Seifert (1977)l. However, there is generally no natural choice of a time function

for a stably causal space-time.

Let f : M -+ W be a smooth function such that the gradient V f is always

timelike. If y : (a, b) -+ M is a future directed nonspacelike curve with nonvan-

ishing tangent vector yl(t), then g(Of (y(t)), yl(t)) = yl(t)(f) must either be

always positive or always negative. Thus f must be either strictly increasing

or strictly decreasing along y. It follows that f must be strictly increasing or

decreasing along all future directed nonspacelike curves. Hence f or - f is a

smooth global time function for M. Furthermore, V f must be orthogonal to

each of the level surfaces f -'(c) = {p E M : f (p) = c), c f W, of f . These

level surfaces are hypersurfaces orthogonal to a timelike vector field and are

spacelike, i.e., g restricted to each of these hypersurfaces is a positive definite

metric. Since the gradient of f is nonvanishing and df is an exact 1-form, it

follows that M is foliated by the level surfaces {f-'(c) : c E W). Each non-

spacelike curve y of M can intersect a given level surface a t most once since f

must be strictly increasing or decreasing along y.

One of the most important causality conditions which we will discuss in

this section is global hyperbolicity. Globally hyperbolic space-times have the

important property, frequently invoked during specific geodesic constructions,


that any pair of causally related points may be joined by a nonspacelike geo-

desic segment of maximal length.

Definition 3.15. (Globally Hyperbolic) A strongly causal space-time (M,g) is said to be globally hyperbolic if for each pair of points p, q E M ,

the set Jf (p) n J-(q) is compact.

A distinguishing space-time (M,g) is causally simple if J+(p) and J-(p)

are closed subsets of M for all p E M. It then follows that

Proposi t ion 3.16. A globally hyperbolic space-time is causally simple.

- Proof. Suppose q E J+(p) - J+(P) for some p E M. Choose any r E I+(q).

- Since I - ( r ) is open and q E J+(p), it may be seen that r E I+(p) by taking

a subsequence {q,) 2 J+(p) with q, -+ q and using the fact that p 6 q, and

qn << r imply p << r. Consequently, q E Jt(p) n J - ( r ) - ( J+(p)n J- ( r ) ) . But

this is impossible since J+(p) fl J - ( r ) is compact hence closed. 0

Globally hyperbolic space-times may be characterized using Cauchy sur-

faces. A Cauchy surface S is a subset of M which every inextendible non-

spacelike curve intersects exactly once. (Some authors only require that every

inextendible timelzke curve intersect S exactly once.) It may be shown that

a space-time is globally hyperbolic if and only if it admits a Cauchy surface

[cf. Hawking and Ellis (1973, pp. 211-212)). Furthermore, Geroch (1970a) has

established the following important structure theorem for globally hyperbolic

space-times [cf. Sachs and Wu (1977b, p. 1155)l.

T h e o r e m 3.17. If (M, g) is a globally hyperbolic space-time of dimension

n, then M is homeomorphic to W x S where S is an (n - 1)-dimensional

topological submanifold of M, and for each t , { t ) x S is a Cauchy surface.

The proof of this theorem uses a function f : M -+ W given by f(p) =

m(J+(p))/m(J-(P)) where m is a measure on M with m ( M ) = 1. The level

sets off may be seen to be Cauchy surfaces as desired, but f is not necessarily

smooth.

A time function f : M --+ W will be said to be a Cauchy time function if

each level set f -'(c), c E W, is a Cauchy surface for M. In studying globally


hyperbolic space-times, it is helpful to use Cauchy time functions rather than

arbitrary time functions.

In a complete Riemannian manifold, any two points may be joined by a

geodesic of minimal length. Avez (1963) and Seifert (1967) have obtained a

Lorentzian analogue of this result for globally hyperbolic space-times.

Theorem 3.18. Let (M, g) be globally hyperbolic and p < q. Then there

is a nonspacelike geodesic from p to q whose length is greater than or equal to

that of any other future directed nonspacelike curve from p to q.

It should be emphasized that the geodesic in Theorem 3.18 is not necessarily

unique. This result will also be discussed from the viewpoint of the Lorentzian

distance function in Section 6.1.

It is natural to consider how continuity properties of a given candidate for

a time function on a given space-time are influenced by the causal structure

of the space-time. We have just indicated how Geroch (1970a) used past and

future volume functions to obtain the globally hyperbolic topological splitting

theorem (Theorem 3.17) and how stably causal space-times may be character-

ized in terms of the existence of a (continuous) global time function [cf. Hawk-

ing (1968), Seifert (1977), Hawking and Sachs (1974)l. It is often stated that

an auxiliary "additive measure H on M which assigns positive volume H[U] to

each open set U and assigns finite volume HIM] to M . . ." may be employed

in this context. For such a measure, it has been asserted that a distinguishing

space-time (M,g) is reflecting (cf. Definition 3.20), hence causally continuous,

if and only if the past and future volume functions t- and t+ associated to

any such measure H on M are continuous global time functions.

J. Dieckmann (1987, 1988) has noted that the above assertion fails to be

valid unless certain further regularity properties are imposed on the measure

H . In particular, one needs to restrict attention to measures that are not

positive on the boundaries of chronological futures and pasts (cf. Definition

3.19). The need for this restriction is best understood by means of an example.

Let (M, g) be four-dimensional Minkowski space-time, and let m be any Borel

measure for M with m(M) finite and with m(d(I+(p))) = m(b(l-(p))) = 0

for all p in M. Such a measure may be constructed using a partition of unity


and local volume forms as follows [cf. Hawking and Ellis (1973, p. 199), Geroch

(1970a, p. 446), Dieckmann (1988, p. 860)]. Let w be the usual volume form

for all of M , and let {p,) be a partition of unity subordinate to a covering

{Un) with each Un being a simple region and with

Then let m be the Borel measure associated (by integration) to the 4-form

It is easily seen that m ( d ( ~ + ( ~ ) ) ) = m(d(1-(p))) = 0 for any p in M .

Now define a measure H for M as follows. Put 0 = (0,0,0,0) and

Then even though (M,g) is certainly causally continuous, the past volume

function t-(p) = H[I-(p)] fails to be continuous at 0. The difficulty is that H

assigns measure one to the point 0. Thus, H assigns positive measure to light

cones containing this point.

This counterexample led Dieckmann (1987, 1988) to investigate more pre-

cisely, for a subclass of probability measures for a given space-time, how the

past and future volume functions are related to space-time causality. We will

summarize certain of these results. We begin with an arbitrary spacetime

(M,g) and, following Dieckmann, abstract the crucial properties of the Borel

measure m whose construction was sketched above.

Definition 3.19. (Admissible Measure) A Borel measure m on (M, g) is said to be admissible provided that

(3.1) m(U) > 0 for all nonempty open sets U,

(3.2) m(M) < +oo, and

(3.3) m(d(If (p))) = m(d(I-(p))) = O for all p in M .

Clearly, the above measure H fails to satisfy condition (3.3) and thus fails

to be admissible.


The past (respectively, future) volume functions (associated to the measure

m) are given respectively by

and

for all p in M.

The crucial importance of property (3.3) is that it implies

and

m (m) = 41- (PI)

for all p in M .

If the given space-time happens to be totally vicious, so that I-(p) =

I+(p) = M for all p in M, then t- and t+ take on the constant values m(M)

and -m(M), respectively. More generally, in the presence of certain causality

violations the past and future volume functions are only weakly increasing

along future causal curves and may also fail to be continuous. Hence the

volume functions do not, in general, define generalized time functions in the

sense of Definition 3.23 below without imposing some causality conditions on

the space-time in question.

For convenience, we will employ the notational convention used in O'Neill

(1983) that p < q if there exists a (nontrivial) future directed nonspacelike

curve from p to q. Thus, p < q if p < q and p # q. The usual transitivity

relationships (i.e., r << p, p < q, and q << s together imply r << q and p (< s)

yield that the volume functions t- and tf are (not necessarily continuous)

"semi-time functions" in the sense that for any p, q in M

(3.8) p < q implies t- (p) 5 t- (q) and t + (PI 5 t+(q).

3.2 CAUSALITY THEORY O F SPACE-TIMES 69

It is also interesting that the usual continuity properties of a measure imply

that t- (respectively, t+) is lower semicontinuous (respectively, upper semi-

continuous).

We now introduce a causality condition important in Hawking and Sachs

(1974) and in Dieckmann (1987, 1988).

Definition 3.20. (Reflecting) A space-time (M,g) is said to be past

reflecting (respectively, future reflecting) a t q in M if for all p in M

(3.9) I+(P) 2 I+(q) implies I-(p) C I-(q),

respectively,

(3.10) I-(p) _> I-(q) implies ~ ' ( p ) I + ( ~ )

and is said to be reflecting at q if it satisfies both conditions. The space-time

is said to be reflecting if it is reflecting at all points.

It is well known that for space-times which fail to be reflecting, the past and

future volume functions may fail to be continuous [cf. Figure 1.2 in Hawking

and Sachs (1974, p. 289)]. Even more strikingly, Dieckmann obtained the

following more precise relationship between continuity and reflexivity. In this

result, properties (3.6) and (3.7) for the admissible measure are crucial.

Proposit ion 3.21. Let (M, g) be a (not necessarily distinguishing) space-

time, and let t- and t+ be the past and future volume functions associated to

an admissible Bore1 measure. Then

(1) t- is continuous at q iff (M, g) is past reffecting at q, and

(2) t+ is continuous a t q iff (M, g) is future reflecting a t q.

Moreover, from his proof of a version of Proposition 3.21, Dieckmann is able

to conclude that the set of points a t which the volume function t- (respectively,

t+) fails to be continuous is a union of null geodesics without past (respectively,

future) endpoints. Especially, a space-time may not fail to be reflecting a t

isolated points, as had already been noted in Vyas and Akolia (1986).

Thus, reflexivity settles the question of the continuity of the volume func-

tions independent of choice of admissible measure. The "time function" aspect


o f being strictly increasing along future directed nonspacelike curves employs

the following lemma [cf. Dieckmann (1987, p. 47)).

L e m m a 3.22. Let ( M , g ) be an arbitrary {not necessarily distinguishing)

space-time, and let t - be the past volume function associated to an admissible

Borel measure. Suppose that p, q in M satisfy p < q and I - (p ) # I - (9) . Then

Lemma 3.22 has the immediate consequence that ( M , g) is chronological i f

and only i f some past (or future) volume function is strictly increasing along

all future timelike curves. More importantly, with Lemma 3.22 in hand the

following proposition may now be obtained with the help o f future set tech-

niques. For clarity, let us first make precise the notion o f "generalized time

function" as employed in the present context.

Definit ion 3.23. (Generalized Time Function) A (not necessarily con-

tinuous) function t : ( M , g ) -+ W is said to be a generalized time fisnction i f ,

for all p, q in M ,

(3.12) p < q implies t (p) < t(q).

Thus t is strictly increasing along all future nonspacelike curves but is not

required to be continuous.

Proposit ion 3.24. Let t - (respectively, t+) be the past (respectively, fu-

ture) volume function on ( M , g ) associated to an admissible Borel measure.

Then

(1 ) ( M , g ) is past distinguishing i f f t- is a generalized time function; and

(2) ( M , g ) is future distinguishing i f f t+ is a generalized time function.

Now it has been established that "causal continuity" (i.e., ( M , g ) is distin-

guishing and I+ and I - are outer continuous) is equivalent to "reflecting" and

"distinguishing" [cf. Hawking and Sachs (1974, p. 292)). Hence, Propositions

3.21 and 3.24 imply the following characterization o f causal continuity in terms

o f volume functions.

3.2 CAUSALITY THEORY O F SPACE-TIMES

T h e o r e m 3.25. The following are equivalent:

(1 ) The space-time ( M , g ) is causally continuous.

(2 ) For any (and hence all) admissible Borel measures, the associated vol-

ume functions t - and t+ are both continuous time functions.

The condition o f "causal continuity" implies the "stable causality" condi-

tion but not conversely. Stable causality has the characterization that ( M , g)

admits some continuous global time function (which will not, in general, be a

volume function associated to some admissible Borel measure).

W e now state the characterization of global hyperbolicity in terms of volume

functions [implicit in Geroch (1970a)l which is given in Dieckmann (1987).

T h e o r e m 3.26. Let t - (respectively, t+) be the past (respectively, future)

volume functions associated to an admissible Borel measure m for the space-

time ( M , g). Define t : ( M , g) -+ R by

for p in M . Then the following are equivalent:

(1 ) The space-time ( M , g) js globally hyperbolic.

(2 ) The past and future volume functions t - , t+ are continuous time

functions, and for any inextendible future directed nonspacelike curve

y : (a , b) -+ ( M , g ) we have lim,,bt+(~(u)) = l imu4at-(y(u)) = 0.

(3 ) The function t given by (3.13) is a continuous time function, and for

any inextendible future directed nonspacelike curve y, range(to y ) = W. (4 ) For all a in R, the set t - ' ({a)) is a Cauchy surface.

The next proposition gives sufficient conditions for a space-time to be causal

in terms o f the volume functions [cf. Dieckmann (1987)l.

Proposition 3.27. Let t be a past or future volume function associated to

an admissible Borel measure for the given space-time ( M , g ) . Suppose further

that

(1 ) p << q implies t ( p ) < t (q) for all p, q in M (which implies chronology),

and


(2) for each complete null geodesic ,f3 : R -+ (M, g), there exist u l , uz in W such that t(P(u1)) < t(P(u2)).

Then (M, g) is causal.

In Vyas and Joshi (1983), a discussion is given of how causal functions sim-

ilar to (3.5) may be related to the ideal boundary points and singularities of

a space-time. An interesting list of 71 assertions on causality to be proved

or disproved (together with answers) has been given in Geroch and Horowitz

(1979, pp. 289-293). We now give a diagram (Figure 3.3) indicating the rela-

tions between the causality conditions discussed above [cf. Hawking and Sachs

(1974, p. 295), Carter (1971a)l.

3.3 Limit Curves and the C O Topology on Curves

Two different forms of convergence for a sequence of nonspacelike curves

{y,) have been useful in Lorentzian geometry and general relativity [cf. Pen-

rose (1972), Hawking and Ellis (1973)l. The first type of convergence uses

the concept of a limit curve of a sequence of curves, while the second type

uses the C0 topology on curves. For arbitrary space-times, neither of these

types of convergence is stronger than the other. However, we will show that

for strongly causal space-times, these two forms of convergence are closely

related. This relationship will be useful in constructing maximal geodesics in

strongly causal space-times (cf. Sections 8.1 and 8.2).

Definition 3.28. (Limit Curve) A curve y is a limit curve ofthe sequence

(7,) if there is a subsequence (y,) such that for all p in the image of y, each

neighborhood of p intersects all but a finite number of curves of the subsequence

{y,). The subsequence (7,) is said to distinguish the limit curve y.

In general, a sequence of curves (7,) may have no limit curves or may

have many limit curves. This is true even if the curves (y,) are nonspace-

like. Furthermore, even in causal space-times a limit curve of a sequence of

nonspacelike limit curves is not necessarily nonspacelike. For example, the

curve ~ ( u ) = (0,0, u) in Carter's example (cf. Figure 3.2) is not nonspacelike

although it is a limit curve of any sequence {y,) of inextendible null geodesics

3.3 LIMIT CURVES AND THE C0 TOPOLOGY ON CURVES 73

globally hyperbolic

causally simple I 4

causally continuous

C I stably causal 1

1 strongly causal

C I distinguishing

4 I causal 1

I chronological 1 FIGURE 3.3. This diagram illustrates the strengths of the causal-

ity conditions used in this book. Global hyperbolicity is the most

restrictive causality assumption that we will use.

contained in the set t = 0. Recently, the phrase "cluster curve" has been sug-

gested as a more mathematically precise term for the convergence in Definition

3.28.

In contrast, we have the following result for strongly causal space-times.

Lemma 3.29. Let (M, g) be a strongly causal space-time. If y is a limit

curve of the sequence {y,) of nonspacelike curves, then y is nonspacelike.


where hij are the components of h with respect to the local coordinates

XI , . . . , 2,. Since Ixi'l 5 K1, the length Lo(y I [tl, ta]) satisfies

where H is the supremum of lhijj on the compact set for 1 5 i, j 5 n. Thus,

any nonspacelike curve from the level set f -l(tl) to the level set f -'(t2) which

lies in U has length bounded by n ~ ~ / ~ K1 Itl - t21. Furthermore, covering

(M, g) by a locally finite cover of sets with the properties of U and (V, x) above,

it follows that any nonspacelike curve of (M, g) defined on a compact interval of

R must have finite length with respect to h. Thus every nonspacelike curve of

(M, g) may be given a parametrization which is an arc length parametrization

with respect to h. Also, an inextendible curve y which has an arc length

parametrization with respect to h must be defined on all of W because do is

complete (cf. Lemma 3.65).

We now state a version of Arzela's Theorem which may be established using

standard techniques [cf. Munkres (1975, Section 7.5)].

Theorem 3.30. Let X be a locally compact Hausdorff space with a count-

able basis, and let (M, h) be a complete Riemannian manifold with distance

function do. Assume that the sequence {f,) of functions fn : X -+ M is

equicontinuous and that for each s o 6 X the set Un{fn(xo)) is bounded with

respect to 4. Then there exist a continuous function f : X + M and a sub-

sequence of {f,) which converges to f uniformly on each compact subset of

X.

Using Arzela's Theorem, we may now obtain the next proposition, given in

Hawking and Ellis (1973, p. 185), which guarantees the existence of limit curves

for a sequence (7,) of nonspacelike curves having points of accumulation.

Proposition 3.31. Let {y,) be a sequence of (future) inextendible non-

spacelike curves in (M, g). If p is an accumulation point of the sequence {y,),

then there is a nonspacelike limit curve y of the sequence {yn) such that p E y

and y is (future) inextendible.

Proof. We will give the proof only for inextendible curves since the proof

for future inextendible curves is similar.


Let h be an auxiliary complete Riemannian metric for M with distance

function 4 as above, and give each y,, an arc length parametrization with

respect to h. Then the domain of each y, i s R as each curve is assumed to

be inextendible. Shifting parametrizations if necessary, we may then choose

a subsequence {y,) of (7,) such that ym(0) + p as m + co since p is an

accumulation point of the sequence y,. Using the fact that each y, has an

arc length parametrization with respect to h, we obtain

for each m and tl,t2 E R. Thus the curves {y,) form an equicontinu-

ous family. Furthermore, since y,(O) + p, there exists an N such that

do(y,(O),p) < 1 whenever m > N. This implies that for each fixed to E R,

the curve ym 1 [-to, to] of the subsequence lies in the compact set {q E M :

do(p, q) 5 to + 1) whenever m > N. Hence the family {y,) satisfies the hy-

potheses of Theorem 3.30, and we thus obtain a (continuous) curve y : R -+ M

and a subsequence {yk) of the subsequence {y,) such that {yk) converges to

y uniformly on each compact subset of R. Clearly, yk(0) + p = y(0). The

convergence of {yk) to y also yields the inequality do(y(tl), y(t2)) 5 Itl - t21

for all t l , t2 E W. It remains to show that y is nonspacelike and inextendible.

To show that y is nonspacelike, fix t l E R and let U be a convex normal

neighborhood of (M,g) containing y(t1). Choose 6 > 0 such that the set

{q E M : do(y(tl),q) < 6 ) is contained in U . If tl < t2 < t l + 6, then (3.16)

and the uniform convergence on compact subsets yields that for all large k, the

set ~ k [ t l > t2] lies in U . Using yk(t1) ~ ( t l ) , ~ k ( t 2 ) --' ~ ( t 2 ) 1 ~ k ( t 1 ) <U ~ k ( t 2 ) for all large k, and the fact that U is a convex normal neighborhood, we obtain

that y(t1) y(t2). Thus y ] [tl,t2] is a future directed nonspacelike curve

in U [cf. Hawking and Ellis (1973, Proposition 4.5.1)]. It follows that y is a

future directed nonspacelike curve in (M, g).

It remains to show that y is inextendible. We will give the proof only of the

future inextendibility since the past inextendibility may be proven similarly.

To this end, assume that y is not future inextendible. Then y(t) -+ qo f M

as t -+ m. Let U' be a convex normal neighborhood of qo such that U' is a

compact set contained in a chart (V, x) of M with local coordinates (xl, . . . , x,)


such that f = x1 : U' -+ W is a time function for U'. An inequality of the

form of (3.15) shows that if y I [tl, m ) C U', then no nonspacelike curve in U'

from the level set f -I( f ( ~ ( t l ) ) ) to the level set f -'( f (qo)) can have arc length

with respect to h greater than some number 6' > 0. On the other hand, for

sufficiently large k we must have yk[tl+ l , t l +6' +2] 5 f -l([f (y(tl)), f (qo)]).

Since the length LO(yk[tl + l , t l + 6' + 21) = 6' + 1 for all k, this yields a

contradiction. (3

Even if all of the inextendible nonspacelike curves of the sequence {y,)

are parametrized by arc length with respect to a complete Riemannian met-

ric h, the limit curve y obtained in the proof of Proposition 3.31 need not

be parametrized by arc length. This is a consequence of the fact that the

Riemannian length functional, while lower semicontinuous, is not upper semi-

continuous in the topology of uniform convergence on compact subsets. Even

though the curve y constructed in the proof of Proposition 3.31 need not be

parametrized by arc length, the curve y will still be defined on all of R pro-

vided each y, is inextendible. Furthermore, if (M,g) is strongly causal, the

Hopf-Rinow Theorem and Proposition 3.13 imply that do(y(O), y(t)) --t co as

I t -+ co. Here, do denotes the complete Riemannian distance function in-

duced on M by h as above. An alternative treatment of the technicalities of

Proposition 3.31, closer in spirit to that given in Hawking and Ellis, may be

found in O'Neill (1983, p. 404) in the section on "quasi-limits."

In the globally hyperbolic case, a stronger version of Proposition 3.31 may

be obtained.

Corollary 3.32. Let (M, g) be globally hyperbolic. Suppose that {p,) and

{q,) are sequences in M converging to p and q in M respectively, with p < q,

p # 4, and p, < q, for each n. Let y, be a future directed nonspacelike curve

from p, to q, for each n. Then there exists a future directed nonspacelike

limit curve y of the sequence (7,) which joins p to q.

Proof. Let h be an auxiliary complete Riemannian metric on M with length

functional Lo. Choose a finite cover of the compact set J+(p) n J-(q) by con-

vex normal neighborhoods Ul, U2, . . . , Uk, each of which has compact closure

and such that no nonspacelike curve which leaves any U, ever returns to that


Ui. As in the proof of Proposition 3.31, there exists a number N, for each i such

that each nonspacelike curve y : [a, b ] -+ Ui has length less than N, with re-

spect to h [cf. equation (3.15)]. Thus if U = UIU.. . u U ~ and N = Nl +. . -+Nk,

every nonspacelike curve y : [a, b] + U must satisfy Lo(?) 5 N.

Extend each given nonspacelike curve y, to a future inextendible nonspace-

like curve, also denoted by y,. We may assume that each y, : [0, co) -+ M has

been parametrized by arc length with respect to h [cf. equation (3.15)]. Thus

if U = Ul U...UUk and Proposition 3.31 is applied to {y,) with accumulation

point p of {y,(O) = p,), then there exist a future inextendible nonspacelike

limit curve y : [0, m ) -+ M with $0) = p and a subsequence {y,) of {y,)

such that y, -+ y uniformly on compact subsets of [0, a). Using y,(t,) = q,

for 0 < t, 5 N and q, -+ q, we conclude that y passes through q for some

parameter value T which satisfies 0 < T I N. It follows that y 1 10, T] is a

nonspacelike limit curve of {y, 1 10, t,]) which joins p to q. 0

We now consider convergence in the C0 topology [cf. Penrose (1972, p. 49)].

Definition 3.33. (Convergence of Curves in C0 Topology) Let y and all curves of the sequence {y,) be defined on the closed interval [a, b]. The

sequence {y,) is said to converge to y in the C0 topology on curves if ?,(a) -+

y(a), yn(b) --+ y(b), and given any open set V containing y, there is an integer

N such that y, C V for all n 2 N.

Any space-time contains a sequence {y,) that has a limit curve y, yet {y,)

does not converge to y in the C0 topology. For, let a, P : [ O , 1 ] -+ M be any

two future directed timelike curves with a([O, I]) n P([O, I]) = 0. Set

Then {y,) does not converge to either a or P in the C0 topology. However, the

subsequence (72,) (respectively, {y2,-1)) of {-/n) converges to a (respectively,

p) in the C0 topology. A space-time which is not strongly causal may also

contain a sequence {y,) of nonspacelike curves which has a nonspacelike limit

curve y, yet no subsequence {y,) of {y,) converges to 7 in the C0 topology on

3 LORENTZIAN MANIFOLDS AND CAUSALITY

remove -- FIGURE 3.4. A causal space-time (M,g), in which a sequence

{y,} o f nonspacelike curves may have a limit curve 7 yet fail t o

have a subsequence which converges to y in the C0 topology on

curves, may be formed from a subset o f Minkowski space as shown.

curves. This is illustrated in Figure 3.4 [cf. Hawking and Ellis (1973, p. 193)

for a discussion of the causal properties o f this example].

Conversely, a sequence o f nonspacelike curves (7,) may converge in the C0

topology to some nonspacelike curve y but fail t o have y as a limit curve. This

may be seen on the cylinder M = S1 x W with the Lorentzian metric ds2 =

d8dt. Let y, be the segment on the generator 8 = 0 given by yn(t) = (0 , t )

for 0 5 t 5 1 and for all n. I f y is the piecewise smooth nonspacelike curve


obtained by going around the circle on the null geodesic t = 0 and then up

the generator B = 0 from t = 0 to t = 1 , then (7,) converges t o y in the C0

topology, but y is not a limit curve o f {y,) (c f . Figure 3.5).

In strongly causal space-times, however, these two types o f convergence are

almost equivalent for sequences of nonspacelike curves [cf. Beem and Ehrlich

(1979a, p. 164)]. A more precise statement is given by the following result.

Proposition 3.34. Let ( M , g ) be a strongly causal space-time. Suppose

that (7,) is a sequence o f nonspacelike curves defined on [a, b ] such that

yn(a) -+ p and yn(b) --+ q. A nonspacelike curve y : [a, b] -+ M with y(a) = p

and y(b) = q is a limit curve o f {y,} i f f there is a subsequence {y,} o f (7,)

which converges to y in the C0 topology on curves.

Proof. (+) W e may assume without loss o f generality that y and {y,) are

all future directed curves. Let V be any open set with y V . Cover the

compact image of y with convex normal neighborhoods W l , W 2 , . . . , W k such

that each Wi V and no nonspacelike curve which leaves Wi ever returns to

Wi. There exists a subdivision a = to < tl < . . . < t j = b o f la, b ] such that

for all 0 5 i 5 j - 1, each pair y(t,), y(t,+l) lies in some Wh. Here h = h ( i )

and 1 < h( i ) 5 k for all i. Let {y,} be a subsequence that distinguishes

y as a limit curve. For each m , let p(0, m ) = y,(a) and p ( j , m ) = y,(b).

Furthermore, for each fixed i with 0 < i < j , choose p(i, m) E ym such that

{ p ( i , m ) } converges to $ti). Since y(ti+l) lies in the causal future o f y(t,) and

M is strongly causal, the point p(i + 1, m) lies in the causal future o f p(i, m)

for all m larger than some N l . Also, there is some N2 such that p(i, m) and

p ( i + l , m ) l i e i n W h ( i ) f o r a l l O ~ i ~ j - l a n d m ? Nz. L e t N = m a x { N l , N z ) .

The portion o f ym joining p ( i ,m) to p(i + 1, m) must lie entirely in Wh(;) for

m 2 N because no nonspacelike curve can leave W h and return. It follows

that ym 5 W l U . . . U W k V for all m > N as required.

(e) Let {y,} be a subsequence o f {y,} converging to y in the C 0 topology on

curves. Define A = { to E [a, b] : each point o f y I [a, to] is a limit point o f the

given subsequence }. W e wish to show that A = [a, b] . Clearly, ym(a) -+ y(a)

implies a E A. I f r = sup{to : to E A ) , then for each a 5 t < 7 the point

y ( t ) is a limit point o f the subsequence (7,). To show r E A we assume

3 LORENTZIAN MANIFOLDS AND CAUSALITY

FIGURE 3.5. In chronological space-times, a sequence of non-

spacelike curves {y,) may converge to the nonspacelike curve y in

the C0 topology on curves, and yet y may fail to be a limit curve

of any subsequence (7,) of {y,). The curves y, are segments on

the line 0 = 0 from t = 0 to t = 1. The curve y goes around the

cylinder once, then traverses y,.

T > a and let i t k ) be a sequence with t k -+ T-. Each neighborhood U(y(7))

of y ( r ) is also a neighborhood of y(tk) for sufficiently large k and hence must

intersect all but a finite number of curves of the subsequence {y,). Thus y ( r )

is a limit point of {y,), and A must be a closed subinterval of [a, b]. Assume

that 7 < b. Using the strong causality of (M,g), we may find a convex normal

neighborhood V of y ( r ) such that no nonspacelike curve of (M, g) which leaves

V ever returns. Letting V be sufficiently small we may assume that (V, gj,) is

globally hyperbolic and that f : V --+ W is a Cauchy time function for (V, gIv)

with f (V) = R and f ($7)) = 0. We may also assume y(b) # V. Then each

inextendible nonspacelike curve of (M, g) which has a nonempty intersection

3.4 TWO-DIMENSIONAL SPACE-TIMES 83

with V must intersect each Cauchy surface f-'(s) exactly once. Fix s with

0 < s < m, and define x(s) = y n fP'(s). This intersection exists because

y ( r ) E V and y(b) $ V. Since y ( r ) and y(b) are limit points of {y,), the

curves y, must have a nonempty intersection with f-'(s) for all sufficiently

large m. Set x,(s) = y, n f-'(s) for all such m. In order to verify that

x,(s) 4 x(s), we observe that for each neighborhood W of y the points

x,(s) must lie in W fl f -'(s) for all large m (cf. Figure 3.6). This shows

that each x(s) is a limit point of the subsequence {y,). Consequently, the

set A contains numbers greater than T, in contradiction to the definition of

T. We conclude A = [a, b] which shows y is a limit curve of the subsequence

{yrn). 0

Let y : [a, b] + (M,g) be a nonspacelike curve in a strongly causal space-

time (M,g). Choose a compact subset K of M such that y 2 Int(K). Let the

nonspacelike curves which are contained in K be given the C0 topology. It

is known [cf. Penrose (1972, p. 54)] that the Lorentzian arc length functional

L(y) [cf. Chapter 4, equation (4.l)] is upper semicontinuous with respect to

the C0 topology on curves fcf. Busemann (1967, p. lo)]. This is the analogue

of the well-known result that the Riemannian arc length functional is lower

semicontinuous.

R e m a r k 3.35. Let (M,g) be strongly causal, and let y be a given non-

spacelike curve in (M, g). If the sequence {y,) of nonspacelike curves converges

to y in the C0 topology on curves, then

L(y) L lim sup L(y,).

3.4 Two-Dimensional Space-times

In this section we consider the topological and causal structures of two-

dimensional Lorentzian manifolds. Using the pair of null vector fields generated

by the tangent vectors to the two null geodesics passing through each point

of M, we show that the universal covering manifold of any two-dimensional

Lorentzian manifold is homeomorphic to R2. We then show that any two-

dimensional Lorentzian manifold homeomorphic to R2 is stably causal. In


FIGURE 3.6. In the proof of Proposition 3.34 the globally hyper-

bolic neighborhood V of y(7) has a Cauchy time function f : V --+ R

with f ($7)) = 0. For all large m the curves ym must intersect

the Cauchy surface f-'(s) at a single point xm(s). If W is any

neighborhood of y, then xm(s) E W n fF1(s) for all large m. We

may choose W such that W n f -'(s) is as small a neighborhood of

Z(S) = y n f -'(s) in f -l(s) as we wish. Thus xm(s) -+ x(s), and

x(s) must be a limit curve of the subsequence {y,).

particular, every simply connected two-dimensional Lorentzian manifold is

causal. Thus no Lorentzian metric for R2 has any closed nonspacelike curves.

It should also be noted that two- (but not higher) dimensional Lorentzian man-

ifolds have the property that (M, -g) is also Lorentzian. This is sometimes

3.4 TWO-DIMENSIONAL SPACGTIMES 85

useful in obtaining results about all geodesics in (M,g) from results valid in

higher dimensions only for nonspacelike geodesics.

Let (M, g) be an arbitrary two-dimensional Lorentzian manifold, and fix a

point p E M. Choose a convex normal neighborhood U(p) based at p, and

consider the following method of assigning local coordinates to points in U(p)

sufficiently close to p. Let the two null geodesics yl and 72 through p be

given parametrizations yl : (-el, €1) -+ U(p) and 72 : (-e2, €2) -' U(p) with

yl(0) = yz (0) = p. For each point q E U(p) sufficiently close to p, the two null

geodesics through q will intersect yl and 7 2 in U(p) at unique points yl(to)

and y2(so) respectively. Assign coordinates (to, so) to q. In these coordinates

the null geodesics near p are contained in sets of the form t = to or s = so.

We have established

Lemma 3.36. Let (M, g) be a two-dimensional Lorentzian manifold. Then

each p E M has local coordinates x = (XI, 22) with x(p) = 0 such that each null

geodesic in this neighborhood is contained in a set of the form x1 = constant

or $2 = constant.

Suppose X is a future directed timelike vector field on M. Then at each p E

M, there are two uniquely defined future directed null vectors nl , n2 E T,M

such that X(p) = nl + n2. Clearly, a sufficiently small neighborhood U(p) of p

may be found such that nl and n2 may be extended to continuous null vector

fields XI, X2 defined on U(p) with X(q) = Xl(q) + X2(q) for all q E U(p). If

M is simply connected, we now show X1 and X2 can be extended to all of M.

Proposition 3.37. Let (M,g) be a simply connected Lorentzian manifold

of dimension two. Then two smooth nonvanishing null vector fields X1 and

X2 may be defined on M such that X1 and X2 are linearly independent a t

each point of M.

Proof. Since M is simply connected, (M,g) is time orientable. Thus we

may choose a smooth future directed timelike vector field X on M.

Fix a base point po E M, and let X(po) = nl + 7x2 as above. Given any

other point q E M, let y : [0,1] -4 M be a curve from po to q. There is exactly

one way to define continuous null vector fields X1 and X2 along y such that


Xl(0) = n l , Xz(0) = n2, and X(y(t)) = Xl(t) + Xz(t) for all t E [O,l]. If

77 : [O, 11 -+ M is any other curve from po to q, then y and 7 are homotopic since

M is assumed to be simply connected. Hence if Yl and Y2 were null vector

fields along q with Yl(0) = n l and E(0 ) = n2, we would have Yl(1) = X1(l)

and Yz(1) = Xz(1) by standard homotopy arguments. Thus this construction

produces a pair of continuous vector fields XI and X2 on M which are linearly

independent at each point. C1

Corollary 3.38. Let (M, g) be any two-dimensional Lorentzian manifold.

Then the universal Lorentzian covering manifold (z,~) of (M, g) is homeo-

morphic to R2.

Proof. Since % is simply connected and tw~dimensional, is homeornor-

phic to EC2 or S2. But since the Euler characteristic of S2 is nonzero, S2 does

not admit any nowhere zero continuous vector fields.

Recall that an integral curve for a smooth vector field X on M is a smooth

curve y such that yl(t) = X(y(t)) for all t in the domain of y [cf. Kobayashi

and Nomizu (1963, p. 12)]. The following result is well known [cf. Hartman

(1964, p. 156)].

Proposi t ion 3.39. Let X be a smooth nonvanishing vector field on R?,

and let y : (a, b) -, R2 be a maximal integral curve of X . Then y(t) does not

remain in any compact subset of R2 as t + a+ [or t + b-).

Now assume that ( M , g ) is a Lorentzian manifold homeomorphic to R2. Let XI , X2 be the null vector fields on M given by Proposition 3.37. Clearly,

each null geodesic of (M,g) may be reparametrized to an integral curve of

X1 or X2. Equivalently, the integral curves of XI and X2 are said to be null

pregeodesics. Suppose y : (a, b) + M is an inextendible null geodesic which

may be reparametrized to an integral curve of XI. If y(t1) = y(t2) for some

t l # t2, then since both yl(tl) and yli(tz) are scalar multiples of Xl(y(tl)), it

follows from geodesic uniqueness that y is a smooth closed geodesic. However,

this is impossible by Proposition 3.39. Thus Proposition 3.39 has the following

corollary.

3.4 TWO-DIMENSIONAL SPACGTIMES 87

Corollary 3.40. If (M, g) is a Lorentzian manifold homeomorphic to R2,

then (M, g) contains no closed null geodesics. Moreover, every inextendible

null geodesic y : (a, b) + M is injective and hence contains no loops.

A family F of inextendible null geodesics is said to cover a manifold M

simply if each point p f M lies on exactly one null geodesic of F. Suppose

(M, g ) is a Lorentzian manifold homeomorphic to W2. Then the integral curves

of the null vector field X1 (respectively, Xz) given in Proposition 3.37 may be

reparametrized to define a family Fl (respectively, F2) of geodesics on M . Each

family F, covers M since X,(p) # 0 for i = 1, 2 and all p E M . Furthermore,

since exactly one integral curve of X, passes through any p E M , each family

F, covers M simply. Consequently, Proposition 3.37 implies [cf. Beem and

Woo (1969, p. 51)]

Proposi t ion 3.41. Let (M, g) be a Lorentzian manifold homeomorphic to

R2. Then the inextendible null geodesics of (M,g) may be partitioned into

two families F1 and F2 such that each of these families covers M simply.

Let y : (a, b) -+ M be an inextendible timelike curve, and let c : (cu,P) -+

M be an inextendible null geodesic. Obviously, in arbitrary two-dimensional

Lorentzian manifolds, y and c may intersect more than once. However, if M

is homeomorphic to W2, y and c intersect in a t most one point [cf. Beem and

Woo (1969, p. 52), Smith (1960b)l.

Proposi t ion 3.42. Let (M, g) be a Lorentzian manifold homeomorphic to

R2. Then each timelike curve intersects a given null geodesic at most once.

Proof. Let co be an inextendible future directed null geodesic in M , which

we may assume belongs to the family Fl defined by the null vector field XI as

above. Suppose that a is a future directed timelike curve in M which intersects

co twice (possibly a t the same point). We may then find a, b f R with a < b

such that u(a), u(b) lie on co and u(t) 4 for a < t < b. Since u is timelike, u

is locally one-to-one. Hence if u / [a, b] is not one-to-one, u contains a t worst

closed timelike loops. Using one of these loops. it is possible to find a , /3 f R

with a < a < ,B < b and a second null geodesic cl E Fl such that u 1 [a, /3] is

one-to-one, u (a ) and u(p) lie on cl, and u(t) 4 cl for cu < t < p.


geodesic

FIGURE 3.7. In a Lorentzian manifold homeomorphic to R2, the

timelike curve y is assumed to cross the null geodesic Q at y(a)

and y(b). The null geodesic cl enters W at y(tl) and first leaves at

y(t;) where t i > t l .

We will show below that if y : [a, b] -, M is an injective future directed

timelike curve, there is no c E FI such that ~ ( a ) and y(b) lie on c but y(t) $ c for a < t < b. If the original timelike curve a 1 [a, b ] is injective, this

argument applied to a 1 [a, b] yields the desired contradiction. If a I [a, b ] is

not injective but intersects at a(a) and a(b), then this argument applied to

cl and a 1 [a, p] yields the desired contradiction.

3.4 TWO-DIMENSIONAL SPACE-TIMES 89

Thus the theorem will be established if we show that it is impossible to find

an injective future directed timelike curve y : [a, b] --+ M with y(a) and y(b)

on q and y(t) $ GJ for a < t < b. Traversing y from y(a) to y(b) and then the

portion of q from ?(b) to y(a) yields a closed Jordan curve which encloses a set

W with 'ii;i compact (cf. Figure 3.7). Let U be a convex normal neighborhood

based on y(a). Choose t l with a < t l < b and y(t1) E U. Let cl be the null

geodesic in Fl passing through y(tl). Since cl may be reparametrized to be

an integral curve of XI and cl enters W at y(tl) , it follows by Proposition

3.39 that cl leaves W a t some point y(ti) with t i > tl. As q intersects y at

y (b), we must have t i < b. In particular, itl, ti] C (a, b). Hence we have found

a closed interval [tl,t;] 2 (a,b) such that y([tl,t;]) 2 and y intcrsccts the

null geodesic cl E Fl at y(tl) and ?(t:) (cf. Figure 3.7).

We may now form a second closed Jordan curve by traversing y from tl to

t i followed by the portion of cl from y (t i) to y (tl). Repeating the argument

of the preceding paragraph, we obtain a closed interval [tz, t i] C (tl,t',) such

that the timelike curve y 1 [tz, t i] intersects a null geodesic c2 in the family

Fl at y(t2) and y(ti) and such that y(t2) is contained in a convex normal

neighborhood of y(tl). Inductively, we can construct a nested sequence of

intervals [tk+1, C (tk,tjc) such that ~ ( t k + ~ ) lies in a convex normal neigh-

borhood of y(tk) and y I [ t k + l r t ~ + l ] intersects a null geodesic ck+l € Fl at

Y(tk+l) and ~ ( t : + ~ ) . Moreover, the intervals [tk, t i ] may be chosen such that

n E l [ t k , t i ] = {to) for some to E (a,b). We thus have constructed two se-

quences tk 1 to and t: 1 to such that the timelike curve y intersects a null

geodesic in Fl at both y(tk) and y(tjc) for each k 2 1. But this is impossible

by Proposition 3.4. Hence the geodesic y : [a, b ] -+ M intersects at most

once.

Theorem 3.43. Let (M, g) be a Lorentzian manifold homeomorphic to R2.

Then (M, g) is stably causal.

Proof. Recall that g E Lor(M) is stably causal if there is a fine C0 neigh-

borhood U of g in Lor(M) such that all metrics in U are causal. Since strongly

causal implies causal, it will thus follow that all metrics in Lor(M) are stably

causal if all metrics in Lor(M) are strongly causal. Thus to prove the theorem,


it is enough to show that if g is any Lorentzian metric for M , then (M, g) is

strongly causal.

Thus suppose that g is a Lorentzian metric for A4 such that (M, g) is not

strongly causal. Then there is some p E M such that strong causality fails at

p. Let (U, x) be a chart about p, guaranteed by Lemma 3.36, such that the

null geodesics in U lie on the sets x l = constant and x2 = constant. Since

strong causality fails a t p, there are arbitrarily small neighborhoods V of p

with V C U and timelike curves which begin at p, leave V , and then return to

V. By Proposition 3.42, there are no closed timelike curves through p. Thus

if y is a future directed timelike curve with y(0) = p which leaves V and then

returns, we have y ( t ) # p for all t > 0. Since the null geodesics in U through p

are given by x1 = 0 and by 2 2 = 0 in the local coordinates x = (XI , x2) for U ,

it follows that y may be deformed to intersect one of the null geodesics through

p upon returning to V (cf. Figure 3.8). Hence y intersects a null geodesic in

Fl or F2 twice, contradicting Proposition 3.42.

Corollary 3.44. No Lorentzian metric for R2 contains any closed non-

spacelike curves.

A different proof of the result that any simply connected Lorentzian two-

manifold is strongly causal may be found in O'Neill (1983). For n > 3,

Lorentzian metrics which are not chronological and hence not strongly causal

may be constructed on Rn.

For every two-dimensional Lorentzian manifold (M,g), there is an associ-

ated Lorentzian manifold (MI -g). The timelike curves of (MI -g) are the

spacelike curves of (M, g) and vice versa. Using M = R2 and applying Corol-

lary 3.44 to (M, -g) we obtain

Corollary 3.45. No Lorentzian metric for R2 contains any closed spacelike

curves.

If (M, g) is two-dimensional and both (MI g) and (M, -g) are stably causal,

then using techniques given in Beem (1976a), one may show there is some

smooth conformal factor 52 : M --, (0 ,m) such that the manifold (M, Rg) is

geodesically complete. This yields the following corollary.

3.4 TWO-DIMENSIONAL SPACE-TIMES

FIGURE 3.8. (M,g) is a two-dimensional space-time such that

strong causality fails a t p. There is a future directed timelike curve

y which starts a t p, later returns close to p, and crosses one of the

null geodesics through p.

Corollary 3.46. Let (M,g) be a Lorentzian manifold homeomorphic to

R2. Then there is a smooth conformal factor 0 : M -+ (0,m) such that

(M, 52g) is geodesically complete.

There are examples of twudimensional space-times such that no global con-

formal change makes them nonspacelike geodesically complete (cf. Section 6.2).

Thus, Corollary 3.46 cannot be extended to all two-d~mensional space-times

by covering space arguments. The recent monograph by Weinstetn (1993)

contains many further interesting results on Lorentzian surfaces inspired in

part by Kulkarni's (1985) study of the conformal boundary for such a surface

[cf. Smyth and Weinstein (1994)l.

-


3.5 The Second Fundamental Form

Let N be a smooth submanifold of the Lorentzian manifold (M,g) . If

i : N + M denotes the inclusion map, then we may regard TpN as being a

subspace of TpM by identifying i,,(TpN) with TpN. Let go = z*g denote the

pullback of the Lorentzian metric g for M to a symmetric tensor field on N .

Under the identification of T p N and i ep(TPN) , we may also identify go at p

and g ( T,N x TpN for all p f N . This identification will be used throughout

this section.

Definition 3.47. (Nondegenerate Submanifold) The submanifold N of

( M , g ) is said to be nondegenemte if for each p E N and nonzero v E T p N ,

there exists some w f TpN with g(v, w) f 0. If, in addition, g 1 TpN x TpN is

positive definite for each p E N , then N is said to be a spacelike submanzfold.

If g I TpN x TpN is a Lorentzian metric for each p E N , then N is said to be

a timelike submanifold.

For the rest of this section, we will suppose that N is a nondegenerate

submanifold. Thus for each p E N , there is a well-defined subspace T k N of

T p M given by

T ~ N = { v ETpM : g(v,w) = 0 for all w E T p N )

which has the property that T k N n T p N = (0). Consequently, there is a well-

defined orthogonal projection map P : TpM + TpN. The connection V on

( M , g) may be projected to a connection V 0 on N by defining V g Y = P ( V x Y )

for vector fields X , Y tangent to N . It is easily verified that V 0 is the unique

torsion free connection on ( N , go) satisfying

X ( ~ ~ ( y l z ) ) = ~ O ( V % Y ? ~ ) + g ~ ( ~ l v % z )

for all vector fields X, Y, Z on N . The second fundamental form, which mea-

sures the difference between V and V O , may be defined just as for Riemannian

submanifolds [cf. Hermann (1968, p. 319), Bolts (1977, p. 25, pp. 51-52)].

Definition 3.48. (Sewnd Fundamental Form) Let N be a nondegener-

ate submanifold of (M,g) . Given n E T ~ N , define the second fundamental

3.5 THE SECOND FUNDAMENTAL FORM 93

form Sn : TpN x TpN -, W in the direction n as follows. Given x , y E T p N ,

extend to local vector fields X, Y tangent to N , and put

Define the second fundamental form S : T i N x T p N x TpN -+ W by

Given n E T i N , the second fundamental form operator L, : TpN + TpN is

defined by g(L,(x), y) = Sn(x, y) for all x, y E T p N .

It may be checked that this definition of S,(x, y) is independent of the

choice of extensions X , Y for x , y E TpN and also that S, : TpN x TpN -+ W is a symmetric bilinear map. Furthermore, S : T i n r x TpN x TpN -, W is

trilinear for each p E N .

Lemma 3.49. Let N be a nondegenerate submanifold of (M,g) . The

second fundamental form S = 0 on N iff V x Y = V g Y for all vector fields X

and Y tangent to N .

Proof. Obviously, Definition 3.48 implies that if V x Y = V$Y for all vector

fields tangent to N , then S = 0.

Now suppose S = 0. Let p E N be an arbitrary point. We then have

g ( V x Y I p - V $ Y l p , n ) = 0 for all n E T i N and vector fields X, Y tangent to

N . Since g T p N x T p N is nondegenerate, g 1 T i N x T i N is also nondegen-

erate. Thus V x Y I p and V $ Y l p have the same projection onto T k N . Since

TpM = TpN @ T k N , we have O X Y I P = V $ Y l p , as required. 0

The second fundamental form may be used to characterize totally geodesic

nondegenerate submanifolds of ( M , g). A submanifold N of ( M , g) is said to be

geodesic at p E N if each geodesic y of ( M , g) with y(0) = p and ~ ' ( 0 ) E TpN

is contained in N in some neighborhood of p. The submanifold N is said to be

totally geodesic if it is geodesic at each of its points. The following proposition

is the Lorentzian analogue of a well-known Riemannian result Icf. Hermann

(1968, p. 338), Cheeger and Ebin (1975, p. 23)].


Proposit ion 3.50. Let N be a nondegenerate submanifold of (M, g). Then

N is totally geodesic iff the second fundamental form S satisfies S = 0 on N .

Proof. Given that S = 0 on N , Lemma 3.49 implies that VxY = V$Y

for all vector fields X , Y tangent to N. Let c : (-6, c) --+ M be a geodesic in

(M,g) with cl(0) = v E TpN for some p E N. Also let y : (-45) --+ N be

the geodesic in (N,go) with yl(0) = v. Since Vy,yl = V:,yl = 0, the curve y

is also a geodesic in (M,g). Set 11 = min{c,6). From the uniqueness of the

geodesic in (M, g) with the given initial direction v, we have c(t) = y(t) for all

t E (-7, 7). Hence y 1 ( - q , ~ ) N as required.

Conversely, suppose N is totally geodesic in (M, g). Let p E N be arbitrary.

Given n E T i N and x E TpN, let c : J -, N be the geodesic (in both M

and N ) with c'(0) = x. Extend cl(t) to a vector field X tangent to N near p.

We then have S(n, x, x) = g(V,yXlp, n) = g(V,lcl(0), n) = g(0, n) = 0. By

polarization, it follows that S(n, x, y) = 0 for all x, y E TpN. Hence S = 0 on

N. 0

As will be seen in Chapter 12, the second fundamental form plays an im-

portant role in singularity theory in general relativity.

3.6 Warped Produc t s

If (M, g) and (H, h) are two Riemannian manifolds, there is a natural prod-

uct metric go defined on the product manifold M x H such that ( M x H, go) is

again a Riemannian manifold. Bishop and 07Neill(1969) studied a larger class

of Riemannian manifolds, including products, which they called warped prod-

ucts. If (M,g) and (H, h) are two Riemannian manifolds and f : M -+ (0, oo)

is any smooth function, the product manifold M x H equipped with the met-

ric g @ f h is said to be a warped product and f : M + (0, m) is called the

warping functzon. Following Bishop and O'Neill, we will denote the Riemann-

ian manifold ( M x H I g @I fh) by M x f H. Bishop and O'Neill (1969, p. 23)

showed that M x f H is a complete Riemannian manifold if and only if both

(M,g) and (H, h) are complete Riemannian manifolds. Utilizing this result,

they were able to construct a wide variety of complete Riemannian manifolds

of everywhere negative sectional curvature using warped products.

3.6 WARPED PRODUCTS 95

In this section, we will use warped product metrics to construct Lorentzian

manifolds and will then study the causal structure and completeness proper-

ties of this class of Lorentzian manifolds. The theory for Lorentzian mani-

folds differs from the Riemannian theory somewhat, since the product of two

Lorentzian manifolds (M, g) and (HI h) has signature (-, -, T, . . . , +) and

hence is not Lorentzian. Nevertheless, warped product Lorentzian metrics

may be constructed from products of Lorentzian and Riemannian manifolds.

In particular, this product construction may be used to produce examples of

bi-invariant Lorentzian metrics for Lie groups (cf. Section 5.5). A treatment of

warped products of semi-Riemannian (not necessarily Lorentzian) manifolds,

including a calculation of their Riemannian and Ricci curvature tensors, is

given in O'Neill (1983).

Throughout this section, we will let T : M x H -+ M and 7 : M x H -,

H denote the projection maps given by ~ ( m , h) = m and ~ ( m , h) = h for

( m , h ) ~ M X H.

Definition 3.51. (Lorentzian Warped Product) Let (M,g) be an n-

dimensional manifold (n 2 1) with a signature of (-, +, . . . , +), and let (H, h)

be a Riemannian manifold. Let f : M -+ (0, cm) be a smooth function. The

Lorentzian warped product M x f H is the manifold % = M x H equipped

with the Lorentzian metric ?j defined for v, w E T~~ R by

Definition 3.52. (Lorentzian Product) A warped product M x f H with

f = 1 is said to be a Lorentzian product and will be denoted by M x H .

R e m a r k 3.53. One may also obtain Lorentzian manifolds by considering

warped products H x f M , where (H, h) is a Riemannian manifold, (M, g) is a

Lorentzian manifold, and f : H --+ (0, a) is a smooth function. The universal

covering manifold of anti-de Sitter space (cf. Section 5.3) is an example of a

space-time important in general relativity which may be written as a warped

product of the form H x f M with H Riemannian and M Lorentzian but not

as a warped product of the form M x f H of Definition 3.51. We will only treat

warped products of the form M x f H in this section.

96 3 LORENTZIAN MANIFOLDS AND CAUSALITY f 3.6 WARPED PRODUCTS 97

We begin our study of the causal properties of warped products with the From the hierarchy of causality conditions given in Figure 3.3, we then

following lemma. obtain

Lemma 3.54. The warped product M x f H of ( M I g) and ( H , h) may be Corollary 3.56. Let ( H I h ) be an arbitrary Riemannian manifold, and let time oriented iff either ( M , g) is time oriented (if dim M 2 2) or ( M , g) is a M = (a , b) with -cm 5 a < b 5 +a begiven the negative definite metric -dt2. one-dimensional manifold with a negative definite metric. For any smooth function f : M -t (0, a), the warped product ( M x f H, 3 ) is

Proof. Suppose that M x f H is time orientable. If dimM = 1. then chronological, causal, distinguishing, and strongly causal.

( M , g ) has a negative definite metric by Definition 3.51. Now consider the

case dim M > 2. Since M x f H is time orientable, there exists a continuous

timelike vector field X for M x f H. Since f > 0 and h is positive definite, we

then have g(r,X, r , X ) 5 g ( X , X ) < 0. Thus the vector field n,X provides a

time orientation for ( M , g).

Conversely, suppose first that dim M 2 2 and ( M , g) is time oriented by the

timelike vector field V. Then V may be lifted to a timelike vector field 7 on

M x H which satisfies r,v = V and 77,V = 0. Explicitly, fixing j j = (m, b) E

M x H , there is a natural isomorphism

Thus we may define V at j j by setting V(p) = ( V ( m ) , Ob) using this isomor-

phism to identify TF (M x H ) and TmM x TbH. It is immediate from Definition - -

3.51 that 3 ( V , V ) = g(V, V ) < 0. Hence 7 time orients M x f H as required.

Now consider the case dim M = 1. It is then known that M is diffeomorphic

to S 1 or R. In either case, let T be a smooth vector field on M with g(T, T ) =

-1. Defining T(P) = (T(r(jj))lO,(ii)) as above, we have 71*T = 0, SO that T time orients B. Note also in the case that M = S1 the integral curves of T in - M are closed timelike curves. Thus M is not chronological. 0

Lemma 3.55. Let ( H I h) be an arbitrary Riemannian manifold, and let

M = (a, b) with -cc 5 a < b 5 +cm be given the negative definite metric -dt2.

For any smooth function f : M --, (0, a), the warped product ( M x f H , g ) is

stably causal.

Proof. The projection map n : M x H --+ M W serves as a time func-

tion. El

In the proof of Lemma 3.54 above, we have seen that if M = S1, then the

warped product (S1 x f H , g ) fails to be chronological and hence fails to be

causal, distinguishing, or strongly causal.

We now list some elementary properties of warped products that follow

directly from Definition 3.51. A homothetic map F : ( M I , g l ) -+ (M2, 92) is a

diffeomorphism such that F'(g2) = cgl for some constant c. We remark that

some authors only require homothetic maps to be smooth and not necessarily

one-to-one.

Remark 3.57. Let M x f H be a Lorentzian warped product.

(1) For each b 6 H , the restriction r/v-l(b) : 77-'(b) --+ M is an isometry

of q-l(b) onto M .

(2 ) For each m f MI the restriction qlT-l(m) : T-'(m) --+ H is a homo-

thetic map of r - ' ( m ) with homothetic factor I / f (m).

(3) If v E T ( M x H ) , then g(n,v,r,v) 5 g(v, v ) . Thus n , : T p ( M x H ) -+

TT(p)M maps nonspacelike vectors to nonspacelike vectors, and r maps

nonspacelike curves of M x f H to nonspacelike curves of M .

(4) Since l g ( n , v , ~ ~ v ) / > lg(v,v)/ if w 6 T(M x H ) is nonspacelike, the

map r is length nondecreasing on nonspacelike curves (cf. Section 4.1,

formula (4.1) for the definition of Lorentzian arc length).

(5) For each (m,b) E M x H , the submanifolds r-'(m) and q-vb) of

M x f H are nondegenerate in the sense of Definition 3.47.

(6) If q5 : H -+ H is an isometry, then the map @ = 1 x q5 : M x f H -+

M x f H given by @(m, b) = (m,q5(b)) is an isometry of M x f H.

(7) If $ : M -+ M is an isometry of M such that f o ?C, = f , then the map

Q =$J x 1 : M x f H -+ M x f H given by Q(m,b) = (?C,(m),b) is an


isometry of M x f H. Thus if X is a Killing vector field on M (i.e.,

Lxg = 0) with X ( f ) = 0, then the natural lift X of X to M x f H

given by x ( p ) = (X(n(p)), Oq(*)) is a Killing vector field on M x f H.

Lemma 3.58. Let M x f H be a Lorentzian warped product. Then for

each b E H , the leaf v-l(b) is totally geodesic.

Proof. Since the map n : M x f H -+ M is length nondecreasing on non-

spacelike curves and since nonspacelike geodesics are locally length maximiz-

ing, it follows that any nonspacelike geodesic of 77-'(b) (in the metric induced

by the inclusion q-'(b) 5 M x f H ) is a geodesic in the ambient manifold

M x f H. Thus the second fundamental form vanishes on all nonspacelike vec-

tors in T(q-'(b)). Since any tangent vector in T(q-'(b)) may be written as a

linear combination of nonspacelike vectors in T(q-'(b)), it follows that the sec-

ond fundamental form vanishes identically. Hence, 77-'(b) is totally geodesic

by Proposition 3.50. 0

In view of Corollary 3.56, we may now restrict our attention to studying

the fundamental causal properties of time oriented Lorentzian warped products

( M x H,i j ) with dim M 2 2.

Lemma 3.59. Let p = (pl, pz) and q = (ql, q2) be two points in M x f H

with p << q (respectively, p < q) in (M x f H , g ) . Then pl << ql (respectively,

Pl < q1) in (M,g).

Proof. If y is a future directed timelike (respectively, nonspacelike) curve

in M x f H from p to q, then n o y is a future directed timelike (respectively,

nonspacelike) curve in M from pl to ql.

While n : M x f H -+ M takes nonspacelike curves to nonspacelike curves,

.rr does not preserve null curves. Indeed, it follows from Definition 3.51 that if

y is any smooth null curve with q,y(t) # 0 for all t , then g(z,y(t), T, y(t)) < 0 for all t.

For points p and q in the same leaf q-l(b) of M x f H, Lemma 3.59 may be

strengthened as follows.


[isometric to M]

[homothetic to H]

FIGURE 3.9. Let (m, b) be a point of the warped product M x f H.

Then the projection map n restricted to q-l(b) is an isometry onto

M, and the projection map 77 restricted to ?rp'(m) is a homothetic

map onto H .

Lemma 3.60. If p = (pl, b) and q = (91, b) are points in the same leaf

~ - l ( b ) of M X I H , then p q (respectively, p Q q) in (M x f H , g ) iffpl << ql

(respectively, pl ,< ql) in (M, g).

Proof. By Lemma 3.59, it only remains to show that if pl << ql (respectively,

pl < ql) in (M,g), then p << q (respectively, p ,< q) in (M x f H,ij) . But if

yl : [O, 11 --, M is a future directed timelike (respectively, nonspacelike) curve

in M from pl to ql, then y ( t ) = (yl(t), b), 0 5 t 5 1, is a future directed

timelike (respectively, nonspacelike) curve in M x f H from p to q.


Lemma 3.60 implies that each leaf 77-'(b), b E H , has the same chronology

and causality as (M,g). In particular, Lemmas 3.59 and 3.60 imply that

( M x f H , g ) has a closed timelike (respectively, nonspacelike) curve iff (M,g)

has a closed timelike (respectively, nonspacelike) curve. Hence

Proposition 3.61. Let (M, g) be a space-time, m d let (H, h) be a Rie-

mannian manifold. Then the Lorentzian warped product (M x j H , g ) is

chronological (respectively, causal) iff (M, g) is chronological (respectively,

causal).

A similar result holds for strong causality.

Proposition 3.62. Let (M, g) be a space-time, and let (H, h) be a Rie-

mannian manifold. Then the Lorentzian warped product (M x j H,ij) is

strongly causal iff (MI g) is strongly causal.

Proof. We first show that if the space-time (M, g) is not strongly causal a t

pl , then ( M xf H , g ) is not strongly causal a t p = (pl, b) for any b E H. Since

(M, g) is not strongly causal at pl, there is an open neighborhood Ul of pl in

M and a sequence {yk : [O,1] -+ M ) of future directed nonspacelike curves

with yk(0) -+ pl and yk(1) -+ pl as k -+ m , but yk(1/2) $! U1 for all k. Define

uk : I0,1] -+ M x H by ak(t) = (yk(t), b ) Let Vl be any open neighborhood

of b in H, and set U = Ul x Vl in M x H. Then U is an open neighborhood of

p = (pl, b) in M x f H , and {ak) is a sequence of nonspacelike future directed

curves in M x f H with ak(0) -+ p and ak( l ) -+ p as k -+ m , but ak(1/2) $ U

for all k. Thus ( M x f H , g ) is not strongly causal a t p.

Conversely, suppose that strong causality fails a t the point p = (pl, q l ) of

( M x j H,ij). Let (xl,. . . ,x,) be local coordinates on M near pl such that g

has the form diag(-1, + I , . . . ,+I) at pl, and let (x ,+~, . . . , on) be local coor-

dinates on H near ql such that f h has the form diag(f1,. . . , +1) a t ql. Then

(51,. . . , x,, x,+l,. . . , x,) are local coordinates for M x f H near p. Further-

more, Fl = x1 and Fz = xl o .ir are (locally defined) time functions for M

near pl and for M x f H near p, respectively. The failure of strong causal-

ity a t p implies the existence of a sequence yk : I O , l ] -+ M x f H of future

directed nonspacelike curves with yk(0) -+ p and yk( l ) -+ p as k -+ m, but

3.6 W A R P E D PRODUCTS 101

Fz(yk(1/2)) 2 E > 0 for all k and some parametrization of ~ k . Choose a

neighborhood W of pl in M such that W is covered by the local coordinates

(XI, . . . , xi) above and such that sup(F1 (r) : r E W ) 5 ~ / 2 . The curves

.rr 0 yk are then future directed nonspacelike curves in ,nil with .ir o ~ ~ ( 0 ) -+ pl ,

.ir 0 yk(l) -+ pl, and .ir o yk(1/2) $! W . Hence W and {T o yk) show that strong

causality fails a t pl in (M, g) required. I?

In Proposition 3.64 we prove the equivalence of stable causality for (M, g)

and ( M x j H, ij) for dim M 2 2. From this proposition and the last two

propositions, it follows that the basic causal properties of (M x f H,i j ) are

determined by those of (M, g).

Remark 3.63. If g < gl on M , then there is a smooth conformal factor

R : M -+ (0, m) such that Rgl(v, 21) < g(71,~) for all nontrivial vect.ors which

are nonspacelike with respect to g.

Proposition 3.64. Let (M, g) be a space-time and (H, h) a Riemannian

manifold. Then the Lorentzian warped product (M x f H , ij ) is stably causal

iff (M, g) is stably causal.

Proof. In this proof we will use the identification Tp(M x H ) 2 Tp, M x TbH

for a l l p = (pl,b) E M x H .

Assuming that ( M x f H, 3 ) is stably causal, there exists ij, E Lor(M x H)

such that ij < gl and ?jl is causal. If b is a fixed point of H , then we may

assume without loss of generality that ?jlI,-,(,) is nondegenerate since ?jI,-,(,)

is nondegenerate. Setting = ijll,-l(b, and using T/,-,(~) to identify 77-'(b)

with M , we obtain a metric gl E Lor(M) such that .irl,-l(b) is an isometry

of (7-'(b),:~) onto (M,gl). Notice that since ( M x H,G1) is causal, the

space-time (q-l(b),&) is causal, and hence (M,gl) is also causal. To show

g < g1 on M , we choose a nonzero vector vl E Tpl M such that g(v1, vl) 5 0.

If Ob denotes the zero vector in TbH, then g(v,v) = g(vl,vl) 5 0. where

v = (vl, Ob) E Tpl M x TbH. Since ij < gl, we obtain ij,(v, v) = gl(vl, v l ) < 0.

Hence g < gl, and (M, g) is stably causal.

Conversely. we now assume that ( M g) is stably causal. Let gl E Lor(M)

be a causal metric with g < gl. By Remark 3.63 we may also assume that


g1 ( v l , v l ) < g(v1, v l ) for all vectors vl # 0 which are nonspacelike with respect

to g. Since Proposition 3.61 implies that g1 = gl @ f h is a causal metric on

M x H , it suffices to show that 3 < 3,. To this end, let v = ( v l , v z ) be a

nontrivial vector o f T p ( M x H ) which is nonspacelike with respect to g. Then

since 3(v , v ) = g(v1, v l ) + f ( 4 2 1 ) ) . h(v2,v2) I 0 and f ( ~ ( v ) ) . h(v2, vz) > 0

with v2 # 0 , the nontriviality of v implies that vl # 0 and g(v l , v l ) 5 0. Thus

31(v, u) = gi(ui ,v i ) + f ( 4 ~ ) ) . h ( v 2 , ~ 2 ) < g(v1,vl) + f ( n ( v ) ) . h ( v 2 , ~ 2 ) I 0

which shows that 3 < 3, and establishes the proposition.

Geroch's Splitting Theorem (cf . Theorem 3.17) guarantees that any globally

hyperbolic space-time may be written as a topological product R x S where

S is a Cauchy hypersurface. Geroch's result suggests investigating conditions

on ( M , g ) and ( H , h ) which imply that the warped product ( M x f H , g ) is

globally hyperbolic. These conditions are given for dim M = 1 and dim M 2 2

in Theorems 3.66 and 3.68, respectively. In order to prove these results, it is

first necessary to show that a curve in a complete Riemannian manifold which

is inextendible in one direction must have infinite length.

Lemma 3.65. Let ( H , h ) be a complete Riemannian manifold, and let

y : [O, 1) -+ H be a curve of finite length in ( H , h). Then there exists a point

p E H such that y ( t ) -4 p as t -+ I - .

Proof. Let do denote the Riemannian distance function induced on H by

the Riemannian metric h. Let L = Lo(?) be the Riemannian arc length of y ,

and set K = { q E H : do(y(O),q) 5 L) . The Hopf-Rinow Theorem [cf. Hicks

(1965, pp. 163-164)] implies that K is compact. Fix a sequence (t,) in [O,1)

with t , -+ 1. Since d(y(O), y ( t ) ) 5 L ( y 1 [O,t]) 5 L for t E [0,1), we have

y[O, 1) C K. Thus by the compactness o f K , the sequence {y( t , ) ) has a limit

point p E K. I f lirnt,i- y ( t ) # p, there would then exist an E > 0 such that y

leaves the ball {m E M : d(p, m) 5 E ) infinitely often. But this would imply

that 7 has infinite length, in contradiction.

The following theorem may be obtained from the combination o f Corollary

3.56 and Lemma 3.65. The proof, which is similar to that o f Theorem 3.68,

will be omitted.


Theorem 3.66. Let ( H , h) be a Riemannian manifold, and let M = (a , b)

with -m 5 a < b I +co be given the negative definite metric -dt2. Then

the Lorentzian warped product ( M x f H , g ) is globally hyperbolic i f f ( H , h )

is complete.

Theorem 3.66 may be regarded as a "metric converse" to Geroch's splitting

theorem. I f f = 1 is assumed, so that the warped product ( M x f H , g ) is

simply a metric product ( M x H , g @ h) , Theorem 3.66 may be strengthened

to include geodesic completeness (c f . Definition 6.2 for the definition o f geodesic

completeness).

Theorem 3.67. Suppose that ( H , h ) is a Riemannian manifold and that

R x H is given the product Lorentzian metric -dt2 @ h. Then the following

are equivalent:

(1) ( H , h ) is geodesically complete.

(2 ) ( R x H , -dt2 @ h ) is geodesically complete

(3 ) ( R x H, -dt2 @ h ) is globally hyperbolic.

Proof. W e know that ( 1 ) i f f (3 ) from Theorem 3.66. Thus it remains to

show (1) i f f (2). But this is a consequence o f the fact that all geodesics o f

IR x H are either (up to parametrization) o f the form (At, c ( t ) ) , (Ao, c ( t ) ) , or

(At, ho), where A, A. E R are constants, ho E H , and c : J -+ H is a unit speed

geodesic in H . D

Suppose that a space-time ( M , g) o f dimension n 2 3 satisfies the timelike

convergence condition (i.e., has everywhere nonnegative nonspacelike Ricci

curvatures) and satisfies the generic condition (i.e., each inextendible non-

spacelike geodesic contains a point where the tangent vector W satisfies the

equation xF,d=l WCWdW[aRblcd[eWf l # 0 [cf. Chapter 21). Then i f ( M , g ) has

a compact Cauchy surface, the space-time ( M , g ) is geodesically incomplete.

Thus Theorem 3.66 may not be strengthened for arbitrary warped products

to include geodesic completeness. The "big bang" Robertson-Walker cosmo-

logical models (c f . Section 5.4) are examples o f globally hyperbolic warped

products which are not geodesically complete.


In contrast, let (W x H, -dt2 6+ h ) be a product space-time of the form

considered in Theorem 3.67. Fix any bo E H. Then y( t ) = ( t , bo) is a timelike

geodesic with R(y l ( t ) , v ) = 0 for all v E T-,(t)(W x H ) for each t E W. Thus

(R x H, -dt2 @ h ) fails to satisfy the generic condition.

If dim M = 1 and M is homeomorphic to W, we have just given necessary

and sufficient conditions for the warped product M x f H to be globally hyper-

bolic. If M = S 1 , we remarked above that (M x f H , g ) is nonchronological no

matter which Riemannian metric h is chosen for H. Thus no warped product

space-time ( S 1 x f H, 3 ) is globally hyperbolic.

We now consider the case dim M 2 2.

Theorem 3.68. Let ( M , g) be a space-time, and let (H, h) be a Riemann-

ian manifold. Then the Lorentzian warped product ( M x f H , g ) is globally

hyperbolic iff both of the following conditions are satisfied:

( 1 ) ( M , g) is globally hyperbolic.

(2) ( H , h ) is a complete Riemannian manifold.

Proof. (+) Suppose first that ( M x f H , g ) is globally hyperbolic. Fixing

b E H , we may identify ( M , g ) with the closed submanifold 77-'(b) = M x {b)

since the projection map r : 7/-1(b) -+ M is an isometry, Lemma 3.60 implies

that under this identification, the set J f (p l ) n 3-(91) in M corresponds to

77-l(b) n J+((pl, b)) n J - ( ( Q , b)) in ( M x f H ) for any pl and ql in M . Since

?-l(b) is closed and ( M x f H, 3 ) is globally hyperbolic, 77-'(b) n J f ( ( p l , b) ) n J-((91, b)) is compact in ( M x f H). Hence J+(pl) n J-(ql) is compact in

M . Because ( M x f H , 3 ) is globally hyperbolic, it is also strongly causal.

Thus ( M , g ) is strongly causal by Proposition 3.62. Hence ( M , g) is globally

hyperbolic as required.

Now we show that (M x f H,G) globally hyperbolic implies that ( H , h ) is

a complete Riemannian manifold. We will suppose that ( H , h ) is incomplete

and derive a contradiction to the global hyperbolicity of ( M x j H , g ) . For

this purpose, fix any pair of points pl and ql in M with pl << ql, and let

7 1 : [0, L] --t M be a unit speed future directed timelike curve in M from

pl to ql. Set a: = sup{f(y l ( t ) ) : t E [O, L ] ) where f : M -+ ( 0 , m ) is the

3.6 WARPED PRODUCTS

FIGURE 3.10. In the proof of Theorem 3.68, the curve c : [0, P ) -+

H is a geodesic which is not extendible to t = /3 < m. The curves

T ( t ) = ( y l ( t ) , c ( t ) ) and p(t) = ( y l ( L - t ) , c ( t ) ) are inextendible

nonspacelike curves in ( M x f H, 3 ) and hence do not have compact

closure.

given warping function. Since y1([0, L ] ) is a compact subset of M , we have

o < a < m .

Assuming that ( H , h ) is not complete, the Hopf-Rinow Theorem ensures the

existence of a geodesic c : [0,/3) -+ H with h(cl(t) , c l ( t ) ) = l / a which is not

extendible to t = /3 < m. By changing c(0) and reparametrizing c if necessary,

we may suppose that 0 < /3 < L/2. Define a future directed nonspacelike curve

7 : [0, /3) -+ M x H and a past directed nonspacelike curve 5 : [0, /3) -+ M x H

by ~ ( t ) = (y l ( t ) , c( t ) ) and ;I;(t) = (yl ( L - t ) , c( t ) ) , respectively. For each t with 0 5 t < /3, we have y l ( t ) << y l ( L - t ) in ( M , g ) since t < L - t . Hence


by Lemma 3.60, we have (y l ( t ) , c ( t ) ) << ( y l ( L - t ) , c ( t ) ) in M x f H . Thus

(p l , c(1)) ,< ~ ( t ) << T ( t ) < (91, c(0)) for all 0 5 t < P (c f . Figure 3.10). It

follows that ?([O, P ) ) is contained in J+ ( ( p l , c(0))) n J - ((ql , ~ ( 0 ) ) ) . Since

c = r] o 7 does not have compact closure in H , the curve 7 : [O, P ) -+ M x f H

does not have compact closure in J+ ((pl , c(0))) n J - ((ql , ~ ( 0 ) ) ) . But since

( M x f H , g ) is globally hyperbolic, the set J + ( ( ~ ~ , c ( o ) ) ) n J - ( ( q l , c ( ~ ) ) ) is

compact, in contradiction.

(e) Suppose now that ( M , g ) is globally hyperbolic. Assuming that the

warped product ( M x f H , i j ) is not globally hyperbolic, we must show that

( H , h ) is not complete. Since ( M , g ) is strongly causal, ( M x f H , i j ) is also

strongly causal by Proposition 3.62. Hence since ( M x f H , i j ) is not globally

hyperbolic, there exist distinct points (p l , b l ) and (p2, b2) in M x f H such

that J+((pl , b l ) ) n J-((pz, b2)) is noncompact. There is then a future directed

nonspacelike curve y : [O, 1) -+ J+((pl, b l ) ) n J-((p2, b2)) which is future inex-

tendible in ( M x f H . 3 ) . Let y ( t ) = (ul ( t ) ,u2( t ) ) , where u1 : [O, 1) --+ M

and u2 : [O, 1) -+ H . Then u1 : [O, 1 ) -+ M is a future directed non-

spacelike curve contained in J+(pl) n JW(p2). Since ( M , g ) is globally hy-

perbolic, J+(pl) n JW(p2) is compact. Hence i f we set a0 = i n f { f ( m ) : m E

J+(pl) n J-(pz)), then c r ~ > 0. Also since (M,g) is strongly causal, no future

directed, future inextendible, nonspacelike curve may be future imprisoned in

the compact set J+(pl) n J-(p2) (c f . Proposition 3.13). Hence there exists a

point r E J+(pl) n J-(pz) with lim,,l- u l ( t ) = r . W e may then extend u1

to a continuous curve u1 : [O, 11 -+ M by setting u l (1 ) = r. Since the curve

y = (u1,u2) was inextendible to t = 1, it follows that u z ( t ) cannot converge

to any point of H as t -+ I-. By Lemma 3.65, either ( H , h ) is incomplete or

u2 has infinite length. As u1 : [ O , l ] -+ M is a nonspacelike curve defined on a

compact interval, ul has finite length in (M,g) . Since f ( u l ( t ) ) > cro > 0 for

all t E [0, I ] and

it follows that u2 has finite length in ( H , h). Thus ( H , h ) is incomplete as

required. U


Cauchy surfaces may be constructed for globally hyperbolic Lorentzian

warped products as follows.

Theorem 3.69. Let ( H , h ) be a complete Riemannian manifold. Let

( M x f H , g ) be the Lorentzian warped product of ( M , g ) and ( H , h ) .

(1 ) I f M = (a , b) with -m < a < b 5 +co is given the metric -dt2, then

{ p l ) x H is a Cauchy surface o f ( M x f H , i j ) for each pl E M .

(2 ) I f ( M , g ) is globally hyperbolic with Cauchy surface S1 , then S1 x H

is a Cauchy surface of ( M xf H , g ) .

Proof. Since the proofs o f (1 ) and (2) are similar, we shall only give the proof

o f (2) . In this case, S1 x H is an achronal subset of ( M x j H , 3 ) . To show

S1 x H is a Cauchy surface, we must show that every inextendible nonspacelike

curve in M x f H meets S1 x H. Now given (pl,p2) E ( M x H ) - (S1 x H ) ,

either every future directed, future inextendible, nonspacelike curve in ( M , g)

beginning at pl meets S1 or every past directed, past inextendible, nonspacelike

curve starting at pl meets S1. Since the two cases are similar, we will suppose

the former holds and then show that every future directed, future inextendible,

nonspacelike curve y : [O, 1) --+ M x f H with y(0) = (pl ,pz) meets S1 x H.

Thus suppose that y : [O,l) -+ M x f H is a future directed, future inex-

tendible, nonspacelike curve with y(0) = (pl ,pz) which does not meet S1 x H .

Decompose y ( t ) = (u l ( t ) ,ua( t ) ) with ul : [O, 1 ) -+ M and u2 : [O, 1) + H .

Since S1 is a Cauchy surface for ( M , g) and ( M , g ) is globally hyperbolic,

the set J+(pl) O J-(S1) is compact [cf. Beem and Ehrlich (1979a, p. 163)).

As in the proof o f Theorem 3.68, the strong causality o f ( M , g) implies that

there exists a point r E J+(pl) n J-(S1) with limt,l- u l ( t ) = r. Since

J+(pl) n J - ( S l ) is compact, the warping function f : M -+ (0, co) achieves a

minimum cro > 0 on J+(pl) n J- (S1) . As in the proof o f Theorem 3.68, this

then implies that u2 : [O, 1) -+ H has finite length. Since ( H , h ) is complete,

by Lemma 3.65 there exists a point b f H with l imthl- u2(t) = b. Setting

y(1) = ( r , b), we have then extended y to a nonspacelike future directed curve

y : [O, 11 -+ M x f H , contradicting the inextendibility o f y. Hence y must

meet S1 x H as required. 0


We now consider the nonspacelike geodesic completeness of the class of

Lorentzian warped products of the form = (a, b) x f H with g = -dt2 @

f h. Here a space-time is said to be null (respectively, timelike) geodesically

incomplete if some future directed null (respectively, timelike) geodesic cannot

be extended to be defined for arbitrary negative and positive values of an affine

parameter (cf. Definitions 6.2 and 6.3). Since we are using the metric -dt2

on (a, b), the curve c(t) = (t, yo) with yo E H fixed is a unit speed timelike

geodesic in (z, 3 ) no matter which warping function is chosen. Consequently,

if a > -oo or b < +co, then ( M , g ) is timelike geodesically incomplete for

all possible warping functions f . Moreover, if a and b are both finite and if y

is any timelike geodesic in z = (a , b) x f H, then L(y) 5 b - a < oo. Thus

if a and b are finite, all timelike geodesics are past and future incomplete.

Nonetheless, if the warping function f is chosen suitably, (%,g) may be null

geodesically complete even if a and b are both finite. This will be clear from

the proof of Theorem 3.70 below.

If z = W x f H with 3 = -dt2 @ fh , then any timelike geodesic of the

form c(t) = ( t , yo) is past and future timelike complete. However, warped

product space-times = W x f H may be constructed for which all nonspace-

like geodesics except for those of the form t + (t, yo) are future incomplete.

One such example may be given as follows. Busemann and Beem (1966) stud-

ied the space-time = = {(x, y) € W2 : y > 0) with the Lorentzian metric

ds2 = y-2(dx2 - dy2). Busemann and Beem (1966, p. 245) noted that all

timelike geodesics except for those of the form t + (t, yo) are future incom-

plete. Setting t = In y, this space-time is transformed into the Lorentzian

warped product R x f R with 3 = -dt2 @ fdt2, where f (t) = e-2t. Since the

map F : ( S , ds2) -+ (W x f W,g) given by F(x, y) = (x, ln y) is a global isome-

try, all timelike geodesics of (W x f W, B) except for those of the form t -+ (t, yo)

are future incomplete. It will also follow from Theorem 3.70 below that all

null geodesics are future incomplete. Similarly, if (Wn, h) denotes Rn with the

usual Euclidean metric h = dxI2 + dxZ2 + . . - + dxn2, then (W x f Wn, ?j) with

g = -dt2 @ f h and f (t) = e-2t is a space-time with all nonspacelike geodesics

future incomplete except for those of the form t -+ (t, yo).


In order to study geodesic completeness, it is necessary to determine the

Levi-Civita connection for a Lorentzian warped product metric. For this pur-

pose, we will consider the general warped product ( M x f H , g e fh) where

f : M -+ (0, co), (H, h) is Riemannian, and (M, g) is equipped with a metric of

signature (-, +, . . . , +). Let V1 denote the Levi-Civita connection for (MI g)

and V2 denote the Levi-Civita connection for (H, h). Given vector fields

XI, Yl on M and X2, Y2 on H , we may lift them to M x H and obtain the vec-

tor fields X = (XI, 0) + (0, X2) = (XI, X2) and Y = (Yl, 0) + (0, Yz) = (Yl , Y2)

on M x H. Recall that the connection 7 for ( M x f H , g @ f h ) is related to

the metric 3 = g @ f h by the Koszul formula

[cf. Cheeger and Ebin (1975, p. 2)]. Using this formula and setting 4 = In f ,

we obtain the following formula for 7 for X and Y as above:

Here grad4 denotes the gradient of the function 4 on (M,g), and we

are identifying the vector VklYl 1, E T,M with the vector (V&,Yl, 0,) E

T(,,,)(M x HI, etc. We are now ready to obtain the following criterion for null geodesic incom-

pleteness of Lorentzian warped products z = (a, b) x f H [cf. Beem, Ehrlich,

and Powell (1982)l. Throughout the rest of this section, let wo denote an

interior point of (a, b).

Theorem 3.70. Let z = (a, b) x f H be a Lorentzian warped product

with Lorentzian metric = -dt2 @ f h where -co 5 a < b 5 +m, (H, h) is

an arbitrary Riemannian manifold, and f : (a, b) -+ (0, m). Set S(t) = m. Then if lim,-.+ JtwO S(s)ds respectively, limt+a- ~ ( s ) d s ] is finite, every [ future directed null geodesic in ( X , ? j ) is past [respectively, future] incomplete.

Proof. Let yo be an arbitrary future directed null geodesic in (?i?,?j). We

may reparametrize yo to be of the form y(t) = (t,c(t)), where y is a smooth


null pregeodesic. Accordingly, there exists a smooth function g(t) such that

[cf. Hawking and Ellis (1973, p. 33)]. However, since yl(t) = d l%[ , +cl(t) and

?j(yl, yl) = - 1 + g(cl, c') = 0, we obtain using formula (3.17) that

Equating terms with a d l & component, we obtain the formula

Thus ~ 7 t y 1 1 t = (1/2)[ln f (t)]' y'(t) = [lnS(t)ll yl(t). If we define p : (a, b) -+ R

by

~ ( t ) = Jr s ( ~ ) d ~ , wo

then pl(t) = S( t ) > 0 so that p-' exists. Moreover, from the classical theory

of projective transformations we know that the curve yl(t) = y o p-'(t) =

(p-'(t), c o p-l(t)) is a null geodesic [cf. Spivak (1970, pp. 6-35 ff.)]. Let

A = lim p(t) and B = lim p(t). t-a+ t-b-

Since p is monotone increasing, we have that p : (a, b) -+ (A, B ) is a bijection.

Hence p-' : (A, B) -+ (a, b) and thus yl = y o p-l : (A, B ) --+ ?@. Therefore if

A is finite, yl is past incomplete, and if B is finite, yl is future incomplete as

required.

It is immediate from Theorem 3.70 that if a and b are finite and the warp

ing function f : (a, b) -+ (0, m ) is bounded, then (?@,g) is past and future

null geodesically incomplete. Thus, assuming that a and b are finite, one-

parameter families ( a , g ( s ) ) = (?@, -dt2 @ f(s)h) of past and future null

geodesically incomplete space-times may easily be constructed. Choosing the

one-parameter family of functions f (s) : (a, b) 4 (0, m ) suitably, the curve


s + g(s) = -dt2 @ f (s)h of metrics will not be a continuous curve in ~ o r ( M )

in the fine C' topologies. Thus the space-times (2, g(0)) and ( M, g(s)) may

be far apart in ~ o r ( M ) for s # 0.

Notice that if the Riemannian manifold (H, h) is geodesically incomplete,

then = (a, b) x H may be null geodesically incomplete even if both integrals

in Theorem 3.70 diverge. On the other hand, if the completeness of (H, h) is

assumed, the following necessary and sufficient condition for the null geodesic

incompleteness of '7i? = (a, b) x H may be obtained from the proof of Theorem

3.70.

Remark 3.71. Let ?@ = (a, b) x f H be a Lorentzian warped product with

Lorentzian metric 3 = -dt2 @ fh , where (H, h) is a complete Riemannian

manifold and - m 5 a < b 5 +m. Let S(t) = as above. Then

( M , 3 ) is past (respectively, future) null geodesically incomplete if and only if

lim LwO S(S) ds is finite (respectively, t-a+

In Powell (1982), a more comprehensive study is made of nonspacelike ge-

odesic completeness of Lorentxian warped products, beginning with the ob-

servation that a geodesic in M x f H projects to a pregeodesic in (H, h) and - continuing with the observation that if 7 and ,B are unit speed geodesics in

(H, h) and (y, 5) is a pregeodesic of M x f H, then (y, g) is also a pregeodesic

of M x f H . Then Powell shows that if the Lorentzian warped product M x f H

is timelike (respectively, null or spacelike) geodesically complete, then (H, h)

is Riemannian complete and further (M,g) is timelike (respectively, null or

spacelike) complete.

A second aspect of Powell (1982) is the study of timelike geodesic com-

pleteness for warped products of the form 2 = (a, b) x f H with metrics

g = -dt2@ f h , corresponding to Theorem 3.70 and Remark 3.71 above for null

completeness. As remarked above, the completeness of the timelike geodesics

of the form T(t) = (t, yo) for some yo E H, which Powell terms "stationary,"

is entirely dependent on whether a = -co or a is finite, and/or b = +co or b

is finite. Also, if a and b are both finite, then all timelike geodesics are hoth

past and future incomplete independent of choice of warping function. How-


ever, for a suitable warping function, even if a = -oo and b = +m, it can

be arranged for all non-stationary timelike geodesics to be incomplete. More

precisely, Powell shows that if (H, h) is complete and wo E (a, b), then all fu-

ture directed non-stationary timelike geodesics are future (respectively, past)

complete if and only if

4 (t) dt = +m, respectively, J w O (a) dt = +m. L (mi)

From this result and our above results on null completeness, relationships may

be derived between null and timelike completeness for this class of warped

products. (In general, these types of geodesic completeness are logically inde-

pendent (cf. Theorem 6.4). In particular, null completeness of R x f H does

not imply timelike completeness.) However, if the warping function f for - M = R x ~ f H is bounded from above and (H, h) is Riemannian complete, then

the timelike and null completeness are equivalent.

In singularity theory in general relativity, conditions on the curvature ten-

sor of (%,-ji) which are discussed in Chapters 2 and 12, called the generic

condition and the timelike convergence condition, are considered. These two

conditions guarantee that if a nonspacelike geodesic y may be extended to

be defined for all positive and negative values of an affine parameter and

d i m z > 3, then y contains a pair of conjugate points. Hence these cur-

vature conditions may be combined with geometric or physical assumptions,

such as ( 2 , g ) is causally disconnected or ( M , g ) contains a closed trapped

set, to show that ( 2 , g ) is nonspacelike geodesically incomplete (cf. Section

12.4). Since ( M , g ) satisfies the generic condition and strong energy condi-

tion if all nonspacelike Ricci curvatures are positive, it is thus of interest to

consider conditions on the warping function f of a Lorentzian warped product

which guarantee that ( z , g ) has everywhere positive nonspacelike Ricci cur-

vatures. The assumption d i m m > 3 made in singularity theory is necessary

for null conjugate points to exist since no null geodesic in any two-dimensional

Lorentzian manifold contains a pair of conjugate points.

We now give the formulas for the curvature tensor R and Ricci curvature

tensor Ric for the warped product space-time ( M x f H , g ) where 3 = g @ fh.


As above, let V' [respectively, V2] denote the covariant derivative of (M,g)

[respectively, (H, h)]. Also let r$ = In f and recall that grad4 denotes the

gradient of 4 on (M,g). As before, we will decompose tangent vectors x in

TF(M x H) as x = (x1,x2). Let R' [respectively, R ~ ] denote the curvature

tensor of (M,g) [respectively, (H, h)]. Given tangent vectors X I , yl E TpM,

define the Hessian tensors H+ and h+ by

and

We will also write 1 1 grad + / I 2 = g(grad 4, grad 4). Using the sign convention

for the curvature tensor and substituting from formula (3.17), one obtains the

formula

where x, y, 2 E Tlp,,)(M x H). . Suppose now that dim M = m and dim H = n. To calculate the Ricci

curvature at jj = @,q) E M x H, let {el,e2,. . . , em) be a basis for TpM

with g(el,el) = -1, g(ej,ej) = 1 for 2 < j < m, and g(ei,ej) = 0 if i # j .

Also, let {em+l,. . . ,en+,) be a 7j-orthonormal basis for T,H. Then for any

x, y E TF(M x H), we have


The d'Alembertian 0 4 of 4 may also be calculated as

Using (3.21), it then follows that

dim H - g(x2, 1/21 [ io4(p ) + 7

dim H -- dim H 2 h + ( x 1 , ~ 1 ) - - - p 1 ( 4 ) ~ 1 ( 4 )

where x = ( x l , y l ) , y = ( y l , y2) E T(,,,) ( M x H ) , and ~ i c l and Ric2 denote

the Ricci curvature tensors of ( M , g ) and ( H , h ) , respectively.

W e now restrict to the case hl = (a , b) x f H with warped product metric

i j = -dt2 @ fh. In this case, O4(t ) = -4I1(t) and 11 gad$(t)l12 = -[4'(t)I2.

Thus we obtain from (3.22) for 3 = (0, v ) E T(t,,)((a, b) x H ) that

(3.23) Ric(i?, i?) = R ~ c ~ ( v , v ) + g(v, v)

I f x = bldtl, + v E T(,,,)((a, b) x H ) with v E T,H, we obtain

1 dim H (3.24) Ric(x, x ) = ~ i c ' ( v , v ) + g(v, v ) {5411(t) + - 4 14'(t)12)

dim H - --[4'(t)12) dim H . 4

Both bracketed terms in formulas (3.23) and (3.24) will be positive provided

that

(3.25) -[4'(t)I2 dim H < 2d1'(t) < -[4'(t)12

for all t E (a , b). Thus i f Ric2(v,v) 2 0 for all v E T H and condition (3.25)

holds, the space-time (v,ij) will have everywhere positive Ricci curvatures.

A globally hyperbolic family o f such space-times is provided by warped prod-

ucts = = (0, co) x f H , where ( H , h ) is a complete Riemannian manifold of


nonnegative Ricci curvature and 3 = -dt2 @ f h with f ( t ) = t' for a fixed

constant r 6 R satisfying (21 dim H ) < r < 2. I f ( H , h ) is taken to be W 3 with

the usual Euclidean metric and r = 413, we recover the Einstein-de Sitter

universe of cosniology theory [cf. Hawking and Ellis (1973, p. 138), Sachs and

W u (1977a, Proposition 6.2.7 ff.)].

W e may also obtain the following condition on 4 = In f for positive non-

spacelike Ricci curvature i f the Ricci tensor of ( H , h ) is bounded from below.

Proposition 3.72. Let 3 = (a , b) x f H with n = dim H 2 2, g = -dt2

fh, and 4 = In f . Suppose that there exists some constant X E R such that

Ric2(v, v ) > Xh(v,v) for all v E T H . Then i f

(3.26) 24I1(t) < min {- (q!~ ' ( t ) )~ , 4(n - l)-'Xe-@'('))

for all t E (a , b), the Lorentzian warped product (li?, 3 ) has everywhere posi-

tive nonspacelike Ricci curvature.

Proof. It suffices to show that Ric(x, x ) > 0 for all nonspacelike tangent

vectors x o f the form x = dldt + v E T ( M x H ) , v E T H . Since 3 (x , x) 5 0

and i j(a/dt , d l d t ) = - 1, we have p = g(v, v ) 5 1. Hence 0 5 ,B 5 1. Then

h(v, v ) = Be-$ and we obtain from (3.24) that

Thus Ric(x, x ) > 0 provided 4" < G(P) for all /3 E [O,l], where

Calculating G1'(P), one finds that G1(,B) does not change sign in [0, I]. Thus

G(,B) obtains its minimum on [ O , l ] for p = 0 or /3 = 1. Hence Ric(x, x) > 0

provided that 4" < min{G(O), G(P)) , which yields inequality (3.26).

W e now consider the scalar curvature o f warped product manifolds o f the

form = R x f H , 3 = -dt2 @ f h . W e will let n = dim H below. Given

( t ,p ) E M, choose el E T,H for 1 < j < n such that i f Zl = (0, e,) E T(t,p)%, - -

then { d l & = (d ld t , O,),El, . . . ,En) forms a 3-orthonormal basis for T( t ,p)M.


Hence {mel,. . . , m e , } forms an h-orthonormal basis for TpH. Thus

if r : 2 -+ W and TH : H -+ W denote the scalar curvature functions of ( 2 , 3 )

and (H, h) respectively, we have

and

Now formulas (3.23) and (3.24) above simplify to

and

for 1 5 j 5 n. Consequently, we obtain the formula

1 1 r( t , p) = - r ~ (p) + n+I1(t) + (n2 + n) [+'(t)12. f (4

Recalling that $(t) = In f (t), this may be rewritten as

where dim H = n as above. In particular, in the case that n = 3 as in general

relativity, we obtain the simpler formula

Example 3.73. With the formulas of this section in hand, we are now

ready to give an example of a 1-parameter family gA of nonisometric Einstein

metrics for Wn+' such that for X = 0, (Wn+l,go) is Minkowski space-time of

dimension n + 1. Let (Wn, h) be Euclidean n-space with the usual Euclidean

metric h = dx12 + dx22 + - . . +dzn2, and put MA = lipn+' = W x f Wn with the

3.7 SEMI-RIEMANNIAN LOCAL WARPED PRODUCT SPLITTINGS 117

Lorentzian metric ijx = -dt2 @ eXth, i.e., f (t) = ext. By Theorem 3.70, for all

X > 0 the space-time (Rn+',?jx) is future null geodesically complete but past

null geodesically incomplete, and for all X < 0, the space-time (Rn+1,3x) is

past null geodesically complete but future null geodesically incomplete. Using

formulas (3.28), (3.29), and (3.30), we obtain

and

Thus if X # 0, (zA,?jx) is an Einstein space-time with constant positive scalar

curvature.

Example 3.74. Let li?x = (0,co) x f x3, where gA = -dt2 @ f h with

f (t) = At, X > 0, and h the usual Euclidean metric on LR3. It is then immediate

from formula (3.31) that r(gX) = 0 for all X > 0. Since d( t ) = ln(Xt), it may

be checked using formulas (3.28) and (3.29) that (z~, 3,) is neither Ricci flat

nor Einstein for any X > 0. Also we have for any X > 0 that

for all t > 0. It follows that the space-times ( z A , ?jA) are "inextendible across"

(0) x R3 (cf. Section 6.5). Also, ( R x , g x ) is future null geodesically complete

by Theorem 3.70.

3.7 Semi-Riemannian Local Warped Produc t Spli t t ings

In each of the exact solutions to Einstein's equations which are presented

as warped product manifolds, the warped product decomposition emerges as a

natural mathematical expression of assumed physical symmetries. Moreover,

formulas for warped product curvatures (cf. Proposition 3.76) indicate that any

semi-Riemannian manifold (M, g) must possess certain measures of symmetry

and flatness in order to be (locally or globally) isometric to a warped product

118 3 LORENTZIAN MANIFOLDS A N D CAUSALITY

B x F. In this section we identify geometric conditions on a semi-Riemannian

manifold (M,g) which are necessary and sufficient to ensure that (M,g) is

locally isometric to a warped product B xf F. We shall call such a local

isometry a 'Llocal warped product splitting."

There are a number of different "splitting" or decomposition theorems

throughout differential geometry. For example, the splitting theorem of Geroch

(1970a), Theorem 3.17 in this chapter, demonstrates that a globally hyperbolic

space-time may be written as a particular type of topological product but not

necessarily as a metric product. By contrast, a well-known result of de Rham

[cf. Kobayashi-Nomizu (1963, p. 187)] asserts that a complete simply con-

nected Riemannian manifold which has a reducible holonomy representation

is isometric to a Riemannian product. Along very different lines, Chapter 14

provides the following Lorentzian analogue of the Cheeger-Gromoll Splitting

Theorem: if (M, g) is a space-time of dimension n 2 3 which (1) is globally

hyperbolic or timelike geodesically complete, (2) satisfies the timelike con-

vergence condition, and (3) contains a complete timelike line, then (M, g) is

isometric to a product (R x V, -dt2 $ h ) , where (V, h ) is a complete Riemann-

ian manifold. Since a product manifold is trivially a warped product, either

of these last two results clearly provides sufficient conditions to ensure that a

manifold is globally isometric to a warped product. However, the examples

(S1 x H, 3) of non-globally hyperbolic warped product space-times discussed

in Section 3.6 indicate that causal assumptions such as global hyperbolicity

are not necessary for the existence of a global warped product splitting.

The global splitting question typically involves rather delicate topological

considerations; our focus in this section will be on the simpler local warped

product splitting question: given a semi-Riemannian manifold (M,g) and a

point p E M, what conditions are necessary and suficient for the existence of

a n open neighborhood U of p such that the submanzfold (U,g lu) is isometric

to a warped product B x F ? The following assumption will be needed.

Convention 3.75. It will be assumed throughout the remainder of this

section that the neighborhood U mentioned above is a connected, simply con-

nected, open set.

3.7 SEMI-RIEMANNIAN LOCAL WARPED P R O D U C T SPLITTINGS 119

The geometry of a warped product B x f F is expressed through the ge-

ometries of the base (B,gB) and fiber ( F , g F ) and various derivatives and

integrals of the warping function f E 5(B). We will consider warped products

M = B xf F where both (B, gs) and (F, gF) may be semi-Riemannian man-

ifolds (thus generalizing the Lorentzzan warped products of Definition 3.51).

The symbols T : M -+ B and a : M -+ F denote the standard projections.

As in portions of Section 3.6, we will find it convenient to consider the square

root S of the warping function f . Throughout this sectzon, we wzll adhere to

the conventzon that S ( b ) = for b E B where M = B x f F denotes a

warped product wzth metric tensor

The function S will be called the root warpzng functzon.

As defined in Section 3.6, the lzft of f E 5(B) to a function j E S(M) is

defined by the formula 7 = f o T. The lifted function will simply be denoted

by f as well, when no ambiguity results. A vector field V E X(B) is lifted to

M by defining E X(M) in such a manner that at each (p, q) E M, T/(p, q)

is the unique vector in the tangent space T(,,,]M such that both d ~ ( ? ) = V

and do(?) = 0. Similar definitions apply for lifts from F to M. We shall

also denote lifted vector fields without the tildes, and we write V E C(F) to

denote a vector field on M lifted from F. More generally, vectors tangent to

leaves B x q are called honzontal while those tangent to fibers p x F are called

vertzcal. Lifts of covariant tensors on B and F are now defined in the obvious

way through the use of the pullbacks T* and a*.

Let D denote the Levi-Civita connection on M, and use V to denote the

Levi-Civita connections on both B and F. The curvature tensors on B x f F =

M may be characterized through their actions on lifted horizontal and vertical

vector fields. In the following proposition, the symbols BR and F~ denote the

lifts to M of the Riemannian curvature tensors on B and F , respectively. The

symbol H S will be used to denote 5, the lift to M of the Hessian of S. Note - that in general H S ( X , Y) = H'(x, Y) only for horizontal vector fields X , Y.

For notational simplicity, the bracket notation ( , ) will be occasionally used


to denote the metric g on M; the metrics on B and F will always be denoted

by g~ and g ~ .

Some basic curvature formulas for warped product manifolds are now given

in Proposition 3.76 for ease of reference. The standard reference for this ma-

terial is O'Neill (1983).

Proposition 3.76. Let the semi-Riemannian warped product M = B x f F

have Riemannian curvature tensor R, Ricci curvature Ric, and root warping

function S = a. Assume X, Y, Z E C ( B ) and U, V, W E C(F).

(1) If h f 5(B), then the gradient of the lift h o 7r of h to M = B x f F is - the lift to M of the gradient of h on B, i.e., grad (71) = grad h.

IV

(2) DxY f L(B) is the lift of VxY on B, i.e., D ~ F = (VxY).

(3) DxV = D v X = (9) V.

(4) Ric(X, Y) = l?.ic(X, Y) - ($) H'(x, Y) where d = dim F.

(5) Ric(V, X ) = 0.

(6) Ric(V, W) = Ric(V, W) - (V, W)Sd

where Sn = 9 + (d - 1)-, d = dimF, and AS is the

Laplacian of the root warping function S on B.

Consider first the local warped product splitting question in dimension two.

Assume a semi-Riemannian surface (M,g) is a warped product so that in the

appropriate local coordinates (yl, y2) adapted to B and F , the metric is given

by

(3.34) g = cldyl 8 dyl + c2[s(y1)I2dy2 8 dy2

where ci = r t l , i = 1,2. It is immediate that d/dy2 is a local Killing field (recall

the elementary fact that a coordinate vector field d/dxk is Killing if and only

if & = 0 for all i, j). Each (d/dyl)-integral curve is a geodesic of M (more

generally, each leaf B x q of a warped product is a totally geodesic submanifold).

Thus d/dy2 restricted to y is a Jacobi field for each such coordinate geodesic y,

and of course, {dl&', d/dy2) = 0 along y. Further, direct calculation shows

that d/dyl is ir-rotational-that is, curl(d/dyl) = 0, where the curl of a vector

field is the skew-symmetric (0,2) tensor defined through the formula

(3.35) [curlX](Y, 2 ) = (DUX, 2 ) - (Y, D z X )


for all vector fields X , Y, and 2. The unit vector field i (d/dy2) does not have

vanishing curl, in general. These observations lead to the following result.

Lemma 3.77. Given a two-dimensional semi-Riemannian manifold (M, g)

and a point p E M, p has a neighborhood U such that (U, g lu) is isometric to

a warped product if and only if there exists a non-vanishing Killing field on

an open neighborhood of p.

Proof. The local existence of a nonvanishing Killing field about any point

of a two-dimensional semi-Riemannian warped product was noted above.

Conversely, suppose there exists a neighborhood U about p E M having a

nonvanishing Killing field V. If the Killing field is null, then it is well known

that (U, g lu) is isometric to (a portion of) Minkowski two-space RT and hence

is (trivially) a warped product.

If the Killing field V is not null, then we may complete the classical con-

struction of local geodesic (or Fermi) coordinates (xl, x2) [cf. do Carmo (1976)]

such that V = d/dx2 on U and x1 measures g-arc length along the geodesics

orthogonal to the integral curves of V. In these coordinates the metric assumes

the form

ds2 = €1 dzl @ dxl + €2 F'(xl, x2)dx2 8 dx2

where ci = &I, i = 1,2. Since V = d/dx2 is Killing, we must have = 0,

yielding

ds2 = €1 dxl 8 dxl + €2 F'(x1)dx2 8 dx2

and leading to the desired local warped product representation. 0

In generalizing the preceding result to higher dimensions, the existence of a

Killing field V must be supplemented by an integrability condition which allows

the construction of leaves B x q orthogonal to V. This integrability condition

holds trivially in dimension two but need not hold in higher dimensions where,

in general, a Killing field need not be irrotational. It is also necessary to

include the additional assumption that the Killing field be nonnull.

Recall that a space-time (M, g) is called static if there exists on M a nowhere

zero timelike Killing field X such that the distribution of (n- 1)-planes orthog-

onal to X is integrable. The following result parallels the formal construction


and a : (x1,x2,. . . ,xn) --t (0,0,. . . ,O,xn). For arbitrary X,Y f X(U) with

coordinate basis expansions X = Xi(d/dxi), Y = Yi(d/dxi), we have

The "dimensional dual" of the preceding result asks for conditions necessary

and sufficient to ensure that (M, g) is locally isometric to a warped product

B x f F with dim B = 1. One answer to this question involves a construction

which bears strong similarities to the formal development of the Robertson-

Walker cosmological models [cf. O'Neill (1983)l. Recall that a signal feature

of Robertson-Walker space-time is the presence of a proper time synchroniz-

able geodesic observer field U . Since the observer field U is irrotational, the

infinitesimal rest spaces of U may be integrated to provide local rest spaces.

Through a construction quite similar to that of the preceding lemma, it is pos-

sible to verify the following [cf. Easley (1991)]: given an n-dimensional (n 2 3)

semi-Riemannian manifold and point p E M I the point p has a neighborhood

U M such that (U,glu) is isometric to a warped product B x j F with

dim B = 1 and dim F = (n - 1) if and only if (1) there exists an irrotational

unit vector field U (either spacelike or timelike) on a neighborhood of p such

that (2) the flow J[r induced by U acts as a positive homothety on the local

(n - 1)-dimensional submanifolds which are everywhere orthogonal to U near

p. Condition (2) is of course equivalent to a number of different geometric

conditions expressible in terms of curvature and the connection.

The local splitting problem in the context of Ricci flat (M,g) has an ex-

tremely restricted class of solutions. We consider first the only case in dimen-

sion four which will not be subsumed under more general results, namely, the

case when M is locally isometric to a warped product with base and fiber each


of dimension two. The following computational lemma is a necessary prelim-

inary step. In the following result, the symbols "K and B K will be used to

denote the sectional curvatures of F and B, respectively. Since these objects

are bnctions on the surfaces F and B, they may be unambiguously lifted to

M as well.

Proposition 3.79. Let M = B x f F be a 4-dimensional semi-Riemannian

warped product with dim B = dim F = 2 and root warping function S = fl. For M to be Ricci fiat, it is necessary and sufficient that

(1) F have constant sectional curvature F K ,

(2) = s A s + gB(grad S, grad S) = FK on B, where F~ is the

constant value from (1) and A denotes the Laplacian on B, and

(3) Dx (grad S ) = (p ) X for all X E L(B) .

Further, if M is Ricci flat, then

A s - B K o n B , (4) -3- -

and the sectional curvatures and root warping function are related as follows:

(5) K . S3 = 2 C,w on B, where CM is a const ant, and

(6) FK = B K . S2 + g~ (grad S, grad S) on B.

Proof. Using Proposition 3.76, we see that M is Ricci flat if and only if

(3.36) Ric(X, Y) - ~H'(X, Y) = 0 and

(3.37) Ric(V, W) - (V, W)S$ = 0

for all X , Y E C ( B ) and V, W E C(F), where SQ = * and A

denotes the Laplacian on B.

On the semi-Ftiemannian surface B we have

with the analogous formula holding on the surface F (here the symbol Ric

denotes the Ricci tensor on B and not its lift to M). Projecting equations

(3.36) and (3.37) onto B and F respectively, we see that M is Ricci flat if and


only if the following two conditions hold:

(3.38) B~ .gB(X,Y) = $H'(x,Y), and

(3.39) F K . g ~ ( v W) = s ~ ~ F ( v , W)S'

for all X , Y E X(B) and V, W E X(F). Equation (3.39) must hold as we

project from each fiber p x F into F. Since S is constant on fibers we see that

equation (3.39) will hold if and only if

F~ = constant, and

f 2 ~ g = S A S + g ~ ( g r a d S , g r a d S ) = F ~ OnB.

Rewriting the Hessian as HS(x , Y) = gB(Vx(grad S), Y), equation (3.38)

is equivalent to

which in turn holds if and only if

Vx(grad S ) = ( B ~ . $) X for all X E X(B).

That conditions (I), (2) and (3) are necessary and sufficient for M to be Ricci

flat now follows. Proposition 3.76-(2) is used to obtain the lifted form of

condition (3).

From condition (3) it follows that for a local orthonormal frame field Eo, El

on B with ~i = gp,(Ei, Ei),

Thus the Laplacian of S is given by

By combining (3.40) with condition (21, we see that

(3.41) F~ = K s 2 + g~ (grad S, grad S )


where F~ denotes the constant value of the sectional curvature of F. Differ-

entiating both sides of equation (3.41) produces

0 = s 2 ( x B K ) + B ~ 2 ~ ~ ~ + 2gB(vx(grad S) , grad S)

= S 2 ( x B K ) + B ~ 2 ~ ~ ~ + S . B ~ ( ~ ~ )

= s 2 ( x B ~ ) + 3 . B ~ ~ ( ~ ~ ) .

Thus, 3 . *K(xS)

X(BK) = - ( ) for all x t X(B)

yielding

and this holds for all lifts X E C(B). It follows that S3 . B K is constant on B

(and hence its lift is constant on M).

It is now possible to characterize all (2 x 2) Ricci flat warped products

M = B x j F with base B of constant curvature.

Corollary 3.80. If M = B x f F is a four-dimensional Ricci flat warped

product with base B a surface of constant curvature, then M is simply a

product manifold B x G where B and G are flat two-manifolds. Thus the

metric on M is semi-Euclidean.

Proof. Assume B has constant curvature. Proposition 3.79-(5) shows that

the root warping function S, and hence also the warping function f , must be

constant on B, and thus M may be v~ewed as a product manifold. Equations

(2) and (4) of Proposition 3.79 now imply that F and B are flat. O

The following result deals with (2 x 2) warped products B x f F where the

curvature B~ is not constant.

6 Proposition 3.81. Let M = B x j F be a Ricc~ flat semi-Riemannian 1 B warped product with dim B = dim F = 2, and assume that the sectional cur-

$ vatureBK of the base B isnot constant. IfgradS isnonnuil and (grad S)I, # 0 F


a t a given point p E B, then there exist local coordinates (t, r ) on a neighbor-

hood U C B of p such that the following conditions hold on U.

(1) The metric on B has coordinate expression ds2 = E(r)dt2 + G(r)dr2,

where G(r) = ( F K - %)-I , E(T) = iG- ' , CM is a constant, and

F K is the constant value of the sectional curvature on F.

(2) The root warping function has the form S(t, T) = T.

(3) The sectional curvature on B is given by B ~ ( ~ , t) = 9. The sign of E in condition (1) depends upon the signature of B.

Proof. Assume grad S is nonnull and (grad S)I, # 0. By continuity, we may

find a neighborhood U of p on which the gradient of S is non-vanishing. Let

co = S-'(k) n U, k E W+, denote a single level curve of S in U. Consider

geodesics intersecting co orthogonally; these geodesics are also integral curves

of grad S, and the orthogonal trajectories of these geodesics are level sets of

S. Applying the classical geodesic coordinate construction, we introduce local

coordinates (u, u) such that the geodesics are the u = constant curves and the

orthogonal trajectories are the u = constant curves. In these coordinates the

metric assumes the form

Rescale coordinates, introducing r = T(U) and t = t(v) such that

r(u) = S(u) and Da,at(a/at) = 0.

Thus, r merely traces the values of the warping function S, and t is an affine

coordinate along the geodesics. It should be noted that this will imply T > 0.

In (t, r ) coordinates we have metric tensor components

with the root warping function given simply by

3.7 SEMI-RIEMANNIAN LOCAL WARPED P R O D U C T SPLITTINGS 129

Furthermore, since

the gradient of S is given in (T, t ) coordinates by

Equation (5) of Proposition 3.79 now shows that the curvature on the surface

B is a function solely of r and is given by

Now gB(grad S, grad S ) = &, so formula (6) of Proposition 3.79 yields

We have determined the metric coefficient G:

It remains to determine the form of the metric coefficient E ( t , r ) . We first

show that E is a function only of the variable r and then derive the function

E(r).

Note that

so that in particular = 0. However, since (t, r ) are orthogonal coordinates,

we have 1

E . G . ~ ; , = l ~ t . ~ .

It follows that Et = 0 and E is a function solely of r .


Now using Proposition 3.79 with grad S = &&, we obtain

S d Dalat (& g) = ( B K . Z) z, or equivalently

1 CM a giving Dalat ( a l d r ) = - - 7-2 at'

G . C M d a t 1 = 7 z.

However, in the orthogonal ( t , T ) coordinate system, we have

It follows that

Er - ~ C M = G(T)- and thus

E T-2 '

With the substitution y = F~ - %, this becomes

In I El = I J 3, giving ?f

ln]El = f In lyl + c, or finally

.=fa. ( F K _ + )

where a = eC is positive. In orthogonal coordinates with e = and

g = m, the curvature is given by the classical formula

Since E and G are functions of r only, this reduces to the following simple

formula. We are ignoring the case-by-case analysis of the signs involving the


absolute values since the end result will show such considerations to be unnec-

essary.

Since B K = 9, we see that a = 1, and the sign in this final term is deter-

mined by the signature of B and the sign of G ( r ) . We have therefore shown

that

Observe that dt is a local nonvanishing Killing field on U in the above lemma,

which is valid for spaces of arbitrary signature. Let us now fix the signature of

all spaces under consideration to the Lorentzian signature (-, +, +, +). Corol-

lary 3.80 and Proposition 3.81 now have the following import: a Ricci flat

(2 x 2) warped product M is completely specified (locally) by the choice of

base ( B , g B ) and the constants F~ and CM in Proposition 3.79. Since we are

working locally, we may assume the fiber F of constant sectional curvature F K

is a subset of

(1) the sphere S 2 ( p ) if " K = $, (2) Euclidean space Ift2 if F~ = 0, or

( 3 ) hyperbolic space H 2 ( p ) if F K = - 7. Theorem 3.82. Assume ( M , g) is a Ricci Aat four-manifold of Lorentzian

signature (-, +, +, +), and let p E M be any point of M. The point p has a

neighborhoodU such that (U, g l u ) is isometric to a (2 x 2) warped product with

both parameters (CM, F ~ ) positive and nonconstant sectional curvature B K r k on B if and only if M is locally isometric to an open subset of Schwarzschild

CM "space-time with mass Mo = -- T ( F K ) ~ '

i Proof. Assume M admits such a local warped product structure. Since we

b a are working locally, we may assume the fiber F is the 2-sphere of radius 1 r JF7;;'


Let du2 denote the canonical metric on the unit 2-sphere. Proposition 3.81

implies the metric may be expressed on U in the form

with E(r) = ( F ~ - %). Upon making the change of variables u = --$-- and v = tm, the metric

F K

becomes

thus demonstrating the first claim. The converse is now clear. 13

This result should be contrasted with Birkhoff's Theorem [cf. Hawking and

Ellis (1973, p. 372)j on spherically symmetric solutions to Einstein's vacuum

field equations, which also offers an alternate proof of Theorem 3.82.

Theorem (Birkhoff). Any C2 solution of Einstein's empty space equa-

tions which is spherically symmetric in an open set V is locally equivalent to

part of the maximally extended Schwarzschild solution in V.

The typical construction of Schwarzschild space-time [cf. O'Neill (1983)l

follows the proof of Birkhoff's theorem. A number of strong physical assump

tions, not the least of which is spherical symmetry, lead one inevitably to

Schwarzschild space-time as the unique model. What we have shown is that

a Ricci flat space-time which possesses enough symmetry to be expressed lo-

cally as a (2 x 2) warped product must have spherical, planar, or hyperbolic

symmetry. In the first case we obtain the conclusion to Birkhoff's Theorem:

M is locally isometric to a portion of Schwarzschild space-time.

The following result may be established by an argument similar to the one

employed in the proof of Proposition 3.79 [cf. Easley (1991)].

Theorem 3.83. Let M = B x f F be a semi-Riemannian warped product

with dim B = 1 and dim F = n 2 2. For M to be Ricci flat, i t is necessary

and sufficient that the following two conditions hold.

(1) The root warping function S satisfies (grad S, grad S ) = Co, a constant.


(2) (F, g ~ ) is an Einstein manifold with Ric = 2(grad S, grad S)gF =

CO. gF.

This result indicates that a Ricci flat manifold ( M , g ) can admit only one

of a restricted class of local warped product splitting at p f M if we require

dim B = 1. If dim F = 2, for example, it follows that F is a surface of constant

curvature F K since FRic = F ~ g ~ in this case. If we also assume F K = 0,

we have S = Jf = constant, and the warped product is merely a standard

product manifold with a semi-Euclidean metric.

If dim F = 3, it follows that the Einstein manifold ( F , ~ F ) has constant

curvature [cf. Petrov (1969, p. 77) 1. If (F, g ~ ) is Riemannian, we may consider

F a subset of

(1) the sphere S3(r) if F~ = 5' (2) Euclidean space IR3 if FK = 0, or

(3) hyperbolic space ~ ~ ( r ) if F~ = - 7 1

with analogous restrictions holding if g~ is indefinite. An exact solution to

Einstein's equations which uses F = S3 is the spatially homogeneous Taub-

NUT model [cf. Hawking and Ellis (1973, p. 170)].

CHAPTER 4

LORENTZIAN DISTANCE

With the basic properties of Riemannian metrics in mind (cf. Chapter I) , it

is the aim of this chapter to study the corresponding properties of Lorentzian

distance and to show how the Lorentzian distance is related to the causal

structure of the given space-time. We also show that Lorentzian distance

preserving maps of a strongly causal space-time onto itself are diffeomorphisms

which preserve the metric tensor.

While there are many similarities between the Riemannian and Lorentzian

distance functions, many basic differences will also be apparent from this chap-

ter. Nonetheless, a duality between "minimal" for Riemannian manifolds and

"maximal" for Lorentzian manifolds will be noticed in this and subsequent

chapters.

4.1 Basic Concepts and Definitions

Let (M, g) be a Lorentzian manifold of dimension n > 2. Given p, q E M

with p < q, let fl,,, denote the path space of all future directed nonspacelike

curves y : [0, I] + M with $0) = p and $1) = q. The Lorentzian arc length

functional L = L, : Clp,q -' R is then defined as follows [cf. Hawking and Ellis

(1973, p. 105)]. Given a piecewise smooth curve y E Q p , , , choose a partition

0 = to < t l < .. . < t,-l < tn = 1 such that y 1 (t,.t,+l) is smooth for each

i = 0 , 1 , . . . , n - 1. Then define

It may be checked as in elementary differential geometry [cf. O'Neill (1966,

pp. 51-52)] that this definition of Lorentzian arc length is independent of

136 4 LORENTZIAN DISTANCE

FIGURE 4.1. The timelike curve y from p to q is approximated

by a sequence of curves {y,) with yn + y in the C0 topology but

L(yn) --t 0.

the parametrization of y. Since an arbitrary nonspacelike curve satisfies a

local Lipschitz condition, it is differentiable almost everywhere. Hence the

Lorentzian arc length L(7) of 7 may still be defined using (4.1). Alternative

but equivalent definitions of L(y) for arbitrary nonnull nonspacelike curves

may be given by approximating y by C' timelike curves [cf. Hawking and Ellis

(1973, p. 214)] or by approximating y by sequences of broken nonspacelike

geodesics {cf. Penrose (1972, p. 53)j. The Lorentzian arc length of an arbitrary

null curve may be set equal to zero.

Now fix p, q E M with p << q. If y is any timelike curve from p to q, then

L(y) > 0. On the other hand, y may be approximated by a sequence (7,)

of piecewise smooth "almost null" curves y, : [O ,1 ] + M with yn(0) = p and

yn(l) = q such that yn + y in the Co topology, but L(yn) -+ 0 (cf. Figure 4.1).

This construction shows, moreover, that given any p, q E M with p << q, there

4.1 BASIC CONCEPTS AND DEFINITIONS 137

are curves y E Q,,q with arbitrarily small Lorentzian arc length. Hence, the

infimum of Lorentzian arc lengths of all piecewise smooth curves joining any

two chronologically related points p << q is always zero. However, if p and q lie

in a geodesically convex neighborhood U, the future directed timelike geodesic

segment in U from p to q has the largest Lorentzian arc length among all

nonspacelike curves in U from p to q. Thus it is natural to make the following

definition of the Lorentzian distance function d = d(g) : M x M + W U {m)

of (M, g).

Definition 4.1. (Lorentzian Distance Function) Given a point p in M ,

if q 9 J+(p), set d(p, q ) = 0. If q E J+(P), set d(p, q) = sup{L,(y) : y E n,,,).

From the definition. it is immediate that

(4.2) d(p, q) > 0 if and only if q E I'(p).

Thus the Lorentzian distance function determines the chronological past and

future of any point. However, the Lorentzian distance function in general fails

to determine the causal past and future sets of p since d(p,q) = 0 does not

imply g E J+(p) - I+ (p). But at least if q E J+(p) - Ii(p), then d(p, q) = 0.

We emphasize that the Lorentzian distance d(p, q) need not be finite. One

way that the condition d(p, q) = m may occur is that tirnelike curves from p

to q may attain arbitrarily large arc lengths by approaching certain boundary

points of the space-time. In Figure 4.2, two points with d(p,q) = co are

shown in a Reissner-Nordstrijm space-time with e2 = m2 [cf. Hawking and

Ellis (1973, p. 160)].

A second way Lorentzian distance may become infinite is through causality

violations. Recall that a space-time is said to be vicious at the point p E M

if I+(p) n I-(p) = M and totally vicious if I+(p) n I V ( p ) = M for all p E M.

Lemma 4.2. Let (M, g) be an arbitrary space-time.

(1) I fp E If(p), thend(p,p) = co. Thusforeachp E M , eitherd(p,p) = 0

or d(p,p) = co.

(2) (M, g) is totally vicious iff d(p, q) = m for all p, q E M.

(3) If (M, g) is vicious a t p, then (M,g) is totally vicious.


Proof. (1) Suppose p E I+(p). Then we may find a closed timelike curve

y : [0,1] -+ M with y(0) = y(1) = p. Since y is timelike, L(y) > 0. If

u, E R , , is the timelike curve obtained by traversing y exactly n times, then

L(u,) = nL(y) -+ ca as n --t ca. Thus d(p,p) = m .

(2) Suppose (M,g) is totally vicious. Fix p, q E M, and let n > 0 be any

positive integer. Since p E I+(p), we may find yl E R,,, with L(yl) 2 n by

part (1). Since q E I+(p), there is a timelike curve 7 2 from p to q. Then

y = yl * yz E R,,, is a timelike curve with length L(y) = L(y1) + L(yz) > n.

Hence d(p, q) = m.

Conversely, suppose d(p, q) = m for all p,q E M. Fixing r E M , we have

d(r,p) > 0 and d(p,r) > 0 for all p E M. Thus by (4.2), it follows that

I + ( r ) n I-(r) = M.

(3) Inspired by Lemma 4.2-(2), T. Ikawa and H. Nakagawa (1988) proved

(3) using the Lorentzian distance function. Subsequently, B. Wegner (1989)

noted that the following elementary argument yields the desired result. Let

q E M = I+(p)nI-(p). Then q E I+(p) so I-(p) C I-(q) by the transitivity of

<. Similarly, q E I-(p) implies I+(p) 5 I+(q). Hence, M = I+(p) n I-(p) C + - 0

By Definition 4.1, if I+(p) # M then there are points q E M with d(p, q) = 0

but p # q. Hence unlike the Riemannian distance function, the Lorentzian

distance function usually fails to be nondegenerate. Indeed, we have seen that

d(p, p) > 0 is possible. But if (M, g) is chronological, then d(p, p) = 0 for all

p E M. Also, the Lorentzian distance function tends to be nonsymmetric.

More precisely, the following may be shown for arbitrary space-times.

Remark 4.3. If p # q and d(p, q) and d(q,p) are both finite, then either

d(p,q) = 0 or d(q,p) = 0. Equivalently, if d(p,q) > 0 and d(q,p) > 0, then

d(p, 9) = 4 9 , P) = a.

Proof. If d(p, q) > 0 and d(q,p) > 0, we may find future directed timelike

curves yl from p to q and yz from q to p, respectively. Define a sequence (7,)

by yn = yl * (72 * y ~ ) ~ E Rp,q. AS n --+ m , L(yn) -+ m , whence d(p,q) is

infinite. Similarly, d(q, p) = m .


FIGURE 4.2. A Reissner-Nordstrom space-time with e2 = m2 is

shown. By taking timelike curves y from p to q close to Jf and

3-. we can make L(y) arbitrarily large. Thus d(p, q) = m , which

means an accelerated observer may take arbitrarily large amounts

of time in going from p to q.

A further consequence of Definition 4.1 is that if y : [O, ca) -+ (M, g) is any

future directed, future complete, timelike geodesic in an arbitrary space-time

(M, g ) , then limt,c, d(r(O), y(t)) 2 limt,m L(y I 10, t ] ) = m. By contrast,

complete Riemannian manifolds (N,go) may contain (nonclosed) geodesies

a : [0, m ) (NIgo) for which sup{do(u(0),u(t)) : t 2 0) is finite. Fur-

ther assumptions are needed for Riemannian manifolds to guarantee that

limt,, d(u(O), u(t)) = co for all geodesics a : [O, m) --+ (N, go) [cf. Cheeger

and Ebin (1975, pp. 53 and 151)].


While the Lorentzian distance function fails to be symmetric and nondegen-

erate, at least a reverse triangle inequality holds (cf. Figure 1.3). Explicitly,

(4.3) If P < r < q, then d(p, q) 2 d(p, r ) + d(r, q).

We now discuss some properties of the Lorentzian distance that make it a

useful tool in general relativity and Lorentzian geometry. First, the Lorentzian

distance function is lower semicontinuous where it is finite [cf. Hawking and

Ellis (1973, p. 215)].

Lemma 4.4. For Lorentzian distance d, if d(p,q) < m, p, -+ p, and

qn -+ q, then d(p,q) 5 liminf d(p,,q,). Also, if d(p,q) = m, p, -+ p, and

q, -+ q, then limn,, d(p,, q,) = m.

Proof. First consider the case d(p, q) < m. If d(p, q) = 0, there is nothing

to prove. If d(p,q) > 0, then q E I+(p) and the lower semicontinuity follows

from the following fact. Given any 6 > 0, a timelike curve y of length L > d(p, q) - €12 from p to q and sufficiently small neighborhoods Ul of p and U2

of q may be found such that y may be deformed to give a timelike curve of

length L' 2 d(p, q) - c from any point r of Ul to any point s of U2.

Suppose now that d(p,q) = oo but liminfd(p,,q,) = R < m. Since

d(p, q) = m there exists a timelike curve y from p to q of length L(y) > R + 2. This implies that there exist neighborhoods Ul and Uz of p and q,

respectively, such that y can be deformed to give a timelike curve of length

L' > R + 1 from any point r of Ul to any point s of U2. This contradicts

lim inf d(p,, q,) = R. 0

In general, the Lorentzian distance function fails to be upper semicontinu-

ous. We give an example of a space-time (M, g) containing an infinite sequence

{p,) with p, -+ p and a point q E I+(p) such that d(p,,q) = m for all large

n but d(p, q) < ca (cf. Figure 4.3).

For globally hyperbolic space-times, on the other hand, the Lorentzian

distance function is finite and continuous just like the Riemannian distance

function.

Lemma 4.5. For a globally hyperbolic space-time (M,g), the Lorentzian

distance function d is finite and continuous on M x M.

4.1 BASIC CONCEPTS AND DEFINITIONS

FIGURE^.^. L e t M b e { ( x , y ) : O < y < 2 ) - { ( x , l ) : - 1 5 x 5 1 )

with the identification (x, 0) - (x, 2) and the flat Lorentzian metric

ds2 = dx2 - dy2. Let p = (O,O), q = (0,1/2), and p, -+ p as shown.

Then p, E I+(pn) and hence d(pn,pn) = m for all n. For large n

we have q E I+(p,) and thus d(p,,q) = m. On the other hand,

d(p, q) = 112 which yields d(p, q) < lim inf d(p,, q). This space-

time is not causal. However, the distance function may also fail to

be upper semicontinuous in causal space-times (cf. Figure 4.6).

Proof. To prove the finiteness of d, cover the compact set J+(p) n J-(q)

with a finite number of convex normal neighborhoods B1, B2, . . . , B, such that

no nonspacelike curve which leaves any Bi ever returns and such that every

nonspacelike curve in each Bi has length at most one. Since any nonspacelike

curve y from p to q can enter each Bi no more than once, L(y) 5 m. Hence

d(p1 q) 5 m.

If d failed to be upper semicontinuous at (p,q) E M x M, we could find

a 6 > 0 and sequences ipn) and {q,) converging to p and q respectively,

such that d(p,, q,) 2 d(p, q) + 26 for all n. By definition of d(p,, q,), we

may then find a future directed nonspacelike curve y, from p, to q, with

L(y,) 2 d(p, q) + 6 for each n. By Corollary 3.32, the sequence {y,) has a


nonspacelike limit curve y from p to q. By Proposition 3.34, a subsequence

{y,) of {y,) converges to y in the C0 topology. Hence L(y) 2 d(p, q) + 6 by

Remark 3.35. But this contradicts the definition of Lorentzian distance. Thus

d is upper semicontinuous a t (p, q). U

We now define the following distance condition [cf. Beem and Ehrlich (1977,

Condition 4)].

Definition 4.6. (Finite Distance Condition) The space-time (M, g) sat-

isfies the finite distance condition if d(g)(p, q) < m for all p, q f M.

Lemma 4.5 then has the following corollary.

Corollary 4.7. If (M, g) is globally hyperbolic, then (M,g) satisfies the

finite distance condition and d(g) : M x M + R is continuous.

If (MI g) is globally hyperbolic, all metrics in the conformal class C(M, g)

are globally hyperbolic. Hence all metrics in C(M,g) satisfy the finite distance

condition. We will examine the converse of this statement in Section 4.3,

Theorem 4.30.

Since the given topology of a smooth manifold coincides with the metric

topology induced by any Riemannian metric, it is natural to consider the sets

{m f I+(p) : d(p, m) < E) for a Lorentzian manifold. However, as Minkowski

space shows, these sets do not form a basis for the given manifold topology

(cf. Figure 4.4). Indeed, this same example shows that no matter how small

E > 0 is chosen, the sets {m E J f (p) : d(p, m) 5 E) may fail to be compact

and fail to be geodesically convex as well as failing to be diffeomorphic to the

closed n-disk.

The sphere of radius E for the point p f M is given by K ( ~ , E ) = {q E M :

d(p,q) = E). This set need not be compact. However, the reverse triangle

inequality and (4.2) imply that K(p, E) is achronal for all finite E > 0 and all

p~ M .

In arbitrary space-times, neither the future inner ball

4.1 BASIC CONCEPTS AND DEFINITIONS

FIGURE 4.4. The set B + ( p , ~ ) = {q E I+(p) : d(p,q) < E)

in Minkowski space-time does not have compact closure, is not

geodesically convex, and does not contain p. Furthermore, sets

of the form B+(p, E) do not form a basis for the manifold topol-

ogy. But in general, if (M,g) is a distinguishing space-time with

a continuous Lorentzian distance function, then a subbasis for the

manifold topology is given by sets of the form B+(p, E) and B-(p, E)

[cf. Proposition 4.311. Hence these sets do form a subbasis for the

given topology of Minkowski space-time.

nor the past inner ball

B-(P, €1 = {q f I-(P) : d(q, P) < €1

need be open. On the other hand, when the distance function d : M x M -+

R U {m) is continuous, these inner balls must be open. In Section 4.3 we will

show that for distinguishing space-times with continuous distance functions,

the past and future inner balls form a subbasis for the manifold topology.


A different subbasis for the topology of any strongly causal space-time

(M,g) with a possibly discontinuous distance function d = d(g) : M x M -+

WU{co) may be obtained by using the outer balls O+(p, E) and 0-(p, E) rather

than the inner balls.

Definition 4.8. (Outer Balls O+(p, E), 0-(p, e)) The outer ball O+fp, E)

[respectively, 0 - ( p , ~ ) ] of I+(p) [respectively, I-(p)] is given by

respectively,

O-(P,E) = {q E M : d(q,p) > E)

(cf. Figure 4.5)

Since the Lorentzian distance function is lower semicontinuous where it is

finite, the outer balls O+(p, E) and 0-(p, E) are open in arbitrary space-times.

The reverse triangle inequality implies that these sets also have the property

that if m, n f O+(p, E ) [respectively, m, n E 0-(p, E)] and m ,< n, then any

future directed nonspacelike curve from m to n lies in Ot(p, E ) [respectively,

0-(p, E)]. Moreover, we have

Theorem 4.9. Let (M, g) be strongly causal. Then the collection

forms a basis for the given manifold topology.

Proof. Let m f M be given, and let U be any open neighborhood containing

m. We may find a local causality neighborhood Ul with m E Ul C U, i.e.,

no nonspacelike curve which leaves Ul ever returns. Choose pl,p2 E Ul with

PI << m (< p2 such that I+(pl)fl I-(pz) C Ul. By the chronology assumptions

on PI and p2, we have d(p1, m) > 0 and d(m,pz) > 0. Choose constants el, t2

with 0 < €1 < d(p1,m) and 0 < €2 < d(m,pz). Then m E O + ( p l , ~ l ) n 0-(pz, €2). Since 0' (PI, €1) C I+(pl) and 0 - ( p z , ~ z ) G I-(pz), we also have

O+(PI, €1) n 0-(pz, €2) E I+(pl) n I-fpz) c UI c u as required.


FIGURE 4.5. The outer balls Of (p, E) = {q E M : d(p, q) > E)

and 0-(p, E) = {q f M : d(q,p) > E) are open in arbitrary space-

times. Furthermore, O+(p, <) and 0-(p, <) are always subsets of

I+(p) and I-(p), respectively. If (M,g) is strongly causal, the

outer balls O+(p, E ) and 0-(p, E) with p E M and E > 0 arbitrary

form a subbasis for the manifold topology.

For complete Riemannian manifolds, any two points may be joined by a

minimal (distance-realizing) geodesic segment. We now examine the dual of

this property for space-times.

In Hawking and Ellis (1973, p. 110), a timelike geodesic y from p to q is said

to be maximal if the index form of y is negative semidefinite. This definition

implies that if the geodesic y is not maximal, there exist variations of y which

yield curves from p to q "close" to y having longer Lorentzian arc length than

y. If y is maximal in this sense, however, no small variation of y keeping p

and q fixed will produce timelike curves u from p to q with L(u) > L(y).


Nonetheless, there may still exist a timelike geodesic a1 in M from p to q

("far" from y) with d(p,q) = L(o1) > L(y). Thus maximality as defined

by Hawking and Ellis does not imply "maximality in the large." To study

"maximality in the large," we adopt, in analogy to the concept of minimality

in Riemannian geometry, a definition of maximality valid for all curves in the

path space O,,, [cf. Beem and Ehrlich (1977, Definition I)]. The motivation

for our definition is Theorem 4.13 below [cf. Beem and Ehrlich (1979a, p. 166)]

and its applications, particularly the construction of geodesics as limit curves

of sequences of "almost maximal" curves in Chapter 8 and the definition of

the Lorentzian cut locus in Chapter 9.

Definition 4.10. (Maximal Curve) Let p, q E M with p < q, p # q. The

curve y E Op,, is said to be maximal if L(y) = d(p,q).

An immediate consequence of the reverse triangle inequality (4.3) is

Remark 4.11. If y : [O, 11 -+ M in R,,, is maximal, then for all s , t with

0 5 s < t 5 1, we have d(y(s), y(t)) = L(y j [s, t]) .

The following result, stated somewhat differently in Penrose (1972, Propo-

sition 7.2), is the analogue of the principle in Riemannian geometry that "lo-

cally" geodesics minimize arc length [cf. Bishop and Crittenden (1964, p. 149,

Theorem 2)].

Proposition 4.12. Let U be a convex normal neighborhood centered at

point p E M. For q E J f ( p ) , Jet jjij denote the unique nonspacelike geodesic

c : [O, 11 -+ U in U with c(0) = p and c(l) = q. If y is any future directed

nonspacelike curve in U from p to q with L(y) = d(p, q), then y coincides with

jjij up to parametrization.

Proof. For q E I+(p) and d(p,q) > 0, Penrose (1972, p. 53) shows, using a

synchronous coordinate system, that if y is any causal trip in U from p to q

other than ?5-Q, then L(y) < L(jjij) = d(p, q). This may be obtained equivalently

using the Gauss Lemma (cf. Corollary 10.19 of Section 10.1). Hence the result

is established if d(p, q) > 0.

Suppose now that d(p, q) = 0, and let y be any nonspacelike curve in U from

p to q. Then L(y) 5 d(p, q) = 0. Thus y : [O,1] -+ M is a null curve. Suppose


that y(t) $ Int(@). Let yl be the unique null geodesic in U from p to y(t),

and let yz be the unique null geodesic in U from y(t) to q. By Proposition

2.19 of Penrose (1972, p. 15), y l * ~ ~ is either a smooth null geodesic or p << q.

Since d(p, q) = 0, p << q is impossible. Hence yl * 7 2 is a smooth null geodesic

which, by convexity of U , must coincide with ?5-Q up to parametrization. 17

Proposition 4.12 has the following important consequence.

Theorem 4.13. If y E Op,, satisfies L(y) = d(p,q), then y may be

reparametrized to be a smooth geodesic.

Proof. Fix any point y(t) on y. We may find 6 > 0 such that a con-

vex neighborhood centered at y(t + 6) contains y([t - 6, t + 61). By Remark

4.11 the curve y I [t - 6, t + 61 is maximal. Thus Proposition 4.12 implies that

y I [t - 6, t + 61 may be reparametrized to be a smooth geodesic. As t was

arbitrary, the theorem now follows.

As an illustration of the use of Definition 4.10 and Theorem 4.13, we give a

simple proof of a basic result in elementary causality theory [cf. Penrose (1972,

Proposition 2.20)] that is usually obtained by different methods.

Corollary 4.14. If p < q but i t is not the case that p << q, then there is a

maximal null geodesic from p to q.

Proof. The causality assumptions on p and q imply that d(p, q) = 0. Now

let y be a future directed nonspacelike curve from p to q. By definition of

Lorentzian distance, d(p,q) 2 L(y) 2 0. Thus L(y) = d(p,q) = 0 and y is

maximal. By Theorem 4.13, y may be reparametrized to a smooth geodesic

c : \O, 11 --t M from p to q. Since L(c) 5 d(p, q ) = 0, the geodesic c must be a

null geodesic.

Note in Corollary 4.14 that since the null geodesic is maximal, it cannot t contain any null conjugate points to p prior to q [cf. O'Neill (1983, p. 404)j. A I sometime useful special case of Corollary 4.14 occurs when p = q is assumed.

k- i; In this case, one may deduce that if the space-time (M,g) is chronological I t but not causal, then there exists a smooth null geodesic 0 : [O, 11 -i (M, g)

/ with P(0) = @(I) and p ( 1 ) = Ap(0) for some A > 0. (If p ( 1 ) and p ( 0 )


were not proportional, then ,O could be deformed near P(0) to be future time-

like, contradicting the condition d(m, m) = 0 for all m in M, since (M, g) is

chronological [cf. Proposition 2.19 of Penrose (1972, p. 15)]. Even more dra-

matically, a closed timelike curve could be produced to violate chronology [cf.

Proposition 10.46 of O'Neill (1983, pp. 294-295)].)

As a second application of the elementary properties of the distance func-

tion, we give a proof of the existence of a smooth closed timelike geodesic

on any compact space-time having a regular cover with a compact Cauchy

surface. Using infinite-dimensional Morse theory, it may be shown [cf. Klin-

genberg (1978)] that any compact Riemannian manifold admits a t least one

smooth closed geodesic. However, the method of proof relies crucially on the

positive definiteness of the metric and thus is not applicable to Lorentzian

manifolds. Nonetheless, one may obtain the following theorem of Tipler by

direct methods [cf. Tipler (1979) for a stronger result].

Theorem 4.15. Let (M,g) be a compact space-time with a regular cov-

ering space which is globally hyperbolic and has a compact Cauchy surface.

Then (M, g) contains a closed timelike geodesic.

Proof. Since M is compact, there exists a closed, future directed, timelike

curve y : [0,1] + M. Set p = y(0) = y(1). Let T : G -+ M denote the given

covering manifold, and let 7 : [0, I] + M be a lift of y, i.e., ~ o y ( t ) = y(t) for all

t E [0, 11. Then 7 is a future directed timelike curve in z. Put pl = y(0) and

p2 = r(1). Then the global hyperbolicity of implies pl and p2 are distinct

points which cannot lie on any common Cauchy surface. Since T : G -+ M

is regular, there must be a deck transformation 11, : G -t % taking pl to pz

[cf. Wolf (1974, pp. 35-38, 60)j. Choose a compact Cauchy surface S1 of G containing pl, and define S 2 = $I(&). Since (%,7j) is globally hyperbolic, the - - distance function d = d(Z) : M x M + WU (m) is finite-valued and continuous.

Thus we may define a continuous function f : S1 -t W by f (s) = d(s, $(s)).

Since f 071) > 0, we have A = sup{d(s, +(s)) : s E S1) > 0. Moreover, since

S1 is compact, A < w and there exists an rl E S1 with d(rl111,(rl)) = A. Let

E : [0, I] -+ M be a timelike geodesic segment with Z(0) = TI, Z(1) = $(rl),

and L(E) = d(rl,$(rl)) = A. This geodesic exists since ( z , ~ ) is globally


hyperbolic. Because 5 = ~ ' g , it follows that c = ?r oE : [0, I] -+ M is a timelike

geodesic. Since Z(0) = r l and E(1) = +(rl) , we also have c(0) = c(1). If c

were not smooth at c(O), we could deform c to a timelike curve u : [O, 11 + M - with L,(o) > L(c), a(0) = u(1) E ~ ( S I ) , which lifts to a curve 3 : [O, 11 -+ M

with a(0) E S1 and Z(1) = q(Z(0)) E S2 . But then L z (a) = L,(u) > L,(c) =

L a (Z) = A, in contradiction. 0

More recently, G. Galloway (1984b) has obtained, by elementary geometric

arguments, the existence of a closed timelike geodesic from any stable nontrivial

free homotopy class of closed timelike curves on an arbitrary compact space-

time. Galloway's approach is based on geodesic convexity methods which were

earlier used to obtain a result, basic to the development of global Riemannian

geometry during the early part of this century, given first by J. Hadamard for

surfaces and then by E. Cartan for general Riemannian manifolds of higher

dimension. Their result concerns the existence, within any nontrivial free ho-

motopy class of curves on a compact Riemannian manifold, of a shortest curve

in the given homotopy class, which must then be a nontrivial smooth closed

geodesic. From a directly geometric viewpoint, certain basic ideas involved in

the existence of this geodesic are the following.

Let Lo > 0 denote the infimum of all lengths of curves in this homotopy

class. Choose a minimizing sequence {ck) with lim L(ck) = LO in the given free

homotopy class. By compactness and convexity radius arguments, one covers

the given manifold by a finite number of geodesically convex neighborhoods

(each having compact closure in a larger geodesically convex neighborhood).

Using these neighborhoods in succession, each ck may be approximated by

a piecewise smooth geodesic, or equivalently, may be viewed as an N-tuple

of successive points pl(k),p2(k), . . . ,pN(k), where a uniform bound needs to

be given for the number N of points required. Since a geodesic segment in

a convex neighborhood is the shortest curve between any two of its points,

this approximation procedure produces a shorter curve than ck, but one which

is also still homotopic to ck, and hence is in the given free homotopy class.

By compactness, a diagonalizing argument gives points pi, pz, . . . , p ~ which

are limits of a subsequence of all of the above sequences. Joining p, to p,+l


successively produces a piecewise smooth geodesic c in the given free homotopy

class which realizes the minimal length Lo. Now the geodesic c must in fact be

smooth a t the pi's, or an even shorter closed curve in the free homotopy class

could be produced by the usual "rounding off the corners" procedure. Since

Lo > 0, this closed geodesic c must be nontrivial, i.e., not a "point curve." A

detailed discussion of the steps involved in the above argument may be found

in Spivak (1979, p. 358).

Now in carrying these ideas over to the space-time setting, it is clear that

"minimal geodesic segment" should be replaced by "maximal timelike geodesic

segment." Technical difficulties arise, however, since the set of unit timelike

tangent vectors based at a given point is not a compact set. Hence, problems

can arise with timelike tangent vectors tending toward a null direction when

trying to do subsequence arguments. Equivalently, in the above context, it is

necessary to prevent timelike geodesic segments joining pi(k) to pi+l(k) from

converging to a null geodesic segment from pi to pi+l when the diagonalization

procedure is carried out. Indeed, Galloway (198613) gives an example of a

compact space-time which contains no closed timelike geodesics but which

contains a closed null geodesic.

In view of these difficulties, Galloway (1984b) considers timelike free ho-

motopy classes which are "stable" for a given compact space-time (M,go).

Here a given free timelike homotopy class C for (M, go) is said to be stable if

there exists a "wider" Lorentzian metric g for M , i.e., go < g in the sense of

the discussion following Remark 3.14, such that if L, denotes the Lorentzian

arc length for (M,g), then the given timelike homotopy class C satisfies the

condition

sup L,(c) < +m. C€E

Galloway (1984b) shows that this concept of stability gives the control needed

to force convergence arguments to be successful and hence obtains the theorem

that for any compact Lorentzian manifold, each stable free timelike homotopy

class contains a longest closed timelike curve which is of necessity a closed

timelike geodesic. Galloway also shows how his result may be used to recover

Tipler's Theorem 4.15 given above and gives a criterion for a free homotopy

4.2 DISTANCE PRESERVING AND HOMOTHETIC MAPS 151

class of closed timelike curves to be stable. Finally, we note that in Galloway

(198613) it is shown by covering space arguments that every compact two-

dimensional Lorentzian manifold contains a closed timelike or null geodesic.

Here one uses dimension two in the essential way that a closed timelike curve for

(M, g) corresponds to a closed spacelike curve (hence a spacelike hypersurface)

for (M, -g), which is also a Lorentzian manifold since dim(M) = 2. Thus,

certain techniques in general relativity involving spacelike hypersurfaces may

be applied to the existence problem.

4.2 Distance Preserving a n d Homothet ic M a p s

Myers and Steenrod (1939) and Palais (1957) have shown that i f f is a dis-

tance preserving map of a Riemannian manifold (Nl, gl) onto a Riemannian

manifold (N2, g2), then f is a diffeomorphism which preserves the metric ten-

sors, i.e., f *gz = 91. In particular, every distance preserving map of (Nl,gl)

onto itself is a smooth isometry. In this section we give similar results for

Lorentzian manifolds following Beem (1978a).

Recall that a diffeomorphism f : (M1,gl) + (M2,gz) of the Lorentzian

manifold (M1,gl) onto the Lorentzian manifold (M2,gz) is said to be homo-

thetic if there exists a constant c > 0 such that gz(f,v, f,w) = cgl(v, w) for

all v, w E TpMl and all p E MI. In particular, if c = 1, then f is a (smooth)

isometry. The group of homothetic transformations is important in general

relativity since it has been shown to be the group of transformations which

preserves the causal structure for a large class of space-times [cf. Zeeman

(1964, 1967), Gobel (1976)j.

We will let dl denote the Lorentzian distance function of (Ml, gl) and d2

denote the Lorentzian distance function of (M2, 92) below. The distance ana-

logue of a smooth homothetic map is defined as follows.

Definition 4.16. (Distance Homothety) A map f : ( M I , gl) --+ (M2,gz) is said to be distance homothetic if there exists a constant c > 0 such that

d2(f(p), f(q)) = cdl(p,q) for all p,q E 1M. If c = 1, then f is said to be

distance preserving.


It is important to note that for arbitrary Lorentzian manifolds, distance

preserving maps are not necessarily continuous. For if (M,g) is a totally

vicious space-time, we have seen that d@, q) = 03 for all p, q E M [cf. Lemma

4.2-(2)]. Hence any set theoretic bijection f : M + M is distance preserving

but need not be continuous.

T h e o r e m 4.17. Let ( M I , g l ) be a strongly causal space-time, and let

(M2,g2) be an arbitrary space-time. If f : (Ml ,g l ) -+ (M2, g 2 ) is a dis-

tance homothetic map (not assumed to be continuous) of Ml onto M2, then f

is a smooth homothetic map. That is, f is a diffeomorphism, and there exists

a constant c > 0 such that f *g2 = cg1. In particular, every map of a strongly

causal space-time ( M , g) onto itself which preserves Lorentzian distance is an

isometry.

Corollary 4.18. If ( M I g) is a strongly causal space-time, then the space of

distance homothetic maps of ( M , g) equipped with the compact-open topology

is a Lie group.

Proof of Corollary 4.18. Since ( M , g ) is strongly causal, this group coincides

by Theorem 4.17 with the space of smooth homothetic maps of M onto itself

which preserve the time orientation. But this second group is a Lie group. CI

The proof of Theorem 4.17 will be broken up into a series of lemmas.

L e m m a 4.19. Let ( M l , g l ) and (Mz,g2) be space-times, and consider a

map f : ( M l , g l ) --+ (M2,g2) which is onto but not necessarily continuous. If

f is distance homothetic, then

Proof. First (1) holds since d2(f (p), f (9)) = cdl(p, q), and P << q [respec-

tively, f (p) << f (q)] iff dl(p, q) > 0 [respectively, d2(f ( P I , f ( 9 ) ) > 01. Since (1)

implies p << r << q iff f (p) << f ( r ) << f (q), statement (2) foll~ws. 0

The importance of (2) stems from the fact that if ( M , g) is strongly causal,

the sets {I+(p) n I - (q) : p, q E M ) form a basis for the topology of M . Recall


that a map f : Ml + M2 is said to be open if f maps each open set in MI t o

an open set in M2.

Lemma 4.20. Let ( M I , g l ) be strongly causal and let (M2,g2) be an ar-

bitrary space-time. If f is a (not necessarily continuous) distance homothetic

map of ( M I , g l ) onto (M2,g2), then f is open and one-to-one.

Proof. The openness of f is immediate from part (2) of Lemma 4.19. It

remains to show that f is one-to-one. Assume there are distinct points p and

q of MI with f ( p ) = f(q). Let U(p) be an open neighborhood of p with

q $! U(p) and such that no nonspacelike curve intersects U(p) more than once.

Choose r l , 7-2 E U(p) with rl << p << 7-2. Clearly, q 6 I f ( r l ) n I - ( r 2 ) . It follows

from Lemma 4.19 that f (r1) << f (p) = f ( q ) << f (r2) implies rl << q << 7-2.

This yields q f I+(r l ) n I-(r2) which is a contradiction. O

Applying Lemma 4.20 to f and f-' we obtain

Proposi t ion 4.21. Let ( M I , g l ) be strongly causal, and let ( M 2 , g2) be an

arbitrary space-time. Let f be a (not necessarily continuous) map of MI onto

M2. I f f is distance homothetic, then f is a homeomorphism and (M2,g2) is

strongly causal.

Proof. The relation f -' is a function since f is one-to-one by Lemma 4.20.

Furthermore, f -' is continuous since Lemma 4.20 shows f is an open map.

In order to complete the proof it is sufficient to show M2 is strongly causal

since Lemma 4.20 will then imply f -' is an open map, whence f is continuous.

Given pi E M2, let p = f-'(p'). If r' << p' << q', then Lemma 4.19 applied to

the distance homothetic map f -' yields f -'(rl) << p << f - l (q l ) . Let U1(p') be

an open neighborhood of p'. Choose V1(p') 2 U'(pl) with the closure of V'(pl)

a compact set contained in an open convex normal neighborhood W'(p l ) of p'.

We may assume that (W' (p f ) ,g2 I,,(,,)) is globally hyperbolic. Let {r;) and

(9;) be sequences in V1(p') such that r; -+ P I , q; --, p', and r; << << q;

for all n. Assume the strong causality of M2 fails at pl. This means that for

each n, the set I+(r;) n I-(9;) cannot be contained in the convex normal

neighborhood W1(p') because otherwise the sets I f (r;) n I-(9;) would give

arbitrarily small neighborhoods of p' which each nonspacelike curve intersects


at most once. Choose a sequence o f points (2;) contained in the boundary

o f V t (p ' ) with zk E I+(&) n I-(qh) for each n. The sequence (26) has an

accumulation point z because the closure o f V t (p t ) is compact. Furthermore,

f E f - l ( I+(rh)nI-(qh)) = I+(f - ' ( rL))nI- ( f The continuity

o f f -' implies that f -l(r;) -+ p and f -+ p. The strong causality o f

MI yields that the sets I+( f - l ( rk ) ) n I - ( f - l (q&)) are approaching the point

p. Thus, f - I ( & ) -+ p which means f - l ( z ) = p = f -'(pt). This contradicts

the one-to-one property of f - l . Consequently, M2 must be strongly causal,

and the proposition is established. 13

Consider the strongly causal space-time M . Given p E M , let U(p) be

a convex normal neighborhood of p. The set U(p) may be chosen so small

that whenever q, r E U(p) with q < r , the distance d(q, r ) is the length o f the

unique geodesic segment a(q , r ) from q to r which lies in U(p). Furthermore,

U ( p ) may be chosen such that i f q, z,.r E U(p) with q << z << r , then the

reverse triangle inequality d(q, r ) 2 d(q, z ) +d(z , r ) is valid with strict cquality

i f and only i f z is on the geodesic segment from q to r in U(p). Thus, timelike

geodesics in a strongly causal space-time are characterized by the space-time

distance function, and it follows that distance homothetic maps take timelike

geodesics to timelike geodesics.

Lemma 4.22. I f f is a distance homothetic map defined on a strongly

causal space-time, then f maps null geodesics to null geodesics.

Proof. Let U(p) be a convex normal neighborhood o f p as in the above

paragraph, chosen sufficiently small such that f (U(p)) lies in a convex nor-

mal neighborhood o f f ( p ) . Let a ( q , r ) be a null geodesic in U(p) . Choose

q, - q and r , -- r with q, << T, for all n. Proposition 4.21 then im-

plies that f (q,) -+ f (q) and f (r,) -+ f ( r ) . The map f takes the time-

like geodesic a(q,,r,) with endpoints q, and r, to the timelike geodesic

a( f (q,), f (r,)). Since the geodesics a(q,, r,) converge to a (q , r ) and the

geodesics a( f (q,), f (r,)) converge to a( f (qf , f ( r ) ) , it follows that f maps

r, a ( f (q)i f ( r ) ) .


Proof of Theorem 4.17. The fact that f is a diffeomorphism follows from

a result proved by Hawking, King, and McCarthy (1976) which states that a

homeomorphism which maps null geodesics to null geodesics must be a diffeo-

morphism. Since Ml and M2 are strongly causal, for each p E Ml there exists

a convex normal neighborhood Ul (p) such that for q E U1 (p) with p << q, the

lengths o f the timelike geodesics a(p , q) joining p to q and a( f (p ) , f (q ) ) join-

ing f (p ) to f (q) are given by dl(p, q) and d2( f ( p ) , f ( q ) ) , respectively. Using

d2(f ( P ) , f (q ) ) = cdl(p ,q) , it follows that f maps gl onto the tensor c-2gz.

It is well known that i f a complete Riemannian manifold is not locally flat,

then it admits no homothetic maps that are not isometries [cf. Kobayashi and

Nomizu (1963, p. 242, Lemma 2)] . An essential step in the proof consists o f

showing for arbitrary complete Riemannian manifolds that any homothetic

map which is not an isometry has a unique fixed point. This may be done

by using the triangle inequality for the Riemannian distance function and the

metric completeness o f any geodesically complete Riemannian manifold.

In view o f Theorem 4.17 above, it is then o f interest to consider the analo-

gous question of the existence o f nonisometric homothetic maps o f a Lorentzian

manifold [cf. Beem (1978b)j. In what follows, we will use the standard ter-

minology o f proper homothetic map for a homothetic map which is not an

isometry.

W e first note that R2 with the Lorentzian metric ds2 = dxdy provides an

example of a globally hyperbolic geodesically complete space-time that admits

a fixed-point free, proper homothetic map. For fixing any ,b' # 0 and choosing

any c > 0, the map f (x: y) = ( x -k p, cy) is a fixed-point free homothetic

map with homothetic constant c. Thus the existence o f a fixed point for a

proper homothetic map must be assumed for geodesically complete Lorentzian

manifolds unlike the Riemannian case.

Now suppose f is a proper homothetic map o f a space-time ( M , g ) such

that f (p ) = p for some p E M . Then f,p : T,M -+ T,M has at least one

nonspacelike eigenvector [cf. Beem (1978b, p. 319, Lemma 3)]. This eigenvec-

tor may be null, however. For example, composing the Lorentzian "boost"


isometry F of (W2, ds2 = dx2 - dy2),

F(x, y) = (x cosh t + y sinh t, x sinh t + y cosh t)

with t > 0 fixed, and a dilation T(x, y) = (cx, y) with c > 0 and c # 1 yields a

proper homothetic map f of Minkowski space-time fixing the origin such that

f*(,,,, has null vectors for eigenvectors.

But if f,p : T p M -+ T p M is a proper homothetic map which has a timelike

eigenvector with eigenvalue X < 1, it may be shown that (M, g) is Minkowski

space-time [cf. Beem (197813, p. 319, Proposition 4)]. Also, iff is a homothetic

map with a fixed point p such that all eigenvalues of f,p are real and all have

absolute value less than one, then (M,g) is Minkowski space-time [cf. Beem

(197813, p. 316, Theorem I)].

We now give an example of a n o d a t space-time admitting a global homo-

thetic flow. Let M = W3 with the metric g = ds2 = exzdxdy + dz2. Thus

are tangent vectors a t (x, y, z), we have

It may then be checked that while (M, g) is not flat, the map 4, : (W3, ds2) -+

(W3, ds2) given by

&(x, y, z) = (etx, e-3ty, e?z)

is a proper homothety with g(&v, q5t.w) = e-2tg(v, w) for each fixed nonzero

t .

We now show, however, that this space-time is null geodesically incomplete.

Let X = d/dx, Y = d/by, and Z = d/az. Then all inner products vanish

except for g(X, Y) = ezz/2 and g(Z, Z) = 1; furthermore, [X, Y] = [X, Z] = 0.

Hence using the Koszul formula

4.3 LORENTZIAN DISTANCE FUNCTION AND CAUSALITY 157

we obtain the following formulas for the Levi-Civita connection of ( R ~ , ds2):

x x x VxY = --ex"Z, V x Z = -X, and V y Z = -Y.

4 2 2

Thus the only nonzero Christoffel symbols are I'il = z, r:2 = I':, = -$ex",

and I'i3 = I'i, = = I?j2 = $. Hence if y(t) = (x(t), y(t), ~ ( t ) ) is a geodesic,

the usual system of second order differential equations

for y reduces to the following system:

It may finally be checked that the curve y : (-1, w) + (LR3,ds2) defined by

y(t) = (ln(1 + t), 0 , l ) satisfies this system of differential equations and hence

is the unique null geodesic in (W3,ds2) with yl(0) = d / d ~ l ~ ~ , ~ , , ) . Thus this

space-time is null geodesically incomplete.

4.3 The Lorentzian Distance Function and Causality

In this section we study the relationship between the continuity and finite-

ness of the Lorentzian distance function d = d(g) : M x M -+ W U {a) for

(M, g) and the causal structure of (M, g). The most elementary properties,

extending Lemma 4.2 above, are summarized in the following lemma. Re-

call that Lor(M) denotes the space of all Lorentzian metrics for M. The C0

topology on Lor(M) was defined in Section 3.2.

Lemma 4.23.

11) d(p,q) > 0 iff 9 f I+(P). (2) The space-time (M, g) is t o t d y vicious iff d(p, q) = co for all p, q E M .


(3) The space-time (M, g) is chronological iff d is identically zero on the

diagonal A(M) = {(p,p) : p E M ) of M x M.

(4) The space-time (M,g) is future [respectively, past] distinguishing iff

for each pair of distinct p, q E M , thcrc is some x E M such that

exactly one of d(p, x) and d(q, x) [respectively, d(x, p) and d(x, q)] is

zero.

( 5 ) The space-time (M,g) is stably causal iff there exists a neighborhood

U of g in the fine C0 topology on Lor(M) such that d(gl)(p,p) = 0 for

all g' E U and p E M.

Proof. Similar to Lemma 4.2 and Remark 4.3.

Recall that the Lorentzian distance function in general fails to be upper

semicontinuous. Thus the continuity of d(g) should have implications for the

causal structure of (M,g). An example is the following result first stated in

Beem and Ehrlich (1977, p. 1130). Here d is regarded as being continuous at

(p, q) E M x M with d(p,q) = co because d(p,, q,) -+ w for all sequences

p, -+ p and q, -+ q (cf. Lemma 4.4).

Theorem 4.24. Let (M,g) be a distinguishing space-time. If d = d(g) :

M x M + R U {w) is continuous, then (M, g) is causally continuous.

Proof. We need only show that I+ and I- are outer continuous. Assume - I+ is not outer continuous. There is then some compact set K C M - I+(p)

and some sequence p, -+ p such that K n I+(p,) # 0 for all n. Let q, E

K n I+(pn) and let {qm) be a subsequence of {q,) such that {q,) converges

to some point q of the compact set K. Then q, -+ q and q, E I+(p,)

imply there must be a sequence {q;) converging to q such that qk E I+(p,) - for each m. Since M - I+(p) is an open neighborhood of q, there is some

r E M - I+(p) with q << r . For sufficiently large m we then have q i << r

and hence p, << qk << r . Thus d(p,, r ) 2 d(p,, qk) + d(qk, r). Using the

lower semicontinuity of distance and the causality relation q << r, we obtain

0 < d(q,r) 5 liminf d(qk,r) . Consequently, d(p,,r) 2 d(q,r)/2 > 0 for

all sufficiently large m. However, since r 4 I+(p), we have d(p,r) = 0, and

hence d(p,r) # lirnd(p,,r). Thus if d is continuous, I+ is outer continuous.


FIGURE 4.6. Let (M,g) denote Minkowski space-time with the

point r deleted. Choose p, q E M such that in Minkowski space-

time the point r is on the boundary of I+(p) n I-(q) as shown.

Let {q,) be a sequence of points approaching q with q << q, for

each n. There is a smooth conformal factor 0 : M -+ (0, m) such

that R = 1 on I+(p) n I-(q) and yet d(Qg)(p, q,) 2 2d(g)(p, q) for

each n. The function R will be unbounded near the deleted point

T. Since d(g)(p, q) = d(Rg)(p,q) < liminf d(Rg)(p,q,), the ca~lsally

continuous space-time (M, Rg) has a Lorentzian distance function

which is discontinuous at (p, q) E M x M .

A similar argument shows that I- is outer continuous. Thus continuity of d

implies that (M, g) is causally continuous. fl


The following example shows that the converse of Theorem 4.24 is false. Let

(M, g) denote Minkowski space-time with a single point removed. The space-

time (M,Rg) will be causally continuous for any smooth conformal factor

R : M -+ (0,m). However, R may be chosen such that d = d(Rg) is not

continuous (cf. Figure 4.6).

We now turn to a characterization of strongly causal space-times in terms

of the Lorentzian distance function. The definition of convex normal neigh-

borhood was given in Section 3.1. Given any spacetime (M, g), we have

Definition 4.25. (Local Distance Function) A local distance function

(D, U) on (M, g) is a convex normal neighborhood U together with the distance

function D : U x U -+ R induced on U by the space-time (U, gIu).

More explicitly, given p, q E U, then D(p, q) = 0 if there is no future di-

rected timelike geodesic segment in U from p to g. Otherwise, D(p, q) is the

Lorentzian arc length of the unique future directed timelike geodesic segment

in U from p to q.

We will let I+(p, U) (respectively, J+(p, U)) denote the chronological (re-

spectively, causal) future of p with respect to the space-time (U, glU).

Lemma 4.26. Let (M, g) be a space-time, and let U be a convex normal

neighborhood of (M, g). Assume that D : U x U -t IF% is the distance function

for (U, glu). Then D is a continuous function on U x U and D is differentiable

on U+ = {(p, q) E U x U : q E I+(p, U)).

Proof. Given p,q E U with q E J+(p,U), let cpq : [0, I] -+ U denote the

unique nonspacelike geodesic segment with cpq(0) = p and %,(I) = q. We

then have D(p, q) = [-g(ci,(~), C ~ , ( O ) ) ] ~ / ~ and D(P, 9)' = [-g(ciq(0), c;,(O))].

Rom the differentiable dependence of geodesics on endpoints in convex neigh-

borhoods, it is immediate that D is continuous on U x U and that D is differ-

entiable on U+. U

Minkowski space-time shows that D fails to be differentiable across the null

cones and thus fails to be smooth on all of U x U. It is not hard to see that the

local distance function (D, U) uniquely determines the Lorentzian metric g on

U. Consequently if {U,) is a covering of M by convex normal neighborhoods


with associated local distance functions {(D,, U,)), then {(D,, U,)) uniquely

determines g on M.

We now characterize strongly causal space-times in terms of local distance

functions [cf. Beem and Ehrlich (1979c, Theorem 3.4)].

Theorem 4.27. A space-time (M, g) is strongly causal if and only if each

point r E M has a convex normal neighborhood U such that the local distance

function (D, U) agrees on UX U with the distance function d = d(g) : M x M -+

R u {m).

Proof. If (M, g) is strongly causal and r E M, then there is some convex nor-

mal neighborhood U of r such that no nonspacelike curve which leaves U ever

returns. The local distance function for U then agrees with d = d(g) 1 (U x U).

Conversely, assume that strong causality breaks down at some point r E M.

Let U be a convex normal neighborhood of r such that D(p, q) = d(p, q)

for all p, q f U. There exists a neighborhood W C U of r such that any

future directed nonspacelike curve y : (0,1] -+ U with y(1) E W and y past

inextendible in U contains some point not in J f (W, U). Since strong causality

fails to hold at r, there is a future directed timelike curve y, : [O, 11 --+ M with

T' = yl(0) E W, y1(1/2) 4 U, and yl( l ) E W. By construction of W, there

is some point p E 71 n U with p $2 J+(r1, U). Hence D(rl ,p) = 0. However,

d(rr,p) > 0 since d(rr,p) is at least as large as the length of yl from r' to

p. Thus D(rl,p) # d(rr,p). Taking the contrapositive then establishes the

theorem. (7

Corollary 4.28. If (M, g) is strongly causal, then d is continuous on some

neighborhood of A(M) = {(p,p) : p E M) in M x M. Also, given any

point m E M, there exists a convex normal neighborhood U of m such that

d 1 (U x U) is finite-valued.

We now give a characterization of globally hyperbolic space-times among

all strongly causal space-times using the Lorentzian distance function. For

this purpose, it is first necessary to show that the usual definition of globally

hyperbolic may be weakened.


Lemma 4.29. Let ( M , g) be a strongly causal space-time. If J+(p)n J-(q)

has compact closure for all p, q E M , then ( M , g) is globally hyperbolic.

Proof. I t is only necessary to show J+(p) n J-(q) is always closed. Assume

T E (J+(p) n J- (q ) ) - (J+(p) n J-(q)) . Choose a sequence {T,) o f points

in J+(p) n J-(q) with r , -+ r . For each n let yn : [O, 1) --, M be a future

directed, future inextendible, nonspacelike curve with p = y,(O) and r, E y,,

q E y,. By Proposition 3.31, there is some future directed, future inextendible,

nonspacelike limit curve y : [O, 1) -+ M of the sequence (7,). Furthermore,

p = y(0). The limit curve y cannot be future imprisoned in any compact subset

o f M because ( M , g ) is strongly causal (c f . Proposition 3.13). Consequently,

there is some point x on y with x @ J+(p) fl J-(q). The definition o f limit

curve yields a subsequence (7,) o f {y,) and points x, E y, with x , -t x.

Since x @ J+ (p) n J - (q) , we have x , 4 J+(p) n J-(q) for all large m. Using

y, C J+(p), it follows that x , @ J-(q) for large m. Hence q lies between p

and x, on y, for large m. Let y [ p , x ] (respectively, y,[p, x,]) denote the

portion o f y (respectively, 7,) from p to x (respectively, x,). By Proposition

3.34 we may assume, by taking a subsequence o f {y,[p, x,]) i f necessary,

that {y,[p,xm]) converges to y [ p , x ] in the C0 topology on curves. Hence

q E y,[p, x,] for large m implies q E y[p, x ] . Also r , -+ r and T, 6 q which

yield r E y [ p , q ] . Thus r E J+(p) n J-(q), in contradiction. O

Recall from Definition 4.6 above that a space-time ( M , g ) is said to satisfy

the finite distance condition i f and only i f d(g)(p, q) < m for all p, q E M . This

condition may be used to characterize globally hyperbolic space-times among

strongly causal space-times [cf. Beem and Ehrlich (197913, Theorem 3.5))

Theorem 4.30. The strongly causal space-time ( M , g) is globally hyper-

bolic i f f (M,gr ) satisfies the finite distance condition for dl g' € C ( M , g).

Proof. I t has already been remarked that i f ( M , g ) is globally hyperbolic,

then all metrics in C ( M , g) satisfy the finite distance condition (c f . Corollary

4.7).

Conversely, assume that ( M , g) is not globally hyperbolic. Lemma 4.29

implies that there exist p,q E M such that J+(p) n J- (q ) does not have


compact closure. Let h be an auxiliary geodesically complete positive definite

metric on M , and let do : M x M -+ B be the Riemannian distance function

induced on M by h. The Hopf-Rinow Theorem implies that all subsets o f M

which are bounded with respect to & have compact closure. Thus J+(p) n J- (q ) is not bounded. Hence, for each n we may choose p, E J+(p) n J-(q)

such that do(p,p,) > n. Choose p' and q' with p' < 1, choose y, to be a future directed timelike curve from p' to pn such that

yn[1/2, 3/41 5 { r E M : n-1 < da(p, r ) < n ) . For each n > 1, let R, : M -+ IW be a smooth function such that O,(x) = 1 i f x 4 { r : n - 1 < &(p, r ) < n ) and

such that the length of yn[1/2, 3/41 is greater than n for the metric R,g. Let

R = n, R,. This infinite product is well defined on M since for each x E M

at most one o f the factors R, is not unity. Then d(Rg) (p', p,) > n for each

n > 1. Hence d(flg)(pl, 9') = as d(Rg)(p1, q') L d(Rg)(pl, pn) + d ( R g ) ( ~ n , q') for each n. 0

W e now turn to the proof that for distinguishing space-times with continu-

ous distance functions, the future and past inner balls form a subbasis for the

given manifold topology. Recall that

and

Thus defining f i : M -+ R for i = 1,2 by f i(q) = d(p,q) and f2(q) = d(q,p),

we have B+(P, E ) = f ~ ' ( ( 0 , E ) ) and B-(p, E ) = f ~ ' ( ( 0 , E ) ) . Hence i f ( M , g) has

a continuous distance function, the inner balls B+ (p, E) and B- (p , E ) of M are

open in the manifold topology.

Proposition 4.31. Let ( M , g) be a distinguishing space-time with a con-

tinuous distance function. Then the collection

forms a basis for the given manifold topology o f M


Proof. The above arguments show that sets of the form B+(p, c l ) n B V ( q , c2)

are open in the manifold topology. Thus given an arbitrary point r E M

and an arbitrary open neighborhood U ( r ) o f r in the manifold topology, it is

sufficient to show that there exist p, q E M and €1, €2 > 0 with T E B+(p, el) n B-(q, € 2 ) U(r ) .

Theorem 4.24 yields that ( M , g ) is causally continuous and hence also

strongly causal. Thus we may choose a convex normal neighborhood V of

r with V U ( r ) such that no nonspacelike curve which leaves V ever returns

and such that d : V x V + W U {oo) is finite-valued (cf . Corollary 4.28). Fix

p, q E V with p << r << q. Then r E I+(p) n I - (q) C V since no nonspace-

like curve from p to q can leave V and return. Letting €1 = d(p,r) + 1 and

€2 = d(r, q) + 1, we obtain

which establishes the proposition. 0

W e conclude this section with a characterization o f totally geodesic timelike

submanifolds in terms o f the Lorentzian distance function. An analogous result

holds for submanifolds o f (not necessarily complete) Riemannian manifolds

[cf. Gromoll, Klingenberg, and Meyer (1975, p. 159)].

Let ( M , g) denote an arbitrary strongly causal space-time. Suppose that

i : N + M is a smooth submanifold and set i j = i*g. Recall that ( N , i j ) is said

t o be a timelike submanifold of ( M , g) i f 3 l p : TpN x TpN + W is a Lorentzian

metric for each p f N . As usual, we will identify N and i ( N ) . Let Z , L

and 2, d denote the arc length functionals and Lorentzian distance functions

of ( N , i j ) and ( M , g ) , respectively. Then i f y is a smooth curve in ( N , i j ) , we

have Z(r) = L(y) . Note also that i f q E I+(p, N ) , then p << q in ( M , g), and

i f q E J+(p, N ) , then p < q in ( M , g ) . Thus it follows immediately from the

definitions o f d and d that

- (4.4) d (m, n) 5 d(m, n) for all m, n E N .

Wi th this remark in hand, we are ready to prove the following result.


Proposition 4.32. Let ( N , 3 ) be a totally geodesic timelike submanifold

of the strongly causal space-time ( M , g ) . Then given any p E N , there exists

a neighborhood V of p in N such that d 1 ( V x V ) = 2 I (V x V ) .

Proof. First, let W be a convex neighborhood of p in ( M , g ) such that every

pair o f points m , n E W are joined by a unique geodesic of ( M , g) lying in W

and i f m 6 n, then this geodesic is maximal in ( M , g ) . We may then choose

a smaller neighborhood Vo of p in M with Vo C W such that i f V = Vo n N ,

then V is contained in a convex normal neighborhood U o f p in N with U

contained in W .

Suppose first that m , n E V and n E J+(m, N ) . Since V C U , there exists

a nonspacelike geodesic y o f (N,ij) in U from m to n . Also, as N is totally

geodesic, y is a nonspacelike geodesic in ( M , g). Because y is contained in U

and U E W , y is maximal in (M,g) . W e thus have d ( m , n ) > Z ( y ) = L(T) =

d(m, n ) . In view o f (4.4), we obtain d(m, n ) = d(m, n ) as required.

It remains to consider the case that m , n E V and n # J+(m, N ) . Thus - d(m, n ) = 0 by definition. Suppose that d (m, n ) > 0. Then there exists a

timelike geodesic yl o f ( M , g ) in W from m to n. On the other hand, since

m , n E U , there exists a geodesic 7 2 o f ( N , i j ) from m to n lying in U which

must be spacelike since n # J+(m, N ) . Since (N,?j) is totally geodesic, 7 2 is

also a spacelike geodesic o f ( M I g) from m to n which lies in U C W . Thus we

have distinct geodesics yl and 72 in W from m to n , in contradiction. Hence

d ( m , n ) = 0 =Z(m,n) as required. Cl

W e now prove the converse of Proposition 4.32.

Proposition 4.33. Let ( N , i j ) be a timelike submanifold of the strongly

causal space-time ( M , g ) . Suppose that for all p E N there exists a neighbor-

hood V o f p in N such that d / ( V x V ) = 2 I (V x V ) . Then ( N , g ) is totally

geodesic in ( M , g).

Proof. It suffices to fix any p E N and show that the second fundamental

form S, vanishes at p (cf . Definition 3.48). Since any tangent vector in TpN

may be written as a sum o f nonspacelike tangent vectors, it is enough to

show that S,(v, w) = 0 for all nonspacelike tangent vectors in T p N . Also, as


Sn(- v, w) = -Sn(v, w), it suffices to show that Sn(v, w) = 0 for all future

directed nonspacelike tangent vectors in T,N.

Thus let v E T,N be a future directed nonspacelike tangent vector. Let

y denote the unique geodesic in (N,?j) with y'(0) = v. Also let V be a

neighborhood of p in N on which the distance functions a and d coincide.

Choose t > 0 such that if m = y(t), then m E V and z(p, m) = Z(y 1 [0, t ] ) is

finite. We then obtain

d ( ~ , m ) L L(T / 10, t ] ) = Z(y 1 10, t ] ) = a(p, m).

But since m E V we have d(p,m) = z(p,m), whence L(y I [O,t]) = d(p,m).

Hence y I [O, t ] is a geodesic in (M,g) by Theorem 4.13. Thus we have shown

that if u E TpN is any future directed tangent vector, the geodesic in (M,g)

with initial direction u is also a geodesic in (N,?j) near p. Therefore Sn(u, u) =

0 for all future directed nonspacelike tangent vectors. Since the sum of two

nonparallel future directed nonspacelike tangent vectors is future timelike, it

follows by polarization that S,(u, w) = 0 for all future directed nonspacelike

tangent vectors u, w E TpN, as required. 0

Combining Propositions 4.32 and 4.33 yields the following characterization

of totally geodesic timelike submanifolds of strongly causal space-times in

terms of the Lorentzian distance function.

Theorem 4.34. Let (M, g) be a strongly causal space-time of dimension

n 2 2, and suppose that (N, i'g) is a smooth timelike submanifold of (M,g),

i.e., ?j = i*g is a Lorentzian metric for N . Then (N,?j) is totally geodesic

iff given any p E N, there exists a neighborhood V of p in N such that the

Lorentzian distance functions 2 of ( N , 3 ) and d of ( M , g ) agree on V x V.

4.4 Maximal Geodesic Segments and Local Causality

We have seen in this chapter that certain causality conditions placed on a

space-time are related to pleasant local or global behavior of the Lorentzian

distance function. For instance, we saw that if (M,g) is globally hyperbolic,

then all metrics conformal to g for M are not only continuous but also satisfy

the finite distance condition as well. Further, the finite distance condition for

4.4 MAXIMAL GEODESIC SEGMENTS, LOCAL CAUSALITY 167

all metrics conformal to a given strongly causal metric for a space-time forces

the conformal class of space-times to be globally hyperbolic. If a space-time

is strongly causal, then we noted that for any given p in A4 there exists a

neighborhood U of p in which the local distance function of (U, glu) coincides

with the global Lorentzian distance function on U x U. Hence the Lorentzian

distance function is forced to be finite-valued on U x U.

For the purposes of this section, it will be convenient to reformulate Defi-

nition 4.10 slightly as follows.

Definition 4.35. (Mmimal Segment) A future directed nonspacelike

curve c : la, b] -, (M, g) is said to be a maximal segment provided that

L(c) = d(c(a), c(b)) and hence L(c 1 [ s , t ] ) = d(c(s), c(t)) for all s , t with

a s s l t l b .

Evidently, the Lorentzian distance function restricted to the image of a

maximal segment must be finite-valued and continuous. In the terminology

of Definition 4.35, the previous Proposition 4.12 may be rephrased as follows.

Suppose that (M, g) is strongly causal. Given p in M , let U be a local causality

neighborhood of p, i.e., a causally convex neighborhood about p which is also

a geodesically convex normal neighborhood. Then any nonspacelike geodesic

segment lying in U is a maximal segment in the space-time (M, g).

In Chapter 14 the Lorentzian Splitting Theorem for timelike geodesically

complete (but not necessarily globally hyperbolic) space-times is studied.

Since global hyperbolicity is not assumed and hence the global continuity

and finite-valuedness of the space-time distance function may not be taken

for granted, it is necessary to make a careful study of the space-time distance

function and asymptotic geodesics in a neighborhood of a given timelike line.

What emerged in a series of papers, as summarized in the introduction to

Chapter 14 which will not be repeated here, is that the existence of a global

maximal timelike geodesic has important implications for the distance function

and the Busemann function of the timelike line in a neighborhood of the given

line [cf. Eschenburg (1988), Galloway (1989a), Newman (1990), Galloway and

Horta (1995)l. Especially, Newman (1990) made a thorough study of the geo-

desic geometry in the case that timelike geodesic completeness, but not global


hyperbolicity, is assumed. In this section we give certain elementary prelim-

inaries which will be germane to Chapter 14 but which fit into the spirit of

this chapter. It is interesting that the reverse triangle inequality plays an im-

portant role, hence this material is decidedly non-Riemannian. Also the lower

semicontinuity of the Lorentzian distance function for an arbitrary space-time

(cf. Lemma 4.4) is useful here.

Note as an immediate first example that if a space-time (M,g) contains

a single maximal segment, then (M,g) cannot be totally vicious, since the

Lorentzian distance function is finite-valued on the particular segment while

totally vicious space-times satisfy d(p, q) = +m for all p, q in M. Newman

(1990) noted the more interesting consequence that the existence of a maximal

timelike segment c : [a, b] -4 (M,g) implies that strong causality holds a t all

points of c((a, b)). Since strong causality is an open condition [cf. Penrose

(1972, p. 30)], this thus yields an open neighborhood U of c((a, b)) for which

strong causality at q is valid for all q in U . Hence, not only does the existence

of a local causality neighborhood in a strongly causal space-time guarantee

the local existence of maximal segments, but a kind of converse holds: the

existence of a maximal timelike segment implies that some local region of

(M,g) containing all interior points of the given maximal timelike segment

must be strongly causal.

For our use in Chapter 14, these basic consequences of the existence of

maximal segments will be treated in the present section. We begin with a

maximal null segment.

Lemma 4.36. Let c : [O, 11 -4 (M,g) be a maximal null geodesic segment,

and let I(c) denote the domain of c extended to be a future inextendible null

geodesic emanating from c(0). Then

(1) For any s, t with 0 5 s < t 5 1 and r E J+(c(s)) fl J-(c(t)), we have

hence T lies on c(I(c));


(2) For any p, q in J+(c(s)) n J-(c(t)) with p ,< q, we have d(p, q) = 0;

and

(3) Chronology holds at all points of J+(c(O)) n J- ( ~ ( 1 ) ) .

Proof. (1) By assumption, c(s) < r < c(t) and d(c(s), c(t)) = 0, so that the

reverse triangle inequality d(c(s), c(t)) 2 d(c(s), r ) +d(r, c(t)) implies equation

(4.5). Further, since c(s) < r < c(t) is assumed, there exist future causal curves

cl from c(s) to r and cz from r to c(t) which by (4.5) must both be maximal

null geodesic segments. Since c(s) and c(t) are not chronologically related,

the concatenation of cl and cz must constitute a single null geodesic by basic

causality theory [cf. Penrose (1972, Proposition 2.19)], whence r E c(I(c)).

(2) This is immediate from the reverse triangle inequality applied to

(3) Condition (3) now follows since if chronology fails to hold, then d(p,p) is

infinite which contradicts (2) applied with q = p.

We should caution that I(c) is not necessarily equal to [O, +m) (cf. Lemma

7.4). In the case of a maximal timelike segment, the reverse triangle inequality

yields the finiteness of Lorentzian distance from points in some neighborhood

of the segment despite the possible general lack of finiteness of distance for

chronologically related points (cf. Figure 4.2).

Similar arguments to those used in Lemma 4.36 yield the following finiteness

of the distance function in the causal hull of any causal maximal segment.

Lemma 4.37. Let c : [O,1] -, (M,g) be a causal maximal segment. Then

(1) Given any p, q in J+(c(O)) n J - ( ~ ( 1 ) ) with p 6 q, the distance d(p, q)

is finite.

(2) Chronology holds at all points of J+(c(O)) n J - ( ~ ( 1 ) ) .

Having dealt with chronology, let us now consider the stronger requirement

of causality. The cylinder M = S' x W with metric ds2 = dBdt contains closed

null geodesics c : [0, +m) -t M which satisfy d(c(O), c(t)) = 0 for all t 2 0.

Hence, the existence of a maximal null geodesic does not imply more than

local chronology.


Lemma 4.38. Let c : [0, I] + (M, g) be a maximal timelike segment. Then

causality holds a t all points of c([O, 11).

Proof. Suppose that cl is a closed nonspacelike curve beginning and ending

at c(t1) with tl > 0. Since chronology holds at c(t1) by Lemma 4.37, cl must

be a maximal null segment (cf. Corollary 4.14). Consider the composite causal

curve y = (c I [0, tl]) * cl from c(O) to c(tl). Since 7 is a causal curve from

c(0) to c(tl) which is a timelike followed by a null geodesic, first variation

"rounding the corner" arguments produce a causal curve yl from c(0) to c(tl)

which is longer than y, hence longer than ~110, t l] [cf. O'Neill (1983, p. 294)).

But this contradicts the maximality of c 1 [O, tl].

If t l = 0, apply the samc type of argument to the composition of the closed

null geodesic cl followed by c 1 [O,l].

Now we turn to a result with a somewhat more difficult proof first given

in Newman (1990, p. 166) and an alternate proof suggested in Galloway and

Horta (1995). In the course of the proof, it will be helpful to employ a charac-

terization of the failure of strong causality given by Kronheimer and Penrose

[cf. Penrose (1972, p. 31)j.

Lemma 4.39. Let p E (M, g). Then strong causality fails a t p if and only

if there exists a point q E J - ( p ) with q # p (which may be chosen arbitrarily

closely to p) such that x <( p and q << y together imply x << y for all x, y in

(M,g).

Proposition 4.40 [Newman (1990)j. Let c : [0, a] + (M, g) be a rnax-

imal timelike geodesic segment. Then for any to with 0 < to < a, strong

causality holds a t p = c(t0).

Proof. For convenience, we will assume that c is parametrized by unit speed

and also, given p = c(to), take q E J-(p) in Lemma 4.39 sufficiently close to p

that q E I+(c(O)). Now assume that strong causality fails to hold a t p.

We first need to establish that d(q,p) = 0, whence q is the initial point

of a maximal null segment from q to p. Suppose that d(q,p) > 0. Choose a

sequence {y,) C I+(q) with y, --, q and a sequence {t,) with 0 < t, < to

and t, -4 to. Put x, = c(t,), whence {x,) C I-(p) and x, -+ p. By the


lower semicontinuity of distance at (q,p), we have d(y,, x,) > 0 for some n

sufficiently large. Hence y, << x,. On the other hand, by Lemma 4.39 we

also have x, << y,. Thus x, << x,, which contradicts Lemma 4.37-(2). Thus

d(q,p) = 0 and the geodesic X from q to p is a maximal null segment.

We now show how a timelike curve from c(O) to c(a) of length greater than

a may be constructed, contradicting the maximality of the timelike segment c.

Consider first the concatenation ,O = X * c I [to, a] which is a future causal curve

from q to c(a). Since /3 consists of a null followed by a timelike geodesic and

L(0) = L(c I [to, a]) = a - to, rounding the corner at p produces a causal curve

from q to c(a) of length greater than L(0) = a - to. (Take a convex normal

neighborhood V centered at p, and join a point close to p on the null geodesic

X to c(to + 6), for 6 sufficiently small, by a timelike geodesic segment lying in

V.) Hence, there exists a constant E > 0 such that d(q, c(a)) > (a - to) + 46.

By Lemma 4.4, we may find a neighborhood U 5 I'(c(0)) of q such that

for every q' in U. Choose n sufficiently large that y, E U and also that

x, = c(t,) satisfies L(c 1 [0, t,]) = d(c(O), c(t,)) = t, 2 to - E. Fixing this n,

let cl be a future timelike curve from x, to y, guaranteed by Lemma 4.39.

Also given y,, in view of inequality (4.6) we may find a causal curve c2 from

y, to c(a) with L(cz) 2 a - to + 26. Now let y = c 1 jO, t,] * cl * c2, which is a

future causal curve from c(0) to c(a). On the other hand, we have the length

estimate

L(r) 1 L(c I [O, t,l) + L(cz)

Z (to - E) + (a - to + 26)

But this inequality contradicts the maximality of the timelike segment c.

Hence, strong causality must hold at c(to). 0

CHAPTER 5

EXAMPLES OF SPACE-TIMES

In this chapter we present a variety of examples of space-times. Some

of these space-times are important for physical as well as mathematical rea-

sons. In particular, Minkowski space-time, Schwarzschild space-times, Kerr

space-times, and Robertson-Walker space-times all have significant physical

interpretations.

Minkowski space-time is simultaneously the geometry of special relativ-

ity and the geometry induced on each fixed tangent space of an arbitrary

Lorentzian manifold. Thus Minkowskian geometry plays the same role for

Lorentzian manifolds that Euclidean geometry plays for Riemannian mani-

folds. Minkowski space-time is sometimes called fE~t space-time. But more

generally, any Lorentzian manifold on which the curvature tensor is identically

zero is flat.

The Schwarzschild space-times represent the spherically symmetric, empty

space-times outside nonrotating, spherically symmetric bodies. Since suns

and planets are assumed to be slowly rotating and approximately spherically

symmetric, the Schwarzschild space-times may be used to model the gravita-

tional fields outside of these bodies. These space-times may also be used to

model the gravitational fields outside of dead (i.e., nonrotating) black holes.

The usual coordinates for the Schwarzschild solution outside a massive body

are ( t , r, 0, &), where t represents a kind of time and r represents a kind of

radius [cf. Sachs and Wu (1977a, Chapter 7)]. This metric has a special radius

r = 2m associated with it. Points with r = 2m correspond to the surface of a

black hole. I t was once thought that the metric was singular at r = 2m, but

it is now known that the usual form of the Schwarzschild metric with r > 2m

may be analytically extended to points with 0 < r < 2m. In fact, there is a

174 5 EXAMPLES OF SPACE-TIMES

maximal analytic extension of Schwarzschild space-time [cf. Kruskal (1960)l

which contains an alternative universe lying on the "other side" of the black

hole.

The gravitational fields outside of rotating black holes apparently corre-

spond to the Kerr spacetimes [cf. Hawking and Ellis (1973, pp. 161, 331),

Carter (1971b), O'Neill (1995)j. These spacetimes represent stationary, ax-

isymmetric metrics outside of rotating objects. The Kerr and Schwarzschild

space-times are asymptotically flat and correspond to universes which are

empty, apart from one massive body. Thus while these metrics may be rea-

sonable models near a given single massive body, they cannot be used as large

scale models for a universe with many massive bodies.

The usual "big bang" cosmological models are based on the Robertson-

Walker space-times. These space-times are foliated by a special set of space-

like hypersurfaces such that each hypersurface corresponds to an instant of

time. The isometry group I ( M ) of a Robertson-Walker space-time (M,g)

acts transitively on these hypersurfaces of constant time. Thus Robertson-

Walker universes are spatially homogeneous. Furthermore, they are spatially

isotropic in the sense that for each p E M, the subgroup of I ( M ) fixing p is

transitive on the directions at p which are tangential to the hypersurface of

constant time through p. In our discussion of Robertson-Walker space-times,

we will use Lorentzian warped products Mo x f H , as described in Section

3.6. The cosmological assumptions made about Robertson-Walker universes

imply that (H, h) is an isotropic Riemannian manifold. Hence the classifica-

tion of two-point homogeneous Riemannian manifolds yields a classification

of all Robertson-Walker space-times. We also show how the results of Sec-

tion 3.6 may be specialized to construct Lie groups with bi-invariant globally

hyperbolic Lorentzian metrics.

5.1 Minkowski Space-time

Minkowski space-time is the manifold M = Rn together with the metric

5.1 MINKOWSKI SPACE-TIME

FIGURE 5.1. Let (M, g) be Minkowski space-time. The null cone

at p has a future nappe and a past nappe. The future [respectively,

past] nappe is also the horismos E+(p) [respectively, E-(p)] of p.

The chronological future I f (p) is an open convex set bounded by

E+(p). In more general space-times, I+(p) may fail to be convex

but is always open.

This space-time is time oriented by the vector field d/dxl. It is also glob-

ally hyperbolic and hence satisfies all of the causality conditions discussed in

Section 3.2.

The geodesics of Minkowski space-time are just the straight lines of the

underlying Euclidean space Rn. The affine parametrizations of these geodesics

in Minkowski space are even proportional to the usual Euclidean arc length

parametrizations in Rn. The null geodesics through a given point p in Minkow-

ski space form an elliptic cone with vertex p. The future directed null geodesics

starting at p thus form one nappe of the null cone of p. This nappe forms the

boundary in Rn of an open convex set which is exactly the chronological future

I+(p) of p. In Minkowski space, the causal future J+(p) of p is the closure of

5 EXAMPLES OF SPACGTIMES

FIGURE 5.2. The unit sphere K ( p , 1 ) corresponding to p is half of

a hyperboloid of two sheets. It is not compact, and p does not lie

in the convex open set bounded by K ( p , 1) .

I + ( p ) . The future horismos E f ( p ) = J + ( p ) - I + ( p ) is the nappe of the null

cone of p corresponding to the future (cf. Figure 5.1).

Minkowski space-time is a Lorentzian product (i.e., a warped product in

the sense of Definition 3.51 with f = 1). If W is given the negative definite

5.1 MINKOWSKI SPACE-TIME 177

FIGURE 5.3. Two-dimensional Minkowski space-time with one

point q removed is shown. The future horismos E + ( p ) of p is an

"L" shaped figure consisting of a half-closed line and a half-open

line segment. The causal future J + ( p ) is the union of I t ( p ) and

E+(p) . Notice that J f (p) is not a closed set nor is J+(p) equal to

the closure of If ( p ) .

metric -dt2 and Wn-' is given the usual Euclidean metric go, then ( R n =

R x Rn-l, -dt2 @go) is the n-dimensional Minkowski space-time.

Consider two points p = (pl ,p2, . .. ,pn) and q = ( q l , 92,. ... qn) in Minkow-

ski space-time. The chronological relation p << q holds whenever pl < ql and

( P I - q ~ ) ~ > (p2 - q2)2 i- . . - + (p , - qn)2 in R. If p << q, then the distance

from p to q is given by

n

( P I - q d 2 - ) i = 2 (pi - qi)2

The "unit sphere" in Minkowski space-time centered a t p is then K ( p , 1) =

! { q E A4 : d(p , g ) = 1) . However, this set is actually one sheet of a hyperboloid

1 of two sheets (ct Figure 5.2). t

. .

5 EXAMPLES OF SPACE-TIMES 5.2 SCHWARZSCHILD .4ND KERR SPACE-TIMES

i +

/--------

i- FIGURE 5.5. The Penrose diagram for Minkowski space-time is

i- shown.

FIGURE 5.4. Minkowski space-time is conformal to the open set Minkowski space-time and many other important space-times may be r e p

enclosed by the two null cones indicated. The vertices i+ and i - cor- resented by Penrose diagrams. A Penrose diagram is a two-dimensional r e p

respond to timelike infinity. All future directed timelike geodesics resentation of a spherically symmetric space-time. The radial null geodesics

go from i- to i+. The sets 3+ and 3- represent future and past are represented by null geodesics a t 145O. Dotted lines represent the origin

null infinity. Topologically, 3+ and 3- are each W x Sn-2. The (T = 0) of polar coordinates. Points corresponding to smooth boundary points

intersection of the two null cones is a set which is identified to a (cf. Section 12.5) which are not singularities are represented by single lines.

single point iO. The point i0 is called spacelike infinity. Double lines represent irremovable singularities (Figure 5.5; cf. Figure 4.2 of a Reissner-Nordstrom space-time with e2 = m2 for an example).

If we remove a point from Minkowski space-time, then it is no longer 5.2 Schwarzschild and Kerr Space-times

causally simple and hence no longer globally hyperbolic (cf. Figure 5.3).

I t is possible to conformally map all of Minkowski space-time onto a small In this section we describe the four-dimensional Schwarzschild and Kerr

open set about the origin. This is illustrated in Figure 5.4 [cf. Penrose (1968, solutions to the Einstein equations. Let R4 be given coordinates (t, r, 8, +),

p. 178), Hawking and Ellis (1973, p. 123)j. where (r, B,4) are the usual spherical coordinates on R3. Given a positive

180 5 EXAMPLES O F SPACE-TIMES

constant m, the exterior Schwarzschild space-time is defined on the subset

r > 2m of W4, a subset which is topologically W2 x S2. The Schwarzschild

metric for the region r > 2m is given in (t, r, 8 ,4) coordinates by the formula

Each element of the rotation group SO(3) for W3 induces a motion of the

Schwarzschild solution. Given qb E S0(3) , a motion ?C, of Schwarzschild space-

time may be defined by setting q ( t , r,$, 4) = (t, $(T, @, $)). Thus a t a fixed

instant t in time, the exterior Schwarzschild space-time is spherically symmet-

ric. The metric for this space-time is also invariant under the time translation

t -+ t + a. The coordinate vector field a/at is a timelike Killing vector field

which is a gradient, and the metric is said to be static. This space-time is

also Ricci flat (i.e., Ric = 0). Using the Einstein equations (cf. Chapter 2),

it follows that the energy-momentum tensor for the exterior Schwarzschild

space-time vanishes. Thus this space-time is empty.

The exterior Schwarzschild space-time may be regarded as a Lorentzian

warped product (cf. Section 3.6). For let M = {(t, r ) E W2 : r > 2m) be given

the Lorentzian metric

and let H = S2 be given the usual Riemannian metric h of constant sectional

curvature one induced by the inclusion S2 -+ W3. Define f : M --+ W by

f ( t , r ) = r2. Then (M x f H,ij) is the exterior Schwarzschild space-time,

where i j = g $ f h.

Physically, the exterior Schwarzschild solution represents the gravitational

field outside of a nonrotating spherically symmetric massive object. Compar-

ison with the Newtonian theory [cf. Einstein (1916, p. 819), Pathria (1974,

p. 217)] shows that m can be identified with the gravitational mass of the

massive body. The solution is not valid in the interior of the body.

The above form of the exterior Schwarzschild metric appears to have a

singularity at r = 2m. However, this is not a true singularity; the exterior

5.3 SPACES O F CONSTANT CURVATURE 181

Schwarzschild solution may be analytically continued across the surface given

by the equation r = 2m.

Kruskal(1960) investigated the maximal analytic extension of Schwarzschild

space-time. Suppressing 8 and 4, the following two-dimensional representation

of this maximal extension may be given (cf. Figure 5.6).

The gravitational field outside of a rotating black hole will not correspond

to the Schwarzschild solution. The generally accepted solutions of the Einstein

equations for rotating black holes are Kerr solutions. In Boyer and Lindquist

coordinates (t, r, 8 ,4 ) the Kerr metrics are given by [cf. Hawking and Ellis

(1973, p. 161), O'Neill (1995)l

where p2 = r2 + a2 cos2 0 and A = r2 - 2mr + a2. The constant m represents

the mass, and the constant m a represents the angular momentum of the black

hole [cf. Boyer and Price (1965), Boyer and Lindquist (1967)l. Tomimatsu

and Sato (1973) have given a series of exact solutions which include the Kerr

solutions as special cases.

5.3 Spaces of Constant Curva tu re

It is known that two Lorentzian manifolds of the same dimension which have

constant sectional curvature K are locally isometric [cf. Wolf (1974, p. 69)].

Thus any Lorentzian manifold of constant sectional curvature zero is locally

isometric to Minkowski space-time. In this section we will consider Lorentzian

model spaces which have constant nonzero sectional curvature.

We first define W: to be the standard semi-Euclidean space of signature

(-, . . , , -, +, . . . , +), where there are s negative eigenvalues and n - s positive

eigenvalues. Hence the semi-Euclidean metric on W: is given by

5 EXAMPLES OF SPACCTIMES

FIGURE 5.6. The Kruskal diagram for the maximal analytic ex-

tension of the exterior Schwarzschild space-time is shown. The ex-

tended space-time is the connected nonconvex region I U I1 U I' U 11'

bounded by the hyperbola corresponding to r = 0. The points of

this hyperbola are the true singularities of this space-time. The

lines a t f 45' separate the space-time into four regions. Region

I corresponds to the exterior Schwarzschild solution. Region I1 is

the "interior" of a nonrotating black hole. Region I' is isometric to

region I and corresponds to an alternative universe on the "other

side" of the black hole. There is no nonspacelike curve from region

I to region 1'.

5.3 SPACES OF CONSTANT CURVATURE 183

In particular, IR; is n-dimensional Minkowski space-time. We also define for

r > 0 [cf. Wolf (1974, Section 5.2)]

and

Topologically, S;" is R' x Sn-I and Hi' is S' x Rn-' [cf. Wolf (1974, p. 68)).

The semi-Euclidean metric on w;" (respectively, R;+') induces a Lorentzian

metric of constant sectional curvature K = r-' (respectively, K = -r-2) on

Sy (respectively, HT). The space-time Sf. is a Lorentzian analogue of the

usual Riemannian spherical space of radius r and has positive curvature T - ~ .

The universal covering manifold H;" of H;" is topologically Rn and is thus

a Lorentzian analogue of the usual Riemannian hyperbolic space of negative

curvature -rW2.

Definition 5.1. (de Sitter and anti-de Sitter Space-times) Let S;" and

Hf. be defined as above. Then SF- is called de Sitter space-time, and the

universal covering H r of H r is called (universal) anti-de Sitter space-time.

Remark 5.2.

(1) Sf . is simply connected for n > 2 and .rrl(S:) = Z.

(2) Sf. is globally hyperbolic and geodesically complete.

(3) H f . is nonchronological since y (t) = (T cos t , r sin t , 0 , . . . ,0) is a closed

timelike curve. Also @, while strongly causal, is not globally hyper-

bolic.

The de Sitter space-time represented in Figure 5.7 may be covered by global

coordinates (t,x,B,$) with -m < t < cm, 0 < x < T , 0 < B 5 T , and

0 < $ 5 27r. Here t is the coordinate on IR and (x, B,$) represent coordinates

on S 3 Icf. Hawking and Ellis (1973, pp. 125, 136)]. In these coordinates, the

metric for de Sitter space-time of constant positive sectional curvature l / r 2 is

given by

184 5 EXAMPLES OF SPACGTIMES

FIGURE 5.7. The n-dimensional de Sitter space-time with positive

constant sectional curvature r-2 is the set -x12 +x22 +- . . +x:+~ =

r2 in Minkowski space-time R;'~. The geodesics of S," lie on the

intersection of S," with the planes through the origin of R;".

This may be reinterpreted as a Lorentzian warped product metric (cf. Section

3.6) as follows. Let f : W + (0, m) be given by f (t) = r2 cosh2(t/r), and

let S3 be given the usual complete Riemannian metric of constant sectional

curvature one. Then the de Sitter space-time described in local coordinates

as above is the warped product (W x S3, -dt2 @ fh).

Universal anti-de Sitter space-time of curvature K = -1 may be given

coordinates (t', r , 8,4) for which the metric has the form

[cf. Hawking and Ellis (1973, pp. 131, 136)). Regarding -(dt')2 as a nega-

tive definite metric on W and dr2 + sinh2(r)(a2 + sin2 B as the complete

5.4 ROBERTSON-WALKER SPACGTIMES 185

Riemannian metric h of constant negative sectional curvature -1 on the hy-

perbolic three-space H = W3, this space-time may be represented as a warped

product of the form (R x f H, - f dt2@h), where the warping function is defined

on the Riemannian factor H (cf. Remark 3.53).

5.4 Robertson-Walker Space-times

In this section we discuss Robertson-Walker space-times in the framework

of Lorentzian warped products. These space-times include the Einstein static

universe and the big bang cosmological models of general relativity. In order

to give a precise definition of a Robertson-Walker space-time, it is necessary

to first recall some concepts from the theory of two-point homogeneous Rie-

mannian manifolds and isotropic Riemannian manifolds.

Let (H, h) be a Riemannian manifold. Denote by I ( H ) the isometry group

of (H, h) and by & : H x H -+ R the Riemannian distance function of (H, h).

Definition 5.3. (Homogeneous and Two-Point Homogeneous Manifolds)

The Riemannian manifold (H, h) is said to be homogeneous if I ( H ) acts tran-

sitively on H , i.e., given any p,q f H , there is an isometry 4 E I ( H ) with

4(p) = q. Further, (H, h) is said to be two-point homogeneous if given any

PI, ql, p2,92 E H with do(~1,ql) = do(p2, 4, there is a n isometry 4 E I ( H )

with +(PI) = p2 and 4(ql) = 92.

Since it is possible to choose p, = q, for z = 1 ,2 , a two-point homogeneous

Riemannian manifold is also homogeneous. Two-point homogeneous spaces

were first studied by Busemann (1942) in the more general setting of locally

compact metric spaces. Wang (1951, 1952) and Tits (1955) classified two-point

homogeneous Riemannian manifolds.

Notice that in Definition 5.3 it is not required that (H, h) be a complete

Riemannian manifold. Nonetheless, homogeneous Riemannian manifolds have

the important basic property of always being complete.

Lemma 5.4. If (H, h) is a homogeneous Riemannian manifold, then (H, hf

is complete.


Proof. By the Hopf-Rinow Theorem, it suffices to show that ( H , h ) is

geodesically complete. Thus suppose that c : [a, 1) -+ H is a unit speed

geodesic which is not extendible to t = 1. Choosing any p E H , we may find

a constant a > 0 such that any unit speed geodesic starting at p has length

1 2 a. Set 6 = min{a/2, ( 1 - a ) / 2 ) > 0. Since isometries preserve geodesics, it

follows from the homogeneity o f ( H , h ) that any unit speed geodesic starting

at c(1 - 6) may be extended to a geodesic o f length 1 2 26. In particular,

c may be extended to a geodesic c : [a, 1 + 6 ) -4 H , in contradiction to the

inextendibility o f c to t = 1. 0

Remark 5.5. It is important to note that the conclusion o f Lemma 5.4 is

false in general for homogeneous Lorentzian manifolds [cf. Wolf (1974, p. 95),

Marsden (1973)l.

W e now recall the concept o f an isotropic Riemannian manifold. Given

p 6 ( H , h ) , the isotropy group I p ( H ) o f ( H , h ) at p is the closed subgroup

I p ( H ) = {4 E I ( H ) : $(p) = p ) o f I ( H ) consisting of all isometries of ( H , h )

which fix p. Given any 4 E I p ( H ) , the differential #,, maps TpH onto TpH

since 4 (p ) = p. As h(4,v, &v) = h(v , v ) for any v E T p H , the differential q5,,

also maps the unit sphere SpH = {v E T,M : h(v , v ) = 1) in T,H onto itself.

Definit ion 5.6. (Isotropic Riemannian Manifold) A Riemannian man-

ifold ( H , h) is said to be isotropic at p i f I p ( H ) acts transitively on the unit

sphere SpH o f T p H , i.e., given any v , w E S,H, there is an isometry 4 E I,(H)

with 4,v = w. The Riemannian manifold ( H , h ) is said to be isotropic i f it is

isotropic at every point.

W e now show that the class o f isotropic Riemannian manifolds coincides

with the class of two-point homogeneous Riemannian manifolds [cf. Wolf (1974,

p 289)].

Proposit ion 5.7. A Riemannian manifold ( H , h ) is isotropic iff it is two-

point homogeneous.

Proof. Recall that & denotes the Riemannian distance function of ( H , h).

First suppose that ( H , h ) is isotropic. Then for each p E H and each inex-

tendible geodesic c : (a , b) -+ H with c(0) = p, there is an isometry 4 E I,(H)

5.4 ROBERTSON-WALKER SPACE-TIMES 187

with $,cl(0) = -cJ(0). Hence by geodesic uniqueness, g!~(c(t)) = c(- t ) for

all t E (a , 6). This implies that the length of c 1 (a , 0] equals the length o f

c 1 [0,6). Since p may be taken to be any point o f the geodesic c, it follows

that a = -m and 6 = f m . Thus ( H , h ) is geodesically complete. Hence by

the Hopf-Rinow Theorem, given any two points pl, p2 E H , there is a geodesic

segment Q o f minimal length &(pl,pz) from pl to p2. Let p be the midpoint

o f Q. As ( H , h ) is isotropic, there is an isometry # E I,(H) which reverses Q.

I t follows that # ( P I ) = p2. Hence ( H , h ) is homogeneous. It remains to show

that i f P I , ql, p2, q2 E H with & ( P I , ql) = do(p2, q2) > 0 are given, we may

find an isometry 4 E I ( H ) with # ( P I ) = p2 and #(ql) = q2. Choose minimal

unit speed geodesics cl from pl to ql and c2 from p2 to 92. Since ( H , h ) is

homogeneous, we may first find an isometry 1I, E I ( H ) with lI,(pl) = p2. Then

as ( H , h ) is isotropic, we may find 77 E I,,(H) with v*((+ o c1)'(0)) = ~ ' ( 0 ) .

I t follows that 4 = 71 0 $ is the required isometry.

Now suppose that ( H , h ) is two-point homogeneous. Fix any p E M , and

let U be a convex normal neighborhood based at p. Choose a > 0 such that

expp(v) E U for all v E T,H with h(v ,v) 5 a . Now let v? w E TpH be any pair

o f nonzero tangent vectors with h(v , v ) = h(w, w) < a/2. Set ql = expp v and

q2 = exp, w. Then ql, q2 E U and d(p, q l ) = = = d(p, q2).

Since ( H , h ) is two-point homogeneous, there is thus an isometry $ E I ( H )

with 4 (p ) = p and $(ql) = q2. I t follows that $,v = w. The linearity o f

qap : TpH --t TpH for any 11 E I p ( H ) then implies that I,(H) acts transitively

on SpH. Thus ( H , h ) is isotropic at p. As the same argument clearly holds for

all p E H , it follows that ( H , h ) is isotropic as required.

Corollary 5.8. Any isotropic Riemannian manifold is homogeneous and

complete.

R e m a r k 5.9. ( 1 ) The two-point homogeneous Riemannian manifolds are

well known [cf. Wolf (1974, pp. 290-296)]. In particular, the odd-dimensional

two-point homogeneous (hence isotropic) Riemannian manifolds are just the

odd-dimensional Euclidean, hyperbolic, spherical, and elliptic spaces [cf. Wang

(1951, p. 47311.

(2) Astronomical observations indicate that the spatial universe is approxi-


mately spherically symmetric about the earth. This suggests that the spatial

universe should be modeled as a three-dimensional isotropic Riemannian mani-

fold. Hence the possibilities are limited to the Euclidean, hyperbolic, spherical,

and elliptic spaces. However, if one only assumes local isotropy, there are more

possibilities [cf. Misner, Thorne, and Wheeler (1973, pp. 713-725)].

(3) Any three-dimensional isotropic Riemannian manifold (H, h) has constant

sectional curvature, and also dim I ( H ) = 6 [cf. Walker (1944)l.

We are now ready to define Robertson-Walker spacetimes using Lorentzian

warped products and isotropic Riemannian manifolds.

Definition 5.10. (Robertson-Walker Space-time) A Robertson-Walker

space-time (M, g) is any Lorentzian manifold which can be written in the form

of a Lorentzian warped product (Mo x f H,g) with Mo = (a ,b) , -co 5 a < b 5 -too, given the negative definite metric -dt2, with (HI h) an isotropic

Riemannian manifold, and with warping function f : Mo -+ (0, m).

In the notation of Section 3.6, we thus have g = -dt2 @ f h and Mo x f H

is also topologically the product Mo x H. Letting du2 denote the Riemannian

metric h for H and defining S(t) = m, the Lorentzian metric g for Mo x f H

may be rewritten in the more familiar form

The map n. : Mo x f H --+ R given by x(t ,z) = t is a smooth time function

on Mo x f H so that the Lorentzian manifold Mo x f H , of Definition 5.10

actually is a (stably causal) spacetime. Also each level surface n.-l(c) of the

map T : Mo x f H --+ Mo c R is an isotropic Riemannian manifold which is

homothetic to (HI h). Furthermore, the isometry group I ( H ) of (HI h) may be

identified with a subgroup T(H) of I(Mo x f H ) as follows. Given 4 E I(H),

defbe E F(H) by $(T, h) = (r,$(h)) for all (r, h) E Mo x H. With this

definition, F(H) restricted to the level surfaces s-'(c) of s acts transitively on

each level surface.

Since all isotropic Riemannian manifolds are complete, Theorem 3.66 im-

plies that all Robertson-Walker spacetimes are globally hyperbolic. From

5.4 ROBERTSON-WALKER SPACE-TIMES 189

Theorem 3.69 we also know that every level surface n.-'(c) = {c) x H is a

Cauchy surface for Mo x f H.

Next to Minkowski space Rn = R x Rn-' itself, the Einstein static universe

is the simplest example of a Robertson-Walker space-time.

Example 5.11. (Einstein Static Universe) Let Mo = R with the neg-

ative definite metric -dt2, and let H = Sn-' with the standard spherical

Riemannian metric. I f f : R -+ (0, m ) is the trivial warping function f = 1,

then the product Lorentzian manifold M = Mo x H = Mo x f H is the n-

dimensional Einstein static universe. If n = 2, then M is the cylinder R x S'

with Aat metric -dt2 + do2. If n 2 3, then this metric for M = R x Sn-' is

not flat since Sn-' has constant positive sectional curvature K = 1.

For the rest of this section we restrict our attention to four-dimensional

Robertson-Walker spacetimes. By Remark 5.9, these are warped products

Mo x f H , where (H, h) is Euclidean, hyperbolic, spherical, or elliptic of di-

mension three. In the first two cases, H is topologically R3. In the third case

H = S3, and in the last case H is the real projective three-space RP3. We

thus have the following

Corollary 5.12. All four-dimensional Robertson-Walker spacetimes are

topologically either R4, R x S3, or R x RP3.

Also by Remark 5.9, the sectional curvature K of (H, h) is constant. If K is

nonzero, the metric may be rescaled to be of the form ds2 = -dt2 + S2(t)du2

on M so that K is either identically $1 or -1. This is the form of the metric

usually studied in general relativity.

In physics, cosmological models are built from four-dimensional Robertson-

Walker space-times assumed to be filled with a perfect fluid. The Einstein

equations (cf. Chapter 2) are then used to find the form of the above warping

function S2(t). Among the models this technique yields are the big bang

cosmological models [cf. Hawking and Ellis (1973, pp. 134-138)]. These models

depend on the energy density p and pressure p of the perfect fluid a s well as

the value of the cosmological constant A in the Einstein equations. In the big

bang cosmological models, the inextendible nonspacelike geodesics are all past

190 5 EXAMPLES O F SPACE-TIMES

incomplete. The stability of this incompleteness under metric perturbations

will be considered in Section 7.3. Astronomical observations of clusters of

galaxies indicate that distant clusters of galaxies are receding from us. This

expansion of the universe suggests the existence of a "big bang" in the past and

also suggests that the universe is a warped product with a nontrivial warping

function rather than simply a Lorentzian product. Observations of blackbody

radiation support these ideas [cf. Hawking and Ellis (1973, Chapter lo)].

5.5 Bi-Invariant Lorentzian Metrics o n Lie Groups

The purpose of this section is to show how Theorems 3.67 and 3.68 of

Section 3.6 may be used to construct a large class of Lie groups admitting

globally hyperbolic, bi-invariant Lorentzian metrics.

We first summarize some basic facts from the elementary theory of Lie

groups. Details may be found in a lucid exposition by Milnor (1963, Part IV)

or at a more advanced level in Helgason (1978, Chapter 2). A Lie group is a

group G which is also an analytic manifold such that the mapping (g, h) -+

gh-I from G x G -+ G is analytic. This multiplication induces left and right

translation maps L,, R, for each g E GI given respectively by L,(h) = gh

and Rg(h) = hg. A Riemannian or Lorentzian metric ( , ) for G is then said

to be left invariant (respectively, right invariant) if (L,*v, L,,w) = (v, w)

(respectively, (Rgtv, R,,w) = (v, w)) for all g E G and v, w E TG. A metric

which is both left and right invariant is said to be bi-invariant. By an averaging

procedure involving the Haar integral, any compact Lie group may be given

a bi-invariant metric [cf. Milnor (1963, p. 112)]. In fact, the Haar integral

may be used to produce a bi-invariant Riemannian metric for G from any left

invariant Riemannian metric for G. Any Lie group may be equipped with a

left invariant Riemannian (or Lorentzian) metric by starting with a positive

definite inner product (respectively, inner product of signature n - 2) ( , )I, on the tangent space T,G to G at the identity element e E G, then defining

( , )I,:T,GxT,G+ W by

5.5 LIE GROUPS: BI-INVARIANT LORENTZIAN METRICS 191

Thus any compact Lie group is furnished with a large supply of bi-invariant

Riemannian metrics.

On the other hand, while (5.1) equips any Lie group with left-invariant

Lorentzian metrics, the standard Haar integral averaging procedure used for

Riemannian metrics fails to preserve signature (-, +, . - . , +), so it cannot be

used to convert left-invariant Lorentzian metrics into bi-invariant Lorentzian

metrics.

But we will see shortly that a large class of bi-invariant Lorentzian metrics

may be constructed for noncompact Lie groups of the form R x G, where G is

any Lie group admitting a bi-invariant Riemannian metric.

Before giving the construction, we need to discuss product Lie groups briefly.

Let G and H be two Lie groups. The product manifold G x H is then turned

into a Lie group by defining the multiplication by

It is immediate from (5.2) that if u = (g, h) E G X H , then the translation maps

L,, R, : G x H -+ G x H are given by L, = (L,, Lh), and R, = (R,, Rh),

i.e., L,(gl, hl) = (L,gl, Lhhl)> etc. Recall that T,(G x H) 2 T,G x ThH.

It is straighforward to check that for any u E G x H and any tangent vector

t = (v, w) f T,(G x H) 2 TgG x ThH, one has

and

Now if ( , )l is a Lorentzian metric for G and ( , j2 is a Riemannian

metric for H , the product metric (( , )) = ( , )1 @ ( , )2 is a Lorentzian

metric for G x H . Explicitly, recalling Definition 3.51, we have for tangent

vectors El = (vl, wl) and t2 = (vz, w2) in T,(G x H ) the formula

X ((Ei,E2)) = (v1,~2)1 + (~1,w2)2. i

It is then immediate from (5.3) and (5.4) that if ( , is a bi-invariant I

Lorentzian metric for G and ( , )2 is a bi-invariant Riemannian metric for i

H , then (( , )) is a bi-invariant Lorentzian metric for G x H. To summarize, P


Proposition 5.13. Let (G, ( , be a Lie group equipped with a bi-

invariant Lorentzian metric, and let (H, ( , )2) be a Lie group equipped with

a bi-invariant Riemannian metric. Then the product metric (( , )) = ( , ) I @

( , )z is a bi-invariant Lorentzian metric for the product Lie group G x H.

Hence (G x H, (( , ))) is a Lorentzian symmetric space and, in particular, is

geodesically complete.

Proof. I t is only necessary to prove the last statement which is a standard

fact in Lie group theory. Recall that we must show that for each a E G x H, there exists an isometry I,, : G x H -+ G x H which fixes a and reverses the

geodesics through a. That is, if y is a geodesic in G x H with y(0) = u, we

must show that I,(y(t)) = y(-t) for all t. This is equivalent to showing that

I,. : T,(G x H ) -+ T,,(G x H ) is the map I,,(E) = -[ and also implies that

Is2 = Id.

We will follow the proof given in Milnor (1963, pp. 109, 112). First, if we

denote the identity element of G x H by e and define a map I, : G x H -t G x H

by Ie(a) = u-' , then I,, : T,(G x H ) -+ Te(G x H ) is given by I,-(v) = -v.

Thus I,. : Te(G x H ) -+ T,(G x H ) is an isometry of T,(G x H). To see that

I,. is an isometry of any other tangent space T,(G x H ) + To-I(G x H ) and

hence that I, : G x H 4 G x H is an isometry, we simply note that

Since (( , )) is bi-invariant, all left and right translation maps are isometries.

Then as

Ie. I,, = R,;l 1,Ie. IeLu;l 1, and I,, 1, is an isometry of T,(G x H), it follows that I,, : T,(G x H ) --+

To-I(G x H ) is also an isometry. The map I, : G x H -+ G x H is thus the

required geodesic symmetry at e.

We define the geodesic symmetry I, for any a E G by setting I,, =

R,I,R,-1. Since R, and R,-1 are isometries by the bi-invariance of (( , ))

and we have just shown that I, is an isometry, it follows that I, : G x H -,

G x H is an isometry, and obviously I,(a) = a since I,(h) = ah-'a. Finally,


for any [ E T,,(G x H) we have

~ u . < = ~ u . (1,. ( ~ ~ ; l [ ) )

= R,,. (- R,:.<) since C E T, (G x H )

= -R,.R,;IJ

= - (R,R,,-I), ( = -[.

Thus I, reverses geodesics at a as required. We have therefore shown that

G x H is a symmetric space.

It may be shown that any symmetric space is geodesically complete as

follows. Let y be a geodesic in M, and set p = y(0). Supposing that q = y(A)

is defined, one may derive the formula [cf. Milnor (1963, p. log)]

provided that y(t) and y(t + 2A) are defined.

Thus if y is defined originally on an interval y : [O, X] -+ G x H , y may be

extended to a geodesic ;j: : [O,2X] -+ G x H by choosing q = $4'2) and putting

y(t) = I,Ip(y(t - A)) for t E [A, 24 . It is then clear that y may be defined on

(-co, co). Thus (G x H, (( , ))) is geodesically complete. 0

Now Proposition 5.13 has the apparent defect that the existence of Lie

groups (G, ( , ) I ) equipped with bi-invariant Lorentzian metrics is assumed.

It will now be shown how such Lie groups may be constructed by taking

products of the form (R x G, -dt2 @ ( , )) where (G, ( , )) is a Lie group

equipped with a Riemannian hi-invariant metric.

The Lie group structure on (R, -dt2) we will use is that induced by the

usual addition of real numbers. Accordingly, we will write (a, b) I-+ a + b for

the Lie group "multiplication" despite our use of the product notation above

for the group operation. Here R is the analytic manifold determined by the

chart t : IW -+ W, t(r) = r. Let a/at denote the corresponding coordinate vector

field on R. The left and right translation maps La, R, : R -+ lR are given by

L,(r) = a + r and R,(r) = r+a . It is easy to check that if v = X dl&/, E T,W, then L,*v and R,-v in T,+,(R) are given by L,*v = Ra-v = X d/&l,+,. Hence

194 5 EXAMPLES OF S P A C S T I M E S

-dt2(L,*v, La.v) = -dt2(~,-v , R,.v) = -A2 = -dt2(v,v) SO that -dt2 is left

and right invariant.

Let (G, ( , )) be a Lie group with a bi-invariant Riemannian metric. By

the proof given in Proposition 5.13, G is a complete symmetric space. Also

using (5.3) and (5.4), it is easily seen that the metric (( , )) = -dt2 @ ( , )

is a bi-invariant Lorentzian metric for W x G. (Here, if El = (A1 a/&],, vl)

and E2 = (Az dldtl, ,v2) with vl, v2 E TSG, the inner product ((El, &)) =

-A1A2 + (vl,v2).) Since (G, ( , )) is a complete Riemannian manifold, the

product (R x G, (( , ))) is globally hyperbolic by Theorem 3.67. We have

obtained

Theorem 5.14. Let (R, -dt2) be given the usual additive group structure

and let (G, ( , )) be any Lie group equipped with a bi-invariant Riemannian

metric. Then the product metric (( , )) = -dt2 @ ( , ) is a bi-invariant

Lorentzian metric for the product Lie group W x G. Thus (R x G, (( , ))) is

a geodesically complete, globally hyperbolic space-time.

Much research inspired by E. Cartan's work was done on Lie groups, ho-

mogeneous spaces, and symmetric spaces equipped with indefinite metrics be-

fore causality theory had assumed such a prominent role in general relativ-

ity. Thus most of this work was carried out not for Lorentzian metrics in

particular but rather for general semi-Riemannian metrics of arbitrary sig-

nature. Rather than attempting to give an exhaustive list of references, we

refer the reader to the bibliography in Wolf's (1974) text. Much of this re-

search has been concerned with the problem of classifying all geodesically

complete semi-Riemannian manifolds of constant curvature (the "space-form

problem"). Two papers dealing with semi-Ftiemannian Lie theory have been

written by Kulkarni (1978) and Nomizu (1979). Nomizu's paper deals specifi-

cally with Lorentzian metrics, considering the existence of constant curvature

left-invariant Lorentzian metrics on a certain class of noncommutative Lie

groups.

A more general treatment of the class of Lie groups which admit left invari-

ant Lorentzian metrics of constant sectional curvature was then given in Barnet

(1989). Also more recently, research in the Lie theory of semigroups, as re-


ported in Hilgert, Hofmann, and Lawson (1989) or Lawson (1989), has sparked

renewed interest in the causality and differential geometry of Lorentzian Lie

groups on the part of the semigroup community. A representative research

paper where causality and Lie semigroups are considered is Levichev and

Levicheva (1992). Additional results for left invariant Lorentzian metrics on

Lie groups of dimension three have recently been obtained by Cordero and

Parker (1995b).

CHAPTER 6

COMPLETENESS AND EXTENDIBILITY

We mentioned in Chapter 1 that the Hopf-Rinow Theorem guarantees the

equivalence of geodesic and metric completeness for arbitrary Riemannian

manifolds. Further, either of these conditions implies the existence of min-

imal geodesics. That is, given any two points p,q E M, there is a geodesic

from p to q whose arc length realizes the metric distance from p to q. If M

is compact, it also follows from the Hopf-Rinow Theorem that all Riemann-

ian metrics for M are complete. In the noncompact case, Nomizu and Ozeki

(1961) established that every noncompact smooth manifold admits a complete

Riemannian metric. Extending their proof, Morrow (1970) showed that the

complete Riemannian metrics for M are dense in the compact-open topology

in the space of all Riemannian metrics for M [cf. Fegan and Millman (1978)l.

In the first three sections of this chapter we compare and contrast these

results with the theory of geodesic and metric completeness for arbitrary

Lorentzian manifolds. In Section 6.1 a standard example is given to show

that geodesic completeness does not imply the existence of maximal geodesic

segments joining causally related points. Then we recall that the class of

globally hyperbolic space-times possesses this useful property. In Section 6.2

we consider forms of completeness such as nonspacelike geodesic completeness,

bounded acceleration completeness (b. a. completeness), and bundle complete-

ness (b-completeness) that have been studied in singularity theory in general

relativity [cf. Clarke and Schmidt (1977), Ellis and Schmidt (1977)l. We also

state a corollary to Theorem 8 of Beem (1976a, p. 184) establishing the exis-

tence of nonspacelike complete metrics for all distinguishing space-times. In

Section 6.3 we discuss Lorentzian metric completeness and the finite compact-

ness condition.

198 6 COMPLETENESS AND EXTENDIBILITY

In the last three sections of this chapter we discuss extensions and local ex-

tensions of space-times. Since extendibility is related to geodesic completeness,

extendibility plays an important role in singularity theory in general relativity

[cf. Clarke (1973, 1975, 1976), Hawking and Ellis (1973), Ellis and Schmidt

(1977)l. In particular, one usually wants to avoid investigating space-times

which are proper subsets of larger space-times since such proper subsets are

always geodesically incomplete.

A space-time (M1,g') is said to be an extension of a given space-time

(M,g) if (M,g) may be isometrically embedded as a proper open subset of

(MI, g'). A space-time which has no extension is either said to be inextendible

[cf. Hawking and Ellis (1973)l or maximal [cf. Sachs and Wu (1977a, p. 29)).

A local extension is an extension of a certain type of subset of a given

space-time. In general, local inextendibility (i.e., the nonexistence of local

extensions) implies global inextendibility. Since questions of extendibility nat-

urally relate to the boundary of space-time, in Section 6.4 we briefly describe

the Schmidt b-boundary and the Geroch-Kronheimer-Penrose causal bound-

ary. In Section 6.5 two types of local extensions are defined and studied. If

a Lorentzian manifold has no local extensions of either of these two types,

it is shown to be inextendible. We also give a local extension of Minkowski

space-time which shows that while b-completeness forces a space-time to be

(globally) inextendible, b-completeness does not prevent a space-time from

having local extensions.

In Section 6.6 local extensions are related to curvature singularities. For

example, if (M,g) is an analytic space-time such that each timelike geodesic

y : [O, a) -+ M which is inextendible to t = a is either complete (in the sense

that a = m) or else corresponds to a curvature singularity, then (M,g) has

no analytic local b-boundary extensions. Also, the a-boundary of Scott and

Szekeres (1994) and its role in classifying singularities is discussed.

6.1 Existence of Maximal Geodesic Segments

The purpose of this section is twofold. First, we recall that for arbitrary

Lorentzian manifolds, geodesic completeness does not imply the existence of

6.1 EXISTENCE OF MAXIMAL GEODESIC SEGMENTS 199

FIGURE 6.1. The universal cover M = {(x,t) : -x /2 < r < 7r/2) of two-dimensional anti-de Sitter space is shown. The metric

is given by ds2 = sec2x (-dt2 + dx2). The points p and q are

chronologically related in M , yet no maximal timelike geodesic in

M joins p to q since all future directed timelike geodesics emanating

from p are focused at r.

maximal geodesic segments joining causally related pairs of points. Second,

we discuss the important and useful fact that distance realizing geodesics do

exist for the class of globally hyperbolic space-times.

The universal covering manifold (M, g) of two-dimensional anti-de Sitter

space provides an example that geodesic completeness does not imply that

every pair p, q E M with p << q may be joined by a timelike geodesic y with

L(y ) = d ( p , q) (cf. Figure 6.1). %call that if y is any future directed timelike


curve from p to q with L(y) = d(p, q), then y may be reparametrized to a

timelike geodesic (cf. Theorem 4.13). Thus this same example shows that

geodesically complete space-times exist which contain points p << q such that

L(y) < d(p, 4) for all y f O,,,.

The space-time (M, g) may be represented by the strip M = {(x, t ) E R2 :

--7r/2 < x < r /2 ) in B2 with the Lorentzian metric ds2 = sec2 x (-dt2 + dx2)

[cf. Penrose (1972, p. 7)j. The points p and q in Figure 6.1 satisfy p (< q, yet

all future timelike geodesics emanating from p are focused again a t the future

timelike conjugate point T. Thus there is no timelike geodesic in M from p to

q. Hence there is no maximal timelike geodesic or maximal timelike curve from

p to q. Even more strikingly, it should be noted that an open set U C I+(p)

of points of the type of q as in Figure 6.1 may be found with the property

that none of the points of U are connected to p by any geodesic whatsoever,

despite the geodesic completeness of this space-time.

We now consider which space-times do have the property that every pair

of points p,q E M with q E J+(p) may be joined by a distance realizing

geodesic. If M = R2-((0, 0)) with the Lorentzian metric ds2 = dx2-dy2, then

p = (0, -1) and q = (0,l) are points in M with d(p, q) = 2 > 0 which cannot

be joined by a maximal timelike geodesic. [The desired geodesic would have

to be the curve y(t) = (O,t), -1 5 t 5 1, which passes through the deleted

point (0, O).] On the other hand, this space-time is chronological, strongly

causal, and stably causal. Thus it is reasonable to restrict our attention to the

class of globally hyperbolic space-times. For these space-times, Avez (1963)

and Seifert (1967) have shown that given any p, q E M with p < q, there is a

geodesic from p to q which maximizes arc length among all nonspacelike future

directed curves from p to q (cf. Theorem 3.18). In the language of Definition

4.10, this may be stated as follows.

Theorem 6.1. Let (M, g) be globally hyperbolic. Then given any p, q E M

with q E J+(p), there is a maximd geodesic segment y E i2,,,, i.e., a future

directed nonspacelike geodesic y from p to q with L(y) = d(p,q).

We sketch Seifert's (1967, Theorem 1) proof of this result [cf. Penrose (1972,

Chapter 6)]. Since (M,g) is globally hyperbolic, it may be shown that if

6.1 EXISTENCE O F MAXIMAL GEODESIC SEGMENTS 201

p < q, the nonspacelike path space a,,, is compact. On the other hand, since

(M,g) is strongly causal, the arc-length functional L : 52,,, --t R is upper

semicontinuous in the C0 topology (cf. Section 3.3). Thus there exists a curve

yo E a,,, with L(yo) = sup{L(y) : y E Q,,,). It follows from the variational

theory of arc length that if yo is not a reparametrization of a smooth geodesic,

a curve a E R,,, with L(a) > L(y) may be constructed, in contradiction.

Alternatively, if L(y0) = sup{L(y) : y E a,,,), then L(y0) = d(p,q) by the

definition of Lorentzian distance. Hence Theorem 4.13 implies that yo is, up

to reparametrization, a smooth geodesic.

In the case that p << q, the maximal curve yo may also be constructed using

the results of Section 3.3. Let h : M --t B be a globally hyperbolic time function

for (M, g). Choose to with h(p) < to < h(q). Then K = J+(p)fl J-(q)flh-'(to)

is compact, and any nonspacelike curve from p to q intersects K. By definition

of Lorentzian distance, we may find a curve y, E O,,, with

for each positive integer n. Let rn E x n K . Since K is compact, a subsequence

{T,(~)) converges to T E K. By Corollary 3.32, there is a nonspacelike limit

curve yo of the sequence {yn(j)) passing through T and joining p to q. Since

(M, g) is strongly causal, a subsequence of {y,(,)) converges to yo in the C0

topology by Proposition 3.34. Using Remark 3.35 and condition (6.1), we

obtain L(yo) > d(p, q). Hence by the definition of distance, L(yo) = d(p, q),

and yo may be reparametrized to a smooth geodesic by Theorem 4.13. If p ,< q

and d(p, q) = 0, we already know that there is a maximal null geodesic segment

from p to q by Corollary 4.14.

In connection with Theorem 6.1, it should be noted that global hyperbolicity

is not a necessary condition for the existence of maximal geodesic segments

joining all pairs of causally related points. For let M = {(x, y) E R2 : 0 < x < 10, 0 < y < 10) be equipped with the Lorentzian rr~etric it inherits as an

open subset of Minkowski space. Since the geodesics in M are just Euclidean

straight line segments, it is readily seen that maximal geodesics exist joining

any pair of causally related points. However, if p = (1 , l ) and q = (1,9), then


J+(p) n 5-(q) is noncompact. Hence this space-time, while strongly causal,

fails to be globally hyperbolic.

6.2 Geodesic Completeness

We showed in Theorem 4.9 that for space-times which are strongly causal,

the Lorentzian distance function may be used to construct a subbasis for the

given manifold topology. Nonetheless, the sets {q E M : d(p, q) < R) fail

t o form a basis for the given manifold topology. Thus geodesic completeness

rather than metric completeness of spacetimes has usually been considered

in general relativity.

Let (M, g) be an arbitrary Lorentzian manifold.

Definition 6.2. (Complete Geodesic) A geodesic c in (M, g) with affine

parameter t is said to be complete if the geodesic can be extcnded to be

defined for -m < t < moo. A past and future inextendible geodesic is said

to be incomplete if it cannot be extended to arbitrarily large positive and

negative values of an affine parameter. Future or past incomplete geodesics

may be defined similarly.

An affine parameter t for the curve c is a parametrization such that c(t) sat-

isfies the geodesic differential equation V,.c1(t) = 0 for all t [cf. Kobayashi and

Nomizu (1963, p. 138)]. It is necessary to use the concept of an affine param-

eter since null geodesics, which have zero arc length, cannot be parametrized

by arc length. If s and t are two affine parameters for c, it follows from the

geodesic differential equations that there exist constants a , /3 E IR such that

s(t) = a t + P for all t in the domain of c. Hence completeness or incompleteness

as defined in Definition 6.2 is independent of the choice of affine parameter. In

particular, if c is an inextendible timelike geodesic parametrized by arc length

(i.e., g(cl(t), cl(t)) = -1 for all t in the domain of c), then c is incomplete if

L(c) < m. Even if L(c) = m, it may happen that c is incomplete. This occurs,

for example, when the domain of c is of the form (a, co), where a > c o .

Certain exact solutions to the Einstein equations in general relativity, like

the extended Schwarzschild solution, contain nonspacelike geodesics which be-

come incomplete upon running into black holes. Even though the existence of

6.2 GEODESIC COMPLETENESS 203

incomplete, inextendible, nonspacelike geodesics does not force a space-time

to contain a black hole, these examples suggest that nonspacelike geodesic

incompleteness might be used as a first order test for "singular space-times"

[cf. Hawking and Ellis (1973, Chapter 8), Clarke and Schmidt (1977), Ellis

and Schmidt (1977)j. Thus it is standard to make the following definitions in

general relativity. Recall that a geodesic is said to be inextendible if it is both

past and future inextendible.

Definition 6.3. (Geodesically Complete) A space-time (M,g) is said

to be timelike (respectively, null, nonspacelike, spacelike) geodesically wm-

plete if all timelike (respectively, null, nonspacelike, spacelike) inextendible

geodesics are complete. The space-time (M, g) is said to be geodesically com-

plete if all inextendible geodesics are complete. Also, (M, g) is said to be time-

like (respectively, null, nonspacelike, spacelike) geodesically incomplete if some

timelike (respectively, null, nonspacelike, spacelike) geodesic is incomplete. A

nonspacelike incomplete space-time is said to be a geodesically singular space-

time.

It was once hoped that timelike geodesic completeness might imply null ge-

odesic completeness, etc. However, Kundt (1963) gave an example of a space-

time that is timelike and null geodesically complete but not spacelike complete.

Then Geroch (1968b, p. 531) gave an example of a space-time conformal to

Minkowski two-space and thus globally hyperbolic which is timelike incomplete

but null and spacelike complete. Also Geroch remarked that modifications of

Kundt's and his examples gave space-times that were (1) incomplete in any

two ways but complete in the third way, (2) spacelike incomplete but null and

timelike complete, and (3) timelike incomplete but spacelike and null complete.

Then Beem (1976~) gave an example of a globally hyperbolic space-time that

was null incomplete but spacelike and timelike complete. These results may

be summarized as follows.

1 T h e o r e m 6.4. Timelike geodesic completeness, nu11 geodesic complete-

ness, and spacelike geodesic completeness are all logically inequivalent.

6 COMPLETENESS AND EXTENDIBILITY

FIGURE 6.2. Shown is Geroch's example of a space-time glob-

ally conformal to Minkowski two-space, which is null and spacelike

geodesically complete but timelike geodesically incomplete. Here

the positive t axis may be parametrized to be an incomplete t i m e

like geodesic since b(O, t) + 0 like tT4 as t -+ a.

In order to illustrate the constructions used in the proof of Theorem 6.4,

we now describe Geroch's example of a space-time which is null and spacelike

complete but timelike incomplete. Let (IR2, gl) be Minkowski two-space with

global coordinates (x, t) and the usual Lorentzian metric gl = ds2 = dx2 - dt2.


Conformally change the metric gl to a new metric g = $gl for W2, where

4 : R2 --+ (0 , co) is a smooth function with the following properties (cf. Figure

6.2):

(1) $ (x , t )= 1 i f x s -1 o r x > 1;

(2) $(x, t) = $(-x, t) for all (x, t ) f W2; and

(3) On the t axis, d(0, t ) goes to zero like t-4 as t -4 a.

Since g is conformal to gl, the space-time (W2, g) is globally hyperbolic, and

null geodesics still have as images straight lines making angles of 45" with the

positive or negative x axis. By property (2), the reflection F ( x , t) = (-x, t)

is an isometry of Since the fixed point set of a n isometry is totally

geodesic, the t axis may be parametrized as a timelike geodesic. By condition

(3), this geodesic is incomplete as t + m. Thus (W2, g) is timelike incomplete.

But every null or spacelike geodesic which enters the region -1 5 x 5 1

eventually leaves and then remains outside this region. Thus condition (1)

implies that (EX2, g) is null and spacelike complete.

We now consider the converse problem of constructing geodesically complete

Lorentzian metrics for paracompact smooth manifolds. In order to preserve

the causal structure of the given space-time, we restrict our attention to global

conformal changes rather than arbitrary metric deformations.

For Riemannian metrics, Nomizu and Ozeki (1961) showed that an arbitrary

metric can be made complete by a global conformal change. On the other hand,

space-times exist with the property that no global conformal factor will make

these space-times nonspacelike geodesically complete. A two-dimensional ex-

ample with this property has been given by Misner (1967). In this example

there are inextendible null geodesics which are future incomplete and future

trapped in a compact set [cf. Hawking and Ellis (1973, pp. 171-172)j. Any

conformal change of this example will leave these null geodesics pointwise fixed

and future incomplete. Thus one may not establish an analogue of the Nomizu

and Ozeki result for arbitrary space-times.

However, the existence of nonspacelike complete Lorentzian metrics has

been shown for space-times satisfying certain causality conditions. Seifert

(1971, p. 258) has shown that if (M, g) is stably causal, then M is conformal


to a space-time with all future directed (or all past directed) nonspacelike

geodesics complete. Also Clarke (1971) has shown that a strongly causal space-

time may be made null geodesically complete by a conformal factor. Beem

(1976a) studied space-times with the property that for each compact subset

K of M, no future inextendible nonspacelike curve is future imprisoned in

K. (Recall that the nonspacelike curve y is said to be future imprisoned in

K if there exists to E R such that y(t) E K for all t 2 to.) If (M,g) is

a causal space-time satisfying this condition, then there exists a conformal

factor Sl : M -t (0, GO) such that (M, Slg) is null and timelike geodesically

complete [Beem (1976a, p. 184, Theorem 8)]. This imprisonment condition is

satisfied if ( M , g ) is stably causal, strongly causal, or distinguishing. Hence

we may state the following result.

T h e o r e m 6.5. If ( M , g ) is distinguishing, strongly causal, stably causal,

or globally hyperbolic, then there exists a smooth conformal factor R : M -,

(0, m) such that the space-time (M, Rg) is timelike and null geodesically com-

plete.

I t is an open question as to whether Theorem 6.5 can be strengthened to

include spacelike geodesic completeness as well (cf. Corollary 3.46 for space-

times homeomorphic to R2).

Suppose that a space-time is defined to be nonsingular if it is geodesically

complete. Then "no regions have been deleted from the space-time manifold"

of a nonsingular space-time [Geroch (1968b, Property I)]. But Geroch (196813,

Property 2) suggested a second condition that nonsingular space-times should

satisfy, namely, "observers who follow 'reasonable' (in some sense) world lines

should have an infinite total proper time." Here a "world line" is a timelike

curve in (M,g). Then Geroch (1968b, pp. 534-540) constructed a geodesi-

cally complete space-time which contains a smooth timelike curve of bounded

acceleration but having finite length. Thus this example fails t o satisfy Ge-

roch's Property 2 even though all timelike geodesics have infinite length by the

geodesic completeness.

Accordingly, in addition to geodesically incomplete space-times, further

kinds of singular space-times have been studied in general relativity. In the rest


of this section, we will discuss two of these additional types of completeness,

b.a. completeness (bounded acceleration completeness) and b-completeness

(bundle completeness).

The concept of b.a. completeness stems from the preceding example of Ge-

roch. For the purpose of stating Definition 6.6, we recall than any C2 time-

like curve may be reparametrized to a C2 timelike curve y : J --+ M with

g(yl(t), yl(t)) = -1 for all t f J.

Definition 6.6. (Bounded Acceleration) A C2 timelike curve y : J -+ M

with g(yl(t), yl(t)) = -1 for all t E J is said to have bounded acceleration if

there exists a constant B > 0 such that Ig(V,iyl(t), V , J ~ ' ( ~ ) ) I 5 B for all

t E J.

Here V is the unique torsion free connection for M defined by the metric g

[cf. Section 2.2, equations (2.16) and (2.17)). In particular, if y is a geodesic,

then y has zero and hence bounded acceleration. The requirement that y be

C2 makes it possible to calculate V,jyl.

Definition 6.7. (b.a. complete space-time) A space-time (M, g) is said

to be b.a. complete if all future (respectively, past) directed, future (respec-

tively, past) inextendible, unit speed, C2 timelike curves with bounded acceler-

ation have infinite length. If there exists a future (or past) directed, future (or

past) inextendible, unit speed, C2 timelike curve with bounded acceleration

but finite length, then (M,g) is said to be b.a. incomplete.

Geroch's example (1968b, pp. 534-540) shows that geodesic completeness

does not imply b.a. completeness. Furthermore, Beem (1976c, p. 509) has given

an example to show that even for globally hyperbolic space-times, geodesic

completeness does not imply b.a. completeness. Trivially, b.a. completeness

implies timelike geodesic completeness. On the other hand, the example of

Geroch given in Figure 6.2 may be modified by changing the sign of the met-

ric tensor to show that b.a. completeness does not imply spacelike geodesic

completeness.

A stronger form of completeness, b-completeness, does imply geodesic com-

pleteness and hence overcomes this last objection to b.a. completeness. The


concept of b-completeness, which was first studied for Lorentzian manifolds by

Schmidt (1971), is intuitively defined as follows [cf. Hawking and Ellis (1973,

p. 259)). First, the concept of an affine parameter is extended from geodesics

to all Ci curves. Then a space-time is said to be b-complete if every C1 curve

of finite length in such a parameter has an endpoint.

We now give a brief discussion of b-completeness. First, it is necessary to

discuss the concept of a generalized afine parameter for any C1 curve y : J --+

M. Recall that a smooth vector field V along y is a smooth map V : J --t TM

such that V(t) E T7(tlM for all t E J. Such a smooth vector field V along

y is said to be a parallel field along y if V satisfies the differential equation

Vy,V(t) = 0 for all t E J (cf. Chapter 2).

A generalized affine parameter p = p(y, El, E2 , . . . , En) may be constructed

for y : J -+ M as follows. Choosing any to E J, let {el, e2, . . . ,en} be any

basis for Ty(t,lM. Let Ei be the unique parallel field along y with E,(to) = ei

for 1 5 i 5 n. Then {El(t), E2(t), . . . , En(t)) forms a basis for T7(tlM for

each t E J. We may thus write yl(t) = Cy=l Vi(t)E,(t) with Vi : J -+ R

for 1 5 i 5 n. Then the generalized affine parameter p = p(y, El,. . . , E n ) is

given by

The assumption that y is C1 is necessary in order to obtain the vector fields

{El, E2 , . . . , E n ) by parallel translation. It may be checked that y has finite

arc length in the generalized affine parameter p = p(y, El,. . . , En) if and

only if y has finite arc length in any other generalized affine parameter p = -

p ( y , E l , . . . , E n ) calculated from any other basis {E,):=, for TMI, obtained

by parallel translation along y [cf. Hawking and Ellis (1973, p. 259)j. Hence

the concept of finite arc length with respect to a generalized affine parameter

is independent of the particular choice of generalized affine parameter. I t thus

makes sense to make the following definition.

Definition 6.8. (b-complete space-time) The space-time (M, g) is said

to be b-complete if every C1 curve of finite arc length as measured by a gen-

eralized affine parameter has an endpoint in M.

6.3 METRIC COMPLETENESS 209

Suppose y : J + M is any smooth geodesic. Taking El(t) = yl(t) in the

above construction, for any choice of E2,. . . , E n we have generalized affine

parameter p(y, E l , E2 , . . . , En)(t) = t. Hence b-completeness implies geodesic

completeness. It is also known that b-completeness implies b.a. completeness.

Geroch's example (cf. Figure 6.2) with the sign of the metric tensor changed

shows that there are globally hyperbolic space-times which are b.a. complete

but not b-complete. Thus b.a. completeness does not imply b-completeness.

6.3 Met r i c Completeness

The Hopf-Rinow Theorem for Riemannian manifolds (N, go) implies that

the following are equivalent:

(1) N with the Riemannian distance function do : N x N + [O, a) is a

complete metric space, i.e., all Cauchy sequences converge.

(2) (N, do) is finitely compact, i.e., all do-bounded sets have compact clo-

sure.

(3) (N, go) is geodesically complete.

Here a set K in a Riemannian manifold (N,go) is said to be bounded if

sup{do(p, q) : p, q E K } < co. By the triangle inequality, this is equiva-

lent to the condition that K be contained inside a closed metric ball of finite

radius.

In Section 6.2 we considered the geodesic completeness of Lorentzian man-

ifolds. In this section we shall consider Lorentzian analogues of conditions (1)

and (2) above. From the very definition of Lorentzian distance [i.e., d(p, q) = 0

if q $! J+(p)], it is clear that attention should be restricted to timelike Cauchy

sequences.

Busemann (1967) studied general Hausdorff spaces having a partial order-

ing with properties similar to those of the chronological partial ordering p << q

of a space-time. Also, Busemann supposed that these spaces, which he called

timelike spaces, were equipped with a function which behaves just like the

Lorentzian distance function of a chronological space-time restricted to the set

{(p, q) E M x M : p < q}. For this class of nondifferentiable spaces, Busemann

observed that the length of continuous curves could be defined and, moreover,


FIGURE 6.3. Shown is a sequence {x,) in Minkowski two-space

(IR2, ds2 = dx2 -dy2) with 2, >> p for all n, d(p, z,) + 0 as n -+ co,

but such that {x,) has no point of accumulation.

the length functional is upper semicontinuous in a topology of uniform con-

vergence [cf. Busemann (1967, p. lo)]. Busemann's aim in studying timelike

spaces was to develop a geometric theory for indefinite metrics analogous to

the theory of metric G-spaces [cf. Busemann (1955)j. In particular, Busemann

studied finite compactness and metric completeness for timelike spaces in the

spirit of (1) and (2) of the Hopf-Rinow Theorem.

Beem (1976b) observed that Busemann's definitions of finite compactness

and metric completeness for timelike G-spaces may be adapted to causal space-

times. First, timelike Cauchy completeness may be defined for causal space-

times as follows.


Definition 6.9. (Timelike Cauchy Complete) The causal space-time

(M,g) is said to be timelike Cauchy complete if any sequence {x,) of points

with x, << x,+, for n, m = 1,2 ,3 , . . . and d(z,, x,+,) 5 B, [or else x,+, << x, and d(x,+,, x,) 5 B,] for all m 1 0, where B, -+ 0 as n -+ oa, is a con-

vergent sequence.

For Riemannian manifolds, finite compactness may be defined by requiring

that all closed metric balls be compact. On the other hand, we have noted

above (cf. Figure 4.4) that thesubsets {q E J+(p) : d(p, q ) 5 e ) of a space-time

are generally noncompact. Thus the Riemannian definition must be modified.

One possibility is the following [cf. Busemann (1967, p. 22)].

Definition 6.10. (Finitely Compact) The causal space-time (M,g) is

said to be finitely compact if for each fixed constant B > 0 and each sequence

of points {x,) with either p << q < x, and d(p, x,) 5 B for all n, or x, < q << p

and d(z,,p) < B for all n, there is a point of accumulation of {x,) in M.

It may be seen that without requiring p << q < x, (or x, < q << p) for some

q E M in Definition 6.10, Minkowski space-time fails to be finitely compact

(cf. Figure 6.3).

For globally hyperbolic space-times, a characterization of finite compactness

more reminiscent of condition (2) above for Riemannian manifolds may be

given.

Lemma 6.11. Let (M,g) be globally hyperbolic. Then (M, g) is finitely

compact iff for each real constant B > 0, the set {x E M : p << q 6 x, d(p, x) 5 B) is compact for any p,q E M with q E I+(p) and the set {x E M : x < q << p, d(x,p) 5 B) is compact for any p,q E M with p E I+(q).

Proof. This now follows easily because the sets J+(q) are closed and the Lor-

entzian distance function is continuous since (M, g) is globally hyperbolic.

Minkowski space-time is both timelike Cauchy complete and finitely com-

pact. More generally, it may be shown that these concepts are equivalent for

all globally hyperbolic space-times [cf. Beem (1976b, pp. 343-344)l.


Theorem 6.12. If (M, g) is globally hyperbolic, then (M, g) is finitely com-

pact iff (M, g) is timelike Cauchy complete. Also, if (M,g) is globally hyper-

bolic and nonspacelike geodesically complete, then (M, g) is finitely compact

and timelike Cauchy complete.

Remark 6.13. Even for the class of globally hyperbolic space-times, finite

compactness, or equivalently, timelike Cauchy completeness, does not imply

timelike geodesic completeness. Indeed, Geroch's example given in Figure 6.2

is a timelike geodesically incomplete, globally hyperbolic space-time which is

finitely compact.

At times, it is important to consider the completeness of a submanifold as

well as that of the given manifold. Conditions which are sufficient to guarantee

the completeness of submanifolds of Lorentzian manifolds are, in general, more

complicated than corresponding conditions for submanifolds of Riemannian

manifolds [cf. Beem and Ehrlich (1985a,b), Harris (1987, 1988a,b, 1994)j. If

(H, h) is a complete Riemannian manifold and F : M + H is an embedding

of M with F(M) a closed subset of H, then M is a complete Riemannian

manifold using the induced metric. The converse of this result is false. For

example, let the curve c : (0, +m) -+ W2 be given by c(t) = (t,sin(l/t)). This

curve has an image which is a complete submanifold of the usual Euclidean

plane, but this image clearly fails to be a closed subset of R2.

Unlike the Riemannian case, closed embedded submanifolds of Lorentzian

manifolds need not be complete. To see this, it suffices to take a curve in

the Minkowski plane which is asymptotic to a null line quickly enough in at

least one direction along the curve. For example, the following map [cf. Harris

is an embedding of W1 as an incomplete closed spacelike submanifold of the

(x, y) plane with the usual Minkowski metric 77 = dx2-dy2. Harris (1988a) has

studied the completeness of embedded and immersed spacelike hypersurfaces of

Minkowski space. Among other things, he has shown that if such an immersed

hypersurface in (n + 1)-dimensional Minkowski space (i.e., Ln+') is complete,


then its image must be a closed achronal subset which is diffeomorphic to

Rn and it must be embedded as a graph of a function defined on a spacelike

hyperplane. A more detailed resume of Harris's work on spacelike completeness

together with additional references is given in Harris (1993, 1994).

Separate investigations had earlier been conducted in the case of constant

mean curvature spacelike hypersurfaces S of Ln+' which are closed subsets of

Ln+' in the Euclidean topology. For this class of space-times, Cheng and Yau

(1976) showed that the condition of constant mean curvature Ho implies that

S is complete in the induced Riemannian metric. (A complete treatment of the

issue of the achronality en route to the proof of completeness is given in Harris

(1988a, pp. 112, 118). Also, Harris (1988a, p. 118) showed that the hypothesis

of bounded principal curvatures may be substituted for constant mean curva-

ture.) Further, Cheng and Yau (1976) showed that constant mean curvature

Ho implies that the length of the second fundamental form is bounded by

n /No[ and also that S has nonpositive Ricci curvature. In particular, in the

case of a maximal (i.e., Ho = 0) spacelike hypersurface, this estimate shows

that the second fundamental form is trivial. Hence these results of Cheng and

Yau (1976) combine with earlier work of Calabi (1968) for n = 3 to yield the

result that the only maximal spacelike hypersurface which is a closed subset

of Minkowski space is a linear hyperplane. Nishikawa (1984) investigated the

more general case of a locally symmetric target manifold satisfying the timelike

convergence condition and a condition that the sectional curvature of all non-

degenerate two-planes spanned by a pair of spacelike vectors be nonnegative.

Nishikawa showed that a complete maximal spacelike hypersurface in such a

target space-time would be totally geodesic.

In the case of nonzero constant mean curvature, Goddard (1977b) car-

ried out perturbation calculations for hyperboloids in Minkowski space (and

also for appropriate submanifolds of de Sitter space-time) which suggested

that perhaps all entire, constant mean curvature, spacelike hypersurfaces of

Minkowski space should be hyperboloids. This issue was thoroughly inves-

tigated by Treibergs (1982), who took the starting point that in the case of

positive mean curvature, an entire spacelike hypersurface could be realized as

214 6 COMPLETENESS AND EXTENDIBILITY 6.4 IDEAL BOUNDARIES 215

the graph of a convex function. Now for an entire convex function f whose

graph is a spacelike hypersurface, Treibergs (1982, p. 51) defines the projective

boundary values o f f at infinity, Vj, as

f (7.~1 Vj(x) = lim -. r++m T

Treibergs defines two entire, constant mean curvature, spacelike hypersurfaces

to be equivalent if they have the same projective boundary values a t infinity.

Treibergs then proves that the set of such equivalence classes coincides with

convex homogeneous functions whose gradient has norm one whenever defined.

Treibergs further shows that constant mean curvature spacelike hypersurfaces

with given projective boundary data a t infinity are highly nonunique, because

arbitrary finite perturbations of the given light cone (at infinity) may be made

producing different f (x), hence different spacelike hypersurfaces, each strongly

asymptotic to its own perturbed light cone, yet all having the same projective

boundary behavior V(x) a t infinity. Thus the geometry for entire, constant

(but nonzero) mean curvature, spacelike hypersurfaces turns out to be more

complicated than the maximal (No = 0) case.

6.4 Ideal Boundaries

In this section we give brief descriptions of the b-boundary and the causal

boundary for a space-time. Further details may be found in Hawking and Ellis

(1973, Sections 6.8 and 8.3) or Dodson (1978).

The b-boundary of a space-time (M,g) will be denoted by dbM. This

boundary is formed by defining a certain positive definite metric on the bundle

of linear frames L ( M ) over M , taking the Cauchy completion of L(M), and

then using the newly formed ideal points of L(M) to obtain ideal points of

M . The b-boundary is particularly useful in telling whether or not some

points have been removed from the space-time. Somewhat unfortunately, the

b-boundary often consists of just a single point [cf. Bosshard (1976), Johnson

(1977)l. This boundary is not invariant under conformal changes and also

is not directly related to the causal structure of (M,g). A discussion of the

merits and demerits of the b-boundary and geodesic incompleteness may be

found in the review article of Tipler, Clarke, and Ellis (1980).

Recall that a curve y : [O,a) --t M is said to be b-incomplete if it has

finite generalized affine parameter (cf. Section 6.2). .4ny curve y : [O,a) --, M

which is both b-incomplete and inextendible to t = a defines a point of dbM

corresponding to y(a). In Minkowski space-time, generalized affine parameter

values along a curve can be made to correspond to Euclidean arc length. Thus

Minkowski space-time has an empty b-boundary, and each b-incomplete curve

in Minkowski space-time has an endpoint in the space-time.

The causal boundary of a space-time (M,g) will be denoted by dcM. This

boundary is constructed using the causal structure of the space-time. Thus it

is invariant under conformal changes. We will only be interested in considering

this boundary for strongly causal space-times.

The causal boundary is formed using indecomposable past (respectively, fu-

ture) sets which do not correspond to the past (respectively, future) of any

point of M . A past (respectively, future) set A is a subset of M such that

I-(A) C A (respectively, I+(A) A). The open past (respectively, future)

sets are characterized by I-(A) = A (respectively, If (A) = A). An indecom-

posable past set (IP) is an open past set that cannot be written as a union

of two proper subsets both of which are open past sets. An indecomposable

future set (IF) is defined dually.

A terminal indecomposable past set (TIP) is a subset A of M such that

(1) A is an indecomposable past set, and

(2) A is not the chronological past of any point p E M.

A terminal indecomposable future set (TIF) is defined dually. The causal

boundary bcM is formed using TIP's and TIF's after making certain identifica-

tions which will be described below [cf. Hawking and Ellis (1973, pp. 218-221)].

These identifications allow the topology of M to be extended to M * = MU&M

in such a way that the causal completion of M is Hausdorff, provided that

(M, g) satisfies certain more restrictive causality conditions such as stable

causality.

The use of TIP's and TIF's to represent ideal points of the causal boundary

of (M, g) is illustrated in Figure 6.4.


- D

FIGURE 6.4. The ideal point p in 8,M is represented by the termi-

nal indecomposable past set A, and the ideal point ?j is represented

by the terminal indecomposable future set B. The point 5: is repre-

sented by both the set C, which is a TIP, and the set D, which is a

TIF.

We now show that a TIP may be represented as the chronological past of a

future inextendible timelike curve. This result is due to Geroch, Kronheimer,

and Penrose (1972, p. 551).

Proposition 6.14. A subset W of the strongly causal space-time (M, g) is

a TIP iff there exists a future directed and future inextendible timelike curve

y such that W = I-(?).

Proof. Assume there is a future inextendible timelike curve y with W =

I - (7 ) . Using the strong causality of (M,g) , it follows that if W is an IP, then

W is a TIP. To show W is an IP, assume that W = Ul U U2 for nonempty

6.4 IDEAL BOUNDARIES 217

open past sets Ul and U2 such that neither is a subset of the other. Choose

rl E U1 - U2 and 7-2 E U2 - Ul . There must exist points r: t y such that

r , E I - ( r : ) for i = 1,2 because Ul U U2 = I - (7 ) . However, whichever Ui contains the futuremost of r; and r; must then contain all four of r l , 7-2, T: ,

and T;. This contradicts either the definition of rl or of r2.

On the other hand, assume that W is a TIP. If p is any point of W , then

W = [ W n I + ( p ) ] u ( W - I+(p)] and thus W = I - ( W n I f ( p ) ) u I - ( W - I+ ( p ) )

since W is a past set. Since W is an IP, either W = I - ( W n I+(p)) or

W = I - ( W - I+(p)). Consequently, as p $ I-(W - I+(p)) , we have W =

I - ( W n I+(p)). Thus, given any q # p in W , there must be some point r in

W which is in the chronological future of both p and q. Inductively, for each

finite subset of W , there exists some point of W in the chronological future

of each point of the subset. Now choose a sequence of points {p,) which

forms a countable dense subset of W . We will define a second sequence {q,)

inductively. Let qo be a point of W in the chronological future of po. If qi for

i = 1,2 , . . . , k - 1 has been defined, then choose q k to be a point of W in the

chronological future of pk and of q, for i = 1,2,. . . , k - 1.

Finally let y be any future directed timelike curve which begins a t qo and

connects each qi to the next qi+1. Clearly, each p, lies in I - ( y ) and I-(?)

W . Using the openness of W and the denseness of the sequence {p,), it follows

that W = I - ( y ) as required. 0

In space-times which are not strongly causal, there may exist future directed

and future inextendible timelike curves y such that I - ( y ) is the chronological

past of some point [i.e., I - ( 7 ) is not a TIP]. Consider, for example, the cylinder

W1 x S 1 with the flat metric ds2 = dt dB and the usual time orientation with

the future corresponding to increasing t. The lower half of the cylinder W =

{ ( t , 8 ) : t < 0) is an IP which can be represented as I - ( y ) for a future directed

and future inextendible timelike curve y. However, W is not a TIP since W

can be represented as the chronological past of any point on the circle t = 0.

By restricting our attention to strongly causal space-times, the IP's which are

not TIP'S are in one-to-one correspondence with the points of M. The dual

statement holds for IF'S which are not TIF1s.


We now define M (respectively, M) to be the collection of all IP's (respec-

tively, IF'S). Fhrthermore, let ~d = M U M / N, where for each p E M

the element I-(p) of M is identified with the element I+(p) of M. The map

I+ : M -+ MI given by p -+ I+(p) then identifies M with a subset of Ms.

Using this identification, the set Mu corresponds to M together with all TIP'S

and TIF's.

In order to define a topology on Mu, first define for any A E M the sets ~ i n t and Aext by

= {V E M : v n A # 0 )

and

AeXt = {V E M : V = I - (W) implies I+(w) 9 A).

Similar definitions of BInt and BeXt are made for any B E M. A subbasis

for a topology on Mfl IS then given by all sets of the form Alnt, AeXt, BRnt,

and Bext. The sets Alnt and BInt are analogues of sets of the form I+(p) and -

I-(p), respectively. The sets A"* and BeXt are analogues of M - I+(p) and -

M - I - (p), respectively.

Now in Geroch, Kronheimer, and Penrose (1972), it is proposed to obtain

a Hausdorff space M * = M U a,M from Mu, with the topology given as

above, by identifying the smallest number of points of MI necessary to obtain

a Hausdorff space M*. Equivalently, it is proposed that M * = M U 8,M

should be taken to be the quotient M I I R ~ , where Rh is the intersection of all

equivalence relations R on MI such that M ~ / R is Hausdorff.

Unfortunately, Szabados (1988) and Rube (1988, 1990) independently ob-

served that the assumption of strong causality for the given space-time (M, g)

was not sufficient to ensure that the minimal equivalence relation Rh exists,

and hence the Geroch-PenroseKronheimer choice of topology may not be

used to induce a Hausdorff causal completion M * for a general strongly causal

space-time. Szabados points out two difficulties that may arise for general

strongly causal space-times if the topology given via {Atnt, Aext, BInt, Bext)

is chosen. First, inner points corresponding to the embedding of M into M@

via the map I+ and pre-boundary points are in general only TI-separated, but

not Tz-separated, with this choice of topology. Second, an example is given to

6.5 LOCAL EXTENSIONS 219

show, for a nonspacelike curve y with endpoint q in M , that the induced curve

I+ o y does not necessarily have a unique endpoint in M. Both Rube (1988)

and Szabados (1988) suggest stronger causality conditions on the underlying

space-time (M, g) which will ensure that the equivalence relation Rh proposed

as above will exist, and hence this construction will produce a Hausdorff causal

boundary d,M for M identified with its image in M * under the mapping If.

The simplest such condition to impose is that (M,g) be stably causal.

6.5 Local Extensions

In this section, extendibility and inextendibility of Lorentzian manifolds are

defined. Also, two types of local extendibility are discussed. Most of the results

of this section hold for Lorentzian manifolds which are not time orientable as

well as for space-times.

Definition 6.15. (Extension) An extension of a Lorentzian manifold

(M,g) is a Lorentzian manifold (MIlg') together with an isometry f : M -+

M' which maps M onto a proper open subset of MI. An analytic extension

of (M, g) is an extension f : (M, g) -+ (MI, g') such that both Lorentzian

manifolds are analytic and the map f : M -+ M' is analytic. If (M,g) has no

extensions, it is said to be inextendible.

Suppose that the Lorentzian manifold (M, g) has an extension f : (M, g ) - (M',gl). Since M' is connected and f ( M ) is assumed t o be open in MI, it

follows that

- where f ( M ) denotes the closure of f (M) in M'. Because d(f (M)) # 0 and

the isometry f : M -+ M' maps geodesics in M into geodesics in M' lying

in f (M), it is easily seen that (M,g) cannot be timelike, null, or spacelike

geodesically complete. Recalling that b-completeness and b.a. completeness

both imply timelike geodesic completeness (cf. Section 6.2), we thus have the

following criteria for Lorentzian manifolds to be inextendible.


Proposition 6.16. A Lorentzian manifold ( M , g ) is inextendible if i t is

complete in any of the following ways:

(1 ) b-complete;

(2) b.a. complete;

(3 ) timelike geodesically complete;

(4) null geodesically complete; or

(5 ) spacelike geodesically complete.

We now define two types of local extensions [cf. Clarke (1973, p. 207), Beem

(1980), Hawking and Ellis (1973, p. 59)].

Definition 6.17. (Local Eztension) Let ( M , g) be a Lorentzian manifold.

(1) Suppose y : [0, a ) --t M is a b-incomplete curve which is not extendible to

t = a in M. A local b-boundary extension about 7 is an open neighborhood

U C_ M of 7 and an extension (U', g ' ) of (U, gl,) such that the image of y in

U' is C0 extendible beyond t = a.

(2 ) A local extension of ( M , g ) is a connected open subset U of M having

noncompact closure in M and an extension (U', g') of (U, gl,) such that the

image of U has compact closure in U'.

Remark 6.18. This definition of local extension differs from the corr*

sponding definition of local extension in Hawking and Ellis (1973, p. 59) in

that U is required to be connected in Definition 6.17-(2) but not in Hawking

and Ellis (cf. Figures 6.5 and 6.6).

We now investigate the relationships between these two types of local exten-

sions. An arbitrary space-time may contain a b-incomplete curve 7 : 10, a ) -+

M which is not extendible to t = a , yet y[O, a ) has compact closure in M.

However, Schmidt has shown that such space-times contain compactly im-

prisoned inextendible null geodesics [cf. Schmidt (1973), Hawking and Ellis

(1973, p. 280)]. On the other hand, if ( M , g ) contains no imprisoned non-

spacelike curves, and ( M , g ) has a local b-boundary extension about y , we

now show this same extension yields a local extension.


FIGURE 6.5. Let y : [O, a ) -+ M be a b-incomplete curve which

is not extendible to t = a in the space-time (M,g) . Assume that

there is an isometry f : (U, glu) -+ (U',gl) which takes y to a curve

f o y having an endpoint p in U'. Then f o y may be continu-

ously extended beyond t = a. Thus ( M , g) has a local b-boundary

extension about r.

Lemma 6.19. If ( M , g) is a space-time with no imprisoned nonspacelike

curves and if ( M , g) has a local b-boundary extension about y, then (M, g)

has a local extension.

Proof. Suppose that f : (U, g l l l ) + (U', g') is a local b-boundary extension

about y. Then f o y : 1 0 , ~ ) 4 U' is extendible, and f o y ( t ) converges to some

p E U' as t --+ a. Let W' be an open neighborhood of f j in U' with compact

closure in U'. Choose to E [O,a) such that f o y ( t ) E W' for all to 5 t < a.

Set Vl = f-'(W'), and let V be the component of Vl in U which contains

the noncompact set y I [to,a). Since U is open in M, the set V is a connected

open set in M with noncompact closure in M . Also f ( V ) has compact closure

in U' since f ( V ) C W' Thus f Jv . (V, glv) -+ (U', g ' ) 1s a local extension of

(M,g) . 0


FIGURE 6.6. Let U be a connected open subset of M having non-

compact closure in M . A local extension is defined to be an isom-

etry f : (U, glu) -+ (U', g') such that f (U) has compact closure in

U'. Minkowski space-time shows that even real analytic b-complete

space-times may admit analytic local extensions (cf. Example 6.21).

Thus a space-time may admit local extensions but not admit local

b-boundary extensions.

We now show that both types of local inextendibility imply global inex-

tendibility.

Proposition 6.20. If the Lorentzian manifold (M, g) has no local exten-

sions of either of the two types in Definition 6.1 7, then (M,g) is inextendible.

Proof. Suppose (M, g) has an extension F : (M, g) + (M1,g'). Let p E

B(F(M)), and choose a geodesic a : [O, I] --t (M1,g') with o(0) E F(M)

and a(1) = p. Since F ( M ) is open in M' and 5 4 F(M) , there exists some

to E (0, I] such that u(t) E F (M) for all 0 < t < to but a(to) 4 F(M). Then

the curve 7 = F-' 0 a][o,to) : [0, to) -+ M is b-incomplete, inextendible to


t = to in M , and has noncompact closure in M. Taking U = M , U' = MI, and

f = F in (1) of Definition 6.17, it follows that (M,g) has a local b-boundary

extension about y. Taking W to be any open subset about p with compact

closure in M', U' = M', and U to be the component of F- ' (W) containing y,

we obtain a local extension Flu : (U, glU) -t (M', 9'). 0

The next example shows that Minkowski space-time has local extensions.

Since Minkowski space-time is b-complete, this example shows that even

though b-completeness is an obstruction to global extensions, it is not an

obstruction to local extensions (cf. Proposition 6.16). This example is unusual

in that it does not correspond to a local extension of M over a point of either

the b-boundary a b h l or the causal boundary d,M. I t is a local extension of a

set which extends to i0 (cf. Figure 5.4).

Example 6.21. Let ( M = Rn,g) be n-dimensional Minkowski space-time

and let M' = W x Tn-l , where Tn-' = {(82,83,. . . ,On) : 0 5 8, < 1) is the

(n - 1)-dimensional torus (using the usual identifications). We may define a

Lorentzian metric g' for M' by g' = ( d ~ ' ) ~ = -dt2 + d822 $ . . . + don2. Then

( M , g) is the universal Lorentzian covering space of (M ' , g') with covering map

f : (M, g) -+ (M', g') given by

f (xl, . . . , xn) = (XI, x2(mod 1), xa(mod I) , . . . , xn(mod 1)).

Fix ,B > 0 and consider the curve y : [I, m) 4 M given by

Then f o y : [I, ca) -+ M' is a spiral which is asymptotic to the circle t = B3 =

. . = 8, = 0 in M'. Let U be an open tubular neighborhood about y in M

such that U is contained in some open set {(XI, . . . , x,) E Rn : 0 < XI < a ) for

some fixed a > 1 and such that f 1, : U -t M' is a homeomorphism onto its

image (cf. Figure 6.7). Intuitively, the set U must be chosen to be thinner as

s -+ c=a in order to satisfy the requirement xl > 0 for (XI, . . . , x,) E U. While

U does not have compact closure in Minkowski space-time, f ( U ) does have

compact closure in M ' since f (U) is contained in the compact set [O, a] x Tn-l.


FIGURE 6.7. Minkowski space-time has analytic local extensions.

Let M = Wn be given the usual Minkowskian metric g, and let Tn-I

be the (n - 1)-dimensional torus with the usual positive definite

flat metric h. Let M' = W x Tn-' be given the Lorentzian product

metric g' = -dt2 8 h. Then (M,g) is the universal covering space

of (MI, g'), and the quotient map f : M -+ M' is locally isometric.

Choose U to be an open set in M about y(s) = (s-P, s, . . . , 0),

7 : [I, m) -+ M, such that f ] U is one-to-one and f ( U ) has compact

closure in Mi. Then f 1~ : (U,gu) -+ (M1,g') is an analytic local

extension of Minkowski space, but this extension is not across points

of dcM and not across points of dbM.

6.6 SINGULARITIES 225

Thus f : (U, glU) --, (M', g') is an analytic local extension of Minkowski space-

time. Notice that if y~ : [0, a) --+ U is any curve with noncompact closure in

M , then f o yl cannot be extended to t = a in MI.

Let d M denote an ideal boundary of M (i.e., d M represents either dbM

or d,M). A point q E d M is said to be a regular boundary point of M if

there exists a global extension (MI, g') of (M, g) such that q may be naturally

identified with a point of MI. A regular boundary point may thus be regarded

as being a removable singularity of M.

Let y : 10, a) -+ M be an inextendible curve such that y(a) corresponds

to an ideal point of M. The curve y is said to define a curvature singularity

[cf. Ellis and Schmidt (1977, p. 916)] if some component of Ra6cd;el,,, , ,ek is not

C0 on [0, a] when measured in a parallelly propagated orthonormal basis along

y. A curvature singularity is an obstruction to a local b-boundary extension

about y because if there is a local b-boundary extension about y, then the

curvature tensor and all of its derivatives measured in a parallelly propagated

orthonormal basis must be continuous and hence converge to well-defined limits

as t -+ a-. A related but somewhat different notion is that of strung curvature

singularity which may be defined using expansion 8 along null geodesics. The

notion of strong curvature singularity has been useful in connection with cosmic

censorship [cf. Kr6lak (1992), Kr6lak and Rudnicki (1993)l.

A b-boundary point q E dbM which is neither a regular boundary point

nor a curvature singularity is called a quasi-regular singularity. Clarke (1973,

p. 208) has proven that if y : [0, a) -+ M is an inextendible b-complete

curve which correspo,lds to a quasi-regular singularity, then there is a local

b-boundary extension about y. This shows that curvature singularities are

the only real obstructions to local b-boundary extensions.

In general, it can be quite difficult to decide if a given space-time has local

extensions of some type. However, for analytical local b-boundary extensions

of analytic space-times, the situation is somewhat simpler (cf. Theorem 6.23).


For the proof of Theorem 6.23, it is useful to prove the following proposi-

tion about real analytic space-times and local isometries. Recall that a local

isometry F : M -+ M' is a map such that for each p E M , there exists an open

neighborhood U(p) of p on which F is an isometry. Thus local isometries are

local diffeomorphisms but need not be globally one-to-one.

Proposition 6.22. Let (M, g) and (Ml,gl) be real analytic space-times

of the same dimension and suppose that F : M -+ MI is a real analytic map.

If M contains an open set U such that Flu : U + Ml is an isometry, then F

is a local isometry.

Proof. Let W = {m E M : F,v # 0 for all v # 0 in T,M), which is an

open subset of M by the inverse function theorem. Since Flu is an isometry,

U is contained in W. Fix any p E U, and let V be the path connected

component of W containing p. We will establish the proposition by showing

first that Ffv is a local isometry and second that V = M.

Let q be any point of V. Choose a curve y : [0,1] -+ V with $0) = p and

$1) = q. By the usual compactness arguments, we may cover y[O, I] with a

finite chain of coordinate charts (Ul, (U2, +2), . . . , (Uk, &) such that each

U, is simply connected, Flu, : U, -+ MI is an analytic diffeomorphism, p E

Ul C u n v , q E UI;, and U,nU,+l # 0 for each i with 1 I i I k-1. Since UI C UnV, we have g = (Flul)*gl on Ul. Thus g = (FIUl)*gl on UlnU2. Since Uln

U2 is an open subset of U2 and F is a real analytic diffeomorphism of U2 onto its

image, it follows that g = ( FIU2)*gl on U2. Continuing inductively, we obtain

g = (FIUk)*gl on Uk, whence F is an isometry in the open neighborhood Uk

of q. Thus FIv : V -+ MI is a local isometry.

It remains to show that V = M. Suppose V # M. Choose any point

rl E M - V. Let yl : [O, 11 -+ M be a smooth curve with y(0) = p and

y(1) = rl. There is a smallest to E [O, 1) such that T = y(to) E M - V. Then

F restricted to the neighborhood V of yl{O, to) is a local isometry. It suffices

to show that r E V to obtain the desired contradiction. Since r E M - V,

there exists a tangent vector x # 0 in T,M with F,x = 0. Let X be the

unique parallel field along 7 with X(to) = x. Then F, o X is a parallel field

along F o yl : [O, to) -+ MI since F is a local isometry in a neighborhood of


71 : [O,to) -+ M. But since F is smooth, F,x = F,X(to) = lim,,5 F,X(t).

Because F, o X is a parallel vector field for all t with 0 < t < to, it follows

that lim,-,o F,X(t) # 0. Hence Fax # 0. Thus F is nonsingular a t the point

r, whence r E V in contradiction. 0

We are now ready to turn to the proof of Theorem 6.23 on local b-boundary

extensions of real analytic space-times.

Theorem 6.23. Suppose (M, g) is an analytic space-time with no impris-

oned nonspacelike curves, which has an analytic local b-boundary extension

about y : [O,a) + M. Then there are timelike, null, and spacelike geodesics

of finite &ne parameter which are inextendible in one direction and which do

not correspond to curvature singularities.

The proof of Theorem 6.23 will involve two lemmas.

Lemma 6.24. Suppose (M,g) is an analytic space-time with no impris-

oned nonspacelike curves, which has an analytic local b-boundary extension

about y : [O, a ) -+ M. Then (M, g) has an incomplete geodesic.

Proof. Let f : (U, glU) -+ (U1,g') be an analytic extension about y. We

may assume U contains the image of y. Also, f o y is extendible in U'. Thus

f o y(t) -+ p E U' as t -+ a-. Let W' be a neighborhood of p such that

W' is a convex normal neighborhood of each of its points. Then exp;l :

W' -+ T,U' is a diffeomorphism for each fixed x E TV'. Assume to is chosen

with f 0 y(t) E W' for all to < t < a. Set q = y(t0) and T = f(q). Then

H = exp, of;' o exP;' : W' -+ M is analytic and is a t least defined near r .

The map H takes geodesics starting a t r to geodesics starting a t q, and H

preserves lengths along these geodesics. In fact, H agrees with f-' near r .

The map H need not be one-to-one since expq is not necessarily one-to-one.

Because the domain of exp, is a union of line segments starting at the origin

of T,M, the domain V' of H must be some subset of W' which is a union of

geodesic segments starting a t r. Hence the set V' fails to be all of W' only

when expq : T,M -t M is defined on a proper subset of TqM which does not

include all of the image f;' o exp;l(W1). Thus if we show V' # W', there is

some incomplete, inextendible geodesic starting at q. But the analytic maps

6.6 SINGULARITIES 228 6 COMPLETENESS AND EXTENDIBILITY 229

H and f -' must agree on the component off (U) n V' which contains r. This

implies V' # W'. Otherwise, H and f -' would agree on f (U) n W' and hence

on a neighborhood of f o ?[to, a). This yields H o f o y = y for to 5 t < a and

implies y is extendible in M across the point H(p), in contradiction.

We will continue with the same notation in the next lemma. f - Lemma 6.25. The map H : V' + M is a local isometry.

Proof. The space-time (M, g) is analytic, (U', g') is analytic, and H is an-

alytic. Furthermore, H agrees with the isometry f -l near r , and H is defined

on an arcwise connected set V'. Thus Proposition 6.22 implies H is a local

isometry. O

We are now ready to complete the

FIGURE 6.8. In the proof of Theorem 6.23, the map f : (U, glu) -+ Proof of Theorem 6.23. There are three cases to consider corresponding

(U', g') is an isometry which is an analytic local b-boundary exten- to incomplete timelike, null, and spacelike geodesics. We only give the proof

sion about y. The point p E U' is the endpoint of f o y in U'. In for the timelike case. Let U, U', f , etc., be as in Lemmas 6.24 and 6.25.

this figure, f (9) = r, and all points of f o y between r. and p are in Assume without loss of generality that there is some point x E W' such that

the chronological future of x. in a chronological ordering on W', we have x << p and x << f o y(t) for all

to 5 t < a (cf. Figure 6.8).

If x 4 V', let a: be the geodesic segment in W' from r to x. Then H takes

an V' to an inextendible, incomplete, timelike geodesic starting at q. Lemma

6.25 implies that this geodesic does not correspond to a curvature singularity. Hence the analyticity in the hypothesis of Theorem 6.23 cannot be replaced

If x E V', let y = H(x) and define H' = exp, OH,= o exp;' : W' + M. by a Cw assumption.

The map H' is defined on some subset V" of W'. It is a local isometry for the Corollary 6.27. Let (M, g) be an analytic space-time with no imprisoned

same reasons that H is a local isometry, and H' agrees with both H and f -' nonspacelike curves such that each timelike geodesic y : [0, a) --+ M which is

near r. The set V" cannot contain all of f o y for y on to 5 t < a since this inextendible to t = a is either complete (i.e., a = CQ) in the indicated direction

would yield an endpoint H1(p) of y in M. Using x << f o y(t) for to 5 t < a, or else corresponds to a curvature singularity. Then (M, g) has no analytic

we conclude that there is an inextendible incomplete timelike geodesic starting local b-boundary extensions.

at y in M which does not correspond to a curvature singularity. 13

Extensive work has been done on classifying singularities by various differ-

Remark 6.26. There are examples of Cw space-times which are both ent methodologies. Indeed, the monograph by Clarke (1993) may be consulted

geodesically complete and locally b-extendible [cf. Beem (1976c, p. 506)]. to obtain much more detailed information than that which is provided in this


chapter. Our treatment here will be limited to a discussion of the "abstract

boundary" &(M) and its application to the classification of singularities as

given in Scott and Szekeres (1994) and Fama and Scott (1994). (In Clarke

(1993), a somewhat different "A-boundary" obtained by taking the closures of

an atlas of normal coordinate charts is discussed.) To form the a-boundary,

equivalence classes of boundary points of a given smooth manifold under all

possible open embeddings are considered. The a-boundary thus has no depen-

dence on any choice of affine connection or semi-Riemannian metric. When

the manifold is further endowed with an extra structure, like an affine con-

nection or a semi-Riemannian metric, then abstract boundary points may be

classified as regular boundary points, points a t infinity, unapproachable points,

or singularities, with the classification having some dependence upon a par-

ticular curve family C selected, which is associated to the added structure for

M . The abstract boundary also has the feature that if a closed region is re-

moved from a singularity-free semi-Riemannian manifold, then the resulting

semi-Ftiemannian manifold is still singularity-free since only regular boundary

points are introduced by the excision of the closed set.

We will begin by discussing the construction of the a-boundary for a given

smooth manifold M of dimension n. Since no metric is yet involved, we will

adopt the terminology of Scott and Szekeres (1994) and define an enveloped

manifold (MI MI, f l ) to be a pair of (connected) smooth manifolds M , MI of

the same dimension together with a smooth embedding fl : M + MI. Since

both manifolds have the same dimension, f l (M) is an open submanifold of

MI. (A very special type of envelopment would then be provided by a global

extension of a given space-time [cf. Definition 6.151.)

Definition 6.28. (Boundary Point, Boundary Set) A boundary point p

of an enveloped manifold (M, MI, f 1) is a point p f MI - fl (M) such that

every open neighborhood U of p in Ml has non-empty intersection with f l(M),

i.e., p belongs to the topological boundary d(fl(M)) = f l (M) - f l(M). A

boundary set B contained in MI - f l (M) is any set of boundary points for the

enveloped manifold (M, MI, fl). We will use the notation (M, MI, f l , B ) for

an enveloped manifold with distinguished boundary set B.


The set of enveloped manifolds and resulting boundary sets is much too

large an object to use to form a useful boundary. Thus the following relation

is introduced between enveloped manifolds and boundary sets.

Definition 6.29. (Covering Boundary Sets) Let (M, MI , f l , B1) and (M, M2, f2, B2) be two envelopments for M with boundary sets B1 and B2,

respectively. Then the boundary set Bl is said to cover the boundary set B2

if for every open neighborhood Ul of B1 in MI, there exists an open neighbor-

hood U2 of Bz in M2 such that

(6.2) fl (u2 n f 2 ( ~ ) ) c u,.

This requirement may be conveniently checked in terms of sequences as

follows.

Proposition 6.30. The boundary set B1 covers the boundary set B2 iff

for every sequence {pk) in hf such that the sequence { f2(pk)) has a limit point

in B2, the sequence (fl (pk)) is required to have a limit point in B1.

Since the requirement (6.2) is to some degree only a topological restriction,

examples may be given of different envelopments of M = Rn - ((0)) for which

one envelopment produces the boundary set B1 = ((0)) consisting of precisely

one point, but a second envelopment produces the boundary set B2 = Sn-l.

and even more remarkably, B1 covers B2 and B2 covers B1. Thus a single

point which is the boundary set of a given envelopment may cover znfinzte sets

of points which are contained in a boundary set for a second envelopment.

Definition 6.31. (Equivalent Boundary Sets) Let (M, M I , f l , B1) and

(M, M2, f2, B2) be two enveloped manifolds with boundary sets B1 and B2, respectively. Then B1 is said to be equivalent to B2 iff B1 covers B2 and Bz

covers B1.

Denote this equivalence relation on the class of boundary sets for M by

B1 - B2. An abstract boundary set [B] is then an equivalence class of bound-

ary sets. In Fama and Scott (1994), topological properties of boundary set

equivalence are studied. It is noted that if a boundary set B of a given envel-

opment is compact, then all boundary sets B' of all other envelopments which


are equivalent to B must also be compact. In particular, all boundary sets

B equivalent to a single boundary point [ p ] are compact. However, closed-

ness, connectedness, and simple connectedness of boundary sets are unfortu-

nately not invariant under boundary set equivalence. Fama and Scott (1994)

give a detailed discussion of "topological neighborhood properties," including

the connected neighborhood property and the simply connected neighborhood

property, which are preserved by boundary set equivalence.

With all of these preliminaries settled, we may now define the a-boundary

as in Scott and Szekeres (1994).

Definition 6.32. (The a-boundary aa(M)) If p is a boundary point of an

enveloped manifold (M, MI, f l) , then the equivalence class [p ] of the boundary

set {p) under the equivalence relation - of Definition 6.31 is called an abstract

boundary point of M. The set aa(M) of all such abstract boundary points for

all envelopments of the given manifold M is called the abstract boundary or

the a-boundary.

More symbolically, one has that

&(M) = ( I p ] : p is in f l (M) - f l ( M ) for some envelopment (M, Mi , fl)).

In the beginning of this section, the concept of regular boundary point was

defined for space-times in terms of global extensions. A similar definition is

given in Scott and Szekeres (1994) for general semi-Riemannian manifolds.

Now let (M, g) be a semi-Riemannian manifold with a metric tensor g of class

Ck. Let (M,g, MI, f l ) be an envelopment of M . The induced metric tensor

(fcl)* g on f l (M) will be denoted by g when there is no risk of ambiguity.

Definition 6.33. (C1-Extension of (M,g)) A Cl-extension (1 5 1 5 k)

of a Ck semi-=emannian manifold (M, g) is an envelopment of (M, g) by a C1

semi-Remannian manifold (Ml,gl), fi : M -+ M I , such that

This will be denoted by (M, g, MI, gl, fi).

With this definition in hand, the concept of a C1-regular boundary point

may be formulated independent of any choice of curve family for (M, g).


Definition 6.34. (C1-Regular Boundary Point) A boundary point p

of an envelopment (M, g, MI, gl, f 1) is C1-regular for g if there exists a C'

semi-Riemannian manifold (M2, gz) such that f l (M) U {p) C Mz C MI and

(M, g, Mz,gz, f i ) is a CLextension of (M, g).

Further analysis of the boundary points as "singular boundary points" or

"points a t infinity" is possible only if a certain family of curves C with the

bounded parameter property has been specified, and indeed it is natural that

this classification should depend on the choice of the particular curve family

selected. Thus we assume that the smooth manifold M is furnished with a

family C of parametrized curves satisfying the bounded parameter property.

Here a parametrized curve will be taken to mean a C' map y : [a, b) -+ M

with a < b 5 +m and with yl(t) # 0 for all t in [a, b).

Definition 6.35. (Bounded Parameter Property) A family C of param-

etrized curves in M is said to satisfy the bounded parameter property if the

members of C satisfy the following:

(1) Through any point p of M passes a t least one curve y of the family C;

(2) If y is a curve in C, then so is every subcurve of y; and

(3) For any pair of curves yl and 7 2 in C which are obtained from each

other by a monotone increasing C1 change of parameter, either the

parameter on both curves is bounded, or it is unbounded on both

curves.

Examples of suitable families of curves C in our context include: (1) the

family of all geodesics with affine parameter in a manifold M with affine

connection; (2) all C' curves parametrized by "generalized affine parameter"

(cf. Definition 6.8) in an affine manifold; and (3) all future timelike geodesics

parametrized by arc length in a space-time.

As an alternative to Proposition 6.30, boundary sets may be studied using

parametrized curves.

Definition 6.36. (Limit Point, Endpoint of Curve) Let y : [a, b) -+ M

be a parametrized curve. Then

(1) A point p in M is a limit point of the curve y : [a, b) 4 M if there


exists an increasing sequence ti -4 b such that $ti) -4 p.

(2) A point p in M is an endpoint of the curve y if y(t) -+ p as t + b-

(For Hausdorff manifolds, curve endpoints are unique.)

Definition 6.37. (Approaching a Boundary Set) Let y : [a, b) - M be

a parametrized curve and (M, MI, fl , B ) a boundary set in an envelopment of

M .

(1) The parametrized curve y : [a, b) -t M approaches the boundary set B

if the curve f l o y has a limit point lying in B.

(2) The parametrized curve y : [a, b) -+ M has its endpoint in B if the

curve f o y : [a, b) -+ MI has its endpoint in B.

The analogue of Proposition 6.30 above in this setting is then

Proposition 6.38. If a boundary set B1 covers a boundary set B2, then

every curve y : [a, b) 4 M which approaches B2 also approaches B1.

Now we restrict our attention to a semi-Riemannian manifold (M,g) for

which a family C of curves for M with the bounded parameter property has

been selected, and we use the notation (M, g, C) to denote this selection.

Definition 6.39. (Approachable Boundary Point, C-Boundary Point) If

(M,AJ1, f l ) is an envelopment of (M,g,C), then a boundary point p of this

envelopment is approachable, or a C-boundary point, if p is a limit point of

some curve 7 : [a, b) -+ M of the family C. Boundary points which are not

C-boundary points will be called unapproachable.

As a result of Proposition 6.38, this definition passes to the a-boundary:

Definition 6.40. (Approachable a-Boundary Point, Abstract C-Boundary

Point) An abstract boundary point [p] is an abstract C-boundary point, or

approachable, if p is a C-boundary point. Similarly, an abstract boundary point

[ p ] is unapproachable if p is not a C-boundary point.

With the family C specified, the points a t infinity for (M, g, C) may now be

detected by determining whether they may be reached along any curve in C

a t a finite value of the given parameter.


Definition 6.41. (C1-Point at Infinity for (M, g, C)) Given (M, g, C), a

boundary point p of an envelopment (M, g, C, MI , f l ) is said to be a C1-point

at infinity for C if

(1) p is not a C1-regular boundary point (recall Definition 6.34),

(2) p is a C-boundary point, and

(3) no curve in C approaches p with bounded parameter.

Since the family C is assumed to satisfy the bounded parameter property,

the concept of a point a t infinity is Independent of the choice of parametrization

for the curves from C which approach p. Condition (3) says more explicitly

that for no interval [a, b) with b finite is there a curve y : [a, b) -, (M,g) in

the family C and an increasing sequence of real numbers t, -, b such that

A tricky aspect of a point p at infinity for (M,g,C) is that it is possible

that p is covered by a boundary set B of another embedding consisting only of

regular or unapproachable boundary points. In this case, Scott and Szekeres

(1994, p. 238) term the point p at infinity for (M, g,C) a removable point at

infinity. If no such covering by a boundary set of another embedding exists,

then p is termed an essential point at infinity. It may be shown that the

concept of being an essential point a t infinity passes to the abstract boundary,

i.e., these points cannot be transformed away by a change of coordinates. Thus

an abstract boundary point [p ] is termed an abstract point at infinity if it has

a representative in some envelopment which is an essential point at infinity

for that envelopment. Essential points at infinity may cover regular boundary

points of another embedding. Hence, the following final dichotomy is used for

essential points a t infinity: an essential boundary point a t infinity which covers

a regular boundary point is termed a mixed point at infinity. Otherwise, p is

termed a pure point at infinity.

So far the two categories of regular points and points at infinity have been

defined. The final category to be treated is that of singular boundary points.

Here a more subtle viewpoint is taken than that of Definition 6.3 above in

which "geodesic singularity" is defined. According to this previous definition)


if one takes, for example, two points p and q in Minkowski space with p << q

and puts M = 14(p) n If (q) with the induced metric from the inclusion of M

in Minkowski space, then M is a singular space-time. Indeed, every geodesic

of (M,g) is incomplete. Yet the given space-time has a global embedding

in Minkowski space such that after this enlargement, every geodesic becomes

complete. Thus this example should perhaps not really be regarded as a sin-

gular space-time, Definition 6.3 notwithstanding.

Definition 6.42. ((7'-Singular Point) A boundary point p of an envel-

opment (M,g,C, MI, f l ) is said to be C1-singular if

(1) p is not a C1-regular boundary point,

(2) p is a C-boundary point, and

(3) there exists a curve in the family C which approaches p with bounded

parameter.

Thus a singular boundary point for the envelopment is a C-boundary point

which is not C1-regular and is not a point a t infinity. Again, in this case,

a finer subclassification is made. If p is covered by a non-singular boundary

set of a second envelopment, then p is termed a removable singularity. If

not, then p is an essential singularity, and a further classification is made as

follows: if p covers some regular boundary points or points a t infinity of another

embedding, then p is commonly called a mixed or directional singularity. If

no such covering behavior is exhibited, then p is said to be a pure singularity.

Then an abstract boundary point [ p ] in d,(M) may be termed an abstract

singularity if it has a representative which is an essential singularity.

We conclude this section with considerations akin to the more simple con-

cepts of Section 6.2 and "geodesic singularity" of Definition 6.3. First, the

notion of geodesic completeness for a space-time may be extended to that of

C-completeness for the manifold M with distinguished curve family C.

Definition 6.43. (C-Completeness) Given a manifold M with a curve

family C satisfying the bounded parameter property, M is said to be C-complete

if every curve y : [a, b) -+ M in C with bounded parameter, i.e., b < +co, has

an endpoint in M.


Definition 6.44. (C1-Singularity) A semi-Ftiemannian manifold (M, g)

with a distinguished class C of curves satisfying the bounded parameter p r o p

erty has a C'-singularity if there exists an envelopment of M having an es-

sential C1-singularity p, i.e., p is a C1-singular point for some envelopment

of (M,g,C) which is not covered by a CL-non-singular boundary set B of

any other envelopment for (M,g,C). Conversely, (hf,g, C) is said to be C1-

singularity free if it has no C1-singularities, i.e., for every envelopment of M ,

the boundary points are either C"nonsingular (C1-regular boundary points,

C'-points at infinity, or unapproachable boundary points), or C'-removable

singularities.

According to Scott and Szekeres (1994)) any theory of singularities ought

to pass over the following hurdle.

Theorem 6.45. Every compact semi-ftiemannian manifold (M, g, C), with

any family of curves C satisfying the bounded parameter property, is singularity

free.

Proof. A compact manifold M has no non-trivial envelopments (M, MI , f l) ,

for MI is required to be connected yet would contain f l (M) as a compact

open subset. Thus f l (M) would be both open and closed in MI, whence

MI = f l(M). Since M has no envelopments, b,(M) is empty and hence can

contain no singular points.

Theorem 6.46. If (M, g) is a semi-Riemannian manifold which is C-com-

plete for a curve family C satisfying the bounded parameter property, then

(M, g, C) is singularity-free.

Proof. Suppose that (M, g, C) contains a singularity. Then there exists an

envelopment (M, g, C, MI , f i) which has a C-boundary point p E M1 - f l(M).

Hence, there exists a curve y : [a, b) -+ M in C which has p as a limit point

and also b < +m. Now by the assumed C-completeness, y has an endpoint

q in M . But endpoints are unique limit points of curves so that of necessity

p = f l (q), which is impossible since p $ fl(M). O

In Section 6.5, Proposition 6.16, it was noted that various types of complete-

ness such as b-completeness, b.a. completeness, timelike geodesic complete


ness, null geodesic completeness, and spacelike geodesic completeness preclude

global extendibility. In the present context, the analogous statement is the fol-

lowing.

Theorem 6.47. If the semi-Riemannian manifold (M, g) is geodesically

complete, then (M, g) has no regular boundary points.

Complete discussions of all of these boundary points classifications together

with extensive examples, including Taub-NUT space-time as simplified by

Misner (1967) and the Curzon space-time, may be found in Scott and Szekeres

(1994) and Fama and Scott (1994).

CHAPTER 7

STABILITY OF COMPLETENESS AND INCOMPLETENESS

In proving singularity theorems in general relativity, it is important to use

hypotheses that hold not just for the given "background" Lorentzian metric g

for M but in addition for all metrics gl for M sufficiently close to g. Not only

does the imprecision of astronomical measurements mean tha t the Lorentzian

metric of the universe cannot be determined exactly, but also cosmological

assumptions like the spatial homogeneity of the universe hold only approxi-

mately. Nevertheless, if an incompleteness theorem can b e obtained for the

idealized model (M, g) using hypotheses valid for all metrics gl for M in an

open neighborhood of g, then all space-times (M, gl) with gl sufficiently close

to g will also be incomplete. Hence if the model is believed to be sufficiently

accurate, conclusions valid for the model are also valid for t h e actual universe.

Recall that Lor(M) denotes the space of all Lorentzian metrics for a gzven

manzfold M and that Con(M) denotes the quotient space formed by identifyzng

all pozntwise globally conformal metrics gl = Rg2 for M. where R : M +

(0, CQ) is smooth. Let T : Lor(M) + Con(M) denote the natural projection

map which assigns to each Lorentzian metric g for M the set ~ ( g ) = g of

all Lorentzian metrics for M pointwise globally conformal to g. Given g E

Con(M), set C(M, g) = T - ' ( ~ ) C Lor(M). It is customary in general relativity

to say that a curvature or causality condition for a space-time (M,g) is Cr

stable in Lor(M) [respectively, Con(M)], if the validity of the condition for

(M, g) implies the validity of the condition for all gl in a Cr-open neighborhood

of g in Lor(M) [respectively, Con(M)]. More generally, a stable condztion for

a set of metrics zs one which holds on an open subset of such metrics.

After the singularity theorems described in Chapter 8 of Hawking and Ellis

(1973) were obtained, it was of interest to study the Cr stability of conditions


ness, null geodesic completeness, and spacelike geodesic completeness preclude

global extendibility. In the present context, the analogous statement is the fol-

lowing.

Theorem 6.47. If the semi-Riemannian manifold (M, g) is geodesically

complete, then (M, g) has no regular boundary points.

Complete discussions of all of these boundary points classifications together

with extensive examples, including Taub-NUT space-time as simplified by

Misner (1967) and the Curzon space-time, may be found in Scott and Szekeres

(1994) and Fama and Scott (1994).

CHAPTER 7

STABILITY OF COMPLETENESS AND INCOMPLETENESS

In proving singularity theorems in general relativity, it is important to use

hypotheses that hold not just for the given "background" Lorentzian metric g

for M but in addition for all metrics gl for M sufficiently close to g. Not only

does the imprecision of astronomical measurements mean that the Lorentzian

metric of the universe cannot be determined exactly, but also cosmological

assumptions like the spatial homogeneity of the universe hold only approxi-

mately. Nevertheless, if an incompleteness theorem can be obtained for the

idealized model (M, g) using hypotheses valid for all metrics gl for M in an

open neighborhood of g, then all space-times (M, gl) with gl sufficiently close

to g will also be incomplete. Hence if the model is believed to be sufficiently

accurate, conclusions valid for the model are also valid for the actual universe.

Recall that Lor(M) denotes the space of all Lorentzian metrics for a given

manifold M and that Con(M) denotes the quotient space formed by identifying

all pointwise globally conformal metrics gl = Rg2 for M, where R : M +

(0,w) is smooth. Let T : Lor(M) + Con(M) denote the natural projection

map which assigns to each Lorentzian metric g for M the set ~ ( g ) = 6 of

all Lorentzian metrics for M pointwise globally conformal to g. Given g E

Con(M), set C(M, g) = T - ' ( ~ ) C Lor(M). It is customary in general relativity

to say that a curvature or causality condition for a space-time (M, g) is Cr

stable in Lor(M) [respectively, Con(M)], if the validity of the condition for

(M, g) implies the validity of the condition for all gl in a Cr-open neighborhood

of g in Lor(M) [respectively, Con(M)]. More generally, a stable condition for

a set of metrics is one which holds on an open subset of such metrics.

After the singularity theorems described in Chapter 8 of Hawking and Ellis

(1973) were obtained, it was of interest to study the Cr stability of conditions

240 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS

such as the existence of closed trapped surfaces, positive nonspacelike Rcci

curvature, and geodesic completeness, which played such a key role in these

singularity theorems. Geroch (1970a) established the stability of global hy-

perbolicity in the interval topology on Con(M). Then Lerner (1973) made a

thorough study of the stability in Lor(M) and Con(M) of causality and cur-

vature conditions useful in general relativity. In particular, Lerner noted that

the interval and quotient topologies for Con(M) coincide. Hence Geroch's sta-

bility result for global hyperbolicity holds for Con(M) in the quotient topology

and thus automatically holds in Lor(M). In Section 7.1 we define the fine Cr

topologies and the interval topology for Con(M). We then review stability

properties of Lor(M) and Con(M) which were established by Geroch (1970a)

and Lerner (1973). We give examples of Williams (1984) to show that both ge-

odesic completeness and geodesic incompleteness may fail to be stable. These

two properties are C0 stable for definite spaces, but for all signatures (s,r)

with s 2 1 and r > 1 one may construct examples for which these properties

are unstable.

In Section 7.2, using the "Euclidean norm"

induced on ( T M J U , E ) by a coordinate chart (U, x) for M and standard esti-

mates from the theory of systems of ordinary differential equations in Rn, we

obtain estimates for the behavior of geodesics in (U, x) under C1 metric pertur-

bations. In Section 7.3 we apply these estimates to coordinate charts adapted

to the product structure M = (a, b) x f H of a Robertson-Walker space-time

(cf. Definition 5.10) to study the stability of geodesic incompleteness for such

space-times. We show (Theorem 7.19) that if ((a, b) x j H, g) is a Robertson-

Walker space-time with a > -m, then there is a fine C0 neighborhood U(g)

of g in Lor(M) such that all timelike geodesics of each space-time (M, gl) are

past incomplete for all gl E U(g). If we assume b < oo as well, we may obtain

(Theorem 7.20) both future and past incompleteness of all timelike geodesics

for all gl E U(g). A similar result (Theorem 7.23) may be established for null

geodesic incompleteness using the C' topology on Lor(M). Combining these

7.1 STABLE PROPERTIES OF Lor(M) AND Con(M) 241

results yields the C1 stabzhty of past nonspacelzke geodeszc zncompleteness for

Robertson- Walker space-tames.

In the last sectlon of this chapter we consider space-times which need not

have any symmetry properties. We show (Theorem 7.30) that nonzmpnson-

ment zs a suficient condztzon for the C' stabzlzty of geodeszc zncompleteness

[cf. Beem (1994)l. Sufficient condit~ons for the stability of geodesic complete-

ness involve pseudoconvexity as well as nonimprisonment. Here a space is

said to have a pseudoconvex class of geodesics if for each compact subset K

there is a larger compact subset H such that any geodesic segment of the

class with endpoints in K lies entirely in H. Nonzmpnsonment and pseu-

doconvexzty of nonspacelzke geodesics taken together are suficzent for the C'

stabilzty of nonspacelzke geodeszc completeness. This result (Theorem 7.35) im-

plies that nonspacelike geodesic completeness is stable for globally hyperbolic

space-times.

At the end of Section 3.6 we discuss the relationship between the choice

of warping function f : (a, b) + (0, oo) and the nonspacelike geodesic incom-

pleteness of a given Lorentzian warped product space-time (a, b) x j H.

7.1 Stable Properties of Lor(M) and Con(M)

An equivalence relation C may be placed on the space Lor(M) of Lorentzian

metrics for M by defining gl,g2 E Lor(M) to be equivalent if there exists

a smooth conformal factor R : M -+ ( 0 , ~ ) such that gl = Rg2. As in

Chapter 1, we will denote the equivalence class of g in Lor(M) by C(M, g). The

quotient space Lor(M)/C of equivalence classes will be denoted by Con(M).

: There is then a natural projection map T : Lor(M) -+ Con(M) given by t 1 ~ ( 9 ) = C(M, g).

The fine C0 topology (cf. Section 3.2) on Lor(M) induces a quotient topology

on Con(M) as usual. A subset A of Con(M) is defined to be open in this

topology if the inverse image ?-'(A) is open in the fine C0 topology on Lor(M).

Con(M) may also be given the interval topology [cf. Geroch (1970a, p. 447)).

Recall from our discussion of stable causality in Section 3.2 that a partial

ordering may be defined on Lor(M) by defining gl < g2 if gl(v, v) 5 0 implies


g2(u, v) < 0 for all v # 0 in TM. It may then be checked that gl , g2 E Lor(M)

satisfy gl < g2 if and only if g; < g; for all g; E C(M,gl) and g$ E C(M, gz).

Thus the partial ordering < for Lor(M) projects to a partial ordering on

Con(M) which will also be denoted by <. A subbasis for the internal topology

on Con(M) is then given by all sets of the form

where gl and g2 are arbitrary Lorentzian metrics for M with gl < g2. The

quotient and interval topologies agree on Con(M) [cf. Lerner (1973, p. 23)).

Thus intuitively, two conformal classes C(M,gl) and C(M,g2) are close in

either of these topologies on Con(M) if and only if at all points p of M the

metrics gl and g2 have light cones which are close in T,M. A property defined on Lor(M) which holds on a Cr-open subset of Lor(M)

is said to be C' stable. Also, a property defined on Lor(M) which is invariant

under the conformal relation C is said to be confomally stable if it holds

for an open set of equivalence classes in the quotient (or interval) topology on

Con(M). The continuity of the projection map r : Lor(M) -+ Con(M) implies

that any conformally stable property defined on Lor(M) is also C0 stable on

Lor(M). Furthermore, since the fine Cr topology is strictly finer than the fine

CS topology on Lor(M) for r > s , any conformally stable property defined on

Lor(M) is also Cr stable for all r 2 0.

Example 7.1. (Stable Causality) Stable causality is conformally stable

and hence also Cr stable for all r 2 0. Indeed, a metric g E Lor(M) may be

defined to be stably causal if the property of causality is C0 stable in Lor(M)

at g.

A second example of a conformally stable property is furnished by a result

of Geroch (1970a, p. 448).

Theorem 7.2. Global hyperbolicity is conformally stable and hence Cr

stable in Lor(M) for all T 2 0.

It may also be shown that if S is a smooth Cauchy surface for (M, g), there

exists a CO neighborhood U of g in Lor(M) such that if gl E U, then S is a

Cauchy hypersurface for (M,gl) [cf. Geroch (1970a, p. 448)].


The Ricci curvature involves the first two partials of the metric tensor but

110 higher derivatives. Using this fact, Lerner (1973) established the following

result.

Proposition 7.3. If (M,g) is a Lorentzian manifold such that g(v, v) 5 0

and v # 0 in T M imply Ric(g)(v, v) > 0, then there is a fine C2 neighborhood

U(g) of g in Lor(M) such that for all gl E U(g), the relations gl(v, v) 5 0 and

v # 0 in T M imply Ric(gl)(v, u) > 0.

It is well known that compact positive definite Riemannian manifolds are

always complete. In contrast, compact space-times need not be complete

[cf. Fierz and Jost (1965)]. Examples of Williams (1984) show that both

geodesic completeness and geodesic incompleteness are, in general, unstable

properties for space-times. In fact, the examples show that both of these

properties may fail to be stable for compact space-times as well as non-compact

space-times. These instabilities are related to questions involving sprays and

the Levi-Civita map [cf. Del Riego and Dodson (1988)j.

An inextendible closed geodesic c is one which repeatedly retraces the same

image. For spacelike and timelike geodesics this implies the geodesic is com-

plete since these geodesics have constant nonzero speed and thus increase in

affine parameter by the same amount for each circuit of the image. However,

there are closed null geodesics which are inextendible and incomplete. This

incompleteness results from the fact that the tangent vector to a null geodesic

may fail to return to itself each time the geodesic traverses one circuit of the

image. For closed null geodesics, the tangent vector may return to a scalar

multiple of itself where the multiple is different from one. In this case, the

closed geodesic fails to be a periodic map and the domain is an open subset

of W which is bounded either above or below. More precisely, we have the

following result.

Lemma 7.4. Let (M,g) be a semi-Riemannian manifold with an inex-

tendible null geodesic P : (a, b) + M satisfying P(0) = P(1) and p ( 0 ) =

XP/(O). If X = 1, then ,D is complete. If 0 < X < 1, then P is incomplete with

a > -m and b = +m. If 1 < A, then ,D is incomplete with a = -co and

b < +m.


Proof. If X = 1, then P[O, l ] is a closed geodesic which is periodic (i.e.,

p(t + 1) = P(t) for all t), and clearly the domain of ,B is all of R.

If 1 < A, then the speed of /3 increases by a factor of X each time around the

image in the positive direction. In particular, if y(s) = P(Xs), then y(Xml) =

P(1) = P(0) = y(0) and y'(0) = Xfl(0) = fl(1). Thus y(t) = P(1 + t), and

the first circuit of y (i.e., 0 5 t 5 X-') corresponds to the second circuit of ,O

(i.e., 1 < t 5 1 + A-I). Thus a countably infinite number of circuits of P in

the positive direction starting with t = 0 increase the &ne parameter of by

It follows that the domain of P has a finite upper bound, and one finds

b = (1 - < +m. Of course, traversing the geodesic P in the negative

direction starting at t = 0 changes the affine parameter by x:=l An = co,

which yields a = -co.

If 0 < X < 1, then similar arguments show a > -m and b = +m.

We will now consider two examples of Williams (1984). These examples

are for metrics of the form dx dy + f (x)dy2 on the torus. Let S1 x S1 =

M = {(x, y) 10 5 x 5 27r, 0 5 y < 27r) with the usual identifications. Let

f : W1 -+ W1 be a periodic function with period 2a. One may then easily

calculate the Christoffel symbols for the metric g = dxdy + f (x)dy2.

Example 7.5. (Instability of Geodesic Completeness) Let g = dxdy + f (x)dy2. If f is identically zero, one has a flat metric g = dxdy on the

torus M which is geodesically complete. In particular, the closed null geo-

desic corresponding to x = 0 is complete. On the other hand, using fn(x) =

( l ln ) sin(x) one may directly calculate that the closed null geodesic x = 0 of

gn = dx dy + fn(x)dy2 is not complete [cf. Williams (1984)l. This incomplete-

ness results from the fact that the tangent vector to this null geodesic of g,

fails to return to itself each time the geodesic traverses the circle corresponding

to x = 0. Each time around the circle, the tangent vector returns to a scalar

multiple of itself where the multiple is different from one. Lemma 7.4 yields

that this null geodesic is incomplete for each g,. Since each Cr neighborhood


of the original flat metric g has metrics g, for large n , it follows that geodesic

completeness is not stable at the Lorentzian metric g.

This last example may be generalized [cf. Beem and Ehrlich (1987, p. 328)].

If (M, g) is a geodesically complete semi-Remannian manifold which contains

a closed null geodesic, then each CT neighborhood of g contains incomplete

metrics. On the other hand, it has recently been shown that any Lorentzian

metric on a compact manifold which has zero curvature, i.e., R = 0, is geodesi-

cally complete [cf. Carrikre (1989), Yurtsever (1992)l. Hence no deformation

of the flat metric g = dxdy on the torus through flat metrics can produce

geodesic incompleteness, as in Example 7.5.

Example 7.6. (Instability of Geodesic Incompleteness) Williams modi-

fied his previous example somewhat to demonstrate the instability of geodesic

incompleteness. Using f (x) = 1 - cosx and xo with 0 < xo < 27r, one may

show that the null geodesic starting at ($0, yo) with dyldx # 0 at xo will

be incomplete for the metric g = dxdy + f (x)dy2. Williams shows this by

solving for x = x(t) and then obtaining an integral formula for y = y(x)

involving integrating the reciprocal of f(x). However, he also finds that

using f,(x) = 1 - cosx + ( l ln ) one obtains geodesically complete metrics

gn = dxdy + fn(x)dy2 which are arbitrarily close to the incomplete metric g.

Examples 7.5 and 7.6 both involve two-dimensional Lorentzian manifolds.

However, the examples may be slightly modified to give examples which show

geodesic completeness and incompleteness are not necessarily stable for any

metric signature (s, r ) with both s and r positive.

If (MI, gl) and (M2,gz) are semi-Riemannian manifolds, the product mani-

fold MI x M2 may be given the usual product metnc g = gl @g2, and with this

metric the geodesics of the product are of the form (yl(t),y2(t)) where each

factor y,(t) is either a geodesic of (Mz,g,) or else is a constant map. Clearly,

the product MI x M2 is geodesically complete if and only if each (M,,gz) is

geodesically complete.

The semi-Euclidean space of signature (s, r) is the product manifold MI x

M2 where Ml = W8 with g1 = C:=l -dxz2 and M2 = Rr with gz = Cr=l dxt2.


By taking the first manifold (Ml,gl) to be the manifold (M,g) of Exam-

ple 7.5 (respectively, Example 7.6) and taking the second manifold (M2,g2)

to be the semi-Euclidean space of signature ( r - 1,s - I ) , one may obtain

an example MI x M2 which shows that geodesic completeness (respectively,

incompleteness) may fail to be stable for signature (s, r ) whenever neither s

nor T is zero.

The situation for positive definite (i.e., s = 0) and negative definite (i.e.,

T = 0) signatures is quite different. For these signatures, both geodesic com-

pleteness and geodesic incompleteness are C0 stable. For example, if (M, g) is

positive definite, then the set U(g) consisting of all positive definite metrics h

on M with 1 h(v,v) < 4 - < - 4 g(v1v)

for all nontrivial vectors v is a Co-open subset of the collection Riem(M) of

all Riemannian metrics on M. Clearly, U(g) contains the metric g. If h is a

fixed metric in U(g), then any given curve has an h-length Lh and a g-length

L, with 1 -L , 5 Lh 5 2Lg. 2

This inequality shows that either g and h are both complete or else both are

incomplete. It follows that geodesic completeness and geodesic incompleteness

are C0 stable for positive definite spaces. The same reasoning applies in the

negative definite case, and consequently one obtains the following result.

Proposition 7.7. Let (M, g) be either a positive definite or negative def-

inite semi-Riemannian manifold. If (M, g) is geodesically complete, then for

each r 2 0 there is a Cr neighborhood U(g) with each gl E U(g) complete. If

(M, g) is geodesically incomplete, then for each r 2 0 there is a Cr neighbor-

hood U(g) with each gl E U(g) incomplete.

We summarize this last result and the examples of Williams in the following

remark.

Remark 7.8. Geodesic completeness and geodesic incompleteness are sta-

ble properties in the Whitney topology for both positive definite and nega-

tive definite semi-Riemannian manifolds. For all other metric signatures (s, T),

7.2 THE C1 TOPOLOGY AND GEODESIC SYSTEMS 247

there are examples to show that both geodesic completeness and geodesic in-

completeness may fail to be stable for the Whitney Ck topologies for k > 0.

In this chapter the manifold M is always fixed. However, one-parameter

families (MA, g ~ ) of manifolds and Lorentzian metrics have been considered in

general relativity [cf. Geroch (1969)].

7.2 The C' Topology a n d Geodesic Systems

If (M, g) is an arbitrary Lorentzian manifold, then metrics in Lor(M) which

are close to g in the fine C1 topology have geodesic systems which are close to

the geodesic system of g. The purpose of this section is to give a more analytic

formulation of this concept, which is needed for our investigation of the C1

stability of null geodesic incompleteness for Robertson-Walker space-times in

Section 7.3.

We begin by recalling a well-known estimate from the theory of ordinary

differential equations Icf. Birkhoff and Rota (1969, p. 155)]. We will al-

ways use /1x1I2 to denote the Euclidean norm [ELl xi2] of the point x =

(51, 2 2 , . . . ,x,) E Wrn.

Proposition 7.9. Suppose that f = ( f l , . . . , f,) and h = (h l , . . . , h,) are

continuous functions defined on a common domain D R x Wm, and suppose

that f satisfies the Lipschitz condition

Ilf (5-1x1 - f(~15)112 I Lllx - 4 1 2

for all (s, x), ( s , f ) E D.

Let x(s) = (xl(s), . . . , xm(s)) and y(s) = (y~(s ) , . . . , ym(s)) be solutions for

0 5 s 5 b of the differential equations

dx - = f (8, x) and - d~ = h(s, y), ds ds

respectively. Then if 11 f (s, x) - h(s, x) ] I 2 5 e for all (s, x) E D with 0 5 s 5 b,

the following inequality holds for all 0 5 s 5 b:

246 7 STABILITY O F COMPLETENESS AND INCOMPLETENESS

By taking the first manifold (M1,gl) to be the manifold (M,g) of Exam-

ple 7.5 (respectively, Example 7.6) and taking the second manifold (Mz,g2)

to be the semi-Euclidean space of signature (T - 1, s - I), one may obtain

an example MI x M2 which shows that geodesic completeness (respectively,

incompleteness) may fail to be stable for signature ( s , ~ ) whenever neither s

nor T is zero.

The situation for positive definite (i.e., s = 0) and negative definite (i.e.,

T = 0) signatures is quite different. For these signatures, both geodesic com-

pleteness and geodesic incompleteness are C0 stable. For example, if (M, g) is

positive definite, then the set U(g) consisting of all positive definite metrics h

on M with

L, with

If (M, g) is an arbitrary Lorentzian manifold, then metrics in Lor(M) which

are close to g in the fine C' topology have geodesic systems which are close to

the geodesic system of g. The purpose of this section is to give a more analytic

formulation of this concept, which is needed for our investigation of the C1

stability of null geodesic incompleteness for Robertson-Walker space-tlmes in

for all nontrivial vectors v is a Co-open subset of the collection Kem(M) of

all Riemannian metrics on M. Clearly, U(g) contains the metric g. If h is We begin by recalling a well-known estimate from the theory of ordinary

fixed metric in U ( g ) , then any given curve has an h-length LI, and a g-lengt fferential equations [cf. Birkhoff and Rota (1969, p. 155)]. We will al-

ays use IlzIlz to denote the Euclidean norm [C:, xZ2] of the point x = 1 -L, 5 Lh 1 2Lg. 2

This inequality shows that either g and h are both complete or else bot roposition 7.9. Suppose that f = ( f l , . . . , f,) and h = (h l , . . . , h,) are

incomplete. It follows that geodesic completeness and geodesic incomple

are CO stable for positive definite

negative definite case, and consequently one obtains the following result Ilf ( s > x) - f (.,z)llz 5 Lllx - z112

Proposition 7.7. Let (M, g)

inite ,cjemi-fi~annian manifold.

& r 2 0 thexe i s a CY neighborhood . , y,(s)) be solutions for

.\M. 3) & *xm*e- -

k,GT,i *edLz?-s'- J -111-

7.2 T H E C' TOPOLOGY AND GEODESIC SYSTEMS 247

there are examples to show that both geodesic completeness and geodesic in-

completeness may fail to be stable for the Whitney C"opologies for k > 0.

In this chapter the manifold M is always fixed. However, one-parameter

families ( M A , gx) of manifolds and Lorentzian metrics have been considered in

general relativity [cf. Geroch (1969)l.

7.2 T h e C' Topology a n d Geodesic Systems


Now let M be a smooth manifold, and let (U, x) be any coordinate chart

for M. We may obtain an associated coordinate chart

z = ( ~ 1 , ~ 2 , . . - ,xn,xn+lr. - . 1 ~ 2 n )

for TMlu as follows. Let d/dxl, . . . , d/dxn be the basis vector fields defined

on U bv the local coordinates x = (xi,. . . ,xn). Given v E TqM for q E n we may write v = zEl ai $1". Then E(v) is defined to be Z(v)

(xl(q), x2(q), . . . , xn(q),al, a2,. . . ,an). These coordinate charts may then be

used to define Euclidean coordinate distances on U and TMIU. Explicitly,

given p, q E U and v, w f TMIU, set

and

respectively. Also if T 2 0 is given, we will use the notation IJgl -g2\lr,v < 6

gl,g2 E Lor(M) and a positive constant 6 > 0 to mean that calculating wi

the local coordinates (U,x), all the corresponding entries of the two metri

tensors and all their corresponding partial derivatives up to order r are 6clos

at each point of U.

We will denote the Christoffel symbols of the second kind for gl,g2 E

Lor(M) by rJk(gl) and rJk(g2): respectively. Then for a = 1,2, the geodesic

equations in the coordinate chart (U, x) for (M, g,) are given by

for 1 5 i, j, k 5 n, where we employ the Einstein summation convention

throughout this chapter.

7.2 THE C1 TOPOLOGY AND GEODESIC SYSTEMS 249

We will use the notation expq[g,] for the exponential map at q E (M,g,),

a = 1,2. If v E TMIU, then s --+ expq[ga](sv) is the solution of (7.1) in U

with initial conditions (q,v) for (M,g,). In order to apply Proposition 7.9

to these exponential maps, we identify TMlu with a subset of W2" using the

coordinate chart (TMIu , Z) and define f (s, X ) = f (X) and h(s, X ) = h(X)

fi+n(X) = -rjk(gl)~j+n~k+n,

and = -rjk(92)xj+nxk+n

for 15 i , j , k 5 n a n d X = (xl ,..., x2,) E WZn.

Lemma 7.10. Let (U, x) be a local coordinate chart for the n-manifold

M. Let (p, v) E TMIu, and assume that cl(s) = exp,[gl](sv) lies in U for all

0 < s 5 b. Given E > 0, there exists a constant 6 > 0 such that JJv - wl12 < 6

and llgl-g2111,u < 6implythatca(s) = expq[g2](sw) liesinU ford10 5 s < b,

and moreover,

Ibj 0 CI)'(S) - (53 O CZ)'(S)I < E

for all 15 j 5 n a n d 0 5 s 5 b.

Proof. Let f (X) and h(X) be defined as above. Then X(s) = (cl (s), cll(s))

and Y(s) = (cz(s), czl(s)) are solutions to the differential equations dX/ds =

f(Xf and dY/ds = h(Y), respectively. Choose Do to be an open set in TMlu

about the image of the curve X(s) such that DO is compact. Then there exists

a constant L such that f satisfies a Lipschitz condition 11 f (X) - f ( x ) ] / 2 5

L1IX - Xllz on Do.

We may make the term jlX(0) - Y(0)IJ2 in the estimate of Proposition

7.9 as small as required by making Ilu - wll2 small. Furthermore, since the

Chriitoffel symbols depend only on the coefficients of the metric tensor and

on their first partial derivatives, we may make 11 f (X) - h(X)ll2 as small as we


wish on Do by requiring that IJgl - g2jll,u be small. Hence for a sufficiently

small 6 > 0, Proposition 7.9 may be applied to guarantee that c2(s) E U for all

0 5 s 5 b and also to yield the estimate IIX(s) - Y(s)l12 < E for all 0 5 s 5 b.

Consequently, Ixi(s) - yi(s)I < E for all 1 5 i 5 2n and 0 5 s < b. In view of

(7.1), this establishes the desired inequalities. 0

The following slightly more technical lemma, which is needed in Section

7.3, follows directly from Lemma 7.10 by using the triangle inequality and the

continuity of the geodesic solution X ( s ) = (cl(s), cl' (s)).

Lemma 7.11. Let (U, x) be a local coordinate chart on the n-manifold

M. Suppose that cl(v) = exp,[gl](sv) lies in U for all 0 < s 5 b. Let

E > 0 and s l with 0 < s l 0

such that if ilv - wllz < 6, 1/91 - 92111,~ < 6, and ]SO - s l J < 6, the geodesic

c2(s) = exp,[g2](sw) lies in U for all 0 5 s < b, and moreover,

and

for all 1 5 j 5 n. Furthermore, if (xl 0 cl)'(sl) # 0, then the constant 6 > 0 may be chosen

such that

7.3 Stability of Geodesic Incompleteness for Robertson-Walker

Space-times

In this section we investigate the stability in the space of Lorentzian rnetrics

of the nonspacelike geodesic incompleteness of Robertson-Walker space-times

M = (a, b) x f H (cf. Definition 5.10). It turns out, however, that the proof of

the C0 stability of timelike geodesic incompleteness uses only the homogeneity

of the Riemannian factor (H, h) and not the isotropy of (H, h). Accordingly,

7.3 GEODESIC INCOMPLETENESS 251

we will formulate the results in the first portion of this section for the larger

class of Lorentzian warped products M = (a, b) x f H with -00 5 a < b 5 t o o

and (H, h) a homogeneous Riemannian manifold. We will let xl = t denote

the usual coordinate on (a , b) throughout this section.

In order to study the geodesic incompleteness of such space-times under

metric perturbations, it is helpful to use coordinates adapted to the product

structure. Fix p = ( t l ,h l ) E (a,b) x H. Since the submanifold {tl) x H

of M is spacelike, the Lorentzian metric g for M restricts to a positive defi-

nite inner product on the tangent space to this submanifold at p. Identifying

{tl) x H and H, we may use an orthonormal basis for the tangent space to

i t l ) x H at p to define Riemannian normal coordinates xz, . . . , x, for H in a

neighborhood V of hl. Then (XI, x2,. . . , x,) defines a coordinate system for

M on (a, b) x V. By construction, g has the form diag{-1, + I , . . . , +1) at p

in these coordinates. Because the submanifold {tl) x H is not necessarily to-

tally geodesic iff is nonconstant, these coordinates are not necessarily normal

coordinates. Nonetheless, the coordinates (XI, x2, . . . , x,) are well adapted to

the product structure since the level sets xl(t) = X are just {A) x V.

We will say that coordinates (XI, . . . , x,) constructed as above are adapted

at p E M and call such coordinates adapted coordinates. It will also be useful

to define adapted normal neighborhoods.

Definition 7.12. (Adapted Normal Neighborhood) An arbitrary convex normal neighborhood U of (M,g) with compact closure 77 is said to

be an adapted normal neighborhood if U is covered by adapted coordinates

(xl,x2,. . . ,x,) which are adapted at some point of U such that the following

hold:

(1) At every point of U , the components gij of the metric tensor g ex-

pressed in the given coordinates (XI, xz, . . . , x,) differ from the matrix

diag(-1, +I,. . . , +1) by at most 112.

(2) The metric g satisfies g <U 71, where 71 is the Minkowskian metric

ds2 = -2dx12 +dxz2 + ... + dxn2 for U (see the definition of stably

causal in Section 3.2 for the notation g <u ql).


Thus on the neighborhood U of Definition 7.12 the Lorentzian metric g may

be expressed as

where the functions kiJ : U -+ IR satisfy Ikajl I 1/2 for all 1 5 i, j I n. For use in the sequel, we need to establish the existence o f countable chains

{Uk) o f adapted normal neighborhoods covering future directed, past inex-

tendible, timelike geodesics o f the form c(t) = ( t , yo). Since in Definition a 7.14 and in Lemma 7.15 it is possible that a = -co, we adopt the following

convention throughout this section.

Convention 7.13. Let wo denote any fixed interior point o f the interval

( a , b).

Now we make the following definition.

Definition 7.14. (Admissible Chain) Let M = (a , b) x f H denote a Lor-

entzian warped product with metric g = -dt2 @ fh. Fix any yo E H and let

c : (a , wo] -+ ( M , 3 ) be the future directed, past inextendible, timelike geodesic

given by c( t ) = ( t , yo). A countable covering {Uk)p=l o f c by open sets and

a strictly monotone decreasing sequence {tk)& with tl = wo and tk -+ a+

as k -+ co is said to be an admissible chain for c : (a,wo] -+ ( M , g ) i f the

following two conditions hold:

(1) Each Uk is an adapted normal neighborhood containing c(tk) = (tk, yo)

which is adapted at some point of c.

(2) For each k , every future directed and past inextendible nonspaceliie

curve u ( t ) = ( t , a z ( t ) ) with u ( t k ) = ( t k , yo) remains in Uk for all t with

tk+l I t I t k .

Any future directed nonspacelike curve u in ( M , g ) may be given a par-

ametrization of the form u ( t ) = (t, ul ( t ) ) . Thus condition (2) applies to all

future directed nonspacelike curves issuing from (tk, y o ) W e now show that

admissible chains exist.

Lemma 7.15. Let M = (a,b) x f H with a 2 -m and 3 = -dt2 $ fh be a Lorentzian warped product. For any yo E H , the timelike geodesic


c : (a,wo] -4 ( M , i j ) given by c(t) = (t, yo) has a covering by an admissible

chain.

Proof. W e will say that {Uk), { tk) is an admissible chain for c I (9, wo], 9 > a, i f {Uk), { t k ) satisfy the properties o f Definition 7.14 except that tk -+ Bt

as k + oo instead of t k --+ a+ as k -t co. Set

T = inf{8 E [a, wo] : there is an admissible chain

{Uk), I t k ) for c 1 (el, wo] for all 0' 9).

W e must show that T = a.

By taking an adapted normal neighborhood centered at c(wo), it is easily

seen that T < WO. Suppose that T > a. Let U be any adapted normal neigh-

borhood adapted at the point (T , yo) E M . Choose T > T such that all future

directed nonspacelike curves u ( t ) = (t, u l ( t ) ) originating at (r , yo) lie in U for

all T - E I t I T , where E > 0. There exists an admissible chain {U,), {t,) for - C ( ( T + 7 ) / 2 , W O ] with tm < r for some m. Define U, = = U. Extending - the finite chain {U1,U2,. . . , Urn-1, Urn, Um+l), { t l , t2 , . . . , t m - l l t m l ~ - 6) to

an infinite admissible chain yields the required contradiction. 0

W e now show that the subset o f Uk for which property (2) o f Definition 7.14

holds may be extended from the point ( t k , yo) to a neighborhood I t k ) x Vk(yo)

in { t k ) x H . The notation Ilg - glllo,pk < 6 has been introduced in Section

7.2.

Lemma 7.16. Let {Uk) , { tk) be an admissible chain for the timelike ge-

odesic c ( t ) = (t, yo), c : (a,wo] -+ ( M , g ) . For each k , there is a neigh-

borhood Vk(y0) of yo in H such that any future directed nonspacelike curve

a ( t ) = ( t , a l ( t ) ) with ~ ( t k ) E { tk ) x &(yo) remains in Uk for d l t with

tk+l 5 t 5 t k . Furthermore, Vk(yo) and 6 > 0 may be chosen such that

i f gl E Lor(M) and I(g - glllo,uk < 6, then the following two conditions

are satisfied. If y ( t ) = (t,-yl(t)) is any nonspacelike curve of ( M , g l ) with

~ ( t k ) E { tk ) x V ~ ( Y O ) , then

(1 ) y remains in Uk for tk+l 5 t 5 t k ; and

(2) The gl-length o f yl [ t k+~, tk] is at most &n(tk - tk+l).


Proof. First recall that z : M = (a, b) x H --+ W given by ~ ( t , h ) = t serves

as a global time function for M . In particular, the vector field V z satisfies

g ( V z , V z ) < 0 at all points of M . Define g E Lor(M) by

I t follows that g < $ on M so that U2 = { i 2 E Con(M) : g2 < ~ ( 9 ) ) is an

open neighborhood o f C ( M , g ) in Con(M). Let Ul = r-l(U2). Then Ul is

a Co-open neighborhood of g in Lor(M) such that i f gl E Ul , the projection

map z : M --+ W is a global time function for ( M , gl). Hence the hypersurfaces

{ t ) x H , t E (a , b), remain spacelike in ( M l g l ) . Thus any nonspacelike curve

y of ( M , gl ) , gl E Ul , may be parametrized as y ( t ) = ( t , y l ( t ) ) . Thus the

lemma will apply to any nonspacelike curve of ( M , gl ) originating at any point

o f { t k ) x Vk(yO) provided g1 E Ul is sufficiently close to g on Uk.

Let ( x l , . . . , x,) denote the given adapted coordinate system for the adapted

normal neighborhood Uk. In view of condition (2) o f Definition 7.12, we may

find 61 > 0 such that llg - g1110,~, < 61 implies gl < 72 on Uk, where 772 is

the Lorentzian metric on Uk given in the adapted local coordinates by 772 =

-3 dx12 + dx22 + - . . + dxn2. Secondly, since Co-close metrics have close light

cones, it follows by a compactness argument that there exist a neighborhood

Vk(yo) o f yo in H and a constant 62 > 0 such that i f gl E Lor(M) satisfies

llg-glllo,vk < 62 and y ( t ) = ( t , y l ( t ) ) is any future directed nonspacelike curve

of ( M , g l ) with y(tk) E { tk ) x V ~ ( Y O ) , then y( t ) E Uk for tk+l i t I tk.

I t remains to establish the length estimate (2). Set 6 = min{61,62, 1/21.

Suppose that g' E Lor(M) satisfies jig'-gllo,uk < 6, and let y ( t ) = (t, y l ( t ) ) be

any nonspacelike curve o f ( M , g') with y (tk) E { tk) x Vk (yo) , y : [ t k + l , tk] -+ M .

Let L ( y ) denote the length o f y in ( M , g'). Thus

Fkom Definition 7.12 and the choice of the 6's, we have lgi31 5 (1+ 1/2)+1/2 =

2 and Iyil(t)l 5 & for all 1 2 i, j 5 n. Thus, as required,


Assuming now that the Riemannian factor (H, h ) o f the Lorentzian warped

product is homogeneous, we may extend Lemma 7.16 from Uk to [tk+l, t k ] x H.

W e will use the notation lgl - glo < 6, as defined in Section 3.2.

Lemma 7.17. Let ( M , g ) be a Lorentzian warped product with ( H , h )

homogeneous, and let {Uk), { tk) be an admissible chain for c( t ) = ( t , yo),

c : (a , wo] -+ M. For each k , there is a continuous function bk : [tk+l, tk] x H -+

(0, co) such that i f gl € Lor(M) and lg - gllo < 6k on [&+I, tk] x H , then any

nonspacelike curve y ( t ) = ( t , y l ( t ) ) , y : [tk+l,tk] -' ( M , g l ) , joining any point

of {tk+l) x H to any point o f { t k ) x H , has length at most &n(tk - tk i l ) .

Proof. Fix any k > 0. Let 6 > 0 be the constant given by Lemma 7.16

such that i f gl E Lor(M) satisfies Ilgl - gllo,pk < 6, then any nonspacelike

curve y ( t ) = ( t , y l ( t ) ) in ( M , g l ) with y(tk) E { tk ) x &(yo) remains in Uk for

tk+l I t i t k and has length at most &n (tk - &+I) . Also let ( x l , . . . , x,)

denote the given adapted coordinates for Uk.

W e may find isometries {$,)El in I ( H ) such that i f yi = $i(yo) and

V k ( ~ i ) = $i(Vk(yo)), then the sets {l/k(yi))& together with Vk(y0) form a

locally finite covering of H. Let @, : M -+ M be the isometry given by

@i(t , h) = ( t , +i(h)), and set Ui = @i(Uk) for each i. Then the sets {Gi)

cover [tk+l, t k ] X H and ( X I , x2 0 ah1, . . . , x, 0 @, l ) form adapted local co-

ordinates for Ui for each i. Since everything is constructed with isometries,

the constant 6 > 0 that works in Lemma 7.16 for Uk and c( t ) = ( t , yo) works

equally well for each 6i and @i 0 c, provided that the adapted coordinates

( ~ 1 ~ x 2 0 @ i l l . . . , x , 0 a;') are used for Ui. ~f we let bk : [tk+l, tk] x H -+ M

be any continuous function such that ilgl - gllo < hk on [tk+1, t k ] x H implies

that llgl - g110,5i < 6 for each i, then the lemma is immediate from Lemma

7.16 Cl

W e are now ready to prove the C0 stability of timelike geodesic incomplete-

ness for Lorentzian warped products M = (a , b) x f H with a > -m and (H. h )

homogeneous.

Theorem 7.18. Let ( M , g ) be a warped product space-time of the form

M = (a, b) x f H with a > -03, g = -dt2 @ fh, and (H , h ) a homogeneous

256 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS 7.3 GEODESIC INCOMPLETENESS 257

Riemannian manifold. Then there exists a fine CO neighborhood U ( g ) of g 'I'heorem 7-19. Let ( M , g) be a Robertson-Walker space-time o f the form

in Lor(M) o f globally hyperbolic metrics such that all timelike geodesics of M = (a , b) x f H with a > -W. Then there exists a fine C0 neighborhood U ( g )

( M , gl ) are past incomplete for each gl E U(g) . of g in Lor(M) o f globally hyperbolic metrics such that all timelike geodesics

of ( M , g l ) are past incomplete for each gl E U(g). Proof. Fix any yo E M and let c : (a,wo] 4 M be the past inextendible

future directed geodesic given by c(t) = (t, yo). Let {Uk), { tk) be an admissible I f we change the time function on ( M , 9 ) to T I : M -+ R defined by ( t , h) =

chain for c, guaranteed by Lemma 7.15. Also choose 6k : [&+I, t k ] x H 4 (0, a) -t, and apply Lemmas 7.16 and 7.17 to the resulting space-time, we obtain

for each tk according to Lemma 7.17. Let 6 : M -+ (0, CO) be a continuous the exact analogue o f these lemmas for the future directed timelike geodesic c :

function such that 6(q) I &(q) for all q E [tk+i,tk] X H and each k > 0. ["o, b) + ( M , 9 ) given by c(t) = (t, yo) in the given space-time. Hence i f ( M , g )

Set v ~ ( ~ ) = {g l E Lor(M) : lgl - glo < 6). Since global hyperbolicity is a is a Lorentzian warped product M = (a, b) x f H with ( H , h ) homogeneous and

Co-open condition, we may also assume that all metrics in q ( g ) are globally b < oo, the same proof as for Theorem 7.18 yields the C0 stability o f the future

hyperbolic. timelike geodesic incompleteness. Combining this remark with Theorem 7.18

By the first paragraph o f the proof of Lemma 7.16, we may choose a CO then yields the following result.

V2(g) o f g in Lor(M) such that for all gl E V Z ( ~ ) , each hyper- Thorem 7-20. Let ( M , g ) be a Lorentzian warped product o f the form

surface { t ) x H , t E (a, b), is spacelike in (M,gl) . Then every nons~acelike M = (a , b) X f H , g = -dt2 @ f h , with both a and b finite and ( H , h ) ho-

curve y : (a, p) -+ ( M , g ~ ) may be parametrized in the form y( t ) = ( t , yl(t)) . mogeneous. Then there is a fine C0 neighborhood U ( g ) o f g in Lor(M) of

Hence Lemma 7.16 may be applied to all inextendible nons~acelike geodesics globally hyperbolic metrics such that all timelike geodesics o f ( M , gl ) for each

in ( M , g l ) with gl E Vz(g). 91 f U(g) are both past incomplete and future incomplete.

Now U ( g ) = Vl(g)nVz(g) is a fine neighborhood o f in the CO topology. It is interesting to note that while the finiteness of a and b is essential to

Let gl E U(g] , and let y : ( a , p ) -t ( M , gl) be any future directed inextendible the proof o f Theorem 7.20, the proof is independent of the particular choice o f timelike geodesic. W e may assume that { t l ) x H is a Cauchy surface

warping function f : (a, b) -t (0, CO). While the homogeneity o f the Riemann- ( M , g l ) by the arguments o f Geroch (1970a, p. 448), and hence there exists an ian factor ( H , h ) is also used in the proof o f Theorem 7.20, no other geometric so E ( a , p) such that y(s0) E { t ~ } x H. In passing f k m {tk+i) X H t o { tk) r topological property o f ( H , h ) is needed. the gl length of y is at most &n ( t k - &+I), applying Lemma 7.17 for each In general relativity and cosmology, closed big bang models for the uni- k. Summing up these estimates, it follows that the gl length o f (a, so] is at

se are considered [cf. Hawking and Ellis (1973, Section 5.3)]. These models most f i n (tl - a) . Since yl (a, so] is a past inextendible timelike geodesic Robertson-Walker space-times for which b - a < oo and H is compact. finite gl length, it follows that y is past incomplete in ( M , g l ) . 0

ce Theorem 7.20 implies, in particular, the 60 stability o f timelike geodesic

Lerner raised the following question (1973, p. 35) about the Roberts mpleteness for these models.

Walker big bang models ( M , g): under small C2 perturbations o f the me now turn t o the proof o f the C1 stability o f null geodesic incompleteness

does each nonspacelike geodesic remain incomplete? Since the Ftiemannl Robertson-Walker space-times. Taking M = ( 0 , l ) x f W with f ( t ) =

factor ( H , h) of a Robertson-Walker spacetime is homogeneous, we obt )-2 and 3 = -dt2 @ fdx2, it may be checked using the results o f Section

the following corollary t o Theorem 7.18 which settles affirmatively for time that the curve 7 : (-m,O) -t ( M , g ) given by ~ ( t ) = (et, e2t ) is a past

geodesics the question raised by Lerner (1973, P. 35). plete null geodesic. Thus by choosing the warping function suitably, it


is possible to construct Robertson-Walker space-times with a > -m which

are past null geodesically complete. Thus unlike the proof o f stability for

timelike geodesic incompleteness, it is necessary to assume that ( M , g) contains

a past incomplete (respectively, past and future incomplete) null geodesic to

obtain the null analogue o f Theorem 7.19 (respectively, Theorem 7.20). Not

surprisingly, the proof of the C 1 stability of null geodesic incompleteness is

more complicated than for the timelike case since affine parameters must be

used instead of Lorentzian arc length to establish null incompleteness. Also

for the proof o f Lemma 7.22, we need the isotropy as well as the homogeneity

o f ( H , h). Thus we will assume that M = (a , b) x f H is a Robertson-Walker

space-time in the rest o f this section.

Let (V, x l , . . . , x,) denote an adapted normal neighborhood of ( M , g) with

adapted coordinates ( x l , . . . , x,). For the proof o f Lemma 7.22, it is neces-

sary t o define a distance between compact subsets o f vectors that are null for

different Lorentzian metrics for M and are attached at different points o f V .

Recall from Section 7.2 that local coordinates ( x l , . . . , x,) for V give rise to

local coordinates Z = ( x l , . . . ,x,, x,+l,. . . , x2,) for T V = TMIV. Thus given

any q E V , gl E Lor(M), and cu > 0, we may define

S(q,a ,gl ) = { v E T,M : gl(v,v) = 0 and z,+l(v) = -a).

Then S(q, a, gl) is a compact subset o f T,M for any cr > 0 and gl E Lor(M).

Given p,q f V , gl,g2 E Lor(M), and cul,az > 0, define the Hawdorff distance

between S (p , ~ 1 , gl) and S(q, a2,g2) by

The continuity o f the components o f the metric tensor g as functions g,? :

V x V -+ W and the closeness o f light cones for Lorentzian metrics close in the

C0 topology imply the continuity o f this distance in p, a , and g [cf. Busemann

(1955, pp. 11-12)].


Lemma 7.21. Let V be an adapted normal neighborhood adapted at p E

(M,g) . Given a > 0 and E > 0, there exists 6 > 0 such that Ilp - ql/2 < 6,

g1 E Lor(M) with Ilg - g1110,v < 6, and la1 - a / < 6 together imply that

dist(S(91 cul,gl), S(P, a , 9 ) ) < c.

Now let ( M , g) be a Robertson-Walker space-time (a , b) x f H which is

past null incomplete. Thus some past directed, past inextendible null geodesic

c : [O,A) -+ (M,g) is past incomplete (i.e., A < ca). Since ( H , h) is isotropic

and spatially homogeneous, and since isometries map geodesics to geodesics,

it follows that all null geodesics are past incomplete. W e now fix through the

proof o f Theorem 7.23 this past inextendible, past incomplete, null geodesic

c : [0, A ) --+ ( M , g ) with the given parametrization.

Let (wo, yo) = c(0) E M = (a, b) x f H. With this choice o f wo, apply

Lemma 7.15 to the future directed timelike geodesic t -+ ( t , yo), t 5 wo, to

get an admissible chain {Uk), { tk) for this timelike geodesic. Using this choice

of { tk) , we may find sk with 0 = sl < s:, < . . . < sk < ... < A such that

c ( s ~ ) f { tk ) x H for each k. Set Ask = sk+l - sk. AS above, let xl : M -+ W denote the projection map x l ( t , h ) = t on the first factor o f M = (a , b) x f H.

Notice that i f (V, X I , x2,.. . ,x,) is any adapted coordinate chart, then the

coordinate function xl : V -r R coincides with this projection map. I f y is any

smooth curve o f M which intersects each hypersurface { t ) x H o f M exactly

once and y ( s ) E { t ) x H , we will say that l ( x lo y)'(s)l is the $1 speed o f y at

{ t ) x H. In particular, we will denote by ak = / ( X I o c)'(.qk)l the x l speed o f

the fixed null geodesic c : [0, A ) -+ ( M , g) at { t k ) x H for each k .

Lemma 7.22. Let E > 0 be given. Then for each k > 0 , there exists a

continuous function 6k : [tk+l, tk] X H -' ( 0 , m ) with the following properties.

Let gl E Lor(M) with lg - g1]1 < 6k on [tk+l,tk] x H , and let y : [O,B) --+ M

be any past directed, past inextendible, null geodesic with $0) E i t k ) x H

and with x1 speed o f cuk at { tk) x H. Then y reaches { & + I ) x H with an

ncrease in f ine parameter o f at most 2Ask, and moreover, the x1 speed 9 of

y at {tk+l) x H satisfies the estimate


Proof. Let c : [0, A) -+ (M, g) be the given past incomplete null geodesic as

above. Fix k > 0. By the spatial homogeneity of Robertson-Walker space-

times, we may find an isometry $ € I ( H ) such that $ = id x 4 € I (M,g)

satisfies +(c(sk)) = (tk, YO) with yo as above. Since k is fixed during the

course of this proof, we may set p = (tk, yo) without danger of confusion. Put

cl(s) = $ o c(s + sk). Then cl is a past inextendible, past incomplete, null

geodesic of (M,g) with cl(0) f {tk) x H, cl(Ask) E {tk+l) x H , and cl(s) =

exp,[g](sv) for v = $,(cl(sk)). Choose b > 0 with Ask Ask, we have cl(b) € {t) x H

for some t < tk+l. Hence (xi o cl) (Ask) - (xi 0 cl)(b) = tk+i - t > 0. Set

€1 = min{c, tk+l - (xi 0 cl)(b)) > 0.

Now let gl E Lor(M), and let q E Uk n ({tk) x H). Suppose that y : [0, B) --+

(M, gl) is any past directed, past inextendible, null gl geodesic with y(0) = q

and with xl speed a k a t q. Then w = yl(0) f TqM satisfies gl(w, w) = 0 and

xn+,(w) < 0. Moreover, y(s) = expq[gl](sw). Applying Lemmas 7.10 and

7.11 to cl and c2 = y with the constant €1 as above, we may find a constant

60 > 0 with 0 < 60 < Ask such that Ilv - wll2 < 60, 119 - gllll,uk < 60, and

Iso - Ask\ < 60 imply that

(I) I(XI 0 c ~ ) ( s ) - (21 0 ca)(~)I < €1 < tk+l - (XI 0 ~ l ) ( b )

for all s with 0 5 s 5 b and

(2) 1 - €1 < 1 ("I O C2)'(s0) 1 < 1 + €1. (21 O cl)'(Ask)

Setting s = b in (I), we obtain ](XI ocl)(b) -(xi oc2)(b)l < tk+l- (xi ocl)(b)

from which (xl o c2)(b) < tk+l. Hence there exists an s1 with 0 < s' < b su

that (xl o c2)(s1) = tk+l. But then s' < b < 2Ask, which shows that th

increase in affine parameter of c2 in passing from {tk) x H to {tk+l) x H

less than 2Ask provided that 60 is chosen as above.

For any geodesic q ( s ) = exp[gl](sw) with gl and w 60-close to g an

as above, let s' denote the value of the f i n e parameter s of c2 such t

x1 o c2(s1) = tk+l. AS 60 -+ 0, the corresponding value of s' must appro

Ask by Lemma 7.10. Thus by continuity we may choose b1 with 0 < 61 <


such that for any geodesic c2(s) = exp[gl](sw) with gl and w both 61-close to

g and v, we have 21 0 c~(s ' ) = tk+l for some st f [Ask - bo, Ask + bo]. Hence

as 61 < 60, we may apply estimate (2) above with so = s' to obtain

We now need to extend these estimates from a neighborhood of v E T,M

to a neighborhood of S(p, a k , g). TO this end, note that since (a, b) x f H =

M is a warped product of H and the one-dimensional factor (a, b) with f :

(a, b) -, R, it follows that I (H) acts transitively on S(p, ak,g). Thus given

any t € S(p, ak,g), we may apply the previous arguments using the same

admissible chain {Uk), {tk) to find a constant 61(t) > 0 such that if w E T M

satisfies Ilw - 4 2 < 61(z), .rr(w) E Uk n ({tk) x HI, llgl - 9lll,uk < bl(z), and

4 s ) = exp[gl](sw) has xl speed a k at {tk) x H , then c2 has an increase

in affine parameter of at most 2Ask in passing from {tk) x H to {tk+l) x H

and satisfies estimate (7.4). Using the compactness of S(p, ak,g), we may

choose null vectors vl, v2, . . . , v, E S(p, ak, g) such that S(p, a k , g) is covered

by the sets {w f S(p,ak,g) : I I w - vrnI12 < 61(vrn)) for m = 1,2,. . . , j. Set

b2 = mini61 (v,) : 1 5 m 5 j}. By Lemma 7.21 we may find a constant b3 with

0 < 63 < 62 such that if I ~ P - 4\12 < 63, llgi - 9111,~~ < 63, and w E S(P, ak, gi),

then Ilw - vrnl12 < 61(vm) for some m. Hence b3 has the following properties.

If y : [0, B) -+ (M, gl) is any past inextendible, past directed, null geodesic of

(M,gl) such that Ilgl-gl(l,vk < 63, ~ ( 0 ) ({tk)xH)n{q f Uk : II~--9112 < 63) where p = (tk, yo), and y has xl speed a k at y(O), then the conclusions of the

theorem apply to y. Since I (H) acts transitively on H , we may extend this

result from ({tk) x H ) n {q f Uk : Ilp - 9112 < 63) to all of {tk) x H just as

in the proof of Lemma 7.17. The function 6k : [tk+l,tk] x H -+ (0, a) may be

constructed exactly as in Lemma 7.17. 0

With Lemma 7.22 in hand, we are now ready to prove the C1 stability of

t null geodesic incompleteness for Robertson-Walker space-times. Since

bertson-Walker space-times are isotropic and spatially homogeneous, past

incompleteness of one inextendible null geodesic implies past incompleteness

of all null geodesics. Thus the stability theorem may be formulated as follows.


T h e o r e m 7.23. Let ( M , g) be a Robertson-Walker space-time containing

an inextendible null geodesic which is past incomplete. Then there is a fine C'

neighborhood U(g) o f g in Lor(M) o f globally hyperbolic metrics such that all

null geodesics of ( M , gl ) are past incomplete for each gl f U(g).

Proof. Let M = (a, b) x f H and let c : [0, A ) + ( M , g ) be the given inex-

tendible past incomplete null geodesic. Wi th wo = xl(c(O)), let {Uk), { t k ) ,

{ s k ) , and { a k ) be chosen as in the paragraph preceding Lemma 7.22. Let

{Pk) be a sequence of real numbers with 0 < Pk < 1 for each k such that

112 < n L l ( 1 - P k ) < 1. Thus for each m 2 1, we have

For each k 2 1, we apply Lemma 7.22 with E = Pk to obtain a continuous

function bk : i tkf l , tk] X H -' (0, m) with the properties o f Lemma 7.22.

Choose a continuous function 6 : M + (0, m) such that for each q in M we

have 6(q) < bk(q) for ail k with q in the domain of 6k. Let Ul(g) = {gl E

Lor(M) : lgl - gjl < 6 ) . Also choose a C1-open neighborhood Uz(g) o f g in

Lor(M) such that all metrics in U2(g) are globally hyperbolic and such that

each hypersurface { t ) x H is spacelike in (M,g l ) for all t E (a , b) and any

gl E U2(g) (cf . Lemma 7.16). Set U(g) = Ul(g) n Uz(g).

Now suppose that gl E U(g) and that y : [0, B ) + M is any past directed

and past inextendible null geodesic o f ( M , gl). Reparametrizing y i f necessary,

we may assume that x l (y (0 ) ) = tk for some k 2 1 and that y has xl speed

C Y ~ at { t k ) x H. By Lemma 7.22, y changes in d n e parameter by at most

2Ask in passing from {tk) x H to x H. In order to apply Lemma

7.22 to y as y passes from {tk+l) x H to {tk+2) x H , it may be necessary

to reparametrize y at {tk+l) x H to have xl speed ak+l at {tk+l) x H.

Nonetheless, i f Bk+' denotes the xl speed of y at {tk+l) x H , we have 1 - Pk < lek+l/~~k+ll < 1 +Pk from Lemma 7.22. Thus the xl speed of y at {&+I) x H

cannot be less than ( 1 - r C j k ) ~ k + ~ . Hence the affine parameter o f y increases

in passing from {tk+l) x H to {tk+2) x H by at most 2(1 - Pk)-lAsk+l.

Arguing inductively, it may be seen that y increases in affine parameter by at

most AS^+^ nfzA(l- ~ k + % ) - ' in passing from {tk+l) x H t o (tk+~+l) x H.

7.4 SUFFICIENT CONDITIONS FOR STABILITY 263

Using inequality (7.5), we thus have that y increases in affine parameter by at

most 4ASk+, in passing from {tk+l) x H to {tk+l+l) x H. Since ELl Ask = A ,

it follows that the total afFine length B o f y is less than 4A. Since 4 A < m, it

follows that y is past incomplete as required. [3

By reversing the time orientation, we may obtain the analogue o f The-

orem 7.23 for Robertson-Walker space-times having future incomplete null

geodesics. Thus Theorem 7.23 implies the following result.

Theorem 7.24. Let ( M , g) be a Robertson-Walker s p a c e t ~ m e containing

an inextendible null geodesic which is both past and future incomplete. Then

there is a fine C' neighborhood U(g) of g in Lor(M) o f globally hyperbolic

metrics such that all null geodesics of ( M , gl) are past and future incomplete

for each gl E U(g) .

W e now obtain two stability theorems for nonspacelike geodesic incomplete-

ness by combining Theorems 7.19 and 7.23 and by combining Theorems 7.20

and 7.24. The first o f these theorems applies to all big bang models, and the

second theorem applies to the closed big bang models.

T h e o r e m 7.25. Let ( M , g) be a Robertson-Walker space-time of the form

(a, b) x f H , where a > -m. Assume that ( M , g) contains a past incomplete

and past inextendible null geodesic. Then there is a fine C' neighborhood

U(g) o f g in Lor(M) o f globally hyperbolic metrics such that all nonspacelike

geodesics o f ( M , g l ) are past incomplete for each gl E U(g) .

Theorem 7.26. Let ( M , g) be a Robertson-Walker space-time o f the form

(a, b) X / H , where both a and b are finite. Assume that ( M , g ) contains an

inextendible null geodesic which is both past and future incomplete. Then

there is a fine C' neighborhood U(g) o f g in Lor(M) o f globally hyperbolic

metrics such that all nonspacelike geodesics o f ( M , g l ) are both past and future

incomplete for each gl E U(g) .

k 7.4 Sufficient Conditions for Stability

In this section we show that nonimprisonment is a sufficient condition for the

C' stability o f incompleteness [cf. Beem (1994)l. Sufficient conditions for the


stability o f completeness involve both nonimprisonment and pseudoconvexity.

W e first state two lemmas which follow from standard results (cf . Lemmas 7.10

and 7.11).

Lemma 7.27. Let ( M , g) be a given semi-Riemannian manifold, fix a ge-

odesic y : (a, b) -+ M of ( M , g ) , and let W = W ( y I Itl,t2]) be a neighborhood

o f y I [ t l , t z ] . There is a neighborhood V o f y1(tl) in T M and a constant E > 0

such that i f llg - gllll,w < E and i f c is a geodesic of gl with c l ( t l ) E V , then

the domain of c includes the value t 2 and c( t ) E W for all tl < t 5 t2.

Lemma 7.28. Let ( M , g ) be a given semi-Riemannian manifold, fix a ge-

odesic y : (a, b) -+ M of ( M , g ) , and let W = W ( y I [tl , t2]) be a neighborhood

o f y 1 [ t l , t2] . Given any neighborhood Vl o f y l ( t l ) in T M , there is a constant

E > 0 and a neighborhood fi of y1(t2) in T M such that i f llg - glll1,w < 6 and

i f c is a geodesic of gl with c1(t2) f V2, then cl(t l) E Vl and c(t) E W for all

tl < t < t2. Furthermore, i f y is timelike (respectively, spacelike), then V2 and

E > 0 may be chosen such that each v E 7 2 o f each such metric gl is timelike

(respectively, spacelike). I f y is null, then E > 0 may be chosen such that each

such metric gl has some null vectors in V2.

W e now define partial imprisonment and imprisonment.

Definition 7.29. (Imprisonment and Partial Imprisonment) Let y :

(a , b) -+ M be an endless (i.e., inextendible) geodesic.

(1) The geodesic y is partially imprisoned as t -+ b i f there is a compact

set K M and a sequence { t j ) with t , -+ b- such that y ( t j ) E K for

all j .

(2) The geodesic y is imprisoned i f there is a compact set K such that the

entire image of y is contained in K.

In other words, y is partially imprisoned in K as t 4 b i f either y ( t ) E K

for all t sufficiently near b or i f y leaves and returns to K an infinite number of

times as t + b-. An imprisoned geodesic is clearly partially imprisoned, but for

general space-times one may have some partially imprisoned geodesics which

fail t o be imprisoned. The stability o f incompleteness result (Theorem 7.30)

will hold for geodesics which fail to be partially imprisoned in the direction


of incompleteness. In contrast, the stability o f (nonspacelike) completeness

(Theorem 7.35) result will require the nonimprisonment o f all nonspacelike

geodesics as well as the pseudoconvexity of nonspacelike geodesics.

Let M be given an auxiliary, positive definite, and complete Riemannian

metric. This metric has a complete distance function do : M x M -+ W, and the

Hopf-Rinow Theorem [cf. Hicks (1965)l guarantees that the compact subsets

o f M are exactly the subsets which are closed and bounded with respect to

do. The proof of Theorem 7.30 [cf. Beem (1994)l will use a sequence of n-

dimensional annuli {W,) [i.e., sets bounded by spherical shells with respect

to the distance do which are centered at a fixed point y(to)]. The approach

involves requiring the perturbed metric gl to be sufficiently close to the original

metric g on the sequence {W,). For a fixed gl, one constructs a sequence o f

geodesics cj o f gl such that a given c, is either tangent or almost tangent to

the original y at a certain point y ( t j ) o f W j . A limit geodesic o f the sequence

will be the desired geodesic c o f gl.

Theorem 7.30. Let ( M , g) be a semi-Riemannian manifold. Assume that

( M , g ) has an endless geodesic y : (a, b) -+ M such that y is incomplete in

the forward direction (i.e., b # oo). I f y is not partiaI1y imprisoned in any

compact set as t -+ b, then there is a C 1 neighborhood U ( g ) o f g such that

each gl in U(g) has at least one incomplete geodesic c. Furthermore, i f y is

timelike (respectively, null, spacelike) then c may also be taken as timelike

(respectively, null, spacelike).

Proof. Choose some to in the interval (a, b). We will construct sequences

t,, D,, and L,. Let tl and Dl be chosen with Dl > 1 and t l > to such

that do(y( to) ,y( t l ) ) = Dl and do(y(to), y ( t ) ) > Dl for all tl < t < b. In

other words, y 1 [to, b) leaves the closed ball o f radius Dl for the last time at

t l . The existence o f t l follows using the fact that y is not partially imprisoned

as t -+ b. Set Lo = 0 and L1 = 1 + sup{do(y(to), y ( t ) ) 1 to < t < t l ) . Notice

that on the interval [to, t l ] the geodesic y remains within the closed ball o f

radius L1 - 1 about y(t0). I f t l , . . . , t,-1, Dl , . . . , D,-1, and Lo, L1, . . . , L,-1

have been defined, then t, and D, are defined by letting D, > L, - 1 i- 1 and

requiring both do(y(to), y(t,)) = D, and &($to), ? ( t ) ) > D, for all t, < t < b.


Existence again follows using the nonimprisonment hypotheses. Define L j =

1 + sup{do(y(to),r(t)) It0 < t < tj). Note that limy(tj) does not exist as

j -+ co, and hence t j -+ b as j -+ w . Set Wl = {x E M I &(y(to), x) < L1),

W2 = {X E M I &(y(to), X) < L2), and Wj = {X E M I Lj-2 < do(y(to), X) < Lj) for j > 2. The sets Wj are n-dimensional annuli with respect to the

distance do. We now define a sequence of positive numbers {cj) and a sequence

of open sets {V,) of TM. Let Vo be an open neighborhood of yl(to) in TM with

compact closure, and assume that the closure of Vo does not contain any trivial

vectors. If g(yl(to), yl(to)) > 0 [respectively, g(yl(to), yl(to)) < 01, we assume

without loss of generality that all v E 70 satisfy g(v, v) > 0 [respectively,

g(v, v) < 01. Assume that &, Vl, . . . , T/;-l have been defined. Use Lemma 7.28

to obtain an open set V, = T/;(yl(tj)) of yl(tj) in T M and a positive number

cj such that if Ilg-gljll,p < cj where P = Wj, and if c is a geodesic of gl with

cl(tj) E Vj, then cl(tj-1) E y-l and c(t) E Wj for all tj-l 5 t 5 tj. Lemma

7.28 implies that if the original geodesic y is timelike (respectively, spacelike),

then we may assume each v in Vj is timelike (respectively, spacelike) for gl. If

y is null, then we may assume that gl has some null vectors in V,. Lemma 7.27

implies we may assume that if c is a geodesic of such a gl with cl(tj) E vj, then the domain of c contains tj+l. We may also assume without loss of

generality that the diameter with respect to & of K(V,) is less than 112 and

cj+l < cj for all j . Notice that points of M are in a t most two consecutive

sets of the sequence Wj. It follows that there is a continuous positive-valued

function E : M -+ (0 ,w) with ~ ( x ) < ej for all j with x 6 Wj. Let gl satisfy

119 - gllll,M < ~ ( x ) . We will construct a sequence of geodesics of gl. Assume

first that the original geodesic y is either timelike or spacelike. For each j,

let cj(t) be the geodesic of gl which satisfies cjl(tj) = yl(tj). Notice that if

y is timelike (respectively, spacelike), then each cj is timelike (respectively,

spacelike). If the original geodesic y is null, then we choose c j such that

cjl(tj) E Vj and cjl(tj) is null. The above construction yields that cjt(tk) E Vk

for all 0 5 k 5 j. Since Vo has compact closure and this closure does not

contain any trivial vectors, one obtains a nontrivial vector v in To and a

subsequence { m ) of {j) such that cml(to) -+ v. The constructions of Vo and


the sequence {cj) yield that v is timelike (respectively, null or spacelike) for

gl if the original geodesic y is timelike (respectively, null or spacelike) for g.

Let c be the endless (i.e., inextendible) geodesic of gl with cl(to) = v. Note

that each geodesic c, satisfies h ( t j ) E .rr(V,) for all 0 5 j 5 m. Also, for

each t in the domain of c, cm(t) converges to c(t) and cml(t) converges to cl(t)

as m -+ w . By construction, cl(tj) f 6 implies c(tj+l) exists. Furthermore,

ckt(tj+1) f V,+I for all k > j + 1 yields cl(tj+l) 6 vj+l. Thus, c is defined for

all values of t j , and &(y(to), c(tj)) -+ oo as j -+ w . It is easily shown that the

limit c is not partially imprisoned in any compact set as t + b. It only remains

to show that c cannot have any domain values above b (i.e., c(b) does not exist).

Assume the domain of c contains b. Since c is continuous, the set c([to, b])

would be compact in contradiction to the condition do(yo(to), c(tj)) -+ co. 0

An interesting aspect of the construction used in the above proof is that the

final geodesic c of the metric gl has the same value b for the least upper bound

of its domain as the original geodesic y of the metric g. In essence, this is

due to cjl(tj) E T/, implying cjl(to) E Vo and to the fact that do(cj(tj),cj(to))

diverges to infinity. Notice that the methods used in the proof of Theorem

7.30 show that if gl is another semi-Riemannian metric on M and if gl is close

to g in the C' sense on a neighborhood of y , then gl has a corresponding

geodesic which is incomplete. In other words, the proof of Theorem 7.30 only

really requires that g and gl be metrics which are C1 close near y. Thus,

if (M, g) has a finite number N of incomplete geodesics and none of these

are partially imprisoned in any compact set, then one may construct a C1

neighborhood U(g) of g such that each gl in U(g) has at least N incomplete

geodesics. On the other hand, one may construct space-times which have all

geodesics complete except for a single geodesic which is incomplete and which

is not partially imprisoned [cf. Beem (1976c)j. Thus, the existence of a single

incomplete geodesic does not imply that there is more than one such geodesic.

Corollary 7.31. If (M,g) is a strongly causal space-time which is causally

geodesically incomplete, then there is a C1 neighborhood U(g) of g such that

each gl in U(g) is nonspacelike geodesically incomplete.


Proof. This follows from Theorem 7.30 using the fact that for strongly

causal space-times, no causal geodesic is future or past partially imprisoned

in any compact set. 0

It is known that no null geodesic in a twc-dimensional Lorentzian manifold

has conjugate points. However, a chronological space-time in which all null

geodesics have conjugate points is strongly causal and has dimension at least

three. Thus, we obtain the following corollary.

Corollary 7.32. Let (M, g) be a chronological space-time of dimension at

least three which is causally geodesically incomplete. If (M, g) has conjugate

points along all null geodesics, then there is a C1 neighborhood U(g) of g such

that each gl in U(g) is nonspacelike geodesically incomplete.

Let (a, b) x f H be given the metric g = -dt2 + f (t)dcr2. If either a or b is

finite, then (a, b) x f H is timelike geodesically incomplete. Note that (a, b) x H

is necessarily stably causal since f (t , y) = t is a time function. Using Corollary

7.31 and the fact that stably causal space-times are strongly causal, we obtain

Corollary 7.33. Assume (H, h) is a positive definite Riemannian manifold,

and let the warped product (a, b) x f H be given the Lorentzian metric g =

-dt2 @ fh . If either a # -ca o rb # +ca, then there is a fine C1 neighborhood

U(g) of g such that each metric gl in this neighborhood is timelike geodesically

incomplete.

This last corollary applies to a much more general class of spacetimes

than the Robertson-Walker space-times because we have not made symmetry

assumptions on (H, h). Of course, for the special case of Robertson-Walker

space-times we have already established the stronger conclusion that there is

a C0 neighborhood U(g) with each gl in U(g) having all timelike geodesics

incomplete (cf. Theorem 7.18).

If K is a subset of Wn, then the conuez hull K H of K is the union of all

Euclidean line segments with both endpoints in K. It is well known that the

convex hull of a compact set is again a compact set. Pseudoconvexity is a gen-

eralization of this property to manifolds. In Definition 7.34 below, we assume

the class of geodesic segments under consideration to be nonspacelike. One


can, of course, consider this property for other classes such as the class of all

null geodesic segments, all spacelike segments, etc. Furthermore, pseudocon-

vexity and disprisonment have also been applied to sprays [cf. Del Riego and

Parker (1995)l.

Definition 7.34. (Nonspacelike Pseudoconuexity) We say (M, g) has a

pseudoconuex nonspacelike geodesic system if for each compact subset K of M ,

there is a second compact set H such that each nonspacelike geodesic segment

with both endpoints in K lies in H.

The following [cf. Beem and Ehrlich (1987, p. 324)] gives sufficient condi-

tions for the stability of nonspacelike completeness.

Theorem 7.35. Let (M,g) be a Lorentzian manifold which has no im-

prisoned nonspacelike geodesics and which has a pseudoconvex nonspacelike

geodesic system. If (M, g) is nonspacelike geodesically complete, then there is a

C1 neighborhood U(g) ofg in Lor(M) such that each gl f U(g) is nonspacelike

complete.

Pseudoconvexity is a type of "internal" completeness condition in somewhat

the same sense that global hyperbolicity is such a condition. The next propo-

sition shows that nonspacelike pseudoconvexity is a generalization of global

hyperbolicity. An example of a nonspacelike pseudoconvex space-time which

fails to be globally hyperbolic is given by the open strip {( t , x) : 0 < x < 1) in

the Minkowski plane.

Proposition 7.36. If (M,g) is a globally hyperbolic space-time, then

(M, g) has a pseudoconvex nonspacelike geodesic system.

Proof. Let K be a compact subset of M. For each p E K choose points q

and r with q in the chronological past I-(p) of p and r in the chronological

future If (p) of p. Then U(p) = I+(q) n I-(T) is an open set containing p.

Since (M, g) is globally hyperbolic, the set U(p) has compact closure given by

J + ( q ) n J-(T). Cover the compact set K with a finite number of open set

U(p,) = I+(q,) n I-(r,) for i = 1 ,2 , . . . , k, and let H be the union of the k2

compact sets of the form J+(q,) n J-(r,) for 1 5 i, j 5 k. It is easily seen that


all nonspacelike geodesic segments with both endpoints in K must lie in the

compact set H.

Since globally hyperbolic space-times are strongly causal, they have no im-

prisoned nonspacelike geodesics. Consequently, Theorem 7.35 and Proposition

7.36 yield the following corollary.

Corollary 7.37. Let (M, g) be a globally hyperbolic space-time. If (M, g)

is nonspacelike geodesically complete, then there is a C1 neighborhood U(g)

of g in Lor(M) such that each gl E U(g) is nonspacelike complete.

Minkowski space-time is globally hyperbolic and geodesically complete.

Furthermore, its (entire) geodesic system is pseudoconvex. In other words,

given any compact set K , there is a larger compact set H such that any ge-

odesic segment (timelike, null, or spacelike) with endpoints in K must lie in

H. In fact, one may take H to be the usual convex hull of K. The next

proposition gives the C1 stability of global hyperbolicity, nonimprisonment,

completeness, and inextendibility for Minkowski space-time. The C1 stability

of global hyperbolicity follows from the stronger C0 stability result of Geroch

(1970a). The stability of nonimprisonment and of geodesic completeness (

all geodesics) follows from Beem and Ehrlich (1987, pp. 324-325). See a

Beem and Parker (1985, p. 18). The stability of inextendibility follows fro

the stability of completeness using Proposition 6.16 which guarantees th

geodesically complete space-times are inextendible.

Proposition 7.38. There is a C1 neighborhood U(7) of n-dimension

Minkowski space-time (M, 7 ) such that for each metric gl in this neighbor-

hood:

(1) (M, gl) is globally hyperbolic;

(2) No geodesic of (M, gl) is imprisoned;

(3) (M, gl) is geodesically complete; and

(4) (M, gl) is an inextendible space-time.

CHAPTER 8

MAXIMAL GEODESICS AND CAUSALLY

DISCONNECTED S P A C S T I M E S

Many basic properties of complete, noncompact Riemannian manifolds stem

from the principle that a limit curve of a sequence of minimal geodesics is

itself a minimal geodesic. After the correct formulation of completeness had

been given by Hopf and Rinow (1931), Rinow (1932) and Myers (1935) were

able to establish the existence of a geodesic ray issuing from every point of

a complete noncompact Riemannian manifold using this principle. Here a

geodesic y : [0, co) --t (N,go) is said to be a ray if y realizes the Riemannian

distance between every pair of its points. Rinow and Myers constructed the

desired geodesic ray as follows. Since (N, go) is complete and noncompact,

there exists an infinite sequence (p,) of points in N such that for every point

p E N, do(p,p,) -+ CXI as n -+ m. Let y, be a minimal (i.e., distance realizing)

unit speed geodesic segment from p = y,(O) to p,. This segment exists by

the completeness of (N,go). If v E T,N is any accumulation point of the

sequence {ynl(0)) of unit tangent vectors in TpN, then y(t) = exp, tv is the

required geodesic ray. Intuitively, y is a ray since it is a limit curve of some

subsequence of the minimal geodesic segments (7,). The existence of geodesic

rays through every point has been an essential tool in the structure theory of

both positively curved [cf. Cheeger and Gromoll (1971, 1972)] and negatively

curved [cf. Eberlein and O'Neill (1973)l complete noncompact Riemannian

manifolds.

A second application of this basic principle of constructing geodesics as

limits of minimal geodesic segments is a concrete geometric realization for

complete Riemannian manifolds of the theory of ends for noncompact Haus-

dorff topological spaces [cf. Cohn-Vossen (1936)l. An infinite sequence (p,)

of points in a manifold is said to diverge to infinity if, given any compact sub-

272 8 MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES 8.1 ALMOST MAXIMAL CURVES AND MAXIMAL GEODESICS 273

set K , only finitely many members of the sequence are contained in K. If a disconnecting two divergent sequences is said to be causally disconnected. It

complete Riemannian manifold (N, go) has more than one end, there exists a is evident from the definition that causal disconnection is a global conformal

compact subset K of N and sequences {p,) and {q,) which diverge to infinity invariant of C(M, g). Applying the principle of Section 8.1, we show that if the

such that 0 < &(p,, q,) --+ oo and every curve from p, to q, meets K for each strongly causal space-time (M,g) is causally disconnected by the compact set

n. Let -y, be a minimal (i.e., distance realizing) geodesic segment from p, to K, then (M, g) contains a nonspacelike geodesic line y : (a, b) -4 M which in-

q,. Since each y, meets K , a limit geodesic y : R -+ M may be constructed. tersects K. That is, d(y(s), y(t)) = L(y I [s, t]) for all s , t with a < s t < b.

Moreover, y is minimal as a limit of a sequence of minimal curves. Then This result is essential to the proof of the singularity theorem 6.3 in Beem

"y(-m)" corresponds to the end of N represented by { p , ) and "y(+m)" to and Ehrlich (1979a), as will be seen in Chapter 12. We conclude this chapter

the end represented by (9,). In particular, a complete Riemannian manifold by studying conditions on the global geodesic structure of a given space-time

with more than one end contains a line, i.e., a geodesic y : (-co, +m) N (M,g) which imply that (M,g) is causally disconnected. In particular, we

that is distance realizing between any two of its points. show that all two-dimensional globally hyperbolic space-times are causally

disconnected. Also, it follows from one of these conditions and the existence

Motivated by these Riemannian constructions, we study similar existence of nonspacelike geodesic lines in strongly causal, causally disconnected space- theorems for geodesic rays and lines in strongly causal space-times. From times that a strongly causal space-time containing no future directed null the viewpoint of general relativity, it is desirable to have constructions that geodesic rays contains a timelike geodesic line.

are valid not only for globally hyperbolic subsets of space-times, but also for

strongly causal space-times. However, if we only assume strong causality, it 8.1 Almost Maximal Curves and Maximal Geodesics not true in general that causally related points may be joined by maximal g e

odesic segments. Thus a slightly weaker principle for construction of maximal The purpose of this section is to show how geodesics may be constructed

geodesics is needed for Lorentzian manifolds than for complete Riemannian as limits of "almost maximal" curves in strongly causal spacetimes. In both

manifolds. Namely, in strongly causal space-times, limit curves of sequences o constructions, the upper semicontinuity of Lorentzian arc length in the C0

"almost maximal" curves are maximal and hence are also geodesics. In Secti topology on curves for strongly causal space-times and the lower semicontinu-

8.1 we give two methods for constructing families of almost maximal curv ity of Lorentzian distance play important roles. The strong causality of (M, g )

whose limit curves in strongly causal space-times are maximal geodesics. T is used in our approach here so that convergence in the limit curve sense and in

strong causality is needed to ensure the upper semicontinuity of arc length the CO topology on curves are closely related (cf. Proposition 3.34). The first

the C0 topology on curves and also so that Proposition 3.34 may be appli nstruction may be applied to pairs of chronologically related points p, q with

In Section 8.2 we apply this construction to prove the existence of past and (p, q) < a. While this approach is therefore sufficient to show the existence

ture directed nonspacelike geodesic rays issuing from every point of a stron of nonspacelike geodesic rays in globally hyperbolic space-times [cf. Beem and

causal space-time. In Section 8.3 we study the class of causally disconnec Ehrlich ( 1 9 7 9 ~ ~ Theorem 4.2)], it is not valid for points at infinite distance.

space-times. Here a space-time is said to be causally disconnected by a cordingly, for use in Sections 8.2 and 8.3, we give a second construction

pact set K if there are two infinite sequences {pn) and {qn) , both divergi ich may be used in arbitrary strongly causal space-times. In Section 14.1 a

infinity, such that p, < q,, p, # qn, and all nonspacelike curves from Pn ghtly different approach to constructing maximal segments from sequences of

meet K for each n. A space-time (M, g) admitting such a compact K caus onspacelike almost maximal curves is presented [cf. Galloway (1986a)J. Here,

274 8 MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES

the use of uniform convergence in a unit speed reparametrization with respect

to an auxiliary complete Riemannian metric is employed to dispense with the

global requirement of strong causality which we assume in this chapter.

Let (M, g) be an arbitrary space-time and suppose that p and q are distinct

points of M with p < q. If d(p,q) = 0, then letting y be any future directed

nonspacelike curve from p to q, we have L(y) 5 d(p, q) = 0. Hence L(y) =

d(p, q) and 7 may be reparametrized to a maximal null geodesic segment from

p to q by Theorem 4.13. Thus suppose that p << q, or equivalently, that

d(p, q) > 0. If d(p, q) < co as well, then by Definition 4.1 there exists a future

directed nonspacelike curve y from p to q with

for any E > 0. Of course, inequality (8.1) is only a restriction on L(y) provided

E < d(p, q). In this case, we will call y an almost maximal curve.

We note the following elementary consequence of the reverse triangle in-

equality.

Remark 8.1. Let y : [O, 11 -+ M be a future directed nonspacelike curve

from p to q, p # q, with

Then for any s < t in [O,l], we have

Proof. Assume that L ( y I [s, t ]) < d(y(s), y(t)) - c for some s < t in [O,l].

Then

in contradiction. 0

8.1 ALMOST MAXIMAL CURVES AND MAXIMAL GEODESICS 275

We are now ready to give an example of the principle that for strongly

causal space-times, limits of almost maximal curves are maximal geodesics.

Strong causality is used here (with all curves given a "Lorentzian parametriza-

tion") since convergence in the limit curve sense and in the C0 topology are

closely related for strongly causal space-times but not for arbitrary space-

times (cf. Section 14.1).

Proposit ion 8.2. Let (M,g) be a strongly causal space-time. Suppose

that p, -+ p and q, -+ q where p, < q, for each n and 0 < d (p , q) < m. Let

y, : [a, b ] -+ M be a future directed nonspacelike curve from p , to q, with

where 6 , --, 0 as n -+ m. If y : [a, b] --+ M is a limit curve of the sequence

{m) with y(a) = p and y(b) = q, then L(y) = d(p,q). Thus y may be

reparametrized to be a smooth maximal geodesic from p to q.

Proof. First, y is nonspacelike by Lemma 3.29. Second, by Proposition

3.34, a subsequence {y,) of {y,) converges to y in the C0 topology on curves.

By the upper semicontinuity of arc length in this topology for strongly causal

space-times (cf. Remark 3.35), we then have

k using the lower semicontinuity of Lorentzian distance (Lemma 4.4). But by

definition of distance, d(p,q) 2 L(y). Thus L(y) = d(p, q) and the last state-

[ ment of the proposition follows from Theorem 4.13. El

We now consider a second method for constructing maximal geodesics in

strongly causal space-times (M,g) which may be applied to points at infinite

Lorentzian distance. For this purpose, we fix throughout the rest of Chapter

8 an arbitrary point po E M and a complete (positive definite) Riemannian

metric h for the paracompact manifold M. Let do : M x M -+ R denote the


Riemannian distance function induced on M by h. For all positive integers n ,

the sets - B n = {m E M : do(po,m) F n)

are compact by the Hopf-Rinow Theorem. Thus the sets {B, : n > 0) form a

compact exhaustion of M by connected sets. For each n, let

denote the Lorentzian distance function induced on B, by the inclusion B, (M, g). That is, given p f B,, set d [Bn](p, q) = 0 if q $! J + ( ~ , R), and for

4 E J+(~ ,B, ) , let d [Bn](p,q) be the supremum of lengths of future directed

nonspacelike curves from p to q which are contained in B,. It is then immediate

that d q) 5 d(p, q) for all p, q E B,. However, d [B,] is not in general

the restriction of the given Lorentzian distance function d of (MI g) to the set - B, x B,. Nonetheless, for strongly causal space-times these two distances

coincide "in the limit."

Lemma 8.3. Let (M, g) be strongly causal. Then for all p, q E M , we have

d(p, 9) = lim d [Bn](p, 9).

Proof. Since d [Bn](p,q) 5 d(p,q), the desired equality is obvious in the

event that d(p, q) = 0. Thus suppose that d(p, q) > 0. By definition of

Lorentzian distance, we may find a sequence {yk) of future directed nonspace-

like curves from p to q such that L(yk) -+ d(p, q) as k -+ m. (If d(p, q) = m,

choose (yk) such that L(yk) > k for each k.) Since the image of yk in M is

compact and the Riemannian distance function do : M x M -+ W is continuous

and finite-valued, there exists an n(k) > 0 for each k such that yk C Bj for all j 2 n(k). Thus d(p, q) = lim L(yk) 5 limd [Bn](p, q). Hence as

d [Bn](p, q) 5 d(p, q) for each n, the lemma is established.

It will be convenient to introduce the following notational convention for

use throughout the rest of this chapter.

Notational Convention 8.4. Let y be a future directed nonspacelike

curve in a causal space-time. Suppose p = ~ ( s ) and q = y(t) with s < t and

p # q. We will let y[p, q ] denote the restriction of 7 to the interval [s, t] .

8.1 ALMOST MAXIMAL CURVES AND MAXIMAL GEODESICS 277

For strongly causal space-times, the Lorentzian distance function d and the

d [Bn]'s are related by the following lower semicontinuity.

Lemma 8.5. Let (M,g) be strongly causal. If p, -+ p and q, -+ q, then

d ( ~ , 9) < lim inf d [Bn] (pn qn) .

Proof. If d(p, q) = 0, there is nothing to prove. Thus we first assume that

0 < d(p, q) < a. Let 6 > 0 be given. By definition of Lorentzian distance and

standard results from elementary causality theory [cf. Penrose (1972, pp. 15-

16)], a timelike curve y from p to q may be found with d(p, q) - 6 < L(y) 5 d(p, q). Since y is timelike and L(y) > d(p,q) - E , we may find TI, rz E y with

d(p, q) - 6 < L(y[rl,rz]) and p << TI << r2 << q. Since I - (TI) and I+(r2) are

open and pn -+ p, qn -+ q, we have p, << r l << rg << qn for all n sufficiently

large. Also y C Bn, p, E J-(rl, B,), and q, E J+(rz, B,) for all n sufficiently

large. Consequently, d(p, q) - 6 < L(y[rl, m]) 5 d [Z,](p,, q,) for all large n.

Since e > 0 was arbitrary, we thus have d(p, q) 5 lim inf d [Bn](pn, q,) in

the case that 0 < d(p, q) < m . Assume finally that d(p, q) = co. Choosing

timelike curves yk from p to q with L(yk) > k for each k, we have for each

k that d [B,](p,, q,) 1 k - 6 for all n sufficiently large as above. Hence

limd [B,] (pn, qn) = co as required. 13

Since we are assuming that (M,g) is strongly causal but not necessarily

globally hyperbolic, it is possible that the Lorentzian distance function d :

M x M --t W U {m) assumes the value +w. Nonetheless, for any given Bn the distance function d [B,] : Bn x B, -+ R U {m) is finite-valued. This is

a consequence of the compactness of the Bn and the compactness of certain

subspaces of nonspacelike curves in the C0 topology on curves [if. Penrose

(1972, p. 50, Theorem 6.5)]. Moreover, this compactness also implies the

existence of curves realizing the d [B,] distance for points p, q E Bn with

E J+(P, Bn).

Lemma 8.6. Let (M,g) be a strongly causal space-time and let n > 0 be

arbitrary. If q E J+(~,B,) , then d [Bn](p,q) < XI and there exists a future

directed nonspacelike curve y in B, joining p to q which satisfies L(y) =

d[B,l(p,q).


Proof. By definition o f the distance d [B,], i f d [Bn] (p ,q ) = 0 and q E

J+(~,B,) , then there exists a future directed nonspacelike curve y in B, from p to q with L ( y ) 5 d [Bn](p,q) = 0. Hence L ( y ) = d [Bn](p,q) as required. Thus we may suppose that d q) > 0. Again by definition

o f d [ & I , we may find a sequence {yk ) o f future directed nonspacelike curves

from p to q with L(yk) -t d [ R ] ( p l q ) . ( I f d [B,](p, q) = oo, choose yk with

L(yk) > k for each k.) Since 3, is compact and ( M , g ) is strongly causal,

there exists a future directed nonspacelike curve y in B, joining p to q with

the property that a subsequence {y,} o f { y k ) converges t o y in the C O topology

on curves by Theorem 6.5 o f Penrose (1972, pp. 50-51). But then using the

upper semicontinuity o f arc length in the C0 topology on curves, we have

d [B,](p, q) = lim L(y,) 5 L ( y ) which implies the finiteness o f d [B,](p, q).

Since L ( y ) <_ d [ R ] ( p , q ) from the definition, we also have d (B,](p,q) = L(y)

as required.

Now let p, q E M with p < q, p # q, be arbitrary. Choose any nonspace-

like curve yo from p to q. Since the image of yo is compact in M and the

Riemannian distance function is continuous, we may find an N > 0 such that

yo is contained in BN. Hence q E J+(p ,Bn) for all n 2 N . Thus using

Lemma 8.6 we may find a future directed nonspacelike curve y, from p to q

with L(y,) = d [B,](p, q) for each n 2 N . For C0 limit curves o f the sequence

{y,), we then have the following analogue o f Proposition 8.2.

Proposition 8.7. Let ( M I g) be strongly causal and let p, q f M be dis-

tinct points with p < q. For all n > 0 sufficiently large, let yn be a future

directed nonspacelike curve from p to q in B, with L(y,) = d [Bn](p, q). If -y

is a nonspacelike curve from p to q such that {y,} converges to y in the C0

topology on curves, then L ( y ) = d(p, q) and hence y may be reparametrized

to a maximal geodesic segment from p to q.

Proof. Using Lemma 8.5 and the upper semicontinuity of arc length in the

C 0 topology on curves in strongly causal space-times, we have

8.2 GEODESIC RAYS IN CAUSAL SPACE-TIMES 279

d(p, q) L lim inf d [Bn] ( P , q)

= lim inf L(yn)

L lim sup L(yn) L L ( y )

L d ( ~ ! q)

as required. C3

Now let p, q be distinct points o f an arbitrary strongly causal space-time

with p < q and let a sequence {y,} of nonspacelike curves from p to q be

chosen as in Proposition 8.7. While a limit curve y for the sequence {y,) with

y(0) = p may always be extracted by Proposition 3.31, we have no guarantee

that y reaches q unless ( M , g ) is globally hyperbolic. Indeed, i f d(p, q) = co,

then there is no maximal geodesic from p to q, and hence no limit curve y o f

the sequence (7,) with y(0) = p passes through q. Thus the hypothesis that y

joins p to q in Proposition 8.7 together with the conclusion o f Proposition 8.7

implies that d(p, q) < co. On the other hand, the condition that d(p, q) < co does not imply that any limit curve y o f (7,) with y(0) = p reaches q when

( M , g ) is not globally hyperbolic. Examples may easily be constructed by

deleting points from Minkowski space-time.

8.2 Nonspacelike Geodesic Rays i n Strongly Causal Space-times

The purpose o f this section is to establish the existence o f past and future

directed nonspacelike geodesic rays issuing from every point o f a strongly causal

space-time ( M , g).

Definit ion 8.8. (Future and Past Directed Nonspacelike Geodesic Rays)

A future directed (respectively, past directed) nonspacelike geodesic ray is a

future (respectively, past) directed, future (respectively, past) inextendible,

nonspacelike geodesic y : [O,a) -t ( M , g ) for which d(y(O), y ( t ) ) = L ( y 1 [0, t ] )

(respectively, d (y ( t ) , y (0)) = L ( y I [0, t ] ) ) for all t with 0 5 t < a.

The reverse triangle inequality then implies that a nonspacelike geodesic

ray is maximal between any pair o f its points.

Using Lemmas 8.5 and 8.6 we first prove a proposition that will be needed

not only for the proof o f the existence o f nonspacelike geodesic rays, but also


for the proof of the existence of nonspacelike geodesic lines in strongly causal,

causally disconnected spacetimes in Section 8.3. Let and d [B,] : x - B, -+ R be constructed as in Section 8.1.

Proposition 8.9. Let (M, g) be a strongly causal space-time and let K be

any compact subset of M. Suppose that p and q are distinct points of M such

that p < q and every future directed nonspacelike curve from p to q meets K.

Then at least one of the following holds:

(1) There exists a future directed maximal nonspacelike geodesic segment

from p to q which intersects K.

(2) There exists a future directed maximal nonspacelike geodesic which

starts at p, intersects K, and is future inextendible.

(3) There exists a future directed maximal nonspacelike geodesic which

ends at q, intersects K , and is past inextendible.

(4) There exists a maximal nonspacelike geodesic which intersects K and

is both past and future inextendible.

Proof. Let yo be any future directed nonspacelike curve in M from p to q.

Since K U yo is compact, there exists an N > 0 such that K U yo is contained

in Bn for all n 2 N. Hence q E J+(P,B,) for all n 2 N. Thus by Lemma

8.6, for each n >_ N there exists a future directed nonspacelike curve y, in Bn joining p to q with L(yn) = d [B,](p, q). By hypothesis, each yn intersects

K in some point r,. Since K is compact, there exists a point T E K and a

subsequence {r,) of {r,) such that r, -+ r as m + m. Extend each curve

ym to a past and future inextendible nonspacelike curve which we will still

denote by ym. By Proposition 3.31, there exists an inextendible nonspacelike

limit curve y for the subsequence (7,) such that y contains T. Relabeling if

necessary, we may assume that {y,) distinguishes y.

Now the limit curve y may contain both p and q, only p, only q, or neither

p nor q. These four cases give rise respectively to the four cases (1) to (4)

of the proposition. Since the proofs are similar, we will only give the proof

for the second case. Thus we assume that y : (a, b) -+ M contains p = y(to)

but not q. We must show that y 1 [to, b) is maximal. To this end, let x be an

arbitrary point of y 1 [to, b). Since 1%) distinguishes y, we may find points

8.2 GEODESIC RAYS IN CAUSAL SPACE-TIMES 281

x, E y, with x, -+ x as m -+ w. Passing to a subsequence {yk) of {y,)

if necessary, we may assume by Proposition 3.34 that yk[p,xk] converges to

y[p, x] in the C0 topology on curves (recall Notational Convention 8.4). Since

y[p, x] is closed in M and q # y, there exists an open set V containing y[p, x]

with q # V . Since yk[p,xk] -+ y[p,x] in the C0 topology on curves, there

exists an Nl > O such that yk[p,xk] 2 V for all k 2 Nl. Hence q @ yk[p,xk]

for all k 2 NI . Thus yk[p,xk] E yk[p, q] for all k 2 N1 which implies that

L(yk[p,xk]) = d [Bk] (P ,~k) for all k > N1. By Lemma 8.5 and the upper

semicontinuity of arc length in the C0 topology on curves for strongly causal

spacetimes, we have

d(p, x) < lim inf d [Bk](p, xk)

= lim inf L(yk[p, ~ k ] )

5 f i m s u ~ L ( y k \ ~ , xk]) 5 L(y[p, XI). Since L(y[p, x]) < d(p,x) by definition of Lorentzian distance, we thus have

d(p, x) = L(y[p, XI) as required.

For globally hyperbolic space-times, case (1) of Proposition 8.9 always ap-

plies because J+(p) n J-(q) is compact and no inextendible nonspacelike curve

is past or future imprisoned in a compact set. However, space-tirnes which

are strongly causal but not globally hyperbolic and which have chronologically

related points p << q to which exactly one of cases (2) to (4) applies may be

constructed by deleting points from Minkowski space-time.

With Proposition 8.9 in hand, we are now ready to prove the existence of

past and future directed nonspacelike geodesic rays issuing from each point

of a strongly causal space-time. By the usual duality, it suffices to show the

existence of a future directed ray at each point.

Theorem 8.10. Let (M, g) be a strongly causal space-time and let p E M

be arbitrary. Then there exists a future directed nonspacelike geodesic ray

y : [O,a) -+ M with y(0) = p, i.e., d(p,y(t)) = L(y 1 [ O , t ] ) for all t with

O l t < a .

Proof. Let c : [0, b) --+ M be a future directed, future inextendible, timelike

curve with c(0) = p. Since (M,g) is strongly causal, c cannot be future


imprisoned in any compact set (cf. Proposition 3.13). Thus there is a sequence

{t,) with t , --+ b such that do(p,c(tn)) --+ co as n co. Set q, = c(t,) for

each n.

We now apply Proposition 8.9 to each pair p,qn with K = {p). Thus

for each n, either (1) there is a maximal future directed nonspacelike geodesic

segment from p to q,, or (2) there is a future directed, future-inextendible, non-

spacelike geodesic ray starting at p. If case (2) occurs for some n, we are done.

Thus assume that for each n there is a maximal future directed nonspacelike

geodesic segment y, from p to q,. Extend each y, to a future directed, future

inextendible, nonspacelike curve, still denoted by y,. By Proposition 3.31, the

sequence (7,) has a future directed, future inextendible, nonspacelike limit

curve y : [O,a) + M with y(0) = p. Relabeling the q, if necessary, we may

suppose that the sequence {y,) itself distinguishes y.

It remains to show that if x E 7 with p # x is arbitrary, then L(y[p, x]) =

d(p, x). Since {y,) distinguishes y, we may choose x, E y, for each n such

that x, -+ x as n + co. Choose a subsequence (7,) of {y,) by Proposition

3.34 such that {~m[p,xm]) converges to y[p,x] in the C0 topology on curves.

Hence there exists an N > 0 such that y,[p,x,] C BN for all m 2 N.

Since BN is compact and do(p, q,) + co, there exists an Nl 2 N such that

q, $ BN for all m 2 Nl. Hence L(ymip, x,]) = d(p, x,) = d [B,](p, 2,) for

all m 2 N1. Using Lemma 8.5 and the upper semicontinuity of Lorentzian arc

length in the C0 topology on curves, we then obtain

d(p, x) 5 lim inf d x,) = lim inf L(y,[p, x,])

5 limsupL(ym[p, xm1) 5 L(y[p,xI)

whence d(p, x) = L(y[p, x]) as in Proposition 8.9. 0

8.3 Causally Disconnected Space-times a n d Nonspacelike

Geodesic Lines

In this section we define and study the class of causally disconnected space-

times. Our definition of this class of spacetimes is motivated by the geometric

realization, discussed in the introduction to this chapter, of the ends of a

8.3 DISCONNECTED SPACE-TIMES, NONSPACELIKE LINES 283

noncompact complete Riemannian manifold by geodesic lines [cf. Freudenthal

(1931) for the original definition of ends of a noncompact Hausdorff topological

space]. Recall that an infinite sequence in a noncompact topological space is

said to diverge to infinity if given any compact subset C, only finitely many

elements of the sequence are contained in C.

Definition 8.11. (Causally Disconnected Space-time) A space-time is

said to be causally disconnected by a compact set K if there exist two infinite

sequences {p,) and {q,) diverging to infinity such that for each n, p, < q,,

p, # q,, and all future directed nonspacelike curves from p, to q, meet K. A

space-time (M, g) that is causally disconnected by some compact K is said to

be causally disconnected.

Note first that if k # n, then pk is not necessarily causally related to q,

or p,. Also the compact set K may be quite different from a Cauchy surface

(cf. Theorem 3.17) and non-globally hyperbolic, strongly causal space-times

may be causally disconnected even though they contain no Cauchy surfaces.

An example is provided by a Reissner-Nordstrom space-time with e2 = m2

(cf. Figure 8.1).

It is immediate from Definition 8.11 that if ( M , g ) is causally disconnected

and gl E C(M, 9) is arbitrary, then (M,gl) is causally disconnected. Thus

causal disconnection is a global conformal invariant.

We have previously used a more restrictive version of the concept of causal

disconnection in which we assumed in addition to the conditions of Definition

8.11 that 0 < d(p,, q,) < co for each n [cf. Beem and Ehrlich (1979a, p. 171,

1979c)l. With this additional condition our previous definition was, in general,

conformally invariant only for the class of globally hyperbolic space-times.

We now give the following definition.

Definition 8.12. (Nonspacelike Geodesic Line) Let (M,g) be an ar-

bitrary space-time. A past and future inextendible, future directed, non-

spacelike geodesic y : (a, b) 4 M is said to be a nonspacelike geodesic line if

L(y ] IS, t ] ) = d(-y(s),r(t)) for all s, t with a < s < t < b.


FIGURE 8.1. A Penrose diagram for a Reissner-Nordstrom space-

time with e2 = m2 containing a causally disconnected set K and

associated divergent sequences { p , ) and {q,) is shown. Thii s p a c e

time contains no Cauchy surfaces because it is not globally hyper-

bolic.


We now establish the existence of nonspacelike geodesic lines for strongly

causal, causally disconnected space-times. This result will be an important

ingredient in the proof of singularity theorems for causally disconnected s p a c e

times in Chapter 12.

Theorem 8.13. Let ( M , g ) denote a strongly causal space-time which is

causally disconnected by a compact set K. Then M contains a nonspacelike

geodesic line y : (a, b) -+ M which intersects K.

Proof. Let K, {p,), and {q,) be as in Definition 8.11. Applying Proposition

8.9 to K , p,, and q, for each n, we obtain a future directed nonspacelike

geodesic 7, intersecting K at some point r, and satisfying at least one of the

cases (1) to (4) of Proposition 8.9. If case (4) holds for any y,, then we are

done. Thus assume that no y, satisfies case (4). Hence at least one of cases (I),

(2), or (3) holds for infinitely many n. Since the proofs are similar, we will only

give the proof assuming that case (2) holds for infinitely many n. Passing to

a subsequence if necessary, we may suppose that condition (2) holds for all n.

Since K is compact, there exists a subsequence {r,) of {r,) such that r, -+ r

as m --+ m. Extend each y, past p, to get a nonspacelike curve, still denoted

by y,, that is past as well as future inextendible for each m. By Proposition

3.31, the sequence {y,) has a future directed, past and future inextendible,

nonspacelike limit curve y with r f y. Relabeling if necessary, we may assume

that {y,) distinguishes y. We will prove that y is the required nonspacelike

line. To prove this, it suffices to show that if z , y f y are distinct points with

x Q r 6 y, then L(y[x, y]) = d(x, y). Since {y,) distinguishes y, we may find

points x,, y, E ym such that x, + x and y, --+ y as m --+ m. Passing to

a subsequence {yk} of (7,) if necessary, we may suppose by Proposition 3.34

that yk [xk, yk] converges to y [x, y] in the fl topology on curves. Since y [x, y]

is compact in M , there exists an N > 0 such that y[x, y] C Int(BN). By

definition of the C0 topology on curves, there is then an Nl 2 N such that

yk[zk, yk] C Int(BN) for all k 2 Nl. Since {pk) diverges to infinity and BN is

compact, there is an Nz > Nl such that pk $ BN for all k 2 Nz. Consequently,

xk comes after pk on yk for all k 2 N2 so that yk[xk,yk] is maximal for all


k > N2. We thus have

d(x, y) 5 lim inf d(xk, yk) = lim inf L(yk[zk, yk])

5 l i m s u ~ L ( ~ k [ ~ k , ~ k ] ) 5 L(Y[x, Y]) < ~ ( x , Y ) .

Hence d(x, y) = L(y[x, y]) as required. 0

We now give several criteria, expressed in terms of the global geodesic struc-

ture, for globally hyperbolic space-times and for strongly causal space-times

to be causally disconnected. In particular, we are able to show that all two-

dimensional globally hyperbolic spacetimes are causally disconnected. Also

one of our criteria (Proposition 8.18) together with Theorem 8.13 implies that

if a strongly causal space-time (M,g) has no null geodesic rays, then (M, g)

contains a timelike geodesic line.

Recall that an inextendible null geodesic y : (a, b) -+ (M, g) is said to be a

null geodesic line if d(y(s),y(t)) = 0 for all s, t with a < s 5 t < b.

Proposition 8.14. Let (M,g) be strongly causal. If (M, g) contains a null

geodesic line, then (M, g) is causally disconnected.

Proof. Let c : (a, b) --+ (M, g) be the given null geodesic line. Choose d with

a < d < band put K = {c(d)). Choose sequences {s,), {t,) with s, < d < t, and s, -+ a, t, + b. Put p, = c(s,) and q, = c(t,). Since c is both future and

past inextendible and (M, g) is strongly causal, both {p,) and {q,) diverge to

infinity by Proposition 3.13. Now because c is a maximal null geodesic, any

future directed nonspacelike curve a from p, to q, is a reparametrization of

cl [s,, t,], whence a meets K as required. (First, a may be reparametrized

to be a smooth future directed null geodesic segment a : [O, 11 --+ M with

a(0) = p,, a(1) = q, by Theorem 4.13. If a l ( l ) # Xcl(tn) for some X > 0,

then q,+~ E I+(pn). But this contradicts d(p,, qn+l) = 0 since c is maximal.

Hence, a'(1) = Xc1(tn) for some X > 0. But then since (M, g) is causal, a must

simply be a reparametrization of c 1 [s,, t,].) D

Proposition 8.14 implies that Minkowski space-time, de Sitter space-time,

and the Friedmann cosmological models are all causally disconnected. Ako,

the Einstein static universe (cf. Example 5.11) shows that there are globally

8.3 DISCONNECTED SPACSTIMES, NONSPACELIKE LINES 287

hyperbolic, causally disconnected spacetimes which do not have null geodesic

lines. Thus the existence of a null geodesic line is not a necessary condition

for a globally hyperbolic space-time to be causally disconnected.

Evidently, the strong causality in Proposition 8.14 may be replaced by any

other nonimprisonment condition which guarantees that both ends of c diverge

to infinity, together with the requirement that (M, g) be causal.

In the next proposition we will give a sufficient condition for a strongly

causal space-time (M,g) to be causally disconnected. For the proof of this

result (Proposition 8.18), it is necessary to recall some additional concepts

from elementary causality theory. A subset S of (M, g) is said to be achronal

if no two polnts of S are chronologically related. Given a closed subset S of (M,g), the future Cauchy development or domazn of dependence D f (S)

of S is defined as the set of all points q such that every past inextendible

nonspacelike curve from q intersects S. The future Cauchy honzon H+(S) is

given by Hf (S) = D+(S) - I-(D+(S)). The future honsmos E+(S) of S is

defined to be E+(S) = J+(S) - I+(S). An achronal set S is said to be future

trapped if E+(S) is compact. Details about these concepts may be found in

Hawking and Ellis (1973, pp. 201, 202, 184, and 267, respectively).

For the proof of Proposition 8.18 we also need to use a result first obtained

in Hawking and Penrose (1970, p. 537, Lemma 2.12). This result is presented

somewhat differently during the course of the proof of Theorem 2 in the text of

Hawking and Ellis (1973, p. 266). In the proof of this theorem, it is assumed

that dim M > 3 and that (M,g) has everywhere nonnegative nonspacelike

Ricci curvatures and satisfies the generic condition [conditions (1) and (2) of

Theorem 21. However, it may be seen that in the proof of Lemma 8.2.1 and

the following corollary in Hawking and Ellis (1973, pp. 267-269), it is only

necessary to assume that (M, g) is strongly causal to obtain our Lemma 8.15

and Corollary 8.16. We now state these two results for completeness.

Lemma 8.15. Let A be a closed subset of the strongly causal space-time

(M,g). Then H+(Ef (A)) is noncompact or empty.

From this lemma, one obtains as in Hawking and Penrose (1970, p. 537) or

Hawking and Ellis (1973, pp. 268-269) the following corollary.


Corollary 8.16. Let (M,g) be strongly causd. If S Ss future trapped in

(M,g), i.e., E+(S) is compact, then there is a future inextendible timelike

curve y contained in D+(E+(S)).

It will also be convenient to prove the following lemma for the proof of

Proposition 8.18.

Lemma 8.17. Let (M, g) be strongly causal. If E+(p) is noncompact, then

E+(p) contains an infinite sequence {q,) which diverges to infinity.

Proof. If E+(p) is closed, this is immediate since a closed and noncompact

subset of M must be unbounded with respect to do. Thus assume that E+(p)

is not closed. Then there exists an infinite sequence {x,) C E+(p) such that

xn -t x 4 E+(p) as n -4 co. Since x, E E+(p), we have d(p, x,) = 0 and hence

as x, E J+(p), there exists a maximal future directed null geodesic segment

yn from p to x, for each n. Extend each /, beyond x, to a future inextendible

nonspacelike curve still denoted by y,. By Proposition 3.31, the sequence {y,)

has a future inextendible, future directed, nonspacelike limit curve y : [O, a) +

M with y(0) = p. We may assume that the sequence {y,) itself distinguishes

y. If x E y, then x E J+(p). Since d(p,x) 5 liminfd(p,z,) = 0, we then

have x E J+(p) - I+(p) = E+(p), in contradiction to the assumption that

x 4 E+(p). Thus x f y. We now show that y[O, a) is contained in E+(p). To

this end, let z E y be arbitrary. Since {y,) distinguishes y, we may find z, E y,

such that z, -t z as n -+ a. By Proposition 3.34, there is a subsequence {yk)

of {?,) such that 7k[P, zk] converges to yip, z] in the c0 topology on curves.

Since x 4 7, we may find an open set U containing 7[p, z] such that x f n. Since xk -t x, it follows that zk comes before xk on yk for all k sufficiently

large. Thus y[p, zk] is maximal and d(p, zk) = 0 for all k sufficiently large.

Hence d(p, z) 5 liminf d(p, zk) = 0. Since z was arbitrary, we have thus shown

that d(p, z) = 0 for all z E y. Since y is a nonspacelike curve, 7 is then a

maximal, future directed, future inextendible geodesic ray. Letting {t,) be

any infinite sequence with t, -4 a- and setting q, = y(t,) gives the required

divergent sequence. D

8.3 DISCONNECTED SPACGTIMES, NONSPACELIKE LINES 289

With these preliminaries completed, we may now obtain a sufficient condi-

tion for strongly causal space-times to be causally disconnected. Minkowski

space-time shows that this condition is not a necessary one. Recall that a future directed, future inextendible, null geodesic y : [0, a) --+ M is said to be

a null geodesic my if d(y(O), y(t)) = 0 for all t with 0 _< t < a.

Proposition 8.18. Let (M,g) be strongly causal. If p E M is not the origin of any future [respectively, past] directed null geodesic ray, then (M, g)

is causally disconnected by the future [respectively, past] horismos E+(p) =

J+(p) - I+(p) [respectively, E- (p) = J- (p) - I- (P)] of p.

Proof. We first show that the assumption that p is not the origin of any

future directed null geodesic ray implies that E+(p) is compact. For suppose

that E+(p) is noncompact. Then there exists an infinite sequence {q,) C E+(p) which diverges to infinity by Lemma 8.17. Since q, E E+(p), we have

d(p, q,) = 0 for all n. As qn E Jf (p), there exists a future directed null

geodesic yn from p to q, by Corollary 4.14. Extend each yn beyond qn to a

future inextendible nonspacelike curve, still denoted by 7,. Let y be a future

inextendible nonspacelike limit curve of the sequence (7,) with y(0) = p,

guaranteed by Proposition 3.31. Using Proposition 3.34 and the fact that

the qn's diverge to infinity, it may be shown along the lines of the proof of

Theorem 8.10 that if q is any point on y with q # p, q 2 p, then L(y [p, q]) =

d(p, q). Thus y may be reparametrized to a null geodesic ray issuing from p,

in contradiction. Hence Ef (p) is compact.

We now show that E+(p) causally disconnects (M,g). Since E+(p) is com-

pact, the set {p) is future trapped in M. Thus by Corollary 8.16, there is a

future inextendible timelike curve y contained in D+(Ef (p)). Extend y to a

past as well as future inextendible timelike curve, still denoted by y. From

the definition of D+(Ef (p)), the curve y must meet E+(p) at some point r .

Since E+(p) is achronal and y is timelike, y meets E+(p) at no other point

than r. Now let {p,) and (9,) be two sequences on y both of which diverge

to infinity and which satisfy p, << r << q, for each n (cf. Proposition 3.13). To

show that {p,), {q,), and E+(p) causally disconnect (M,g), we must show

that for each n, every nonspacelike curve X : [O, 11 -+ .hri with X(0) = p, and


X(1) = q, meets E+(p). Given A, extend X to a past inextendible curve 5 by

traversing y up to p, and then traversing X from p, to q, (cf. Figure 8.2). As

qn E D f (E+(p)), the curve 5 must intersect Ef (p). Since y meets E+(p) only

a t T, it follows X intersects E f (p). Thus {p,), {q,), and K = E+(p) causally

disconnect (M, g) as required. 13

Combining Theorem 8.13 and Proposition 8.18, we obtain the following

result on the geodesic structure of strongly causal space-times with no null

geodesic rays. Examples of such space-times are the Einstein static universes

(Example 5.1 1).

Theorem 8.19. Let (M, g) be a strongly causal space-time such that no

point is the origin of any future directed null geodesic ray Then (M, g) contains

a timelike geodesic line.

Proof. By Proposition 8.18, (M,g) is causally disconnected. Thus (M,g)

contains a nonspacelike line by Theorem 8.13. By hypothesis, the line must

be timelike rather than null. 0

An equivalent result may be formulated using the hypothesis that no point

is the origin of any past directed null geodesic ray.

Using Propositions 8.14 and 8.18, we are now able to show that all two-

dimensional globally hyperbolic space-times are causally disconnected. We

first establish the following lemma.

Lemma 8.20. Let (M, g) be a two-dimensional globally hyperbolic space-

time. If E+(p) [respectively, E-(')] is not compact, then both of the future

[respectively, past ] directed and future [respectively, past] inextendible null

geodesics starting at p are maximal.

Proof. Assume E+(p) is noncompact for some p E M. Let cl : 10, a ) -+ M

and c:! : [0, b) + M be the two future directed, future inextendible, null

geodesics with cl(0) = cz(0) = p. Suppose that cl is not maximal. Setting

to = supit E [0, a) : d(p, cl (t)) = 0), we have 0 < t o < a since cl is not maximal

and (M,g) is strongly causal (cf. Section 9.2). Let q = cl(t0) and choose

t, --i t g with t, < a for each n. By the lower semicontinuity of Lorentzian


FIGURE 8.2. In the proof of Proposition 8.18, the set E*(p) is

shown to causally disconnect (M,g) if p is not the origin of any

future directed null geodesic ray. The timelike curve y intersects

E+(p) in the single point T, and any nonspacelike curve X from p,

to q, must meet E+(p).

distance, w-e have d(p, q) = 0. Since cl(t,) E I+(p) and (M, g) is globally

hyperbolic, there is a maximal future directed timelike geodesic y, from p to

292 8 MAXIMAL GEODESICS, DISCONNECTED SPACETIMES

cl(tn) for each n. The sequence (7,) has a limit curve y which is a nonspacelike

geodesic, and y joins p to q by Corollary 3.32. Ikthermore, d(p, q) = 0 implies

that y is a null geodesic. Since dim M = 2, y is either a geodesic subsegment of

cl or a subsegment of cz. If y 5 cz, then cz passes through q at some parameter

value tb. In this case E+(p) = {cl 1 [O, to]) U {cz 1 [O, tb]) is compact and we

have a contradiction.

Assume y cl and let U(q) be a convex normal neighborhood about q.

Let U(q) be chosen so small that no nonspacelike curve which leaves U(q)

ever returns. The inextendible null geodesics of U(q) may be divided into

two disjoint families Fl and F2, each of which cover U(q) simply (cf. Section

3.4). Let Fl be the class which contains the null geodesic cl n U(q). Let c3

be the unique null geodesic of F2 which contains q. For some fixed large n

the timelike geodesic segment yn from p to cl(tn) must intersect c3 at some

point r. We must have q < r since if r < q, then p << r would yield p << q,

which is false. However, q < r, q = cl n cs, q < cl(tn), and r E cs imply

r $ cl(tn), [cf. Busemann and Beem (1966, p. 245)]. This contradicts the fact

that r << cl(tn) because r comes before cl(tn) on the timelike geodesic 7,.

This establishes the lemma. D

Theorem 8.21. All two-dimensional globally hyperbolic space-times are

causally disconnected.

Proof. Let (M, g) be a two-dimensional globally hyperbolic space-time. If

E+(po) is compact for some po E M, then each future directed null geodesic

starting at po has a cut point and thus fails to be globally maximal (cf. Section

9.2). Thus (M,g) is causally disconnected by Proposition 8.18. Now suppose

E+ (p) is noncompact for each p E M. Assume c : (a, b) -, M is a future

directed inextendible null geodesic. Let s , t with a < s 5 t < b be arbitrary.

Applying Lemma 8.20 at the point p = c(s), we find that c 1 [s, b) is maximal

and thus d(c(s), c(t)) = L(c] [s, t]) . This implies that c is maximal and hence

c is a null line. Using Proposition 8.14, it follows that (M,g) is causally

disconnected. O

8.3 DISCONNECTED SPACGTIMES, NONSPACELIKE LINES 293

We obtain the following corollary to Theorems 8.13 and 8.21.

Corollary 8.22. Let (M, g) be any globally hyperbolic space-time of di-

mension n = 2. Then (M, g) contains a nonspacelike line.

CHAPTER 9

THE LORENTZIAN CUT LOCUS

Let c : 1 0 , ~ ) -+ N be a geodesic in a complete Riemannian manifold

starting at p = c(0). Consider the set of all points q on c such that the portion

of c from p to q is the unique shortest curve in all of N joining p t o q. If this

set has a farthest limit point, this limit point is called the cut point of p along

the ray c. The cut locus C(p) is then defined to be the set of cut points along

all geodesic rays starting at p. Since nonhomothetic conformal changes do not

preserve pregeodesics, the cut locus of a point in a manifold is not a conformal

invariant.

The cut locus has played a key role in modern global Riemannian geometry,

notably in connection with the Sphere Theorem of Rauch (1951), Klingenberg

(1959, 1961), and Berger (1960). The notion of cut point, as opposed to the

related but different concept of conjugate point, was first defined by Poincari.

(1905). An observation of Poincark (1905) important in the later work of

Rauch, Klingenberg, and Berger was that for a complete Riemannian manifold,

if q is on the cut locus of p, then either q is conjugate to p, or else there

exist at least two geodesic segments of the same shortest length joining p to

q [cf. Whitehead (1935)l. Klingenberg (1959, p. 657) then showed that if q is

a closest cut point to p and q is not conjugate to p, there is a geodesic loop

at p containing q. Klingenberg used this result to obtain an upper bound for

the injectivity radius of a positively curved, complete Riemannian manifold

in terms of a lower bound for the sectional curvature and the length of the

shortest nontrivial smooth closed geodesic on N.

The importance of the cut locus in Ftiemannian geometry suggests inves-

tigating the analogous concepts and results for timelike and null geodesics in

space-times. The central role that conjugate points, which are closely related

296 9 T H E LORENTZIAN CUT LOCUS

to cut points, have played in singularity theory in general relativity (cf. Chap

ter 12) supports this idea. While there are many similarities between the

Riemannian cut locus and the locus of timelike cut points in a space-time,

there are also striking differences between the Riemannian and Lorentzian cut

loci. Most notably, null cut points are invariant under conformal changes.

Thus the null cut locus is an invariant of the causal structure of the space-

time (M,g). This may be used [cf. Beem and Ehrlich (1979a, Corollary 5.3)]

to show that if (M,g) is a Riedmann cosmological model, then there is a C2 neighborhood U(g) in the space C ( M , g) of Lorentzian metrics for M globally

conformal to g such that every null geodesic in (M,gl) is incomplete for all

metrics gl E U(g). Because there are intrinsic differences between null and

timelike cut points, we prefer to treat these cases separately. One such differ-

ence is that unlike null cut points, timelike cut points are not invariant under

global conformal changes of Lorentzian metric.

In Section 9.1 we consider the analogue of Riemannian cut points for time-

like geodesics. In the Lorentzian setting, timelike geodesics locally maxi-

mize the arc length between any two of their points. Thus the appropri-

ate question to ask in defining timelike cut points is whether the portion

of the given timelike geodesic segment from p to q is the longest nonspace-

like curve in all of M joining p to q. This may be conveniently formulated

using the Lorentzian distance function. Let y : [O,a) + M be a future di-

rected, future inextendible, timelike geodesic in an arbitrary space-time. Set

to = sup{t E 10, a) : d(y(O), y(t)) = L(y I [0, t])). If 0 < to < a, then y(t0) is

said to be the future timelike cut point of y(0) along y. The future timelike cut

point y(to) then has the following desired properties: (i) if t < to, then y I [O, t ]

is the only maximal timelike curve from y(0) to y(t) up to reparametrization;

(ii) y 1 [O,t] is maximal for any t 5 to; (iii) if t > to, there exists a future

directed nonspacelike curve a from y(0) to y(t) with L(a) > L(y 1 [O,t]); and

(iv) the future cut point y(to) comes at or before the first future conjugate

point of y(0) along y.

Since many of the theorems for Riemannian cut points are true only for

complete Riemannian manifolds, it is not surprising that the more "global"

9 THE LORENTZIAN CUT LOCUS 297

results given in the rest of Section 9.1 often require global hyperbolicity. What

is essential here is to know that chronologically related points at arbitrarily

large distances may be joined by maximal timelike geodesic segments. Even so,

the proofs are more technical for space-times than for Riemannian manifolds.

Instead of using the exponential map directly as in Riemannian geometry, it

is necessary to regard a sequence of maximal timelike geodesics as a sequence

of nonspacelike curves, extract a limit curve, take a subsequence converging

to the limit in the C0 topology (by Section 3.3), and finally, using the upper

semicontinuity of arc length, prove that the limit curve is maximal and hence a

geodesic. This technical argument, which is isolated in Lemma 9.6, yields the

following analogue of Poincark's Theorem for complete Riemannian manifolds.

If (M, g) is a globally hyperbolic space-time and q is the future cut point of p

along the timelike geodesic segment c from p to q, then either one or possibly

both of the following hold: (i) q is the first future conjugate point to p; or (ii)

there exist a t least two maximal geodesic segments from p to q.

In Section 9.2 we study null cut points. Even though null geodesics have zero

arc length, null cut points may still be defined using the Lorentzian distance

function. Let y : [O,a) -+ M be a future directed, future inextendible, null

geodesic with p = y(0). Set to = supit E [O,a) : d(p, y(t)) = 0). If 0 < to < a,

then $to) is called the future null cut point of y(0) along y. The null cut

point, if it exists, has the following properties: (i) y is maximizing up to and

including the null cut point; (ii) there is no timelike curve joining p to y(t) for

any t 5 to; (iii) if to < t < a, there is a timelike curve from y(0) to y(t); and

(iv) the future null cut point comes at or before the first future conjugate point

of y(0) along y. For globally hyperbolic space-times, it is true for null as well

as timelike cut points that the analogue of Poincark's Theorem for complete

Riemannian manifolds is valid. Thus if (M,g) is globally hyperbolic and q is

the future null cut point of p = y(0) along the null geodesic y, then either one

or possibly both of the following hold: (i) q is the first future conjugate point of

p along y; or (ii) there exist at least two maximal null geodesic segments from

p to q. We conclude Section 9.2 by using null cut points to prove singularity

theorems for null geodesics following Beem and Ehrlich (1979a, Section 5).

298 9 THE LORENTZIAN CUT LOCUS 9.1 T H E TIMELIKE CUT LOCUS 299

The nonspacelike cut locus, the union of the null and timelike cut loci of Definition 9.2. Let T-IM = {v E T M : v is future directed and g(v,v) =

a given point, is studied in Section 9.3. For complete Riemannian manifolds, -1). Given p E M , let T-lMlp denote the fiber of T-l M at p. Also given

if q is a closest cut point to p, then either q is conjugate to p or there is a v E T-1M, let c, denote the unique timelike geodesic with cV1(0) = v.

geodesic loop based a t p passing through q. The globally hyperbolic analogue It is immediate from the reverse triangle inequality that if y : [0, a) -+ M is

(Theorem 9.24) of this result has a slightly different flavor, however. If (M, g) a future directed nonspacelike geodesic and d(y(O), y(s)) = L ( y 1 [O, s]), then is a globally hyperbolic space-time and q E M is a closest (nonspacelike) cut d(y(O),y(t)) = L ( y 1 [O,t]) for all t with 0 I t 5 s. Also, the reverse triangle point to p, then q is either conjugate to p or else q is a null cut point to p. inequality implies that if d(y(O), y(s)) > L(y 1 [0, s]), then d(y(O), y(t)) > Thus there is no closest nonconjugate timelike cut point to p. We also show L(y 1 [O,t]) for all t with s < t < a. Hence the following definition makes

that for globally hyperbolic space-times, the nonspacelike and null cut loci are

closed (Proposition 9.29). It can be seen (Example 9.28) that the hypothesis Definition 9.3. Define the function s : T-lM -, R u {m) by s(v) = of global hyperbolicity is necessary here. supit 2 0 : d(r(v), c,(t)) = t).

In Section 9.4 we treat null and timelike cut points simultaneously, but

nonintrinsically, using a different tool than the unit timelike sphere bundle We may first note that if d(p,p) = m , then s(v) = 0 for all v E T-lM with

of Section 9.1. This approach, developed in collaboration with G. Gallow r(v) = p. Also S(V) > 0 for all v E T-1M if (M,g) is strongly causal. The

while non-intrinsic, enables the hypothesis of timelike geodesic completenes number s(v) may be interpreted as the "largest" parameter value t such that

to be deleted from Proposition 9.30. c, is a maximal geodesic between c,(O) and c,(t). Indeed from Lemma 9.1 we

know that the following result holds.

9.1 The Timelike Cut Locus Corollary 9.4. For 0 < t < s(v), the geodesic c, : [0, t ] -+ M is the unique

Recall that a future directed nonspacelike curve y from p to q is said t maximal timelike curve (up to repararnetrization) from c,(O) to c,(t).

be maximal if d ( p , q) = L(y). We saw above (Theorem 4.13) that a maxim The function s fails to be upper semicontinuous for arbitrary spacetimes as

future directed nonspacelike curve may be reparametrized to be a geode may easily be seen by deleting a point from Minkowski space. But for timelike

We also recall the following analogue of a classical result from Rieman geodesically complete space-times we have the following proposition.

geometry. The proof may be given along the lines of Kobayashi (1967, p. Proposition 9.5. Let v f T-1M with s(v) > 0. Suppose either that using in place of the minimal geodesic segment from pl to p2 in the Rieman (v) = +m, or ~ ( v ) is finite and c,(t) = exp(tv) extends to [0, s(v)]. Then s is proof the fact that if p << q and p and q are contained in a convex norm pper semicontinuous at v f T-1M. Especially, if (M, g) is timelike geodesi- neighborhood, then p and q may be joined by a maximal timelike geo ally complete, then s : T-1M -+ W U { ~ ) is everywhere upper semicontinuous. segment which lies in this neighborhood.

Proof. It suffices to show the following. Let v, -+ v in T-lM with (s(vn)) Lemma 9.1. Let c : [O, a] -+ M be a maximal timelike geodesic segm onverging in R U {a). Then s(v) 2 lim s(v,). If s(v) = m, there is nothing

Then for any s , t with 0 5 s < t < a , the curve c 1 [s, t ] is the unique maxi prove. Hence we assume that s(v) < lims(vn) = A and s(v) < co and geodesic segment (up to parametrization) from c(s) to c(t). rive a contradiction.

Before commencing our study of the timelike cut locus, we need to defi We may choose 6 > 0 such that s(v) + 6 < A is in the domain of c,

the unit future observer bundle T-IM [cf. Thorpe (1977a,b)]. d also assume that s(v,) 2 s(v) + 6 = b for all n. Let c, = tun. Since

300 9 THE LORENTZIAN CUT LOCUS 9.1 THE TIMELIKE CUT LOCUS 301

b < s(v,), we have d(n (v,), c,(b)) = b for all n. Since v, 4 v, we have by lower and q = c,(A + 6). Since v, 4 v and c, is defined for some parameter values semicontinuity of distance that d(r(v), ~ ( b ) ) 5 liminf d(r(v,), ~ ( b ) ) = b. beyond A + 6, the geodesics c, must be defined for some parameter values

Thus d(x(v), c,(b)) N. of arc length. On the other hand, d(n(v), ~ ( b ) ) 2 L(q, 1 [0, b]) so that Now c, I [O, b,] cannot be maximal since b, > s(v,). Because M is globally d(r(v),q,(b)) = L(c, I [0, b]) = b. Hence s(v) 2 b = s(v) + 6, in contra- hyperbolic and c,(O) << c,(b,), we may find maximal unit speed timelike diction. 0 geodesic segments y, : [O,d(p,,q,)] M from p, to q,. Set w, = yn1(O).

Since c, ( [0, s(v)) is a maximal geodesic and thus has no conjugate points, it In order to prove the lower semicontinuity of s for globally hyperbolic space-

is impossible for v to be a limit direction of {w, : n 2 N). Thus the maximal times, it will first be useful to establish the following lemma.

geodesic c, joining p to q given by Lemma 9.6 applied to {w,) is different from Lemma 9.6. Let (M, g) be a globally hyperbolic space-time, and let {p,) c,. This then implies that s(v) < A + 6, which contradicts A + 6 < s(v).

and {q,) be two infinite sequences of points with p, p and q, --+ q where

p << q. Assume c, : [O,d(p,, q,)] 4 M is a unit speed maximal geodesic The following example of a strongly causal space-time which is not globally

segment from p, to q,, and set v, = ~ ' ( 0 ) E T-lM. Then the sequence {v,) hyperbolic shows that the hypothesis of global hyperbolicity is necessary for

has a timelike limit vector ur f T-l M. Moreover, c,,, : [0, d(p, q)] --+ M is a the lower semicontinuity of the function s in Proposition 9.7. Let W2 be given

maximal geodesic segment from p to q. the usual Minkowski metric ds2 = dx2 - dy2, and let (M,g) be the space-

time formed by equipping M = W2 - {(l,y) € R2 : 1 < y < 2) with the Proof. By Corollary 3.32, there is a nonspacelike future directed limit curve

induced metric (cf. Figure 9.1). Let p = (O,O), p, = (l/n,O), v = bldyl,, and c of c, from p to q. By Proposition 3.34 and Itemark 3.35, we have L(c) '2

v, = b/dylpn for all n 2 1. Then v, --+ v as n -+ m. Also let y(t) = (0,t) limsup L(%) = limd(~,, q,) = d(p,q) > 0. Thus as d(p, q) 2 L(c), it follows

and yn(t) = (p,, t) for all t 2 0. Conformally changing g on the compact set that L(c) = d(p, q) > 0. Hence Theorem 4.13 implies that the curve c may

C shown in Figure 9.1 which is blocked from I+(p) by the slit {(I, y) E It2 : be reparametrized to be a maximal timelike geodesic segment from p to q.

1 < y 5 21, we obtain a metric Zj for M with the following properties. First Finally, w = c'(0) / [-g(cl(0), c'(o))]'/~ is the required tangent vector.

y is still a maximal geodesic in (M,Zj) so that s(v) = +m. But for each n

We are now ready to prove the lower semicontinuity of s for globally hype there exists a timelike curve on passing through the set C and joining p, to

bolic space-times. a point q, = ( l l n , y,) on yn with y, 5 4 such that L(a,) > L(ynIpn, q,]).

Hence y,[p,, q,] fails to be maximal for all n so that s(v,) 5 4 for all n. Thus Proposition 9.7. If (M,g) is globally hyperbolic, then the function

s is not lower semicontinuous. Note that (M, g ) is strongly causal but (M, Zj) T- lM --+ W U {co) is lower semicontinuous.

is not globally hyperbolic since J + ( ( l , 0)) n J-((1,3)) is not compact. The

Proof. It suffices to prove that if v, -+ v in ThlM and s(v,) --+ A analogous construction may be applied to n-dimensional Minkowski space to W U {m}, then s(v) 5 A. If A = oo, there is nothing to prove. Thus supp roduce a strongly causal n-dimensional space-time for which the function s

A < m. Assuming s(v) > A, we will derive a contradiction. ils to be lower semicontinuous.

Choose 6 > 0 such that A + 6 < s(v). (Since A + 6 < s(v), the Globally hyperbolic examples also may be constructed for which the func- c,(A + 6) exists even if G, does not extend to s(v).) Define b, = s(v, tion s is not upper semicontinuous. However, this discontinuity may occur at and let No be such that b, < s(v) for all n 2 No. Put c, = cvn, pn = ~n a tangent vector v E T-lM in a globally hyperbolic space-time only if s(v) is

302 9 THE LORENTZIAN C U T LOCUS

finite and ~ ( t ) does not extend to t = s(v). Combining Propositions 9.5 and

9.7, we obtain the following result.

Theorem 9.8. Let ( M , g ) be globally hyperbolic, and suppose for v E

T- lM that either s (v) = +co, or s (v ) is finite and c,(t) = exp(tv) extends to

[0, s (v)] . Then s is continuous at v E T- lM. Especially, i f ( M , g ) is globally

hyperbolic and timelike geodesically complete, then s : T-1M 4 B U {CQ} is

everywhere continuous.

W e are now ready to define the timelike cut locus.

Definition 9.9. (Future and Past Timelike Cut Loci) The future timelike

cut locus in T,M is defined t o be r+(p) = {s(v)v : v E T-lMIp and 0 < s(v) < co). The future timelike cut locus C , f (p ) of p in M is defined to be

C t ( p ) = exp,(r+(p)). I f 0 < s(v) < co and c,(s(v)) exists, then the point

c,(s(v)) is called the future cut point o f p = c,(O) along c,. The past timelike

cut locus C;(p) and past cut points may be defined dually.

W e may then interpret s (v) as measuring the distance from p up to the

ture cut point along c,. Thus Theorem 9.8 implies that for globally hyperboli

timelike geodesically complete spacetimes, the distance from a fixed p E

t o its future cut point in the direction v E T-lMlp is a continuous function

v.

W e now give Lorentzian analogues of two well-known results relating cut a

conjugate points on complete Riemannian manifolds. The following proper

o f conjugate points is well known [Hawking and Ellis (1973, pp. 111-116

Lerner (1972, Theorem 4(6))].

Theorem 9.10. A timelike geodesic is not maximal beyond the first co

jugate point.

In the language o f Definition 9.9 this may be restated as follows.

Corollary 9.11. The future cut point o f p = c,(O) along c, comes no la

than the first future conjugate point o f p along c,.

Utilizing this fact, we may now prove the second basic result on cut an

conjugate points in globally hyperbolic space-times.

9.1 THE TIMELIKE CUT LOCUS

FIGURE 9.1. A strongly causal space-time ( M , 5) is shown with unit timelike tangent vectors v, which converge t o v, but with

s (v ) = +m and s(v,) 5 4 for alln 2 1. Hence s : T - I M -, R ~ { c o }

is not lower semicontinuous.

Theorem 9.12. Let ( M , g) be globally hyperbolic. I f q = c( t ) is the future

cut point o f p = c(0) along the timelike geodesic c from p to q, then either one

or possibly both o f the following hold:

(1 ) The point q is the first future conjugate point o f p along c.

304 9 THE LORENTZIAN CUT LOCUS

(2) There exist a t least two future directed maximal timelike geodesic seg-

ments from p to q.

Proof. Without loss of generality we may suppose that c = c, for some

v E T-lM and thus that t = d(p, q) = s(v). Let {t,) be a monotone decreasing

sequence of real numbers converging to t. Since c(t) E M, the points c(tn) exist

for n sufficiently large. By global hyperbolicity, we may join c(0) to c(tn) by

a maximal timelike geodesic c, = c,,, with v, E T-1 MIp. Since t, > t = s(v),

we have v # v, for all n. Let w E T-lM be the timelike limiting vector for

{v,) given by Lemma 9.6. If v # w, then c and c, are two maximal timelike

geodesic segments from p to q.

It remains to show that if v = w, then q is the first future conjugate point

of p along c. If v = w, then there is a subsequence {v,) of {v,) with v, -+ v.

If v were not a conjugate point, there would be a neighborhood U of v in

T-lMI, such that expp : U -+ M is injective. On the other hand, since c, and

c 1 [0, t,] join c(0) to c(t,) and v, -+ v, no such neighborhood U can exist.

Thus q is a future conjugate point of p along c. By Corollary 9.11, q must be

the first future conjugate point of p along c.

Theorem 9.12 has the immediate implication that for globally hyperbolic

spacetimes, q E C:(p) if and only if p E Cc(q).

The timelike cut locus of a timelike geodesically complete, globally hyper-

bolic space-time has the following structural property which refines Theorem

9.12. We know from this theorem that if q f C$tp) and q is not conjugate

to p, then there exist at least two maximal geodesic segments from p to q.

Accordingly, it makes sense to consider the set

Seg(p) = { q E C$(p) : there exist at least two future directed

maximal geodesic segments from p to q).

Since s : TP1M -+ W U (m) is continuous by Theorem 9.8 and maximal

geodesics joining any pair of causally related points exist in globally hyperbolic

space-times, it may be shown that Seg(p) is dense in C,f(p) for all p E M. The

proof may be given along the lines of Wolter's proof of Lemma 2 for complete

9.2 THE NULL CUT LOCUS 305

Riemannian manifolds [cf. Wolter (1979, p. 93)]. The dual result also holds

for the past timelike cut locus C c (p).

9.2 The Null Cut Locus

The definition of null cut point has been given in Beem and Ehrlich (1979a,

Section 5) where this concept was used to prove null geodesic incompleteness

for certain classes of space-times. Let y : [0, a) -+ (MI g) be a future directed

null geodesic with endpoint p = $0). Set to = supit E [0, a) : d(p, y(t)) = 0).

If 0 < to < a , we will say y(t0) is the future null cut point of p on y. Past null

cut points are defined dually. Let C$(p) [respectively, CG (p)] denote the future

[respectively, past] null cut locus of p consisting of all future [respectively,

past] null cut points of p. The definition of Cz(p) together with the lower

semicontinuity of distance yields d(p, q) = 0 for all q E C2(p). We then define

the future nonspacelike cut locus to be C+(p) = C$(p) U Cz(p). The past

nonspacelike cut locus is defined dually. For a subclass of globally hyperbolic

space-times, Budic and Sachs (1976) have given a different but equivalent

definition of null cut point using null generators for the boundary of I+(p).

The geometric significance of null cut points is similar to that of timelike

ut points. The geodesic y is maximizing from p up to and including the cut

oint ?(to). That is, L ( y 1 [O,t]) = d(p,y(t)) = 0 for all t 5 to. Thus there

no timelike curve joining p to y(t) for any t with t 5 to. In contrast, the

eodesic y is no longer maximizing beyond the cut point y(t0). In fact, each

oint y(t) for to < t < a may be joined to p by a timelike curve.

Utilizing Proposition 2.19 of Penrose (1972, p. 15) and the definition of

aximality, the following lemma is easily established.

Lemma 9.13. Let (M, g) be a causal space-time. If there are two future

irected null geodesic segments from p to q, then q comes on or after the null

cut point of p on each of the two segments.

The cylinder S1 x W with Lorentzian metric ds2 = dedt shows that the

assumption that (M,g) is causal is needed in Lemma 9.13.

We next prove the null analogue of Lemma 9.6.


Lemma 9.14. Let (M, g) be globally hyperbolic, and let p, q be distinct

points in M with p 6 q and d(p,q) = 0. Assume that p, -+ p, q, - q and

p, < q,. Let c, be a maximal geodesic joining p, to q, with initial direction

v,. Then the set of direction {v,) has a limit direction w, and c, is a maximal

null geodesic from p to q.

Proof. Using Corollary 3.32 we obtain a future directed nonspacelike limit

curve X from p to q. Since d(p, q) = 0, the curve X must satisfy L(X) = 0. It

follows that X may be reparametrized to a maximal null geodesic.

We may now obtain the null analogue of Theorem 9.12.

Theorem 9.15. Let (M,g) be globally hyperbolic and let q = c(t) be the

future null cut point of p = c(0) dong the null geodesic c. Then either one or

possibly both of the following hold:

(1) The point q is the first future conjugate point of p along c.

(2) There exist a t least two future directed maximal null geodesic segments

from p to q.

Proof. Let v = c'(O), and let t, be a monotone decreasing sequence of real

numbers with t, -+ t. Since q E M, we know that c(t,) exists for all sufficiently

large n. Since (M, g) is globally hyperbolic, we may find maximal nonspacelike

geodesics c, with initial directions v, joining p to c(t,). By Lemma 9.14 the

set of directions {v,) has a limit direction w. If v # w, then the geodesic c,

is a second maximal null geodesic joining p to q as d(p, q) = 0. If v = w, then

q is conjugate to p along c. Since d(p,q) = 0, q must be the first conjugate

point of c (cf. Theorem 10.72). D

We now show how the null cut locus may be applied to prove theorems

on the stability of null geodesic incompleteness. The key ideas needed for

this application are first, that many physically interesting space-times may be

conformally embedded in a portion of the Einstein static universe (cf. Exa

ple 5.11) that is free of null cut points and second, that null cut points

irivariant under conformal changes of metric. A discussion of global con

ma1 embeddings of anti-de Sitter space-time and the Friedmann cosmologi


models in the Einstein static universe is given in Penrose (1968) [cf. Hawking

and Ellis (1973, pp. 132, 141)].

We now digress briefly to give an explicit computational discussion of the

well-known fact that null pregeodesics are invariant under global conformal

changes. An indirect proof of the conformal invariance of null cut points may

also be given using the Lorentzian distance function.

Recall that a smooth curve y : J -+ M is said to be a pregeodesic if y can

be reparametrized to a smooth curve c which satisfies the geodesic differential

equation VC~cr(t) = 0. Also recall the following definition.

Definition 9.16. (Global Conformal Dzffeomorphism) A diffeomorphism f : (MI, gl) --+ (M2, g2) of MI onto M2 is said to be a global conformal diffeo-

morphism if there exists a smooth function R : MI -+ R such that

The space-time (M1,gl) is said to be globally conformally diffeomorphic to an

open subset U of (M2,gz) if there exists a diffeomorphism f : M1 -+ U and a

smooth function R : Ml -r R such that

The purpose of using the factor e20 rather than just a positive-valued

smooth function in Definition 9.16 is to give the simplest possible formula

relating the connections V for (M1,gl) and 9 for (M2, 92). Explicitly, it may

be shown that if f*g2 = e20gl where j : (M1,gl) --+ (Mz,g2) is any smooth

function, then

: for any pair X, Y of vector fields on MI. Here "gradRn denotes the gradient

j vector field of C2 with respect to the metric g1 for MI. : Using formula (9.1), it is now possible to show that if y : J --+ MI is a null b geodesic in (MI, gl) , then a = f 0 y : J -+ M2 is a null pregeodesic in (M2,g2). h

308 9 T H E LORENTZIAN CUT LOCUS

The crux of the matter is that because yl(t) is null and V,q' = 0, formula

(9.1) simplifies to

?bjal(t) = 2y1(t)(R)a'(t)

for all t E J. Note, however, that if y were a timelike geodesic, then the factor

gl (y', y') f* (grad R) in formula (9.1) would prevent f o y from being a timelike

pregeodesic in (M2, g2).

Lemma 9.17. Let f : (MI, gl) -+ (M2, g2) be a global conformal diffeo-

morphism of MI onto M2. Then y : J -+ (MI, gl) is a null pregeodesic for

(MI, gl) iff f o y is a null pregeodesic for (M2,gz).

Proof. Since f -' : (M2, g2) + (MI, gl) is also a conformal diffeomorphism,

it is enough to show that if y : J + MI is a null geodesic, then a = f oy is a null

pregeodesic of (M2, 92). That is, we must show that a can be reparametrized

to be a null geodesic of (M2,g2). But it has already been shown that

for some smooth function f : J --t W. Just as in the classical theory of

projectively equivalent connections in Riemannian geometry, however, formula

(9.2) implies that a is a pregeodesic [cf. Spivak (1970, pp. G37 ff. )].

We are now ready to prove the conformal invariance of null cut points

under global conformal diffeomorphisms f : (M1,gl) -r (M2, g2). Notice that

if (MI, gl) is time oriented by the vector field XI, then (M2, g2) is time oriented

either by X2 = f*X1 or -X2. If M2 is time oriented by X2, then f maps future

directed curves to future directed curves and thus future (respectively, past)

null cut points to future (respectively, past) null cut points in Proposition

9.18. On the other hand, if M2 is time oriented by -X2, then f maps future

(respectively, past) null cut points to past (respectively, future) null cut points

since f maps future directed curves to past directed curves.

Proposition 9.18. Let f : (MI, gl) --t (M2, g2) be a global conformal

diffeomorphism of (Ml,gl) onto (M2,gz). Let y : [O,a) --t (MI, gl) be a

null geodesic of MI. If q = $to) is a null cut point of p = y(0) along the


geodesic y, then f (q) is a null cut point of f (p) along the null pregeodesic

f o r : I0,a) -+ (M2,92).

Proof. I t suffices to assume that q is the future null cut point of p along

y and that (M2,g2) is time oriented so that f o y is a future directed null

curve in M2. Let a : [0, b) + (M2,g2) be a reparametrization of f o y to a

future directed null geodesic guaranteed by Lemma 9.17 with f (p) = a(0).

Then f (q) = u(t1) for some tl E (0, b). Let d, denote the Lorentzian distance

function of (M,, g,) for i = 1,2.

We show first that d2(a(0), o(t)) = 0 for any t with 0 < t 5 t l . For suppose

d2(a(O),u(t)) # 0 for some t with 0 < t 5 t l . Then we may find a future

directed nonspacelike curve P in M2 from a(0) to a(t) with Lg,(P) > 0. Thus

f -' o p is a future directed nonspacelike curve in MI from p to f -l (a(t)) with

L,, (f -' o p) > 0. Since f (q) = cr(tl), we have f-l(u(t)) = y(t2) for some

t2 5 to. Hence dl(p, y(t2)) 2 L,, (f -' op) > 0, which contradicts the fact that

dl(p, y(t2)) = 0 since t2 5 to and y(t0) = q was the future null cut point to p

along y.

We show now that d2(u(O),a(t)) # 0 for any t > tl. This then makes

f (q) = u(t1) the future null cut point of f (p) along a as required. To this

end, fix t > tl. There is then a t2 > to so that f-'(a(t)) = y(t2). Since

y(t0) = q is the future null cut of p along y, we have dl (p, y(t2)) > 0. Hence

there is a future directed nonspacelike curve a from p to y(t2) with L,, (a) > 0.

Then f o a is a future directed nonspacelike curve from f (p) to u(t) Hence

d2(f(p),a(t))2Lg,(f oa)>Oasrequired. !J

With Proposition 9.18 in hand, we may now apply the null cut locus to study

null geodesic incompleteness. Recall that a geodesic is said to be incomplete if

it cannot be extended to all values of an affine parameter (cf. Definition 6.2).

Let R and Ric denote the curvature tensor and Ricci curvature tensor of

(M,g), respectively. Recall that an inextendible null geodesic y will satisfy

the generic condition if for some parameter value t, there exists a nonzero

tangent vector v E T,(t)M with g(v, yl(t)) = 0 such that R(v, yl(t))y'(t) is

nonzero and is not proportional to yl(t) (cf. Proposition 2.11, Section 2.5). In

particular, if Ric(v, v) > 0 for all null vectors v E TM, then every inextendible

310 9 T H E LORENTZIAN CUT LOCUS 9.3 T H E NONSPACELIKE CUT LOCUS 311

null geodesic of (M,g) satisfies the generic condition. In Section 2.5, it was (M, g) is globally conformally diffeomorphic to some portion of the subset M' noted that dim M 2 3 is necessary for the null generic condition to be satisfied. of the Einstein static universe, then d l null geodesics of (M, g) are incomplete. Thus we will assume that dim M 2 3 in the following proposition.

Since Minkowski space-time is free of null cut points, the above result re- Proposition 9.19. Let (M,g) be a space-time of dimension n > 3 such mains valid if we replace M' with Minkowski space-time.

that all inextendible null geodesics satisfy the generic condition and such that Friedmann space-times are used in cosmology as models of the universe. In Ric(v, v) 2 0 for all null vectors. If (M, g) is globally conformally diffeomorphic these spaces it is assumed that the universe is filled with a perfect fluid having to an open subset of a space-time (MI, g') which has no null cut points, then zero pressure. We will also assume that the cosmological constant A is zero for all null geodesics of (M, g) are incomplete. these models. These space-times are then Robertson-Walker spaces (cf. Sec-

Proof. Proposition 4.4.5 of Hawking and Ellis (1973, p. 101) shows that if tion 5.4) with p = A = 0. These space-times may be conformally embedded in

the Ricci curvature is nonnegative on all null vectors, then each complete null the subset M' of the Einstein static universe defined above [cf. Hawking and

geodesic which satisfies the generic condition has conjugate points (cf. Propo- Ellis (1973, pp. 139-141)). Furthermore, Ric(g)(v, v) > 0 for all nonspacelike

sition 12.17). Since maximal geodesics do not contain conjugate points, we vectors in a F'riedmann cosmological model (M, g). By Proposition 7.3, there

need only show that all null geodesics of (M, g) are maximal to establish their is a C2 neighborhood U(g) of g in C(M,g) such that Ric(gl)(v, v) > 0 for all

incompleteness. Assume that y is a future directed null geodesic from p to q gl E U(g) and all nonspacelike vectors v in (M, gl). Since R ~ C ( ~ I ) ( V , v) > 0

which is not maximal. Then there is a timelike curve from p to q. Since con- implies that the generic condition is satisfied by all null geodesics in (M, gl),

formal diffeomorphisms take null geodesics to null pregeodesics and timelike Corollary 9.20 yields the following result.

curves to timelike curves, the image of y must be a nonmaximal null geodesic Corollary 9.21. Let (M, g) be a Friedmann cosmological model. There in (MI, g'). This implies that (M', g') has a null cut point and hence yields a is a c2 neighborhood U(g) of g in C(M, g) such that every null geodesic in contradiction. (M, gl) is incomplete for all gl E U(g).

Recall that the four-dimensional Einstein static universe (Example 5.11) is

the cylinder x2+y2+~2+w2 = 1 embedded in R5 with the metric induced from 9.3 The Nonspacelike C u t Locus

the Minkowski metric ds2 = -dt2 +dx2 +dy2 + dz2 +dw2. The Einstein static Recall the following definitions. universe is thus B x S3 with a Lorentzian product metric. The geodesics and

null cut points are easy to determine in this space-time. The null cut locus of Definition 9.22. (Future and Past Nonspacelike Cut Loci) The future

the point (t, x, y, t, w) merely consists of the two points ( t i n , -x, -y, -2, -w) nonspacelike cut locus C+(P) of p is defined as C+(p) = C:(P) U C$(P). The

past nonspacelike cut locus is C-(p) = CL(p) U Cg(p), and the nonspacelike Consequently, the subset cut locus is C(p) = C-(p) U C+(p).

M ' = ( ( t , s , y , z , w ) : O < t < n , X ~ + ~ ~ + Z ~ + W ~ = ~ )

We mentioned in Chapter 8 that a complete noncompact Riemannian man- has the property that C N ( ~ ) n M' = 0 for all p € M'. Proposition 9.19 th ifold admits a geodesic ray issuing from each point. Thus at each point there has the following consequence. a direction such that the geodesic issuing from that point in that direction

Corollary 9.20. Let (M, g) be a space-time such that all null geod as no cut point. For strongly causal space-times, the Lorentzian analogue of

satisfy the generic condition and such that Ric(v, v ) >_ 0 for alI null vectors is property follows similarly from the existence of past and future directed


nonspacelike geodesic rays issuing from each point. This may be restated as

the first half of the following proposition.

Proposition 9.23. (1) Let (M,g) be strongly causal. Then given any

point p E M, there is a future and a past directed nonspacelike tangent vector

in T,M such that the geodesics issuing from p in these directions have no cut

points.

(2) Let (M,g) be globally hyperbolic. Given any point p f M , p has no

farthest nonspacelike cut point.

Proof. As we have already remarked, part (1) follows immediately from

Theorem 8.10. For the proof of part (2), assume q E M is a farthest cut

point of p. Then q is a cut point along a maximal geodesic segment y from

p to q. Choose a sequence of points {q,) such that q << q, for each n and

q, --+ q. Since (M,g) is globally hyperbolic, there exist maximal timelike

geodesic segments c, : [O, d(p, q,)] + M from p to q, for each n. Extend each

c, to a future inextendible geodesic. Since q is a farthest cut point of p and

d(p, q,) 2 d(p, q) +d(q, q,) > d(p, q), for each n the geodesic ray c, contains no

cut point of p. The sequence {G) has a limit curve c which is a future &rected

and future inextendible nonspacelike curve starting at p by Proposition 3.31.

By passing to a subsequence if necessary, we may assume that {c,) converges

to c in the topology on curves (cf. Proposition 3.34). Using the global

hyperbolicity of (M,g) and the fact that q, -+ q, we find that q E c. If

r E c and r, E c, with r, -+ r , then d(p,r) = limd(p,r,). Using the upper

semicontinuity of arc length for strongly causal space-times, we find that the

length of c from p to r is at least as great as limsupd(p, r,) = limd(p, r,) =

d(p, r). Thus c is a maximal geodesic ray. Since q is a cut point of p on 7,

the geodesics c and y are &stinct maximal nonspacelike geodesics containing

p and q. Either Lemma 9.1 or Lemma 9.13 now yields a contradiction to the

maximality of c beyond q. 0

We now turn to the analogue of Klingenberg's observation that, for Rie

mannian manifolds, if q is a closest cut point to p and q is nonconjugate to p,

then there is a geodesic loop a t p passing through q. For Lorentzian manifolds,

however, a different result is true. If {q,) 5 Cz(p) converges to q E C$(p),

9.3 THE NONSPACELIKE CUT LOCUS 3 13

then d(p, q,) -4 0 in a globally hyperbolic space-time since the Lorentzian

distance function is continuous. Thus it is not unreasonable to expect that,

for globally hyperbolic space-times, there is no closest tirnelike cut point q to

p that is nonconjugate to p.

Theorem 9.24. Let (M,g) be a globally hyperbolic space-time, and as-

sume that p E M has a closest future (or past) nonspacelike cut point q. Then

q is either conjugate to p, or else q is a null cut point of p.

Proof. Let q be a future cut point of p which is a closest cut point of p

with respect to the Lorentzian distance d. Assume q is neither conjugate

to p nor a null cut point of p. Then p << q, and by Theorem 9.12 there

exist a t least two future directed maximal timelike geodesics cl and cz from

p to q. Let y : [O,a) -+ M be a past directed timelike curve starting at

q. By choosing a > 0 sufficiently small, we may assume the image of y lies

in the chronological future of p. Then p << y(t) << q for 0 < t < a implies

d(p, q) > d(p, y(t))+d(y(t), q) > d(p, y(t)) using the reverse triangle inequality.

Since q is a closest cut point, the point y(t) comes before a cut point of p on

any timelike geodesic from p to y(t). Thus any timelike geodesic from p to

y(t) is maximal. Since q is not conjugate to p along cl, there is a timelike

geodesic from p to y(t) near cl for all sufficiently small t. Similarly, there

exists a timelike geodesic from p to y(t) near cz for all small t. The existence

of two maximal timelike geodesics from p to y(t) implies y(t) is a cut point of

p and yields a contradiction since d(p, y(t)) < d(p, q). O

For Riemannian manifolds, the set of unit tangent vectors in any given tan-

gent space is compact. It is then an immediate consequence of the Riemannian

analogues of Propositions 9.5 and 9.7 above that the cut locus of any point in

a complete Riemannian manifold is a closed subset of M [cf. Gromoll, Klin-

genberg, and Meyer (1975, pp. 170-171), Kobayashi (1967, pp. 100-101)]. For

Lorentzian manifolds, the timelike cut locus is not a closed subset of M in

general as the Einstein static universe (Example 5.11) shows. However, for

globally hyperbolic space-times it may be shown that the nonspacelike cut

locus and the null cut locus of any point are closed subsets of M [cf. Beem and

Ehrlich ( 1 9 7 9 ~ ~ Proposition 6.5)]. It may be seen by Example 9.28 given below

314 9 THE LORENTZIAN CUT LOCUS 9.3 THE NONSPACELIKE CUT LOCUS 315

that the assumption of global hyperbolicity is necessary for the nonspacelike

cut locus to be a closed subset of M. An "intrinsic" proof for Lorentzian man-

ifolds is more complicated than the proof for Riemannian manifolds as a result

of the lack of unit null tangent vectors.

We now turn to the proof of the closure of the nonspacelike and null cut loci

for globally hyperbolic space-times. We first establish a technical lemma. Here

the assumption that expp(v) = q is essential to the validity of the conclusion

of this lemma.

Lemma 9.25. Let (M, g) be a strongly causal space-time. Assume p < q, and q, + q # p. If expp(vn) = q,, exp,(v) = q, and the directions of the

nonspacelike vectors v, converge to the direction of v, then u, + v.

Proof. Choose numbers a, > 0 such that the vectors w, = a,v, converge

to v. Then the sequence of points r, = expp(wn) must be defined for large n

and must converge to q. Since (M, g) is causal, there is only one value t , of t

such that expp(tnwn) = q,, namely t, = a;'. Using r, -+ q, q, -+ q, and the

strong causality of (M, g) near q, we find that t, -+ 1. Since v, = t,w,, the

sequence {u,) converges to v.

Fix a point p E M. A tangent vector v E T,M is a conjugate point to p

in T,M if (exp,), is singular at v. The conjugate points to p in T,M must

form a closed subset of the domain of exp, because (exp,), is nonsingular on

an open subset of T,M. A point q f M is conjugate to p along a geodesic c if

there is some conjugate point v E T,M such that exp, v = q and c is (up to

reparametrization) the geodesic expp(tv). If q f M , p < q, and q is conjugate

to p along a nonspacelike geodesic, then we will say q is a future nonspacelike

conjugate point of p. Past nonspacelike conjugate points are defined dually.

If (M,g) is a causal space-time, then p cannot be in its own future or past

nonspacelike conjugate locus because a nonspacelike geodesic passes through

p a t most once and (expp), is nonsingular at the origin of T,M.

Remark 9.26. It is possible even in globally hyperbolic space-times for a

point to be conjugate to itself along spacelike geodesics, however. This happens

in any Einstein static universe of dimension n 2 3.

Lemma 9.27. Let (M, g) be a globally hyperbolic space-time. Then the

future (respectively, past) nonspacelike conjugate locus in M is a closed subset

of M.

Proof. Assume that {q,) is a sequence of future nonspacelike conjugate

points of p with q, + q. Then p # q, p < q, and p < q, for each n. Let

v, E T,M be a nonspacelike vector such that (exp,), is singular at v, and

exp,(v,) = q,. Then c,(t) = exp,(tv,) is a future directed, future inextendible

geodesic starting a t p and containing q,. The geodesics c, must have a limit

curve y by Proposition 3.31. Since (M, g) is globally hyperbolic, y must contain

q. Using the strong causality of (M,g) and the fact that y is a limit curve

of nonspacelike geodesics, we find that y is itself a nonspacelike geodesic. Let

v be the unique (nonspacelike) vector tangent to y at y(0) = p such that

expp(v) = q. Since y is a limit curve of {c,), there must be some subsequence

{m) of {n) such that the directions of the vectors urn converge to the direction

of v. Lemma 9.25 then implies that v, -+ v. Since the vectors urn belong to

the total conjugate locus of p in T,M and this conjugate locus is a closed

subset of the domain of exp,, the vector v is a conjugate point of p in T,M.

Because v is a future directed nonspacelike vector, the point q = expp(v) is a

future nonspacelike conjugate point of p in M. This establishes the closure in

M of the future nonspacelike conjugate locus. The dual proof shows that the

past nonspacelike conjugate locus is also closed. Ci

At this juncture we stress that Lemma 9.27 reflects a specifically non-

Riemannian phenomenon. Contrary to well-established folk lore, the (first)

conjugate locus, even for a compact Riemannian manifold, need not be closed

in general [cf. Magerin (1993)l. In the space-time case we are rescued by the

non-imprisonment property that a nonspacelike geodesic must escape from any

compact subset of a globally hyperbolic space-time in finite affine parameter.

On the other hand, examples may be constructed of stably causal space-times

which do not have closed nonspacelike conjugate loci by deleting appropriate

subsets from the four-dimensional Einstein static universe.

Example 9.28. Let M = S1 x IR = ((9,t) : 0 5 0 5 2 ~ , t E R) with

the flat metric ds2 = do2 - dt2. This is the two-dimensional Einstein static

316 9 THE LORENTZIAN C U T LOCUS

universe which is globally hyperbolic. There are no conjugate points in this

space-time. If p = (0, O), the future null cut locus C$(p) consists of the single

point (T, n). The future timelike cut locus Cz(p) consists of {(n, t) : t > T} and is not closed. On the other hand, C+(p) = C$(p) U C$(p) is closed.

If we delete the two points (f n/4,277) from M, we obtain a strongly causal

space-time (M', glM,) which is not globally hyperbolic. In (MI, glM,) the

future nonspacelike cut locus C+(p) is not closed.

Proposition 9.29. Let (M,g) be a globally hyperbolic space-time. For

any p E M, the sets C$(p), Ci(p) , C+(p), and C-(p) are all closed subsets

of M. In particular, the null cut locus and the nonspacelike cut locus ofp are

each closed.

Proof. Since the four cases are similar, we will show only that C$(p) is

closed in M . To this end, let q, E C$(p) and q, -+ q. Since (M,g) is

globally hyperbolic, q # p and q E Jt(p). From the definition of C$(p) we

see that d(p, q,) = 0. Using the continuity of Lorentzian distance for globally

hyperbolic space-times (Lemma 4.5), it follows that d(p,q) = limd(p, q,) = 0.

Let 7 be any nonspacelike geodesic from p to q. Then d(p, q) = 0 implies that

y is a maximal null geodesic and that the cut point to p along y cannot come

before q.

There are two cases to consider. Either infinitely many points q, are future

nonspacelike conjugate to p or else no q, is future nonspacelike conjugate to p

for large n. In the first case we apply Lemma 9.27 and find that if q is future

nonspacelike conjugate to p, then if this conjugacy is along y, q is the cut point

along y because the cut point along y comes before or a t the &st conjugate

point along y. If q is a future nonspacelike conjugate point to p along some

other nonspacelike geodesic y', then y' must be null, and Lemma 9.13 shows

that q is the cut point along both y and y'.

Assume now that q, is not future nonspacelike conjugate to p for all suffi-

ciently large n. Theorem 9.15 implies that for large n there exist a t least two

maximal null geodesics from p to q,. Thus for large n there are two future

directed null vectors v, and w, with expp(vn) = exp,(w,) = q,. The future

directed nonspacelike directions a t p form a compact set of directions. Thus

9.3 THE NONSPACELIKE CUT LOCUS 317

the directions determined by {v,) and {w,) have limit directions v and w

respectively. If v and w determine different directions, we apply Lemma 9.14

to obtain two maximal null geodesics from p to q. In this case q is a cut point

of p. On the other hand, v and w may determine the same direction. If this is

so, we first note that there are constants a > 0 and b > 0 such that av = bw

and expp(av) = expp(bw) = q. Applying Lemma 9.25, it follows that for some

subsequence (m) of (n) we have urn -+ av and w, + av. However v, # w,

and expp(vrn) = expp(wrn) then yield that (exp,), must be singular at av.

Thus q is conjugate to p along y(t) = exp,(tav), and since d(p, q) = 0 we find

that q is the cut point of p along y(t). !I

Surprisingly, the above result holds without any assumptions about the

timelike or null geodesic completeness of (M,g). In particular, the null cut

locus and the nonspacelike cut locus in globally hyperbolic space-times are

closed by Proposition 9.29, even though the function s : T T I M -+ R U {co)

may fail to be upper semicontinuous.

It is well known [cf. Kobayashi (1967, pp. 100-101)] that using the cut locus,

it may, be shown that a compact Riemannian manifold is the disjoint union

of an open cell and a closed subset (the cut locus of a fixed p E M) which

is a continuous image (under expp) of an (n - 1)-sphere. Thus the cut loci

inherit many of the topological properties of the compact manifold itself. For

Lorentzian manifolds, cut points may be defined (using the Lorentzian distance

at least) only for nonspacelike geodesics. Hence a corresponding result for

space-times must describe the topology not of all of M itself, but only of J+(p)

for an arbitrary p E M. To obtain this decomposition with the methodology of

this section, we need to assume that (M, g) is timelike geodesically complete

as well as globally hyperbolic, so that the function s : T - I M --t R U {co)

defined in Definition 9.3 will be continuous and not just lower semicontinuous

(cf. Propositions 9.5 and 9.7).

Recall also that the future horismos E+(p) of any point p E M is given by

E+(p) = J+(p) - I+(p) and that C+(p) denotes the future nonspacelike cut

locus of p.

318 9 THE LORENTZIAN CUT LOCUS 9.4 THE NONSPACELIKE CUT LOCUS REVISITED 319

Proposition 9.30. Let (M, g) be globally hyperbolic and timelike geodesi- cut loci for globally hyperbolic spacetimes could not draw on the continuity

cdly complete. For each p E M the set J+(p) - [C+(P) U E+(P)] is an open properties of the s-function established in Proposition 9.7. Rather, we relied

cell. on a more indirect proof method which called upon the characterization of

PTOO~. Let B = J+(p)- [C+(p)uE+(p)]. Then q E B if and only if there is a nonspacelike cut points given in Theorems 9.12 and 9.15. In this section wc

maximal future directed timelike geodesic which starts at p and extends beyond will give a more elementary method to derive the closure of the nonspacelike

q. Thus B = I+(p) - C+(p) which shows that B is open. Now T-'MI, = {v E cut loci for globally hyperbolic spacetimes which treats both null and timelike

TpM : v is future directed and g(v,v) = -1) is homeomorphic to lRn-l. Let cut points simultaneously but is non-intrinsic. The treatment given in this

H : T-'MIP + IRn-' be a homeomorphism, and define 3 : Itn-' -+ R U {m) section has been worked out in discussions with G. Galloway.

by 3 = s o H-'. There is an induced homeomorphism of B with {(x,t) E We have noted in Section 9.3 just prior to Example 9.28 that Magerin (1993)

wn-1 x R : 0 < t < ~ ( x ) ) defined by q + (H(v), t), where v is the vector has given examples for Riemannian spaces that show that, contrary to folk lore,

in T-IM such that exp,(sv) is the unique (up to reparametrization) maximal the first conjugate locus of a point in a compact Riemannian manifold need not

geodesic from p to q, and expp(tv) = q with v f T-IM. Let f : [0, m] + [o, 11 be closed. Thus for completeness, we present an elementary proof that the cut

be a homeomorphism with f (0) = 0 and f (m) = 1. Then the map (x, t ) -4 locus of any point in a complete Riemannian manifold is closed, nonetheless.

(x, f ( t ) / f ( ~ ( 2 ) ) ) shows B is homeomorphic to Etn-' x (0, I), which establishes Theorem 9.31. Let (N, go) be a complete Riemannian manifold, and let

the proposition. p in N be arbitrary. Then the cut locus C(p) of p in N is a closed subset of

Since I+(p) J+(p ) , Proposition 9.30 yields some indirect topological in-

formation about I+(p). The Einstein static universe shows that I+(P) need not Proof. We have the basic fact that the distance to the cut point in unit be an open cell even if (M, g) is globally hyperbolic. However, (I+(p), glI+(p)) direction v, which we will denote as is customary by s : SN -, W u {rn) is globally hyperbolic whenever (M, g) is globally hyperbolic. Thus I+ (p) may corresponding to Definition 9.3 for unit timelike vectors, is continuous pro- be expressed topologically as a product I+(p) = s x W, where S is an (n - 1)- vided (N, go) is complete. Now let {q,) G C(p) be given with qn -+ q in M. dimensional manifold (cf. Theorem 3.17). Represent q, = expp(s(vn)vn) with llvnll = 1. Since the set of unit vectors in

9.4 T h e Nonspacelike C u t Locus Revisited TpM is compact, we may assume by passing to a subsequence if necessary that

un -, v f TpM. By continuity of the s-function, lims(vn)vn = s(v)v. Since

In Sections 9.2 and 9.3 we treated the null and timelike cut loci separate1 (N,go) is geodesically complete, exp,(s(v)v) is defined. Hence

because the intrinsic unit observer bundle T-1M was used as the prim technical tool for investigating the timelike cut locus. This approach s q = lim qn = lim exp,(s(vn)vn) = exp,(s(v)v).

the drawback, however, that if {q,) is a sequence of future timelike cut poi to p and q, = exp(s(vn)v,) converges to a null cut point q to p, then Therefore, q is a cut point to p.

(9.3) lim s(vn) = 0. Now Galloway suggested a non-intrinsic way to deal with the nonspacelike

cut locus in such a manner as to bring an analogue of (9.4) to bear. The As a result of the technical differences in handling the null and timelike wo facts that (i) future inextendible nonspacelike geodesics are not trapped

loci, the proof in Proposition 9.29 of the closure of the nonspacelike and n m compact sets in strongly causal spacetimes, and (ii) a modified s-function

I 320 9 THE LORENTZIAN C U T LOCUS

defined for all future causal directions to be constructed below is lower semi-

continuous for globally hyperbolic space-times suffice to ensure that the basic

ideas o f the above proof are valid in the space-time setting.

Fix throughout the rest o f this section an auxiliary complete Riemannian

metric h for the space-time ( M , g). Let

U M = {v E T M : h(v, v ) = 1 and v is future directed ).

Then we define the modified s-function on U M , which will be denoted by s l ,

as would be expected.

Definition 9.32. Define the function sl : U M + W U { m ) by

where c,(t) = exp(tv) as before.

Evidently, for timelike v in UM

and further it is not difficult to see that the semicontinuity proofs given for s

and T-lM apply equally well to sl and U M with minor modifications. Also,

i f ( M , g) is strongly causal, then s l (v ) > 0 for any v in U M .

Proposition 9.33.

(1) Let v E UM with s l (v ) > 0. Suppose either that s l ( v ) = +m, or

s l (v ) is finite and cv(t) extends to [O,sl(v)]. Then sl is upper semi-

continuous at v in U M . Especially, i f (M,g) is future nonspacelike

geodesically complete, then sl : U M -+ JR U { m ) is everywhere upper

semicontinuous.

(2) I f ( M , g) is globally hyperbolic, then sl : U M --t R U { m ) is lower

semicontinuous.

Before giving the elementary proof o f the closure o f the nonspacelike cut

locus for globally hyperbolic space-times, it is helpful t o note the following

technical result. I t is not required a priori in this result that either of a or b is

finite. Similar issues are addressed in a more general context and on a more

systematic basis in Lemma 11.20 and the proof o f Theorem 11.25.

9.4 THE NONSPACELIKE CUT LOCUS REVISITED 321

L e m m a 9.34. Let ( M , g ) be a globally hyperbolic space-time. Suppose

that c, : [0, b,] -+ ( M , g) is a sequence o f future nonspacelike maximal geodesic

segments with v, = cnl(0) in U M and c,(O) = p. Put q, = cn(bn), and

suppose that q, --t q in M , b, -+ b > 0 in W U { m ) , and v, -+ v in U M . Let

c : 10, a ) + ( M , g) denote the maximally extended nonspacelike limit geodesic

given by c(t) = exp,(tv). Then

(1) b < a, so that b is finite and positive,

(2 ) c(b) = q, and

(3 ) s1(v) 2 b.

Proof. ( 1 ) Suppose a 5 b. Fix any q' with q << q'. Then for any s with

0 < s < a, we have c,(s) defined for all n sufficiently large, and c,(s) E I-(9')

for all such large n. Since v, -+ v and cn(s) is defined for all sufficiently large n,

we have c(s) = limc,(s), whence c(s) E J-(9'). Thus the future inextendible,

nonspacelike geodesic c 1 10, a) is contained in the compact set J+(p) n J-(q') ,

which is impossible since ( M , g) is strongly causal. Thus b < a, and b = limb,

must be finite as well.

(2) Since b < a, we find that c(b) is defined and q = lim q, = lim c,(b,) = c(b)

by uniform convergence.

(3) Since the geodesic segments c, 1 10, b,] are assumed to be maximal and b =

limb,, the limit segment c I [0, b] must also be maximal, whence s l (v ) 2 b. O

T h e o r e m 9.35. Let (M,g) be globally hyperbolic. Then the future non-

spacelike cut locus and the future null cut locus o f any point p in M are closed

subsets o f M .

Proof. The proof may be given in a uniform fashion either for (q,) a se-

quence o f null cut points to p or a sequence o f nonspacelike cut points to p, so

we will not specify. Suppose qn -+ q in M . Choose v, E U M with

and put b, = sl(v,). Because {w E UM : w E T p M ) is compact, by taking

successive subsequences we may suppose that v, --t v in U M and that b, -+ b

in W U { m ) . Put c( t ) = expp(tv). By Proposition 9.33-(2), we have b 2 s l ( v ) ,


so that b > 0. By Lemma 9.34, b is finite? c(b) = q, and s l ( v ) 2 b. Hence

s l (v) = b, and thus q is a cut point of the required type.

G. Galloway has also remarked that with non-intrinsic methods, the hy-

pothesis of "timelike geodesic completeness" may be removed from Proposition

9.30.

Proposit ion 9.38 (Galloway). Let (M, g) be globally hyperbohc. Then

for each p in M the set B = Ji(p) - [C+(p) U E+(p)] is an open cell.

Proof. -4s before, B = I + ( p ) - C+(p). Fix c > O sufficiently small that if

we set V = ( v E T M , h ( v , v ) = c, v future timelike), then C - expp(V) is

contained in B. Also, fix a diffeomorphism f : U + Rn-l. Define a projection

map r, : B i U by letting ?(m) denote the point of intersection of U with

the unique future timelike gcodcsic in M from p to m. Now choose a second

auxiliary complete Riemannian metric hl just for the open subset B of M.

considered as a manifoid in its own right. For each q in U, let "14 denote the

g-geodesic cexp;~(q) reparametrized as an hl-unit speed curve in B with

efq(0) = q and parametrized with the same orientation as the corresponding

g-geodesic. By Lemma 3.65. y, : W -+ 3 for each q in U. Now define t : B -+ R

by the requirement that

(t(m)) = m

Then we may define a homeomorphism F : B -+ R x Rn-' by F(m) =

. f ?

CHAPTER 10

MORSE INDEX THEORY ON EORENTZIAN MANIFOLDS

Given a nonspacelike geodesic y - [O. a ) -+ M In a causal space-tlme, Xe

have seen in Chapter 9 that if ?(to) is the future cut point to 7 ( 0 ) along 2 , then

for any t < t o the geodesic segment y I [O, t ] is the longest nonspacclike curve

from ~ ( 0 ) to y(t) In all of M . We could ask a much less stringent quest~on:

among all nonspacelike curves u from $0) to "((to) sufficiently "closc" to 7,

is L(y ) 2 L(u)? If so, y(to) comes at or before the first future conjugate

point of y(0) along y. The crucial difference here is between "in all of M"

for cut points and "close to y" for conjugate points. The importance of t h ~ s

distinction is illustrated by the fact that while no two-dimensional space-time

has any null conjugate points, all null geodesics in the two-dimensional Einstein

static universe have future and past null cut points.

Since only the behavior of "nearby" curves is considered in studying con-

jugate poirits, it is natural to apply similar techniques from the calculus of

variations to geodesics in arbitrary Rlernannian manifolds and to nnnspace-

like geodesics in arbitrary Lorentzian manifolds. To ~ndicate the flavor of the

Lorentzian index theory. we sketch the Ivlorse Index theory of an arbitrary

(not, necessarily complete) Riemannian manifold (iV, go). Let c : [a, b j -+ N be a fixed geodesic segment and consider a one-parameter family of curves a,

starting s t c(a) and ending at c(b). More precisely, let a : 'a, bl x ( - c , E ) --, AT

be a continuous, piecewise smooth map with C Y ( ~ . 0) = ~ ( t ) for all t E [a, b ] ,

and &(a, s ) = c(a), a(b, s) = c(b) for all s f (-t, t) Thus each neighbor~ng

clirve a, . t 4 cr,(t) = a(t, s ) is piecewlse smooth. The vanatzon vector field

(or s-parameter denvatzve) V of this deformation is given by

324 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS

Since c is a smooth geodesic, one calculates that

and the second variation works out to be

d2 -(b(ffs))1 ds2 S=O

b

bo(V1, V') - go(R(V, c1)c', V) - c'(go(V, cl))l I t d t =a

b + 90(VvV,c1) I,.

This second variation formula naturally suggests defining an zndez form

on the infinite-dimensional vector space V$ (c) of plecewise smooth vector fields

X along e orthogonal to e with X(a) = X(b) = 0 It IS then shown that

necessary and sufficient cond~tlon for e to be free of conjugate points is tha

I ( X , X ) > 0 for all nontr~vial X E V&(c).

T h ~ s suggests that for a geodeslc segment c : [a, b] -+ N wlth conjugat

points to t = a, the index Ind(c) of c with respect to I : Vg'(c) x V$(c) -+

should be defined as the supremum of dimensions of all vector subspaces o

V&(c) on which I is negative definite. Even though e ( c ) is an infinit

dimensional vector space, the Morse Index Theorem for arbitrary Riemannia

manifolds asserts that.

(1) Ind(c) is finlte; and

(2) Ind(ef equals the geodesic index of c, l.e., the number of conjugat

points along c counted wlth multlphattes.

More precisely, if we let Jt(c) denote the vector space of smooth vector fie1 c - 0 w i Y along c sat~sfying the Jacob1 differential equation Y" + R(Y, c') ' -

boundary conditions Y ( a ) = Y(t) = 0, then (2) is equivalent to the formula

(3) Ind(c) = dlm Jt (c).

10 MORSE INDEX THEORY ON LORENTZIAN RlANIFOLDS 325

Also, c has only finitely many points in (a, bj conjugate to c(a).

With the Morse Index Theorem in hand, the homotopy type of the loop

space for complete Riemannian manifolds may now be calculated geornetri-

cally [cf. Milnor (1963, p. 95)]. One obtains the result that if (N,go) is a

complete Riemannian manifold and p, q E N are any pair of points which are

not conjugate along any geodesic, then the loop space fl(p,,) of all contiriuous

paths from p to q equipped with the compact-open topology has the homotopy

type of a countable CW-complex which contains a cell of dimension X for each

geodesic from p to q of index A.

The purpose of Sections 10.1 and 10.3 is to prove the analogues of (1)

through (3) for nonspacelike geodesics in arbitrary space-times Let c : [a, b] -,

&I [respectively, P : [a.b] -+ M ] denote an arbitrary timelike [respectively,

null] geodeslc segment in ( M , g ) . Let V , (c ) [respectively, V$(P)] denote the

infinite-dimensional vector space of piecewise smooth vector fields Y along c

[respectively, P] perpendicular to c [respectively, P] with Y ( n ) = Y(b) = 0.

The timelike index form I : q ( c ) x Vo-(c) -+ R may be defined by

in analogy with the Riemannian index form. It may also be shown that c . [a. b] - A4 has no conjugate points in (a. b) if and only if I . *(c) x V$(c) i

R is negative definite.

Similarly, an index form I : V&(P) x @(P) + R may be defined by

r(X, Y) = - J ,~[~ (x ' , Y ' ) - g(R(X . P1)P1, Y)] dt. But since P is a null geo-

desic, g(P1, @') = 0. Consequently, vector fields of thc form V(t) = f (t)P1(t)

with f : [a. b] -+ R piecewise smooth and f ( a ) = f (b) = O are always

in the null space of I : V;(f?) x e(P) -- R yet never give rise to null

conjugate points. One way around this difficulty is to consider the quo-

tient bundle Xo(P) of V$(P) formed by identifying YI and Y2 in V$(/3) if

Yl - Y2 = fP/ for some piecewise smooth function f . [a. b ] --+ W. The in-

dex form I : Vo-(@) x Vo-(6) -+ R may also be projected to a quotient index

form 7 : Xo(Pj x Xo($) 4 R. It may then be shown that the null geo-

desic segment ,B : [a, b ] -+ M has no conjugate points in [a, b ] if and only if


- I : Xo(P) x Xo(P) -+ ?It is negatlve definite [cf. Hawking and Ellis (i973. Sec-

tion 4.5), Bolts (1977, Chapters 2 and 4)j. In the case of a congruence of null

geodesics, the procedure of forming the quotient bundle has also been utilized

in general relativity since the 1960's. and the term "screen space" has been

ernployed [cf. Robinson and Trautman (1983) for a recent reference].

Let Jt (c) [respectively, Jt (P)] denote the space of Jacob1 fields along c [ r e

spectively, ,Dl with Y ( a ) = Y ( b ) = 0. Define the index Ind(c) of c [respecti.rrely,

Ind(B) of ,!?I to be the supremum of dinlensions of all vector subspaces of V$(c)

[respectively, Xo(B)] on which I [respectively, -I] is positive definite. We es-

tablish the Morse Index Theorem

Ind(c) = dim Jt (c) t E ( a , b )

and

for the timelike geodesic c : [a, b ] -+ M and the null geodesic ,5' . [a, b ] -+ n/f in

Sections 10.1 and 10.3, respectively. The reason for ronsidering the timelike

and null cases separately is that the quotient index form f : Xo(j?) x Xo(P) -+ R must be used to obtain the Null, but not the Timelike. Morse Index Theorem.

In Section 10.2 we study the theory of timelike loop spaces for globally

hyperbolic space-cimes, summarizing some results of Uhlenbeck (1975). Since

completeness is necessary to develop the Riemannian loop space theory. it is

not surprising that global hyperbolicity is needed for the Lorentzian theory.

Yet a significant difference occurs between the Lorentzian and Riemannian

loop spaces. Let (.&?,go) be a complete Riemannian manifold with positive

Ricci curvature bounded away from zero. Thus N is compact by Myers' The-

orem. It is shown in Milnor (1963, Theorem 19.6) that if p and q in such

a Riemannian manifold are nonconjugate along any geodesic: then the loop

space O(p,p) has the homotopy type of a CW-complex having only finitely

many cells in each dimension. But the loop space may still be an ~nfinzte

CW-complex as is seen by the example of Sn with the usual complete metric

of constan: sectional curvature. On the other hand. if (M,g) is an arbztrary

10.1 THE TIMELIKE MORSE INDEX THEORY 327

globally hyperbolic (hence noncompart) space-time, and p, q M are any two

points with p << q such that p and q are not conjugate along any nonspacelike

geodesic, then the timelike loop space C(,,,) has only f int felg many cells

This striking difference between the Lorentzian and Riemannian loop spaces

is a result of the following basic difference between Lorentzia~ and Riemannian

manifolds. If y is any nonspacelike curve from p to q ir an arbitrary space-

time and d ( p , q ) 1s finite. then y has bounded Lorentzian arc length since

L(y) 5 d ( p , q). In contrast. any curve a in the Riemannian path space saticifies

Lo(a) 2 do(p, q), hence has arc length bounded from below but not above

10.1 The Timelike Morse Index Theory

In this section we give a detailed proof of the Morse Index Theorem lor

timelike geodesics in an arbitrary space-time. Several different approacl~eb to

Morse index theory for timelike geodesics under various causality conditions

have been published by Uhlenbeck (1975), Woodhouse (1976), Everson and

Talbot (1976). and Beem and Ehrlich (1979~). Were we give a treatment which

parallels the proof of the Morse Index Theorem for an arbitrary bcmannian

manifold in Gromoll, Kllngenberg, and hfeyer (1975, Sections 4.5 and 4.6).

A sim~lar treatment of the results in this section through Theorem 10 22 has

been given in Bolts (1977).

Let (M, g) be an arbztrary space-time of dimens~on n 2 2 throughout this

section. We will denote by ( , ) the Lorentzian metric g IR this section. Also.

all timelike geodesics c : [a, b] 4 M will be assumed to have unit speed, 1.e..

(cl(t), c1(t)) = -1 for all t E [a, b ] .

Definition 10.1. (Paecewzse Smooth Vector Fzeld) A pzece.iiilse smooth

vector field Y along (the timelike geodesLC; c [a, b ] - Jf is a coniin~ioiis

map Y : [a.b] -* I M . with Y(t) E Tc(t+' for all t E [a. h ] , such that there

exists some finite partition a = to < t l < < t k m l < t k = h of [a, b j so that

Y/[,, ,,+,\ : [t,,t,+l] 4 TM is smooth for each z with 0 5 2 i, k - 1. Let VL(c)

denote the R-vector space of all piecewise smooth vector fields Y along c with

{Y(t). c l j t ) ) = 0 for all 0 < t 5 b. Also let fib'(c) = {Y t VL:c) Y(a\ =

Y(b) = 0 ) and Ar(c(t)) = ( u E T,(L)&f. (v ,c i ( t ) ) = 0).


G~ven Y E V L ( c ) , there is a finite set of points I. (a , b) such that Y 1s

differentiable on [a, b] - lo. Define Y' : [a, b] + T M by Y 1 ( t ) = V,tY(t) for

t E [a, b] - I. and extend to points t , E lo by secting

Y 1 ( t i ) = lim Y 1 ( t ) t-t;

Thus Y' is extended to [a, bj by left continuity but is not necessarily continuous

at points of Io.

Remark 10.2. Since cl(t) is timelike, N(c( t ) ) is a vector space of dimension

n-1 consisting of spacelike tangent vectors, and thus (v E N(c(t)) : (v: w) 5 1)

is compact.

Definition 10.3. (Conjugate Point) Let c : [a, b] + M be a timelike

geodesic. Then t l , tz E {a, b ] with tl # t2 are conjugate with respect to c

if there is a nontrivial smooth vector field J along c with J ( t l ) = J ( tn ) =

0 satisfying the Jacobi equation J" + R(J,cl)c' = 0. Were J' denotes the

covariant derivative operator on vector fields along c induced by the Levi-

Civita connection of ( , ) on M. Also tl E (a , b) is said to be a conjugate

poznt of the geodesic segment c : [a, b] --+ M if a and tl are conjugate along

c. The geodesic segment c is said to have no conjugate points if no tl E (a: b ]

is conjugate to t = a along c. A smooth vector field J along c satisfying the

differential equation J1' + R(J, c1)c' = 0 1s called a Jacob% fieEd along c

We now define the Lorentzian index form I : V L ( c ) x V L ( c ) 4 R.

Definition 10.4. (Lorentzian Index Form) The indez form I : V'fc) x

V L ( c ) --t R is the symmetric bilinear form given by

for any X : Y E V L ( c )

If X E V L ( c ) is smooth, we also have

10.1 THE TIMELIKE LIORSE INDEX THEORY 329

Thus if Y E VoL(c) and X is smooth, one obtains the formula

5

(10.3) (X" + R ( X , cl)c', Yj d t

linking the index form to Jacobi fields.

Remark 10.5. Given a piecewise smooth vector field X E V L ( c ) . we may

choose a partition a = t o < t l < < tk-I < t k = b such that X / It,, is smooth for each i = 0, l . 2 , . . . , k - 1. Let

A,, ( X I ) = X1(a+),

A,, ( X I ) = -X1(b-), and

A , ( X I ) = lirn X 1 ( t ) - lim X 1 ( t ) t-t: t-t;

for i = 1,2, . . . , k - 1. It may then be seen by applying (10.2) on each subin-

terval (t,, that

This is the form of the second variation formula given in Hawking and Ellis

(1973. p. 108) and Bblts (1977. pp. 86-87) [cf. Cheeger and Ebin (1975, p. 21)]

In order to give geometric applications using the index form, it is useful to

make the following standard definition.

Definition 10.6. (Variatzon) Let c : [a, b] -+ M be a smooth curve. A

vanatzon (or homotopy) of c is a smooth mapping a [a, b ] x ( - E . E ) -+ M .

for some F > 0, with cr(t 0 ) = c ( t ) for all t E :a, b j The variation cu is said to

be a proper vanatton if cr(a.9) = c(o) and cr(b, s) = c(h) for all s E (-E, E ) A

continuous mapping a : [a, b] x ( - E . e ) -+ M is said to be a pteceui7se f m 00th.

va7.~atzon, of c if there exists a finite partition a = to < t i < . . . < tk -1 < t b = h

of [a. b ] such that cr I [t,, t,+l] x ( - E , E ) is a smooth variation of c / [t,, t,-l] for

eachz=0,1 ,2 ,..., k - 1 .

If we set a,(t) = cu(t.s) in Definition 10 6. then for a smooth variation a,

each curve a, : [a, b ] -+ M is a smooth curve and thus :he mapping s - cu,


is a deformation of the curve c through the "neighboring curves' a,. If a

is a piecewise smooth variation, the neighboring curves a, will be 2iecewise

smooth. If a is a proper variation, all of the neighborlng curves a, begin at

c(a) and end at c(b). It is customary in defining variations of timelike curves

to ratr ict consideration to variations having the additional property that all

neighboring curves or, : [a, b] --, M are timelike. However, if we use Definition

10.6, this restriction is unnecessary by the following lemma.

Lemma 10.7. Let a : [a, b ] x ( - 6 , ~ ) -, A4 be a piecewise smooth variation

of the tirnelike geodesic segment c : [a, b ] -+ Al. Then there exists a constant

O > 0 depending on a such that the neighboring curves a, are timelike for all

s with Is] 5 6

Proof We first suppose that a is a smooth variation. Choose any el with

0 < €1 < E. Then a is differentiable on the compact set [a,b] x [-el,el] by

Definition 10.6. Hence by definition of differentiability, a extends to a smooth

mapping of a larger open set containing [a, b] x [-el, el]. Since c is a timelike

geodesic, the vectors cl(a+) and cl(b-) are timelike. It follows from this and

the extension of N to an open set containing [a, h ] x [-el, el] that there exists

a constant El > 0 such that the tangent vectors a,'(ai) and cuS1(b-) to the

xeighboring curves a, are timelike for all s with (s j < bl.

Suppose now that no 6 > 0 can be found such that all the curves a, are

timelike for Is] < 6. Then we could find a sequence s, --+ 0 such that the

curves a,,, failed to be timelike. Thus there would be t, E [a,b] so that

g (orin (t,), a:, (t,)) 2 0 for each n. Since [a, b ] x [-el. el] is compact, the

sequence ((t,, 5,)) has a poirit of accumulation ( t , s). Since s, - 0 , this

point must be of the for111 (t,O). and also the existence of 61 above shous

that t # a, b. But then as g (a:,, (t,), a,'n (t,)) 2 0 for each n, it follows that

g(cJ(t) , cl(t)) >_ 0, in contradic~ion to the fact that c was a timelike gecdesic

segment. Thus we have seen that ~f a : [a. b ] x (-E,E) 4 M is a smooth

variation of the timelike geodesic c : [a, b] 4 M , then there is a constant 5 > 0


such that a,'(af) and a,'jb-) are timelike vectors for ail 1s 5 6 and the curves

a , are timelike for all 1s 5 6.

Now let (Y : [a, b j x (-E.E) -+ M be a piecewise smooth variation of the

timeilke geodesic c : [a, b ] -+ lif. There is by Defin~tion 10 6 a finite partition

a = to < t l < ... < tk = b such that a 1 [t:, t,+( x ( - E , F ) 4 hf is a smooth

variation of c / :t,, t,+l]. By the above paragraph. we may find a constant 6, > 0

such that for all s with Is/ 5 E,, the tangent vectors as1(t:) and a,'(t;+,) are

timelike and a, I [t,, t,,~] 1s a timellke curve. Taking b = min(6~. b l , . . . , 6k-1)

shen yields the required 6.

Remark 10.8. There is no result corresponding to Lemma 10.7 for varia-

tions of null geodesics (cf. Definition 10.58 ff.).

The index form may now be related to variat~ons of timelike geodesic seg-

ments c : [a, b] -, ht as follows. Given Y E V ~ ( C ) , define the canonzcal proper

variation a : [a, b ] x (-6, E) -+ M of c by setting

(10.5) 4 t . s) = exp,(z)(sy(t)).

I t should first be noted that slnce c([a,G]) is a compact subset of M , and

the differential of the exponential map, exp,-. is nonsingular at the origin of

T,M for all p E M, it is possible given c([a, b] ) to find an E > 0 such that

exp,(,)(sY(t)) is defined for all s with Is/ 5 c and for each t E [a, b]. Secondly,

fro111 Definition 10.1 it follows that a( t , s) defined as in (10.5) 1s a piecewise

sn~ooth variation of c. Nence given Y t VoL(c), we know from Lemma 10.7

that there exists some constant 5 > 0 such that all the neighborlng curves

a, : t -+ a(t , s) are timelike for all s with -6 < s < 6. Given an arbitrary smooth variation a . [a, bj x ( -e , E ) -, M of c : [a, b ] -+

M . the van'atzon vector field V of a is defined 50 be the vector field V(t) along

c given by the formula

More precisely, lettlng a l a s be the coordinate vector fieid on [a, b ] x (-e, e)

corresponding to the s parameter, the variation vector field is given 'ny

332 10 MORSE INDEX THEORY ON LORENTZIAN MAXIFOLDS

For a piecewise smooth variation a : [a. b ] x (-E, E ) --+ M , one obtains a

continuous piecewise smooth varlatlon vector field as follows. Let a = to < t2 < < < tk = b be a partit~on of [a, b ] such that cu / It,. t,+l] x (-E,E)

IS smooth for z = 0,1,. . . , k - 1. Given t E [a, b], choose an index z such that

t, < t < t,+l, and set

~ ( t ) = [ a I it,, t,+ll x (-E, ~11 ,

The canonical variation (10.5) has the property that each curve s -+ a ( t , s) 1s

just the geodesic s -i exp,(,)(sY(t)) which has initial direct~on Y(t) at s = 0.

Hence the variation vector field of the canonical variation (10.5) is just the

glven vector field Y E V$ (c).

If we put L(s) = L(a,) = L(t -+ a ( t , s)), then L'(0) = y \ 8 = o = 0 since

c IS a smooth tlmelike geodesic, and

(10.8) L"(0)=2L(s) =I(Y,Y). d Z Lo

Thus if I(Y, Y) > 0 for some Y E Vg'(c), then the canonical proper variation

a ( t , s) defined by (10.5) using Y will produce timelike neighboring curves a,

joining c(a) to c(b) with L(a,) > L(c) for s sufficiently small. Thus if the

timdike geodesic c : ;a, b] -i M is maximal [i.e., L(c) = d(p,q)], then I . V$(c) x V&(c) --+ W must be negative semidefinite. Before proving the Morse

Index Theorem for tirnelike geodesics, we must establish the following more

precise relationship between corljugate points, Jacobi fields, and the index

form. First, the null space of the index form on V$ (c) consists of the smooth

Jacobi fields in VO-(c), and second, c has no conjugate points on [a, b] if and

only if the index form is negative definite on V&(c).

We first derive an elementary but important consequence of the Jwobi

differential equation.

Lemma 10.9. Let c : [a, b ] --+ M be a timelike geodesic segment an

Y be any Jacob] field along c. Then (Y(t), cl(t)) 1s an affine function of

(Y ( t ) , cl(t)) = a t + /3 for some constants a, p E R.

Proof. First (Y, c')' = (Y', c'} + (Y, c") = (Y', c'), as c" = V,lcr =

along the geodesic c. Differentiating again, we obta~n (Y, c')" = (Y", d )


(Y', c"} = (Y", c') = -(R(Y, cl)c', c') = 0 by the skew symmetry of the

Riemann-Ghristoffei tensor Thus (Y: c') is an affine function.

Corollary 10.10. If Y is any Jacob1 field along the timelike geodesic

c : [a,b; --+ IM and Y ( t ~ j = Y(tz) = 0 for distinct t ~ . t n E [a,bj, then

Y E VL(c).

Corollary 10.11. If Y E @ ( c ) is a Jacobi field; then Yf E VL(c).

Using the canonical variation, we are now ready to derive the following

geometric consequence of the existence of a conjugate point to E (a, b) to t = a

along c.

Proposition 10.12. Suppose that the timel~ke geodes~c c : [a. b ] -+ M

contains a conjugate point to E (a, b) to t = a along c Then there exists a

piecewise smooth proper variation a, ofc such that L(a,) > L(c) for all s # 0.

Thus c : [a, b] --+ M is not maximal.

Proof. In view of (10.5), (10.8), and Lemma 10.7, it is enough to construct

a piecewise smooth vector field Y E V$(c) with I(Y, Y) > 0 and let a be

the canonical variation associated with Y. To this end, let Yl be a nontrivial

Jacobi field along c with YI (a) = Yl (to) = 0. By Corollary 10.10, Yl E VL(c).

Hence as Yl(a) = &(to) = 0, we have YI' E VL(c) by Corollary 10.11. Since

Yl(to) = 0 and Yl is a nontrivial Jacobi field. it follows that YI1(to) is a

(nonzero) spacelike tangent vector.

Let I( , ): denote the restriction of the index form to cj [a, s ] , that IS.

Then since Yl is a Jacobi field, we have from (10.2) of Definition 10.4 that

r any Z E Vi(c).

We are now ready to construct a piecewise smooth vector iield Y 6 V$(c)

ith IfY, Y) > 0. Let $J : [a, b] --, IR be a smooth function with @(a) = @(b) =

0 and +(to) = 1. Also let Z1 be the uniqae smooth parallel vector Seld along


c with &(to) = -fil(to). Then Z = ~ Z 1 E V ~ ( C ) . Define a oneparamete1

family Y, E @(c) by

YI(~) + €Z(t) i f a s t t o , K(t) =

t:Z(t) if to < t < b.

Then using (10.9) we obtain

I ( K , K ) = I(Y,, Y,)2 + I(Y,, yE)fo

= I(Y1 + €2, Yl + E Z ) ~ + I(&, EZ):~

= I(Y1, fi)? + 2tI(Yl, Z)? + C~I(Z,Z)? + t21(2, z) i0

= - (Yll, YI)I: - 26 ( ~ 1 ' . z)(: + E~I(Z, 2).

Since Yl(a) = Yl(to) = 0, this simplifies to

I ( X . Y,) = -2€(Y11(to), Z(t0)) + €21(2. 2)

= 2E 11 Yjl(fJo) / I 2 E'I(Z, 2).

As Yll(to) is a (nonzero) spacelike tangent vector and I (Z , Z) is finite, it follows

that there is some t: > 0 such that I ( K , K ) > 0. Put the required vector field

Y = YE for this value of E . @

We now turn to an important characterization of Jacobi fields in terms of

the index form. The same characterization holds for Riemannian spaces with

an identical proof. It is important to note that the index form characterizes

smooth Jacob1 fields among all pzecewzse smooth vector fields in VoL(c) and

not just among all smooth vector fields in Vg'(c).

Proposition 18.13. Let c : [a, b ] -+ ( M , 9) be a timclike geodesic segment.

Then for Y E Vo-(c), the following me equivalent:

(1) Y is a (srnootl-i) Jacobi field along c.

(2) I (Y, Z) = 0 for all Z E V$ (c) .

Proof. First (1) + (2) is immediate, since for smooth vector fields Y and

arbitrary Z the index form may be written as

I (Y.Z)=-(Y1,z) / :+ (Y1l+R(Y,e')e',Z)dt. i"


If Y is a Jacobi field. I(Y, Z) = - (Y', 2) 1:. Thus l(Y, Z) vanishes for all

z E VoL(c).

To show (2) =+ (1). we first note that since c is a tirnelike geodesic segment

and Y E v ( c ) , we have (Y', c') = 0 and (Y" + R(Y, c')cl, cl) = 0 at all points

where Y is differentiable. Taklng left-hand limits. we have (Y1(t,), cl(t,)) = 0

also at the finitely many polnts of d~scontinuity a = to < tl < , . < tk = b of

Y. By contmuity, the right-hand limit lim,+,;(Y1(t),c'(t)) = 0 as well. Hence

the vectors Ot*(Y1) defined in Remark 10.5 also satisfy (At,(Y').cl(t,)) = 0

By (10.4) of Remark 10.5, the index form may be calculated as

k b

(10.10) I(Y. Z) = X ( O t , (Y'), Z(t,)) + (Y" + R(Y, cl)c', 2 ) dt ,=o

Let p : [a, b] -+ [0, 11 be a smooth function with d( to) = +(tl) = . . . = Q(tk) =

0 and #(t) > 0 elsewhere. Then the vector field Z1 = d(Y" + R(Y, c')cl) is In

V$(c) and Zl(t,) = 0 for all 2. Since it 1s assumed that I(Y. 2) = 0 for all

Z E V&(c), we obtain from (10.10) that

As 21 is a spacelike vector field, smooth except at the t,'s, and ~ ( f ) > 0 if

t 4 ita) . we obtain Y1'(t) + R(Y(t). cl(t))c'(t) = 0 if t d it,). Thus Y is a

piecewise Jacobi field and formula (10.10) reduces to

Recalling from above that (At,(Y').cl(t.)) = 0 for each 2, a vector field Z2 E

V$(c) may be constructed with Z2(t,) = At, (Y') for 1. = 1,2. . . , k - 1. Then

we have k-1

0 = I(Y, 22) = /lAl%(Y1) l 2 2=1

Since all of the tangent ~ectors in this sum are spacelike. it follows that

At,(Y1) = 0 for z = 1,2, . . . , k - 1. This :hen implles that Y1 has no breaks

at the t,'s. Since for any t E [a, bj there IS a .lniqu3 Jacobi field along c

336 10 MORSE INDEX THEORY Oh' LORENTZIAN MANIFOLDS

with Y(t) = v and Y 1 ( t ) = tour? it follows that the Jacobi fields Y / [ti, ti+,] fit

together to form a smooth Jacobi field.

In view of Propositions 10.12 and 10.13, it should come as no surprise that

the negative definiteness of the Lorentzian index form should be relzted to the

absence of conjugate points just as the positive definiteness of the Riemannian

index form is guaranteed by the nonexistence of conjugate points [Gromoll,

Klingenberg, and Meyer (1975, p. 145)j. The negative semidefiniteness of the

index form in the absence of conjugate points has been given in Hawking

and Ellis (1973, Lemma 4.5.8). It has been noted in Bolts (1977. Satz 4.4.5)

and Beem and Ehrlich (1979c, p. 376) that the negative definiteness of the

index form in the absence of conjugate points follows "algebraically" from the

semidefiniteness just a s in the proof of positive definiteness for the Eemannian

index form. In order to give a proof of the negative semidefiniteness of the

Lorentzian index form in the absence of conjugate points, we need to obtain

the Lorentzian analogues of several important results in Riemannian geometry

[cf. Gromoll, Klingenberg, and Meyer (1975, pp. 132, 136-137, 140)].

For the purpose of constructing Jacobi fields, it is useful to introduce some

notation for the identification of the tangent space T,(TpM) with TpM its

by "parallel translation in T,M."

Definition 10.14. (Tangent Space T,(T,M)) Given any p E M an

v E TpM, the tangent space T,(T,M) to the tangent space T'M at v is give

by

T,(T,M) = (4, : R -+ T,M)

where

&(t) = v i r t u .

Then T,(TpM) may intuitively be identified with T,M by identifying th

image of 9, in TpM with the vector w. More formally, let 1 be given th

usual manifold coordinate chart x ( r ) = r for all r E W, i.e., s = id. The

B/dz is a vector field on R. Since 4, : R -. TpM and &(0) = v, we hav

+,, : TOR -+ T, (T,hf). We may then make the following definition.


Definition 10.15. (Canonical Isomorphzsm) The canonzcal isomo7;iihssm

r, : TpM -+ T, (T,M) is given by

where, as in Definition 10.14,

In particular, let v = Q,, the zero vector in the tangent space T,M. Then

4, : R -+ TpM is the curve &(t) = tu: in Top(TpiZif). and we find that

ro,(w) = 4,. Thtis To,(T,M) is often canonically identified with T,M itself

by identifying the vector w E TpM and the map 4,. If p E M and v E T p M ,

then since exp, : T'M -t M . the definition of the differentlal gives

In particular, for v = Op we have

exp,, : To, (TpM) --t T,IM

since expp(O,) = p. If E = ~ o p (v) E Top (TpM) and we define Q : R - T, bf hy

q(t) = tu, then exp,,(b) = exp,.(q5, d/dzlo), where a/& is the usual bask

bector field for TR defined above. Thus we obtain

hus exp, orv = I ~ T , M . This fact 1s commonly stated as "the differential .%'

the exponential map at the origin of TpJVi 1s the identity."

The following proposition shows how the differential of the exponeritial map

ay be used to construct Jacobi fields.


Proposit ion 10.16. Let c : [ O , b] -, M be a geodesic with c(0) = p. Let

w E T,M be arbitrary. Then the unique Jacobi field J aiong c with J(O) = 0

and J1(0) = w is given by

J(t) = ex?,- ( t ~ i ~ ~ ( ~ ) ~ ) .

Prooj. Set v = cl(0). We may find an E > 0 so that the smooth variati

a : [a, b ] x (-E, E ) + M of c : [a, b] -+ M given by a(t, s) = exp,(t(v + sw

is defined. Since this is a variation of c whose s-parameter curves t + a(t, s

are geodesics, it follows that the variation vector field of this deformation i

a Jacobi field. Since a, D/dsl(,,,) = expp* (rt(v+sw) (tw)), the variation vecto

field is just J ( t ) = ex?,. (rt,tw) = exp,. (t ~ ~ , w ) . Since a(0, s) = c(O) for a1

s, we have J(0) = 0, and a calculation also gives J1(0) = w Icf. Gromoll

Klingenberg, and Meyer (1975, p. 132)].

As in the Riemannian theory, it is now possible to prove the Gauss Eemm

using Proposition 10.16. We first need to put a natural inner product ( ( , ) on T,(T'M) using the given Lorentzian metric ( , ) on TpM and the canonic

isomorphism.

Definition 10.17. (Inner Product on T,(T,M)) The inner product

(( , )) : Tv(Xp~v) x Tv(TpAd) -+ R

associated with the Lorentzian metric ( , ) for M is given by

((a, b ) ) = (7,'(a),.r,-l(b))

for any a, b E T,(TpM).

Theorem 10.18 (Gauss Lemma). Let v E T'M be a t ange~~t vect

in the domain of definition of the exponential mapping and let a = T,(%)

T,(T,M), Then for any b E T,(TpM), we have

(10.12 ) ((a, b:: = (exp,. a , exp,. b),

Thus the exponential map is a "addial isometry."

10.1 THE TIMELIKE MORSE INDEX THEOW 339

Proof. If d(t) = tz, then a = dl(l) . If c is the geodesic c(t) = exp,(tv) =

exppc+(t), we then have expp.a = cl(l) . Also set w = rV-l(b) E T,M. Let

Y be the unique Jacobi field along c with Y(0) = 0 and Y1(0) = w. By Proposition 10.16, we know that Y ( t ) = exppw ( t ~ ~ , w ) . 1x1 particular, Y ( l j =

expp, (rvw) = expp. (b).

From Definition 10.17, we have ((a,b)) = (.r;'(a),r;'(b)) = ( v , ~ ) . Hence

the Gauss Lemma IS proved if we show that (z., w) = (cl(l), Y(1)). But by

Lemma 10.9, the function f (t) = (cl(t),Y(t)) = cyt + /3 for some constants

a , P E R. Since Y(0) = 0, we have O = O and j ( t ) = t f l ( 0 ) = t(cf(0), Y1(0)) =

t(v, w). In particular, ( c l ( l ) , Y(1)) = f (1) = (L, w) % required. D

The Gauss Lemma has the following geometric consequences. The proofs of

these corollaries, which are given along the lines of Gromoll. Klingenberg, and

Meyer (1975, pp. 137-138) in Bolts (1977, pp 75-77), will be omitted. The

use of the Gauss Lemma here replaces the use of a synchronous coord~nate

system in Penrose (1972, p. 53).

Corollary 10.19. Let U be a convex normal neighborhood in M, and let

c : [O, lj -, U be a future directed timel~ke geodesjc segment from p = c(O) to

q = c(1) in U. Then if P : [0, 11 - U is any future directed timelike piecewise

srnooth curve from p to q, we have L(P) 5 L(c), and L(O) < L(c) unless is

ju s t a reparametrization of c.

The basic idea of the proof is that since lJ is convex, B and c may be lifted

to rays : [0,1] + TpM and F: [O, 11 -, T,M. with F(t) = tcl(0) Then the

Gauss Lemma may be appl~ed to compare P I = expp* OF and d = exp,. o Z and hence the lengths of P and the geodesic c.

An alternative formulation of Corollary 10.19 is also given in Bolts (1977.

pp. 75-77) as follows.

Corollary 10.20. Let v E T,M be a timelike tangent vector In the domain

of definition of exp,. Let : [0,1] + T,M be the curve +(t) = tv. Let + : ;0, lj + TpM be a piecewise smooth curve with d(0) = q(0) and +(1) = d(1)

such that exp, 3w : j0,11 --$ M is a future directed nonspaceljke curve Then


L(exp o+) 5 L(exp 04); and moreover,

provided that there is a to E (O,1] such that the component b of +'(to) per-

pendicular to ~+(~,)($(to)/ll$(t~)/l) satisfies expp.b # 0.

We are now ready to show that if the timelike geodesic segment c has

no conjugate points (recall Definition 10.3), then the length of c is a local

maximum.

Proposit ion 10.21. Let c : [a, b] 4 M be a future directed timelike geo-

desic segment with no conjugate points to t = a and let a : [a, b] x (-E, 6) - M

be any proper piecewise smooth variation of c. Then there exists a constant

6 > 0 such that the neighboring curves a, : [a, b] -+ M given by cr,(t) = a(t , s)

satisfy L(cr,) 5 L(c) for ail s with Is/ < 6. Also L(a,) < L(c) ~f 0 < 1st < 6

unless the curve a, is a reparmetrizat~on of c.

Prooj. Reparametrize cr to a variation n : [O.P] x ( - c , ~ ) M . By

Lemma 10.7, there is an €1 > 0 such that all the neigllboring curves a,

of a 1 [0,/3] x (-tl,el) are tirnelike. We may then restrict our attention to

I [O,P] x ( - ~ 1 , ~ 1 ) .

Set p = c(0) and let 4 : [O, $ 1 4 TpM be the ray +(t) = tcl(0). Since

c has no conjugate points, expp has maximal rank at p(t) E TpM for each

t E [O, p ] . Thus by the inverse function theorem, there is a neighborhood of

+(t) in TpM which is mapped by exp, diffeomorphically onto a neighborhood

of c(t) in M. Since +([O, PI) is compact in TpM, we can find a finite partition

0 = to < t l < - - . < t k = P and a neighborhood U, > 6([t3, t,+l]) in T,M for

each J = 0,1,. . . , k - 1 such that h, = exp I : U, 4 M is a diffeomorphism p u3

of U3 onto its image. By continuity, we may then find a constant 6, > 0

such that a([t,, t,+l] x (-6,, 6,)) exp,(U,) for each j = 0,1, . . . : li - 1. Set

6 = min(61,62,. . . ,6k).

Then we may define a pierewise smooth map Q : [0,/3] x (-6.6) -+ T,M

with exgp o@ = N and @(t. 0) = d(t) for all t E [O. 41 as follows. Given (t, s) E

IO,@] x [-6,6]. choose j with t E [t,,t,+l], and set @(t, s) = (h,)-"a(t, s)).


Corollary 10.20 then implies that L(a,) = L(expp oQ) 5 L(exp, od) = L(c)

for each s with Is1 5 6, and equality holds only if a, is a reparametrization of

c. C

With Proposition 10.21 in hand, we may now show that the negative defi-

niteness of the index form on V$(c) is equivalent to the assumption that c has

no points conjugate to t = a along c in [a, b] .

Theorem 10.22. For a given future directed tirnelike geodesic segment

c : la, b ] M , the following are equiva!ent:

(1) The geodesic segment c has no conjugate points to t = a in (a, b ] .

(2) The index form I : V/(c) x V$(c) -, R is negative definite.

Proof. ( 1 ) 3 (2) Suppose Y # 0 in V$(c) and I(Y. Y) > 0. Let cr(t. s) =

exp,(,)(sY(t)) be the canonical variation associated to Y . Then we have

L1(0) = 0, L1'(0) = I ( Y . Y ) > 0, so that for all s # 0 sufficiently small,

L(cY,) > L(c). Bu& this then contradicts Proposition 10.21. Hence if Y # 0,

then I (Y. Y) 5 0, so that the index form 1s negative semidefinlte.

It remains to show that if Y E V&(c) and I(Y, Y) = 0. then Y = 0. To this

end, let Z E V$(c) be arbitrary. By Remark 10.2, we have Y - t Z E V$(c)

for all t E R. Hence I(Y - tZ, Y - t Z ) 5 0 for all t E W by the negative

semidefiniteness of the index form just established. Since I(Y - tZ, Y - tZ) =

-2tI(Y, Z) + t21(Z, Z), it follows that I(Y, Z ) = 0. As Z was arbitrary. this

then implies by Proposition 10.13-(2) that Y is a Jacobi field. Since c has no

conjugate points, Y = 0.

(2) + (1) Suppose Y is a Jacobi field with Y (a ) = Y(tl) = 0, a < tl 5 b. By

Corollary 10.10, Y E VL(c). Extend V to a nontrivial vector field Z E V$(c)

by setting

Then I (Z , 2) = I(Z. 2): + I(Z, Z)& = - (Z', z)/: + 0 = 0.

A consequence of Theorem 10.22 that is crucial to the proof of the Timelike

Morse Index Theorem is the following maximality pro?erty of Jacobi fields with

respect to the index form for timelike geodesic segments without conjugate


points. This result is dual to the minimality of Jacobi fieids with respect to the

index form for geodesics without conjugate points in Riemannian manifolds.

Theorem 10.23 (Maximality of Jacobi Fields). Let c : [a, b] + M

be a timelike geodesic segment with no conjugate points to t = a, and let

J E VL(c) be any Jacobi field. Then for any vector field Y E VL(c) with

Y # J , and

(10.13) Y(u) = J(a) and Y(b) = J(b),

we have

Proof. The vector field W = J - Y E V&(c) by (10.13) and W # 0 as

Y # J by hypothesis. By Theorem 10.22. we thus have I(W, W) < 0. Now

calculating i(W, W) we obtain

I(W, W) = I(Y, Y) - 2I(J, Y) + I (J , J )

= I(Y,Y) + 2 (J', Y)/! - (J', J)J;.

Since Y(a) = J (a ) and Y(b) = J(b), we have (J',Y)I: = (J'. J)/:. Thus

I(W,W) = i (Y ,Y ) + 2 { J 1 , ~ ) l b , - (J', J)/:

= I(Y,Y) + (J', J)I: = I(Y, Y) - I ( J , J ) .

As i(W, W) < 0, this establishes (10.14).

Now that Theorem 10.23 is obtained, a Morse Index Theorem may be estab-

lished. First we must define the ~ndex of any timelike geodeslc c : [a, b] --+ M.

The definition given makm sense because Vo-(c) is a vector space.

Definition 10.24. (Indez and Extended Indez) The index Ind(cf and

the extended zndez Indo(c) of the timelike geodesic c : [a, b ] -+ M are defined

by

Ind(c) = lub(dim A : A is a vector subspace of ~ & ( c )

and Il A x A is positive definite),

10.1 THE TIMELIKE MORSE INDEX THEORY

Indo(c) = iuWdii11A . A is a vector subspace of v$(c)

and I / .A x A is posltlve sem~definite).

Also let Jt(c) denote the W-vector space of sclooth Jacobi fields Y along c with

Y(a) = Y( t ) = 0. a < t 5 b.

Mre now relate Ind(c) to Indo(c) and establish their finiteness in Proposition

10.25. The maximality of Jacobi fields with respect to the index form for

timelike geodesics without conjugate points plays a key role in the proof of

this proposition. The basic idcas involved in the proof of Proposition 10.25

and Theorem 10.27 are due to Marston Morse (1934)

Proposition 10.25. Let c : [a, b ] -+ M be a future directed timellke geo-

desic segment. Then Ind(c) and Indo(c) are finite and

(10.15) Indo(c) = Ind(c) + dim Jb(c).

Proof. The proof follows the usual method of dpproxirnating V$(c) by

finite-dimensional ~ec to r spaces of piecewire srriooth Jacob! fields. To this

end, dioose a finite partition a = to < t l < . . . < t k = b so that c 1 [t,, t,+l]

has no conjugate poirlts to t = t , for each z with 0 5 z 5 k - 1. Let J ( t , ) denote

the subspace of Vj'(c) consisting of all Y E V&(c) suck1 that Y / [t,,t,+l] is a

Jacobi field for each z with 0 5 z < k - 1. Since c / [t,, t,+l] has no conjugate

points for each 2, it follows that dim J{t,) = (n - l ) (k - 1).

Mre now define the approximation 4 : ~ & ( c ) 4 J{tz) of V,'-(c) by J { t , ) as

follows. For X E V&(c) let (pX) I [t,,t,+l] be the unique Jacobi field along

c / [tl,tB+l] with (q5X)(tE) = X(tb) and (4X)(t,+l) = X(t,,l) for each z with

0 ( i 5 k - 1. Thus X is approximated by a piecewise smooth Jacob] field

OX such that (dX)(tt) = X(t t ) at each t,, 0 < 5 k This approximat~on is

useful in t h ~ s context a s 9 is iqdex nondecr~asing KIore explicitly, Q 1 . J { t , )

is just the identity map, so that I ( X , X) = I(4X. q X ) ~f X E J{t,). On the

other hand, if X $ J j t , ) then the inequality

344 10 MORSE INDEX THEORY ON LORENTZIAX MANIFOLDS

may be obtained by applying Theorem 10.23 to each subinterval It,, t,+l] and

summing.

We estabiish the following sublemma which shows the finiteness of Inda(c)

and Ind(c) and also allows us to replace Vo-(c) by J{ti) in calculating these

indexes.

Srrblemrna 10.26. Let Indb(c) and Indl(c) denote tbe extended index and

index, respectively, of I1 J( t i ) x J(ti). Then

Indo(c) = Indb(c) and Ind(c) = Indl(c).

Hence Indo(c) and Ind(c) are finite.

Proof. First it is easily seen by the uniqueness of the Jacobi field J along

c ] [t,,t,+l] with J(t,) = v and J(t,+l) = w that ihe map 4 . VOL(c) -+ J{t,)

is R-linear. That is, if XI , X2 E V$(C) and a,P E R, then @(ax1 + PX2) =

aQ(Xl) + P+(Xa). Thus + maps a vector subspace of Vd(c) to a vector

subspace of J{t,).

In order to establish the sublemma, we first show that if A is any subspace

of q ( c ) on which I I A x A is positive semidefinite, then & I A : -4 + J(t,) is

injective. Thus suppose X E A and &(X) = 0. If X E J(t,), then + ( X ) = X,

so that X = 0. If X 4 J(t,), then I(+(X),p(X)) > I (X ,X) by (10.16).

Thus $X = 0 implies that I (X, X) < 0 which contradicts the assumption that

I ] A x A is positive semidefinite Thus if +X = 0: then X = 0.

Now that we know that c$ is injective on subspaces of V$(C) on which I is

positive semidefinite, we may prove the sublemma. For if A is a subspace of

V/(c) on which 11 A x A is positive semidefinite, inequality (10.16) implies

that the index form of J ( t , ) is positive semidefinite on the subspace d(A) of

J{t,). Since q5 is injective on such a subspace from above, dim A = dim $(A) .

Hence Indb(c) 2 Indo(c). However, since J( t , ) is a vector subspace of q ( c ) ,

we have Ind&(c) 5 Indo(c). Thus Indo(c) = Indb(c). The same argument

shows Ind(c) = Ind(cl). 3

To conclude the proof of Proposition 10.25, we must establish the equal-

ity Indo (c) = Ind(c) + dim Jb(c). To this end, we choose a second partition


a = so < s l < . . . < s,, = b so that {sl,sa, .. .,sm-3} n (t1,t2,. . . . t r - l ) = Q and c / [s,, s,+lj has no conjugate points for each i with 0 5 z < m - 1. Lct

J{s ,) denote the vector subspace of V ~ ( C ) consisting of all vector fields Y such

that Y 1 [s:, sz4 11 is a Jacobi fieid for each z with 0 < i < m - 1. Since the two

partitions (3,) and {t,) are distinct except for a = so = to and b = s, = tk,

it fallows that

J(tJ n JGE) = ~ ~ ( 4

where .Jb(c) denotes the vector space of all (smooth) Jacobi fields J along c

with J (a) = J ( b ) = 0. By Corollary 10.10, we have Jb(c) V$(c). Also, if

X E J(s,) but X $4 Jb(c), we have by (10.16) that

Applying the proof of Sublemma 10.26 to the partition {s,) of [a! b ] , we may

choose a vector subspace BA of J{s,) with Il Bh x BA positive semidefinite

and with Indo(c) = dim B;. Since dim BA = Indo(c) < m, it follows that Jb(c)

is a vector subspace of BA. By (10.17) and the proof of Sublemma 10.26, the

map

bjBA : BA - J{t,}

is injective. Thus if we set Bo = &(B&), we have d!m Bo = dimBA = Indo(c).

Since Bo is a finite-dimensional vector space and Jb(c) is a vector subspace,

we may find a vector subspace B of Bo such that Bo = B 9 Jb(c), where 9

denotes the direct sum of vector spaces

We claim now that TI B x B is positive definite By construction. we

know that Il Bh x B; is positive semidefinite. Also if 0 # Z E B and we

represent Z = $(X) with X E Bb, then X 4 .ib(c). (For if X E Jb(c) . we

have b(X) = X so that Z = 4(X) = X E J ~ ( c ) also, which is impossible since

B r Jb(c) = {O) by construction.) Hence I (Z . Z ) = I($X, 4X) > I (X ,X) ,

the last inequality by (10.17). Thus as I Bh x B& is positive semidefinite,

we obtain I(Z, 2) > I (X ,X) 2 0 so that I ( Z , Z ) > 0 . This shows that

I1 B x B is positive definite. With the notation of SubIernma 10.26, we then

have Indl(c) 2 dim B.

346 10 MORSE IXDEX THEORY ON LORENTZIAN MANIFOLDS

From the direct sum decomposition Bo = B @ Jb(c). we obtain the equality

Indo(c) = dim Bo = dim B + dim Jb(c),

and we also know that Indl(c) 2 dim B. Thus the proof of Proposition 10.25

will be complete if we show that Indl(c) 5 dirn B.

To establish this inequality, suppose B' J(t,) is a vector subspace with

I j B' x 3' positive definite and dim B' = Indl(c). Suppose that dim B' > dim B. Then I is positive semidefinite on B' @ &(c) SO that dim(B1@ Jb(c)) 5 Indo(c). On the other hand, since dim B1 > dim B we obtain

This contradiction shows that dimB1 5 dimB, whence Indl(c) 5 dim B as

required. 13

We are now in a position to prove a Morse Index Theorem for tirnelike geo-

desic segments. The proof we give here 1s modeled on the proof for Riemannian

spaces given in Gromoll. Klingenberg, and bfeyer (1975, pp. 150-152).

Theorem 10.27 (Timelike Morse Index Theorem). Let c : [a, b ] 4

A4 be a timelike geodesic segment, and for each t E [a, b] let Jt(c) denote the

R-vector space of smooth Jacobi fields Y along c with Y (a) = Y (t) = 0. Then

c has only finitely many conjugate points, and the index Ind(e) and extended

index Indo(c) of the index form I : V&(c) x V ~ ( C ) --+ W are given by the

formulas

(10.18) Ind(c) = dim Jt(c) tE(a ,b )

and

(10.19) Indo(c) = dim Jt(c), t ~ ( a , b j

respectively.

Proof. We first show th2t xCtE(a,bj dim Jt(c) is a finite sum. We know that

dim Jt(c) 2 1 if arid only if c(t) is a conjugate point of t = a along c. We may

also define embeddings

i : Jt(c) - I#(.)

10.1 THE TIMELIKE MORSE INDEX THEOXY

for each t E [a, b ] by

for a 5 s 5 t , i(Y) (s) =

0 for t 5 s 5 b.

Evidently dirn Jt(c) = dirni(Jt(c)) for any t 6 (a. b ]

Recall that Indo(c) is finite by Proposition 10.25 Thus to show that c has

only finitely many conjugate points, it suffices to prove that if I t l , t2, . . tk) is

any set of pairwise distinct conjugate points t o t = a along c, then k 5 Indo(c).

To this end, set Ag = z ( J t , ( c ) ) for each j = 1,2, . . . , k and A = A1 B . . . @ Ak.

Then A is a vector subspace of V&(c), and ilecon~posing Z E A as Z =

c:,~ X3ZJ with A3 E W, Zg 6 A3, we obtain I (Z, 2) = C,,l A,AII(Z,, 21).

But if t3 _< t l , we obtain I(Z2, Zl) = I(Zj. 21): + I (Z3. Zl)Zi = -(&I, ZJ)l; + I(0, &);, = 0 using (10.21, Zl 6 At, and Z,(a) = ZJ(tl) = 0. Hence i ( Z . 2 ) = 0

from the symmetry of the index form. Thus I I A x A is positive semidefinite.

Hence

as required. Therefore c : [a, b ] -+ A4 has only finitely many conjugate points

in (a. b] , which we denote by tl < tz < ... < t , Except for t E ( t l , ta , , t,), we have dim Jt(c) = 0, and thus CtE(a,b dim .7t(r) i.; a finite sum

Since Jndo(c) = Ind(c) + dim Jb(c) from Proposition 10 25, !t is enough to

establish the equality

Indo(c) = dim Jt(c)

* to prove Theorem 10.27. Let Z denote the integers with the discrete topology

and define f , fo : (a , b] -+ Z by f (t) = ind(c 1 [a, t ] ) arid fo(t) = Indo(c / [a, t ] ) .

We now show that (10.20) holds if we establish the left ~oritinuity of f arid

the right continuity of So. liere we mean that iirntnrt f ( t n ) = f ( t ) a11d

Emtnlt fo(tn) = fo(t). Using (10.15) it follows that f (tj- fo(t) = - aim Jt(c) =

0 if t $! ( t i , t z , . . . , t,). Assuming we have shown f is left continuous and f U

is right cont~nuous, we also have f (t3+l) = fO(t3) f ~ r each 3 = 1,2 , . . . ,T - 1.


Thus

By Theorem 10.22 the index form is negative definite if c has no conjugate

points, so that f ( t ) = 0 for all t < t l . Hence f(t l) = 0 since f is left

continuous. Thus CtE(a,bl dim Jc(c) = fo(tr). Since fo is constant on [t,, b],

we have fo(t,) = jo(b). Hence

which establishes (20.20).

We have thus reduced the proof of the Morse Index Theorem to showing

that f is left continuous and fo is right continuous. First note that f and fo

are nondxreasing (i.e., f ( t ) 1 f (8) if t 2 s). For suppose we fix t l , t2 E (a, b]

with tl 5 tz. Let cl = c 1 [a, t l ] and 6.2 = cl [a,tz]. We then have an R-linear

embedding i : v$(c~) + ~ d ( c 2 ) given by

This map has the property that I(Y, Y) = I(i(Y), i(Y)), where the indexes are

calculated with respect to cl and c2, respectively. Thus if A C @(cl) is a vec-

tor subspace on which the index form of cl is positive (semi) definite, then i ( A )

is a vector subspace of V&(cz) on which the index form of c2 is positive (serni)

definite and dim A = dim i (A) . Thus f (tl) = Ind(c I [a, tl]) < Ind(c I [O, tz]) =

f jt2) and similarly fo(tr ) 5 Soft2). Thus fo and f are nondecreasing.

To obtain the continuity properties of f and fo, we fix an arbitrary E

(a , b] and study the continuity off and fo at ;using the same approximation

techniques as in the proof of Sublemma 10.26. Since c([a, b ] ) is a compact

subset of M , there is a constant 5 > 0 such that for any sl,sz E la, b] with

Is1 - szl < 6 and sl 5 8 2 , the geodesic segment c 1 [sl, s2] has no conjugate


- points. Choose a partition a = to < tl < . .- < t k - l < tk = t (unrelated

to the above enumeration of conjugate points) such that It, - tgtll < S for

each z = 0,1,2,. . . , k - 1. Let J C [a, b ] be an open interval containing - t with It - tk-11 < E for ail t E J . FOT each t E J , let J"(c,) denote the

finite-dimensional R-vector subspace of V&(c/ [a, t]) consisting of all vector

fields Y E 'ybi(cj[a,t]) such that Yl[t,,t,+i] for each j = 0.1.. . . k - 2

and Y I [tk-1, t] are Jacobi fields. By Sublemma 10.26, f ( t ) is the index of I

restricted to J"(ct) x y(ct) and fo(t) is the extended index of 1 restricted to

T(ct) x T(ct).

Now let E = N(c(t1)) x N(c(t2)) x . . - x N ( ~ ( t k - ~ ) ) The set E is ciosed

since each N(c(t,)) = (v E T,(t,)M : (v,cl(t,)) = 0) is a closed set of space-

like tangent vectors. We may define a Euclidean product metric (( , )) : E x E -+ R by ( ( v , ~ ) ) = ~f iT , ' (v , , w,) for v = (vi, 212, - . , , uk-1). w =

(ul, w2 ,... ,wk-l) E E. Then by Remark i0.2, S = {v E E : lltrll = 1) is

compact.

If Y E T(ct), then Y ( a ) = Y(t) = 0 by definition. Also since c j [t,,t,+l]

has no conjugate points, for any v E N(c(t,)) and w f N ( c ( ~ , + ~ ) ) there is a

unique Jacobi field Y along c with Y(t,) = v and Y(t,+l) = ZL. Since (Y,cl)l,

is an aEne function of t and (v. cf(t,)) = (w, ~ ' ( t , + ~ ) ) = 0. it follows that

(Y, c') 1, = 0 for all t. Hence the map

4t : ?(ct) + E

defined for t E J by

is an isomorphism. For each t t J we also have a quadratic form Qt : E x E -4

R given by QQt (u, v) = I(#;' (u), #tf (v)). Sublemma 10.26 implies that jo(t)

the extended index of the quadratic form &t on E x E and f ( t ) is the index

o f Q t o n E x E f o r e a c h t ~ J .

Each Qt may then be used to define a map Q . E x E x J -+ W given

by Q(u. v, t) = Qt(u, v). We want to show that Q is continuous in order to

prove that f and fo have the d ~ i r e d continuity properties. To this end, let


B = (Y / [a,tkPl] : Y E Z(c,)). Then B is isomorphic to E via the mapping

4 : B -+ E given by

Thus

where and YVat are the Jacobi fields along c given by

and

YV,t = $';'(v) I [ t k - 1 , t l .

Since the map (u, v) -+ I(p-'(u). $-'(v)) from E x E 4 W is a bilinear form,

the map ( u , ~ , t ) + I($-1(u),6-1(v)) is certainly continuous. By Proposition

10.16, the map (u,v,t) 4 (Xu,t,~:,,)/:k-l is continuous. This establishes the

continuity of Q : E x E x 3 -+ W. We are finally ready to show that fo is right continuous and f is left con-

tlnuous at the arbitrary t € (a, b]. Since Sublemma 10.26 implies that f

is finite-valued, we may choose a subspace A of E with dimA = f(t) and

Q ( u , u , t ) > 0 for all u E A, u f 0 Since Q : E x E x .J -+ R is contirnions

and S1 = S n A = (u E A : l/ull = 1) is compact, there is a neighborhood 30

of Fin J such that Q(zl.u, t ) > 0 for all t E Jo and u E S1. Hence Qt 1 A x A

is positive definite for dl t E Jo. Thus f(t) 2 f ( r ) for all t f Jo. Since f is

nondecreasing, it follows that f ( t ) = f (z) for all t E Jo with t 5 z Hence f 1s

left continuous.

It remains to show the right continuity of fo at Let (s,) be a sequence

In J with s, > t for all n and s, -+ Since Jo is nondecreasing and integer-

valued, we may suppose that fo(s,) = k for all n. By the monotonicity of

fo, we then have f o ( t ) 5 k. It thus rernains to &how that f o f t ) > k . To

accomplish this, choose for each n a k-dimensional subspace A, of E such

10.1 THE TIMELIKE MO=E INDEX THEORY 351

that Q,,, / A, x A, is positive semidefinite. Let (a' (n), a2(n), . . . . ak(n)) be an

orthonormal basis of A, for each n. Thus al(n), uz(n), . . . . ak(n) are contained

in the compact subset S of E. By the compactness of S, we may assume

a,(n) -+ a, 6 S for each j By continuity of the innel product, it follows

that the vectors (51, ar , . . . ,a,) form an orthonormal subset of S. Let A =

span{al, az. . . . , ak), which is thus a k-dimensional subspace of E. Given u E k k A. we may write u = A3a3. Let u(n) = C,=l Xla,(n). Obviously

~ ( n ) --I u as n -+ oo. Thus using the continuity of Q : E x E x J - R. me

obtain

QZ(u,u) = lim Q(u(n),u(n).s,) 2 0 11-00

since Q,- 1 A, x A, is positive semidefinite for each n. Hence Qt I A x A is

positive semidefinite. Thus fo( t ) 2 dim A = k as required. O

We cnnclude this section with an application of the Timelike Morse Index

Theorem to the structure of the cut locus of future one-connected, globally

hyperbolic space-times [cf. Beem and Ehrlich (1979c, Section 8)j.

Definition 10.28. (Future One-Connected Space-Tzme) A space-time

( M , g ) is said to be juizlkre one-connected if for all p,q E M, any two future

directed timelike curves from p to q are homotoplc through smooth future

directed timelike curves with fixed endpoints p and q.

This concept, a Lorentzian analogue for simple connectivity, has been stud-

ied in Avez (1963), Smith (1960a), and Flaherty (1975a, p. 395). The van-

ishing of the Lorentzian fundamental group implies that ( M , g ) is future orie-

connected. However, the simple connectivity of M as a topological space does

not imply that ( M , gj is future one-connected, as the following example of Ge-

roch shows. Consider B3 with coordinates (x, y, t ) and the Lorentzian metric

ds2 = -dt2 + dz2 + dy2. Let T = ((z, 0,O) E W3 : x 7: 0) and set M = B3 - T

with the induced Lorentzian metric from (W3, ds2) Then M is simply con-

nected. On the other hand, let p = (2,0, -1) and q = (2.0,1). Then p and q

may be joined by future directed timelike curves yl and lying on opposite

sides of the positive z axis (cf. Figure 10.1) But -/I and 72 are not homotopic

through fcture directed timelike curves starting at p and ending at, q since such

352 10 MORSE INDEX THEORY ON LORENTZISN MANIFOLDS

FIGURE 10.1. A space-time M = W3 - T which is simply con-

nected but not future one-connected is shown. The future directed

timelike curves yl and 7 2 from p to q are not homotopic through

timelike curves with endpoints p and q.

a homotopy would have to go around the point (0. 0 , 0), which would introduce

spacelike curves.

Note that if (M, g) is future one-connected. then the path space of smooth

timelike curves from p to q is connected. Thus Lemma 4.11-(2) of Cheeger

and Ebin (1975, p. 85) and the standard path space Morse theory [cf. Everson

and Talbot (1976), Uhlenbeck (1975), Woodhouse (1976)] imply the followmg

proposition.


proposit ion 10.29. Let (M,g) be future one-connected and globally hy-

perbolic. F i x p 6 M and suppose that every future directed timelike geodesic

segment starting at p has index zero or index greater than or equal to two.

Given q E I+@) such that q is not conjugate to p along any future directed

timelike geodesic from p to q, there is exact1.y one future directed timelike ge-

odesic from p to q of index zero, namely, the unlque maximal geodeslc from p

to q.

We are now ready to prove the Lorentzian analogue for globally hyperbolic,

future one-connected space-times of a theorem of Cheeger and Ebin (1975.

Theorem 5.11) on the cut locus of a complete Riemannian manifold which

generalized a theorem of Crittenden (1962) for simply connected Lie groups

w ~ t h bi-invariant Riemannian metrics. The global hyperbolicity is used in The-

orem 10.30 to guarantee the existence of maximal geodesic segments joining

chronologically related points. Here the order of a conjugate point t = to to

p along the t~rnelike geodesic y with $0) = p is defined as dim J,, (y) lrecali

Definition 10.241.

Theorem 10.30. Let (M, g ) be future one-connected and globally hyper-

bolic. Suppose that for p € A4, the first future conjugate point ofp along every

timelike geodesic y with y(0) = p is of order tu-o or greater. Then the future

timelike cut locus of p and the locus of first future timelike co~~jugate po~n t s of

p coincide. Thus all future timelike geodesics from p maximize up to the first

future conjugate point.

Proof. All geodesics will be unit speed and future timelike during the course

of this proof. It suffices to show that ~f y : [O, t ] -+ M wlth y(0) = p has index

zero and y(t) is not conjugate to p along y 1 10, t ', then y 1s maximal. Since

the set of singula points of expp is closed, it follows from Theorem 10.27 that

there exist E I , E ~ > 0 such that if <(u,y1(O)) < €1 and c < € 2 , then the future

timelikc geodesic c- : [O, t + €1 --t M nit11 al(0) = u is of index zero. Here Q

denotes angle measure using an auxiliary Ttiernailrlidn metric.

By Sard's Theorem, we may find a sequence of points { p , ) E, I t ( p ) with

p, -4 y ( t ) such that every timelike geodesic segment from p to pz has concon-

jugate endpoints. Thw, by hypothesis, every such segment has index zero or

354 10 MORSE INDEX THEORY ON LORENTZIAK MANIFOLDS

Index two or greater. By Proposition 10.29, there is a unique timelike maximal

geodesic segment 7% from p to p, of index zero.

Since (exp,), is nonsingular at t?'(O), for ? suEcie~ltly large there are geo-

desic segments 7% from p to p, with 7% -+ y. Since 6(Tt1(0), yf(D)) -+ 0, these

segments have index z ~ r o for large i. It follows that for i sufficiently large, - y, = y, and hence 7, is maximal. Thus 7 is maximal as a limit of maximal

geodesics. t

10.2 The Timelike Path Space of a Globally Hyperbolic

Space-time

In this section we discuss the Morse theory of the path space of future

directed timelike curves joining two chronologically related points in a globally

hyperbolic space-tirne following Uhlenbeck (1975) [cf. also Woodhouse (1976)j.

Both of these treatmerits are modeled on Milnor's exposition (1963, pp. 8&92)

of the Morse theory for the path space of a complete Riemannian manifold in

which the full path space is approximated by piecewise smooth geodesics. A

different approach has been given to the Morse theory of nonspaeelike curves

in globally hyperbolic space-time by Everson and Talbot (1976, 1978). They

use a result of Clarke (1970) that any four-dimensional globally hyperbolic

space-time may be isometrically embedded in a high-dimensional hlinkowski

space-time to give a Wilbert manifold structure to a subclass of timelike curves

in M .

We now turn to Uhlenbeck's treatment of the Morse theory of the path

space of piecewise smooth timelike curves joining points p << q of any globally

hyperbolic space-time (M, g) of dimension n 2 2

Definition 10.31. (The Tzrnelzke Path Space C!,,,)) Given p, q E. ( M , g)

with p << q, let C(,,,) denote the space of future directed piecewise smooth

timelike curves from p to q, where two curves which differ by a parametrization

are identified.

While C:p,q) is not a manifold modeled on a Banach space, it does possess

tangent spaces consisting of piecewise smooth vector fields along the given

piecewise smooth curve assumed to be parametrized by arc length. Functionah

10.3 HYPERBOLIC SPACE-TIME: TIMELIKE PATE SPACE 355

F . C(p,,) -t R may then be considered from the pomt of vlew of the calculus

of variations. Thus a cntzcal poznt of F is a polnt at ujhlch all first variations

vanish, and a cntzcal value is the image under F of a critical polnt. The

functional F on C(,%,) is said to be a homotopzc A4orse jknctzon i f for any

b > a which is not a critical value of F, the topological space F-I ( -m, b) 1s

homotopically equivalent to the space F-"-w. a) with celis adjoined, where

one cell of dimension equal to the index of the cor~esponding critical polnt 1s

adjoined for each critical point of F with critical talue in (a , b).

We will show in this section that the Lorentzian arc length functional is a

homotopic Morse function for C(,,,) provided that p and q are nonconjugate

along any nonspacelike geodesic. This result is analogous to Morse's result

[cf. Milnor (1963, Theorems 16.3 and 17.3)) for complete Riemannian mani-

folds. Namely, if p and q are any two dlstinct points not conjugate along any

geodesic, then the space R(,,,) of piecewise smooth curves from p to q has the

homotopy type of a countable CW-complex with a cell of dimension X for each

geodesic from p to q of index A.

Given that p and q must be nonconjugate along any geodesic for L to be

a Morse function, it is of interest to know that such pairs of points exist. As

for Riemannian spaces, conjugate points in an arbitrary Lorentzian manifold

may be viewed as singularities of the differentia! of the exponential mapping.

Hence Sard's Theorem [cf. Hirsch (1976, p 69)] may be applied. Here a subset

X of a manifold is said to have measure zevo if for every chart (li. 4): the set

O(U n X ) E R" has Lebesgue measure zero in Wn, n = dim A 4 Also a subset

of a manifold is said to be residual if it contains the intersection of countably

many dense open sets. A residual subset of a complete metric space is dense

by the Baire Category Theorem [cf. Kelley (1955, p. 200):. Sard's Theorem

then implies the following result,

Theorern 10.32. Let ( M , g ) be a globally hyperbolic space-time, and let

p E M be arbitrary. Then the set of points of M conjilgate to p along some

geodesic has measure zero. Thus for a residual set of q E M , p and q are

nonconjugate along ail geodesics between them.


Recall the following properties of the Lorentzian arc length functional L :

C(p,q) -" R. First, from the cdculus of variations, the critical points of L on

C(p,,) are exactly the future dlrected timelie geodesic segments from p to q

with parameter proportional to arc length. Second, the Timelike Morse Index

Theorem (Theorem 10.27) implies that the index of a critical point of L is

just its index rn a geodesic, i.e , the number of conjugate points t o p along the

geodesic, counting multiplicitim, from p up to but not including q.

In order to show that L is a homotopic Morse function, it is necessary to a p

proximate C(p,p) by a subset such that there exists a retraction of C(p,q)

onto -W(,,,) which increases the length functional L. This step corresponds to

the finite-dimensional approximation of the loop space in Riemannian Morse

theory. The corresponding Lorentzian approximation makes crucial use of the

global hyperbolicity of (n/i,g), as will be apparent from Lemma 10.34 below.

First it is useful to make the following definition.

Definition 10.33. (Timelike Chain) Let p. q E ( M , g ) with p << q. A

finite collection (xl,x2,. . . , x3) of points of A4 is said to be a timelike chazn

fvom p to q if p << x1 <( xz << . . . <C xJ << q.

The next lemma follows from the existence of convex normal neighborhoods

(cf. Section 3.1), the compactness of J f (p) n J-(q) in a globally hyperbolic

space-time, and Theorem 6.1.

Lemma 10.34. Let (M, g) be a globally hyperbolic space-time with a fixed

smooth globally hyperbolic time funct~on f : M -+ R. Let p, q E M with p < q

be given. Then there exists ( t l , tz , . , . , tk) with f ( p ) < tl < tz... < tk < f ( q )

satistying the following properties:

( 1 ) If z E f - -'(f(P), tl] m d p < x, then there is a unique maximal future

directed nonspacelike geodesic segment from p to x;

(2) For each i with 1 5 i 5 k - 1, if x E f-'(tz) and y E f-'(t,,t,+l]

with p < x ,< y < q. then there is a unique maximal future directed

nonspaceiike geodesic segment from z to y:

(3) If y E. f-l[tk. f ( q ) ) and y G q, then there is a unique maximal future

dlrected nonspacelike geodesic segment from g to q;

10.2 HYPERBOLIC SPACGTIME: TIMELIKE PATH SPACE 357

(4) In paxticular. if {zl, x2,. . . , s k ) is any timefike chain from p to q with

x, E f-l(t,) for each i = 1.2, . . . , k , then there is a unique maximal

future directed tlrnelike geodesic segment from p to sl , zk to q, and s,

to x , ,~ for each E with 1 5 i 5 k - 1.

Since p , q E M with p < g are given, we may now fix i t l , t 2 , . . . . t k ) sat-

isfying the conditions of Lemma 10.34. Denote by S, the Cauchy surface

S, = f -I(&) for each i = 1,2, , . . , k. We may now define a space McP,4) of

broken timelike geodesics to approximate C(p,q) from the point of view of the

arc length functional.

Definition 10.35. (Tzmeizke Guwe Space M(,,,)) Let M(PPq be the

space of all continuous curves y : [O, 11 -+ M such that y(0) = p, y (1 ) = q. and

y ( i / ( k t 1)) E St for each i = 1.2,. . . . k and such that the restricted curve

7 1 [ i / ( k + 1) . (z -I- 1)/(k + I ) ] is a future direczed timellke geodeslc for each

i = 0 , 1 , . . . . k.

Since each y / [ i / ( k + I ) , (i + 1 ) / ( k + l ) ] must be the unique maximal seg-

ment joining its endpoints by Lemma 10.34, it follows that

(10.21) J W ( ~ , ~ ) = { ( Z ~ , T Z , . . . , zk) : x, 6 S, for 1 < i 5 k ,

p <C 2 1 , x, < s,+l for each 1 s i 5 k - 1, arid xp, << y ) .

Since chronological future and past sets arc open. M(p,g) viewed as in (10 21)

is an open submanifold of S1 x Sz x - - x &. Let 7 i ~ : M(p,q) + S2 be the

projection map given by 7 i , ( ~ ~ , x 2 , . . . . x k ) = z1 for each z = 1,2. . . , k . S~nce

I+(p) n I - ( q ) h a compact closure by the global hyperbolicity of M , and

~ , ( h f ( ~ , ~ ) ) c I+(p) n I - (q ) , ~t follows that T~(A~(,,,)) has compact closure In

S, for each i = 1,2.. . . . k. A length-nondecreasing retraction of C(p,q) onto hf(p,,) may now be estab-

lished along the lines of Riemannian Morse theory [cf M~lnor (1963, p Ql)]

Proposit ion 10.36. There exists a retraction Qx, 0 < X 5 1, of C(,,,)

onto hf(p,q) which is Lorentzian arc length nondecreasing.

Proof. Let y E C(,,,; be arbitrary W e may suppose that y is parametrized

so that f ( y ( t ) ) = t where f : M -+ B is the globally hyperbolic time function


fixed in Lemma 10.34. Tnus y : [f (P), f (q)] - M For each X E [O. I], define

Qx(y) : [f (P), f (411 -" M as follol-s Set B = (1 - X ) f (PI + X f (q) , and put

to = f (p) and tk+: = f (4). If P satisfies t , < 0 5 tl+l for i with O < 'L j k, let

Q x ( ~ ) ( t ) be the unique broken timelike geodesic joining the point p to r ( t l ) ,

y(tl) to y(tz). . . . , y(t,-1) to y(t,), and y(t,) to yfP) successiveiy for t 5 p. and let Q:,(r)(t) = y(t) for t 2 p (cf. Figure 10 2). I t is ~mmediate from the

definltlon that Qo(y) = y and that Ql(y) E M(,,,). Since Q~(y) [y ( t , ) ,~ (P) ] is

maximal from y(t,) to r (P) by Lemma 10.34, we have

where d denotes the Loreiltzian distance function of (M, g) and we have used

Notational Convention 8.4. Similarly, since Qx(y) is the unique niaximal ge-

odesic segment from y(t3) to y(t,+l) for each j with 0 5 j < i - 1, we have

L($x(y)[y(t,),y(t,+~)]) 1 L(y I [t,,t,+,]) for each j with 0 L j I i - 1. Sum-

ming, we obtain L(Q:,(y)[p, y(P)]) 1 L(y I [f ( P ) , PI). Since Qx(y)(t) = ;dt) for

t 2 p, we thus have L(Qx(y)) 2 L(y). Moreover, it is clear from the above

argument that L(QA(y)) = L(y) for all X if and only if y is a broken time-

like geodesic from p to q with breaks possible only at the s , ' ~ . In particular.

Q:, ( IV~,,,) = Id for all X E [O,1]. Finally, the continuity of the map X --, Qx

is clear from the differentiable dependence of geodesics on their endpoints in

convex neighborhoods.

As in Uhlenbeck (1975, p. 791, we denote by L, = L 1 A4(,,,) the restriction

of the Lorentzian arc length functional to the subset M(p,q) of C(p,q). Just as

in the Riemannian case [cf. Milnor (1963. Theorem 16.2)], it can be seen that

L, : Id(,,,) 4 R is a faithful model of L : C(,,,) 4 W in the following sense

Proposition 10.37. lf(-k?. g) isglobally hyperbolic, then the critical points

of L, = L 1 Af[,,,) are smooth timelike geodesic segments from p to q. These

rritical points are nondegenernte iffp is not conjugate to q dong any timelike

geodesic from p to q. Moreover, the index of each critical point is the same in

C(,,,) and M(,,,). namely, the index by conjugate points.

Proof. Identify M(p,q) with k-chains (zl, xz, . . . xk) from p to q with z, .E S,

for each i, as in (10.21). Set so = p and xk+l = q. Let y, : [0, I] -r M

10.2 HYPERBOLIC SPACE-TIME: TIMELIKE PATH SPACE 359

FIGURE 10.2. In the proof of Proposition 10 36. the curve Qx(y)

is used to approximate the given curve 7 .

denote the unique maximal timelike geodesic segment from r, to rz+l for

z = 0,1. . . , k. Then

k r l 1

L. ( r l , TZ, . . . , I*) = C J j - ( -hf j t ) . yti,'(i)) d t . 2 = 0 0

Suppose we have a smooth deformation (xl ( t ) . s2(t), . . , sk ( t j ) of the given

chain {xl , 22, . . . . xk}. Then since each t -r x, ( t ) is a curve in SZ, the vari-


ation vector field I/ of the deformation must lie in T,,(S,) at x,. Also as

we are deforming the glven chain, which represents a piecewlse smooth time-

like geodesic, through plecewise smooth timelike geodesics with discontinuities

only at the St's, the space of deformations of (xl, 22, , xk) may be den-

tified with all vector fields Y along is1. 12,. . , xi,) so that Y (z,) E Tx3 (S,)

for z = 1,2, . , k . Y ( p ) = Y ( q ) = 0, and Y 17, is a smooth Jacobi field

along y, for each z = 0,1, , k. By choice of the S,'s as in Lemma 10.34,

no y, . [0, l] -- Af has any conjugate points. Tllus given any v, E T,* (St )

and zu, E T,,,, (S,+lJ, there is a unique Jacobi field J along [O. 1, -+ iM

with J(0) = ?it and J(1) = w,. Thus the space of 1-jets of deformations of

the given chain isl, x2, , sk) may slmply be identified with the Cartesian

product T,, (SI) x Tx,(Sz) x x T,, (Sk).

Let u : [a, b ] --+ M be a smooth future d~rected timelike curve parametrized

so that J-(al(s). al(s)) = A is constant for all s E ( a , b) and so that ol(a+)

and ol(b-) are timelike tangent vectors Let n : [a, h ] x (-c.c) + M be a

smooth variation of a through nonspacelike curves with variation vector field

V . If A * : [a. b ] + itl denotes the curve a,(t) = o( t , s) , then the first variation

formula for ~ ' ( 0 ) = (d/ds)L(~~.)],=o is given by

6 Thus if o is a timelike geodesic. ~ ' ( 0 ) = (-l/A)(V,

\'Vc now apply (10.22) to calculate the first variation of a proper deformation

a of an element {s l ,zz , . ,zk) E M(P,,,). As before, let y, : [O. 11 -+ M be

tlmelike geodesics from x, to r,+l for 2 = 0,1, k Then {xl, x2 , xk) IS

represented by the piecewise smooth timelike geodesic y = -0 * yl * ., . * y k

Let V be the variation vector field of a along y and set y, = V(rr,) E T,, (S,)

As we mentioned above the piecewise smooth Jacobi field V along y may then

be identified with (yl,y2, . l /k) f TT, (5'1) x ... x T l k (Si,). Restricting V to

each y, and applying formula (10.22), we obtain the first variation formula

10 2 HVPERSOLIC SPACE-TIME TIMELIKE PATH SPACE 361

where

for L, = L 1 -44(,,,). Rom (10.23), it is a standard argument to see that

6L,(yl,y2,. . . . yk) = 0 for all (yl, yz.. . . . yk) E 'Iz, (5'1) x . . . x Tzls(Sk) if and

only if the tangent vectors

7i1(1) and 7:+1(0) 1/-(7%1(1)-711(1)) J- kL'+1(0)? pf%'-l (0))

in T,> (M) have the same projection into the subspace T,, (S,) of T,, (M) for

each i = 1: 2,. . . , k. This then implies that y = yo x y, * . . . * yk can be

reparametrized to be a smooth timelike geodesic from p to g.

Thus we have seen that the critical points of L, : M(,,,) -+ R and L :

C(,,,) -+ R coincide and are exactly the smooth timelike geodesics from p to q.

By our proof of the Timelike hlorse Index Theorem (Theorem 10.27) by ap-

proximating V&(c) by spaces of piecewise smooth Jacobi fields (cf. Sublemma

10.26), it follows that the indices of L, : -M(,,,) -t R and L : C( ,,,, i R

coincide. The Timelike Morse Index Theorem (Theorem 10.27) then implies

that a critical point c is degenerate if and only i f p and q are conjugate along

c (cf. also Proposition 10.13).

With Proposition 10.37 in hand, the followiilg result rnay now be estab-

lished.

Proposition 20.38. Let ( M g) be globally hyperbolic. If p and g are

nonconjugate along any nonspacelike geodesic. then L : C(p,,) -+ lR and L, . M(,,,) i R have only a finite number of critical points. ln partielljar, there

are only finitely many future directed timelike geodesies from p to q.

ProoJ. By Proposition 10.37, it is enough to show that L, : ~%f(,,~) + R

has only finitely many critical points. Suppose L, has infinitely many critical

points. Then there would be an infinite sequence of smooth timelike geodesics

from p to q. Since each ? T ~ ( M ( ~ , ~ , ) has compact closure in S,, the h ' s

have a subsequence which must then converge to a timelike or null geodesic c


from p to q. Since c is a limit of an infinite sequence of geodesics from p to q,

it follows that p is conjugate to q along c, in contradiction. 0

Another interesting property of L, : M(,,,) -+ R is given by the following

result.

Proposition 10.39. L, : &I(,,,) - B assumes its maximum on every

component of hf(,,,).

Proof. While M(,,,) is an open submanifold of SI x Sz x . . x Sk, its closure - iM(,,,) is compact in S1 x S2 x . . . x Sk. Also

where xo = p and xk+l = q for any timelike chain {xl, x2,. . . , xn) from p to q.

Thus as the Lorentzian distance function is finite valued for globally hyperbolic

space-times, it follows that L, is bounded on each connected component U

of Ad(,,,). Let (7,) be a sequence in the connected component U of M(p,q)

with L,(y,) -+ sup(L,(y) : y E U) as n -+ a. By compactness of Z(,,,) in S1 x Sz x - . . x Sk, the sequence {y,) has a limit curve y, in SS, x

$2 x . x Sk. Since {y,) was a maximizing sequence for L, ] U, it follows that

L,(y,) 2 L,(n) for any a E g, and L,(y,) > 0. If y, is represented by the

element (sl, 22,. . . , xk) E S1 x . . . x Sk, we have p < zl < 3.2 < . . . ,< r k 6 q,

and if a is the unique maximal geodesic from 2% to x,+l guaranteed by Lemma

10.34, then each 7% is either null or timelike. If some y, is null. it follows from

the first variation (10.23) that y, may be deformed to a curve y E jl with

L, (y) > L, (7,). This then contradicts the maximality of L, / at y,. Thus,

in fact, the Iimit element r,, E U . U

Again, consider LW(~,,) .ca a subset of S1 x . . . x Sk. Let P, : T,(M) --+ T, (St)

denote the orthogonal projection map for any x E S, and each i = 1,2.. . . , k .

The set M(p,q) regarded as an open subset of S1 x - x 31, may thus be given

a Riemannian metric induced from the Lorentzian metric restricted to the

spacelike Cauchy hypersurfaces S1, Sz, . . . , Sb It then follows from formula

(10.23) that at the point of &I(,>,) represented by the broken timelike geodesic

10.2 HYPERBOLIC SPACE-TIIvIE: TIMELIKE PATH SPACE 363

y = 70 * 71 * . . - * */kt with each y, parametrized on [O. 11 arid A,, B, s above.

the gradient of L. is given by the formula

Using this formula. Uhlenbeck (1975, pp. 80-81) then establishes the following

lemma.

Lemma 10.40. Let /3 : (a, b) -+ M(,,,) be a maximal integral curve for

grad L,. Then b = oo and limt,, P( t ) lies in the cri~ical set of L,.

With the behavior of grad L, established, ~t now follows [Uhlenbeck (1975,

p. 81)] that if p and 4 are nonconjugate along any timelike geodesic. then

L, : M(,,,) -+ R is a Morse function for A4(,,,). The main result now follows

from this fact and Propositions 10.36 and 10.38.

Theorem 10.41. Let (M, g) beglobally hyperbolic, and let p, q he any pair

of points in (34, g) with p << q and such that p and q are not conjugate along

any nonspaceiike geodesic from p to q. Then there are only finitel-v many future

directed timelike geodesics in (M, g) from p to q, and the arc length functional

L : C(,,,) -+ W is a homotoplc Morse function. Thus ~f b > a are any two

noncritical values of L, then L-I( -m, b) is homotopically equivalent to the

space L-'(-03, a ) with a cell attached for each smooth timellke geodesic y

from p GO q with a < L(-f) < b. where the dimens~on of the attached cell is

the (geodesic) index of y Thus C(,,,) hds the hornotopy type of a finite CW-

complex with a cell of d~mension X for each srnooth future directed timellke

geodesic 2 from p to q of index A.

Note that in Theorem 10.41 the topology of Ci,,,) is not related to the

given rnanifold topology. But Uhlenbeck (1975, Theorem 3) has shown for a

class of globally hyperbolic spacetimes sat~sfying a metric growth condition

[cf. Uhlenbeck (1975. p. 7 2 ) ] that the hamotopy of the loop space of M itself

may be calculated geometrically as follows. Let (M, g) be a globally hyperbol:~

space-time satisbing Chlenbeck s metric growth condition. Then there is a


class of smooth timelike curves y : [0. 03) + M with the following property

For any such timelike curve y, there is a residual set of points p E M such

that the loop space of Al is homotopic to a cell compiex with a cell for each

null geodmc from p to y, where the dimension of the cell corresponds to the

conjugate polnt index of the geodes~c.

It will be clear from the proof of Proposition 10.42 below that the finiteness

of the homotopy type of the timelike loop space C(,,,) in Theorem 10.41 follows

from the assumption that p is not conjugate to q dong any null geodesic

together with the fact that L(7) 5 d ( p , q ) < co for a l ly E R(,,,). For complete

Riemannian manifolds (N.go) on the other hand, it is known from work of

Serre (1951) that if iV is not acyclic [i.e., H,(N;Z) + 0 for some i > 01. then

the loop spaces f l ( , ,q) are infinite CW-complexes for all p, q E N . Thus if p

and q are nonconjugate along any geodesic, there are infinitely many geodesics

c, : [O, 11 4 N from p to q. Since p and q are nonconjugate along any geodesic

and La(?) > do(p, q) > 0 for all y E a(,,,), it may be seen that Lo(&) -+ oo

asn-+w.

For completeness, we now give a different proof from Proposition 10.38 of

the existence of only finitely many critical points for L : C(,,,) + P. Instead of

using the finite-dimensional approximation of Cb,,) by W,,,). we work directly

with C(,,,) using the existence of nonspwelike limit curves as in Section 3.3.

Proposition 10.42. Let ( M , g) be globally hyperbolic, and suppose that

p, q E M with p << q are chosen ssuch that p and q are not conjugate dong

any future directed nonspacelike geodesic. Then there are only finitely many

cimelike geodesics from p to q.

Proof. Suppose that there are infinitely many future directed timelike ge-

odesic segments c, : [0, l] A4 in C(p,g). Using Corollary 3.32 and the

arguiients of Section 3.3 we obtain a nonspacelike geodesic c : [O, 11 + A4 with

c(0) = p and c(1) = q which is a limit curve of the sequence (h) and such

that a subsequence of {c,) converges to c in the CO topology on curves. Since

a subsequence of the pairwise distinct tangent vectors ( ~ ' ( 0 ) ) converges to

cl(0), it follows that q is conjugate t o p along c, in contradiction. El

10.3 T H E NULL MORSE INDEX THEORY 365

Suppose that it is only assumed that p and q are not conjugate along any

timelike geodesic in the hypotheses of Prcpositioc 10.42. If there are infinitely

many tirxelike geodesics {c,) in C(p,q), we then have L(c,) -+ 0 as n - m,

and the limit curve c in the proof of Proposition 10.42 is a, null geodesic such

that q is conjugate t o p along c. In particular, c contains a future null cut point

t o p (cf Section 9 2) Thus, Theorem 10.41 or the proof of Proposition 10.42

also yields the following result which applies. in particular, to the Friedmann

cosmological models with p = A = 0.

Corollary 10.43. Suppose that (M,g) is globally hy-perboljc and that

p,q E M are chosen such that p << q, p is not conjugate to q along an^;

timelike geodesic, and the future null cut locus of p in M is empty, Then there

are oniy finitely many timelike geodesics from p to q.

Prooj. If there were infinitely many timelike geodesics from p to q, there

would be a null geodesic c from p to q such that q is conjugate to p along c.

But then p contains a future null cut point along c, in contradiction. R

If d imM = 2, then (M,g ) contains no null conjugate points (cf. Lemma

10.45). Thus we also have the following result.

Corollary 10.44. Suppose that (M, g) is any two-dimensional globally

hyperbolic space-time and that p. q E M are chosen such that p < q and q is

not conjugate to p along any ~imelike geodesic. Then there are only finitely

many timelike geodesics from p to q.

10.3 The Nu11 Morse Index Theory

Thi9 section is devoted to the proof of a Morse Iedex Theorem for null geo-

desic segments @ : [a, h ] + (M,gj in an arbitrary space-time. The appropr~ate

index form we will use. however, 1s not the standard index form defined on

piecewise smooth vector fields orthogonal to 0' but rather its proj~ction to the

quot~ent bundle formed by identifving vector fields differing by a miiltipie of

p'. The tdea to use the quotient bundle as the domain of definition for the null

index form is zmplicitly contained in the discussion of variatmn of arc length

for null geodesics in Hawking and E!hs (1973, Sectlon 4.5) and is further de-


veloped in Bolts (1977) [cf. Robinson and Trautman (1983)]. The first part

of this section develops the basic theory of the index form along the lines of

Chapters 2 and 4 of Bolts (1977) In the second part of this section, we give

a detailed proof of the Morse Index Theorem for null geodesics sketched In

Beem and Ehrlich (1979d).

Instead of working with the energy functional. Uhlenbeck (1975, Theo-

rem 4.5) has constructed a Morse theory for nonspacelike curves in globally

hyperbolic space-times as follows. Choosing a globally hyperbolic splitting

M = S x (a, b) as in Theorem 3.17. Uhlenbeck projected nonspacelike curves

y ( t ) = ( c l ( t ) , cZ(t)) onto the second factor and showed that the functional

J ( 7 ) = S[c2'(t)l2dt y~elded an index theory.

It should be mentioned at the outset that the null index theory, unlike the

timelike index theory, is interesting only if dim M 2 3 for the following reason

Lemma 10.45. No null geodesic ,# in any two-dimensional Lorentzian man-

ifold has any null conjugate poin~s.

Proof. Let p : (a, b) + ( M , g ) be an arbitrary null geodesic. Suppose that

J is a Jacobi field along /3 with J(tl) = J ( t 2 ) = O for some t I # t2 in ( a , b).

Just as in the proof of Lemma 20.9, we have ( J , ,@)I' = - ( R ( J : ,B')P1, 9') = 0

so that ( J ( t ) , P1 ( t ) ) = 0 for all t E (a, b). Since spacelike and null tangent

vectors are never orthogonal when dimM = 2, the space of vector fields Y

along j3 perpendicular to @' is spanned by P' itself. Hence J(tf = f ( t )P1( t )

for some smooth function f : (a, b) + R. The Jacobi equation then becomes

0 = J" + R(J. @)P' = f"P1 + fR(P' , P')P = f1I/?' by the skew symmetry of

the curvature tensor in the first two slots. Hence f"( t ) = 0 for all t E (a, b)

Since J ( t i ) = J ( t z ) = 0, it follo~vs that f = 0. Thus J = 0 as required.

In view of Lemma 10.45. we will let ( M , g ) be an arbitrary space-time of

&rnension n 2 3 throughout this section and let 0 . (a, b ] -+ M be a fixed

null geodesic segment in -44. If we let V1(P) denote the W-vector space of

all piecewise smooth vector fields Y along ,!3 with (Y(t),P1(t)) = 0 for all

t E [a, b ] , then pl( t ) and t,B1(t) are both Jacobi fields in V1(P). k t h e r ,

10.3 THE NULL MORSE INDEX THEORY 367

suppose we consider an index form I : Vj(P) x Vt(f i ) -+ R defined by

I ( X 7 Y ) = - / [(x', Y' ) - ( R ( X , 01)/3' .Y)] dt

analogous to the index Lorm (10.1) for timelike geodesics In Section 10.1. Then

A = {f ( t )P i ( t ) 1 f . [a, b ] -+ R is a smooth function with f ( a ) = J ( b ) = 0) 1s

an infinite-dimensional vector space such that I (Y . Y) = 0 for all Y E A. Thus

the extended index of /3 defined using the index form I : Yo'-(@) x V:($) -+ R

is always infinite. Also while I ( f P', Y) = 0 for any Y E VZ(0) and f 9' E A.

the vector field fP1 is not a Jacobi field unless f" = 0. Thus the relationships

we derived for the index form of a timelike geodesic in Section 10.1 linking

Jacobi fields, conjugate points, and the definiteness of tlie index form fail to

hold for I : voA(a) x V,-(B) - R

The crux of the difficulty is that P' and tP1 are both Jacobi fielda in ~ ' ( 0 ) . Thus by ignoring vector fields in A, it is possible to define an ~ndex form for

null geodesics nlcely related not only to the second variation formula for the

energy functional but also to conjugate points and Jacobi fields. This may

be accomplished by working with the quotlent bundic VL(/3)/[/3'] rather than

V1(P) itself Using this quotient space, an index form 7 may be defined so

that ,9 has no conjugate points if and only if? is negative definite [cf. Hawking

and Ellis (1973, Proposition 4.5.11). Bolts (1977, Satz 4.5.5)j. Also, a Morse

Index Theorem for null geodesic segments in arbitrary space-timcs may be

obtained for the iridex form 7 [cf. Beern and Ehrlich (1979d)j.

Since we are interested in studying conjugate points along null geodesics, it

is important to note that Lemma 10 9 and Corollar~es 10.10 and 10.11 carry

over to the null geodesic ease with exactly the same proofs

Lemma 10.48. Let /3 : [a, b ] -+ (M,g ) be a null geodesic segment. and

let Y be any Jacobi field along P. Then (Y ( t ) , y(t)) is an afine function of f

Thus i f Y ( t l ) = Y(t2) = 0 for distinct t l , t z E [a. b j . then ( Y ( t ) , f l ( t ) ) = 0 for

all t E [a, b ] .

Accordingly, we may restrict our attention to the following spaces of vector

fields.

368 15 MORSE INDEX THEORY ON LORENTZIAK MANIFOLDS

Definition 10.47. Let V L ( P ) denote the R-vector space of all piecewise

sinooth vector fields Y along P with ( Y ( t ) , f l ( t ) ) = 0 for all t E [a, b] . Let

V$ (0) = ( Y E v'@) : Y (a) = Y {b) = 0). Also set N(P(t) ) = (v E Tp(tjM :

(v. pl(t)) = 01, and let

= I.) N(P(t) ) . a s t s b

For any Y E V L ( @ ) , the vector field Y' E V L ( P ) may be defined using

left-hand limits just as in Section 10.1. Since /3 is a smooth null geodesic.

P1( t ) E N ( p ( t ) ) for all t E [a. b ] . Thus, following Bolts (1977, pp. 39-44). we

make the follo\ving algebraic construction. Since N(P( t ) ) is a vector space and

is a vector subspace of N(P( t ) ) for each t E [a, b] , we may define the quotient

vector space

and the quotient bundle

Elements of G(P(t) ) are cosets of the form v + [P1(t)], where 1; E N ( B ( t ) ) and

the vector subspace [P1(t)] is the zero element of G(P(t)) for each t E [a, b].

Also 7: + [PJ ( t ) ] = w + [@'(t)] in G(P( t ) ) if and only if v = w + XP1(t) for some

X E R. We may define a natural projection map 2.r : :V(P(t)) 4 G(B(t) ) by

(10.27) K ('u) = TJ + [P1(t)].

The projection map 7i on each fiber induces a projection map 2.r : N(P) -+ G(P)

given by r ( Y ) = Y + [P'], i.e., r ( Y ) ( t ) = Y ( t ) + [P1(t)] E G(P(t)) for each

t E [a, b ] .

This quotient bundle construction may be given a (non-unique) geomet-

ric realization as follows. Let n E Tp(o)M be a null tangent vector with


(n, fll(0)) = -1. Parallel translate n along P to obtain a null parallel field

q along with ( ~ ( t ) , P1( t ) ) = -1 for all t E [a, b ] . Choose spacelike tangent

vectors el. en,. . . , en-2 6 Tp(o)M such that (n, e,) = (01(0), e3 j = 0 for ali

j = 1.2, . . . . n - 2 and (e,, e2) = 6,, for all J = 1,2, . . . , n - 2. E x t e ~ d by par-

allel translation to spacelike parallel vector fields El, E2,. . . , En-2 E vJ-(P), and set

n-2

(10.28) CA,E,(~) :A, ER for 1 < j s n - 2 3=1

Then V ( P ( t ) ) is a vector subspace of N(P( t ) ) consisting of spacelike tangent

vectors, and we have a direct sum decomposition

for each t E [a: b ] . Set V ( P ) = lJa<t5b V(D(t) ) , and let %(a) = {Y E V(O) : - Y ( u ) = Y ( b ) = 0). If P'(t) and ~ ( t ) are given, V ( P ) 1s independent of the

particular choice of {el, en, . . . , en-2) in Tp(o)lVI. However. if n E T p p ) M

and hence 17 are changed: a different direct sum decomposition (10.29) may

arise since the given Lore~ltzian metric g restricted to .?-(,B(t)) 1s degenerate.

Nonetheless, we may regard V(P( t ) ) as being a geometric realization of the

quotient bundle G(@) via the map Z --+ Z + (@'I from V(P) :nto G(P). It 1s

easily checked that this map is an isonlorphisin since (10.29) is a d~rect sum

decomposition.

We may dcfine the inverse isomorphism

for each t E la, b ] as follo~vs. Given v E G($(t ) ) , choo~i. any rr E N ( P ( t ) ) with

~ ( x ) = TJ. Decomposing 3: uniquely as x = X/j"(t) + zo with vo E V ( P ( t ) ) by

(10.29), set 6(v) = vo. If any other X I E N(P(t) ) with 7i(zl) = .r, had been

chosen. we would still have X I = pP(t) + vo w ~ t h the same ' ~ ' 0 E V ( B ( t ) ) for

some /I E R. Thus 6 is well defined.

As a first step toward defining the index form on the quotient bundle

G(P), we show how the Lorentzian metric, covariant derivat:ve, and curvature


tensor of (M, g) may be projected to G(P). First, given any u , w f G(P(t)).

choose x, y f N(P(t)) with n ( x ) = v and n(y) = w. Define B(v. w ) by

Suppose we had chosen X I , yl E N(P(t)) with ~ ( 2 1 ) = v and ~ ( y l ) = w.

Then x = z1 + XPf ( t ) and y = yl + p/3'(t) for some X,p E R, arid thus

g(x, Y) = g(al,yi)+Xg(yl. ,3'(t)) f ~ g ( x ~ , P ' ( t ) ) + ~ X g ( P ' ( t ) , P ' ( t ) ) = y(z l ,y l ) . Hence g is well defined. I t is also easily checked that B(v, w) = g(B(u), B(w)) for

all v, zli E G(P(t)). Hence the metric on G(P(t)) may be identified with the

given Lorentzian metric g on V(O(t)). Since g 1 V ( P ( t ) ) x V ( P ( t ) ) is positive

definite for each t 6 [a, b ] . the induced metric 3 : G(P(t)) x G(P(t)) -t R thus

has the following important property.

Remark 10.48. Fhr each t E [a, b ] , the metric jj : G(P(t)) x G(P(t) ) -+ R

is positive definite.

We now extend the covariant derivative operator acting on vector fields

along P to a covariant derivative operator for sections of G(F). We first intro-

duce the following notation.

Definition 10.49. Let X(P) denote the piecewise smooth sections of the

quotient bundle G(P), and let

Given V E X(@, choose X E V'(6) with n ( X ) = V. Set

If XI E V L ( P ) also satisfies: x ( X 1 ) = V , then XI = X + jPf for some f :

[a, b ] + R, and we obtain X I 1 = XI + f1p since P is a geodesic. Thus

x ( X l f ( t ) ) = x ( X 1 ( t ) ) for all t E [a, b] . Hence the covariant, derivative for X ( P )

given by (10.32) is well defined. It may also be checked that this covariant

differentiation is compatible with the metric g for G(P) and satisfies thc usual

properties of a covariant derivative.


Given V E X ( P ) , choose X E V L ( 0 ) with T ( X ) = V . We then have

X ( t ) = f(t)O1(t) + B(V)(t) with 6 as in (10.30). According to (10.32), we may

thus calculate V1( t ) using B(V) as V f ( t ) = 7(VP(B(V)) ( t ) ) . Now B(V) satisfies

(P1(t), O(V)(t)) = ( ~ ( t ) , B(V)(t)) = O for all t E [a, 61. Since 0 is a geodesic and

7 j is parallel along 3, we obtain on diserentiating that (O1(t), VptB(V)( t ) ) =

( ~ ( t ) , V p B ( V ) ( t ) ) = 0 for all t E [a, b ] . Thus 'G?$'O(V) E V(P) . Hence we

obtain

(10.33) 6 ( V f ( t ) ) = (B(V))'(t) for all t E [a, b ]

where the differentiation on the left-hand side is in G(P) and on the right-

hand side is in V(/3). Thus if we identify G(0) and V(P) using @, covariant

differentiation in G(P) and in V(0) are the same.

To project Jacobi fields and the Jacobi differential equation to G(/3). it 1s

necessary to define a curvature endomorphism

for each t E [a, b ] . This may be done as follows. Given v E G(P(t)), choose

any 3: E N(P( t ) ) with ~ ( x ) = v, and set

This definition is easily seen to be independent of the choice of x 2: f (B( t ) )

with ~ ( x ) = v since R(P1, fi l)Pf = 0. if 71 = T ( X ) and u: = n(y) with x, y E

N(P( t ) ) , it follows from (10.31) and (10.34) that

Finally horn the symmetry properties of g(R(x, g)z, w) we obtain

for all u, w E G(P(t))

With these preliminary considerat!ons completed, we are now ready to de-

fine Jacobi classes in G(P) [cf. Boltr (1977, pp 43-44)]


Definition 10.50. (Jacobi Class) A smooth section V E X(P) is said to

be a Jacobi class in G(P) if V satisfies the Jacobi differential equation

where the covariant differentiation is given by (10.32) and the curvature endo-

morphism a by (10.34).

Given a Jacobi class W E X(P) with W ( a ) # [,@(a)] and W(b) # [ P ( b ) ] ,

we will see in the next series of lemmas that there IS a two-parameter family

Jx , of Jacobi fields of the form JA,, = J i- A/? + ptP1 in V i ( P ) , A. /I E R,

with r ( J x ,) = W . Nonetheless, it should be emphasized that given a Jacobi

class W E X(P), there may be no Jacobi field J in any geometric realization

V ( G ) for G ( P ) with z ( J ) = W . On the other hand, there will always exist

a Jacobi field J E VL(P) with r ( J ) = W . But it may be necessary for J

to have a component in ! /? ' I . The reason for this is made precise in Lemma

10.52. In the next series of lemrnas, we will use the geometric realization

VL(p ) = [pl] @ V ( p ) defined in (10.29).

Lemma 10.51. Let W be a Jacobi class of vector fields in G(@). Then

there is a smooth Jacobi field Y E VL(0 ) with K(Y) = W . Conversely, i f Y is

a Jacobi field in VL (P) , then r (Y) is a Jacobi class in G(B).

Proof. The second part of the lemma is clear, for if Y" + R(Y,Pt)j3' = 0,

then [P'] = r ( 0 ) = 7i(Y1' + R(Y, PI)@) = ( a ( Y ) ) l l + z(=(Y), PI)@.

I t remains to establish the first part of the lemma. Given the Jacobi class

W. let Yl be a smooth vector field in the geometric realization V ( P ) with

.ir(Y1) = W . Slnce W 1 ' + ~ ( W , PI)@' = [,@I in X(@), there is a smooth function

f : [a, b ] 4 R such that Yl" + R(Y1,pt)p' = fP1. Let h : [a,b] -. R be a

smooth function with h" = f and set Y = Yl - hp'. Then r (Y) = W , and

f 0' = Yl " + R(&, /'3")P1 = (Y + hfi')" + R ( Y + hot , P1)P'

= Y" + h"P1 + R(V, Pt)P' = Y" + fP'+ R(Y,P1)/3'.

Therefore 0 = Y ' I -+ R(V, ,B')/?' as required. O


A more precise relationship between Jacobi fields in the geometric realiza-

tion V(0) of G ( P ) and Jacobi classes in X(0) is given by the following lemma.

Lemma 10.52. Let W he a Jacobi class in X(P). Then there is a Jacohi

field J E V ( P ) with r ( J ) = W iff the geometric realization 6(W) of W in V(0) satisfies the condition R(O(W),/3')p'/t E V ( P ( t ) ) for all t E [a, b ] .

Proof. If J E V(P) and n ( J ) = W , we have @(W) = J. But as J is a Jacobi

field, we then obtain

since J E V(P).

Now suppose that R ( @ ( W ) , /?')dl E V ( @ ) and J = 0 ( W ) . Then R ( J , /3')P1 =

R(O(W),,B')P1. Since W " + f i(W, P1)j3' = [PI] in G(0), we know that J must

satisfy a differential equation of the form J" + R(J,P1)P1 = fP1 in V L ( P ) .

But if R(0(V),j3')fi1 = R(J,P1)@' E V(P). the vector field J" -R(J,P)/3 ' is in

V ( P ) Hence by the deconlposition (10.29), J" + R(J,P1),D' = 0. 0

For the purpose of studying conjugate points, it is helpful to prove the

following refinement of Lemma 10.51 [cf. Bolts (1977, pp. 43-44 )].

Lemma 10.53. Let W E X(P) be a Jacobi class with W ( a ) = [P1(a)] and

W ( b ) = [/3'(b)]. Then there is a unique Jacobi field Z E vL(P) with ~ ( 2 ) = W

and Z(a) = Z(b) = 0.

Proof. From Lemma 10.51, we know that there exists a Jacobi field Y E

VL(/3) with a ( Y ) = W . However, we do not know that Y ( a ) = Y ( b ) = 0. But

for any constants A, p E R, the vector field Y + XB' + /1t/3' is also a Jacobi field

in V-(a) with 7i(Y + X P t + pt?) = W . Since x ( Y ) = W , W ( a ) = [/.?'(a)],

and W ( b ) = [,B'(b)], we know that Y ( a ) = c1,O1(a) and Y ( b ) = c2,B1(b) for

some constants cl,c2 E W. Choosing X = (cza - c ~ b ) ( b - a)-' and p =

b-I [(clb - c2a) ( b - a)-' - c2], it follows easily that Z = Y + X,Bt - pt9' satlefies

Z ( a ) = Z(b) = 0.

For uniqueness, suppose that Zl is a second Jacobi field in VL(@) with

r ( Z 1 ) = W and Zl ( a ) = 21 (b) = 0. Then X = Z1 - Z is a Jacob) field of the

form X = hp' with X ( a ) = X ( b ) = 0. Since 0 = X" + R(X,f l ) ,B1 = h"O1 i


hR(P1, P1)F = h1'pP', it follows that h is an xffine function. As h ( a ) = h(b) = 0,

we must have iz = 0. Therefore Z 1 = Z as required. C3

With Lemma 10.53 in hand, it is now possible to make the following defi-

nition. Recall also that if J is any Jacobi field along ,f3 with J ( t l ) = J ( t 2 ) = 0

for t l # t2, then J E Vi(p) (cf. Lemma 10.46).

Definition 10.54. (Conjugate Point) Let ,B : [a, b] -+ (M.g ) be a null

geodesic. For tl # t2 in [a, b ] , tl and t2 are said to be conjugate dong /3

if there exists a Jacobi class W # [P'] in X(0) with W ( t l ) = [B1( t l ) ] and

W(t2) = [pl( t2)] . Also t l E (a, b ] is said to be a conjugate point of ,B if t = a

and tl are conjugate along 8. Let

Jt(P) = (Jacobi fields Y along P : Y ( a ) = Y(t) = 0)

and

- J@) = (Jacobi classes W E X(P) : W ( a ) = [P1(a)] and

W(t) = IB1(t)j).

The11 J t (P) C vi(@) and tl and t 2 are conjugate along f l in the sense of

Definition 10.54 if and only if there exists a nontrivial Jacobi field J along /3

with J( t1 ) = J( t2 ) = 0. Also we obtain the following important result from

the uniqueness in Lemma 10.53.

Corollary 10.55. The natural projection map T : J t@) - J,(,L?) is an

isomorphism for each t E (a, b ] . Thus Tt(P) is finite-dimensional, and also

dim Jt(@ = dim &(p) for all t E (a, b].

We are now ready to study the index form of the null geodesic /3 : la, b ] --+

(M,g) . For the geometric interpretation of the index form, ~t will be useful

to introduce a functionai analogous to the arc length functional for timelike

geodesics, namely the energy functional. The reason for using energy rather

than arc length is simply that the derivative of the function f (2) = f i does

not exist at x = 0, but J--g(,l3'(t), ,@(t)) = 5 for all t E [a, b] if /3 is a null

geodesic.


Definition 10.56. (Energy Functzon) Let y : [ c , d ] -t (M,g) be

smooth nonspacelike curve. The smooth mapping E- : [c, d ] -+ R given by

is called the eneqy finct~on of 7, and the number E ( y ) = E,(d) is called ti

energy of y.

The energy of a piecewise smooth nonspacelike curve y is calculated 1

summing the energies over the intervals on which -/ is smooth just as in formu

(4.1) for the arc length functional. If we let Ly( t ) = L(y jc , t ] ) , then tl

Cauchy-Schwarz inequality yields

Also. equality holds in (10.38) if and only if Ily'(t) 1 1 1s constant. Thus equals

holds for null or timelike geodesics.

Recall that V-(8) denotes the space of piecewise smooth vector fields

along ,B that are orthogonal to P', and @(@) = (Y E ~ ' ( 9 ) : Y ( a )

0 and Y ( b ) = 0).

Definition 10.57. (Index F o n ) The index f o n I : v-'(@) x V L ( P ) -

W of the null geodesic P : [a, b ] M is given by

b

(10.39) ((X'. Y ' ) - ( R ( X , P')P1. Y ) ) dt

for any X . Y E V-' ( p ) .

Formula (10.39) may be integrated by parts just as in the timelike c a

(cf. Remark 10.5) to give the alternative but equivalent definition of the i~

dex form used in Hawking and Ellis (1973, p. 114) and Bolts (1977, p. 110

Explicitly,


where a partition a = to < tl < - .. < t k P l < t k = b has been chosen such that

X 1 It,, taTlj is smooth for i = 0: 1 , 2 , . . . , k - 1 and such that

A,, ( X I ) = X' (a f ),

A,, ( X ' ) = -X'(b-),

and

At, (X') = lim X1( t ) - lim X1( t ) t-+t;t t-t;

f o r i = 1 , 2 ,..., k - 1 .

If a is a variation of the null geodesic ,@ such that the neighboring curves

are all null, then all derivatives of the energy functional vanish at s = 0 since

E(a,) = 0 for all s. Thus it is customary to restrict attention to the following

class of variations.

Definition 10.58. (Admisszble Variation) A piecewise smooth variation

cu : [a, b ] x ( - 6 , E) 4 A4 of a piecewise smooth nonspacelike curve /3 : [a, b] -+

M is said to be admissible if all the neighboring curves a, : [a, b ] + M given

by a,(t) = aft, s ) are timelike for each s # 0 in ( - 6 , E ) .

Now suppose that for W E VL(P), there exists an admissible variation

a . [a, b ] x ( - E , E) -+ M with variation vector field a, d / d ~ / ( , , ~ ) = W ( t ) for each

t E [a, b ] . Let E(u,) be the energy of the neighboring curve a , : [a, b] -- M

for each s E ( - E , t). Then d/ds [E(a , ) ] Is=, = 0 since p is a geodesic and

As E(P) = E(ao) = 0. we must have d2/ds2 [E(a , ) ] I,=, > 0. Thus a

necessary condition for W E VL(/3) to be the variation vector field of some

admissible variation a of /3 is that I(TN, W ) 2 0. Note also that if the future

null cut point to P(a) along P comes after P(b). then there are no admissible

proper variations of P (cf. Corollary 4.14 and Section 9.2).

It is immediate from (10.39) that if X E I/dF($') is a vector field of the form

X ( t ) = f (t),B1(t) for any piecewise smooth function : [a, b] -+ R with f (a ) =

f (b) = 0. and Y 6 V&(P) is arbitrary, then I ( X , Y) = 0. Thm the index forrr

I : Vo-(P) x V$($) -7 W is never negative definite and even worse, alwaytys ha

an infinite-dimensional null space. This suggests that the index form should be

projected to G(D) [cf. Ifawking and Ellis (1973. p. 114). Bolts (1977, p. 111)]

Recall that the notation X(@) was introduced for piecewise smooth sections o

G(P), and Xo(P) = {W E X ( P ) : W ( a ) = [P'(a)l, W ( b ) = [P'fb)] 1. Definition 10.58. (Quotient Bundle Index F o n n ) The indea: form

7 : X(p) x X ( p ) 3 R is given by

(10.42) [3(V', W') - 3 @(v, P1)P1, W ) ] dt

where V , W E X(P) and g R, and the covariant differentiation of sectlons of

G(P) are given by (10.31), (10.34), and (10.32). respectively.

Just as for the index form I : V L ( P ) x V L ( p ) 4 IR, formula (10.42) may

be integrated by parts to give the formula

where a partition a = t o < tl < . - . < < tk = 1, has been chosen such that

Vl[t , , t,,l] is smooth for i = 0,1, . . . . k - 1. In particular, if V is a smooth

section of G(P) we obtain

and if V is a Jacobi class in X(P), we have

Also, since vector fields of the form f ( t)P1(t) are in the null spzce of the index

form I . V L ( P ) x Vi(P) -+ R, for any X,Y E vL(,B) with a ( X ) = V and

sr(Y) = W , it follows that

378 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLCS

where the index on the left-hand side is calculated in X ( P ) and on the right-

hand side in V L ( B ) .

We saw in Section 10.1 that for timelike geodesic segments. the null space of

the index form consists preciseljr of the smooth Jacobi fields vanishing at both

endpoints. We now establish the analogous result for the index form 7 on the

quotient space Xo(Y) for an arbitrary null geodesic segment : la, b ] --+ M.

Theorem 10.60. For piecewise smooth vector classes W E Xo(P) , the

following are equimlent:

(1) W is a smooth Jacobi class in Xo(/3).

( 2 ) I ( W, Z) = 0 for all Z 5 Xo (9).

Proof. First. (1) 5 (2) is clear from formula (10.45). For the purpose

of showing (2) + ( I ) , fix the unique piecewise smooth vector field Y In the

geometric realization V ( P ) for G ( P ) given by (10.28) with r ( Y ) = W and

Y ( a ) = Y ( b ) = 0. Let a = to < t1 < . . - < t k = b be a finite partition of [a , b ]

such that Y / [t3,t3+1] is smooth for J = 0 ,1 , . . . , k - 1. Let p : [a, b ] --+ R be

a smooth function with p(t3) = 0 for each j with 0 5 j 5 i, and p ( t ) > 0

otherwise. Set Z = 7r(p(Y1' + R(Y, /3 ' )Pf) ) E Xo(P). Then from (10.43) we

obtain

Remembering that 3 is positive definite (cf. Remark 10.48), we obtain Wtl(tj+ - R ( W ( t ) , P 1 ( t ) ) P ' ( t ) = [,Bf(t)] except possibly at the t3's. Thus JV / [tl , t,+l] is

a smooth Jacobi class for each j. To show that W fits together at the tJ's to

form a smooth Jacobi class on all of [a, b ] , it is enough to show that the vector

field Y in the geometric realization V ( P ) representing W is a 6' vector field

First observe that since W1'(t) R(w ( t ) . /3'(t))P1(t) = [PI (t)] except possiblj

at the t,'s, we have using (10.43) that

k - 1

0 = T(w, Z ) = r;: 3 ( Z ( t , ) . A , ( W 1 ) ) j=1

for any Z E Xo(j3). Since Y E V(P) , we have ( Y ( t ) , P1( t ) ) = ( Y ( t ) , ~ ( t ) : = 0 for

all t E [a, b ] . Thus ( Y r ( t ) , B / ( t ) ) = ( Y 1 ( t ) , 7 ( t ) ) = 0 for t $ ( to , t l . . . . , t k ) . and


it follows by continuity that AtJ ( Y ' ) E V ( P ( t 3 ) ) for each j = 1.2,. . . , k-1. Let

X E V(P) be a smooth vector field urith X ( a ) = X ( b ) = 0 and X ( t 3 ) = A,, ( Y 1 )

for each j = 1 ,2 , . . . , k - 1. Set Z = T ( X ) E X o ( P ) Then we obtain

which yields At, ( Y 1 ) = 0 for 3 = 1,2, . . . , k - I. Hence Y' is continuous at the

t,'s as required. P

Notice that in the first part of the proof of ( 2 ) + (1) of Theorem 10.60, we

do not know that Y" + R ( Y . f l ) Y E V(P) even though Y E V(O). Thus we

cannot conclude that Y is a Jacobi field except at the t,'s. Thls is precisely

the point In passing to the quotient bundle G ( P ) in which W" + z(W, pP')P' lies in G ( P ) by construction and the induced metric g is positive definite on

G ( P ) x G ( P ) . The aim of the next portion of this section is to prove the important result

that the index form 7 : Xo(P) x Xo(/3) -t R is negative definite if and only if

there are no conjugate points to t = a along ,B in ju, b 1. A proof of this result,

which is implicitly given in Propositions 4.5.11 and 4 5.12 of Hawking and

Ells (1973), is given in complete detail in Bolts (1977. pp. 117-123). Because

the proof of negative definiteness differs considerably from the corresponding

proof (cf. Theorem 10.22) for the timelike geodesics, we briefly outline the

proof in the timelike case in order to clarlfy the differences. Recall that w-e

first showed that arbitrary proper piecewise smooth variations a of a timelike

geodesic segment have timelike neighboring curves a, for sufKciently small s It

was then a conseqnence of the Gauss Lemma and the inverse function theorem

that if c : u, b] --+ M is a future direc:ed timelike geodesic segment without

conjugate points. then for any proper piecewise smooth variation a of cj the

neighboring curves a, satisfy L(a , ) 5 L(c ) for all .s isufficiently small This

implied that the lndex form is negative semidefinite In the absence of conjugate

points. Algebra~callv, we were then able to prove the negative definiteness.

Conversely. if c had a conjugate point in (a, b ] we produced a piecewise smooth

vector field Z E Vg'(c) of zero index csing a nontrivial Jacobi field guaranteed

by the existence of a conjugate point to t = a.


For the purpose of the null index theory, however, we need to consider

(1,1) tensor fields not on VL(P) but rather on the quotient bundle G(P). A

(1, l) tensor field 2 = x( t ) : G(P(t)) -+ G(P(t)) is a linear map for each

t E [a, b ] which maps vector classes to vector classes Using the projection

of the covariant derivative to X(P) [cf. equation (10.32)], we may differentiate

piecewise smooth (1,1) tensor fields in G(P(t)) and obtain the formulas

and

provided is nonsingular. Since the projected metric 3 : G(P(-t)) x G(O(t)) +

R is positive definite, we may also define the adjoznt x* = z * ( t ) to the (1,1)

tensor field A(t) for G(P(t)) by the formula

for all vector classes Z, W E G(P(t)) and all t E [a, b ] . We may also define the

composed endomorphism fjz : G(P(t)) --, G(P(t)) by

where 3 is the projected curvature operator given by (10.34). Also the kernel

ker ( X ( t ) ) of the (1.1) tensor field z(t) : G(P(t)) -+ G(P(t)) is the vector

space

-4 (1,1) tensor field x(t) : G(fl(t)) -, G(F(t)) is the zero tensor field, written - A = 0, if z(t)(w) = [P1(t)] for all w E G(P(tf) and all t E [a, b].

With these preliminaries out of the way, we are ready to define Jacobi and

Lagrange tensor fields.

10.3 THE NULL MOXSE INDEX THEORY 383

Definition 10.61. (Jacob2 Tensor Fzeld) A snlooth (1.1) tensor field - A : G(P) G(P) is said to be a Jacobz tensor if 3 satisfies the conditions

and

for all t E [a, b].

Condition (10.56) has the consequence that ~f Y E X(P) is any parallel

vector class along P: then J = A(Y) is a Jacobi class along 8. This follows

since

J" + B(J, O')pl = Z"(Y) + Z A ( y )

= (A" + R;?)(Y) = 0

using Y' = 0 Condition (10 57) has the follow~ng impllcatlon if Yl , Yn-2 -

are linearly Independent parallel sections of X(P). then ~ ( Y I ) . . , A(Yn-2) are

!inearly Independent Jacobi sect~ons in the following sense. If X I . . . , are

real constants such that

for a l l t E [ a , b ] , thenX1 = A 2 =. . ,=X, -2=0 .

The converse of the above construction may be used to show that Jacobi

tensors exist,. Fbr let El. Ez, . . . . En-2 be the spacehke parallel fields spanning

V(P) chosen ?n (10 28). Let z, = .;r(E,) be the corresponding parallel vector - - -

class ~ I I X(P) Also let J 1 , J z , . . . , Jr.-2 be the Jacobi classes along P with

initial conditions 5,(a) = [@'(a)] and TZf (a ) = F,(a) for z = 1.2 , . . . ,n - 2.

A Jacobi tensor 2 satisfying the initial conditions x ( a ) = 0, x1(a ) = Id may

then be defined by requiring that


for each i = 1,2. . . . , n - 2 and extending to all G(P) by linearity. Since the - JI3s are Jacobi classes and the z ' s are parallel classes in X(P), it follows that - A satisfies (10.56). To check that 3 satisfies (10.571, suppose that x ( t ) ( ~ ) = - 1 A (tj(?;) = [P1(t)] for some E G(P(t)). Choose the unique v E V(P(t)) with

~ ( v ) = 'ii, and write v = ~rz x,E,(~). Then 'ii = X,F,(t), and we

obtain [P1(t)] = -A(%) = ~:=;2 ~ ~ 5 % ( t ) and

Thus 7 = ~:_;2 ~ ~ 5 % is a Jacobi class in G(P) with J ( t ) = 5'(t) = [P1(t)].

Hence 7 = [p'j. Thus X,Z,(s) = [P1(s)] for all s E [a, b], contradicting - - -

the linear independence of the parallel classes El, Ez, . . . , Therefore

ker(a(s)) n ker(zl(s)) = [@'(a)] for all s E [a, b] as required.

I t is also useful to note the following result.

Lemma 10.62- ker(x(so)) n ker(xl(so)) = [pl(so)] for some so E [a, b] iff - A satisfies condition (10.57).

Proof. Suppose v E ker(X(s0)) n ker(X'(so)), v # [P1(so)]. Let Y be the

parallel class in T(,/i) with Y(so) = u. Then J = Z(Y) is a Jacobi class

satisfying J(so) = A(u) = {@'(so)] and .J1(.so) = ~ ' ( z J ) = !P1(so)]. Hence

J = [ P I ] from which it follows that Y(t) E ker(z(t)) n ker(xl(t)) for all

t ~ [ a , b ] .

The following two lemmas are not d~fficult to obtain from the relationship

between parallel classes, Jacobi classes, and Jacobi tensors indicated above

icf. Bolts (19771, p. 28 and p. 49)j.

Lemma 10.63. The points @ ( t o ) and $(tl) are conjugate along the null

geodesic P : [a, b -+ M iff there exists a Jacobi tensor x : G(P) -+ G(0) with - A(to) = 0: z'(t0) = Id, and ker(Z(t1)) # (p(tl)]).

Lemrna 10.64. Let /3 : [a,b] -, A4 be a null geodesic without eonju-

gate points to t = a. The= there exists a unique smooth (1, l) tensor field - A : G(P) -+ G(P) satisfying the differential equation x" + RZ = 0 with

10 3 THE NULL MORSE INDEX THEORY 385

given boundary conditions A(a) : G(B(n)) - G(/?(a)) and A(b) . G(B(b)) --+

G(a(b)).

Proof, Let J' denote the space of (1.1) tensor fields in G(0) satisfying the

differential equation 2" i- ?iz = 0. A linear map $ : J' -+ L(G(P(a))) x

L(G(P(b))) may then be defined by Q ( l ) = (z(a),A(b)). Since P : [a, b] -, M

has no conjugate points, Qj is injective. For if +(x) = (0.0) and Y is any

parallel vector class in X(P), then J = X(Y) is a Jacobi class with J ( a ) =

[@'(a)] and J(b) = [Fi(b)] so that J = [$'I. Slnce b is an injective h e a r map

and dim J' = ddim[L(G(P(a))) x L(G(P(b))) 1, it follows that rp is surjective. 13

Even though R = R* and (A1)* = ( A * ) ' , the adjoint 3' of a Jacobi - -*

tensor 3 is not necessarily a Jacobi tensor since (zx)* ji R A in general

Rather, ( R z )* = 2 *R. Nonetheless, Jacobi tensors and thelr adjoints have

the following useful property which is conveniently stated using the Tironskian

tensor field [cf. Eschenburg and O'Sullivan (1976, p 226), Bolts (1977. p. 23))

Definition 10.65. (Wronskian) Let 2 and B be two Jacobi tensors - -

along G(@). Then their Wronskian W(A, B) is the ( I l l ) tensor field along

G(0) given by

It follows from the fact that R* = R and equations (10 51) and (10.56) that - -

if 2 and B are any two Jacobi tensor fields along G(,f3), then [ W ( A ; B)!' = 0. - -

Thus W ( A, B ) is a constant tensor field. It is then natural to consider the

following subclass of Jacobi tensors.

Definition 10.65. (Lagraege Tensor Fzekd) X Jacobi tenbur field 2 along G(P) is said to be a Lagrange tensor field if W ( Z , Z ) = 0.

For the proof of Proposition 10.68, W P need the following consequence of

Definition 10.66.

Lemma 10.67. Let 2 be a Jacobi tensor along G(P). If x(so) = 0 for

some 30 E [a, b], then 2 is a Lagrange tensor, and in particular,

386 10 MORSE INDEX TEEORY ON LORENTZIAN MAXIFOLDS

Proof. We know that W ( 2 , X ) is a constant tensor already. But if Z ( s o ) = - -

0: then z*(so) = 0 also, and iletlce W ( x , X ) ( s o ) = 0. Thus W ( A , A ) = 0 as

required.

JVe are now ready to prove the following important proposition

Proposition 10.68. Let ,i3 : [a, b ] -+ M be a null geodesic without conju-

gate points to t = a in (a , b ] . Tl~eu P(W, W ) < 0 for all W E Xo(P), T.I' # [P'].

Proof. Let ';l be the Jacobi tensor along G(P) with initial conditions X ( a ) =

0 and ~ ' ( a ) = Id. Since ,!3 has no conjugate points, Lemma 10 63 implies that

k e r ( z ( t ) ) = ( [ p l ( t ) ] ) for all t E (a, b ]

Now let W E Xo(/3) be arbitrary. Since 2 is nonsingular in (a , b ] and

W ( a ) = ip1(a)], we may find Z E X ( 0 ) with W = x(Z). By the rules for

covariant differentiation of tensor fields. we obtain

and

Now we are ready to calculate ?(w; W ) . First

h

=-i [ g ( 7 i i ( 2 ) , X 1 ( 2 ) ) -k 2 3(Z1(2) , Z(2 ' ) )

+ i j ( Z ( z ' ) . X ( z f ) ) + B ( ~ ~ ' ( Z ) , A ( Z ) ) ] dt

using (10.56) and (10.61). Now

~ ( X " ( Z ) . ~ ( 2 ) ) = ij([Z1(.Z)]'. ~ ( 2 ) ) - i j ( A ' ( Z 1 j , ~ ( 2 ) )

= [g (Z1 ( z ) , Z(z));l - ~ ( Z ' ( Z ) > [ X ( Z ) 1') - g(Z1(z ' ) ,B(z) )

= [T ( 2 ' ( ~ ) , Z ( z j ) ] ~ - g ( A ' ( z ) , Z ( z l ) ) - 3(A1(z),X'(z))

- 3(A1(z1) ,A(zj) .

10.3 THE NULL MORSE INDEX THEORY

Substituting into the above formula for 7 ( ~ : w) then yields

Now the first term vanishes since W ( a ) = :,!3'(a)] and W(b) = [pl(b)]. Thus

we obtain

using (10.60). If Z i ( t ) = 0 for all t , then Z is parallel along 8 and hence

W = X ( Z ) would be a nontrivial Jacobi vector class along /3 with W ( a ) =

lP1(a)] znd W(b) = [/3'(b)]. Since 3 has no conjugate points in (a. l i ] , this is

impossible. Hence from the positive definiteness of 3 and the nonsingularity

of A ( t ) for t E (a , b ] , we obta~n ?(I.I/, W ) < 0 as required.

This proposition has the well-known geometric consequence in general rel-

ativity that if /3 : [a, b ] -+ (M, g) is free of conjugate points, then no "small"

variation of f l gives a timelike curve from p to q [cf. Hawking and Ellis (1973:

p. 115)].

We are now ready to prove the fnllowing theorem

Theorem 10.69. Let O : [a, b ] -, M be a null geodesic segment Then the

following are equivalent.

(1) The segment ,t3 has no conjugate points to t = a in (a, b ] .

( 2 ) f(W, W ) 4 0 for dl W E Xo(P), W # [pi:.

388 10 MORSE INDEX TWEGRY ON LOFtENTZIAN MANIFOLDS

Proof. We have already shown ( 1 ) + (2) in the proof of Proposition 10.68.

To show (2) + ( I ) , we suppose @(a) is conjugate to P(to) with O < t o 5 b

and produce a nontrivial vector class W E Xo(p) with?(W, W ) 2 0. If ,/3(a) is

conjugate to P(tof, we know that there is a nontrivial Jaeobi cl= Z E X(P)

with

Z(a) = /P1(a)] and Z(to) = [Pf(to)].

Set z(t) a < t L t o , W( t ) = ial(tji to t 5 b.

Then

- I(W, W ) = f (w, W)? -t 7(W, w)!',

= -g(Zi , 2)12 s 0 = 0

using (10.451, Z ( a ) = [PJ(a)] , and Z( t0) = [,fj"(to)]. Thus (2) + ( 1 ) holds.

-4s in the Riemannian and timelike cases (cf. Theorem 10.23); Theorem

10.69 has the following consequence which is essential to the proof of the

Morse Index Theorem for null geodesics.

Theorem 10.70 (Maxirnality of Jacobi Classes). Let ,5 : [a, b j --+ M be a null geodesic segment with no conjugate points t o t = a, and let J E X ( 0 ) be any Jacobi class. Then for any vector class Y E X ( P ) with Y f J ,

(10.63) Y ( a ) = J ( a ) , and Y ( b ) = J ( b )

we have

- (10.64) I ( J ; 3) > f (Y, Y).

Proof. This may be established just as in Theorem 10.23 using (10.45) and

Theorem 10.69. U

Since the canonical variation (10.5) of Section 10.1 applied to a vector fie

W E V'(@ is not necessarily an admissible variation of f i in the sense of D

inition 10.58, Theorem 10.69 do- not imply the existence of a timelike c

10.3 T a E NULL MORSE INDEX THEORY 38s

from @(a) to P(b) provided that t = a 1s conjugate to some to E ( a , 6) along

p. We thus give a separate proof of this result m Theorem 10.72 [cf. Hawking

and Ellis (1973, pp. 115-116), Bolts (1977, pp. 117-121); It 1s first helpfu

to derive conditions for a proper varlstion of a null geodeslc to be admissible

For convenience, we will assume that /3 : [O. 11 -+ M so that the formulas lr

the proof of Theorem 10.72 will be simpler.

Now let a : jO. 1) x ( - t , ~ ) -+ M be a piecewise smooth proper variation o

0 such that each neighboring curve a , ( t ) = a(t, s ) 1s future timelike for s # 0

Let d / d t and a i d s denote the coordinate vector fields on [O, I] x ( - E . e ) , anc

put V = cr,(d/as) and T = a , ( d / a t ) . Then T / ( t , o ) = P1( t ) , and Vl[,,,) is

called the vanatoon vector field of the variation CA of ,B. Since a is a proper

variation, V/(o,o) = V/(, ,) = 0. Also,

since g(T . T)j(,,,) < 0 for s # 0 and g(T, T)j(t,o) = g($( t ) , P ' ( t ) ) = 0. Calcu-

lating,

Thus since ,3 is a geodesic, we obtain

Now f (t) = g(V. T)l( t ,o) is a piecewise smooth function with f ( 0 ) = f ( 1 ) = 0.

Hence if f ( t ) # 0 for some t 6 10, 11, there exists a to E (O,1) with f l ( t o ) > 0.

Then


in contradiction to (10.65). Hence we obtain as a first necessary condition for

a to be an admissible deformation of ,6 that

for all t E [0, I]. Thus V E V-(0). Consequently

for all t E [O, I]. It then follows from (10.67) that the neighboring curves a, of

the variation will be timelike provided that the variation vector field satisfies

(10.66), (10.67). and the condition that d 2 / d s 2 ( g ( T , T))l(t,O) < -c < 0 for all

t E (0: 1) . As above,

Hence

Using the identities

and

ValatTllt ,) = 'ValatP1lt = 0 ,

10.3 THE NULL MORSE INDEX THEORY

we obtain

This calculation yields the following lemma.

Lemma 10.11. A sufficient condition for the plecewise smooth proper vari-

ation CL. [0,1] x ( - e , ~ ) -+ A4 of the null geodes~c D [0, I] -+ M to he an

adrnlsslble varlatlon [I e , the curves cx,(t) = a(t s ) are tlrnellke for s j O] for

a11 s # 0 suficientlysmall i s that the vanation vector field V ( t ) = ~ ~ . , d / d $ l ( ~ , ~ )

satisfies the condltlons

(10.68) g ( ~ ( t ) , p ' ( t ) ) = 0 for all t E {O,I]:

(10.69) d

- ( g ( T T ) ) / ( t , O ) = 0 for f E [O,l:; and d s

for all t E (O,1) at which V is smooth.

We are now ready to prove the desired result [cf. Hawking and Ellis (1973,

p. 115))

Theorem 10.72. Let P : [0,1] -+ M be a null geodesic. If ,!3(to) is conju-

gate to F(0) along /3 for some to E (0. I ) , then there is a timelike curve from

a(o f tool-1).

Proof. We uiill suppose that to > 0 is the first conjugate point of B(0) along

0. It is enough to show that for some t2 wlth to < ta < 1. there is a future

directed timelike curve from $(O) to $(t2). For then we have B(O) < B(tz) <


P(1) whence P(O) < P ( 1 ) Thus there exists a timelike curve from B(0) to

P(1). Since P(to) is conjugate to B(0) dong ,D, there exists a smooth nontrivial

3acobi class W E X(3) with W ( 0 ) = {fl(O)] and W ( t o ) = [$'(to)]. W-e may

write

where I@ is a smooth -vector class along /? with g(%, I&') = 1 and f : [ O , I ] --+ B a smooth function. Since to is the first conjugate point along P, f (0) = f ( to) =

0 , and changing t%' to -T@ if necessary, we may assume that f ( t ) > 0 for

all t E (0,to). Since W is a nontrivial Jacobi class and W(to) = [$'/(to)].

we have Url(to) # [P1(to)]. Thus from the formula W1(to) = f l ( to)I&'(to) + f (to)@'(to) = f'(to)l.i/(to), we obtain fl(to) # 0. Hence we may choose

tl E ( to . 11 such that W ( t ) # $'(t)] and f (f) < O for all t E (to, t l ] .

With tl as above, the idea of the proof 1s now to show that there exists a t2 E

( to , t l ] such that there is an admissible proper variation cu : [0, t2] x ( -e , €1 - M

of ,B I [O, tz] . Then the neighboring curves a, of the variation a will be timelike

curves from P ( O ) to O ( t 2 ) for s $ 0 . This will then imply that F ( O ) < P(1) as

required. To this end, we want to deform W 1 [0, t l ] to a vector class Z 1 [I). t2] - - 11

with 7.j ( 2. Z +a@, ot)@l) > 0 so that if Z E V L ( P ) is an appropriate lift of - Z E X($ / [0, t z ] ) and a is a variation of ,B 1 [O: tz] with variation vector field Z ,

then conditions (10.69) and (10.70) of Lemma 10.71 will be satisfied.

Consider a vector class of the form

with b = - f ( t l )(eatl - I)- ' E R and a > O in R chosen such that

(a2 + glb(h(t) : t E [O, t l ] ) ) > 0,

where

h(;) = g(* + K(I@,B')P', 1,.

10.3 THE NULL lvIORSE INDEX THEORY

Because W is a Jacobi class, we obtain

o = 3 (w", a(w pl)pt , %)

= fIi + 2 ft3 (w,@-) + f g ( W , @) + f 3 (R(*,pl)P'.

= f"+2f1?j(lV'.*)+ fh .

But since g($, @) = a(g(@, I@))' = $(l) ' = 0, we obtain the formllla f" =

- fh. Returning to consideration of the vector class 2, first note that by choice

of the constants a and b, we have z ( 0 ) = [P1(0)] and z ( t l ) = [P1(tl)]. In view - - / I --

of formula (10.70), we wish to have g(Z, Z + R(Z,P1)P1) > 0 also. Setting

r ( t ) = b(eat - 1) + f ( t ) remembering that 7j(lvi/', I%') = 0. and dderentiating

yields ij(Z, Z1' + x (Z , p l )p l ) = r(rU + r h ) = r[beat(n2 + h) - hh + fl ' + f h].

Since f" = - f h. we obtain

Noiv b = - f (tl)(eatl - 1) > O as f ( t l ) < 0. so the expression b(eat[a2 + - - I / - -

h( t )] - h( t ) ) > 0 for all t E [0, t l ] . Thus g(Z , Z + R ( Z , ,LY)O1)It > O provided

r ( t ) > 0. Since f ( t) > 0 for t E (O,to), we have r(t) > 0 for t E (0, to]. By

continuity, there is some t z > to with r ( t ) > 0 for t E [to, t z ) and r ( t z ) = 0. If

t 2 > t l r then in fact t 2 = t l since r ( t l ) = 0 by construction, and we let tz = tl below. If tz < tl , then the vector class Z / 10, tz] will satisfy Z ( t a ) = [Pr(t2)]

- - , I -- since r ( t z ) = 0 and also g ( Z , Z + R ( Z . P')F1)lt > 0 for all t E (0, t a ) We now

work with ,B 1 [O, t2] .

Let 2 E V L ( P I [0, t z ] ) satisfy a ( 2 ) - = z. Since z(0) = [,LY(O)] and Z( t2 ) =

[pt ( t2)] , we have Z(0) = pP1(0) and Z(t2) = XP1(t-,) for some constants p. X E

R. Set Z = 2 - p,8' + [ ( j i - X)/t2]t01. Then Z(0) = Z ( t a ) = 0 and 7i(Z) = 2. Consequently

for all t E (0 , t 2 ) . Chowe a constant E > O SO that

(10.73) e < glb {g (Z1 ' ( t ) + R(z( t ) .P1( t ) )P ' f t ) . Z(t ) ) : t G [?, 21)


which is possible in view of (10.72). Now define a function p : [O: tP] -+ R by

- €t o i t i 2 ,

€ ( t - 2 ) h < t < & 4 - - 4 ,

~ ( t 2 - t ) 9 5L5t2.

We now have a given vector field Z E V&(P / [0, t z ] ) and a given function

,G : (0.t2] -+ R. Recall that we had fixed a pseudo-orthonormal frame field

El. E2,. . . . En-?. 7, @' for ,6 with (? ,p i ) = -1 in (10.28). We now need to

find a proper variation a : [O, tz] x ( - E , E ) --+ M of ,B / [O, t2] satisfying the initial

conditions

and

for all t E [O, t2]. Thus we wish to specify the first and second derivatives of

the curves s -+ a ( t , s) for each t E [O, tzj. The existence of such a deformation

is guaranteed by the theory of differential equations applied to (10.74) and

(10.75) written out in terms of a Fermi coordinate system for the geodesic

defined by the pseudo-orthonormal frame El. E2, . . . , En-2, 7, pl.

Given the proper variation cu of ,O 1 [O, tz] satisfying (10.74) and (10.75) arid

setting T = a,a/dt and V = a,d/ds as above, it follows that

Hence

ost<g, d - (g(Va/asV, p') + Q(V, V')) /(,,o) = ~ ' ( t ) = dt

? s t < ? , y<t<tz

Thus in vlew of (10.73), the variation a of P I [O. ta] satisfies condition (10.70)

of Lemma 10.71. Applying Lemma 10 71, we find that this variation produces

timelike curves a, from P(0) to P(t2) for smail s f 0 as required. U


Corollary 10.73. The null cut point of ,B : [O, a ) -r M comes at or before

the first nul l conjugate point.

We are at last ready to turn to the proof of the T\/iorse Index Theorem for

null geodesics The proof parallels that of the Timelike Morse Index Theorem.

Theorem 10.27. In view of Theorem 10.69, the index ind(0) and the extended

index Indo(@) of p with respect to the index form 7 : Xo(P) x Xo(P) -+ R

should be defined as follows.

Definition 10.7'4. (Index and Extended index) The zndex Ind(P) and

extended znde.c Indo(P) of P with respect to the index form -f : Xo(O) xXo(0 ) --+

R are given by

Ind(p) = lub {dim A : A is a subspace of Xo(p)

and 1 A x A is positive definite )

and

Indo(Oj = lub {dim A : A is a subspace of Xo(P)

and f 1 A x A is positive semidefinite ) ,

respectively.

As a preliminary step toward establishing the null index theorem, we need

the following lemma.

Lemma 10.75. If ,l3 / [5 , t ] is free of conjugate points, then give11 ary v f G(P(s) ) and iij E G(P( t ) ) , there is a unique Jacobi class Z E X(P) with

Z(5) = ij dnd Z(t) = a.

Proof. Let v E V(P(s)) and zu 6 V(P(1)) satisfy ~ ( v ) = Ti and ~ ( w ) = E

By the nonconjugacy hypothesis, there is a unique Jacohi field J 6 VL(/?)

with .7(.9) = a and J ( t ) = a,. Then Z = ~ ( 5 ) is a Jacobi class in X ( 3 ) with

Z(s) = 5 and Z ( t ) = E.

Suppose now that Zl is a second Jacobi class in X(B) with Z l ( s ) = 5

and Zl(t) = E. By Lemma 10.51, there 1s a Jacobi field JI E VL(fi) wlth

7i(JI) = ZI. Sirice Zl(s) = B and Zl ( t ) = a, ~t follows that Jl (s) = v + clfl (s)

398 10 MORSE INDEX THEORY ON LOWNTZIAN MANIFOLDS

contradiction. Hence Ind'(0) < dimB as required, completing the proof of the

proposition. ti

Now that we have obtained Proposition 10.76, it i,i straightforward to see

that the proof of Theorem 10.27 of Section 10.1 may be applied to the index

form f : G(P) x G(P) -t R and the projected positive definite metric ?j to yield

the equalities

and

Since dimjt(/3) = dim Ji(/3) by Corollary 10.55, we have thus established the

following Morse Index Theorem for null geodesics.

Theorem 10.77. Let @ : [a, b] -t M be a null geodesjc in an arbitrary

space-time Let 7 : Xo(0) x Xo(P) -+ lft be the index form on plecewise

smooth sections of the quotient bundle G(P) defined in (10.42). Then 0 hhas only finitely inany conjugate points, and the index Ind(P) and extended index

Indo (P) off : Xo (P) x Xo (P) -t R are related to the geodesic index of the null

geodesic P by the formuias

Ind(P) = dim J t ( @

t € ( a , b )

and

where &(a) denotes the vector space of jacobi fields Y along P with Y ( a ) =

U ( t ) = 0.

Extensions of Theorem 10.77 to the focal case of a null geodesic segment

perpendicular to spacelike endmanifolds have been given irr Ehrlich and Kim

(1989a,b), Studies of spacelike conjugate points have recently been made in

Helfer (1934a,b).

CH.4PTER 11

SOME RESULTS I N GLOBAL LORENTZIAN GEOMETRY

In Chapter I1 we apply the techniques of the preceding chapters to obtain

Lorentzian analogues of two remarkable results in global Riemannian geometry.

The first, the Bonnet-Myers Diameter Theorem, asserts that if a complete Rie-

mannian manifold AT has everywhere positwe Rlccl curvature bounded away

from dero, then N is compact, has finite dlameter. and has finlte fundamen-

tal group. The second result, the Hadamard-Cartan Theorem, states that if

a corilplete Riernannlan manifold has everywhere nonpositlve sectional curva-

ture, then its universal cowring manifold is diffeomorphic to IRr" and thus the

higher hoinotopy groups r,(Ai, *) = (e) for z 2 2. In addition. the un~versal

covering space with the pullback Riemannian metric has the property that any

two points may be joined by exactlj one geodesic, up to reparametrization.

In Section 11.1 we consider the Loreritzian a~lalogue of the Bonnet-Myers

Theorem and in so doing, study the timelike diameter of space-times The

tzmelake dzameter diam(M, g ) of a space-time (M, y) is given by

diam(M. g) = sup(d(p, q) : p, q E A l ) .

Classes of space-times with finlte timellke dlameter, lncludlng the "Wheeler

universes," have been studied in general relativltg [cf. Tipler (1977c p. 500)j.

If a complete Riemannian manifold has finite dlameter, ~t is compact by the

Nopf-Rillow Theorem. Even so, all geodesics have znfinzte length as a result of

the metric completeness. But for spitce-times (M, g ) since L(y) 5 d(p. q) for

all future directed nonspacelike curtes 7 from p to q. evenj timelike geodesic

must satisfy L(r) < diam(M, g). Thus if a space-tlme ( M . g ) has finite timelike

diameter, all timelike geodesics have finite length and hence are incorilplete.

In particular, a space-time (M.g) with finite timelike diameter is timelike

geodesically incomplete.

3 99

400 11 SOME RESULTS IN GLOBAL LORENTZIAX GEOMETRY

Since we have adopted the signature convention (-, +, . . . , +) for the Lor-

entzian metric instead of (+, -, . - . , -), curvature conditions of positive (re-

spectively, negative) sectional curvature for Riemannian manifolds trw-slate

as curvature conditions of negative (respectively, positive) timelike sectional

curvature for Lorentzian manifo!ds.

Using the timelike index theory developed in Section 10.1, we obtain the

fol!owing Lorentzian analogue of the Bonnet-Myers Theorem for complete Rie-

mannian manifolds. Let ( M , g) be a globally hyperbolic space-time with either

(1) all nonspacelike Ricci curvatures positive and bounded away from zero, or

(2) all timelike sectional curvatures negative and bounded away from zero.

Then ( M , g) has finite timelike diameter.

In Section 11.2 we give Lorentman versions of two well-known comparison

theorems in Eemannian geometry, the index comparison theorem and the

Rauch Comparison Theorem. Using the latter of these two results, we are

able to give an easy proof (Corollary 11.12) of the basic fact that in a space-

time with everywhere nonnegative timelike sectional curvatures, the differen-

tial expp* of the exponential map,

is norm nondecreasing on nonsparelike tangent vectors.

In Section 11.3 u-e consider two analogues of the Nadamard-Cartan Theo-

1c space- rem. The first of these is for future one-connected globally hyperbol:

times. The space-time (M.g) is said to be future one-connected if for any

p, q E M with p < q, any two smooth. future directed timelike curves with

endpoints p and q are homotopic through smooth. future directed timelike

curves with endpoints p and q. Using the hlorse theory of the timelike path

space C(,.,) from Sectlon 10.2, it may be shown that if (M,g) is a future

one-connected globally hyperbolic space-time with no nonspacelike conjugate

points, then given any p, q E M with p << q, there is exactly one future directed

timelike geodesic segment (up to parametrization) from p to q (cf. Theorem

11.16).

A Lorentzian manifold is said to be geodesically connected if each pair of

distinct p o i ~ t s is joined by at least one geodesic segment. Global hyperbolic-

11.1 THE TIMELIKE DIAMETER 401

ity guarantees the existence of at least one (maximal length) geodesic segment

joining any two causally related points. However, it does not yield any informa-

tion about points which fail to be causally related. Furthermore, geodesic com-

pleteness does not imply geodesic connectedness for Lorentzian manifolds. In

fact, even compact space-times may fail to be geodesically connected (cf. Ex-

ample 11.23). SufKcient conditions for geodesic connectedness are given in

Proposition 11.22 in terms of disprisonment, pseudoconvexity, and a lack of

conjugate points along all geodesics. In Theorem 11.25 we find these conditions

actually suffice for our second version of the Kadamard-Cartan Theorem This

result yields sufficient conditions for the exponential map at any fixed point to

be a diffeomorphism from its domain in the tangent space onto the manifold

M.

11.1 The Timelike Diameter

Motivated by the concept of the diameter of a complete Riemannian man-

ifold, the following analogue has been considered for arbitrary space-times

[cf. Beem and Ehrlich (1979c, Section 9)].

Definition I f .I. (Timelike Dzameter) The tzmelzke dzameter of the

space-time (M, g), denoted by diam(M, g), is defined to be

diam(M, g) = supid@, q) : p, q G M ) .

A similar concept has been used by Tipler (1977a. p. 17) m studying singu-

iarity theory in general relativity. Physically. the timel~ke diameter represents

the supremum of possible proper times any particle could possibiy experience

in the given space-time. A space-time of finite timelike diemeter is singular

(recall Definition 6.3) in a striking way.

Remark 11.2. If diarn(M, g) < m, then all timelike geodesics have length

less than or equal to diam(M, g ) end are thus incomplete.

Proof. Suppose that c . (a, b) -+ M is a timelike geodehic which satisfies

L(c) > diam(N,g). We may then find s , t E (a; b), 5 < t , such that

402 11 SOME RESULTS IN GLOBAL LORENTZI.4N GEOMETRY

But then

d(c(.s), e ( t ) ) 2 L(c / :s, t 1) > diam(M. g)

which is impossible.

From a physlcal point of view. the most interesting space-times of finite

timelike diameter are the Wheeler universes [cf. Tipler (1977c, p. 500)]. In par-

ticular, the "closed" Fhedmann cosmological models are examples of Wheeler

universes.

For a complete Riemann~an manifold ( N . go), the diameter is finite if the

manifold is compact. In this case. we may always find two points of Ai whose

distance realizes the diameter. On the other hand, for space-times with finite

timelike diameter, the diameter is never achieved.

Proposition 11.3. Let (34,g) be an arbitrary space-time. and suppose

that there exist p ,q E M such that d(p, q) = diam(M, g). Then d(p. q) = oo.

Proof. Suppose d ( p , q ) = diam(.%f:g) < m, Let q' E If ( q ) be arbitrary.

Then

d(p , 9') L d(p: 9 ) + 4% p') > d(p, 9) = diam(M, gl

in contradiction. C

Recalling that globally hyperbolic spacetimes satisfy the finite distance

condition (cf. Definition 4.6), we have the following corollary to Proposition

11.3.

Corollary 11.4. (1) The timelike diameter is never realized by any pair of

points in a space-time of finite timelike diameter.

(2) The timelike diameter is never achieved in a globally hyperbolic spacetime

TVe now prove the Lorentzian analogue (Theorem 11.9) of Bonnet's Theo-

rem and Myers' Theorem for complete Riemannian manifolds [cf. Cheeger and

Ebin (1975. pp. 27-28)], Similar results have been given by Avez (seminar lec-

ture), Flaherty (unpublished), Uhlenbeck (1975, Theorem 5.4 and Corollary

5 4 , and Beem and Ehrlich (1979c, Theorem 9.5). Also, Theorem 11.9 is con-

tained implicitly in stronger results using the Raycl~audhuri equatiori needed

11.1 T H E TIMELIKE DIAMETER 403

for singularity theory in general relativity [cf. Section 12.2 or Hawking and

Ellis (1973. Section 4.4)].

Definition 11.5. (Tzmelzke Two-Plane) A tzmellke two-plane 0 is a two-

dirnenqional subspace of T,M for some p € M which is spanned by a spacelike

tangent vector and a timelike tangent vector.

Recall that the sectional curvature K(o) of the timelike two-plane 0 may

be calculated by choos~ng a basis {v. w) for u consisting of a timelike and a

spacelike tangent vector and setting

Remark 11.6. Rather than considering all sectional curvatures. zt is es-

sential to restrict attention to timelike sectional curvatures for the following

reason. If (M, g) is a space-time of dimension n 1 3 and the sectional cur-

vature function of ( M , g ) is either bounded from above for all nonsingular

two-planes or bounded from below for all nonsingular two-planes, then (M, g)

has constant sectional curvature [Kulkarni (1979)'. Here a two-plane a 1s said

to be nonszngular ~f g(v, v)g(w, w) - [g(v,m)I2 # 0 for some (and hence for

any) basis {v. w) for a. On the other hand, space-times with all tzmelzke sec-

tional curvatures K ( a ) 5 -k2 or a11 timel~ke sectional curvatures K(u) 2 k2

exist. But Harris (1982a) has shown that if all timelike sectional curvatures

are bounded both from above and below, then (iM,g) has corlstant sectional

curvature. Thus no obvious analogue for the Lorentzian sectional curvature

exists for pinched Riemannian rnanifoids [cf. Cheeger and Ebin (1975. p 118)

for Riemannian pinching].

With the signature convention (-: i-, . . . . +) used here for Lorentzian met-

rics, curvature conditions in Riemannian geometry for complete Riemann-

ian manifolds of positive (respectively, negative) sectional curvature tend to

correspond to theorems for globally hyperbolic space-times of negative (re-

spectively, positive) sectional curvature. On the other hand, if we change

this signature convention to (+, -, . . . , -) by setting (M, g) = (M, -g) , then

K(3) = -K(g). Thus Riemannian theorems for pos~tive (respectively. nega-

404 11 SOME RESULTS IN GLOBAL LORENTZlAN GEOMETRY

tive) sectional curvature correspond to Lorentzian theorems for positive (re-

spectively, negative) sectional curvature for (M,tj) [cf. for instance Flaherty

(1975a, pp. 395-396) where the convention (+, -, . . . : -) is used]. But whether

g or ij is chosen as the Lorentzian metric for M , Ricfg) = Ric(6).

It is convenient to isolate part of the proof of Theorem 11.9 in the folfotving

proposition. &call the notation V&(c) from Definition 10.1.

Proposition 11.7. Let (M,g) be an arbitrary space-time of dimension

n 2 2. Suppose that (M, g) satisfies either the curvature condition

(1) Every timelike plane 0 has sectional curvature K ( a ) < -k i 0, or the curvature condition

(2) Ric(g)(v, v) 1 (n - 1)k > O for aI I unit timelike tangent vectors v E

T M .

Then if c : [D, b ] -+ M is any timelike geodesic with L(c) 2 TI&, the geodesic

segment c has a pair of conjugate points.

Prooj. Since curvature condition (1) implies curvature condition (2) by tak-

ing the trace, we will prove that condition (2) implies the desired conclus~on.

For convenience, we will suppose that c : [0, L] --, A4 is parametrized as a unit

speed tinlelike geodesic with length L. Set En(t) = ct(t) for all t E [O, L]. and

let (El, E2, . . . , En-l) be n - 1 spacelike parallel vector fields along c such

that {El (t). Ez(t), . . . , En(t)) fornls a Lorentzian orthonormal basis of T'(l)lM for each t E (0, L]. Set liV,(t) = sin(xt/L)E,(t), so that VnJ, E q ( c ) . Using

equation (10.3) of Definition 10.4, we obtain

Hence

If Ric(c'(t),ct(t)) 2 (n - l)k for all t E [O. L] and L 2 Ti/&, we find that

I(WL, Ws) > 0. Hence I(W,,W,) 2 0 for some i E ( I ,? , . . . ,n - 1). On

11.1 THE TIMELIKE DIAMETER 405

the other band, if c 1 10, Lj is free of conjugate points, then I(W,. W,f < O for

each i by Theorem 10.22. Hence c has a pair of conjugate points if L > T/&,

as required. 13

A slight variant of Proposition 11.7 may also be proved similarly using the

Timelike Morse Index Theorem (Theorem 10.27).

Proposition 11.8. Let (M,g) be an arbitrary space-time of dimension

n satisfying either (or both) of the curvature conditions of Proposition 11.7.

If c : [a, b ] -+ M is any timelike geodesic with L(c) > 7r/&, then t = a is

conjugate along c to some to E (a, b), and hence c is not maximal.

Proof. Using L(c) > n/& and the same vector field ti/; as in the proof of

Proposition 11.7, we obtain this time that

n-1

CI(W%,W%) > 0. %=I

Hence I(Ur2, I/Yd) > O for some i. Thus Ind(c) > O. By the Timelike hIorse

Index Theorem (Theorem 10.27), dim Jt(c) # 0 for some t E (a. 6) . This

completes the proof.

Now we are ready to give the Lorentzian analogue of the Bonnet Myers

Diameter Theorem for complete Riemannian manifolds.

Theorem 11.9. Let (M, g) be a globally hyperbolic space-time of dimen-

sion n satisfying either of the following curmture conditions:

(1) K(o) < -k < 0 for all timelike sectional curvatures K ( o ) .

(2) Ric(%, a ) 2 (n - l)k > O for all unit timelike vectors v E T M .

Then diam(M, g) I 7i./&.

Pmof. Suppose that &am(M,g) > n/&. We may then find p , q E M

with d(p , q) > 7i/& by defin~tion of diam(M,g). S~nce (M,g) 1s globally

hyperbolic, there exists a maximal tlrnellke geodesic segment c : 10, 11 -- M

with c(O) = p, c(1) = q. But as L(c) = d ( p , q ) > n/&, the geodmic segment

c is not maximal by Proposition 11.8, in contradiction. U

It is clear that an analogue of Proposition 11.8 could be obtained for null

geodesics using the null index theory developed in Section 10.3. However. a

406 I1 SOME RESULTS IN GLOBAL LORENTZIAN GEOMETRY

stronger result may be obtained using the Raychaudhuri effect [cf. Hawking

and Ellis (1973, p. 101)]. Thus we refer the reader to Section 12.2 for a dis-

cussion of these results rather than pursuing this analogy any further here

[cf. also Harrls (1982a)l. In addition to extensive results in the general relativ-

ity literature along the lines of Proposition 11.8, obtained using Raychaudhuri

techniques. disconjugacy theory of O.D.E.'s has also been applied in this set-

ting (cf. Tipler (1978), Galloway (1979), Kupeli (19861, just to single out a

few of many possible references).

11.2 Lorentzian Comparison Theorems

For use in Section 11.3 as well as for their own intrinsic interest, we now

present the timelike analogues of two important tools in global Riemannian

geometry: the index comparison theorem [Gromoll, Klingenberg, and Meyer

(1975, p. 174)] and the Rauch Comparison Theorem [Gromoll, Klingenberg,

and Meyer (1975, p. 178) or Cheeger and Ebin (1975, p. 29)]. The results in this

section, except for Corollary 11.12, have been published in Beem and Ehrlich

(1979c, Section 9). Actually, there are two versions of the b u c h Comparison

Theorem, often called Rauch Theorem I and Rauch Theorem 11, that are useful

in global Riemannian geometry [cf. Cheeger and Ebin (1975, Theorems 1.28

and 1.29, respectively)]. The result (Theorem 11.11) given in this section is

the Lorentzian analogue of Rauch Theorem I. Harris (1979, 19S2a) has given

proofs of Lorentzian analogues for both Rauch Theorems I and I1 and using

Rauch Theorem I1 has given a Lorentzian version of Toponogov's Comparison

Theorem [cf. Cheeger and Ebln (1975, p. 42)j for timelike geodesic triangles in

certain classes of space-times. Using this result, Karrls (1979, 1982a) has aiso

obtained a Lorentzian analogue of Toponogov's Diameter Theorem [cf. Cheeger

and Ebin (1975, p. 110) and Appendix A].

In the rest of this section, let (MI, gl) and (A42,92) be arbitrary space-times

with dim n/r, 5 dim &if2. Also let c, : [O, L] -+ M,, z = 1,2, be unit speed future

directed timelike geodesic segments. Throughout this section we will denote

both the index form on ~ ' ( c l ) arid V1(cz) by I. Also, during the proofs we

will denote both Lorentzian metrics gl and g2 by ( , ). The index form I for

11.2 LORENTZIAN COMPARISON THEOREMS 407

a timelike geodesic c and the index Indic) and extended index Indo(c) of c are

defined in Section 10.1, equation (10.1) and Definition 19 24. respectivel?..

TVe first need to define an isomorphism

4 : vl(cl) -+ v ~ ( c ~ )

so that g2(4X(t). q X ( t ) ) = g l ( X ( t ) , X ( t ) ) for all t E 10. L] This may be done

following the usual parallel translation constructron in Riemannian geometry

We first define an rsometry

4t : Tc, :t)fif~ + Tc,(t)Al,

as follows. Let

Pt : Tcl( t )M~ -+ T,,(G)MI

denote the Lorent7ian inner product-preserving isomorphism of parallel trans-

lation along rl . Explicitly, given u € TC,(,)IMI, let Y be the unique parailel

field along cl with Y(t) = v, and set Pt(v) = Y(0) . S~milarly. let

Qt : Tc,(,)hf2 + TC2(0)M2

denote parallel translation along cz. Cnoose an injective Lorentzian inner

product-preserving linear map

i : (T,,(o).MI. l,,(o)) -+ (Tcs(0)1~2. g21cz(o))

where i(cll(0)) = czt(O). Then the map

Qt : (TC,(tpfl. 9llc,(t)) + (Z2, t )n / f , , g21c2!t))

given by Ot = Q,' o i o Pt 1s an isometry since parallel translation preserves

the Lorentzian structures. We may then define the map

4 : vL(cl) + vL (4 as follows. Given X E VL(cl). define OX E vL(c2) by ((OXj(t) = Qt(X(t)).

It follows as in the Riemannian proof that (QX)' = q ( X 1 ) , where the first

covariant differentiation is in M2 and the second is in hif~

Let 6 2 , t ( ~ z ) denote the set of ail timelike planes 0 containing c,'(t) for

i = 1.2. There is then an induced map

4t : Gz,t(cl) -+ G2,t(c2)

defined as follows. If (T E GZ, t (~ l ) we may write g = span(v.cl1(t)) where v

is spacelike. Put &(v) = span{&(u), czl(t)) E G2 t(c2)

408 11 SOME RESULTS IN GLOBAL LOREXTZIAN GEOMETRY

We are now ready to state the timelike version of the index comparison

theorem.

Theorem 11.10 (Timelike Index Comparison Theorem).

Let ( M I , 91) and (M2.92) be space-times with dim MI I dim M2, m d let

cl : [0, p] + MI and c2 : [O,@] -+ M2 be unit speed, future d~rected cimelike

geodesics. Suppose for a21 t with 0 5 t 5 ,O and dl timelike planes a E G2,t(c1)

that the sectional curvature KMl (a) 2 %(dim) . Then for any X E v1(cl)

we have

(1) I(X,X) I I(OX, 4x1; (2) Ind(c1) 5 Ind(c2); and

(3) Indo(s) 5 Indo(c2).

Proof. Recalling that (cl', el') = -1 and (X, cl') = 0, we obtain

Similar formulas hold for QX and czl. Thus

P {-(XI, XI) + ( R ( X , cll)cl', X ) ] dt = I ( X , X). CI

We may now obtain the more powerful Timelike Rauch Comparison The-

orem using the timelike index comparison theorem. Recall Erst that since

c, : [0, L] -+ Mi is a timelike geodaic segment for each i , the vector fields in

VL(c,) are all spacelike vector fields.

Theorem 11.11 (Timelike RaucP1 Comparison Theorem).

Let (&, gl) and (M2,gz) be space-times with dim MI 5 dim A&. Let cl :

[O, L] -- -MI and e2 : [O,L] + M2 be future directed, timelike, unit speed

geodesic segments. Suppose for all t E [O, L] and my a E Gs,t(cl) that

11.2 LORENTZIAN COMPARISON THEOREMS '

Let Yl E VL(c:) and Yz E VL(c2) be Jacobi fields on -211 and respective

satisfying the initial conditions

and

rf cz ha? no conjugate points t o t = 0 in (0) L), then

for all t E (0, L]. In particular. cl has no conjugate points to t = 0 in (0. L )

Proof. This may be given along the lines of Gromoll, Klingenberg, a1

Meyer (1975, pp. 130-161), except for the proof of their inequality (7), p. 1€

which must be modified as follows. Let Z 6 Vi(cs) be the unique Jacobi fie

along cz with Z(0) = 0 and Z(to) = cpYl(to), $ as above. We must show th;

To this end, let cg = CI 110, to] and c4 = c2 / 10. to]. Then

(the above inequality by the t i~el ike index comparison theorem [Thcore 11.1&(1)])

(the above inequal~ty by the rnaximality of Jacobi fields with respect to t: index form in the absence of conjugate points [Theorem 10.231)

410 I1 SOME RESULTS IN GLOBAL LORENTZIAN GEOMETRY

It may also be assumed m Theorein 11.11 that c2 has no conjugate points

in [O, L]. Then cl would also ha%e no conjugate points in (0, L] by Theorem

11.10-(3).

For Riemannian manifolds of nonpositive sectional curvature, by ~qu ipp~ng

the tangent space with the "Aat metrlc" (cf Definition 10.17) it may be shown

thar, the exponentla1 map does not decrease the length of tangent ~ectors

[cf. Blshop and Crittenaen (1964, p. 178, Theorem 2 (i)) for a precise state-

ment]. A slmple proof of this fact may be given uslng the R a ~ c h Comparison

Theorem and conlpdring Jacob1 fields on the given Riemannian manifold to

those in R". We will now use the Timelike Rauch Comparison Theorem to

prove the analogous ~esul t for space-times of nonnegative tlmelike sectional

curvature [cf. Flaherty (1975a- p. 39771. Intuitively. Corollary 11.12 below ex-

presses the fact that if all timelike sectional curvatures of (M,g) are pos~tive,

:hen future directed t~melike geodesics emanating from a given point of lliP

spread apart faster than "corresponding" geodesics in M~nkowski space-time.

Recall that the canonical isomorphis~n r, has been defined in Sect~on 10.1,

Definition 10.15.

Corollary 11.12. Let (M.g) be a space-time with everywhere nonn

tive timelike sectional curvature, and let v E T,M be a given future direct

timelike tangent vector with g(v, z) = -1 Then for any future directed no

spacelike tangent vector zu 6 T'M, the vector b = ~ ( w ) E T, (T,M) satisfi

the mequality

9fexpp, b, exp,. 0) 2 g(w, w) = ( (4 b)).

Proof. We first prove the inequality for b = T,(w) E T,(T,llf) w ~ t h g(v, w)

0 by applying the Timelike Rauch Comparison Theorem with (Ml,gl)

(iC.1, g) and (Mz, gz) identified with Minkowski space-time (P?, go) with n

dimM. Setting cl(t) = exp, tv, let Yl E VL(c) be the unique Jacobi field wi

Yl(0) = 0, YI1(0) = w By Proposition 10.16, we have Yl(l) = exp,. b.

Now let cz : 10. I] -+ Ry be an arbitrary unit speed timelike geodesic, a

choose 5 E N(cz(0)) with g o ( F , ~ > = gl(w,w). Let & E VL(c2) be t

umque Jacobi field with Yz(0) = 0 and V2'(0) = 3. Then Yz(t) = t

11.3 LOREKTZIAN HADAMARD-CARTAN THEOREMS 411

where Pt denote the Lorentzian parallel translation along cz from c2(0) to

cz(t). Applying Theorem 11.11, we obtain

gl(expp. b. exp,* b) = gi(K(1). &(1))

L gofY2(1).Yz(l))

= 5Q(PI (El, PI @)I

= go (727. G) = gl ( w , W) = ((b, b ) )

as required.

Returning to the general case, we may decompose ,w = w1 + w2 where

wl = Xv for some X > 0 and g(v, wz) = 0. Set b, = r,(wz) for i = 1 , 2 SO that

b = bl + 6% Wc now calculate

Applying the Gauss Lemma (Theorem 10.18) to the first two terms, we

obtain

as ((b1,bz)) = g(wl. wz) = 0. Now applying the first part of the proof to the

last term, we have

as required. 0

11.3 Lorentzian Nadarnard-Cartan Theorems

We first prove a basic result linking conjtlgate points and timelike sectional

curvat3xe [cf. Flaherty (1975a, Proposition 2.lj-here Flaherty uses the signa-

ture convention (+, -, . . . , -) for the Lorentzian metric so that his sectional

curvature condition has the opposite sign from ours].

412 11 SOME RESULTS IN GLOB.4L LORENTZIAN GEOMETRY

Proposit ion 11.13. Let (M, g) be a space-time with everywhere nonneg-

ative timelike sectional curmtures. Then no nonspacelike geodesic has any

conjugate points.

Proof. First let c : 10,a) -, iM be an arbitrary future directed unit speed

timelike geodesic. Recall from Corollary 10.10 that if X is a Jaeobi field along

c with X(O) = X(t0) = O for some to E (O,a), then X E VL(c). Thus we

may restrict our attention to Jacobi fields J E vL(c) with J(O) = 0. Since

J E VL(c) and c is a timelike geodesic, J' E VL(c) also. Consider the smooth

function f (t) = g ( J ( t ) , J t ( t ) ) . Differentiating yields

for all t t [O, a) . Now if J(t0) = 0 for some to E [&a), then f (0) = f (to) = 0.

Thus as j is nondecreasing. f ( t ) = 0 for all t € [O, to]. Hence 0 = f'(0) =

g(J'(0). J1(0)) as J (0 ) = 0. from which we conciude that J1(0) = 0. Therefore

J = 0, and no t o f [O, a ) is conjugate to t = 0 along c.

We now treat the case that f l : [0, a ) -+ M is a null geodesic using the null

index form. Let t o E [O, a ) be arbitrary. We will show that 7 : Xo(P [O, to]) x

Xo(P 1 (0, to]) -+ R is negative definite. Hence no s E (0, to] is conjugate to

t = 0 along j3 by Theorem 10.69.

If W E Xo(P) is a smooth parallel vector class, then W = [P'] since W(0) =

[,B1(0)]. Thus if W E Xo(B) is not a smooth parallel vector class, we have

g(W1(s) , W 1 ( s ) ) > O for some s E ( @ t o ) . Also i j (Wr(t) . W 1 ( t ) ) 2 O for all t E

[O, to] since 3 is positive definite. Now consider 3 @ ( ~ ( t ) , j3'(t))Dt(t), W( t ) )

for any fixed t E [O,to]. If W(t ) = [/3'(t)]. then g @ ( ~ ( t ) , ~ ' ( t ) ) f l ' ( t ) W ( t ) ) =

0. Otherwise, we rnay find a spacelike tangent vector w perpendicular to P1(t ) with ~ ( w ) = W ( t ) . Then, using equation (10.35) of Section 10.3 we have

3 (-iE(w(t), p'(t))pl ( t ) , W ( t ) ) = g (R(w , P( t ) )Pt ( t ) , w). Since w is spaceliie,

we may find a sequence of t i d i k e two-plaw 5, = (wn,v,) with w, spacelike,

0, timelike, g(w,,u,) = 0, w, -+ w , and v, -r pl(t) , so that a, -+ (w.P1(t)) .

11.3 LORENTZIAN HADAMARPCARTAN THEOREMS 413

We then obtain by continuity

3 Cfi(w(t>:P1(t))P1(t),w(t)) = 9 (R(w,P1(t))P' t t ) . w)

= lim g(R(wn, vn)vn, w,) i"M

= lim K(wn,vm)g(w,,wn)g(vm.vn) L 0 n-m

since g(w,, w,) > 0, g(v,, v,) < 0. and K(v,, w,) 2 0 for each n. Thus in

either case, 3 @(W( t ) , Pt( t))p '( t) , TV(t)) < 0. Hence, provided W # [@'] we

have - t o

I(W, W ) = Lo [-3wJ, wi) +z(x(w ~ l ) p l , w)ll, dt <

as required.

Motivated by a standard definition in global Riemannian geometry, we make

the following definition.

Definition 11.14. The space-time (M, g) is said to have no future timelzke

conjugate poznts if for any future directed timelike geodesic c : 10, a) 4 ( M , g),

no nontrivial Jacobi field in VL(c) vanishes more than once

In view of Lemma 10.46, similar definitions may be formulated for space-

times with no future null conjugate points or no future nonspacelike conjugate

points. Proposition 11.13 guarantees that if (M,g) is a space-time with ev-

erywhere nonnegative timelike sectional curvature, then ( M , g) has no future

nonspacelike conjugate points.

Lorentzian manifolds with nonnegative timelike sectional curvature or with

no future timelike conjugate points may be characterized in terms of the be-

havior of their Jacobi fields. A similar characterization applies to Riemannian

manifolds [cf. O'Sullivan (1974, Proposition 4)]

Proposition 11.18. i

(If (M,g) has everywhere nonnegative timelike sectiorid curvature iff P

for every Jacobi field Y E VL(c) along any future d~rected timelike

geodesic c.

414 11 SOME RESC'LTS IV GLOBAL LORENTZIAN GEOMETRY

(2) (M, 9) has no future t~rneljke conjugate pornts iff g(Y(t), Y(t)) > 0 for

all t > 0 where Y E VL(c) is any nontrivial Jacobi Geld with Y(0) = 0

along any future directed timellke geodesic c.

Proof. (1) Assume that (Id. g) has everywhere nonnegative timel~ke sec-

tional curvature. Let Y E VL(c) be a Jacobi fieid along the u n ~ t speed timel~ke

geodesic c. Then Y' E VL(c) also, and we obtain

d 2 -(g(Y, Y)) = 2g(Y1, Y') - 2g(R(Y, c1)c', Y) dt2

= 2g(Y1 ,Y ' )+2g(Y ,Y)K(Y ,c ' ) 2 0

Conversely, let {v,u;) be future timelike and spacelike tangent vectors,

respect~vely, spanning an arbitrary timelike two-plane with g(v, v) = - 1.

g(w,u~) = 1 and g(v, w) = 0 Let c(t) = exp(lv) and let Y E VL(c) be

the .Tacohi field with initial conditions Y(0) = w and Y'(0) = 0. Then we have

by hypothesis,

d2 0 5 9 1 = -29 (R(Y(0) c1(0))c1(O),Y(Oi)

t=O

= -29 (R(w, v)v, w) = 2K(v, w)

since the first term in the differentiation vanishes as Y'(0) = 0. Thus K(v, w) 2 0 as required

(2) This is clear from Defiilition 11.14. 0

Using the timelike index theory of Sections 10.1 and 10.2, we now gi

the following version of a Lorentzian Radamard-Cartan Theorem for global1

hyperbolic space-tlmes This proof is similar to the Morse theory proof of t

Hadamard-Cartan Theorem for complete Riemannian manifolds [cf. Lililn

(1963. p. 102)] Recall that a spacetime 1s said to be future one-connect

(Definition 10.28) if any two smooth, future directed timelike curves from p t

q are homotopic through (smooth) future directed timelike curves w ~ t h fixe

endpo~nts p and q.

Theorem 11.16. Let (M, g) be a future one-connected globally hyperbo

space-time with no future nonspacelike conjugate points. Then given

11.3 LORENTZIAK HADAMARD-CARTAN THEOREMS 415

p, q E M with p << q, there is exactly one future directed timelike geodes~c (up

to reparmetrization) from p to q.

Proof Since (M g) is globally hyperbolic, there exists a maximal f2~ture

directed timelike geodesic from p to q. Since there are no future nonspacelike

conjugate points, any future directed geodesic from p to q has index 0 by

Theorem 10.27. Thus the timelike path space C(p,,) has the homotopy type

of a CW-complex with a cell of dimension 0, i.e., a point, for each future

directed timelike geodesic from p to q On the other hand, since hf is future

one-connected, C(,,,) is connected and hence consists of a single point. Thus

there is at most one future directed timelike geodesic from p to q.

d similar result was obtained by Uhlenbeck (1975, Theorem 5 3) for globally

l~yperbolic spacetimes satisfying a metric growth condition [Uhlenbeck (1975.

p. 72) ] and the curvature condition g (R(v. U)W, 0) 5 0 for all future dlrected

null hectors v and vectors w with g(v, 2 ~ ) = 0 at every point of M . Namely, M

can be covered by a space which is topologically Mlnkowski (3.e.. Euclidean)

space.

Flaherty (1975a, p. 398) has also shown that if ()If, g) 1s future 1-connected,

future nonspacelike carnplete, and has everywhere nonnegative timelike sec-

tional curvatures. then the exporlential map exp, regularly embeds the future

cone in TpM at each point p into M. To obtain this result, Flaherty used

a lifting argument to show that urider these hypotheses, if v. w E T,M are

any two future directed timelike tangent vectors with expp v = exp, w, then

v = w. Thus future onoconnected, future nonspacelike co~rlplete space-times

with nonnegative timelike sectional curvatures satisfy the conclusion of Theo-

rem 11.16. On the other hand, Flaherty showed (1975b, p. 200) that any fu~ure

one-connected, future nonspacelike complete space time with everywhere non-

negative timelike sectional curvatures is also globally hyperbolic [cf. Galloway

(1986a, 1989b)j.

h c a l l that in Section 7.4 we applied disprisonment and pseudoconvcxity

to nonspacelike geodesics to obtain sufficient conditions for stability of non-

spacelike geodesic incompleteness (nonspacelike d~sprlsonment suffces) and

the stability of nonspacelike geodesic completeness (nonspacelike disprison-

416 11 SOME RESULTS IN GLOBAL LORENTZIAN GEOMETRY

rnent and nonspacelike pseudoconvexity taken together suffice). In order to get

a Hadamard-Cartan Theorem that applies to the entire space-time, we will

apply these two conditions to the set of all geodesics-not just to nonspace-

like geodesics [cf. Beem and Parker (1989)j. A manifold will be geodesically

disprisoning if each end of every inextendible geodesic fails to be imprisoned

in a compact set. The manifold will be geodesically pseudoconvex if for each

compact set K the convex hull, using geodesic segments joining points of K,

iies in a larger compact set. More precisely,

Definition 11.17. ( Geodesacally Disprasonzng) The spacetime (M. g)

is geodesically ckisprisonzng if, for each inextendible geodesic c : (a, b) - M and any fixed to E (a , b), the images of each of the two maps c I (a, to] and

c I [to, b) fail to have compact closure.

Definition 11.18. ( Geodesically Pseudoconvex) The space-time (M, g)

is geodesically pseudoconvex if for each compact subset K of M there is a

compact set N such that each geodesic segment c : [a, b] -+ M with c(a), c(b) E

K satisfies c([a ,b] ) C W .

By itself, Definition 11.17 allows for the possibility that either (or both) ends

of a given geodesic may be pwtially imprisoned (cf. Definition 7.29). However,

the next lemma shows that if ( M , g ) satisfies both of the above conditions,

then partial imprisonment cannot occur.

Lemma 11.19. Let (M, g) be both geodesirally disprisoning and geodesi-

calIy pseudoconvex. If c : (a. b) -+ M is any inextendible geodesir of M and

K is any compact subset of M . then there are parameter values sl . sz with

a < 51 < s~ < b such that c(t) # K for all a < t < sl a d all s~g < t < b.

Proof. Assume the bmma is false. Then without loss of generality we may

assume there is some geodesic c: (a, b) -+ M and a sequence {tk) satisfying

a < tl < . . . < tk < . . < b, tk + b-, and c ( t k ) E: I( for some compact subset

K of M . Let Ii be a cornpact set such that any geodesic segment of ( M , g )

with endpoints in K lies in W . Then the image of the map c 1 itl, b) lies in the

compact set N, in contradiction to the disprisonment assumption. 0

11.3 LORENTZIAN HAD.4MAtLD-CARTAN THEOREMS 417

Lemma 11.20. Let (M, g) be both geodesically disprisoning and geodesl-

cally pseudoconvex. Let {p,) and (T,) be sequences converging to p and T ,

respectively. If each pair {p,, T,) is joined by a geodesic h, then here is a ge-

odesic c joining p and r . Furthermore, if each c, is nonspizcelike (respectively,

null). then c may be chosen to be nonspacelike (respectively, null).

Proof, Let h be an auxlliarjr Riemannian metric on M . Let K be a compact

set containing p and r as well as all {p,) and {T,). Fbr each n, let c, : [O, b,) -4

M be a geodesic for the metric g which satisfies c,(O) = p,, %(a,) = r,, and

which is not extendible to b,. We may assume without loss of generality that

h(cn l (0 ) , cnJ(0)) = 1. Since h is positive definite, the sphere bundle over K

with respect to h is compact. Consequently, there 1s a subsequence (m) of

(n) and a vector w E T,M with ~ ' ( 0 ) -+ w. Let c: [0, b) -+ M be the unique

geodesic with c'(0) = u: and such that c is not extendible to t = b. Let H be a compact set such that all geodesic segments with endpoints in K lie in

H . Since (M, g ) is disprisoning, there is some 0 < T < b with C(T) $ H. The

construction of W then implies c(t) g' K for all r 5 t < b. Since c,(T) -+ c(T),

it follows that O < a , < T for all suffic~ently large rn. Using the fact that

[O, T ] is compact, we find there is a limit point a of the sequence {a,). Let

{k) be a subsequence of {m) with ak -+ a. Then ck(ak) -+ c ( a ) , and it follows

that c(a) = r = lirn r k . Hence c is a geodesic containing both p and r. If each

c, is nonspacelike (respectively, null), then cl(0) = lim ~ ' ( 0 ) shows c IS also

nonspacelike (respectively, null). U

Definition 11.21. ( Geodesacally Connected) The spacetime (M, g ) is

geodesically connected if for each pair of distinct points p and q there is at

least one geodesic segment joining p to q.

The Nopf-anow Theorem [cf. Hopf and Rinow (1931)] guarantees the geu

desic connectedness of any complete Riemannian man~fold. However, complete

Lorentzian manifolds may fail to be geodesically connected. In fact, for com-

plete Lorentzian manifolds, even if r is in the causal future of p, it may happen

that there is no geodesic from p to r (cf. Figure 6 1). Of course, global hyper-

bolicity yields a (maximal length) geodesic segment joining any two causally

related points. However, geodesic cornpletenms does not imply global hyper-

418 11 SOME Rl3SULTS IN GLOBAL LORENTZlAN GEOMETRY

bolicity. Furthermore. global hyperbolicity does not yield the existence of

geodesic segments between points that far1 to be causally related The next

proposition gives sufficient conditions for the existence of a geodesic segment

between anj two points The hypotheses include disprisonment, pseudocon-

vesity, and a lack of conjugate points but do not require geodesic completeness.

Proposit ion 11.22. Let ( M , g ) be geodesically disprisoning and geodesi-

cally pseudoconvex. If (M, g ) has no conjugate points, t h ~ n (M, g ) is geod~si-

cally connected.

Proof Let p and r be distlnct points of M, and construct an arbitrary curve

a : [O, I ] + M with a ( 0 ) = p and a ( 1 ) = r . Let .r = supis E [O,11 : there

IS a geodesic segment from p to a( t ) for all 0 5 t 5 s). The fact that p lles

in a convex normal neighborhood yields r 3 0. Lemma 11.20 implles that

there is a geodesic segment c. 10, l] --t M wlth c(0) = p and c (1 ) = a(r) Let

X = c l (0 ) Since ( M , g ) has no conjugate points, the exponential map exp,

takes a sufficiently small open neighborhood of X onto an open neighborhood

of c(1) = a ( r ) . Thus, there exist geodesic segments from p to all points of

a sufficiently near ~(7). It follows that T = 1, and hence there is a geodesic

segment from p to T , as desired. O

Compact Riemannian manifolds are always geodeslcally connected. In con-

trast, the next example shows that compact Lorentzian manifolds may fail to

be geodesically connected.

Exsunple 11.23. (Compact space-tzmes need not be geodeszcally connecte

~ e t M = S b SS' = ((t,s) .Oi s 5 l a n d -47; < t I 4 n ) w i t h t h e u s u

identifications Let g = (- cost)dt2 + (2 sint)dtds + ( c o s t ) d s v h ~ s is

Lorentzian metric, and X = cos(tl2) a/& - sin(tl2) B/ds is a unit t ~ m

vector field which tlme orients hf. Using light cones, it is not hard to c

that all geodesics starting on the circle t = 0 must lie in the set -5.rr/2 < t 7x/2. It follows that ( M , g) 1s a compact space-tlme which is not geodesical

connected.

11.3 LORENTZIAN HXDAMARD-CARTAN THEOREMS 41s

Let G be a group of homeomorpl.lisrns acting on the manifold 111. The group

G is said to act freely if f ( p ) = p for some p E M irriplies that j is the identity

e of 6. If G acts freely, then e is the only element of G that has any fixed

points. A discrete group G is said to act properly dascontznzlously jcf Boothby

(1986), Wolf (1974)] if the following hold:

( 1 ) Each p E M has a neighborhood U such that the set { f E G : f ( U ) n U # 0) is finite, and

(2) If f (p) # q for all f E G, then there are neighborhoods V and W of p

and q, respectively, such that j (V) n W = 0 for all f E 6.

Group actions arise naturally in the study of co~rering spaces of manifolds.

Let M be a smooth manifold. A covenng space of iM 1s a tnple (E, T, iM) where

E is a smooth manifold and n : E + M IS a smooth map of E onto il.1 which

is evenly covered [cf. O'Neill (1953)l. Here is sald to be evenly covered l f for

eachp E M there is a neighborhood U of p such that T - ' ( U ) consists of dlsjolnt

open sets (W,) in E and .rr / W, is a diffeomorphisrn of W, onto U for each a.

If E is simply connected, then (E. n, M ) is the unzversal coverzng space of &if.

A covering map T is a lo~a l diffeomorphism, but not all local diffeomorphisms

are covering maps because a local diffeomorphism is not necessarill an even

cover. For example, let M = S' = {(x, y) : ;c2 + y2 = 1) = { z E C : 121 = 1) .

The map f . ( 0 . 4 ~ ) -+ S' defined by f (6 ) = r" is d local diffeomorphism

but fails to be a covering map. The inverse image of e'" (1 ,O) is the single

number 27r. Other polnts have two preimages, and the Inverse image of aniall

connected open set U about the point (1,O) in S1 is three open intervals, one

of which contains the number 27r. Thc map f restricted to the interval of

f-l(U) containing 2n is a diffeomorphism onto U , but f restricted to either

of the other two intervals is not a diffeomorphism onto U. The following topological result will be of use in the proof of Theorem 11 25.

Lemma 11.24. If T : Rn --+ M is the universal covenng space of the

rnanrfold IM, then the fundamental group i71(.44) of M is either tr~vtal or

infinite. If ?rl (M) is trivial, then the covering map .ir :s one-to-one.

Proof The fundamentai group 7r l ( iZ / [ ) of hf acts freely and properly dis-

continuously on the total space Rn of the covering [cf. Kobayashi and Nom~zu

420 11 SOME RESULTS IN GLOBAL LORENTZIAN GEOMETRY 11.3 LOREKTZIAN WADAMARD-CARTAX THEOREMS 421

(1963, p. 61)]. If ax (M) 1s finite, then it is either trivial or else has an element

of prime order. However, a result of Smith (1941, p. 3) yields that any homeo-

morphism of Bn having prime order must fix some point of Rn. But this would

contradict the free action of ~ 1 j i t l ) . Thus, the fundamental group is infinite or

has exactly one element. If the fundamental group is trivial, then each p E -M has exactly one preimage and hence the covering is one-to-one. U

The next theorem is a Lorentzian version of the Hadamard-Cartan Theorem

and is actually valid for any (connected) manifold with an &ne connection

[cf. Beem and Parker (1989)l. The usual completeness assumption used in the

Riemannian version [cf. Kobayashi (1961)] is replaced by disprisonment and

pseudoconvexity assumptions. In some sense, the pseudoconvexity condition

is a type of "internal completeness" assumption which plays a role similar to

global hyperbollcity (cf. Section 7.4). Since global hyperbolicity is a causality

assuimption, it only provides a certain amount of "control" over nonspacelike

geodesics. It should also be recalled that global hyperbolicity, which was used

in the hypothesis of our previous version of a Lorentzian Hadamard-Cartan

Theorem, Theorem 11.16 earlier in this section, implies nonspaceltke geodesic

pseudoconvexity (cf. Proposition 7.36).

Theorem 11.25. Let (M, g) be geodesically disprisoning m d geodesically

pseudoconvex If (M, g) has no conjugate points, then for each point p E M ,

the exponential map exp, : D -4 M is a diffeomorphism from the domain

D T,M of exp, onto M. Consequently, M is diffeomorphic to Rn, M is

geodesically connected, and any two distinct points of M determine exactly

one geodesic.

Proof. The map expp is onto M by Proposition 11.22. Furthermore, the

exponential map at p must be at least a local diffeomorphism from its domain

D onto M since no conjugate points exist. In order to show that expp is a

diffeomorphism. we will first show that each r E M has a finite number of

preimages Secondly, the map expi, will be shown to be a covering map, and

finally, Lemma 11.24 will yield that exp, is one-to-one.

To show that a fixed point r of M has a finite number of preimages in D, iue

begin by choosing a compact set K of M containing p and with r in the interior

of K. Then there is an open neighborhood U(r) of T with r E U(r) c K. Let

H be a compact set such that any geodesic segment wlth endpoints in K lies

in W . Pseudoconvexity guarantees the existence of N. Choose an auxiliary

Riemannian metric h, and let Sh(p) = (V E T,M h(v, u) = 1). Let .I; E Sh(p)

be arbitrary. If c, : [0 ,b) -+ M is the geodesic of g wltn c,'((O = a. then

disprisoriment yields c,(r) $? H for some T. It follows that c,(t) @ K for

all r 5 t < b. Using the continuous dependence of geodesics on the initla1

tangent vector, it follows that there is a neighborhood N(v) of v in Sh(p) such

that each geodesic with initial tangent vector lying in this neighborhood must

have 7 in its domain and rrlust have image points not in K for all t > T . In

part~cular, if c, happens to intersect U ( r ) . then the intersection must occur

for pasameter values not greater than T. Let r(v) = {w E T?M : w =

a v with v E Sh(p) and 0 5 cr 5 r ) . The set I'(v) is compact and lies in the

domain of the local diffeomorphism exp,. Consequently, each point q E U ( r )

has only a finite number of preimages in I'(v). We cover the compact set Sh (p)

with a finite number of neighborhoods N(vi), . . . iV(v,) of the above type for

corresponding vectors vl , . . . . v,. Thus, we obtain a finite number of compact

sets I'(vl). . . . . r(v,) each having a finite number of preimages of any fixed

point q E U(r). Let I' = I'(vl) U. -UI'(z,). By construction, all preimages of

each q E U(r) lie in the compact set I'. Hence we find that r E M has a finite

number of preimages in D 5 T,M.

Assume that exp, 1s not a covering map, There must be some r E M such

that for each open neighborhood U of r , the set expil(U) fails to consisr, of

components each of which is diffeornorphlc to U under expp. Let X, U ( r ) , H.

and I? be as above. Let r have k preimages u l , . . . . uk. Slnce exp, is a local

diffeomorphism, there is an open neighborhood V c U(r) of r diffeomorphic

to an open ball. and corresponding disjoint open neighborhoods T.PPi, . . . , in T,M containing the individual prelrriages of 7 , such that exp, / is a

digeomorphism onto V for each 2. It follows that exp;' (V) consists of more

than Wl U T.Y2 U U K. and this must be -,rue no matter Iiow sinall V 1s

chosen Consequently, there is a sequence r, - r such that the nuriiber of

preimages of each r, is larger than k . This yields a sequence (w,) in T p M

42% 11 SOME RESULTS IN GLOBAL LORENTZIAN GEOMETRY

with exp,(w,) = T, such that no subsequence of (w,) converges to one of the

preimages ul, . . . , u k of r. Since each w, lies in the compact set I?, there is a

subsequence {w3 1 converging to some w in r. Thus. w is in the domain of expP,

and expp(w) = limexp,(w,) = limr, = r , which shows w must be one of the

k preimages of T, in contradiction. We conclude that expp must be a covering

map. The (star shaped) domain D of exp, is some open set in Tpn/l, and D

must be homeomorphic to W". Thus exp, . D --t M is a universal covering

and the number of elements in (M) is equal to the number of preimages of

any point r E M . Lemma 11.24 now yields that T ~ ( M ) is trivial and hence

that exp, : D + A4 is a diffeomorphism. as desired.

- Let a Lorentzian manifold ( M , g ) be given, and assume n : M 4 A4 is

a covering of M by z. One may obtain an induced metric ij on z such

that .ir is a local isometry [cf. Wolf [1974), pp. 41-42]. The geodesics of li;?

project to geodesics of M , and geodesics of M lift to geodesics of ;i?. It may

easily 'happen that the cover is geodesically disprisoning even if the original

base manifold ( M , g) fails to be disprisoning. For example, one could have

Af = S1 x 5" = { ( t , x ) : 0 5 t 5 1, 0 5 5: 5 1) with the usual identifications,

given the flat metric g = -dt2+dz2. Then (x,?j) is the usual two-dimensional

hPvlinkowski spacetlme.

Clearly, Theorem 11.25 may be applied to the cover to obtain information

on the base manifold M.

Corollary 11.26. Let ( M , g ) have a covering space n : --+ 11.1 with

induced metric ?j = 7i'g on z. If this covering space (%: 3) has no conjugate

points, is geodesically pseudoconvex, and is geodesically disprisoning, then

both (=, 3) and (M, g) are geodesically connected. Furthermore, if p is any

point of M , then the exponential map exp, is a covering map.

Proof. Theorem 11.25 yields the geodesic connectedness of the cover To

show that M is geodesically connected, first let p and q be two points of M .

Lift them to points jj and Ti, respect~vely, in z. Then the geodesic segment

from p to Fj projects to a geodesic segment from p to q. Thus the geodesic

connectivity of the cover implies the geodesic connectivity of the base.

11.3 LORENTZIAN RXDAMARD-CARTAX THEOREMS 423

It remains to show that exp, is a covering map. Select a point p E sr - ' (p ) .

Let D, 2 T,M be the domain of exp, and D y 5 TF % be the domain of

expp. The composition a o exp-, , Dy + M is a covering map since exp-, is

a diffeomorphism onto M and .;; is a covering map w:th domain ;i?. Recall

that n is an isometry of any sufficiently smali neighborhood W' of P to a

corresponding ne~ghborhood U of p. Also, each geodesic through p projects to

a geodesic through p, and each geodesic though p lifts (uniquely) to a geodesic

through p. It follows that the differential of the map (n / W)-' at p yields a

diffeomorphism f : D, --+ D F such that expp = K 0 expp o f . Consequently.

exp, : Dp --+ M is a covering map, as desired. C;

Versions of Proposit~on 11.22 and Theorem 11.25 for certain classes of sprays

on manifolds have been given in Del Riego and Parker (1995).

CHAPTER 12

SXNGULARPTIES

A common assumption made in studying Riemannian manifolds is that the

spaces under consideration are Cauchy complete or. equivalently, geodesically

complete. This assumption seems reasonable since a large number of important

Riemannian manifolds are complete.

The situation for Lorentzian manifolds 1s quite different. A large number of

the more important Lorentzian manifolds used as models in general reiativiry

fail to be geodesically complete. Also, the problem of completeness is further

complicated by the fact, observed in earlier chapters, that there are a number

of inequivalent forms of completeness for Lorentzian manifolds,

In this chapter we will be concerned wwlth establ~shing theorems which guar-

antee the nonspacelike geodesic incompleteness of a large class of space-tlmes.

These spacetimes contain at least one nonspacelike geodesic which is both

inextendible and incomplete. Such a geodesic has an endpoint in the causal

boundary &A4 which may be thought of as being outside the space-time but

not a t infinity. For example, if y is a future inextendibie and future incomplete

timelike geodesic which has jj E d,M as a future endpoint, then y corresponds

to the path of a "freely falling" test particle which falls to the edge of the

universe (at p ) in finite time.

It has been known for a long t m e in general relativity that a number of

important space-times are nonspacelike incompiete. Nonetheless, this incom-

pleteness was thought to be caused by the symmetries of these models. Thus

it was felt that nonspacelike completeness w s a reasonable assumption for

physically realistic space-times. The argument for this assumption was based

on physical intuition which was evidently unjustified with hindsight [cf. Tipler,

Clarke, and Ellis (1980, Chapter 4)).

426 12 SINGULARITIES

If (M, g) is an inextendible space-time which has an inextendible nonspace-

like geodesic which is incomplete, then ( M , g) is said to have a singularity. The

purpose of this chapter is to establ~sh several singularity (i.e.. incompleteness)

theorems.

Before beginning our study of singularity theory, we pause to explain why.

from a mathema~ical viewpoint. this theory works for all space-times of di-

mension n > 3 but not for spacetimes of dimension two. It ts simply the

fact noted at the beginning of Section 10.3 that no null geodesic in a two-

dimensional space-time contains any conjugate points. Yet a key argument in

proving singularity theorems is showlng that certain curvature conditions force

every complete nonspacelike geodesic to contain a pax of conjugate points.

Familiarity with the notations and some of the basic properties of Jacobi

fields treated in Sections 10.1 and 10.3 will be assunied in this chapter.

12.1 Jacobi Tens~rs

As we saw in Section 10.3 (Definition 10.61 ff.), Jacobi terisors provide

a convenient way of studying conjugate points. Given a timelike geodesic

segment c : [a, bj -4 M, let N(c(t)) denote the (n - 1)-dimensional subspace of

Tc(t$4 consisting of tangent vectors orthogonal to cJ(t) a s in Definition 10.1.

A (1,l) tensor field A(t) on VL(c) is a linear map

for each t 6 [a, b]. Further. a composit~ endomorphism R.4(t) : N(c(t)) 4

Nfcft)) may be defined by

The adjoint A*(t) of A(t) is defined by requiring that

for all v, w E N(c(t)).

A smooth (1,l) tensor field A(t) on VL(c) is said to be a Jacobz tensorjeld

if

A'/ + RA = 0

12.1 JACOB1 TENSORS 427

and

ker(A(t)) nker(A1(t)) = ( 0 )

for all t E [a, b ] If Y is a parallel vector field along c and -4 1s a J ~ c o b ~ tensor

on VL(c). then the vector field J = A ( Y ) satisfies the d~fferentlal equation

J" - R(J,cJ)c' = 0 and hence 1s a Jacobi field The condition ker(A(t)) 17

ker(ill(t)) = (0) for ail t E [a. b] guarantees that ~f Y 1s ;t nonzero parallel

field along c, then J = A(Y) IS a nontrivial Jacobi field. Suppose that A is a

Jacob] tensor on VL(c) with A(a) = 0. If A(tn)(v) = 0 for some to E (a, b ]

and 0 # v E N(c(to)), then letting Y be the unique parallel field along c with

Y( to ) = v. we find that J = A(Y) is a Jacobi field with J ( a ) = J(t0) = 0.

A Jacobi tensor A IS sald to be a Lagrange tensor field ~f

for all t 6 [a, b]. As in the proof of Lemma 10.67, ~t may be shown that a Jacobi

tensor field A is a Lagrange tensor field if A(to) = 0 for some to E [a. b ] .

Remark 12.1. Let c : ja, b] --. M be a unit speed timelike geodesic, and

suppose El. E2,. . . , En is a parallelly propagated orthonormal basis along c

with En = GI. Then N(c(t)) is the span of El, E2,. . . , En-lr and each Jacobi

vector field J along c which is everywhere orthogonal to c' may be expressed in

terms of El. ET, . . . ,En-]. Thus J may be represented as a column vector with

(n - 1) components. Using this representation. let J, = J,(t) be the column

vector corresponding to the Jacobi field J along c which satisfies J ( t o ) = 0

and J 1 ( f n ) ='E,(to) Let

be the (n - 1) x (n - 1) matrix with J,( t ) for the 7th coli~mn Thic: matrix

A(t) is a representation of a Lagrange tensor field along c Using this same

basis El, Ez, . . . , E,-i, the adjolnt A* (t) is represented by the transpose of

A@). The space of Jacob1 fields whlch vanlsh at to and which have derivatives

orthogonal to c' at to may be identifed wiih the span of the columns of A. Tnus

conjugate points of c(to) along c are exactly the points where det A(t) = 0.

Hence det A(t) has isolated zeroes on the interval [a, b] . Also, the multiplicity

of a conjugate point t = t1 to to along c is just the nullity of A(tl) : iV(c(tl)) -,

N(c(t1)).

The fact that &to) = O and A1(to) = E is essential to the above discussion.

Lagrange tensors along a timelike geodesic c : J + ( M , g) may be constructed

which are singular at distinct to, t l E J , yet c has no conjugate points. Fbr

example, let ( M , g) be IR3 with the Lorentzian metric ds2 = -dz2 + dy2 + dz2

and let c(t) = (t, 0,O). Let El = d/By and E2 = d/Bz. Then if A is the Jxobi

tensor along c with the matrix representation

with respect to El 0 c and E2 o c, we have A' = E and A* = A so that

(A1)*A-A*A1 = 0, and A is a Lagrange tensor. Evidently, A(o)(E~(c(o))) = O

and ~ ( l ) ( E ~ ( c ( l ) ) ) = 0. But c has no conjugate points since (IE3,ds2) is

Minkowski &space.

W-e now define the expansion, vorticity, and shear of a Jacobi tensor A along

the timelike geodesic c : [a, b] + M. As before, E = E(t) will represent the

(1, l) tensor field on vi(c) such that E(t) = Id : N(c(t)) -+ Ar(c(t)) for each

t. Note that this definition of B from A parallels the passage in O.D.E. theory

from the Jacobi equation to the associated Riccati equation.

Definition 12.2. (Expanston. Vortzczty. and Shear Tensors) Let A be

a Jacobi tensor field along a timelike geodesic, and set B = A'A-I at points

where A-I is defined.

(1) The expansion B is defined by

6 = tr(B).

(2) The vorticzty tensor w is defined by

1 w = - (B - B*).

2

(3) The shear tensor 0 is defined by

Using an orthonormal basis of parallel fields for VL(c) and matrix algebra,

it may be shown that

e = tr(AIA-'1 = (det A)-'(det A)'

Thus if A is a Jacobi tensor field with A(&) = 0, A1(to) = E . and IB(t)l -i oo

as t + t l , then detA(tl) = 0 and t = tl is conjugate to to along c.

We now calculate the derivative of B = AIA-' using ( -4 - I ) ' = -A-lAIA-'.

First we have

Using 6' = tr(B) and B = w + o + [O/(n - 1)]E, we obtain

where we have used tr(w) = tr(a) = tr(w0) = 0. Using the orthonormal basis

El, Ezl . . . , En along c with En = cl, we find that

= Ric(cl, c'j.

and - 1 82 6 = - t r ( 3 ) - t r (z2) - t r ( ~ ' ) - -

n - 2' We may calculate t r ( x ) as follows. Let V ( P ) denote the geometric realiza-

tion for G(P) constructed as in (40.28) of Section 10.3: and let (Yl,. . . , Yn-2)

be an orthonormal basis for V(P) at every point of ,B. Extend (Y', . . . , Yn-*) to an orthonormal basis {YI, Yz, . . . , Y,) along P , where Y, is timelike and

,B1 = (Yn-l + y,)/h. Then we have

using the basic properties of the curvature tensor. Consequently, we obtain

This yields the Raychaudhuri equation for Jacobi tensors along null geodesics:

The same reasoning as in the proof of Lemma 12.3 shows we may slmplify - * - I

the above equation when ;? is a Lagrange tensor field ji.e.. when A A =

z'*z). Lemma 12.6. If A is a Lagrange tensor field, then the vorticity tensor Zj

vanishes along PI.

We thus obtain the vorticzty-free Raychaudhuri equation for Lagrange tensor

fields along null geodesics:

12.2 GENERIC AND TIMELIKE CONVERGENCE CONDITIONS 433

12.2 The Generic a n d Tirnelike Convergence Conditions

111 this section we show that if (1Z.l.g) is a space-time of dimemion at

least three which satisfies the generic and tirilelike convergence conditions,

then every complete nonspacelike geodesic contains a pair of conjugate points

[cf. Hawking and Penrose (1970. p. 539)j. The timelike arid null cases are

handled separately. Similar treatments of the ma:erial in this section may be

found in Bolts (1977), Hawking and Ellis (1973, pp. 96-101), and Eschenburg

and 03ullivan (1976).

We may state the definitions of the generic condition and the timelike con-

vergence condition for our purposes in this section ES follows (cf. Propositions

2.7 and 2.11).

Definition 12.7. (Generic Condition) A timelike geodesic c . (n,b) --t

( M , g ) is said to satisfy the generic wndition if there exists some to E (a , b)

such that the curvature endomorphism

is not ~dentlcally zero. A null geodesic /3 : (a, b) --+ ( M . g ) is said to satisfy

the genenc condition if there exists some t o E (a , b) such that the curvature

endomorphism

of the quotient space G(P(to)) is not identically zero. The space-time (M. g) is

said to satisfy the genenc condztaon if each inextendible nonsgacelike geodeslc

satisfies this condition.

In Section 2.5 we have shown that this formuiation of the generic condition

is equivalent to the usual defin~tion given in general relativity; namely. a non-

spacelike geodesic c with tangent vector W satisfie8 the generic condition if

here is some point of c at which

w ~ w ~ u I [ , R ~ ~ ~ ~ ~ ~ % f 0.


Definition 12.8. (Temeizke Convergence Condztzon) A space-tlme sat-

isfies the tzme13e contergence condatzon ~f Ricjv, v) > O for all nonspacelike

tangent vectors v E TA4.

By continuity, the curvature condition of Definition 12.8 is equivalent to

the tzrnelzke conveqence condztzon of Hawking and Ellis (1973, p 95) that

&c(v,v) > O for all timehke v E TM. Hawking and Ellis (1973, p 95) also

call the curvature cond~tlon h c ( w , w) 2 O for all null w E TM the null

convergence condatzon. In Hawking and Ellis (1973, p 89), a Four-dimensional

space-time (1M.g) with energy-momentum tensor T (cf. Section 2.6) is said to

satisfy the weak energy condatzon if T(v , v) > O for all tlmelike v E T M . If

the Einstein equations hold for the four-d~mens~ona! space-time (M, g) and T

with cosmological constant A, then the cond~tion Bc(v,v) 2 O for all timel~ke

c E TM implies that

T(v, t ( y - :_&) 0(". v)

for all timelike v E TM. Hence in Hawking and Ellis (1973, p. 95). the four-

dimensional space-time (M,g) and energy-momentum tensor T are said to

satisfy the strong energy condztzon if T(v, v) > (trT/S)g(v,u) for all timelike

v E TM. When dimM = 4 and A = 0, this nlay be seen to be equivalent

to the condition R~c('u, v) 2 0 for all timelike z E TM (cf. Sectio11 2 6). In

Hawking and Penrose (1970, p 533), the condition Ric(v, v) 2 0 for d l unit

timelike vectors v E T M is called the energy condztaon. In Frankel (1979) and

Lee (1975), the term "strong energy condition'' is provided the same definition

which we have given to "timelike convergence condition" in Definition 12.

A discussion of the physical interpretat~on of these curvature conditions in

general relativity may he found in Hawking and Ellis (1973. Section 4.3)

As we have just noted above, if ( M g) satisfies the timelike convergence co

dttion or the strong energy condition, then (M, g) satisfies the null convergen

condition. In view of several rigidity theorems associated wlth curvature co

dltlons In R~emannian geometry [cf Cheeger and Ebin (1975. pp. v and v~

1s natural to consrder the impllcatlons of the curvature condition Rc(w. wj

for all null vectors w 6 T M . Applying linear algebraic arguments to each ta


gent space. Dajczer and Komizu (198Oa) have obtained the rigidity result that

if dim M > 3 and Ric(w, w) = O for ail null vectors ui E TM, then (Ad, g)

is Einstein, i.e., Ric = Xg for some constant X E W. Thus if ( M , g ) is not

Einstein, there are some nonzero null Ricci curvatures. Suppose further that

(M, g ) is globally hyperbolic with a smooth globally hyperbolic time function

h : &I - R such that for some Cauchy hypersurface S = h-'(to), all null Ritci

curvatures satisfy Ric(g)(w, w) > O ~f 7i(w) E S. If ( M , g ) also satisfies the

null convergence condition, then M admits a metric gl globally conformal to

g such that the globally hyperbolic space-time (M, gl) satisfies the curvature

condition Ric(gl)(w. w) > 0 for all null vectors w E TA4 [cf. Beem and Ehrlich

(1978, p. 174, Theorem 7.l)j.

An essential step in proving that any complete timelike geodesic in a space-

time satisfying the generic and timelike convergence conditions contains a pair

of conjugate points is the following proposition.

Proposition 12.9. Let c : J -t (M, g) be an ~nextendible timelike geodesic

satisfying Ric(cl(t), cl(t)) > 0 for all t E J. Lct A be a Lagrange tensor field

along c. Suppose that the expansion B(t) = tr(A1(t)A-'(t)) has a negative

[respectively, positive] value B1 = B(tl) at ti E J Then det A(t) = 0 for some

t in the interval from t j to t l - (n - l)/B1 [respectively, some t in the interval

from tl - (n - I)/& to tlj provided that t E J.

Proof. Since Q = (detA)'(detA)-', we have detA(t0) = 0 provided that

101 -+ oo as t -+ to. Thus we need only show that I@( -+ oa on the above

intervals. Put n - 1

s1= -. 81

The \orticity-free hychaudhuri equation (12.2) for timelike geodesics and the

condition Ric(cl: c') > 0 yield the inequality

In the case that < 0, integrating this inequality from tl to t > t l we obtain


for t E [tl. t l - s l ) . Hence /O(t)l becomes infinite for some t E ( t l , t l - sl

provided that c(t) is defined. In the case that B1 > 0, we obtain for t

(tl - sl , t l i that n - 1

l + si - t l , Hence lB(t)l again becomes irliinite for some t E [tl - sl.tl) provided that c(t

is defined. D

We now show that a timelike geodesic in a space-time that satisfies t

timellke convergence condition and the generic condition must either be

complete or else have a pair of conjugate points.

Proposition 12.10. Let (M,g) be an arbitrary space-time of d~men

n _> 2. Suppose that c : R -r (M,g) is a complete timelike geodesic wh~

satisfies Itic(c1(t). cl(t)) 2 0 for dl t E B. If R(,,c'(tl))cl(tl) : N(c(tl))

N(c(tl)) is not zero for sorne t l E W, then c has a pair of conjugate point

We first prove four lemmas which are needed for the proof of Propositio

12.10 [cf. BGIts (1977, pp. 30-37)].

Lemma 12.11. Let c : [a, b] -r (M, g) be a timelike geodes~c w ~ t

conjugate points Then there is a unique (I, 1) tensor field A on VL(c)

satisfies the diflerexltial equation A1'+RA = 0 with given boundary co

A(a) and A(b).

Proof. Let S be the vector space of (1,1) tensor fields A on VL(c)

A" + RA = 0, and let L(N(c(t))) denote the set of linear endomorphisms

N(c(t)). Define a linear transformation 4 S -+ L(N(c(a))) x L(N(c(b)))

@(A) = (A(a), .4(b)).

Since d i m s = dim(L(N(c(a)))) + dim(L(N(c(b)))) = 2(n - I ) ~ , ~t IS o

necessary to show that $ is injective in order to prove that Q is an isomorph

and establish the existence of a unique solution A. Assume that $(

(A(a), A(b)) = (0,O). If Y(t) is any parallel vector field along c, then J ( t

A(t)Y (t) is a Jacobi field -with J(a) = J(b) = 0. Thus J = 0. However,

Y( t ) was an arbitrary parallel field, this implies that A(t) = 0, which sh

is injective and establishes the lemma. CI


Now let c . [ t l , w ) -+ (M,g ) be a timelike geodesic without conjugate

points and fix s E (t l ,oo), Then by Lemma 12.11, there exists a unique (1,1)

tensor field on VL(c). which we will denote by D,, satisfying the differential

equation D," + RDs = 0 with initial conditions D,(tl) = E and D,(s) = 0.

As D,(tl) = E, we have ker(D,(tl)) nker(DBi(tl)) = 10). Thus L), is a Jacobi

tensor field (cf. Lemma 10.62). Abo, since D,(s) = 0, it follows that D, is a

Lagrange tensor field. It will also be shown dur~ng the course of the proof of

Lemma 12.12 that if A is the Lagrange tensor field on VL(c) with A(tl) = 0

and A1(tl) = E, then Cls1(s) = -(A*)-'(s)

Lemma 12.12. Let c : [tl, oo) -+ M be a timelike geodesic without con-

jugate points. Let A be the unique Lagrange tensor on VL(c) with A(tl) = 0

and A1(tl) = E. Then for each s E (tl, eo) the Lagrange tensor D, on VL(c)

with Ds(tl) = E and D,(s) = 0 satisfies the equation

for all t E ( t s] . Thus D, ( t ) is nonsingular for t E (t 1, s) .

Proof. Set X ( t ) = A(t) S;(A*A)-'(T)~T. It suffices to show that XI' + RX = 0. X(s ) = D,(s) = 0, and X'(s) = D,'(s).

We first check that X" r RX = 0. Differentiating, we obtain

Hence

438 12 SINGULARITTES

But since -4 1s a nonsinguiar Lagrange tensor, (A')' = A*rllA-I, so that

(A*)-l(A*)'(A*)-l = A'A-l(A")-l , and we obtain

We then have

xl'(t) + R ( t ) X ( t ) = (Atl( t) + ~ ( t ) A ( t ) ] / ' (A*A) -~ ( I ) dr = o t

since A1'(t) + R( t )A( t ) = 0. Thus X satisfies the Jacobi differential equatio

Setting t = s, we obtain

x ( s )= ~ ( s ) l s ( ~ * ~ ) - l ( I ) r l r = o

and

s

@*A)-I(.) d~ - A ( s ) ( A * A ) - l ( ~ ) = -(A*)-'(s).

Thus it remalns to check that Dal(s) = -(A*)- '(s) . But ustng R* = R. w

obtain

[(A*)'D, - A*D,'jl = (A*)I1D, + (A*)'DSJ - (A")'Dsl - A'D,"

= (A f ) "D3 - A* Dgl'

= -A*R8DS +A* RD, = 0.

Thus (A*)'D, - A"D,' is parallel along c. -4t t = t1, the initial condltlo

A(t1) = 0 and A'(t1) = E for A mpiy that Ae( t l ) = 0 and (Ae ) ' ( t l )

(Ar )* ( t l ) = E. Hence as D,(t l) = E, we obtain

((A*)'D, - A*DS1)( t l ) = E.

Hence ((A*)'D, - A'DS1)(t) = E for all t. Setting t = A, we have

E = ((A*)'D, - A*Dal)(s) = - (A"Dar)(s )

which implies that Dal(s) = -(A")- ' (s) = X1(s ) . Therefore since D, and

both satisfy A" + RA = 0 and have the same valum and first derivativ

t = s , the tensors must agree for all t.


Finally, the nonsingularity of D,(t) for t E ( t l , s) foliows from the formula

since (A*A)- ' ( t ) is a positive definite, self-adjoint sensor field for all t > tl . C3

Note that while the integral representation of the Lagrange tensor D, along

c satisfying D, ( t l ) = E and D,(s) = 0 given in Le~nrr~a 12.12 was proven only

for t E ( t l , s]. if c is defined for all t E R and has 110 conjugate points, then

D,(t) is defined for all t E R.

We now show that if c : [a .m) -+ (M,g) is a timelike geodesic without

conjugate points. then the above tensor fields D, converge to a Lagrange tensor

field D as s -+ oo. This construction parallels the construction of stable Jacobi

fields in certain classes of complete Riemannian manifolds without conjugate

points [cf. Eschenburg and O'Sullivan (1976, pp. 227 ff ), Green (1958). E. Ropf

(1948, p. 48)).

Lemma 12.13. Let c : la,co) -+ (1W,g) be a timelike geodesic without

conjugate points. For it1 > a and s E [a. x) - ( t l ) , let D, be the Lagrange

tensor field along c determined by D,(tl) = E and D,(s) = 0 Then D(t ) =

lim,,, D,(t) is a Lagrange tensor field. Furthermore, D(t ) is nonsingular for

ail t with tl < t < m.

Proof. CVe first show that Dal( t l ) has a self-adjoint lir~lit ds s --+ x. Since

D, is a Lagrange tensor, (D,'*D,)(tl) = (D,*D,')(tl). Using D,(l l) = E, we

obtain DS1*(tl) = Dal( t l ) . Thus the limit of DS1(tl) must be a self-adjoint

linear map which we will denote by Dr( t l ) . N ( c ( t l ) ) -+ N ( c ( t l ) ) if it exists.

Consequently, we need only show that for each y E N ( c ( t t ) ) , the value of

g(Dsl(t l)y, y) converges to some value g(D1(tl)y, y).

We ~vil! show that the function s - g(DS1(t1)y, y ) is monotone increasing

for all s with tl < s < oo and is bounded from above by g(Da1( t l )y ,y) to

establish the existence of this limit To this end. assume that tl < r < s

Then by TJernma 12 12 we have


Thus for t E (tl, s) we obtain

where A is the Lagrange tensor field along c satisfying A ( t l ) = 0; A1(t l ) = E,

arid Y ( t ) is the parallel vector field along c with Y ( t l ) = y. Thus for t with

t l < t < T , lt follows that

is given by

Letting t --+ t l and using Y ( t l ) = y and A1(tl) = 27, we then have

u ( o s l ( t ~ ) a 9 ) - g(~ . ' ( t i ) y , 9 ) = g ( [ l g ( ~ * ~ ) - l ( T ) d i ] ti)) Y ( ~ I ) . 1 Since Y is parallel along c. it may be checked by choosing an orthonormal

basis of parallel fields for VL(c) that

Since (A*A)-' = A-~A*-' , this may be written as

which must be positive because (A*)-"(T)Y(T) is a spacelike vector In N(c(

for each T E [r, s]. Thus

g(D8"tl)XA v ) - g(DT1(tl1v. y) > 0,

and the map s -+ g(DS1(tl)y, y) is monotone for all s > t l as required.

12.2 GENERIC AND TIMELIKE CONVERGEKCE CONDITIONS 441

We now show that g(DS1(tl)y, y) < g(Dal(t l)y, y) for all s > t : and any

y E N(c( t l ) ) . Again let Y be the unique pardlei field along c with Y( t l ) = y.

Let J be the piecewise smooth Jacobi field along c / [a. s] given by

DG(t)Y ( t ) for a s t < t l . J ( t ) =

Ds ( t ) Y ( t ) fort1 < t < s.

Also let J, = J 1 [a, t l ] and J, = J l [ t l .s] . Then J ( a ) = J ( s ) = 0 , and J is

well-defined at t = t l since D,(tl) = D,(tl) = E. Using the index form I for

cI [a. s] given in Definition 10.4 of Section 10.1, we obtain

where we have used formula (10.2) of Definition 10,4 and D, ( t ! ) = D, ( t l ) = E.

Since J ( a ) = J ( s ) = 0 and c has no conjugate points in [a, cc). we have

I ( J , J) < 0 by Theorem 10.22. Thus

for all s > t l ; we conclude that the self-adjoint tensor D1(t l ) = lim DS1(t l ) s- -+a

exists.

Now we define D(t ) by setting D(t ) equal to the unique Jacob! tensor along c

which satisfies D(t1) = E and D1(t l ) = lim,,, D,'(tl). Since D ( t ) and D,(t)

both satisfy the differential equation A" + RA = 0 and the initial conditions

of D, approach the initial conditions of D a s s --t w, it follows that D ( t ) =

lim,,, D,(t) and D1(t) = lim,,, DS1(t ) for all t E [a, cc). This inlplies that

the limit D(t) of the Lagrange tensors D,(t) must also be a L- &grange :ensor.

The last statement of the lemma now follows using the representation

w(t) = ~ ( t ) J ~ - ( A * A ) - ~ ( ~ ) d7

and the fact that (A*A)-I ( t ) is a positive definite self-ad~o~nt tensor field for

a l l t> t l . El


Now divide the Lagrange tensors with A(t1) = E along a complete timelike

geodesic c : (-m, 400) -+ ( M , g ) with Ric(cl, c') 2 0 and R( . . c'(tl))c1(tI) # 0 for some tl E R into two classes L+ and L- as follows [cf. Bolts (1977. p. 36).

Hawking and Ellis (1973, p. 98)], Put

L+ = (A : A is a Lagrange tensor with A(tl) = E and

q t l ) = tr(A'(t1)) Z 0)

and

L- = ( A : A is a Lagrange tengor with A(tl) = E and

B(tl) = tr(A1(tl)) 5 0)

Lemma 12.14. Let c : R -+ (34, g) be a complete timel~ke geodesic such

that Ec(cl , c') 2 0 and R( . , c'(t1))c1(tl) # 0 for some t l E R. Then each

A E L- satisfies detA(t) = 0 for sorrle t > t l , and each A E L+ satisfies

detA(t) = 0 for some t < t i .

Proof if A 6 L-, then B(tl) = tr(AJ(t~)A-'(ti)) = tr(A1(ti)) < 0. Using

the vorticity-free Raychaudhuri equation (12.2) for timelike geodesics with

Ric(ct, c') 2 0 and tr(02) 2 0, we find #( t ) < 0 for all t. Thus B(t) < 0 for

all t 1 t l . If 6(to) < 0 for some to > t i , then the result for A fo l lo~~s from

Proposition 12.9. Assume therefore that B(t) = 0 for t > tl. This implies

B1(t) = 0 for t 2 t l which yields tr(a2) = 0. Hence a = 0 for t > tl since

cr is self-adjoint. CTsing B = 0 and the self-adjointness of B, ure thus have

B = o = 0 which by equation (12.1) implies that = -B2 - B' = 0 for

t 2 t l . in contradiction to R(tl) $ 0.

If A E L+, the proof is similar. U

We now come to the

Proof of Proposztzon 12.10. Let c . R -+ (M,g) be a complete timelike

geodesic with Ric(ci(t), c1(t)) > 0 for all t E R and with R( - ,c'(tl))cl(tl) # 0

for some t1 E R. Suppose that c has no conjugate points. Then let D =

lirn,,, D, be the Lagrange tensor field on Vi(c) with D(tl) = E constructed


in Lemma 12.13 Since c I [tl, m) has no conjugate polnts D(t) is nonsingular

for all t 2 t l . Thus D $ L- by Lemma 12.14. Hence D E L- and moreover,

tr D1(tl) > 0 as D (t L-. Since Di(tlf = lirn,,, DSi(tl) there exists an

s > tl such that tr(DBfjtl)) > 0. Hence by Lemma 12.14, there exist a tz < tl and a nonzero tangent vector v E iV(c(t2)) such that D,(tz)(v) = 0. Recall

also from the proof of Lemma 12.12 that D,(s) = 0 but DS1(s) = -(A*)-'(s)

is nonsingular, Therefore, ~f we let Y E vi(c) be the unique parallel field

dong c with Y(t2) = 'u, then J = DD,(Y) is a nontrivial Jacobi field along c

with Y( t2 ) = Y(s) = 0, In contradiction.

Corolary 12.15. Let ( M , g ) be a space-time of dimension n 2 2 which

satisfies the timelike convergence condition and the generic condizion. Then

each timelike geodesic of (M,g) 1s either incomplete or e!se has a pair of con-

jugate points.

We now consider the existence of conjugate points on null geodesics. The

methods and results for null geodesics are much the same as for timelike

geodesics except that it is now necessary to assume that dimitl 2 3 since

dim G(P) = dim M - 2 and null geodesics in two-dimensional space-times are

free of conjugate points First, using the vorticity-free Fhychaudhuri equation

(12.5) for null geodesics, the following analogue of Proposition 12.9 may be

established with the same type of reasoning in the timel~ke case

Proposition 12.16. Let (M. g) be an arbitrary space-time of dimension

7~ > 3. Suppose that ,b : J -+ ( M . g) 1s an inexteridible null geodesic satisfying

Ric(pl(t), pl(t)) > 0 for all t 6 J. Let 2 be a Lagrange tensor field dong P such 1

that the expansion 8(t) = t rV1( t )A- ( t ) ) = /detz(t)]-'\detz(t)]' has the

negative [respectively, positive] vdue 8i = 8( t l ) at tl E J. Then detx( t ) = 0

for some t in the i n t e r d from t l to tl - (n- 2)/81 [respectrvely. some t in the

interval from tl - (n - 2)/iJ1 to ti] provided that t 5 J.

The null analogue of Proposition 12.10 may also be established for all space- - - - times of dimension n 2 3 using A, B, 8, Zr, etc. in place of the corresponding

A, B, 8. a, etc. used in the proof for timelike geodesics.

444 12 SINGULANTIES

Proposition 12.17. Let /3 . R 4 (M,g) be a complete null geodesic with

Ric(/?'(t), ,f3'(t)) 2 0 for all i E IW. If dimM 2 3 and if z(. , p(t))P'(t) :

G(P(t)) -, G(P(tf) is nonzero for some tl E R, then ,f3 has a pmr of conjugate

points.

Combining this result with Corollary 12.15, we obtain tine following theo-

rem.

Theorem 12.18. Let ( IM, g) be a space-time of dimension n 2 3 which

satisfies the timelike convergence cond~tion and the generic condition. Then

each nonspacelike geodesic in (1W, g) is e~ther ~ncomplete or else has a pair of

conjugate points. Thus every nonspacelike geodesic in (M, g) without conju-

gate points is incomplete.

The material presented in this section may also be treated within the frame

work of conjugate points and oscillation theory in ordinary differential equa

tions [cf. Tipler (1977d, 1978), Chicone and Ehrlich (1980), Ehrlich and Kim

(1994)l. In this approach, the Raychaudhuri equation is transformed by a

change of variables to the differential equation

zU(t) + F(t)x(t) = 0

with

F( t ) = A[Ric(y1(t), yl(t)) + 2a2(t)j

where rn = n - 1 if y is timelike and rn = n - 2 if y is null.

12.3 Focal Points

The concept of a conjugate point along a geodesic can be generalized to t

notion of a focal point of a submanifold. Let H be a nondegenerate subrnani

of the space-time (M,g) . At each p E H the tangent space TFW may

naturally identified with the vectors of T,M which are tangent to H at

The normal space T k H consists of all vectors orthogonal to H at p. Sinc

H is nondegenerate, T k H n T,H = 10,) for all p E W . We -will denote t

exponential map restricted to the norma! bundle TiH by expi. Then

vector X E T k H is said to be a focal poznt of H if (expi), is singular at

12.3 FOCAL POINTS 445

The corresponding point expi(x) of M is said t,o be a focal point o j H along

the geodesic segment expi(tX). When W is a single point, then T,IH = T,M,

and a focal point is just an ordinary conjugate point.

Focal points may also be defined using Jacobi fields and the second funda-

mental form [cf. Bishop and Crittenden (1964, p. 225)j. This approach will

be used in this section following the treatment given in Bolts (1977). Jacobi

fields are used to measure the separation (or deviation) of nearby geodesics.

For example, when a point q 1s conjugate to p along a geodesic c, geodesics

which start a t p with initla1 tangent close to c' at p wlll tend to focus a t q up to

second order. They need not actually pass through q but must pass close to q.

In studying submanifolds one may take a congruence of geodesics orthogonal

to the submanifold and use Jacobi fields to measure the separation of geodesics

in this congruence. If p is a focal point along a geodesic c which is orthogonal

to the submanifold H, then some geodesics close to c and orthogonal to H

tend to focus at p. This is illustrated for the Euclidean plane with the usual

positive definite metric in Figure 12.1 and for Lorentzian manifolds in Figure

In Section 3.5 we defined the second fundamental form S, : TpW x T p H -t R in the direction n, the second fundamental form S T ~ U x T,H x T,H -t R

and the second fundamental form operator L, : T,W - TpH (cf. Defini-

tion 3.48). Recall that, S(n, x, y) = S,(x, y) = S,(y, x) and g(L,(x), y) =

S,(z, y) = g(VXY, n) for n E T i H and z, y E TFH. where X and Y are local

vector field extensions of z, y.

In this section we will primarily be concerned w ~ t h the operator L, : TpH -$

T,W. Note that a vector field 7 which is orthogonal to H at all points of H

defines a (1,1) tensor field Lq on H. We will first consider spacelike hypersur-

ces. If the timelike normal vector field 7j on H sat~sfies g(g,7) = -1, then

we may calculate L, as follows.

Lemma 12.19. Let H be a spacelike hypersurface with unit timelike nor-

mal field 7. Ifx is a vector tangent to W , then L,(x) = -V,v

Pmoj. Since g ( ~ , 9 ) = -1, we have 0 = X(~(V, 7)) = 2g(0,9, 7 ) which

bows that V,q is tangent to W . Now if Y is any vector field tangent to

12 SINGCLARITIES 1

FIGURE 12.1. A curve in the Euclidean plane has its centers of

curvature as its focal points. The osculating circle to the curve y

at t = t o is shown. The segment from $to) to the center p of the

osculating circle contains x In its mterior. The point y lies beyond

p on the ray from ?( to ) through p. For some ~nterval to - €1 < t < t o + €1, the closest point of y to x is ?(to). On the other hand. for

some interval to - €2 < t < t o T €2, the farthest pomt of y(t) from

y is ?(to). Furthennore, the straight lines which are orthogonal to

y near $to) tend to focus at p.

H, then g ( ~ , Y ) = 0. Thus O = x(g(q,Y)) = g(V,q, Y ) $ g(7, V,Y

that g(V,Y, q) = g(-V,q, Y ) . Consequently, g(L,(x), Y) = g(V,Y, q )

g(-V,q,Y). Since Y was arbitrary, the result follows O

Given a unit normal field 77 for the spacelike hypersurface N, the collectio

of unit speed timelike g ~ d ~ i c s orthogonal to W with initial direction


at q E H determines a congruence of timelike geodesics. Let c he a timelike

geodesic of this congruence intersecting N at q. and let J denote the variation

vector field along c of a one-parameter subfam~ly of the congruence. Then

J is a Jacob1 field which measures the rate of separation of geodesics in the

one-parameter subfamily from c. Since the geodesics of the congruence are

all orthogonal to N, it may be shown using Lemma 1219 that J satisfies the

initial condition

This suggests the following definition of a focal point to a spacelike hypersur-

face in terms of Jacobi fields.

Definition 12.20. (Focal Point on a Timelike Geodesic) Let c be a time-

like geodesic which is orthogonal to the spacelike hypersurfate N at q. A point

p on c is said to be a Jocal po~nt of H along c if there is a nontrivial Jacobi

field J along c such that J is orthogonal to c', vanishes at p, and satisfies

J ' = -L,J at q,

Suppose that A is a Jacob1 tensor along the timelike geodesic c which sat-

isfies A = E and A' = -L A = -L, at the point q, where c intersects the r )

epacelike Iiypersurface N. Then prior to the first focal point, every Jacob1 field

J orthogonal to c which satisfies J' = -L, J at q may be expressed as J = AY

where Y = Y( t ) is a parallel vector field along c whlch is orthogonal to c.

Since there are n - 1 linearly independent parallel vector fields orthogonal to

c, there is an (n - 1)-dimensional vector space of Jacobi fields along c which

satisfy J' = -L,J at q. We now show that such a Jacob1 tensor A satisfying

A = E and A' = -L, at q is in fact a lagrange tensor field.

Lernrna 12.21. Suppose that A is a Jacobi teilsor field along the timelike

geodesic c . Let c be orthogonal co W at ti, and let ZV be the second funds-

mental form operator on H . If A(t l ) = E and A1(t1) = -L,A(tl), then A is

a Lagrange tensor field.

P~oof . The second fundamental form S, at 77 E T'H 1s symmetric, which

implies that L, is self-adjoint at q since g(L , (x ) , y) = SV( zp g) = S7(g; x ) =

12 SINGULARITIES 12.3 FOCAL POINTS 449

g(L,(?~),s). Thus at t l we have that At(tl) = -L,A(tl) = -L, is self-

adjoint. Hence AS'(tl) = A1(tlj. Using A(tl j = A*(tl) = E, it follows that

A1((t)*A(tl) = A*'(t1).4(tl) = A* (tl)A1jtl). Thus A is a Lagrange tensor field

as required. C

The tensor B = A'A-', expansion 0, and shear o of Lagrange tensors

A satisfying the conditions of Lemma 12.21 may be defined as In Sectlon

12.1, Definition 12.2. As before, the expansion 0 of the iagrange tensor A

along c satisfies the vorticity-free hychaudhuri equation (12.2) for timelike

geodesics. We now establish the following analogue of Proposition 12.9 for

spacelike hypersurfaces.

Proposit ion 12.22. Let (M, g) be an arbitrary space-time of di~nension

n 2 2. Stlppose that c : J --+ ( M , g) is an inextcndible timelike geodesic which

satisfies Ric(cl(t), c'(t)) 2 0 for all t E J and is orthogonal to the spacelike hy-

persurface H at q = c(tlf. If - tr(L,) has the negative [respectively, positive]

value 81 at q: then there is a focal point c ( t ) to W for t in thc interval from

tl to tl - (n - I)/&, [respectively, in the interval from tl - (n - l ) /d i to tl ]

provided that t E J.

Proof, By Lemma 12.21, the Jacobi tensor field A along c with initial con-

ditions A(tl) = E and A1(tl) = -L,(,) is a Lagrange tensor field. Also,

= 8(tl) = - tr(L). Hence by Proposition 12.9 the tensor A is singular on

the interval from t l to t l - (n- I)/@!. Thus the result follows from the remarks

following Definition 12.20.

Fbr the purpose of studying focal points to subman~folds. it will be helpful

to have the second variation formuia for the arc length functional at h a ~ d We

thus give a derivation of the first and second variation formulas for complete-

ness. Consider a piecewise smooth variation a : [a, b ] x (-€, 6) 4 ( M , g) of a

piecewise smooth timelike curve c : [a, b] -+ (M, g). Kerice a(t , 0) = c(t) for

all t E [a. b ] , and there is a finite partition a = to < t l < . . < tk-l < tk = b

such that a 1 [t,-l.t,] x (-6, E ) is a smooth variation of c 1 [t,-t, tE ] for each

i = 1.2 , . . . , k (cf. Definition 10.6). We will also assume that the neighboring

curves a, = a ( . , s ) : [a, b ] -+ (?M,g) are timelike for all s with -c < s < E

FIGURE 12.2. A spacelike submanifold H in a Lorentzian manifold

( M . g ) is shown. Were p is a focal point of H along the geodesic c.

The geodesic segment from p to q contains s in its interior, and 4~

lies beyond p on the geodesic c. All nonspacelike curves "close" to

c[q, x] which join a point of H near q to s have length less than or

equal to c[q, XI. On the other hand, the farthest point of H near q

to y is not q. There are points close to q on H which may be joined

to y by timelike curves longer than c[q, gj. Furthermore, there is at

least one curve y on H through q such that a family of geodesics

orthogonal to W with respect to the given Lorentzian metric and

starting on y near q tend to focus at p up to second order.


(cf. Lemma 10.7). As in Section 10.1, for t with t,...~ < t < t,, we define the

variation vector field V of a: along c by

V ( t ) = (a It,-I, t,] x (-6. E ) ) , - --n ---

Also, set

At , (Y1) = Yi(t:) - Y ' ( t t ) for i = 1,2,. . . , k - 1,

( Y = Y ) and At , (Y t ) = Yt ( t$ )

for any piecewise smooth vector field Y ( t ) along c which is smooth on each

subinterval t , ) .

As in Section 10.1, we will use L(s ) = L(cr,) to denote the length of the

curve t I-+ a ( t , sj. The fir& variation formula for the arc length functional may

then be derived as usual.

Proposition 12.23. Let c . [a, b ] -, (M,g) be a unit speed timelike curve

which is piecewise smooth. I f a: : [a, b ] x ( - 6 . 6 ) -* ( M , g) is a variation of c

through timelike curves, then

Pmof. If L, : (-E, E ) -+ R denotes the arc length runctlon or cu \t.-l,t.j

{s). then L(s) = ~ f = ~ L,(s) and

Thus


On the other hand, since

and

it follows that

Thus we have

which implies that

as required. C1

If c : [a, b ] - hf is a timelike geodesic, then c" = V,fcl = 0 and A,, (c') = O

for all z = 1,2,. . . , k - 1. Thus for a timellke geodesic, the first variation

formula simplifies as follo~.~s.

Corollary 12.24. If c : [a. b ] -, (&I, g ) is a unit speed timehke geodesic

segment and cu : [a, b ] x ( - E , E ) -+ (hd, g ) IS a variatian of c, chen

Now let H be a spacelike hypersurface and assume that c : /a, bi i M is

unit speed timelike curve with c(a) E H In studying focal points of the

manifold H , attention may be restricted to variations a + [a, b ] x ( - E , E ) -+

g) of c which start on H [i.e., a(a, s) E H for all s E (-E, E ) ] and which end

c(b) [i.e., a(b, s ) = c(b) for all s E ( - E , E ) ] . For these variations, V ( b ) = 0

nd V ( a ) is tangent to H at c(a). Proposition 12.23 then yields the following

rst variation formula:

452 12 SINGULAWTIES 12 3 FOCAL POINTS 453

Given a spacelike hypersurface H without boundary in (M,gf , fix a point miation o f c which is smooth on each s e ~ (t,-l, t,) x (-LC) for the part~tion

q E M - H. Consider the collect~on (possibly empty) of all timelike curv a = to < t l < . < tk = b of [a. b ] . Let V(t) = c r , d / d ~ l ( ~ , ~ ) denote the

which join some point of H to q. If this collection contains a longest curve variation vector field of a along c, and set

c : [a, b ] -+ (M.9). then c must be a smooth timeiike geodesic by the usual N ( t ) = V(t) + g(V(t),c'(t))c'(t) arguments that timehke geodesics locally maximize arc length and using the d d first variational formula to see that c has no corners. Thus assume that the + g a,-,a,- a, =.*%I ,,,, ( a. t ) $ 1 ,,,, ,

unit speed timelike geodesic c : [a, b] + (M, g ) is the longest curve from H to

with a(a , s) E H and a(b, s ) = q for all s with - E 5 s 5 E , then using V(b) =

and Corollary 12.24 we have k

+ ) : g ( ~ ( t * ) , At , (N1) ) - 9(V2?,as~, ct)jL. z=o

Lt(0) = g(V(a), c'(a)).

On the other hand, Lt(0) = 0 since c is of maximal length from H to q. Th -5 ( a*-,a*- ; ;)I [-:g ( v ~ / & (@* $1 . Ll* dt.

V ( a ) is orthogonal to cl(a). Since variations a as above may be constructed

with V(a) E T,(,)H arbitrary, it follows that c must be orthogonal to H

e(a). Thus we have obtained the following standard result. Note also that

is necessary for H to have no boundary in order to be sure that the extre

c is perpendicular to N at c(a).

Proposition 12.25. Let H be a spacelike hypersurface without bound

q = c ( b ) $ H which is of maximal length among all timelike curves from H t

q. Then c is a timelike geodesic segment, and c is orthogonal to H at c(a). 9(Va/asValdt~*a/as, a * a / a ) - 9(Va/ata*alas. oa/asa*a/at) [-g(a,d/dt, a,d/&);1/2

g(Valata*a/as, a,a/at)g(Va,dta*a/as, a*d /d t ) have discontinuitia in its derivative at the t-parameter values {tl , tz, . . . ,

[-g(a,d/&, a,d/&)J1/2g(cy,a/dt. cy,d/at) a t which cr may fail to be smooth. Thus the normal component N = a,d

for ( t , s ) E (t,-*, t,) x (-6. E) Furthermore, we have g(a,B/ds,a,d/bt)a,B/dt of V along c may also fail to be smooth at th

terms of N and V [cf. Bolts (1977, pp. 86-90), Hawking and Ellis

p. 108)].

Proposition 12.26. Let c : [a, b ] -, (M.g) be a unit speed tim

odesic segment, and let cr : [a. b ] x (-6, e) + (M, g) be a piece

since c is a geodesic and g(N, a,d/at)l(,,o) = 0 for all t E (taml, t,). Using

these last two calculations, we then obtain

Thus since

it follows from these calculations that


Substituting this expression for g(Valat~,d/ds, ValaBa,d/dt)l(,,o) in (12.8)

and using g(a,d/dt, ~ , a / d t ) j ~ , , ~ ) = -1, we obtain

a ~ o / a ~ v a / a ~ a : a, a*;) /lt ,oj - g ( v a / a t ~ . vaIBt-v) / ( t , O ) .

This yields

L,"(Oj =

a V~/~aVt?/na*- . 0 s a*:) 1 - g ( V B / C ~ N , vajal") ( i ,o)

(t 0)

Also

a

since

Since c(t) = a(t, 0) is a geodesic,

a a d a - at = ~g (Oalasa*z, *.at

! t , O )

Therefore,

The equations

456 12 SINGULARITIES 12.3 FOCAL POINTS 457

and V(t ) = a,B/8sI(t,o) together with L = xFE1 L, yield

since V = nT - g(V, cl)c'. Thus

where

Proposition 12.26 has the following consequence.

Corollary 12.27. Let H be a spacelike hypersurface, and assume tl

c : [a, b] -+ (M. g) is a unit speed timelike geodesic which is orthogonal to

at p = c(a) E H Suppose that a : [a, bj x ( - E , E ) 4 (M,g) is a variation

c such that ~ ( n , s ) 6 N and a(b, s) = q = c(b) for all s with - E < s 5 E .

V = a,d/dsj;t,o) and N = V + g(V , cl)c', then

where L,, is the second fundamental form operator of H at p.

Proof. In view of Proposition 12.26 and the equation

it is only necessary to show that -g(Va/asV, c1)j: is equal to g(L,,(Ar), N). To this end, we first note that a(b, s) = q implies that a , B / d ~ / ( ~ , , ) = O for

all s which yields g(Vala,V.cl)lb = 0. Also a (a , s ) E H for all s with - E < s < E implies that a,d/8sj(,,,) is tangential to N for all s , and hence ,&'(a) =

ry,d/f/a~l~,,~~. Let y(s) = a ( a , s) for all s with - E < s < 6. Extend the vector

N(a) E T p H to a local vector field X along N with X o - ( s ) = a*df8s/(,,,)

for all s with - E < s < E . Then

by Definition 3.48. Also let 77 be a unit normal field to H near p with 71(p) =

cl(a). Then we have

But g(a,b/as,.rl o a)l(,,,) = 0 for all s since a,a/bs)(,,,) is tangential to H . Thus we obtain

as required. Here we have used the fact that since XIp = a,d/dslf,,o),

XI,(g(X, 77)) may be calculated as


In view of Corollary 12.27, the index IH(V, V ) of a vector field V along a

t~melike geodesic orthogonal to a spacelike hypersurface should be defined as

follows [cf. Bolts (1977, p. 94)].

Definition 12.28. (Spacelake Nypersurjace Index Form) Let c : [a, b ] -i

(M.9) be a unit speed timelike geodesic which is orthogonal to a spacehke

hypersurface N at c(a). Assume that Z is a piecewise smooth vector fie1

along c which is orthogonal to c. if Z(a) # 0 and Z(b) = 0, then the zndex

Z *zth respect to H is given by

TH ( z , z ) = I ( z . z ) + g(Lcl(Z), Z)la

where

It-1

where a partition it,) of (a, b] is chosen such that Z is differentiable except at

the t,'s.

Let a : [a, b ] x ( - E , E ) -+ ( M , g) be a variation of the timelike geodesic c, and

assume the variation vector field V = a , d / d ~ j ( ~ , ~ ) satisfies the conditions of

Definition 12.28. Then equation (10.4) of Sect~on 10.1 together with Coroliary

12.27 yield L1'(0) = .fH(V,v). This may also be written

L* * (V) = IH (br, V) ,

thogonai to a spacehke hypersurface H falls to maximize arc !ength to N a

the first focal point. Recall that V L ( c ) consists of plecewlse smooth vect

fieids along c which are orthogonal to c.

Proposition 12.29. Let c . la, b ] -+ M be a unit speed timelike geo


and Z(b) = 0 such that I H ( Z , Z ) > 0. Consequentl~; there are tlmelike curves

from H to c(b) which are longer than c.

Proof, By hypothesis there exists a nontrivial Jacobi field JI along c with

J1 orthogonal to c. Jl ( t k ) = 0 and Jli(a) = -L,,(,) Jl (a ) . Define a piecewise

smooth Jacobi field J along c hy

Since Atb(J1) # 0 , we may construct a smooth vector field V orthogonal to

c such that V'(a) = V ( a ) = V(b) = 0 and g(V(tk), A t = ( J f ) ) = -1. Defiae a

vector field Z in V L ( c ) by

Then ZJ(a) = -Lct(,)Z(a), and the index I N ( Z . 2) is given by

IH(Z, Z ) = I ( Z , Z ) +g(Lc/(Z) , Z)ja

= I ( Z , Z ) + g ( ~ , l ( r - l ~ - r V ) , r - ' ~ - rV)la

= 1(Z, 2) + r-'g(~,t J , J ) 1, = I ( Z , Z ) + r - 2 g ( - ~ ' , J ) I ~ = T - ~ I ( J , J ) + r21(V, V ) - 2i(J. V ) + T - ~ ~ ( - J' J ) 1, = r 2 i ( v V ) - 2I(J, V ) .

Here we have used

Since J is a piecewise smooth Jacobi field, equation (10.4) of Section 10.1

implies that

I ( J , V ) = g(V(tk) , &,(J1)) = -1.

Consequently,

I H ( Z , Z ) = T ~ I ( V . V ] + 2


which shows that for sufficiently small T # 0 the index satisfies

IH(Z, Z) > 0.

This last inequality and the condition Z1(a) = -LC,(,) (Z(a) ) imply that there

exist small variations of c with varlation vector field Z which join W to c(b)

and have length greater than c. Cl

We now turn our attention to focal points along null geodesics which are

orthogonal to (n - 2)-dimensional spacelike submanifolds. If W is a spacelike

(n - 2)-dimensional submanifold, then the induced metric on TpH is positive

definite and the induced metric on T;H is a two-dimensional Minkowski metric

for each p E H . Thus there are exactly two null lines through the origin in

T ~ H . Since the time orientation of (M, g) induces a time-orientation on T ~ H , there are thus two well-defined future null directions in T i H for each p E H.

Locally, we may then choose a smooth pseudo-orthonormd basis of vector

fields El, E2,. . , E, on N such that En-1 and En are future directed

null vectors in T k H for p E H. That is,

9(Et, E J ) = 6,, i f l S z , j < n - 2 ,

g(E*. En-I) = g ( E , En) = 0 i f I < i < n - 2 ,

9(En, En) = g(En-1, En-1) = 0,

and g(13n,En-l)=-1.

The null vector fields E,-l and En defined locally on N give rise to second

fundamental forms LE,-& and LE,, respectively, which are locally defined

(I, 1) tensor fields on H.

If p : /a. b ] --, ( M , g ) is a future directed null geodesic with ,b"(a) = E

[or @(a) = En-* (p)] at p E H, the tangent vectors El ( p ) , Ez(p), . . . . En ( p

p may then be parallel translated along P to give a pseudr>-orthonormal b

along /3 whlch will also be denoted by El, E2, . . . , En. The set of vectors no

to ,BJ(t) is thus the space N(P( t ) ) spanned by El, E2,. . . , En-2, P', and we

form the quotier~r, space G(P(t)) = N(B( t ) ) / [p( t ) ] with corresponding

bundle G(P) as in Section 10.3. If 7(- : N(P) -+ G(P) denotes the pr


map, then r T ~ H : TpH -+ G(P(a)) is a vector space isomorphism. Hence we

may project the second fundamental form operator LEs . T,W --+ T,H to an

operator ZE, : G(P(a)) + G(F(a)) by setting

- LE, = n o LE, (T / TpN)-I for 2 = n - 1, n

Let a : [a, b] x ( - E , E ) --, ( M , g) be a variation of the null geodesic ,O such that

a(a, s) 6 H for all s with -E < s < E . If V = a*d/ds and W = a,6/&, we

will also require that W(a , s ) = E,(a(a,s)) for all s with - E < s < E. Thus

the neighboring curve a ( . , s) starts at a(a, s) on H perpendicular to W and

has as initial direction the null vector En(a(a, s)) for all s.

We now calculate V 1 ( a ) .

Lemma 12.30. Let V = a,d/ds be tangential to N for f = n and s

arbitrary as above. Then V1(a) = -Lp(a) (V(a) ) + XP'(a) for some constant

X E R.

Prooj. Using [V. W ] = [cu,d/ds, a,b/dt] = cu, [a/&, d/6t] = 0, we have

VVW = VwV = V p V at t = a. s = 0.

On the other hand, W(a . s) = E?,(a(a, s ) ) and g(E,,. En) = 0 yield 0 =

V(g(W, W ) ) = 2g(VvW, W ) = 2g(DwV, W ) = 2g(Vp1V, P') at t = a. s = 0.

Thus

Vp(,,V E N(O(a)) = T p H 9 [P1(a)].

Since Lp,(,)(V(a)) E Tp(,)H and 'CiP(,)V E N(,B(a)), ~t thus suffices to

ow that

for every y E T9(,]H to establish the result. To calculate g(Lai( , ) (V(a)) .~) ,

extend y to a local vector field Y along the curve s -+ @(a. s) that 1s tangent

462 12 SINGULAWTIES

to N Then we have

9(Lg~(a)(V(a)) , l l ! = 9 ( ~ V Y I ( , , , ) ? B ' ( ~ ) )

= 9(VvY . Wlj(,,o)

- - a. a/ (dY. W ) ) - g!Y, vv W ) l b 0 ) 8.3 (a,,)

= 0 - g(Y. VV) I ( a , o ) = -.Y(Y~ v~' (a)V)

where we have used a*3/Dbs\(,,o)(g(Y, W ) ) = 0 since g(Y, W) = g(Y. En) = 0

along the curve s + a(a , s ) . O

If V is to be a Jacobi field measuring the separation of a congruence of nu1

geodesics perpendicular to H , then the last result implies that the vector class - V = n(V) along ,l3 should satisfy the Initial condition

- I V (a) = -Z,p(,)V(a)

at p = P(a) E H. Here zp, = x o Lp, o (n / T,H)-' as above This rnotivat

the following definition of focal point along a null geodeslc perpendicular to

{cf. Hawking and Ellis (1973, p. 102), Bolts (1977, p. 58)].

Definition 12.31. (Focal Poznt on a Null Geodesgc) Let H be a spacelik

submanifold of dimension (n - 2) , and suppose that 13 . [a, b ] - (M, g) is

null geodesic orthogonal to H at p = P ( a ) Then to E (a , b ] is said to be

focal poznt of N along ,l3 if there is a nontrivial smooth Jacobi class 7 in G(P

such that ; f t ( a ) = -Zpt(,) J ( a ) and ; f ( t o ) = [BJ( to)] .

We noted above that a t~melike geodesic orthogonal to a spacellke hy

surface falls to be the longest nonspacehke curve to the hypersurface a

the first focal point (cf. Proposition 12.29). A simllar result holds for a n

geodesic which is orthogonal to an (n - 2)-dimensional spacel~ke submanlfol

[cf. Hawking and Ellis (1973, p. 116), Bdlts (1977, p. 123)]. We sta

result as the following propohition.


Proposition 12.32. Let N be a spacelike submanifofd of ( M , g ) of dimen-

sion n - 2, and let p : [a, b ] -+ ( M I g) be a null geodesic orthogonal a t $(a) to

H . If to E (a, b) is a focal point of H along P, then there is a timelike curve

from N to P(b). Thus P does not maximize the distance to N after the first

focal point.

A simple example of a focal point is shown in Figure 12.3.

Using the same type of reasoning as in Proposition 12.22, the following

result may also be established.

Proposition 12.33. Let (hf, g) be a space-time of dimension n 2 3, and

let H be a spacelike submanifold of dimension 12 - 2. Suppose that P : J -+

(M. g) is an inextcndible null geodesic which is orthogonal to H at p = P(tl )

and satisfies the curvature condition Ric(p1(t).,8'(t)) 2 0 for all t E J. If

- tr(Lp.(t,)) has the negative Ircspectively. positive] value fI1 at p, then there is

a focal point to to H alongp in the parameter interval from t l to t l - (n- 2)/01

[respectively, in the parameter interval from t l - (n - 2)/Q1 to t l ] provided

that to E J

A particularly important case occurs when H is a compact spacelike (n-2)-

dimensional submanifold which satisfies the condition (tr LEn) (tr LE,-l) > 0

at each point [cf. Hawking and Ellis (1973. p. 262)) Here if (el, e2,. . . , en-z)

is an orthonormal basis for T,H, then t r LEn may be calculated as

Definition 12.34. (Clo.qea' lhpped Surface) Suppose that N is a com-

pact spacelike submanifold withoiit boundary of ( M , g ) of dimension n - 2.

Let En and En-I be linearly independent future directed null vector fields on

N as above. Assume that L1 and L2 are the second fundamental forms on H

corresponding to En and En-', respectively. Then N is said to be a closed

ce if t r LI and trL2 are bosh elther always pos~tive or always

FIGURE 12.3. Let (M, g ) be three-dimensional Minkowski space-

time with the usual metric ds2 = -dt2+dz2+dy2, Let H be a circle

of radius a in the z-y plane. Then H is a spacelike submanifold

of codimension two. For each q E N , there are exactly two future

directed null geodesics and ,B2 through q which are orthogonal

to H . The focal point of M along Dl is pl = (Q,O, a), and the focal

point along /?z is pz = (0, 0, -a). The point y1 lies on beyond p l .

All of H - {q) is contained in the chronological paqt of g1. Thus

there are timelike curves from points on H arbitrarily close to q to

yl, and pl[q. 5111 does not realize the distance from N to yl.


A related concept is that of a trapped set jcf. Hawking and Penrose (1970

pp. 534-537):. Recall that the future horisrnos of a bet A is defined by E + ( A ) =

J+ ( A ) - I+ (A) .

Definition 12.35. (Trapped St) A nonempty achronal set A is said tc

befutz~re [respectively, past] trapped if E+(A) [respectively, E-(A)] is compact

A trapped set is a set which is either future trapped or past trapped.

In general, a closed trapped surface need not be a trapped set and vice versa

However, in Proposition 12.45 we will show that under certaln condlt~ons, tht

existence of a closed trapped surface implies either null incompleteness or thf

existence of a trapped set. In establishing Proposition 12.45. we will need tht

following corollary to Proposition 12.33.

Corollary 12.36. Let ( M , g ) be a space-tjmo of dimension n 2 3 whicf

satisfies the condition Ric(v,v) > 0 for all null vectors u E TM. If (M,g:

contains a closed trapped surface H, then either ( I ) or (2) or both is true:

(1) At least one of the sets E f ( H ) or E - ( H ) i s compact.

(2) ( M , g) is nu12 incomplete.

Proof. Assume that ( M , g) is ilull complete and that t r L1 > 0 and tr L2 > G for all q E W . Consider all future directed null geodesics which start at some

point of H and have initial direction either or E, at this point. Each

such geodesic contains a geodesic segment which goes from a point q 6 H to

a first focal point p to W. Using Proposition 12.33 and the cornpactness of H ,

it follows that the union of all such null geodesic segments from I1 to a focal

point is contained in a compact set K consisting of null geodesic segrnents

starting on N. Now if T E E i ( H ) , then T can be joined to H by a past

directed null geodesic but not by a past directed timelike curve. Thus r E K.

Hence E + ( H ) 5 K. To show that E i ( H ) is closed let {x,) be s sequence of

points of E f ( H ) . This sequence has a limit point x E K From the definition

of K we have x E J+(H) . i f .G E I f ( N ) , then the open set I + ( H ) must

contain some elements of the sequence {z,), c~ntradicting x, E E + ( N ) for

all n. Thus x 4 I f ( H ) which yields z E E+(H) . This shows that E + ( H ) is a

closed subset of the compact set K and hence is compact.

-3 12 SINGULARITIES d

FIGURE 12.4. The future Cauchy development D+(S) of S con-

sists of all points p such that every past inextendible nonspacelike

curve through p intersects S .

If we assume that (AM, g) is null complete and that t r L1 < Q and tr L2 < 0

for all q E N, then similar arguments show E - ( H ) is compact This establishes

the corollary.

We now define the Cauchy developments of a closed subset S of (

[cf. Hawking and Ellis (1973, pp. 201-205))

Definition 12.37. (Cazlchy Development) Let S be a closed subset o

(M,g) . The future [respectively, past] Cauchy development Di ( S ) [respec

tively, D - ( S ) ] consists of all points p E M such that every past [respective

future] inextendible nonspacelike curwe through p intersects S. The Cauc

deuelopment of S is D ( S ) = D t ( S ) J D - ( S ) (cf. Figure 12.4).

Closed achronal sets have played an important role in causality the

and singularity theory in general relativity. They have the follo

erty [cf. Hawking and Ellis (1973, pp. 209, 268), Hawking and Penr

p 537)).

Proposit ion 12.38. If S is a closed achronal set in the space-time (

then In t (D(S ) ) = D ( S ) - d D ( S ) is globally hyperbolic.

12.4 THE EXISTENCE OF SINGULARITIES 467

12.4 The Existence of Singularities

In this section we give the proofs of several singularity theorems in general

relativity. In part~cular, we prove the maln theorem of Hawking and Penrose

(1970, p. 533). Our approach is somewhat different from Hawking and Penrose

(1970) in that we show ( M , g) is causally disconnected if :t contains a trapped

set and then apply Theorem 6.3 of Beem and Ehrlich (1979a, p. 172).

The basic technique in proving nonspacelike incornpletenws is to first use

physical or geometric assumptions on (M, g ) to construct an inextendible non-

spacelike geodesic which is maximal and h ~ n c e contains no conjugate points.

If ( M , g ) has dimension n > 3 and satisfies the generic condition and time-

like convergence condition, this geodesic must then be Incomplete by Theorem

12.18.

We first show that a chronological space-time with a sufficient number of

conjugate points is strongly causal [cf Hawking and Penrose (1970, p 536),

Lerner (1972, p 41)]

Proposit ion 12.39. If (M, g ) is a chmnological space-time such that each

inextendible null geodesic has apair of conjugatepoints, then ( M , g ) is strongly

causal.

Proof. Assume that strong causality falls at p E M . Let U be a convex

normal neighborhood of p, and let Vk C U be a sequence of neighborhoods

which converge top. Since strong causality fails at p, for each k there is a future

directed nonspacelike curve yk which starts in V', leaves U , and returns to

Vk. Using Proposition 3.31, one may obtain an inextendible nonspacel~ke limit

curve y through p of the sequence ( y k ) . No two points of y are chronologically

elated since otherwise one could obtain a closed tirrlelike curve, contradicting

he fact that (1%, g) is chronological. Thus y is a null geodesic, This yields a

contradiction since by hypothesis each null geodesic has conjugate po~nts and

thus contains points which may be joined by timelike curves. O

Proposition 12.39 and Theorem 12 18 then imply the followifig result.

468 12 SINGULARlTIES

Proposit ion 12.40. Let ( M , g) be a &rono2ogical space-time of dimen-

sion n 2: 3 which satisfies the generic condition and the timelike convergence

condition. Then (M, g) is either strongly causal or null incomplete.

We may now prove the following singularity theorem. The concept of a

causally disconnected space-time has been given in Section 8.3, Definition

8.11.

Theorem 12.41. Let ( M , g) be a chronological space-time of dimension

n 2 3 which is causally disconnected. If (1W.g) satisfies the generic condition

and the timelike convergence condition, then ( M . g) is nonspacelike incomplete.

Pmf, Assume all nonspacelike geodesics of ( M , g) are complete. By Propo-

sition 12.40. the space-tlme ( M , g ) is strongly causal, and by Theorem 12.18

every nonspacelike geodesic has conjugate points. On the other hand. Theo-

rem 8.13 yields the existence of an inextendible maximal nonspacelike geodesic.

But then this geodesic has no conjugate points, in contradiction. O

Recall that the fut.are [respectively, past] horisrnos of a subset S of ( M ,

is given by E f ( S ) = J+(S) - I f ( S ) [respectively, E - ( S ) = J - ( S ) - 1-(S)

The achronal set S is said to be fit'are trapped [respectively, past trapped]

E+(S) [respectively, E - ( S ) ] is compact. We now give conditions under which

the existence of a trapped set irnpl~es causal disconnection.

Proposition 12-42. Let ( M , g ) be a chronological spacetime of dimension

n 2 3 such that each inextendible null geodesic has a pair of conjugate points.

If ( M , g) contains a future [respectively, past] trapped set S , then ( M . g ) is

causally disconnected by E f (S) [respectively, E- (S)]

Proof First, Proposition 12 39 shows that ( M , g) is strongly causal. If S

assumed to be fi~ture trapped, then Corollary 8.16 yields a frlture inextendibl

timelike curve y in the future Cauchy development D f ( E f (S)). Extend

to a future and past inextendible timelike curve in ( M , g ) , still denoted by

Then y Intersects the achronal set E f ( S ) in exactly one point r. As in

proof of Proposition 8.18, we choose two sequences ( p , ) and (q,) on 7 w

diverge to infinity and satisfy p, << T < qg, for each n. To show that {p


(q,). and E 4 ( S ) causally disconnect ( M , g ) , we must show that for each n

every nompacelike curve X : [O, 11 --+ M with X(0) = p, and X(1) = q, meets

E+(S). Given A, extend X to a past inextendible curve 3; by traversing y uy

to p, and then traversing X from p, to q,. As q, E D f ( E + ( S ) ) , the curve - X must intersect E+(S) . Since 7 meets E+(S) only at T , it follows that > intersects E+(S) . This establishes the proposition if S is future trapped. P similar argument n a y be used if S is past trapped 0

Theorem 8.13, Proposition 12.39, and Proposition 12.42 now imply the mair

theorem of Hawking and Penrose (1970, p. 538).

Theorem 12.43. No spacetirne (M, y) of dirnensioi~ n 2 3 can satisfy a1

of the following three require~nents together:

(1) (M, g) contains no closed timelike curves.

(2) Every inextendible nonspacelike geodesic in (M. g) contains a pair o

conjugate points.

(3) There exists a future trapped or past trapped set S in (M, g).

This result of Hawking and Penrose implies the following result which k

more similar to Theorem 12.41.

Theorem 12.44. Let ( M , g ) be a chronologicd space-time of dimensior

n 2 3 which satisfies the generic condition and the timelike convergence condi

tion. If ( M , g ) contains a trapped set, then ( M . g) is nonspacelike incomplete

Recall that a closed trapped surface is a compact spacelike submanifold o

dimension n - 2 for which the trace of both null second fundamental forms L j

and La is either always positive o: always negative (cf Definition 12.34).

Proposition 12.45. Let ( M , g ) be a strongly ca7zsaJ space-time of dimen

sion n 2 3 which satisfies the condition %c(v. v) 2 0 for all null vector,

v E T M . I f (M,g) contains a closed trapped surface N, then at least one o

(1) or (2) is true:

(1) ( M , g) contains a trapped set.

(2 ) ( M , g) is null incomplete.


Proof. Assume that ( M , g ) IS null complete. Corollary 12.36 then lmpiies

that one of E f (H) or E - ( H ) is compact. We consider the case that E f ( H )

is compact and define S by S = E f ( H ) R H. We will show that S is a futxre

trapped set. Notice that S is achronal since E f (HI is achronal and S is

compact as the intersection of two compact sets.

Since E + ( H ) = J + ( H ) - I L ( H ) , the set S will be nonempty if and only

if H contains some points which are not in I+(N). But ~f H were contained

in I f ( H ) , there would be a finite cover of the compact set H by open sets

I+(pl ) . I+(p2), . . . , l+(p,) with all p, E N. I-Iowever, this would ~mply the

existence of a closed tirnelike curve in (M,g ) (cf. the proof of Proposition

3.10) which would contradict the strong causality of (M, g). Hence S # 0.

In order to show that S is a future trapped set, it is sufficient to prove that

E + ( S ) = E+(H). We will demonstrate this by showing that I A ( S ) = I+(H)

and J + ( S ) = J f ( H ) . To this end, cover the compact set H with a finite

number of open sets Ul. U2.. . . , Uk of (M, g) such that each U, is a convex

normal neighborhood and no nonspacelike curve which leaves U, ever returns.

Since H is a spacelike submanifold, we may also assume that each U, rt H is

achronal by choosing the U, sufficiently small. Clearly, I i ( S ) 5 I+(H) since

S E H. To show I f (H) 5 I f ( S ) , assume that q E I + ( H ) - I+(§) . Then there

exists pl E N with pi << q. Now pl E U,(l) n H for some index i (1 ) . Since

q $ I+(S ) , we have pl S and hence pl $ EC(H) . Thus there exists p2 E H

with p2 << pl. Since U,(l) n H is achronal, pa $ U,(l). Now pz E ? N for some i ( 2 ) # i(l). Again q $ I f ( S ) yields p2 $ E+(N) . Thus there exists

p~ E N with p3 << p2. Furthermore. by construction of the sets U, we have

p~ @ U,(l) U Thus p3 E f l H for some i ( 3 ) different from z ( 1 ) a

i (2) . Continuing in thm fashion, we obtain an infinite sequence pl, ~ 2 . ~ 3

in H with corresponding sets U,(*). U+). . . such that i ( j i ) # i(3z 31 # ja This contradicts the finiteness of the number of sets U, of the giv

cover. Hence I C ( S ) = I f ( H ) . I t remains to show that J+ (S) = J+ ( H

J 4 ( S ) C_ J t ( H ) as S C_ H. Thus assume that q E J+(H) - .T+(S). Th

q $ I + ( S ) = I+(H) , and hence there is a future directed null curve R

some point p E H to the point q. Since p E N and :, 6 I f ( H ) as p <


q $ If IN), we have p E Ef ( I f ) Thus p E E-(H) ? N = S which yleldi

q E Jf ( S ) , In contradiction. Thls shows that S 1s a future trapped set.

A similar argument shows that d (M.g) is nuil complete and ~f E - ( H ) i g

compact, then S = E - ( H ) n H IS a past trapped set. Thus the proposition ic

established. 5

It is possible for a trapped set to consist of just a single point. One way

this may arise is as follows [cf. Hawking and Penrose (1970, p. 5431, Hawking

and Ellis (1973. pp. 266-267)]. Let ( M , y) be a space-time of dimension n 2 2

which satisfies the curvature condition Ric(v, c) 2: 0 for all null vectors u E

TM. Suppose that there exists a point p such that on every future directed null

geodesic @ : [O,a) -t ( M , g ) with P(0) = p, the expansion 8 of the Lagrange

tensor fieid Z? on G(P) with x ( 0 ) = 0 and z ' ( 0 ) = E becomes negative for

some t i > 0. Intuitively, each future directed null geodesic emanating from T;

has a point /3(tI) in the future of p at which all future directed null geodesics

are reconverging. Thus provided that, the given null geodesic P can be extended

to the parameter value t l - (n -2 ) /8 ( t i ) . ,D has a future null conjugate point to

t = tl along P. Hence @(t) E I+(p) for all t > tl - ( n - 2) /B( t l ) . Since the set

of null directions at p is a compact set, it follows that E+(p) = J+(p) - I+(p)

is compact provided that ( M , g ) is null geodesically complete. Thus we have

obtained the following result.

Proposition 12.46. Let (M, g) be a space-time of dimension n 2 3 with

Ric(v. v) > 0 for all null vectors 2: E TM. Suppose that there exists a point

p E M such that on every future directed null geodesic P from p = P(O), the expansion $ of the Lagrange tensor field 2 on G(O) with ';i(O) = 0 and - I .4 ( 0 ) = E becomes negatlve for some t l > 0. Then at least one of (1) or (2)

holds:

(1 ) ( p ) is a trapped set, i.e., E+(p) = J+(p) - I+@) is compact;

( 2 ) (A{, gj is null incomplete

We now consider the case of z compact connected spacelike hypersurface S

without boundary in a space-time (M, g). If S is achronal, then E f ( S ) = S S is a trapped set. On the other hand, it may happen that S is not

472 12 SIKGULARiTIES

achronal. In fact, an example may easily be constructed of a compact spacelike

hypersurface S with S 2 I C ( S ) and hence E+(S) = @ (cf. Figure 12.5).

A compact spacelike hypersurface S which is not achronal may be used to

construct an achronal compact spacdike hypersurface in a covering manifold

of M [cf. Geroch (1970), Hawking (1967, p. 194), O'Neill (1983)j. Thus if

a space-time has a compact spacelie hypersurface, then there is a covering

manifold of the given space-time which has a trapped set. But in proving

the nonspacelike incompleten~s of (M, g), we may work with covering mani-

folds just as well as with (M , g) since ( M , g) is nonspacelike incomplete if and

only if each covering manifold of (M, g) equipped with the pullback metric IS

nonspacelike incomplete.

These observations on covering spaces together with Theorem 12.44, Propo-

sition 12.45, and Proposition 12.46 yield the following theorem [cf. Rawkin

and Penrose (1970. p. 544)].

Theorem 12.47. Let (M, g) be a chronological space-time of dimension

n 2 3 which satisfies the generic condition and the timelike convergence co

dition. Then the space-time (M, g) is nonspaceIike incomplete if any of th

following three conditions is satisfied:

(1) ( M , g) has a closed trapped surface.

(2) (M, g) has a point p such that each null geodesic s tar t~ng a t p is recon

verging somewhere in the future (or past) of p.

(3) (M, g) has a compact spacelike hypersurface.

Remark 12.48. Conditions (1) and (2) of Theorem 12.47 are reason

cosmological assumptions. Robertson-Walker space-tirnes with physical1

sonable energy momentturn tensors, positive energy density, and A = 0

closed trapped sxrfaces [cf. Hawkin

(1). There is some astronomical evi

p. 355)j.

12.5 Smooth Boundaries

In this section we consider the relationship between causal disconnec

nompacelike geodesic incompleteness, and points of the causal boundary

12.5 SA4OOTH BOUNDARIES

FIGURE 12.5. Let M = 5% S1 be given the Lorentzian metric

ds2 = dBI2 - dBz2. The set S = {(01, 0) : 1 9 ~ E S1) is a compact

spacelike submanifold of codimension one such that E+(S) = 0. Here S is not a trapped set because - it is not achronal. However.

(M, g) has a covering manifold ( M , i j ) which contains a trapped set

#? which is diffeomorphic to S.

474 12 SINGULARTTIES

(cf. Section 6.4) at which the boundary is differentiable. Many of the more im-

portant spacetimes studied in general relativity have causal boundaries which

are differentiable at a large number of points. For example, the differentiable

part of the causal boundary of Minkowski spacetime consists of JC and 3-

(cf. Figure 5.4). Since these sets correspond to null hypersmfaces, it is tllus

natural to call the points of J+ and 5- null boundary points. Penrose (1968)

has used conformal methods to study smooth boundary points of Minkowski

space-time and other space-times.

In general, we consider a space-time ( M , g) with causal boundary d,M and

let M* denote the causal completion A4 U DcM of (M,g). Placlng various

causality conditions on ( M , g ) (e.g., stable causality) ensures that this com-

pletion may be given a Hausdorff topology such that the original topology on

A 4 agrees with the topology induced on iM as a subset of Mycf . Hawking and

Ellis (1973, pp. 220-2211, Rube (1988). Szabados (1988), or Section 6.41

Assume that j j EcM and let U L = Ut( j j ) be a neighborhood of p in A4

Denote by (U, g) the metric g restricted to the set

A confomak representation of U8(ji) will be a space-time (M'. g') and a hom

omorphic embedding f : U* -t M' such that:

(1) f I U is a smooth map; and

(2) There is a smooth function R : U -+ R such that R > 0 and fig = f *gl

on U (Figure 12.6).

If the conformal representation f : U* -t MI maps U* to a smooth manifo

with boundary, then we will say that jj is a smooth bozlnda9-g poznt.

Definition 12.49. (Smooth Causal Boundary Point) Let U* (p) have

smooth conformal representation f : U* + M' such that f ( U ) is a smo

manifold with a smooth boundary d( f ( U ) ) in M'. Then the point

to be a smooth spacelike (respectively. null, tzmebzke) boundary poznt

corresponding boundary

hypersurface in (MI. 9').

a( f ( U ) ) is a spacelike (respectively, null,

12.5 SMOOTH BOUNDARIES 45

FIGURE 12.6. The space-time ( M , g ) has a causal boundary point

j?, and U*(B) is a neighborhood of p in the causal co~npletion M' =

MU a,M of (M. g). The homeomorphic embedding j : U * --+ M'

is a smooth conformal map on U = Um(p) n M .

If -y : [a, 5) --+ U is a curve in M such that ~ ( t ) --t p E M* - Li as t -+ b, ther

the curve y is said to have the boundary point ,5 E M* as an endpoint, Also i

$j is a smooth spacelike boundary point. then it is not hard to find a compact

set K in M such that all inextendible nonspacelike curves with one endpoint at

must intersect K In fact, K may be chosen as a compact, achronal spacelikg

ypersurface with boundary (cf. Figures 12.7 and 12.8) Furthermore, giver

any neighborhood U*(p) of j3 in M f . the compact set K may be chosen to lit

= u * ( p ) n M .

12 SINGULARITIES

Theorem 12.51. Let ( M . g) be a chronological space-time of dimension

n 2 3 which satisfies the generic condition and the timelike convergence con-

dition. If (M, g) has a smooth spacelike boundary point, then (M, g) is non-

spacelike incomplete. CHAPTER 13

Proof. This follows from Lemma 12.50 and Theorem 12.41. GRAVITATIONAL PLANE WAVE SPACCTIMES

Notice that if (141, g) has one smooth spacelike (respectively, null, timelike)

boundary point, then (M, g) has uncountably many smooth spacelike (respec-

tively, null, timelike) boundary points. For if jj 6 d,hf corresponds to the It is well known and already mentioned in earlier chapters that for general

point z E d( f ( U ) ) under the given conformal representation J : U* -+ M' of space-times, geodesic connectedness and gcodcsic completeness are not well Definition 12.49, then points y f L?(J(U)) close to x wili dso represent smooth

related. Indeed, in Figure 6.1 we have given an example of an elementary spacelike (respectively, null, timelike) boundary points in a,M under the given space-time which is geodesically complete yet have indicated in this figure a conformal representation and d(f (U)) has dimension n - 1 in M'. Hence M point q in I+(p) which is not joined to p by any geodesic whatsoever) let alone contains uncountably many smooth spacelike boundary points. Thus using

by a maximal timelike geodesic. More dramatically, in this same example a the fact that a causally disconnecting set K may be chosen arbitrarily close to whole open neighborhood U about q exists, with U contained in I + ( p ) . yet any smooth spacelike boundary point jj and the result that a limit of maximal

there is no geodesic from p to any point of U . Hence: despite thc fact that curves is maximal in a strongly causal space-time, we may also establish the this space-time is geodesically complete, the exponential map from p fails to following result. map onto open subsets of IT(p). In Section 11.3, geodesic connectedness was

Theorem 12.52. Let (M, g) be a strongly causal space-time of dimension explored from a general viewpoint and related to geodesic pseudoconvexity

n 2 3. Assume that (iM, g) has a smooth spacelike boundary point and sat- and geodesic disprisonment.

isfies the generic condition and the timelike convergence condition. Then for In this chapter we shall instead consider from the viewpoint of this book,

each smooth spacelike boundary point p, there is an incomplete nonspacelike and in particular as an illustration of the nonspacelike cut locus defined in

geodesic which is inextendible and has jj as an endpoint. Thus (iM, g) has an Chapter 9, a class of spacetimes which originated ns wt,rophysical examples

tlncountable number of incomplete, inextendible, nonspacelike geodesics. yet which display a novel, very non-ltiemannian geodesic behavior. Penrose

Pmof. First, it follows from the preceding remarks that (M,g) contains (1965a) was apparently sufficiently impressed with the fact that the focusing

an incomplete nonspacelike geodesic for each smooth nonspacelike boundary behavior of future null geodesics issuing from appropriately chosen points p

point. But we have just noted that if (M, g) contains one spacelike boundary precluded this class of space-times from being globally hyperbolic that he

point, it contains uncountably many spacelike boundary points. chose the title "A remarkable property of plane waves in general relativity" for

(1965a). In a series of articles by Ehrlich and Emch (1992a,b 1993), a detailed

investigation of the behavior of all geodesics issuing from appropriately chosen

p was conducted, and in so doing it was possible to place this class of space-

times precisely in the standard causality ladder as being causally continuous

yet not causally simple. Also, the future nonspacelike cut locus was determined

480 13 GWITATIONAL PLANE WAVE SPACE-TIMES

and seen to coincide with the first future nonspacelike conjugate locus of p.

Unfortunately. since this class of space-time5 was known in advance to fail to

be globally hyperbolic, the theorems of Chapter 9 were not applicable here,

only the basic concepts and definitions.

In Chapters 35 and 36 of Misner, Thorne, and Wheeler (1973), extensive

discussions of gavitational waves as small scale ripples in the shape of space-

time rolling across space-time is given by analogy with water waves a? smali

ripples in the shape of the ocean's surface rolling across the ocean. These

gravitational ripples are supposed to be produced by sources like binary stars,

supernovae, or the gravitational collapse of a star to form a black hole. The

exact plane wave solutions then arise as simplified models for this phenom-

enon, compromising between reality and complexity to paraphrase Misner,

Thorne, and Wheeler (1973). We wili refer the reader to t h s source rather

than discussing the physics of these models in this chapter.

13.1 The Metric, Geodesics, and Curvature

This class of spacetimes, in the form in which we will describe them below,

seems to originate with N. Brinkmann (1925) and sterns from the work of

Einstein jcf. Einstein and Rosen (1937)) These metrics were brought to the

renewed attention of the general relativity community in the late 1950's by

I. Robinson [cf. Bondi, Pirani, and Robinson (1959) for a discussion of this

history]. Let M = B4 with global coordinates (y, z, v, u). Heuristically, we

may think of starting with the usual Midowski coordinates and then setting

I I (13.1) u = - ( t -

45 a) and v = - - ( t + x ) . JZ

In these coordinates, the Minkowski metric 7 has the form

We also make a choice of time orientation which will be convenieni in the

sequel, so that B/Bv is past directed null at all points. Now the class of "plane

fronted waves" may be defined to be those metrics for M = B4 with global

coordinates (y, z , V , u) which may be written in the form

13.1 THE iMETRiC, GEODESICS, AND CUR\.'ATURE 45 1

where H(y, z, u) has an appropr~ate degree of regularity and is nonvanishing

somewhere. The global coordinate u : M -+ R plays an important role in the

differential geometry of this c l a s of space-times and is even more important for

the subclass of gravitational plane waves ~ncroduced below in (13.9). The plane

fronted metrics (13 3) have the interesting property that all have vanishing

scalar curvature but need not be Rcci fiat unless

No general statement can be made about the geodesic completeness of this

class of spacetimes apart from the geodesics lying in the null hyperplanes

On these hyperplanes, the ambient metric (13.3) is degenerate, so these are

not "Lorentzian submanifolds" in the sense of O'Neill (1983) or of our pre-

vious Definition 3.47. Denoting y, Z ,V ,U by 1 ,2 ,3 .4 respectively, the only

non-vanishing Chfistoffel symbols are

and

By direct calculation, it may be verified that any geodesic e : (a , b) --t M

satisfies ( ~ 4 0 c)"(t) = 0. Then in the case that, 24 0 e( t ) = uo for all t , one

obtains that the geodesics in the nail hyperplanes F(710) are precisely stra~ght

lines lying in P(uo) of the form

which are ma~lifestly complete.

These geodesics are either spacelike or null; we will reserve the notation

482 13 GRAVITATIONAL PLANE WAVE SPACETIMES

for the member of the class of future d~rected null geodesics lying in P ( u o )

which passes through the point (yo, zc,O, uo). An aspect of the differential

geometry of plane fronted waves related to this class of null geodesics is the

fact that the vector field gradu = 'Ciu = d/Dv is always a global parallel vector

field, independent of the particular form of W(y, z , u).

Already, it may be seen that the global coordinate u : M -+ R has many

aspects of a time function. First. we set things up in analogy with (13.1) so

that is past directed null. Thus it follows that u is strictly increas~ng

along any smooth future directed timelike curve c( t ) , for

as cl(t) is future timelike and d/dv is past directed nonspacelike. Hence, any

plane fronted wave is chronological. &rthermore, this coordinate function

u is strictly increasing along any smooth future causal geodesic except for

those null geodesics q ( s ) of the form (13.6). For writing an arbitrary geodesic

as ~ ( s ) = ( ~ ( s ) , z ( s ) , ~ ( s ) , u ( s ) ) , the geodesic differential equation reduces as

noted above to ul ' (s) = 0 for this last component. Hence, if u is nonconstant,

it may be rescaled to be u(s ) = s , and

Thus u is indeed strictly increasing along all smooth null geodesics except for

those of the form (13.6) lying in one of the null planes P(uO). Recall that

a chronological space-time which fails to be causal contains a null gcodesic

/? : [0,1] -+ M with P(0) = P(1). But such a segment cannot lie in one of the

planE% P(u0) hecame all null geodesics in these planes are injective by direct

solution of the geodesic PD.E.; hence P must have the form (13.8), but then

the form of the iast coordinate precludes P(0) = p(1) from occurring Thm

the nature of the global coordinate u and the form of the geodesics in the null

planes P(uO) ensure that all members of the class of plane franted waves are

causal space-time.

In Eardley, Isenberg, Illarsden, and Moncrief (1986), it was shown that a

space-time which is a non-trivial vacuum solution of the Einstein equations

and which admits at least one non-homothetic conformal Killing vector field

must be a plane fronted wave.

13.1 THE METRIC, GEODESICS, AND CURVATURE 48:

Definition 13.1. (Gmv~tat ional Plane Wave) The manifold M = R4 together with a plane fronted metric g = q + H(u, y, z)du2 for which H takes

the quadratic form

or in matrix notation

with f ( u ) or g ( u ) not vanishing identically, is said to be a gravatational plane

wave. The wave may he said to be polarezed if g ( u ) = 0. The wave is commonly

termed a sandwa'cli wave if both functions f and g have compact support in u.

I t is interesting that as a consequence of the quadratic nature of (13.9),

or equivalently (13.10), the same linear second order system of differential

equations

arises in studying the Kllhng fields, the Jacobi fields, and the geodesics of

these space-times. C Ahlbrandt has commented that this is to be expected

for the last two of these objects by the classical calculus of variations. given

the quadratic nature of the metric In (13.9) and hence of the assoc~ated art

length functional. Also, all gravitat~onal plane waves are geodesically complete

and satisfy Rie = 0 but are not flat since e~ther f or g 1s assumed to be

nonvanishing. Thus, dny geodesic conjugacy whlch arlses may be attributed

to the shear term rather then the Ricci curvature in equation (12.2) of Sectlon

12.1. It may be checked directly that any future nonspacelike geodesic, other

than one of the complete null geodesics of the form (13.6). experiences some

nonzero timelike sectional curvatures and hence saiisfies the generic cond1t:on

(2.40).

We now turn to a more careful discussion of the geodesics of a plane wave

space-time which are transverse to the null planes P ( u o ) . As previously noted,

such geodesics may be parametrized with U ( S ) = S. or equivalently, as

484 13 GR4VIT-ATIONAL PLANE WAVE SPACE-TIMES

Kow in the case of the gravitational wave, the ChristoRel symbols simplify

to

I = 5 [f '(u) ty2 - z2) + 2g1(u)yz] .

r;* = ril = -rid = ~ ( Z L ) Y I g ( ~ ) ~ ,

and I?:* = I?& = = g(u)y - f (u)z.

Hence, the y and z components of the geodesic P.D E read

(13.13) y"(u) = f(u)y +g(u)z and

(13.14) ~ " ( u ) = - f (u)z

as claimed above. Hence, it 1s ~mrned~ate that if f and g are continuous,

then different~able solutions to (13.13) and (13.14) exist for all values of the

parameter u. Rather than using the geodesic differentlal equatlon to solve

v(u), it is convenient to give the following indirect argument. Suppose we wish

to find the geodesic c(t) with g(cl, c') = A. As cl(u) = (yl(u), zl(u), v'(u) I),

this condition is expressed as

x = q(cf, c') = (7Jf)2 4- (z1)2 + 2vt(u) . 1 + [f (u)(g2 - z2) + 2g(u)yz] - 1. I

or

x = (v1)2 + (z1)2 + (f (u)y + g(t4)z)y + (g(u)y - f(u)z) 2 + 22').

Using (13.13) and (13.14), t h s last equatlon becomes

X = (y')2 + (zf)2 + yl1g + zllz + 22'

so that vl(u) = [-(yyl)' - (zzl)' + A] Integration produces the equation

1 (13.15) v(u) = - [-y(u)y'(u) - z(u)zJ(u) -Xu + C]

2

which shows that slnce y(u) and z(u) are d~fferentiable for all values of u, v

may be calculated for all values of u by employing (13 15). T h ~ s calculatlo

together with our previous knowledge about the geodesics of the null

planes P(zbo), gives an elementary direct proof of the geodesic completen

13.1 THE METRIC. GEODESICS, AND CURVATURE 485

for this class of metrics; an alternate proof may be given based on a knowledge

of the isometry group of this class of space-times. Equaiions (13.12)-(13.14)

reveal that an important first step in understanding the geodesic behavior of

these space-times is studying the system (13.11).

Thorough studies of the Killing vector fields and related isometry groups

of these space-times have been given in Ehlers and Kundt (1962), Krarner,

Stephani, MacCallum, Herit, and Schmutzer (1980), and Ehrlich and Emch

(1992a), among others. Rom t h ~ natttlue of the isometry group, i t follows that

if the behavior of the exponential map at Po = (0.0,O. ug) is understood, then

as >sometries map geodesics to geodesics, the same behavior obtains at an

arb~trary point P with u(P) = uo. With choice of initial co~d~t ions y(uo) =

Z(UO) = v(u0) = 0. equation (13.15) simplifies to

In Emch and Ehrlich (1992a,b), a conjugacy index associated to this system

with the preceding initial condition was discussed. For u, E R, let

(13.17) L(75) = (solutions (y(a). z(u)) to (13.11) with y(uj) = z(u3) = 0).

Definition 13.2. (Conpgacy Index) For distinct real numbers u g < ul,

define the conjugucy indez I(u0, ul) of O.D.E. system (13.11) with respect to

uo and ul to be

which, given our dimension restriction, is either 0. 1, or 2. The pair {uo, ul) is

said to be conjugate provided i(uO. uI) > 0 and astagmatzc conjugate provided

that I(uo u l ) = I. Also, {uo, u l ) is said to be a fir& conjugate pazr provided

that l(uO. 2 ~ ~ ) > 0 and I(vo, T I ) = 0 for all u E (uo, ul). Also. !ntroducc thc

notation

(13.19) Conn(Po. u) = (Q E P(u) . there is a geodesic from Po to Q).

It is a consequence of Propos~t~ons 12.15 and 12.17 that every gravitational

plane wave space-time admits a first conjugate pax. This was esteblished

456 13 GRAVITATIONAL PLANE WAVE SPACE-TIMES

by direct considerat~ons in a symplectic context in Lemma 2.4 of Ehrlich and

Emch (1992a).

In Ehrlich and Emch (1992aj the following facts were established by study-

ing the exponential map and geodesic equation associated to a first astigmatic

conjugate pair uo < UI for an arb~trary plane wave: let PO E P(uo) b ~ . arii-

trary. Then

(1) if uo 5 u(Q) < ~ 1 , then there exists a unique geodesic (parametrize

as discussed above) from PO to Q; hence dim Conn(P0, u) = 3 for all

in [uo.ul).

(2) dimConn(P0. u l ) = 2 and Conn(P0~ul) is in fact a plane in P(ul

every point of which is conjugate to Po by a one-parameter family o

geodesics.

(3) If u(&) > u1, then Q E 14(Po).

(4) If uo 5 u(Q) < U I , then Q E I+(Po) if and only if Q lies on a uniqu

maximal timelike geodesic segment issuing from Po.

Also, the first future nonspacelike conjugate locus and the future non

spacelike cut locus of Po in Conn(&,ul) were seen to coincide. A no

ferentlable example was also given of a space-time of the above form

dimConn(Po,ul) = 1, so this possibility was not ruled out in the consider

ations of Ehrlich and Emch (1992a). Now, the proofs of these results calle

upon the estabhshment of a symplectic framework somewhat outside the view

point of the subject matter of this book Since it was seen in t h ~ s refercnc

that precisely the same behavior obtains in the polarized and nonpolarize

cases, we will restrict our discussion in the next section to the polarized c

for whlch the equations are somewhat less complicated and more easily de

with In order to illustrate the above points.

13.2 Astigmatic Gonjugacy and the Nonspacelike Cut Locus

In thls section we turn to the study of astigmatic or Index 1 conju

and the global geometric behavior of the geodesics issuing from Po As

mentioned, to avoid the complexities of dealing with the general case, which d

not in any way affect the qualitative outcome of the geodesic geometry, we wi

13 2 ASTIGMATIC CONJUGACY, NONSPACELIKE CUT LOCVS 41

restrict our attention to the somewhat simpler case of a polarized gravitation,

wave with g(u) = 0 and also assume f (u) 2 0 Hence the metric takes tk

form

and the geodesic differential equation for geodesics transverse to the null plane

P(u) uncouples as

y'' = f (u) y and

2'' = - f (u)t.

Under the above assumption on f , standard O.D.E. theory shows that equatio

(13.22) admits a nontrivial first conjugate pair solution z(u) with z(u0) - t (u l ) = O and z(u) > 0 for u with uo UO.

Let us denote by yl(u) [respectively, zl(u)] the solution to the 0.D.E

(13.21) [respectively, (13.22)] with initial conditions

(13.23) YI(UO) = zl(vo) = 0, and ?J:(uo) = -7: (uo) = 1.

Since the isometry group of any gravitational plane wave space-tirile a ~ t s :ran

sitivcly on the null planes P(uo) for any uo 6 W, it is sufficient to understanc

the behavior of geodesics issuing from Po = (0.0.0, uO) to understand expl

for any P E M with u(P) = uo. Evidently, any geodesic c(u) issuing from l?

which is not a straight line lying in P(uo) and which satisfies g(cl . c') = X ma:

be described as

for some choice of constants cl, c2 in R, in view of eqdation (13.16). For an!

u with uo 0 Hence, it 1s eas:


to check from (13.24) that not only does Conn(Po,u) = P(u) for all such u,

but also given any Q with u(Q) 6 [uo, ul), there is a unique geodes~c from Po

to 4. Now take any poiilk Q = (c, d. e, 2 ~ ~ ) in Conn(Po. 211). Note that

(13.25) C(UI) = 2

Hence, d = 0. Now cl is determined as cl = ~ / y ~ ( 2 ~ ~ ) , possible slnce yl ( u l ) > 0

Given this determination of cl, we then make v(ul) = e by choosing th

constant X to satisfy

(13.26) A = [2e + c2?4l'(W)lyl(ul)]

1'111 - ~ 0 1

and note that this setup is valid for any c2 E B. Hence. we have not on1

established that

(13.27) Conn(P0,ul) = P(u1) n { z = 01,

but also letting c2 vary over all possible values -m < cz < +co, we have show

that each point & in Conn(P0,ul) is conjugate to P by this oneparamete

family of geodesics as cl arld X = g(c', c') are uniquely determined by Q, but

c2 is undetermined. The "'unbounded'" nature of this one-parameter family

geodesics precludes the geodesic system from being pseudoconvex, and als

prevents this class of space-times from being globally hyperbolic [cf. Penros

(1965a)l.

To recapitulate. we have found that as u increases from u = 710. prlor t

reachlng P(ul) each point Q with u(Q) = a is joined to Po by a unique ge

desic, but when u = u1, the geodesic jo~n set Conn(Po, ul) drops from th

mensions to two dimensions, and as that happens, every point in Conn(P0, u

is conjugate to Po by a one-parameter family of geodes~cs as exhibited expl

itly above

The first future null conjugate locus of Po in the null hyperplane

may now be determined by taking X = 0 in the above discussion. Noting

yl'(ill)/yl(ul) > 0 under our hypotheses, and denoting they and v coordina

of ~ - ~ l = ~ l ~ ~ l ~ ~ l ~ ~ ~ l ~ l

, ul 2

13.2 ASTIGMATIC CONJUGACY, NONSPACELIKE CUT LOCUS 48

Conn(Po , u,) n ( z = 0

FIGURE 13.1. The conjugate locus of Po in Conn(Po. u l ) = P(uo)n

( z = 0) for a first conjugate palr (uO, 2 ~ ~ ) and Po = (0,0,0, uo) is

shown.

by Y and V respectively, we obtain the equation

(13.28) v = -[(~/~)YI'(~I)/YI(~I)]~~, 2 = 0,

for the first future null conjugate locus of Po in Conn(Po,ul). Hence. this

null conjugate locus, Nconj(Po), 1s a planar parabola in Conn(Po.ul). This

same phenomenon is shown to occur for an arbitrary gravitational plane wave

in Ehrlich and Emch (1992a, p. 214) The points in Conn(P3, u l ) whrch lie

on timeliice geodesics issuing from Po are precisely those which are In the

same component of Conn(Po,ul) - Nconj(Po) as the focus of the parabola

while those which lie on spacelike geodesics ~ s s u ~ n g from Po !ie in the other

component (cf. Figure 13.1).


More exactly, the Eollovii~lg result is established for the general plane wave

space-time in Ehrlich and Emch (1992a, p. 214).

Proposit ion 13.3. In the presence of astigmatic conjugacy from Po, the

future null (respective1,vi nonspacelike) cut locus of Po is prec~sely the first fu-

ture null (respect~vely, noespacelike) conjugate locus to Po as descr~bed abovc.

In particular, every point in the con~ponent of Zonn(P0, ul) -Nconj(Po)

lies in the connected component containing the focus of the parabola Nconj

is a future timclike cut point to Po.

Proof We first recall that these space-times have been known at least since

Penrose (1965a) to fail to be globally hyperbolic; thus we may not use the

existence of maximal nonspacelike geodesic segnleritv in this proof. Instead,

it is helpful to develop a version of the basic Proposition 3.4 which is more

adapted to the sublevel sets of the global coordinate function u : M -+ W rather than considering a small convex neighborhood centered at Po. While

is not a time function in the sense of Section 3.2, u does possess the following

two properties as considered in Ehrlich and Em& (1992a, pp. 200-204).

Definition 13.4. (Quasi-Time Fuactzon) A smooth function f : N

on an arbitrary space time (N, g) is said to be a quasi-tzme functzon provid

(I) The gradient V f is everywhere nonzero. past directed nonspaceli

and

(2) Every null geodes~c segment c such that f o c is constant is injectiv

From our description of the geodesics of gravitational plane wave spac

times and the fact that V f = dldv is past directed null, it is evident that t

global coordinate function u : M -+ P satisfies the properties of this definitio

Property (1) implies that a quwi-time function is also a semi-time func

as defined in Section 3.2, but it is the combination of both properties (1)

(2) in this definition which ensure that an arbitrary space-time which ad

a quasi-time function must he causal While a semi-time function pre

chronology from being violated it does not, in general, prevent the emst

of closed nu1 geodesics.

13 2 ASTIGEvfATIC CONJUGACY, NONSPACELIKE CUT LOCUS 491

Returning to the general setting of a plane wave, for any uo, ul with uo < ZLI,

introduce the notation

Arguing along the lines of (13 7 ) , it may be seen that any suchu-strip has the

convexity property that if m, n are contained in S(110, ul) . and y . [O,1] --+ M

is a future directed nonspacelike curve with y(0) = m and y(1) = n, then

Now we restrict our attention to a strip S(uD,ul) for whirh (t~o, Z L ~ ) is a first

conjugate pair. Then by our above considerations. expro is invertible on this

particular strip. Thus the Synge world function cP : S(u0,u1) + R given hy

is differentiable on S(uo, u1). Using the Sgnge world function. define the fol-

lowing partition of S(uo. Ill). Put

N = @-'(I)) = (points in S ( U ~ ! u l ) that are joined to Po

by a null geodesic),

T = {m E S(uo,ul) : +(m) < 0) = {points in S(UO, ul)

which are joined to Po by a "timelike geodesic),

and

8 = {rn E S(UO, %I) : +(m) > 0) = (points in S(uO, ul)

which are joined to Po by a spacelike geodesic)

Even though T may not be contained in a convex normal neignborhood of

Po, we still have VQ, existing and being future t~melike on all of T Now the

prom~sed technical lemma may be stated

13 GRAVITATIOKAL PLANE WAVE SPACE-TIMES

FIGURE 13.2. The null tail NT(P0) in P(uo) for a first astigmatic

conjugate pair uo < u1 is shown.

Points Q on the null tail NT(P0) then fall into one of two categories: first,

those points lying on Nconj(P0) and thus joined to Po by a one-paramet

family of null geodesics; and second, those points Q not lying on Nconjj

which are thus joined to Po by a one-parameter family of spacelike geod

Penrose (1965a) observed from the "noncompact" behavior of the family of n

geodesics of the type (13.24) from Po which refocus at Qo that PQ, happ

to be a limit curve of this family of null geodesics from Po to Qo, and e

more remarkably, this behavior prevents this class of space-times from b

globally hyperbolic. 111 Ehrlicli axid Emch (1992a, p. 218), we noted that

observation may be used as a step in the proof of the following result.

Lemma 13.9. Suppose uo < u1 is a first astigmatic conjugate pair

u(Q) > u1. Then Q E I+(Po).

Proof. First, choose c > 0 such that (al, .ul -t- c] centalm no u-value co

gate to u = 211. Then from the explicit form of the geodesics for gravi~atio

plane waves, it may be calculated that for any Q in the strip S(uI ,u l

there exists a future timeirke geodesic from a point R on the null ray PQ,

13.3 ASTIGMATIC CONJUGACY, ACHRONAL BOUNDARY 495

((0,O, s, ul) : s E R) to 4. But then since R E I+(Po) as Penrose observed,

we have Q E I+(Po) by future set theory, as I+(Po) = (rn E M : I+(m) 2 I"(P0)) (cf. Proposition 3.7 or corollary 3.5). Now glven an arbitrary Q with

u(Q) > u1, simply take any past directed timelike geodesic c(u) lssulng from

Q, and from knowledge of the quasi-time function u, we know for any u2 !n

(u1,ul + c), that S = c(u2) satisfies S << Q and S 6 S(uI ,u I + C) I+(PO).

Hence, Q E I+(Po) as well. U

Now F = I+(Po) is a particular example of a future sct. and the corre-

sponding achronal boundary is given by

The characterization [cf. Penr~se (1972, Section 5 ) ] of a general achronal

boundary in terms of what relatwists call the null geodeszc generators trans-

lates in our simpler context of (13.33) as follows

Theorem 13.10. Let (N,g) be an arbitra~y space-time, arid let B =

a(I+(Po)) for any Po E N. Suppose l? 6 B - (Po). Then there exists a

null geodesic /3 contained in B with future endpoint R which sat~sfies eilher

(1) p : [O, I] + N wjth P(0) = Po andfi(1) = R, or

(2) ,O : ( -a , 11 -+ B is past inextendible and 9(1) = R.

It is possible for a particular point on the achronal boundary B to possess

geodesies of both types (I) and (2) above. The null tail NT(Po) defined above

happens to be a subset of the achronal boundary B = d(I+(Po)). and po~nts R

of NT(Po) which lie on Nconj(Po) as well have null geodesics with endpoint R

of both type (1) and type (2) of Theorern 13.10. On the other hand, points R

of the null tail NT(Po) which do not lie on the null corijugate locus Nccnj(Po)

are future endpoints of a null geodesic 8 only of t jpe (2).

For this simple example of a future set. note that points R which satisfy

alternative (1) lie in the future horismos E f (Po) = J+(Po) - I'(Po) of Po.

Also, it should be noted that if a space-trme (N ,g) is causally simple, so that --

J f ( Q ) = J+(Q) = I+(Q) for any Q in N , then any future set B defined as in

(13 33) sat~sfies B = b(If (Q)) = E+(Qf Thus to get an example of such an

496 13 G W I T A T I O N A L PLAKE WAVE SPACE-TIMES

achfonal boundary B whlch contains polnts R satisfjing altei~iative (2) but

not alternative (I) of Theorem 13.10, it is necessary to consider space-times

which fail to be causally simple. Fortunately, ail gravitational plane waves

are known to fa11 to be causally simple, essentially because of the behamor

mentioned above that the null geodesic ray pq,, apart from its terminai point

Qo, is in J f (Po) but not in Jf (Po).

At present, we know that I+(Po) and B = B(I+(Po)) have the following

properties for the first astigmatic conjugate pair u0 UO.

(2) Tf uo 5 u(R) < u1, then R 6 I+(Po) if and only if R lies on a unique

maximal timelike geodesic segment c with c(uo) = Po and c(u(R)) = R.

(3) If u(R) > 211, then R E I+(Po) automatically.

(4) If u(R) = u1 and R E Conn(P0, ul), then R E I*(Po) if and only if R

lies on a uniquely determined 1-parameter family of maximal timellke

geodesic segments from Po to R.

What is not settled is the qaestion of what other points R in P(ul) which do

not lie on geodesics issuing from Po, i.e., z (R) # 0, are contained in I+(Po),

in simpler terms, what happens to the chronological future of Po as astigmatic

conjugacy occurs when u = u;?

The corresponding facts for the achronal boundary B = d(I+(Po)) are

fol~ows.

(5) If R E B. then u0 5 u(R) < u1. (6 ) Tf uo < u(R) < u1, then R E B if and only if R is the endpoint of

unique maximal null geodesic segment from Po to R.

(7) For R in Conn(Po,uI), it is known that R E B if and only ~f R

contained in the null tail NT(Po).

In the case of B, then, the issue 1s whether B n P(ul) = NT(Po), or equi

lently, whether P(ul) - Conn(Po, ul) 5 I+(Po).

I t may be seen for a subclms of polarized gravitational waves, which

termed "unimodal" in Ehrlich and Emch (1992a) and Ehrlich (1993).

these related ilssues may be settled affirmatively.

13.3 ASTIGMATIC CONJUGACY, ACHRON.4L BOUNDARY 497

Definition 13.11. (U~zmodal Gravztatzonal Plane Wave Metnc) A prr

larized gravitational plane wave metric g = i- f(u)ly2 - z2)du2 is said to

be mimodal ~f f (u) 2 0 and there exist A > 0 and T i 6 R with the following

properties:

(13.35) '?r

( ) < A < - ; 2 and

Requirement (13.35) serves to ensure that in a certain Priifer type trans-

formation for the solution of the 0,D.E. (13.22), a continuous determination

of arctan may be made [cf. Ehrlich and Emch (1992a, p. 188)]. What we need

for our purposes in this section, which is where the assumption of unlmodality

enters into the discussion below, is the following variant of a Sturm Separation

Theorem, which is implicit in the proof of Theorem 2.14 of Ehrlich and Emch

(1992a, p. 187).

Lern~na 13.12. Let uo < ul be a first astigmatic conjugate pair for a given

unimodal gravitational wave. Then there exist 5 > 0 and a strictly increasing

continuous function a : [O, 6) -+ [0, +x) with a(O) = 0 and such that

is a first astigmatic conjugate pair for all E in [O, 6).

Now we are ready for

Theorem 13.13. Let uo < u1 he a first astigmatic conjugate pair for

a unimodular polarized gavitational wat7e. T,et Po = (yo, 20, vo, uo) be an

arbitrary point of the null plane P(u0). If R E P(ul ) does not lie on any

geodesic issuing from Po, then R E It(Po). Hence, the null tail NT(Po) of Po

satisfies


Proof From the nature of the isometry group, ~t suffices to make explicit

calculations with Po = (0.0.0, u0) as above. With this particular cholce of Po,

for any gravitational plane wave metric with g(u) = 0 we have that P(s) =

(0,O. 0. s) is a future directed null geodesic issuing from Po Let P, denote the

point given by P, = P(uo + E) = (0,O. 0, u0 + r ) Further, let y,(u) and re(%)

denote respectively the unique solutions to the IVP's

(13.39)

Y" - f (2519 = 01 Y ( W + E ) = ~ , y ' (uo+e)= l and (13.40)

zl/ + f (U)Z = 0, z(u0-J-e)=O, zl(ugSc) = l

which arise in studying the exponential map from P,. In view of (13.24)

translated to P,, if a point Q = (y, z , v.u) with u > ug + E lies on a geodem

~ ( u ) issuing from P, with g(c'. cl) = A, then

(13.41) zv = _&.I9 y2 - i:(U) z2 + - uo - €), YE(u) zdzo

Now fix R = (a, b. c, ul) in P(ul ) which does not lie on any geodesic issuing

from Po, ensurlng b f 0. If we show that R lies on some future timelik

geodesic issuing from P, for somc E > 0. then R lies on a future null geod

from Po to P,, follouled by a future timelike geodesic hom P, to R, whe

PQ < R as desired.

But the point R will lie on a timelike geodesic from P, for somc c > 0

provided that the inequality

(13.42) Y~'(uI) z*'(uI) b2 2 c < - - yc(ul) ~ r (u1)

is satisfied for some E > 0. Since b2 > 0, this will ensue provided that

(13.43) %'(uI) yc(u1)

is uniformly bounded as E + O while

(13.44) z,/(u1) lim - = -oc

E+O ~ ~ ( 2 ~ 1 )

i3.3 ASTIGMATIC CONJUGACY. ACHRONAL BOUNDARY 499

Choose a constant E > 0 so that 0 5 f (u) < B2 for a with uo - ! 5 ?L < ul t 1.

Also as f (u ) 2 0, we have that y,(u) and g,/(u)/g, JU) are b o ~ h positive for

any ?I. > ?LO -I- E . Hence (13.43) follows by the usual b u c h comparison theory

techniques

The more technical part of the proof is to establish (13.44). Here there are

two cases. both of which occur in explicit examples: first, u i $ supp(f(u)),

and second, ul E supp(f (u)). In either case, let a ( € ) be as in (13.37).

Consider first the case that 7 ~ 1 @ ~11pp(f(u)), and so also UI + a ( € ) cf supp(f (u)) for all E > D and z"(u) = 0 for all ZL > ul . Hence there exist

constants A, so that

for all u > ul. Consequently,

so that limit (13.44) is immediate

Finally, suppose that 111 E supp(f(u)) L. Flaminio has remarked that the

analysis step (13.46) should remain valid because the Jacobi solution Z(U) near

~ t s zero at ul +a(€) is known to be asymptotically linear A detailed calculation

demonwrating thar, this intuition is valid may be given by drawing on the

variant of the Priifer transform technique used ai; in the proof of Theorem

2.14 of Ehrlich and Emch (199%). The IVP's (13.40) in the (16, z) plane are

transformed into the (s, 8) plane as in Theorem 2.14 in order to carry out the

details of the analysis. The crux of the matter is that the transform of zc (u )

to @,(s) enables a uniform bound on 6,'(s) to be given because of the presence

of the term sin[8,(8)] on the right hand side of equation (2.83) of Ehrlich and

Ernch (1932a).

CHAPTER 14

THE SPLITTING PROBLEM IN

GLOBAL EOPeENTZIAN GEOMETRY

An important aspect of global Riemannian geometry over the past 25 years

has been the investigation of rigidity in con~iection with curvature inequalities.

This concept was first widely disseminated in 1975 in the first two paragraphs

of the preface to the iduential monograph by J. Cheeger arid D. Ebin (1975),

Comparison Theorems in Riemannian Geometry:

In this book we study complete riemannian manifolds by devel-

oping techniques for comparing the geometry of a general manifold

M with that of a simply connected model space MH of constant

curvature W. A typicai conclusion is that M retains particular ge-

ornetacal properties of the model space under the assumption that

its sectional curvature KIM, is bounded between suitable constants.

Once this has been established, it is usually possible to conclude

that M retains topological praperties of MH as well.

The distinction between strict and weak bounds on KM is im-

portant, since this may reflect the difference between the geometry

of say the sphere and that of euclidean space. However, it is often

the case that a conclusion which becomes false when one relaxes the

condition of strict inequality to weak inequality can be shown to

fail only under very special circumstances. Results of this nature.

which are known as ngadzty theorems, generally reqmre a delicate

global argument.

A first example is provided by the topological sphere theorem established by

W. Raueh (1951), M. Berger (1960), and W. Klingenberg (1959, 1961, 1962),

among other authors.

502 14 THE SPLITTIKG PROBLEM IN LORENTZIAN GEOMETRY

Topological Sphere Theorern. Let (N. h) be a s~mply connected, com-

plete Riemannian manifold of dimension n 2 2. Suppose the sectional curva-

tures of (N, hj satisfy the curvature pinchag condition

(14.1) d . A < K ( a j < A

for all two-planes o in 6 2 ( M ) , where d satisfies

(14.2) 1

d > ;

and A > O is arbitrary. Then N is homeomorphic to the sphere Sn.

Notice that in this result we have a strict inequality in (14.2) and also in

the left inequality of (14.1) Now the rigidity in this context was first proved

by M. Berger under the weaker pinching hypothesis

(14.3) 1 - A 5 K(cr) 5 A. 4

Berger obtained that either the previous topological conclusion held and that

AT would still be homeomorphic to Sn. or if the previous conclusion faded to

hold, and hence N was no longer homeomorphic to Sn, then A: had not just to

be homeomorphic but had to be zsometnc to a compact rank one Rieinannian

symmetric space.

Here 1s a second example more germane to our context. It follows from the

work of Gromoll and Meyer (1969) that a complete Riemannian manifold

dimension n 2 2 wlth everywhere positive Ricci curvatures

(14 4) Ric(v, v) > 0 for all nonzero v in T M

is connected a t infinity. Rigidity now enters into this situatioil in the followln

manner. Suppose only that Ric(v, v) > O and that (N, h) fads to be connecte

at infinity. Then since (N, h) is assumed to be complete, there ex~sts w

is often termed a (geodesic) lzne, i.e., a complete geodesic c R - (N,

joining any two different ends of IV and minimizing distance between any tw

of its points. But then the Cheeger-Gromoll Splitting Theorem (1 971) m

applied to the line c to establish that (N, h) is zsometnc to a product mani

14 T H E SPLITTING PROBLEM IN LORENTZIriN GEOMETRY 503

Cheeger-Gromoll Spl i t t i~ ig Theorern. Lei (rV, h) be a complete Rie-

niannian manifold of dimension n 2 2 which satisfies the curvature condition

(14.5) R~c(v, z;) 2 O for ail 71 in T A . ~

and which contains a complete geodesic line. Then (iV. h) may he written

uniquely as an isometric product Nl x IWk wl~ere NI contains no lines and W k is given the standard Aat metric.

Thus in this example the curvature rigidity arises as condition (14.4) is

weakened to condition (14.5).

In 1932 Busemann (1932) introduced an analytic method of studying gener-

alizations of the classical horospheres of hyperbolic geometry to a very general

class of metric geometries. In the proof of the Cheeger-Gromoll Splitting

Theorem given in Cheeger and Gromoll (19711, this particular function was

independently rediscovered during the course of the proof as a key ingredient.

At about the same time. the Busemann function had also been seen to be

a useful tool in the study of complete Riemannian manifolds of nonpositive

sectional curvature (cf. Eberlein and O'Ne111 (1973), Eberlein (1973a), where

the function was named the "'Busemann function" in honor of iis discoverer)

This function has played a prominent role continuing up to the present time

in global Riemannian geometry.

In Chapter 12 singularity theorems for spacetimes which satisfy certain

global geometric conditions and certain curvature conditions have been dis-

cussed. A simple prototype may be stated as a reminder.

P ro to type Singularity Theorem. Let ( M : g) be a spacetime of dimen-

sion n > 2 which satisfies the following three conditions:

(1) ( M , g) con tans a compact Cauchy surface;

(2) ( M ; g ) satisfies R ic (~ , V) 2 0 for ail timelike (or ali nonspacelike) v;

and

(3) Every inextendible nonspacelike geodesic satisfies the generic cond~tion

(cf. Definition 12.7 and Theorem 12.18).

Then (M, g) contains an incomplete nonspacelike geodecsic

504 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 505

Now in this result the "strict curvature inequality" is condition (3), which In a more differential geometric vein, S. T. Yau an the early 1980's (unpub-

requires for each inextendible nonspacelike geodesic that a certain curvature lished) proposed the idea of a "rigid singularity theorem.' to use the language

quantlty be nonzero a t some point of the given geodesic. Thus, in this context, of Galloway (1993). This was later stated in Bartnik (1988b) as follows: curvature rigldity would suggest studying spxEr-dimes which satisfy all the Goeecture. Let ( M . g) be a space-time of dimension greater than two conditions of the Prototype Singularity Theorem but condition (3)

Earlier In an essay on singularities in general relativity, Geroch (1970b, (1) contains a compact Cauchy surface, and

pp. 264-265) had suggested for inextendible closed universes satisbing the Ein- (2) satisfies the timelike convergence condition Ric(c, v) > 0 for all timelike

stein equat~ons that, generically. such space-timm should be timelike geodesi- vectors v

cally Incomplete [cf. the m~ddle column in Figure 2 on p. 266 of Geroch Then either ( M , g) 1s timelike geodesically incomplete, or else ( M , g) splits iso-

(1970b)l. Geroch's conjectured v~ewpoint was stated as metrically as a product (R x V, -dt2@h), where (V, h) is a compact Rie~narlnian

Thus, we expect that the diagram for closed universes will be

almost entirely black. There are, however, at least a few white This philosophy apparently prov~ded Yau with the motivation to pose in points there exist closed. geodesically complete flat spacetimes the well-known problem sectlon of the Annals of Mathematics Studies, Vol- . . . Perhaps there are a few other nonsingular closed universes, but ume 102 [cf. Yau (1982)], the problem of obta~nlng the Lorentzian analogue of these may be expected to appear either as lsolated points or at least the Cheeger-Gromoll Ricci Curvature Splitting Theorem stated above. Speclf- regions of lower dimensionality in an otherwise black diagram. ically, Yau posed the question as follows: Show that a space-time (1M,g)

which is t~melike geodesically complete, obeys the t~mellke convergence con- Here Geroch is representing diagammaticauy the space of all solutions to

dition, and contains a complete timelike line, splits as an Isometric product Einstein's equations in the above discussion, with singular space-times repre-

(R x V, -dt2 $ h). Thls problem was then studled by Galloway (1984~) as well sented as a black dot and non-singular, inextendible space-times represented

as by Beem, Ehlich, and Markvorsen Galloway considered spatially closed as a white dot. Further, In Geroch (1970b, p. 288) the following problem was

space-times and employed maximal surface techniques stemming from results

in Gerhardt (1983) and Bartnik (1984). The approach of Beem, Ehrlich and

Prove or find a counterexample: Every inextendible space-time Markvorsen, which was directed toward globally hyperbolic rather than time-

containing a compact spacelike 3-surface and whose stress-energy like geodesically complete space--times, employed entirely different methods in

tensor satisfies a suitable energy condition is either 6-singular or which the use of the Busemann function for a future complete timelike geo-

flat. (The energy condition must be strong enough to exclude the desic ray ' i v s introduced into spacetime geometry. Under the less stringent

Einstein universe. Perhaps one should first prove that the 3-surface t~rnelike sectional curvature assumption K 5 0 a splitting theorem was ob-

cannot be, topologically, a 3-sphere.) tained; combining forces with Galloway yielded Beem, Ehrlich, Markvorscn,

and Galloway (1984, 1985)

Gdloway and Norta (1995) summarize Geroch's ideas here as "spatlall A Riemannian proof contained in Eschenburg and Heintze (1984) for the

closed spacetimes should fail to be slngular only under exceptional circu Cheeger-Gromoll hemannian Splitting Theorem, different than that originally stances." glven in Cheeger and Gromoll (1971). provided a helpful model for obtaining

506 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14 1 THE BUSEMANN FUNCTION 50

results in the space-time context The d'Alembertian operator for a space- obtained by the use of "aimost maximizers" to construct to-rays rather t h a ~

t:me is hvperboiic rather than elliptic. but Cheeger and Gromoll (1971) used relyicg on maximal geodes~cs as in the previous papers and in the classica

analysis methods relying on ellipticity. Next Eschenburg (1988) made a key constructions done for complete Bemannian manifolds [cf. Newman (1990

new observation that if one only studied the geometry in a ne~ghhorhood of the Lemma 3.9)]. Since this method of proof fits rather well with the spirit o

given timelike line, then the hcc i curvature assumption would allow certain Proposition 8.2 of Section 8.1, ~t is this approach to the Lorentzian Splitting

arguments to be successfully modlfied from the K < 0 case to the Ricci curva- Theorem which will be treated here.

ture case, provlded that the space-time was globally hyperbolic and timelike An elementary complication in the proof of the space-tlme splitting theoren

geodesicaiy complete; then a "continuatlon type argument" would allow the occurs in the construction of what would be called in Riemannian geometry ar

splitting obtained in a tubular neighborhood of the given complete timelike ge. "asymptotic geodesic" to a given complete timelike geodeslc through a limltlng

odesic to be extended to the entire space-time. Galloway (1989a) removed the process In the space-time c s e , some care is needed to ensure that the llmiting

assunlption of tirnelike completeness from Eschenburg's work, and finally New- geodesic, which on a priori abstract grounds could be timelike or null, turns

man (1990), rounding out the circle of papers, gave a proof assuming timelike out to be timelike rather than null.

con~pleteness rather than global hyperbolicity. Here the new philosophy was 14.1 The Busemann Function of a Timelike Geodesic Ray that even though nonspacelike geodesic connectibility is not ensured without

the assumption of global hyperbolicity, within a tubular neighborhood of the We begin with a few remarks about the Riemannian Busemann function.

given complete timelike geodesic, the existence of the single complete geodesic Thus let (N, h) be an arbitrary Riemannian manifold with associated distance

enables certain limiting arguments to be made successfully in this tubular function do : N x N -+ R. Recall that even for incomplete Riemannian

neighborhood despite the possible general lack of geodesic connectibility in manifolds, the Riemannian distance function is always continuous and finite-

other regions of the space-time. It should also be recalled that certain el valued. Let c : [O, +m) 4 (N, h) be any future complete unit speed geodesic

mentary consequences of the existence of a maximal timelike geodesic segment my, i e , &(c(O),c(t)) = t for all t 2 0. Then we may consider a Busemann

have already been treated In Section 4.4. function b associated to the geodesic ray c by the formula

All of these results may be summarized in the following simple statement b(q) = t t ~ m [ t - do(c(t)? q)j of the Lorentzian Splitting Theorem = lim Ido(c(O), c( t ) ) - do(c(t), q)].

t-+m LorePltzian Splitting Theorern. Let (M. g) be a space-time of dimensio Since the Riemannian distance function a continuous and the triangle inequal-

76 > 2 which satisfies the following conditions: ity holds, elementary arguments reveal that b(q) exists, 1s finlte-~alued, and (1) (M, g) is either globally hyperbolic or timelike geodesically complete; is continuous. Let us review these arguments to have them firmlv fixed Put (2) ( M , g) sat~sfies the timelike convergence cond~t io~; and f (q, t) = t - do(c(t), q ) , so that b(qj is the iimi: of f (q. t ) a s t approaches +m.

(3) (M , g ) con t a n s a compfete t~melike line. First the triangle inequality yields

Then (M. g) splits isornetnca1ly ;is a product (R x V, -dt2 @ h), where (V, f (4, t ) l do(c(O), q: is a complete lijemmnim mmifold.

and second for any t > s that Many of the arguments given in the references cited above are rather corn

plicsted. Thus in Galloway and Borta (1995), considerable simplifications we f (41 t ) 2 f (9, s).

(14.7) and the finiteness of Riemannian distance, is bounded from above hence versions of the sphtting theoreem with the more desirable timelike convergence

has a limit. Further, one has the elementary estimate Ricci curvature condition were estabiished, an important new realization was

first found in Eschenburg (1988). This was the realization that with the weaker (14.9) If ( ~ , t ) - f (q, t)l 5 ~ o ( P , qf curvature condition, although regularity for the Busemann functions could

which shows the equicontinuity of this family in t and q and passes in the limit not initidly be obtained on large subsets of the given manifold, the existence

of the future complete timelike line ensured good behavior of the Busemann to the inequality

function and of asymptotic geocimics in a tubular neighborhood of the given

(14.10) I~(P) - b(q)l 5 do@> line. This control also sufficed to prove the Lorentz~an Splitting Theorem for

globally hyperbolic, timelike geodaically complete space-tlmes. For instance, which evidently establishes the continuity of the Eemannian Busemann func- in Section 3 of Eschenburg (1988) it was shown that the Busemann function tion. is Lipschitz continuous in a neighborhood of a given timelike geodesic ray.

Recall that for non-globally hyperbolic space-times, the Lorentzian dis- An important aspect of the differential geometry of a complete, noncompact tance function need not be continuous, only lower-semicontinuous (cf. Lemma Riemannian manifold (N,go) is the construction of an asymptotic geodesic ray. 4.4), and also that chronologically related pairs of points p << q may have starting at any p in N, to a given future complete geodesic ray c : 10, +m) -+ d ( ~ , q ) = +m (cf. Figure 4.2 and Lemma 4.2). Further, d(p,q) = 0 automati- (N,go). Let us review this construction to motivate some of the technicalities cally if q is not contained in Jf ( p ) , and the triangle inequality is replaced by which must be overcome in the space-time setting. Take it,) with t, -+ +oo. the reverse triangle inequality, assuming the points involved satisfy appropr By the Bopf-Rinow Theorem, there exists a minimal unit speed geodesic c, ate causality relations. Thus some care is needed in the preliminary analys with h ( 0 ) = p and e,(&(~,c(t,))) = c(t,). Letting v be any accumulation of the space-time Busemann function, especially as one case of the Lorentz point of the sequence of unit tangent vectors (cnl(0)) in T'n/l, put y(t) = splitting problem assumes timelilie geodesic completeness and not global exp,(tv), defined for all t > 0 by the geodesic completeness. As a limit of a perbolicity. Even assuming global hyperbolicity and hence guaranteeing a con- subsequence of the minimal geodesic segments {c, \ 0. d o ( ~ , ~(t ,))]) , c ~rlust be tinuous, finitevalued distance function, examples of space-times conformal globally minimal, recalling that Minkowsk'~ space show that it is possible for the space-time Busemann functio

to assume the value -co or to be discontinuous unless further assumptions (14.11) iim do(p, ~ ( t , ) ) = +oo

made. In Beern, Ehrlich, Markvorsen, and Gailoway (1985), which studied t since the triangle inequality gives the immediate estimate

Lorentzian splitting problem for globally hyperbolic space-times with time1

sectional curvatures K 5 0, it was shown that both Busemam functions do(^, c(tn)) L do(c(0). c(tn)) - &(P, ~ ( 0 ) ) = in - do(p, ~ ( 0 ) ) . and b- associated with the two different ends of a complete timelike geod

line c were continuous on the subset I(c) = 17(c) f7 I+(c), and it was Even though S. T. Uau formulated the Lorentzian splitting problem for

established that I(c) = M. Hence the sectional curvature assumption ens timelike geodesically complete space-times, as recalled in the introduction to

that the Busemann functions associated with a complete timelike line this chapter, the first attacks on this problenl were carried out instead for

continuous on all of M . globally hyperbolic spacetimes, for this is the class of space-times for which

510 14 THE SPLITTING PROELEILI IN LORENTZIAN GEOMETRY 14.1 THE BUSEhlANN FUNCTION 511

maximal geodesic segments exist connecting causally related pairs of points.

Hence, one may a t least start the construction as above by taking maximal

tirnelike unit speed segments c, : {O,d(p, c(t,))] -+ ( M . 9).

Now, however, a problem arises in considering {c,'(O)) in that even tl~ough

the space of futt~re nonspacelike directions at p is compact (despite the fact

that she set of unit future timelike tangent vectors is not compact). the limit

direction v obtained for this sequence might be a null vector rather than a

timelike vector. Dealing with this technicality motivated the introduction of

the t~melike co-ray condition in Beem, Ehrlich. Markvorsen, and Galloway

(1985) as well as the proof that this condition would be satisfied provided that

the timelike sectional curvature assumption K < 0 was imposed in order to

bring Harris's Toponogov niangle Theorem of Appendix -4 to bear.

Subsequent proofs of the Lorentzian Spl~tting Theorem in the globally hy-

perbolic case under the weaker Ricc~ curvature assumption dealt with showing

that the asymptotic geodaic construction would be well behaved in some

neighborhood of the given timelike ray. On the other hand. when Newman

(1990) returned to Yau's original formulation of the problem, he had to deal

with the further complication that the existence of maximal geodesic segments

connecting pairs of causally related points could not be assumed. Thls led to

the necessity of working with sequences of "almost maximal curves" as In

Section 8.1 and also in the treatment of Galloway and Horta (1995) to the

consideration of the "generalized co-ray co~~dition."

In the construction of asymptotic geodcslcs from almost maximal curves as

treated in Galloway and Horta (1995), a somewhat different formulation of

the basic nonspacelike limit curve apparatus is employed than that discussed

in Section 3.3 above. particularly Propositions 3.31 and 3.34. Their approach

ha- the advantage of not requiring the assumption of strong causality since

the topology of uniform convergence on compact subsets, rather than the C0

topology on curves, is used in Proposition 14.3 below.

Fix a complete Riemannian metric h for the space-time ( M : g) throughout

the rest of this section.

Convention 14.1. Given a future inextendible nonspacelike curve y in

(M,g) . y will always be assumed to be reparametrized to be an h-unit speed

curve unless otherwise stated.

Hence, with such a parametrization. the Hopf-Rinow Theorem guarantees

that

7 : [ao. +w) -+ (34. g)

(cf. Lemma 3.65). Similarly, a fiitltr~ causal curve c which is both past and

future inextendible may always be given an h-parametrization

c : (-00, +m) - ( M : g).

Starting with Galloway (1986a) and continuing In Eschenburg and Galloway

(1992) and Galloway and Horta (1995). the following formulation of the Limit

Curve Lemma has been found helpfill.

Lemma 14.2 (Limit Curve Lemma). Let y, : (-oe, +m) --+ ( M , g ) be

a sequence of causal curves parametrized with respect to h-arc length, and

suppose that p E M is an accumulation point of t h e sequence {y,(O)). Then

there exist an inextendible future causal curve y : (-00. +m) --+ ( M , g ) such

that ~ ( 0 ) = p and a subsequence (7,) which converges to y uniformly with

respect to h on compact subsets of B.

As noted in Eschenburg and Galloway (1092). dn advantage to the use of the

h-limit curve reparametrization is the following upper sernicor~tiriuity result.

obtained without the assumption of strong causality (cf. Remark 3.35).

Proposit ion 14.3. If a sequence 7,L : [a, b] -4 (hf, g ) of future causal

curves (parametrized with h-arc length) converges uniformly to the causal

curve y : !a, h ] --+ (M, g), then

L(y) L !im sup L(y,)

Proof, Partition [a, b ] as a = t o < t ; < . . . < t , = b such that each pub-

segment 7 / [t,, t,+l] is contained in a convex normal neighborhood A',. By the

512 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14.1 THE BUSEMANN FUKCTION 513

assumption of uniform convergence, y, it,, is also contained in this neigh- A future directed nonspacelike geodesic ray in the sense of Definition 8.8 borhood N, for all sufficiently large n. Hence the known upper-semicontinmty may then be given an h-reparanletrization as an S-ray with S taken to be

of the Lorentzian arc length functional may be applied to the strongly causal S = <y(O)). Conversely, an S-ray y in the sense of Definition 14.4 may be

space-time (N,, g /N*) to obtain U? / [t,,t ,+~]) 2 l i m s u ~ , ~ - + ~ L(?, 1 it,, t , + ~ ] ) , reparametrlzed to be a nonspacetike geodesic ray in the sense of Definition 8.8 and now summing over z produces inequality (14.12). U

As was noted in Eschenburg and Galloway (1992, pp. 211-212). a second ad- d(y(01,dt j ) < d(S, -4)) = / [O, t j),

vantage of the use of the awdiary h-parameter is that the Busemann function which implies that y may be reparametrized as a Lorentman distance realizing may be defined for a h ture incomplete timelike geodesic ray 7 : 10, a) --+ ( M , g) geodesic and d(y(O), y ( t ) ) is finite for any t > 0. Of course, in the applications even though the finiteness of a does not permit t -+ +oo in the expression S will usually be taken to be a spacelike hypersurface unless S = { ? ( O ) ] . analogous to (14.6) above. But if we reparametrize y with h-arc lengtli, then The first steps toward establishing the regularity of the Busemann function

y : [0, +m) 4 (M. g), and we may put and the Lorentzian distance function in a neighborhood of the given timelike

(14.13) ray may now be taken [cf. Eschenburg and Galloway (!992), Galloway and

b(q) = lim b,(q) T-DI)

where Lemma 14.5. Let y : [O, +GO) 4 (M,g) be an S-ray pardmetrized by

(14.14) br(q) = d(?{O), ~ ( r ) ) - 4 9 % % T I ) h-arc length. Then

and where d denotes the Lorentzian distance function of the given space-time (1) d(p,q) < +GO for all p,q E I-(?) n It(S), and especlaily d(p,p) = O

(M,g) . Note that if q E n/i - I - ( y ) , then d(q, y fr ) ) = 0. Hence if y is future for all p E I-(?) n I+(S);

complete and thus a = -a, we have that b(q) = +co. Thus the space-time (2) For d q E I-(?) f7 I L ( S ) , d(S. q) > 0 is finice.

Busemann function b : M -, [-XI, +m] in general. (3) The Busemmn function b : I-(?) --+ [-GO, +oo] associated to y exists

As recalled above, for general noriglobally hyperbolic space-times, the Lor- and is upper semicontinuous;

entzian distance function can ediibit various pathologies. Nonetheless, as first (4) Thc Busemann function b associated to tile S-ray y is finite-valued and

noted in Eschenburg (1988), the existence of a timelike geodesic ray y give positive on I - ( y ) n I+(S);

finiteness of the distance function and Buseman11 function in I - ( y ) n I+(y(O (5) For p, q E I-(?) n I+(S) with p < q. the Busemmn function satisfies

It will be useful to develop these propesties in the more general context of b(q) 2 b(P) $. d(p, 4) ; and

rays, a s introduced in Eschenburg and Galloway (1992). Thus let S b (6) Suppose q,, = y(t,) and p, = y(s,) with s, 5 t,, and suppose t , - t subset of M, and recall that the distance from q to S is defined as d(S, q) and s, -+ s. Put q = y ( t ) and p = $8). Then d(p, q) = lim d(p,, q,).

sup{db, 9) . P S). Proof. (1) Unless p < q, d(p, q) = 0 automatically. Thus suppose p 9 q.

Definition 14.4. (S-rug zn a Space-tzme) The fkture inextendible caw Take any s E S n J-(p), and choose ro > 0 so that s G p $ q << ~ ( r ) for any

curve y with h-arc length parametrization y : [O, +m) + (M. g ) is said to b ch s and r 2 ro. The reverse triangle inequality then yields

an S-ray if $0) E 5' and 7 maximizes distance to S, l.e., d(s, P ) + d@, q) + d(q,y(r f 5 4 8 , y (r ) )

(14.15) L ( y j [O, a]) = d(S, y (a) ) for all a 2 0. 5 d(S, y(rj) = L ( y 1 [O. r ] ) < +oz

514 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY

In particular. d(p.q) must be finite From Lemma 4.2-(1) we then have

d(p,p) = O for all p E I-(*) n d+(S)

Recalling that L(y 1 [O.r]) = d(y(Q), y ( r ) ) , the above chain of inequalitm

also yields d(s, p) + d(p, q) + d(q, y(r ) ) 5 d(y(O), y(r)) for any s in S n J-(p).

Since d(s,p) = O automatically if s E S - Jm(p) , we obtain from thls 1 s t

mequality

(14.16) d (S , P ) + d(P 9) + 4 % ~b-1) 5 d(y(O), ~ ( 4 )

for p < q and any r 2 1-0 chosen as above

(2) Take p = q in (14.16) to obtain

(14 17) 4% 9) + d(q,y(r)) 5 d(?(O), ?(TI)

for all suEciently large r. Since d(y(0). yjr)) is finite, d(S, q) must also be

finite. Since q E I+(S), d(s, q ) > 0 for some s in S ; hence d(S, q) > 0.

(3) Suppose ,T E I-(?). Thus there exists T > O with x < y(r) . Take any

s > r. Then

4x7 y(s)) L d(z .y ( r ) ) + d(-f(r),?(s)).

and also

d(y(Q), y(s)) = d(4(0), ?(TI) f d(y(r) , ?(§)) .

Hence

bs(z) = d(y(O),y(s)) - 4 2 , r j s ) )

5 d(y (O)*?(~) ) + d(y(r) , y(s) ) - Id(x.y(r)) + d(y(r) , y(s))I

= d(r(O), y(r)) - d(x, ? ( T I ) = b, (x) .

Thus for fixed x E I - (y) , r H b,(x) is decreasing monotonically in r . so t

b(x) = brn,,,, b,(xf exists, possibly with the value -a or +m. By the 1

semicontinuity of Lorentzian distance (cf. Lemma 4 4), each b,(x) is upp

semicontinuous. and hence the limit function b(x) is also upper semicontlnu

(4) Given any z E I-(?) n I+(S), choose s > O so that x < - / (s )

14.1 THE BUSEMANN FUNCTION 515

b,(x) = d(? ( O ) , y ( s ) ) - d(x, y ( s ) ) is finite by (1) applied to p = x , q = y ( s ) ,

since d(y(O), *(s)) = L(y 1 [0, s]) is automatically finite. Hence as r H b,(x) is

monotone decreasing, we have b(x) < +m It remains to rule out b ( z ) = -a.

To this end, take q = x in equation (14.17), and subtract d(z, y(r ) ) to obtain

b,(x) 2 d(S, x) > 0 for any r > s. Taking limits yields

for any x in I - ( y ) n I+(S) .

(5) Choose TO so that p < q 4 y(r) for all r 2 ro. Applying the reverse

triangle inequality to p < q y(r) yields

Let r -+ +m to obtain the desired inequality.

(6) Since y is maximal, we have d(pn, q,) = L(y 1 [s,, t,]) for all 11. Thus

limd(p,,q,) = lim L(y 1 [s,,tn]) = lim L(y 1 [s, t ] ) = d(p,q).

It has been noted above that in considering the constrilction of asymptotic

geodesics, somewhat different techniques must be employed to deal with the

timelike geodesically complete, but not necessarily globally hyperholir, case.

Hence we find certain conventions employed in Galloway and Horta (1995)

to be helpful and will use them below. First, we formalize certain aspects

of Section 8.1, recalling here that unlike Section 8.1, strong causality is not

assumed and also that curves are always given an h-arc length parametrization

in this section unless otherwise stated.

Definition 14.6. (Lirnzt Meximzzzng Sequence of Causal Curves) A se-

quence y, . [an, hn] 4 ( M , g ) of h-arc length parametrized future causal curves

is said to be limzt marcimizzng if

where en + 0 as n -+ +m.


Gwen any p, q in ( M . g) with 0 < d(p, q) < TCO, the definition of Lorentzian

distance implies that a sequence of l im~t maximal curves from p to q may

be constructed. In a manner similar to the proofs In Section 8.1, the lower

semicontinuity of dlstance and the upper semicontlnity of L established in

Proposition 14.3 combine to yield

Proposition 14.7. Suppose that -i, : [a,, b,] --. (M, g) is a limit maxri-

mizing sequence of future directed causal curves that converges uniformly to

a causal curve y : [a, b ] -+ M on some subinterval [a, b j E (),[an, bn]. Then

L(y) = d(y(a), y(b)), and hence y may be reparamecrized as a future directed

maximal geodesic from y (a ) to y(b) .

Proof. Putting y, = yn I [a, b], we have as usual

limsupL(7,) 5 L(y) 5 d(y(a),y(b)) n-+m

5 llm inf 47, (a), yn (b) ) n-+oo 5 lim inf (L(yn) + 6,)

,+too

= lim inf L(y,), n-+w

whence

(14 20) L(y) = hm L(-;/,) = lim d(y,(a),%(b)) = d(r(a),y(b)). n++m n-+m

The following lemma from Eschenburg and Galloway (1992) has been help

in constrricting asymptotic geodesic rays It should be emphasized that unl

(14.11) above for complete R~emannian manifolds, the condition d(z,, p,)

+oo in Lemma 14.8 dam not imply that (p,) divergs to infinity but only t

a, --+ +w.

Lernrna 14.8. Let (2,) be a sequence in (M.g) with z, 4 z. Let

IS(z,) with d(a,, p,) < +m. Let -/, . [0, a,] + (M, g) be a lirnit maxi

sequence of future causal curves with ~ ( 0 ) = z, and y,(a,) =

yn : 10. +m) -+ (M. 9) be any future lnextendible extension of 7,.

that d(p,, zn) + +DO. Then any limit curve y : [0, +co) + (M,g)

sequence {T,) is a nonspacelike geodesic ray starting a t z.

14.1 THE BUSEMANN FUNCTION 517

Proof. It is necessary to show chat a, 4 +a. Suppose not. By passing to

a subsequence, we may suppose that an -+ a < +x. Because the curves y,

are parametrized by h-arc length and h is a complete bemanman metric, it

follows that all y, are contained in a, compact subset S of M. Note also that

in this case we have Lh(yn) = a, -+ a < +w.

On the other hand, we will show that the assumption that d(z,,p,) -+ +DO

implies that Lh (7%) + +oo, yielding the required contradiction. To this end.

put a second auxiliary Riemannian metric ho on M more closely associated to

the given Lorentzian metric g by the following standard construction. Let T

be a unit timelike vector field on (M, g), and define an associated one-form by

r = g(T, .). Put

h 1 ~ = ~ + 2 r @ r = ~ ' + r @ ~ ,

and note that for this second Riemannian metric we have the given Lorentzian

arc length of any causal curve segment c and the ho-arc length related by

Since h and ho are Riemannian metrics and K is compact, there exists a

constant X > 0 such that h(v, c) 2 X2 hO(u, V ) for any v E TM / K, whence

for any curve segment c contained in K. Thus we have

Lh('Y731 2 ALg(%) > Xd(&,pn) - At, --+ +a by hypothesis. Cl

'4 more complex lemma is considered in Escheiiburg and Galloway (1992)

in order to allow for dealing with inextendible timelike rays which are not

necessarily future complete. But in view of the hypotheses involved in the

Lorentzian splitting problem, let us suppose now that y : :O, +oo) -i (M.g)

is a future complete t i d i k e S-ray, hence of infinite Lorentzian length. Fix

z E I-(y) n I+(S). Let z, -, z; and put p, = y(r,) where r, + -co. Then

d(z,,pn) < -kc for all sz suffciently large by Lemma 14.5-(1). On the other

hand, if we fix s > 0 with t << y(s), then for all sufficiently large a we have

518 14 T H E S P L I T T I N G P R O B L E M I N LORENTZIAN GEOMETRY

r , << ~ ( s ) as well. Hence the reverse triangle inequality yields d(r,.p,) >. d(zn,T(s)) + ~ ( Y ( s ) , " / ( T ~ ) ) --+ -too since d(r(s) .y(r ,)) = L(y / Is,r,]f

by hypothesis.

Thus wlth Lemma 14.8 in hand and the above verification, the coxtruction

of asymptotic geodesics to a future complete S-ray, y . 10, +m) + (M, g). may

now be accomplished Simply let {p,) be any sequence of limit maximizing

curves from z, to p,, and let p 10, +oo) --+ (M, g) be any limit curve guar-

anteed by Lemma 14.8 for the extended curves (5,). The use of the auxiliary

complete Riemannian metric h to pwametr~ze the curves by h-arc length is

what ensures the uniform convergence on compact subintervals necessary to

apply Proposition 14.7 and Lemma 14.8.

We will adopt the following terminology of Galloway and Horta (1935).

Definition 14.9. (Co-rays, Generalzted Co-yaps, and Asymptotes)

(i) Any ray p constructed in this fashion will be called a generalzzed co-ray

to :he given future complete S-ray y.

(2) The ray p constructed in this fashion will he called a co-ray to the

given future complete S-ray y if all y, are distance maximizers, I e , L(pn) = d(zn, p,) for all n.

(3) The ray p constructed in this fashion will be called an asymptote if all

p, are distance maxlmlzers and if t, = z for all n

In the space-time setting. the concept and term "co-ray" were first In-

troduced in Beem. Ehrlich, hlarkvorsen. and Galloway (1985), influenced by

terminology in Busemann (1955). Generalized co-rays for space-times were

first employed in Eschenburg and Gailoway (1992)

W-hiie considering limit maximizing sequences of timelike curves. Newm

(1990, Lemma 3.9) noticed an important ~mplication of timelike geodesic c

pieteness which is formulated In Galloway and Horta (1995) as follows.

Proposition 14.10. Let (M,g) be future timelike geodeslcally comple

Suppose p, q are points in (M, g ) mth p < q and d(p , q) < fm.

?;1 . [0, a,] -+ (M, g) be a Iimit rnaxirnizing sequence of future directed caus

curves from p to q. For each n, let Fn . [O, -%a) --t (M. g) be a future in

14.2 CO-RAYS A N D T H E B U S E M A N N F U N C T I O N 519

tendible extension of y,. Then if7 : [O. f a ) --t (.V, g) is a iimit curve of {?,I, either y maximizes from p to q or y is a null ray.

Proof. Let a = sup(a,). The exis~ence of convex neighborhoods centered

at p implies a > 0. By passing to a subsequence if necessary, we mav assume

that a, -. a. Then by Proposition 14.7. the iimit curve y [0, a) 1s a maxima!

nonspaceliko geodesic.

First suppose a < +w. Because we have diosen an auxiliary complete

Riemannian metric and parametrized the y, by l'L-arc length. we know that

y I [0, a ) extends to t = a. Also, we have that y / [O, a] is iriaxirnal, and ?(a) =

lim,,+,y,,(a,,) = q. Since d(p. q) > 0 and 7 is rnaxinial from p to q in this

case, y must be timelike.

Now suppose a = -too. If y is null. then y is the required null raj, and

we are finished. Thus suppose 7 is tirnclike. By the assumption of geodesic

completeness, y has infinite Lorentzian length. As d(p, q) is a finite positive

number. there exists T > 0 such that L(y ] [O, TI) > dfp. q) . Since a, -+ +GO,

equation (14.20) in the proof of Proposition 14.7 implies that for n sufficiently

large, we have T < a,, and hence

But this is impossible because y, is a timelike curve segment from p to q.

whence L(yn) < d (p . q) by definition of Lorentzian distance. Ci

14.2 Co-rays and the Busernann Function

It u7as first noted in Beem. Ehrlich, Markvorsen. and Galloway (1985)

that the acsumption for a globally hyperbolic space-time that all cc-rays con-

tructed to either end of a complete timellke geodesic line y : (-m. +m) +

M,g) turn out to be timelike rather than null lrnplies that the Busemann

functions bi and b- associsted to the two different ends of y are continuous

and also finite-valued on the beta I-(y) and I+(- / ) , respectlveiy. This assump-

tion was called the "timelike co-ray corid~tion.' It was also shown, employing

rris's Lorentzian version of the Toponogov Coii~parison Theorem for time-

geodesic triangles in globally hyperbolic space-times (cf. Appendix A).


that the assumption of everywhere nonpositive timelike sectional curLraturea

ensures that the given globaliy hyperbolic space-time satisfies this timellke

co-ray condition.

In the subsequent works, beginn~ng with Eschenburg (1988), more difficult

studies of the regularity properties of the Busemann function were made as-

suming only the timelike convergence condition on the Iticci curvature; hence

the space-time Toponogov Comparison Theorem could not be brought to bear

on this issue.

Another ingredient in the proof of the Lorentzian Splitting Theorems w

the fortuitous publication of Eschenburg and Weintze (1984) in which a ne

proof method for the Riemannian Cheeger-Gromoll Ricci Curvature Splitti

Theorem was employed. utilizing the technique of smooth upper and lower

support functions for the (a priori only continuous) Busemann functions of

a geodesic line. The origlnal proof method of Cheeger and Grornoll (1971

relied heavily on the ellipticity of the Laplacian and hence did not seem to be

adaptable to the case of the hyperbolic dlAlembertian.

We begin this section with the definition of these support functions as fir

treated in Eschenburg (1988) for the globally hyperbolic case and later treat

In Galloway and Horta (1495) in the current context of timelike geodesic co

pleteness. Let y : [0,+w) -+ ( M , g ) be a future complete S-ray in a spac

time ( M , g), and let b be the associated Busemann function given hy eqilatio

(14.13) and (14.14).

Definition 14.1 1. (Upper Support Fanctzon) Let a : [0, +m) -+ ( M ,

be a timeliie asymptote to 7 with a(O) = p E I - ( y ) f l I+(S) . For each 5 > let

b,,, : ,v -+ [-co, +m]

be defined by

(14.21) bp,s(2) = b(p) + d(p, 4 3 ) ) - d(x , 4 s ) ) .

Lemma 24.12. For any 9 > 0 the function given by (14.21) is

continuous upper support function for b at p, i.e , b,,,(x) 2 b(x) for all x n

p with equizlity holding when z = p.

14.2 CO-RAYS AND THE BUSEMAKN FUNCTION 521

Proof. By definition of an asymptote, there exists a sequence of maxi-

mal timelike geodesic segments a, : [O a,] --+ ( M , g) with a,(O) = p and

a,(a,) = y(r,). where r, -+ i-m as n --+ +m Also, since y is future com-

plete, a, -+ fix; slnce d(p.r(r,)) -+ i m . Let z be In the open neighborhood

U = I - ( a ( s ) ) f'l I+(S) of p for the fixed s > 0. Then for n sufficiently large.

x << a,(s) as well. By lower semicontinaity of distance, there exists 4, 4 0

such that d(x , cu,(s)) 2 d(x, a@))-S, for all large n. Then for all n sufficiently

large. the reverse triangle inequality and the maximality of y imply

By (14.20) also, d(p,a,(s)) -+ d(p ,a(s) ) as 71 -+ +w. Hence taking the

limit in (14.22) yields b(z) - b(p) 5 d(p, a ( s ) ) - d(x .a ( s ) ) , so that

as required. D

As noted by Eschenburg (1988) in the globally hyperbolic case, the expected

behavior of the Busemann function associated to the glven ray y on rays a

mymptotic to y , akin to that previously observed for complete Riemannian

manifolds, may now be obtained.

Corollary 14.1 3. Let y : 10, +m) -+ ( M , g) be a future comple~e S-ray

in the space-time (M,g ) , and let b be the Busemann function associated to

y. If a a 10. +m) -+ (M,g ) is a timelike asymptote to y starting at p in

I-(?) n I + ( S ) , then b(a( t ) ) = d(p, a ( t ) ) + b ( p ) for all t 2 0.

Proof. Taking x = a(t) and fixing any s > t in (14.23) gives

522 14 THE SPLITTING PIEOBLEhI IN LORENTZIAN GEOMETRY

since a is maximal. On the other hand, applying Lemma 14.5-(5) to p and

q = a ( t ) yields

b ( 4 t ) ) L b(p) + 4 ~ . 4 t ) ) . 0

Now we are ready to state the crucial new condition in Galloway and Norta

(1995) which leads to the local regularity of the Busemann function as well as

some control over maximal geodesic connectibility.

Definition 14.14. (Generalized Tzrnelzke Co-ray Condztzon at p) Let

( M , g) be a future tirnelike geodesically complete space-time, and let y

timel~ke S-ray Suppose that p E I - ( y ) n I + ( S ) . Then the generalzzed tzrnelz

co-ray condztzon holds a t p if every generalized co-ray to y starting at p i

tirnel~ke.

Arguments similar to thobe of Lemma 14.15 below show that this cond~tio

is an open condition, i.e , ~f the conditiori holds at a given p E I-(?) n I t (S

then the condition holds in some neighborhood of p.

Lemma 14.15. Suppose (hf, g) is future timelike geodesically comple

and Jet y be a timelike S-ray. Assume that the generalized tlmelike c

condition holds at p E I-(?) n I+(S). Then there exist a neighborhood

p and a constant R > 0 such that for all q E U and for dl r > R, there

a maximal timel~ke geodesic segment from q to y(r) .

Proof. Assuming that the desired conclusion is false, we will show that

generalized timelike co-ray condition cannot hold at the given point p

diagonalizing argument.

Supposing that the conclusion of the lemma is false, we may find { p ,

2-(y) n I+(S) with pn --t p, r, --t +m, and p, << y(r,) such that there

no maximal timelike geodesic segment from p, to y(r,) for ail n. In vie

Lemma 14 % ( I ) , we have 0 < d(p, , , ~ ( r , ) ) < +m for each n. Hence

n we may construct n limit maximizing sequence of future nonspacehke cu

ank [O, ank' + ( M . g)

from p, = ank(0) to y (r,) = ank(ank) with

L ( ~ n k ) 1 d(~nk (O) , ~ n k ( a n b ) ) - enk

14.2 CO-RAYS AND THE BtiSEMANh FUNCTION 523

where limk,,, e,k = 0. Because of the dshumed nonexistence of timelike

maximizers from p, to ?(m), the proof of Proposition 14 10 of Section 14.1

implies :hat l:mk,+, a,k = +m and also that the sequence {ii.,k) converges

to a null ray as k - +m, which we will call 0, . [O, +m) --t ( M , g ) , with in~tial

point .Dn(O) = p,. Hence equation (14.20) forces

for any s > 0.

Now do a diagonalizing procedure on the nonspacelike curves ( a n k ) to find

an increasing sequence k(n) --+ f oo with the property that for 7 , = a,&&(,)

and b, = u,k(,), the associated sequence jVn) satisfies

(1) b, + +o=:

( 2 ) L(7, / [O, 11) < 11% for all n; and

( 3 ) L(7n 1 [O,bn]) 1 d(~n(O).qn(bn)) - 1/n for a11 n.

By ( I ) , (3); arid Proposition 14.7, the sequence {q,) converges to a nonspace-

hke geodesic ray q with q(0) = p. By co~ldition (2) , the ray q must be null.

Thus negating the desired conclusion. we have produced a generalized co-

q to y with q(0) = p which is a null ray. But this contradicts the hypothesis

of the lemma that all co-rays to y at p are tlmelike. C

The next several results from Galloway and Horta (1995) treat the rela-

tionship between limit curve convergence and convergence by initial tangent

irection for timelike geodesic segments and in particular the property that

he timelike co-ray condition ensures that initial tangents of asymptotes con-

tructed in a neighborhood of p are bounded away horn the null cone. Out

these considerations. among others. the local continuity of the B~isemann

nction may then be established in Theorem 14.19,

It is helpful to adopt some mechanism to produce appropriate compact

sets of the noncompact set of unlt timelike tangent vectors. To be consistent

h Chapter 9, we will recall the notat~on T-liLII, from Definition 9.2 and

troduce a new notation following Galloway and Horta of

524 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14.2 CO-RAYS AXD THE BUSEMANN FUNCTION 525

where C > 0 and h is the auxiliary Riemannian metric for ( M , g ) fixed above. But this is contrdictory since h(a'(O),a'(O)) is finite, yet by supposition,

These sets are compact and nonempty for C sufficiently large, and evidently limn,+, h(crnl(0). anf(0)) = +oo.

Corollary 14.17. Let (M, g) be future tlmelike geodesically complete, and

let y be a timeiike S-ray. Suppose the generdized tlmelike co-ray condition

Lemma 14.16. Let (M, g) be future timelike geodesically complete, and let holds at p E I-(7) n If (S). Then there exist a ne~&borhood U of p and a

r be a timelike S-ray. Assume that the generalized timelike co-ray condition constant C > O such that for ail q E U , if a : [0, +m) 4 (M, g) is a co-ray (or

holds at p E I-(?) n I'(S). Then there exist a neighborhood ti of p and asymptote) to y frorn q, parametrized with respect to Lorentzian arc length,

constants R > 0, C > 0 such that for alj q in U and any r > R, :f cx : [0, a) -, then a'(0) E Kc(q) .

(M, g) is any maximal timelike geodesic segment from q to y(r) parametrized The generalized timelike co-ray condition also has implications for the con-

tinuity of the Lorentzian distance function, as noted in Galloway and Norta

Proof Suppose that the lemma fails to hold. Then there exist sequenc

Proposit ion 14.18. Let (M, g) be future timel~ke g~odesically complete,

and let y be a timelike 5'-ray. Suppose the generdized timelike co-ray condition property that h(crn1(0),cun'(0)) --, +oo as n + +m. To apply the limit c holds at p E I-(7) f l I+(S). Then there exist a neighborhood U of p and a

number R > 0 such that for al! r > R the function 6 : U -i [O. fcc) defined

by 6(x) = d(x, yjr)) is continuous on U . produces a nonspacelike geodesic ray Z with Z(0) = p and with Z a co-r

Proof, Recall first that Lemma 14.5-(1) guarantees that 6 is finite-valued y, to which a subsequence of the 6,'s converge. which we may assume is

U. By Lemma 4.4, 6 1s lower semicontinuous, so it only remains to establish given sequence (6,) itself. Since the generalized timelike co-ray condition

Choose U. R, and C to satisfy the conclusions of Lemmas 14.15 and 14.16. speed timelike ray cu. : [O. +a) -i ( M , g). The domain is as indicated beca

any r > R. Suppose 6 is not upper semicontinuous at x in U. Then of the assumption of future timelike completeness.

Choose S > 0 so that 6([O, b ] ) is contained in a convex normal nei d(x, ?(T)). For each n, let a, be a maximal timelike geodesic segment whlch

borhood of p and also is contained in I - (7 ) . Because of this suppositi s parametrized by g-arc length, from x, to y(r) . In view of the conditions

tisfied by U , R, and 6, the initial tangents (anl(0)f- are contained in a d(E,(O), Gn(6)). By equation (14.20) we have E, 4 E. Now using the

ompact subset of T_lMI,. By this fact and the assumption of timelike future timelike completeness and choice of convex normal neighborhood, w

geodesic a frorn x to y(r). But then by equat~on (14.20), we ha%e €,an1(O) = exP2 (E(6)) -, exppl (~(6)) = E al(0).

Hence, a,'(O) -+ al(0) which implies that d(xn(k), Y(T)) = L(an(ic)) + L(a) = d(x, 7(r)).

ontradiction.

526 14 THE SPLITTING PROBLEM Ih' LORENTZIAN GEOMETRY

With these results now established. we are ready for the proof given in

Galloway and Horta (1995) of the continuity of the Busemann function in the

presence of the generalized tlmei~ke co-ray condition. The treatment is inspired

by that in Eschenburg (1988) for the globally hyperbolic case.

Theorem 14.19. Let (M, g) be future timelike geodesically complete, and

let y be a timelike S-ray. Assume that the generalized timelike co-ray condition

holds at p in I-(?) n If (S). Then the Buseiiiann function b associated to 7 is

Lipschitz continuous on a neighborhood ofp.

Proof. It is helpful to use a lemma given in the Appendix to Eschenbur

(1988).

Sub lemma 14.20. Let U be an open convex domain in Wn and f . U

a continuous function. Suppose that for each q in U there is a smooth 1

support function f, defined in a neighborhood of q such that l/d(fq)qll < Then f is Llpschltz continuous with Lipschitz constant L. i.e., for all x , y i

U we have

Proof of Sublemma. First treat the case that n = 1 and thus U is an op

interval and the estimate amounts to supposing that

in U such that by replacing f with - f if necessary, we may suppose t

y, and f satisfy f ( x ) < f(y) and also If(z) - j(y)I > Ljz - yl. G considerations evidently enable one to find an affine function l ( x ) with

slope I1(x) = Lo > L such that f (sf < E(s) and f ( y ) > l(y). Put qo = su

[z, y] : f ( t ) < l ( t ) ) . By continuity of f, evidently fm (qo) = f (qo) = l(qo)

f,, (t) 5 f (t) < l ( t ) for all t < qo sufficiently close to qo. Hence,

l1(q0) = Lo > L, in contradiction to inequality (14.26). Thus the

estimate (14 25) must be valid in the case that n = 1.

14.2 GO-RAYS AND THE BUSEMANN FUNCTION 527

Now consider the case that n > 1. Fixing arbitrary x , y in U , let c(t) =

(1 - t ) x + ty and I = W nc - l (U) . Put g = f o c : I -+ R. Note that In view of

the chain rule, the relevant constant in (14.26) applied to g is L1 = L / jx - y / .

Applying the n = 1 case to g ( t ) with t = 0 and t = 1 yields

required.

Proof of Theorem 14.23. Let y be parametrized by Lorentzian arc length

below. Choose U, R, and C according to Lemmas 14.15 and 14.16. Further.

take U sufficiently small that is compact and is contained in a convex normal

neighborhood ( N : 4) such that Q(U) is convex in Wn, n = dim M. Let q5 =

( x I 7 . . , x,), and let ho denote the associated Euclidean metric for T N given

by n

2=1

Since U has compact closure and h, ho are both R~emann~an metrics, there is

a constant K > 0 such that ho < K h on TU.

Put d,(x) = d(x, y ( r ) ) . It is sufficient to show that the functions b, = r-d,

are Lipschitz continuous on U with the same Lipschitz constant for all T > R. Since

br(x) - b,(y) = d,(y) - d , j ~ ) .

this is equiwalent to obtaining the Lipschitz continuity of the d,, r > R, on

U . From Proposition 14.13 we have the continuity of the d,'s; it thus remains

only to obtain the Lipschitz estimate using Sublemma 14.20.

Let D . N x N + [O,+m) denote the local Lorentzian distance function

for (N, g I N ) as described in Definition 4.25 and Lemma 4 26. Fix any point

q in U and r > R. Let a! be a maximal time!ike geodesic segment from q to

~ ( r ) parametrized by g-arc length, guaranteed by Lemma 14.15. Flx any q' in

U which lies on cu with q # q'. Having made these choices, a candida:e for a

local smooth support function for d, near q may then be defined by


For o in a neighborhood of q, the hnctions z I-+ D(x, q'), and hence f,,,(x), are

smooth by Lemma 4.26. Also the Iocd distance function D generally satisfies

D(x, ql) 5 d(x, q') for any x, q' in N. Thus by the reverse triangle inequality,

fq,r(x) 5 4 x > q') + 4 q 1 > ?(T))

5 d b , ?(TI) = dr(x),

and also

fp,T(q) = D ( ~ ' 4') + d(ql> Y(T)) = d ( ~ l ql) d ( ~ ' l ~ ( r ) ) = dr(q)

using the maximality of a.

I t remains to establish an estimate for the lower support functions fq,r o

the form needed to employ Sublemma 14.20. To this end, put

G=sup(lg,,(x)l:3:EU, l I % , ~ < n )

and l/vl/o = J w ) for w € TqM Since U has compact closure In N,

have G > 0. The basic differential geometry of the distance function d(. , q

from the point q1 prior to the past timelike cut locus of q' provides the gradient

calculation

v (fq,v) = 9(v. al(0))

for any v E T,M (cf. Lemma 14.26 above) Hence,

I d (f,,,) (v)l = Idv, I G lI~'(0)llo llvlio

I G K J ~ ( ~ ~ ' ( o ) , a'(0)) I I ~ I I O

5 G K C ~ Hallo.

Hence, L = G K C ~ has the property that lid ( fq,r), 11 ( L for all q E

Thus d, and hence 13, are Lipschitz continuous on U with Lipschitz const

L = G K C ~ by Sublemma 14.20 for any r > R.

Having presented four results on the pleasing consequences of th

alized timelike co-ray condition, let us now show as in Gdloway and H

(1995) that in the Important special case where S = (-/(0)), and hence

S-ray 1s an ordinary geodesic ray, the generalized timeiike co-ray co

holds in a neighborhood of the given ray, possibly excluding the initial p

y(0). This may be accomplished by &st proving the follo-wing resuit.

14.2 CO-RAYS AND THE BUSEMANN FUNCTIOK 529

Proposition 14.21. Let ( M , g) be future timelike geodesicdly complete,

and let y : [0, t w ) + ( M , g ) be a future directed timelike ray. Then any

generalized co-ray starting at p = y (a), a > 0, must coincide with y.

Proof. Here the limit curve techniques of Section 14.1 will be employed, so

all curves will be given a unit speed parametrization with respect to the fixed

auxiliary Etiemalnian metric h. Let a : [0, +m) -+ (M, g) be a generalized co-

ray to y at p = y ( ~ ) . Thus there exists a limit maximizing sequence of future

directed causal curves a, : 10, a,] 4 ( M , g ) from p, to y(r,) with p, --t p,

r, + +w, and a, -, +oo, which converges to a. Let U be any convex normal

neighborhood of p. Choose any q in U lying on y to the past of p; and take a

sequence {q,} of points on y between q and p such that q, --+ p. Since q, << p

and pk 4 p, by taking a subsequence of (p,) if necessary, we may suppose

that q, << p, for all n and that {p,) C U . Since pn, q, are contained in U and

limp, = lim q, = p, we may connect q, to p, by a timelike geodesic segment

whose length approaches 0 a s n + fw.

Now define a new sequence {on : 10, s,] ( M , g ) ) of causal curves by

following from q to q,, then going along the above timelike geodesic segment

in U from q, to p,. and finally following along a, &om p, to yjr,). Put

qn = an(an) and p, = cr,(b,). Fix r > a so that p, E Im(y(r ) ) and r, > r for

all sufficiently large a. We now need to establish the limit maximality of this

new sequence {o-,}.

Since y is maximal and the an's are limit maximizing, we have

L(mn I LO: snI) = L(cn I [Otan]) + L(mn I [an, bn]) + L(gn I [bn, s,])

L d(q, qn) + L(5n I Ian, bn]) + d ( ~ n , ~ ( r n ) ) - 6 ,

= d(q, P) - djqnl P) + L(G I [an. b,,]) + d(?n, ~ ( r n ) ) - 6,.

On the other hand, the lower semicontinuity of distance at (p, y ( r ) j implies

the foilowing inequality is valid:

= d(p,y(r,)) - 6, with 6, -+ O as n --+ +CG.

530 14 T H E SPLITTING PROBLEM IN LOREKTZIAN GEOMETRY

Combining these two inequalities yields

L(ocT, I [O. sn]) L d(q P ) - d(qn>?) + L(0n I [an, bn]) + d(p, ~ ( r n ) ) - 6n - en

= d(q. y(r,)) - d ( q n , ~ ) + L(an 1 [a,,b,]) - 6, - c,.

Bjr Lemma 14.5-(61, we have d(q,,p) - 0 as n -+ +w. Also by constru

tion, employing the normal neighborhood U , we have L(cr,, I [a,,,b,]) -+ O

n -+ +m Hence {on} is limit maximizing as desired

By Lemma 14.8, any limit curve a of this sequence is a maximal nonspace!~

consist of the portion of y from q to p, followed by CY But then by the u

which implies that a is contained in 3 as desired. 0

the following consequence is obtained from Proposition 14.21.

Corollary 14.22. Let ( M , g ) be timelike geodesically complete, an

y : [0, t o o ) --+ ( M , g) be a future d~rected timel~ke ray Issuing from p =

on an open set c~ntaining y - {pj .

port functions bp,, to the Busemann function given in equation (14.21),

necessary to have some control of the timelike cut locus. This was done in

assumption of timelike gcodeslc completeness rather than global hyperbo

so that the theory of Chapter 9 is not immediately appl~cable

Proposition 14.23. Let (11.1. g ) be future timelike geodesically corn

and let y be a timelike S-ray. Suppose that the general~zed timelike c

condition hdds at p E I-(?) n l+(S), and let a : \O,+m) -+ (M.g)

14.2 GO-RAYS AND THE BUSEMANN FUNCTION 531

Proof By taking unions. it is sufficient to show that for each s in [O,r],

there exi9ts a neighborhood C of a ( s ) which does not meet the past timelike

rut locus of a(r ) . There are three cases to consider s = r; O < s < r; and

s = O

Case (1) s = r . Since N is a timelike ray, strong causal~ty holds at a(r)

by Proposition 4.40. Hence by Theorem 4.27, there exists a convex normal

neighborhood U of a ( r ) such that the global Lorentzian distance function d(g)

agrees with the local distance function (D. U) Thus nonspnrelike geodesics

emanating from the center a ( r ) of this convex neighborhood contain no non-

spacelike cut points within Li, and U itself provides the required neighborhood

Case (2) 0 < s < T . This case, which essentially corresponds to Newman (1990.

Lemma 3.10), IS similar to the s = 0 case but simpler. Thus d e t a ~ h will be

omitted.

Case (3) s = 0 This is the case which makes use of the general~zed timelike

co-ray condition. It is helpful to establish two sublemmas.

Sublemma 14.24. Let the assumptions be as in Proposition 14.23. Then

for any T > 0, the function x H d.,.(z) = d(x. cr(r)) is continuous at p.

Proof of Sublemma 14.24. Let pn -+ p. Starting with Lemma 14.5-(5) and

the lower semicontinuity of distance, we have for the Busemann function b = b,

of y that

b(a(r) ) - b(pn) 2 d ( p n , c*(r))

2 d(p. 4 ~ ) ) - 6,

= b(a(r) ) - b(p) - 6,,

the last equality from Corollary 14.13 Smce the generalized timelike co-ray

condition is assumed to hold at p. the Busemann function h is continuolls a t p

from Theorem 14.19. Hence d(pn, rr(r)) - d ( p , tu(r)) as n --+ +GO. C1

Sublemma 14.25. Let the assumptions be as in Propos~tion 14.23. Then

for each r > 0 there exists a neighborhood U of p such that for each q in U ,

there exists a maximal timelike geodesic segment from q to cr(r).

ProoJ of Sublemma 14.25. Suppose the lemma is false. Then there exlst

T > 0, p, - - p, arid a limit maximizing sequence of past directed causal curves


an,& : [o. a , k j 4 ( M , gf such that for each k, a,,b(O) = a ( r ) and a,,k(a,,k) =

p,. Furthermore for each fixed n, by selecting a suitable subsequence. we

may suppose that as k - +ce we have u,,k -t a,, where a, is either a

past inextendible null ray or a past inextendible timelike ray. As was done

in Lemma 14.15, disgondize to obtain a limit maximizing sequence {qn :

[O, a,] ( M , g) ) with Vn(0) = ~ ( r ) , nn{an) = p,, a, + +m: and with {q,)

converging to a past inextendible ray 7 : [0, +oo) ( M . 9) where q ( 0 ) = a(?).

(Note here that the curves are parametrized with respect to the auxiliary

Riemannian metric h and also that 7 plays no explicit role in the rest of the

proof other than serving as a reference frame for the curves q,.)

Fix ro > r , and consider the past directed nonspacelike curves jj, : [0, c,] -+

( M , g ) obtained by concatenating each q, with tile segment along -a from ro

to T and reparametrizing appropriately (here c, = 1 f a, where I = TO - T ) .

Note that $,(O) = a ( r o ) , il,(l) = a ( r ) , and 17,(c,) = p,. Also put 6, =

d,(p) - d,(p,) -+ 0 as n + i c e by Sublemma 14.24.

Now recalling that the original sequence ( 7 % ) is hmit maximizing, the li

maximality of the new sequence {ij,) may be established. For

L(% I IO, %I) = d(a (T) , a:(T0)1 + L(k I I4GZl)

2 d ( a ( ~ ) , a ( r o ) ) + d ( ~ n , Q ( T ) ) - 6%

= d ( a ( ~ ) , a:(ra)) + d(p , a ( r ) ) - 6, - en

= d(p, a ( % ) ) - 6, - E n

where 6,, E , -+ 0 .

Since (ij,) is limit maximizing, this sequence converges to a past direc

nonspacelike geodeic ray ;j with i j(0) = a ( r o ) Since each 7j, containa

segment of ou from a ( r ) to &(TO), the limit ray must be timelike and fur

must coincide with a back to a ( 0 ) = p. Hence i j passes through p, an - -

constrllction, is contained in Jf(p) = I+(p). This implies, using Pr

3.7-(I), that p is in I f (p ) . whlch contradicts the finiteness of d(p ,p ) .

Proof of Case (3) of Proposztion 14.23: s = r. Let dl timelike ge

etrized with respect to Lorentzian arc length. S

this case. Then there exist r > 0 and p, 4 p suc

14.2 CO-RAYS AND THE BUSEMANN FUNCTION 533

p, is a past timelike cut point to a(r) for all n. Recall from Section 9.1 that

this means that there exists a past directed timelike geodesic

Cn : 10. d(pn: &(TI) + 6,) + (b.3 9)

with ~ ( 0 ) = a ( r ) and ~ ( d ( p , , a ( r ) ) ) = p, such that c , [ 10, d(p,, a ( r ) ) ] is

maximal but that d(&(t ) .a(+)) > L(c, I [ 0 , t ] ) for any t > d(p, ,a(r) ) . Be-

cause a is maximal, p and a ( r ) are not conjugate along a . Hence there is a

neighborhood V of -a1 ( r ) in T,(,)M which is diffeomorphic under expa(,) to

a neighborhood U of p. By re-indexing { p , } and shrinking U if necessary, we

may assume that (p,) 2 U and that C l satisfies the conclusions of Sublemma

14.25, so that there is a maximal timelike segment from a j r ) to each point of

U .

Let c, be a maximal segment from a ( r ) to p, for each n as described above

so that p, is the past timelike cut point, to a(r) along c,. If cnl(0) E V ,

by travelling along c, a little further than p,, we obtain a point q, in U with

q, << p, and such that c, from a ( r ) to q, is not maximal. BJ using Sublernma

14.25, we then obtain a maximal timelike segment y, from a(r) to q, which

must satisfy ynl(0) $ V. In this case, replace p, with q,.

Hence we may assume that we have maximal timelike geodesic segments

c, from a(? ) to p, with %'(Of $ V for any n. By passing to a subsequence

and reasoning as in Sublemma 14.25. ~e obtain a maximal geodesic segment

c from a ( r ) to p. Since a: is also a maxima! timelike segment from a ( r ) to p,

we must have that c is timelike. Also, since ct (0) is a limit point of { c n l ( 0 ) ) ,

we must have cl (0) $8 V. But this then forces CY and c to be two distinct

maximal timelike segments from p to a(?-). But then cu from p to a ( r f 6 ) is

not maximal; contradicting the fact that a was a timelike ray. 0

&examination of the arguments employed in the proof of Proposition 14.23

shows that the following alternative result may be proved about the timelike

cut locus and a maximal timelike geodesic segment [cf. Galloway and Horta

(1995, Proposition 3.9)j. Let y : [O, TO] --+ (M. g) be a maximal timelike ge-

odesic segment in an arbitrary space-time, and suppose that for each 7 with

< T < T O , the distance function


is finite-valued on a neighborhood of y(0). Then for each T with 0 < T < ro,

there exists a neighborhood U of the segment y(j0, r ] ) which does nos meet

the past tirnelike cut locus of yjr).

Now that control of the cut locus has been established, ~t is standard in the

above setting that the distance function a I-+ dr(s) = d(x,cr(r)) is not only

continuous near p as established in Sublemma 14.24 but also is smooth near

p. (This actually follows wizhout needing continuity first since the control of

the timelike cut locus ensures that Lorentzian distance may be expressed in

terms of the length in the appropriate tangent space of exponential Inverses of

points in M [cf. Lemma 4.263.)

We turn to an estirnate (14.29) for the d'alembertian U(d,) = troHdr,

which is an important ingredient in the proof of the splitting theorem. Similar

estimates have been important for a long time in global Riemannian geometry

[cf. Calabi (1957) for an early illustration]. In our current setting with U chosen

as in the proof of case (3) of Proposition 14.23, we first note the following basic

result.

Lemma 14.26. For q ~n U , let to = d(q, ~ ( r ) ) , and let c : 10. to] -- ( M , g )

be the past directed unit speed maximal tirnelike geodesic segmezt with c(0) =

~ ( r ) and c(to) = q. P u t f (x) = d,(x) = d(x, cr(r)). Then

(1) (pad fI(9) = -cl(to).

Hence,

(2) (gradf,gradf) = -1,

and the past directed maximal timelike geodesic segments from ~ ( r ) to poi

of U are integral curves of - grad f in U .

P~oof. Let v E T,M be given. and set c,(s) = expp(sv) For s suEcie

small, we may define y, : 10, d(c,(s), o(r))] -. ( M , g ) to be the unique maxirn

unit speed timelike geodesic segment from a(rl to c,,Is\. Then setting 7

( t) defines -

a variation ...

of the geodesic 1 ,

c through -, , eodesics with

14.2 CO-RkYS AND THE BUSEMANN FUNCTION

for all s. Hence by Corollary 12.24,

for any v. Thus. (grad f)(q) = -c1(to) is future directed tirnelike. and

(grad J, grad f ) = - 1

Lemma 14.26 then implies that we are now precisely in a well-known geo-

metric setup which is studled in Appendix B. We do our curvature calcula-

tions in this framework. Thus let Ul be an open subset of (M.g ) , and let

f : Ul --+ R be a smooth function on Ul which satisfies the eikonal differential

equation (grad f , grad f ) = -1. By Lemma B.1. the integral curves of grad f

are tlmelike geodesics. Moreover, we note in Section B.l that the (wrong way)

Gauchy-Schwarz inequal~ty for timelike tangent vectors Implies that any such

integral curve of grad f is maximal in the space-time (Ul, g /u,) We also note

that curvature calculations lead to the formulas (B.22) and (B.23) relatlng the

Ricci curvature of the integral curves off to the d'Alembertian O( f ) = tr o ~ j ,

Helpful in these calculations is the baslc identity

Iff (v, grad f ) = 0

for any v in T(Ul). since

In view of our desired application for which f = d,, we let c denote an integral

curve in Ul of X = - g a d f. (Thus there are certa~n slgn d~fferecces w ~ t h

(B.22) and (B.23) In Appendix B in which c is an integral curve of grad f )

Recall that V x X = 0 on U1 and also Nf (x, Y) = 0 for any Y in T(Ul ) .


Choose an orthonormal frame {El, E2, . . . , E, = X) along c and neighbor-

ing integral curves. Also, let V be any parallel field along c, also extended to

these nearby integrai curves as a vector field parallel along them as well Then

[V,X] = v v x - V,fV = V V X

since V is parallel along c Hence,

R(V,cl)c' = V v V x X - V,rVvX - VLv,xlX

= -v,~vvx - V(vvx)X.

Thus

(R(V, c')cl, V) = -(Vcl VvX, V ) - (V(o,x)X, V) n-l

= -cl ((VvX, V)) + (VvX, Vclv) - ~ ( ~ v x , E ~ ) ( v E , x , V) ,=I

using V v X = c,"~:(vvx,E,)E~, valid since N ~ ( v , B ~ ) = 0. Recalling that

V,lV = 0 and X = -grad f, we then obtain

d n-l

(R(v, c')cl, v ) = - { ~ f (v, V) 0 C) - C ( ~ f (v, E,)) . dt

j=1

If we put V = E, in this last equation and sum, we obtain

n-1

- RC(C', c') = - ( ~ ( f ) 0 c)' + C (HJ(E, E,:)~ %,?=I

n-l

r - ( W ) c)' 4- C (ilf(~,,~,))~. 2=1

The Cauchy-Schwarz inequality yields

n-1

Thus we are led to the Eccati inequality

14.2 60-RAYS AND TEE BUSEMANN FUNCTION 537

somewhat reminiscent of the Raychaudhuri equation (12.2). Of course, it

has long been known in the theory of conjugate 0 D E 's that such Riccati

inequalities lead to estimates on growth of solutions In this particular case.

we will verify that provzded Ric(cl, c') 2 0. integration techniques like those

already encountered in Proposition 12.9 may be used to obtain the desired

estimate for the d'Alernbertian of f = d,:

for any q in U. Hence, a key aspect of the hypothesis in the Lorentzian Splitting

Theorem that Ric(v, v) 2 0 for all timelike w is to ensure that estimate (14.29)

is valid.

To aid in the derivation of (14.29), introduce the notation +(t) = Dcrd,(c(t));

where c : [O,d(q, a(r))] + ( M , g ) denotes the past directed maximal timelike

geodesic segment from a(r) = c(0) to q = c(d(q, a(r)) j . Put t o = d(q, a(r ) ) .

Under the Ricci curvature assumption, inequality (14.28) impi~es

It may be checked that +(t) --+ -m as t -+ O+ by identifymg @(t) a s the

trace of the second fundamental form of the "distance sphere" of a(r) through

c(t) or by checking for Minkowski space [cf. Beem, Ehrlich, Markvorsen, and

Galloway (1985, p. 38)). Now

in view of (14.30). Integating from t = 0 to t = to yields

so that

Hence. recalling that to = d(q, a(?-)), inequality (14.29) is establislied.

538 14 THE SPLITTIXG PROBLEM IN LORENTZIAN GEOMETRY 14.3 THE LEVEL SETS OF THE BUSEMANN FUKCTION 539

14.3 The Level Sets of the Busemann hnction

In the differential geometry of simply connected, complete Riemannian man-

ifolds of nonpositi~e sectional curvature, the !eve! sets of the Busemann func-

tion are called "horosphcrcs" and have been much studied in differential geome-

try and dynamical systems Indeed, the already existing notion of a llorosphere

for the Poincar6 disk or upper half plane with the usual complete Eemannian

metric of constant curvature -1 seems to have motivated Busemann to define

what is now called the Busemann function [cf. Busemann (1932)) Much later,

in the proof of the Cheeger-Gromoll Riemannian Splitting Theorem. the letel

sets of the Busemann function provided the Riemannian factor in the isometric

splitting This aspect of the Busemann function suggested that perhaps the

Lorentzian Busenlann function should be studied as a means toward proving

a Lorentzian ,.,litting Theorem So far in spacetime geometry, the "hor

spheres" have primarily been employed as a tool In proving the space-ti

splitting theorem

Before returning to Galloway and Horta (1995), it is necessary to introduce

a standard concept from general relativity which has not been previously en-

countered in this book. Let A be an achronal subset of a space-time, i.e., no

two po~nts of A are chronologically related.

Definition 14.27. (Edge of an Achronal Se t ) The edge, edge(A). of th

achronal set A conslsts of all points p in such that every neighborhood i7 o

p contains a timelike curve from I - (p , U ) to I f (p, U ) which does 7 ~ 0 t meet A.

The set A is said to be edgeless if edge(A) = @.

Evidently, edge(A) C 2. O'Neill (1983, pp. 413-415) gives an excellent ex-

position of the basic facts concerning edge(Aj and topological hypersurfaces:

(1) The "achronal identity". - A - A & edge(A);

(2) An achronal set A is a topological hypersurface if and only if

edge(A) = 0, and

(3) An achronal set A is a closed topological hypersurface if end o

Note that being edgeless is somewhat like being "locally" a Cauchy surface

in that future timelike curves starting at points in I-(A) sufficiently close to

A must intersect 3. Galloway and Horta (1995) adopt the following definition of a topological

spacelike hypersurface. Recall that an acausal set fails to contain any causally

related points and is thus also achronal.

Definition 14.28. (Spacelike Hypersurface) -4 subset S of ( M , g) is said

to be a CO-spacelzke hypersurface if for eachp in S. there exists a neighborhood

U of p in M such that S n U is acausal and edgeless in (U,g 1 ~ ) .

The basic "edge theory' previously mentioned ensures that a spacelike hy-

persurface according to Definition 14.28 1s an embedded topological submani-

fold of M of codimension one. A smooth spacelike hypersurface, i.e., a smooth

codlmension one submanlfold with everywhere tlmellke normal, is a spacelike

hypersurface in the sense of Defin~tion 14 28. A somewhat stronger concept IS

that of a partial Cauchy surface.

Definition 14.29. (Partial Cauchy Surface) A subset S of (M, g) is said

to be a partzal Cauchy surface if S is acausal and edgeless in 1M.

In particular, a partial Cauchy surface is a spacelike hypersurface in the

above sense and 1s also closed.

Now we are ready to show that the level sets of the Busemann function of

a tirnelike 5'-ray are locally partial Cauchy surfaces, provlded the generalized

timelike co-ray condition holds [cf. Gallow-ay and Horta (1995, Proposit~on

4.2). Galloway (1989a, Lemma 2.3)].

Proposition 14.30. Let (M, g) be future timelike geodeslcdly complete

and let 7 be a timelike S-ray. Suppose that p E I - (? ) n I+ (S) IS a point of a

level set {b, = c ) where the generdlzed t~melike o-ray condit~on holds Then

there exists a neighborhood U of p such that the set C, = U n (b , = c j is

acausal in M and edgeless in U. In particular, C, is a partial Cauchy surface

in (U,G/U).

1 Proof. Choose an open neighborhood W of I, so that b = by is continuous

I on U by Theorem 14.19, and also such that Lernrnar 14.15 and 14.16 and

540 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14.3 THE LEVEL SETS OF THE BUSEMANV FUNCTION 54

Corollvy 14.17 hold on U. Since b(q) 2 b(p) + d(p , q) by Lemma 14.5, we have the geodesic ray c, : [O. +m) -+ (N. go) by taking the union of distance balls

b strictly increasing along timelike curves, which impl~es that C, is achronal.

Suppose p is an edge point of C, in U. Then there exist s, y E U - C, and Bu= l J % ( c ~ ( t ) ) = U { q e (N,go) d o ( q , k ( t ) ) < t ) . t>o t>O

future timelike curves cl, c2 : [0, 11 -+ U joining x to y such that cl passes The horospheie Ii, associated to c, is then the boundary of the horobal

B,. If H, turns out to be sufficiently differentiable (C2), then there is

normal geodesic variation of c, along H, for which the stable Jacobi tenso

such that b o c2(t0) = 0. Thus cz(t0) lies on C, in contradiction. field D, is a variation tensor field [cf. Eschenburg and O'Sullivan (1980, p. 8

and Lemma 12.13 for the construction of the stable Jacobi tensor field for

maximal null geodesic segment 11 from x to y. By choice of U , there exist In the proofs of various versions of the Lorentzian Splitting Theorem, mostl;

sequence r, + +co such that there are maximal timelike segments a, from in the globally hyperbolic case (and also in separate investigations of hyper

surface families in Riemannian manifolds), various estimates have been rnadt

also the notation b,(x) = d(y(O), y ( r ) ) - d ( x , y ( r ) ) and the definition b( on the Hessian of the diitance function s ++ d(z, a ( t ) ) as t -+ +co where c

limp,+, b,(x). Now an aspect of ow choice of U is that the initial tange is a timelike asymptote to y , en route to obtaining various maximum prin

to timeiike geodesics remain bounded away from the null cone in making t ciples for the Busemann function and its sublevel sets [cf. Eschenburg (1987

asymptotic geodesic construction as a result of the neighborhood Kc( 1988, 1989), Galloway (1989a), Newman (1990)]. The discussion in the pre

Lemma 14.16 and Corollary 14.17. With this control, it follows that by cu vious paragraph shows that this step in the proof of the splitting theoren

the corner of the broken geodesic q u a , and comparing with the corner of qU corresponds to forming the horoball in Riemannian geometry. In particular

there exists E > 0 such that for all n sufliciently large, Riccati comparison techniques to estimate these Hessians have been well ex-

plored for Riemannian hypersurfaces by Eschenburg (1987, 1989), and it i~

~ T , ( Y ) - bTn(x) = d ( x ! ~ ( r n ) ) - d ( ~ > ~ ( h ) ) > E . noted that these techniques apply also to families of spacelike hypersurface

in space-times. Eschenburg, Xarcher (1989). and others take the viewpoint

Letting r , -+ +m; we obtain b(y) - b(x) 2 E. But since x , y E C,, we m

have b(z) = b(y) which furnishes the desired contradiction 0

rather than by using Jacobi equatio


We return to the case of interest in proving the Lorentzian Spllttlng The-

orem and thus implicitly to the "horobalK of a timelike geodesic asymptote

to y as given in Galloway and Horta (1995, Lemma 4 3 and Proposition 4.4)

under the asumption of future timelike geodesic completeness rather than

global hyperbolicity. In this result. all timelike geodesics are parametrized

Lorentzian unit speed curves, not as unit speed curves in the auxiliary R

mannian metric h.

Lemma 14.31. Let (M, g) be future timelikegeodesically complete, and I

y be a timelike S-ray Assume that the generalized timelike co-ray cond~tio

holds at p E I-(?) n I f ( S ) Then there exist a neighborhood U of p an

constants to > 0 and A > 0 such that for each q in U and each timeli

asymptote a to r from q we have

(14.32) f Jd t (v, v) 2 - A (u J. ,v i )

for all v E T,M and t 2 to, where dt = d ( . , a@)) and vi is the projectio

onto the normal space (a'(0))i.

Proof. Let U be a neighborhood of p with compact closure on which t

generalized timelike co-ray condition holds and also Lemmas 14.15 and 14

and Corollary 14 17 hold. By Corollary 14.17, the set of all initial tangents

asymptotes to y starting from points of U is contained in a compact sub

of the unit timelike tangent bundle. Then, by taking U and to sufficient

small, we may ensme that the set (al( t) : 0 5 t < to ) of all tangents to

initial segments of length to of all asymptotes a emanating from points of

contained in a compact subset of the unit timelike tangent bundle. Hence

set of all timelike two-planes Il containing these initial tangents is contdne

a compact subset of the set of all timelike two-planes in G2(M) . Thus, t

of all sectional cl~rvatures of such planes is bounded above by some co

k; K(rI) 5 k .

Now consider B = -Hd" along the segment a 1 10, to]. B = B, corresp

to the second fundamental form of the level sets

14.3 THE LEVEL SETS OF THE BUSEMANN FUNCTION 543

at a(to - s) for 0 < s < to. Moreover. u. ++ B,, u = to - s. obeys the Riccati

equation B,' -+- BU2 + R( . , @')a1 = 0 [cf Eschenburg f 1987, Equation f3))j

Hence given the sectional curvature bound K ( n ) 5 h-, the appropriate Ric-

cati comparison Theorem may be applied je.g., Proposition 2.4 in Eschenburg

(1987)] to deduce the existence of a constant A = A(k) such that for all v in

T,M

J I d i o (w, v) 2 -A (vi, 3').

The lemma now follows since t ++ Hdt(v , u ) is an increasing function of t

[cf. the proof of Lemma 12.131. 0

Before proving the next result and obtaining two corollaries, which are

employed in the proof of the Lorentzian Splitting Theorem, we recall from the

Preface of Protter and Weinberger (1984, p. v) a general philosophy important

in P.D.E.'s of whicii Propositio~~ 14.32 is an example.

One of the most useful and best known tools employed in the

study of partial differential equations is the maximum principle.

This principle is a generalization of the elementary fact of calculus

that any function f (z) which satisfies the inequality fl' > 0 on an

interval [a, b ] achieves its maximum value at one of the endpoints of

the interval We say that solutions of the inequality f" > 0 satisfy

a maximum principle. More generally, functions which satisfy a

differential inequality in a domain D and, because of it, achieve

their maxima on the boundary of D are s a ~ d to possess a maximum

principle.

We now turn to the proof of Proposition 14.32 following Galloway and Horta

(1995, Proposition 4.4). An earlier result for globallq hyperbolic space-times

was obtained in Galloway (1989a, Lemma 2.4), where the opposlte convention

for choice of normal was used for calcalat~on of the mean curvature [cf. Newman

544 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14.3 THE LEVEL SETS O F THE BUSEMANPU' FUNCTION 545

Proposit ion 14.32. Let (M, g) be future timelike geodesically cornplece Let a : [O, 4- oo) -+ ( M , g) be a timelike asymptote to 7 with a(0) = p. Then

and suppose also that Ric(v, v) 2 0 for ail timelike v. Let y be a timelike for each t > 0, recalling Lemma 14.12 and Proposition 14.23.

S-rajr, and let W 5 I-(?) n l+(S) be an open set on which the generalized

timelike co-ray condit~on holds. Let C C W be a connected smooth spacelike b,,t(z) = b(p) + t - d(x. a@))

hypersurface with nonposrtive mean curvature, H z = divc(N) _< 0, where IV is the future pointing unit normal dong C. If the Busemann function b = b, is a smooth upper support function for b at p. Thus the function

attains a minimum dong C, then b, is constant dong C.

Proof. Suppose that b achieves a minimum b(q) = a at q in C. Let U be fc,t = bp,t + ~h

a neighborhood of q on which the generalized timelike co-ray condition hold is a smooth upper support function for f, at p, which implies that j,,t also has

and also Lemmas 14.15, 14.16, and 14.31 hold. Since bl C is continuous, a minimum at p. We will obtain a contradiction by calcula~ing Ax f,,,(p) and

suffices to show that b = a in a neighborhood of q in C. If this is not correc showing that it is negative for e sufficiently small and t sufficiently large.

then there exists a coordinate ball B with B C C n U centered at q such that

8(B) f DO(B), where First we digress to give a generd calculation for a smooth funct~on f :

M + W relating the dd'Alernbertian O(f) of f to the Laplaclan Ax( f 1 C) of f

D0(B) = {z E a(B) : b(x) = a). restricted to the spacelike hypersurface C. Fix p E C, and let (el, ez, . . . , en- )

be an orthonormal basis of spacelike vectors for T,C. Combining the sign Note that b > a on d(B) - DO(B). Also it follows from Lemma 14.31 that ther

conventions of Calloway and Horta (1995) with those in Chapter 3, Defin~t~on exists a constant C > 0 such that

3.43 for the second fundamental form, we ntay define the mean curvature Hc(p)

(14.33) ~ ~ * ( v , v ) 2 -C

for all asymptotes at 2, for all x in B, for all v 21 TT,C with (v, v) 5 1, and n-1 n-1

all t sufficiently large. H ~ P ) = - s ~ ( e , , e , ) = E(V,,N, e , ) t=1 %=I

By choosing B sufficiently small, we may construct a smooth function h

C having the following properties [cf. Eschenburg and Heintze (1984), Newm For the purposes of the calculations of Suble~nma 14.33, we denote the

(1990, p 177)]: adient in (M, g) by v( f ) and the gradient in (C, g lc) by V( f ). Also, let

(1) h(q) = 0; denote the Levi-Civita connection for (M, g), let V denote the Levi-Civita

(2) IjVchj( < 1 on B, where Vc denotes the gradient operator on C; onnection for (C, g j ~ ) , and let H; denote the hessian of f 1 C in (C, g (x).

(3) AGh < -D on B, where D is a positive constant and Ac is the indue Sublemma 14.33. Let C be a spacelike hypersurface of (,if. g), and let Laplacian on C; and

: M + R be a smooth function. Then for any p in C and with the mean

ure H z ( p ) given by (14 351, we have

-

546 14 THE SPLITTING PROBLEM IN LOREKTZIAN GEOMETRY 14.3 THE LEVEL SETS OF THE BUSEhIANN FUNCTION 547

Pmof o j Sublemma (1) We have Recalling inequality (14.29)) we have

n-I n - l

B(f)(P) = Z ( V ( f ) ( p ) . e , ) e , - @(f)(P)>N(P))N(P) W P , t ) ( p ) = -O(dt!(P) 5 7. t=l n-1 Because f,,t has a minimum on C at p, we have Vc( jE , t ) (P) = 0. Also,

= ei(f)ez - W f ) N ( P ) V(bt,,)(p) = -aJ(@) [cf. Lemma 14 261. Noiv using Sublemrna 14.33-(I), z=l

= V ( f 16) - lV( f )N(P) . V z ( f e , t ) ( ~ ) = Vc(bp,t)(p) S E V Z ~ ( P )

(2) Now = v(bp,t) ( P ) + (N(P) , V(bp , ) ( P I ) N ( P ) + E V C ~ ( P ) n-1

U(f )@) = Z g ( e z , e z W f ( e , , e,) + ~ ( N ( P ) . W P ) ) Hf ( N ( P ) , N ( P ) ) = -a'@) - (N(p) . crl(0)) N(p) + ~ V c h ( p )

z = l

n-I Thus, V ~ ( f , , , ) ( p ) = 0 implies that

= H f (ez , e,) - Iff (N , N ) I p . *=I N ( P ) = (N(P) , ~ ' ( 0 ) ) - ~ [ - ~ ' ( 0 ) + f(Vch)(p)j .

Hence we must calculate ~f (e,, e,) using (1). Now Since Hd$(vd t . .) = 0. we also have Hdc(al(0) . . ) = 0 Thus we obtain

Hf(e,.e,) = ('5,,Vf,et) = ( V e s V ( f I C).eZ) - ( a e , N ( f ) N . e z ) Hbppt (AT(p). N(P) ) = e2 (iv(p), ~ ' ( O ) ) - ~ H ~ ~ , ( V ~ h ( p ) , Vch(p) ) .

= (Ve ,V( f 16), e,) - ( e t ( N ( f ) ) N ( ~ ) + Np(f)Ve,N? et) By the (wrong way) Cauchy-Schvlarz inequality,

= Hi(e, ,et) - N p ( f ) (V,,N,e,)

Thus, I(N(p),a'(O))l 2 IlN(P)ll . l l~ ' i@)l l = 1

n-1 Since V z h ( p ) is tangent to C at p and also llVch(p)ll 5 1, u-e may apply U ( f ) ( p ) = A c ( f i WP) - N * ( f ) ):(Ve,N ez) - Hf t N ? N ) l F

r = l inequelity (14 33) to deduce H b p f t (Vch(p), Vch(p) ) < C and hence

= A c ( f I WP) - N,(f)Hc(p) - . f f f ( * i . N ) l p . a ~ ~ p ~ ( l v ( ~ ) , N ( ~ ) ) L ce2.

We now return to the proof of Proposition 14.32. By definition of f,,t Thus under the mean curvature assumption Nc(p) 5 0 and with the choice of

have N(p) as future directed tlmellke, we obtain

A~( f~ , t ) ( f - r ) = A z ( b p , t ) ( ~ ) + E A c ~ ~ P ) n - 1 nz(bt,,>(p) r + cE2

< A~(bp , t )@) - DE- -

By Sublemma 14.33 we have

548 14 THE SPLITTING PROBLEM IK LORENTZIAN GEOMETRY

Then taking t > 0 sufficiently large and E > 0 sufficiently small, we obtain

& 2 ( f t , t ) ( ~ ) < 0,

which is the desired contradiction 3

Proposition 14.32 has the following two corollaries [cf. Galloway and Norta

(1995)], which wlll be used in the proof of the splitting theorem in Section

14.4. Inequality (14.36) and these corollaries will indicate that the superlevel

sets

(4 E M : b,(q) > 4 are in some sense mean convex. This has been studied in the Riemannian case

in Eschenburg (1989).

Corollary 14.34. Let (M,g) be a future timeiike geodesically compl

space-time which obeys the timehke convergence condition Ric(v, v) 2 0

all timelike v, and let y be a timellke S-ray. Let W I - ( y ) f? IT(S) be

open set on which the generalized timelike co-ray condition holds. Let C be

connected xausal smooth spacel~ke hypersurface in W with nonpositive me

curvature. Suppose that C and C, = {b, = c ) n W have a point m com

and that C Jf (C,, W ) . Then

C c C,.

Corollary 14.35. Let ( M , g ) be a future timelike geodesically corn

space-time which sat~sfies the timelikc convergence condition Ric(v, v) 2 all timelike v, and let y be a timelike S-ray. Let W c I-(?) n I f (S) be

open set on which the generalized timelike co-ray condition holds. Let C b

smooth spacelike hypersurface with nonpositive mean curvature whose clo

is contained in IV -4ssume that C is acausal in W and E is compa

edge(C) C {b, 2 c ) , then

c r 1 el.

Section 14.5 of this ch

how the regularity for the Busemann

14.4 TEE PROOF OF THE LORENTZIAN SPLITTING THEOREM 549

a timelike S-ray on the set I - ( y ) n I + ( S ) may be extended to the larger region

1- ( y ) n [Jf (S) U D-(S)]: where D- ( S ) denotes the past Cauchy development

(or past domain of dependence) of S , i.e.,

D-(S) = ( q E M : every future inextendible nonspacelike

curve from q intersects S )

In particular, it is noted that if S is a compact acausal spacelike hypersur-

face in the future timelike geodesically complete space-time (M,g) , then the

generalized timelike co-ray condition holds on I-(-!) n [ J + ( S ) U D-(S)] . As a

consequence of these considerations, the following is obtained in Galloway and

Borta (1995, Corollary 5.6).

Corollary 74.36. Let (M,g ) be a globally hy.~erboIic, future timelike

geodesically cnrnprete space-time which contains a compact spacelike hyper-

surface S, and let y be any S-ray Then 6, : (M g ) -+ (0, +co] is continuous.

Moreover, the level sets C, = (q E M : h,(q) = c ) are partial Catlchy surfaces

in (M, g).

Galloway and Worta (1995) note that the de Sitter space-time provides an

example of ( M , g ) which is globally hyperbolic, geodesically complete, with

compact Cauchy surfaces S , yet contains S-rays y for which I - ( 7 ) # M .

Hence, the Busemann function b, of such a ray y is not finite-valued on all of

M ,

14.4 The Proof of the Lorentzian Splitting Theorem

The results have now been assembled which enable us to present a proof of

the Lorentzian Splitting Theorem in the timelike geodesically complete case.

as originally formulated by S. T. Yau (1982) in problem number 155. A proof

er this hypothesis, rather than that of global hyperbollcity, was first given

Fewman (1990). The proof given in this sectlon follows Galloway and Horta

, in which simplifications are obtained in the previous proofs cited in

oduction to this ckapter based on the systematic use of the generalized


Theorem 14.37. Let (M, g) be a t~melike geodesically complete space-

time which satisfies the t~mel~ke convergence condition Ric(v,v) >_ 0 for all

timelike vectors v. If ( M , g ) contains a timelike line, then ( M , g) is isometric

to (R x S, -dt2 h) where (S, h) is a complete Riemannian manifold.

Earlier work on the Lorentzian Splitting Theorem involving the Busemann

function approach assumed global hyperbolicity rather than timelike geodcsic

completeness, in part; because of the automatic existence of maximal timelike

segments connecting any pair of chronologically related points. Additionally,

the Lorentzian distance function of any globally hyperbolic space-time is al-

ways continuous as well a~ finite-valued. This version of the splitting theorem

is as follows:

Theorem 14.38. Let ( M , g ) be a globally hyperbolic space-time which

satisfies the timelike convergence condition Ric(v, v) > 0 for all timelike vectors

v. If ((M. g) contains a complete timeiike line, then ( M , g ) splits as in Theorem

In this second version, while completeness for all timelike geodesics is n

assumed, timelike complpteness of some maximal timelike geodesic is require

Proof of Theorem 14.37. Let y : I --+ (M.9) be any complete timelike 11

parametrized wlth (y1(t),-y'(t)) = -1. Let -y( t ) = y(-t) denote y wlth t

opposite orientation. As in all the proofs of the various versions of the splitti

theorems, define

and

bf = lim b) and b- = lim b;. r++m T-+w

Put I ( y ) = I+(?) r l I P ( y ) . Since y / [a. +m) and -y 1 la, +m) are tim

rays for all a E R, b+ and b- are defined and finite-valued on I ( y ) .

all the regularity results (and their time duals) hold on a neighborhood

point of y. It will be helpful for use in the sequel to note that Lemma 14 5-

translates into the following: given any p, q in I(?) with p < q, then

14.4 THE PROOF OF THE LORENTZIAN SPLITTING THEOREM 551

and

Now fix q in I ( y ) . Then there exist s. t > 0 so that y(-s) <C q << y ( t ) . Berice.

using the reverse triangle inequality,

whence

Thus B = b+ + b- 2 0 on I(?). Similar calculations show that B ( y ( t ) ) = 0

along the geodesic y(t) itself.

Let U be a convex normal neighborhood of :/((I) such that all the preceding

regularity properties hold for bf and b- on U . Consider the "horospheres'"

S+ = (b+(q) = 0 ) n U and S- = {b-(q) = 0 ) n U . By Proposition 1 4 30, both

Sf and S- are partial Cauchy surfaces in U. each containing y(0) Using the

topological manifold structure of S f . let W be a small coordinate hall in S+ centered at -(O) whose closure is also In S+. By applying the fundamental

existence result of Bartnik (1988a, Theorem 4 1) for spacelike hypersurfaces

with rough boundary data, we obtain a smooth maximal (i.e., Hz = 0) space

like hypersurface C U such that C is acausal in U with compact closure.

edge(C) = edge(W) C S f = (b+(q) = 0 ) n U, and C meets y. By deleting

points if necessary. we will also take C to be defined such that

First apply Corollary 14.35 to conclude that C C {hf ( q ) 2 0) n U But

also. since b-(q) 2 -b f (q) on I ( y ) , we have simultaneously that

Hence, by the time dual of Corollary 14 35 applied to --f, .rvc obtain

552 i 4 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY

Thus

C G jb+(q) L 0) n {b-(4) L 0).

Now since C C (b+(q) 2 0) and points of edge(C) satisfy b+(q) = 0, we obtain

from Corollary 14 34 that b+ = 0 along C. Hence if we let x = y(to) denote

the point of intersection of C and y, we have b+(z) = 0. Thus as B = O on 7,

we obtain b-(x) = 0 as well. This fact and C {b-(9) 2 0) enable the time

dual of Corollary 14.34 to be applied to conclude that C C S-. whence

(14.38) b+ = b- = O along C.

Given any x in C, let a,f : [0, +m) -+ (M, g) denote any timelike asymptote

to yl [O,+CG) issuing from x, and let a; : [O,+oo) -+ (M.g) denote any

tlmellke asymptote to y ] [O. -m) = -7 I [0, +GO) issuing from x. With (14.3

in hand, it is now poss~bie to establish that at any x in C, the asympto

geodesics a; and a: fit together at x to form a smooth maximal geodesi

line, I.e., [a;i.]'(Q) = -[a,]/(O), and further, that through each such x in

there is exactly one tlrnelike asymptote a, to y passing through x. Towa

this end consider the (possibly) broken (at t = 0) geodesic

&(t) for t 2 0,

a;(-t) for t < 0.

Flrst, from Corollary 14.13 we obtain for t 2 0 that

(14.39) bf (a,(t)) = b+(x) + d(x. a,(t)) = O + t = t

and similarly b-(a,(t)) = -t for t < 0. Now take t < 0, and consider (14.3

w ~ t h p = a,(t) = a;(-t) and q = x. We obtain

0 = b+(x) 2 b+(a,(t)) + d(a, (-t), x)

= b+(@X(t)) + (4,

whence bf (a,(t)) 5 t for all t < 0. On the other hand, for m y t < O

B(Q,(~)) = h+(a,(t)) + b-(a,(t))

= hi (a, (t)) + b- (a; (-t))

= b'(a,(t)) - t


SO that as B 2 0 on I ( y ) , b+(a,(t)) 2 t for all t < 0. Combining these two

calculations yields b+(a,(t)) = t for any t < 0, hence for all t in R, recalling

(14.39). An analogous argument using inequality (14.37b) in place of (14.37a)

shows that b-(a,(t)) = -t for any t in R.

We are now ready to show that ax( t ) is globally maxlmal. For take any t l ,

t 2 with t l < 0 < tz. Applying (14.37a) with p = a,(tl) and q = a,(tz) gives

whence

d(ax(tl)ffs(t2)) 5 t 2 - t l .

Automatically,

Hence, d(a,(tl), a,(tz)) = tz - tl for any tl < 0 < t2, which implies that a,

is globally maximal. In addition, note that if a, were broken at t = 0, then

the usual "cutting the corner" argument applied at x = a,(O) would produce

a longer curve from a,(-to) to a , ( t ~ ) than a, / [-to, to] for a small to > 0,

contradicting the maximality of a, just established. Thus we have seen that

for x in C , any future directed asymptote crl to 7 1 10, +m) with ul(0) = x and

any past directed asymptote a2 to -y 1 [O,+m) with (TZ(O) = 3. fit together

at z to form a smooth, maximal timelike geodesic line This is of course the

geometric basis for the splitting theorem to be valid in a tubular neighborhood

of y. as we now establish.

First we establish that both a,f and a; are C-rays for any x in C. Re-

call that for any q in I-(? j [O, +m)) n f + (C), we have 0 < d(C, g) < bi(4)

[cf. inequality (14.18)j. Taking q = a, (t) with t > 0, we obtain

But dfX, cr,(t)) 2 d(c~,(O),a,(t)) = L(a, ' [0, t 1) since a, is maximal. Hence,

d(C,a,(t)) = t = L(a, / [ 0 7 t ] ) for all t > 0, and a$ I [O, +os) is a C-ray as

desired. Dud arguments show that a; is also a X-ray. It is then a consequence

554 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14 4 THE PROOF O F THE LORENTZIAN SPLiTTING THEOREM 555

of basic theory in the calculus of variations that a: and a; are both focal po~nt hypothesis of timelike geodesic completeness, we may then define the normal

free and that a, meets C orthogonally (cf. Propositions 12.25 and 12.29). exponential map E : iR x C -+ (M, g) by E( t , q) = expq(tN(q)). i n view of

Let us ilow sl~ow that direct geometric arguments reveal that for any pan of the properties derived above for the asymptotic geodesics to y issuing from

distinct points p, q in C, yl = a," and 72 = a: do not intersect. For suppose points of C. we already have that E is nonsingular and ~njective ar.d hence a m = yl(s) = yz(t) for some s, t > 0. First, t = s = d(C,m) since both diffeomorphism onto its image.

geodesics are unit speed X-rays. Thus put to = t = s > 0, and note also that By our above calculations for b+(a,(t)) acd h - ( ~ , ( t ) ) , we have that

L(yl 1 10. t o ] ) = L(y2 I [0, to]) = d(X, m). Choose any 5 > 0, and let tl = to + S

and n = yl(tl). Starting with the fact that yl is a X-ray, we obtain h+(E(t.q)) = b+(exp(tN(q))) = t

d(C,n) = L(7l I 10,tlI)

= L(YI! [O, t o ] ) + L h l I [ to , tl]) b-(E(t, q)) = b - ( e ~ p ( t N ( ~ ) ) ) = -t.

= L(72 I [O,tol) + L(7l I [to, ill). Since E is a diffeomorphism onto its image, these equatioils reveal that both

b+ and b- are differentiable on the connected open set fi = E(W x C). Let c = yzlp,,ol * ~ l l ~ ~ ~ , ~ , ~ , a curve from q to n which, by the previous c

Fix x E C, and let a,,, denote the unit timelike geodesic from .L to ? ( r ) lation, has the property that L(c

for r sufficiently large. Then as b,f(x) = r - d(x, y ( r ) ) = r - d,(x), one has we have yll(to) # yzl(to). Hence,

Vbf (x) = -Vd,(x) = -a&(0) (cf. Lemma 14.26). Hence letting r, -+ +m. done using a convex normai neighborhood of m, show that we may construe

and remembering that U has been chosen so that the asymptotic geodesic future causal curve a from q to n with L(a) > L(c). But then L(a) > d(C,

construction is well behaved, we have a,'(O) = limn,+, cuL,,n(0) and thus which is impossible. Thus the future timelike geodesics a:, x E C, do

intersect. Dual arguments reveal Vb+(x) = lim Vb?=(x) = lim -c~j.~,(O) = -aZ1(0), n-TW n-+m

x E C, can intersect either.

Let us now turn to the non-i where a,(t) = E( t , x). More generally,

form yl = a: and 72 = a;, for any p, q E X except possibly at p = q Vb+ (az (t)) = -ayl(t) C. Suppose that there exist s, t > 0 such that yl(s) = yz(t) = m.

b+(m) = b+(yi(s)) = s > 0. On the other hand, taking any t l with 0 < t l r any t 6 W jcf. Eschenburg (1988)], This discussion shows that f = b+ 1s a

we have b+(-y2(tl)) = -tl from our previous work in the course of thih pr ifferentiable function on Ut which satisfies the eikond equation ('17 f. D f) =

Since b+ o yz 1s continuous, there exists t2 > 0 such that b'(yz(t2)) = 0.

this then contradicts the achronality of the set {b+ = 0) . Hence we h s that for any point q In Ul there exists a unique unit speed maximal

established the fact that no two normal geodesics to C intersect, except elike line asymptotic to y passing through q ) Put

the trivial possibility that both issue from the same point p of C and that Zt = E ( ( t ) x C).

intersect precisely at that point or coincide.

%call that C is a smooth spacelike hypersurface. Hence there exist

smooth future pointing unit ~imelike normal field N along C Recalling also


a certain Lagrange tensor field dong any unit timelike geodesic c(t) in U1 asymptotic to y calculates the mean curvature W ( t ) of the leaf of this foliation

at c(t). The Raychaudhuri equation associated to this Lagrange tensor fieid

then translates Into the following evolution inequality [cf. Galioway and Horta

Assuming that N fails to be parallel along the geodesic c and recalling the

curvature assumption Ric(N, N) 2 0, arguments similar to those in Proposi-

tion 12.9 of Chapter 12 show that the mean curvature H(tf to Ct a t c ( t ) must

blow up in finite time. But this contradicts the fact established above that all

such timelike asymptotes c to y are focal point free to Co at c(O). Hence, the

normal vector field N = -Vb+ must be parallel in U l , and the level surfaces

Ct are all totally geodesic (cf. Lemma B.5 of Appendix B). Moreover, the par-

allelism of N also implies that Ul splits locally isometrically by Wu's proof

(1964, p. 299) of the Lorentzian de Rham Theorem. More exactly, let h de-

note the Riemannian metric induced on the spacelike hypersurfxe C = Go by

inclusion in ( M , 9). Then E is an isometry of (R x C, -dt2 $ h) and (Ul. g lU, and hence provides the desired isometric splitting in the neighborhood Ul o

the given timelike line.

We summarize what has been accomplished so far as

Proposition 14.39. Suppose ( M , g) is timelike geodesicdiy complete an

satisfies fic(v. v) 2 0 for every timelike v , Let y : R -4 (M, g) be any complet

timelike line in (hif, g) . Then there exists an open neighborhood U of y (W)

M sucb that

( I ) C = (bt = b- = 0) n U is a smooth embedded acausal hypersurfa

(M, !?I, (2) (U. glu) is isometric to (W x C, -dt2 @ h), where h is the metric ind

on C by inclusion in ( M , g), and

(3) for each q in C, the curve R i- U given by cq(t ) = exp(iA

ma-imai timelike line ~assinp throud a which is ~ v m ~ t o t i c

given timelike line y. - - -

Further, any timelike asymptote to

-y / [a, foo), for any a in R, issuing from any point q in U , coincides

up to parametrization w ~ t h one of these flow !ines c,(t> of Vb+. (Here

N denotes the future directed unlt clmellke normal to C . )

It is now necessary to show that the local splitting given in Proposition

14.39 may be extended to a global splitting. In Eschenburg (1988) this wa;.

carried out by defining tmelike lines ci. ca to be equzvalent if they are joined

by a chain of timelike lines = c l , p2,. . . , /3h = c2 such that each successive

pair /3,, fl,+l forms the boundary of an isometrically immersed flat strip 4 : ( R x [al. aa], -dt2@ds2) -+ ( M , g ) satisfying 4 ( W x {al)) = P3, F3(Rx {a2)) =

pJ+l , and Fj(R x is)) is a timelike line for each s in [al.a2]. Later. following

the publication of Galloway (1989a), Galloway ind~cated to us that an alternate

proof of the global splitting could be given by maximally enlarging the local

hypersurface C of Proposition 14.39 originally obtained in a neighborhood of

a given timelike line. It is this second line of reasoning that will be pursued

next.

It will be useful for this purpose to recall the followirlg elementary rela-

tionship between parallel translation and parallel field&. Suppose that U is a

(path) connected open set and N is a srnootl~ vector field which is globally

parallel on U. i.e., for any q E U and any v 6 T q M , we have V,N = 0. Fix

any point ql in U , and let vl = N ( q l ) E Tq,M. Then for any qr: in U ,

If c : 10.11 + U is a smooth curve with c(0) = 41 and c(1) = qz. (14.44) and v2 E T,,M is obtained by parallel translating zl along c from

t = O to t = 1, then the resulting vector vz is independent of choice of path c from qi to qz.

This basic fact is valid because the parallel field P along c with P(0) = v l

rrtust be given by P(t) = A' o c(l) for all t i11 [O, I]. whence va = P(1) = N(q2)

which depends on N and q2 but not on the particular clloice of smooth ~ u i v e c

m ql to 92. Thus if the field 1%' on U is globally given. no holonorny problems

e encountered in parallel translation along different curves in U .

We turn now to the problem of compatibly enlarging a local splitting of

rm given in Proposition 14.39. Thus let 7 3 IR -+ (M, g) denote a given

i timelike line, and let W be a convex normal neighborhood of po = ~ ~ ( 0 ) such

55% 14 THE SPLITTING PROBLEhl IN LORENTZIAN GEOMETRY

that the regularity properties for timelike asymptote construction are valid on

V/ and the Busemann functions b$ and b, are continuous on W. Let Wo

be a second convex open neighborhood of po with T o 2 W, which is used

to obtain a local isometric splitting E : (B x Co, -dt2 @ hO) -+ (Uo,g) with

Co 2 Wo and E(t,q) = exp(tNo(q)), where No = -Vb$ is globally parallel

on U = E(R x C o ) Let pl E edge(C0) be arbitrary. Take (q,) C Co with

limn,+, E(0, q,) = pi. By continuity, b$(pl) = limn,+, bof(E(0, q,)) = 0,

and similarly bi(pl) = 0. Recalling that Co is totally geodesic and W is

convex, there exist unit speed ho-geodesics c, . 10, a,] + Co with c,(a,) = q,.

Let w in Tpo (CO) be an accumulation vector of {cnl(0)) and put c(t) = exp(tw),

c : 10, a) -+ (Ca, ho ) Since cz(t) = (2t, c(t)). 0 < t < a, is a unit timelike

geodesic in (M, g) and ( M , g) is timelike geodesically complete, it follows that

cz(t) extends to t = a. Hence, the geodesic c : [O,a) + (20, he) extends to a

geodesic ci : 10, a] -+ (&I, g) with cl (a) = pl.

Let P be the unique parallel field along cl : [O, a] -+ (M,g) with P(0) =

yol(0). Put v = P(1) in T,,M, and let yt : R -+ ( M , g ) be given by yl(t) =

expPl (tv). If we take {s,) (0, a ) with s, -+ a and put u,(t) = exp(tP(s,))

14.39, each u, is a maximal timelike line asymptotic to 70. Thus 71 is also

maximal timellke line.

Apply Proposition 14.39 to yl to get a local splitting E : (B x El, -rkt2 $

parallel on Ul, b$(pl) = b;(pl) = 0. We wish to show that Nl(q) = &(q)

all points q of Uo n U1, or equivalently, bf(q) = bof(q) (and thus also b l (q) =

b;(q)) at all polnts q of Uo n UI.

Choose b with 0 < b < a so that q = cl(b) 6 C1 n Co, where cl . [0, a]

(M,g) denotes the g-geodesic in W from go to pl constructed above. Rec


curve u : i0, I] -+ Cl P Co from q = a(0) to m = a(1) Since Pil and No are

both parallel at all points of C1 n Co and Nl(q) = No(q), ~t foilows agaln from

(14.44) that Nl(m) = No(m). Hence. f i = iv, at all points of Cl fl Co.

Now given any n E Ul Ti Uo, use either splitting to find a smooth curve

p(t) = E(t ,m) from a point m in El n Co to n. Since Nl(m) = No(m) and

Nl (n) and No(n) are obtained from Nl (m), No(rn) by parallel translation

dong p, we must have Nl (n) = No (n).

Thus we have obtained Vb$ = vb$ on Ul TiCro. Since bf (q ) = b$(q) for any

q i11 C1 n Co, it follows that bf = bz on Ul n lie. Put C = Co U C1 furnished

with the Riemannian metric h induced bj inclusion in (M. y). Fbrther, put

and define the extended splitting E : (R x C, -dt2 63 h) -+ (Lro U U, , g) by

E( t , q) = e w ( t N(q)).

Kow we ~ndicate how the or~ginal splitting may be extended from the given

local splittmg W x Co -+ Uo .'in all directions." Select sufficiently many points

(PI, pz, . . . , pk) in edge(&) and associated convex neighborhoods Mil , . . . , Wk

such that after determining local hypersurfaces El, C2,. . . , Ck and tubular

neighborhoods Ul, U2,. . . , Uk, that if any pair Up n Uq # 0 (or Cp n Cq # @),

then C, n C, contains a point of the starting hypersurface Co. With this

proviso, it follows as above using (14.44) that the Busemann functions (and

their gradients) obtained from the local splitting8 E, : R x C, -+ U, agree on

all overlaps, so that we may put

and

c = ~ o u ~ l U " ' u ~ k

and define a parallel field on U by N(q) = iVj (q) for q in U3 to obtain a well-

defined isometric splitting E : R x C + (U,g lo-) via E( t , qj = exp(t N(q)).

We now present the more general inductive szep necessary to conclude that

e given local splitting E : (B x Co, -dt2 @ h) -+ (Uo, g) may be globalized.

560 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY g Thus we suppose that an open, smooth, path connected spacelike hypersurface

C with Riemannian metric h ~nduced by lncluslon In (hf,g) is given, wlth a

timelike unit normal field N parallel at all points of C so that the mapping

given by E(t ,q) = expg(t N(q)) is a diffeomorphism onto its image, ns well as

being an isometry, and that the timelike geodesics yq(t) = E(t , q) are timelike

lines for all q in C. Inductively. suppose further that if N is extended to

U by setting N(yq(t)) = yql(t), then N is globally parallel on (U. g). Finally.

although this is not necessary to the argument, suppose that C contains a given

local hypersurface Cg formed by starting with an originally given complete

timelike line yo and that h7 / U t = -Vb:o. as in Proposition 14.39.

Suppose that edge(C) is nonempty. Take any p E edge(E). To complete the

argument. we will show that C can be extended consistently past p to form

an even larger spacelike hypersurface C1 satisfying the above conditions. This

then shows that if we want to carry out "Zorn's Lemma" type arguments, then

Eb/ may be maximally extended to a smooth spacelike hypersurface ( S , h ) w ~ t h

edge(S) = 0, and we may thus obtain the desired global splitt~ng.

Let W & U be a geodeslcally convex open set with p E and W contamed

Inside a larger geodesicallv convex open set. By our prevlous discuss~on, ~f

we select po in W sufficiently close to p and let be a geodesically convex

neighborhood of po with p In d(Wo), then applying the local splitting Propo-

sition 14.39 produces a hypersurface Co contained in C (? Wo with po 6 Co

odesic el . \0,1) -+ ( M , g) with q ( 0 ) = po, q ( l ) = p, and c1([0,1)) &

14.4 THE PROOF OF THE LORENTZIAK SPLITTING THEORElvl 561

i e., using the Busemann function associated to c(t), the parallel field iV may

be consistently extended to C U El.

The crucial remaining step in this procedure is to show that this process just

described is well defined. That is. suppose a different point go E W is selected,

the geodesic r2 . [0, 11 --+ (M,g) with c2(0) = q ~ . cz(1) = p, and c2([0, 1)) S C

is constructed. and ?u2 6 T,M is obtained by parallel translating N(qo) along

c2 from t = 0 to t = 1. It is necessary to establish that wl = wz in Tp&f.

Thus suppose that url # w2 in T,M. Let P, denote the parallel field along

c, : 10, l] -, (M, g) with Pl(0) = N(po) [respectively, Pz(O) = N(qo) ] . Recall

that in fact P,(t) is equally well determined by the value of N(c3(t)) for any

t with O < t < 1. Thus in view of the proof of the local splitting Proposition

14.39, especially the use of Bartnik (1988a). we will replace the given initial

points po, go by points p& on c1 and qh on c2 stifficiently close to p such that

after the associated local splittings

are obtained, centered about ?,b(t) and %b ( t ) respectively, we have p E C;nCb

Fkom our previous studies of the compatib~lity of local spllttings, we also have

~ b ; ' = 0b2f at all points of C; n C;. and part~cularly. PI(p&) = -Vb;(& and I

P~(q6) = -Vb2 (¶A). Thus. in fact, wl = Pl(p) = -Vb:jp) = -'C7bz(p) = wa.

as desired.

Now that the existence of a global isometric splitting ( M , g) = (EX S. -dt2@

h) has been established, the only remaining detail in the proof is to show that

the Riemannian factor (S, h) is forced to be corriplete by the assumed t~mellke

completeness of (M, g). But this is the same type of argument that has already

been used above. Suppose that (S, h ) fails to be Riemannian complete. Then

there exkts an h-unit speed Riemannian geodesic c : (a , b) --t (S; h) where

either n or b is finite, which is inextendible either $0 t = a or to t = tr or

possibly both. Define 7 (a, b) -+ (M,g) by -{(t) = (2t,c(t)), which is a

geodesic in (iM,g) by formula (3.17) with f = 1. Further, g(-yJ(t), y l ( t ) ) =

2 + h(el( t ) ,c l ( t ) ) = -1. Thus y is a timdike geodesic in (A4.g) which is

either past incomplete or future incomplete, contradicting the hypothesis of

timelixe geodesic completeness of (M, g). O

562 14 THE SPLITTING PROBLEM IN LORENTZIAK GEOMETRY

Remark 14.40. It is perhaps amusing to note after the fact that the two

versions of the Lorentzian Splitting Theorem are not so far apart.

(1) Note that under the conclusion of Theorem 14.37. (M,g) must be

strongly causal by corollary 3.56. Alternatively, we know from the

proof of Theorem 14.37 that a complete timelike line passes through

every point of (M, g), hence strong causality also follows from Propo-

sition 4.40.

(2) Since the Riemannian factor (S, h) in Theorem 14.37 turns out to be

complete, we know that (M,g) had also to be globally hyperbolic (as

well as geodesically complete) by Theorem 3.67, (1) =+ (2), (3).

(3) Similarly in Theorem 14.38, the global hyperbolicity of (M, g) ensures

after the splitting is obtained that ( M , g) had to be geodesically com-

plete, also by Theorem 3.67, (3) * (2).

(4) The use of Theorem 4.1 of Bartnik (1988a) enables certain analytic

arguments involving smooth super support functions and subsupport

functions to show that the continuous function b+ o c is a%ne, hence

differentiable, to be dispensed with in the proof presented here. An as-

pect of this other proof method is to observe that by estimatw a k ~ n to

(14.36), these support functions have arbitrarily small second deriva-

tives [cf. Eschenburg and Heintze (1984), Beem, Ehrlich, Markvorsen.

and Galloway (1985, Lemma 5.1), Eschenburg (1988, Lemma 5.2 and

Proposition 6.3)j.

A different application of the Busemann function methods discussed in this

chapter has been given in Andersson and Howard (1994) to obtain results more

closely allied to Harris's Maximal Diameter Theorem A.5 of Appendix A. In

this setting. for example, estimate (14.29) translates into

O(d,) > -(n - 1) tan(a/2 - d,).

Here are two of the results obtained in Andersson and Howard (1994, The

14.5 RTGIDITY OF GEODESIC INCOMPLETENESS 563

Theorem 14.41. Let (M", g ) be a globally hyperbolic space-time which

satisfies the curvature condition %c(P~, v) 2 (n - 1) for any unit tlmelike vector

v. Suppose that (M, g) contains a mlwirnal timelike geodesic segment y :

( - ~ / 2 , ~ / 2 ) -+ ( M , g). Then there exists a comple~e Riemaqnian manifold

(S, h) such that (M, g) is isometric to the warped product

( ( - ~ / 2 , ~ / 2 ) x S, -dt2 + cos2(t) h)) .

As a consequence of Theorem 14.41, Andersson and Howard are able to

obtain a related result to Theorem A.5 of Harris (1982b). The definition of

"globally hyperbolic of order q = 1" is given in dppendix A, Definition -4.3.

Theorem 14.42. Let (M" , g) be a space-time which is globally hyperbolic

cf order 1 and also satisfies the curvature condition Ric(v, v) 2 (n - 1) for any

unit timelike vector v.

(1) If ( M , g) contains a maximal timelike geodesic segment y : r-;, $1 -+

(M, g) of length x joining x to y , then the causal lnterval D = If (z) ri

I - (9) is isometric to the warped product of the hyperbolic model space

(3, h) of constant curvature -1 and dimension (n - 1) and the interval

(-5, z ) , with warped product metric -dt2 - cos2(t) h.

(2) If (M,g) contains a timelike geodesic y : (-oo, +co) -+ ( M , g) such

that each segment y 1 [t, t + T] is maximal, then ( M , g) is isometric to

the universal cover of anti-de Sitter space.

14.5 Rigidity of Geodesic Incompleteness

Now that the space-time splitting theorem has been obtained. we are ready

to return to the question of rigidity of timehke geodes~e incompleteness as

discussed in the introduction to this chapter. The following formulation of

this ques~ion was given in Bartnik (1988b). Bartnik defines a space-tlme to

be cosmological if it is globallj hyperbolic with a compact Cauchy surface and

satisfies the timelike convergence condition Ric(v, v) 2 0 for all timelike v.

564 14 THE SPLITTIXG PROBLEM IN LORENTZIAX GEObIETAY

Co~ecture 14.43 (Bartnik). Let (M. g ) be a cosmoiogical space-time.

Then

(I) (M, g) contains a constant mean curvature Cauchy surface; and

(2) (M, g) is either timelike geodesirally incomplete or else ( M g) splits

isometrically as a product (R x V, -dt2 @ h), where (V h) is a complete

Riemannian manifold.

In Geroch (1966) the following condition was introduced

(14.45) For some p E &f. the set A4 - [If (p) n I-(p)] is compact.

Bartnik (1988b) proves that if condition (14.45) is satisfied, then there is a

constant mean curvature Cauchy surface in (M,g) which passes through p.

Also Bartnik (1988b) notes that if condition (14.45) holds, then either (M,g)

is a metric product or else is timelike geodesically incomplete, extending earlier

work of Galloway (1984~) which required that this condition hold at all p on a

compact Cauchy surface. Bowever. (1) is false in general [cf. Bartnik (1988b)l.

In Beem, Ehrlich, Markvorsen, and Galloway (1985), a rigidity result was

stated which may be obtained as a consequence of the Lorentzian Splitting

Thcorcm along fairly elementary lines, without any causality assumption like

(14.45), but under the weaker sectional curvature hypothesis. -4 complete

proof was published in Ehrlich and Galloway (1990)

Theorem 14.44. Let (M, g) be a space-time with a compact Cauchy sur-

face and everywhere nonpositive timelike sectional curvatures K 5 0. Then

either (M, g) is timelike geodes~cdy incomplete or else (-!. g) splits as a metric

product (B x V. -dt2 $ h), where (V, h) is cornpact

Prooj. Suppose that ( M . g) is timelike geodesically complete. Since (M, g)

is assumed to have a compact Cauchy surface S, it is not difficult to see that

(M, g ) is causally disconnected with K = S. Hence by Theorem 8.13, (M, g)

contains a nonspacelike geodes~c line. If the line is shown to be timeilke, then

the desired splitting follo~vs from the Lorentzian Splitting Theorem. Nence, it

only remains to rule out the alternative that the line is null.

This may be accompiishd by using a coroliary to Harris's Toponogov The-

orem of Appendix A [cf. Ehrlich and Galloway (1990)).

14.5 RIGIDITY OF GEODESIC INCOMPLETENESS 565

Theorem 14.45. Let (A4.g) be a space-time with compact Cauchy sur-

face and with everywhere nonpositive t~melike sectiond curvatures. If ( M , g)

is fl~tiire tjmehke geodesicajly complete, then each future inextendible null

geodesic contains a null cut point.

Pmoj. Assume that there is a future inextendible null geodeslc q : 10, a) -+

( M , g ) such that there is no cut point to p = q(0) along 7. Let (tk} be

an increasing sequence of nonnegative real numbers converging to a. and set

4.4 = rj(tk). Hence d(p, q k ) = 0 for all k since q contains no null cut point to p.

Fix a compact Cauchy surface S containing p, and for each k let A/k be a future

directed timelike geodesic segment from some point r k of S to q k which realizes

d(S, qk). (Her~ce yk is also a rrlaximal geodesic segment.) The sequence irk) has an accumulation point r on S, and thus some subsequence (y,) of (y,)

converges to a future inextendible norispacelike limit curve y with y(0) = r

which by the usual arguments is a maximal geodesic. Also, for each fixed t the

segment y 1 [O, t 1 realizes the distance from S to y ( t ) . klareover, for some t > 0

we have d(S, y ( t ) ) > 0 since y cannot be imprisoned in S. Hence, the limit

geodesic must be timelike rather than null. Note also that y is future complete

by the assumption of future timelike geodesic completeness. Furthermore,

(14.46) r n If (p) = 0.

For suppose y meets I+(p). Then y, meets I i ( p ) for large m. Hence q, E

I f (p ) , and thus d(p, q,) > 0, in contrad~ction.

Now we use the sectional curvature hypothesls to obtain a contradiction

to (14.46). Because y is a timelike limit curve of (y,}. we may choose m

sufficiently large so that q , E I+(r). Lsing I'(q,) I+(p). we then obta~n

I+( r ) n If (pj f 0.

Thus we may fix x in If (r) fl It(p) and let 6 : 10, L] -i ( M , g) be a maximal

timelike geodesic segment from r to z. S~nce K < 0 and 7 is future complete,

eorem 3.1 of Harris (1982b, p. 313), a consequence of Harris's Toponogov

e is some t o with x = b(L) <_ ?(to), which yields

?(to) in contradiction to (14.46).

566 14 THE SPLITTING PROBLEM IN LORENTZPAK GEOMETRY

At the end of Section 14.3, a summary was given of how Galloway and Norta

(1995, pp. 18-20) extend the regularity results for the Busemann function as

given in this chapter for I-(y) f lI+(S) to I-(?) n [J"(S) U D-(S)] . With this

accomplished, Galloway and Worta (1995, Theorem 5.1) are able to improve

upon earlier results in Eschenburg and Galloway (1992, Theorem B).

Theorem 14.46. Let ( M , g) be a space-time which contains a compact

acausal spacelike hmersurface S and satisfies the timelike convergence condi-

tion Ric(v, v) 2 O for all timelike v. If (M, g) is timelike geodesically complete

and contains a future S-ray y and a past S-ray 77 such that I-(y) n 1 + ( ~ ) # 0. then (M, g) splits as in the rigidity conjecture.

An entirely unrelated investigation of rigidity of geodesic incompleteness is

pursued in Garcia-Rio and Kupeli (1995). Suppose (I\.I, 9) is a stably causal

spacetime with smooth time function f : M -t R. i.e., g(V(f) ,V(f) ) < 0.

Rescale the given metric g to produce a new metric g, for M for which

g, (Vc(f),Vc(f)) = -1 by setting g, = -g(V(f), V(f) ) . g. Now the integral

curves of VC(j) are unit speed timelike geodesics (cf. Lemma B.1). Hence,

tirnelike geodesic conipleteness of (M, g,) implies that the flow of f in (M, g,)

is complete, and the flow may be used to produce a metric g(t) on the level

surface f -l(t) from that of (f -l(0), g lf-i(,)). Thus the following result is

obtained.

Theorem 24.47 (Garcia-Rio and Kupeli). Let ( M , g ) be a stably

causal space-time with smooth time function f : M -+ R. Let

Then either

(1) (M, g,) is timelike geodesically incomplete, or

(2) (Ad, g) is conformally difleomorphic to a type of pasametrized warped

product (R x N, -dt2 @ g(t)), where N = f-'(0) and g(t), t E R. is a

one-parameter family of Riemannian metrics for N.

APPENDIX A

JACOB1 FIELDS AND TOPONOGOV'S

TMEOBM FOR LORENTZIAN MANIFOLDS?

One of the important consequences of the Rauch Comparison Theorem

in Riemannian geometry is the Toponogov Triangle Comparison Theorem

[cf. Cheeger and Ebin (1975, pp. 42-49)]. Let M be a complete Riemannian

manifold (metric here and elsewhere in this appendix denoted by (. : . )) with

sectional curvature of all 2-planes a in M satisfying K(cr) 2 H for some num-

ber H. Let (yl,y2,y3) be geodesics in M forming a triangle: 31(0) = yz(0),

yl(L1) = "/3(0), and y2(Lz) = -y3(L3), where L, = L(y,). Suppobe that yz

and 7 3 are minimal geodesics and, in the case that H = q2 (q > 0), suppose

that L, _< r /q for i = 1,2,3. Assume the geodesics 75 are paranietrized by

arc length, and define a3 = (yli(0),y2'(0)) and a 2 = {-yI'(Ll),-~31(0)). For

a triangle ( T l , T 2 , ~ 3 ) , possibly in another manifold. 5 2 and Z3 are defined

similarly.

(1) In the simply connected two-dimensional Riemannian manifold

of constant curvature H , there is a geodesic triangle (7,. T2, y3) with

L(T*) = L,, i = 1,2,3, and 5 2 5 12, E3 5 02. This triangle is determined up to congruence if H < 0, or if H > 0 and all L, < r lq .

(2) In %, let 'J1 and T2 be geodesics with Tl(0) = T2(0), L(7,) = L1.

L(Yz) = L2, and Eag = aag. Let T3 be a minimal geodesic between the

endpoints of and 'J2. Then L(%) >_ L(y3).

It is possible to develop an analogous theorem for Lorentzian geometry: the

program will be sketched here, although the proofs wl!l be omitted Details appear in Harris (1979, pp. 3-41) and Harris (1982a).

fgy Stever, 6. Harris, Department of Mathematics, St Lours Umversity, St Louxs, Mlssourl

568 APPENDIX A JACOBI FIELDS AND TOPOKOGOV'S THEOREM

The first step is to modify the Timelike Rauch Comparison Theorem I

(ci. Theorem 11.11) so that it applies to tirnelike geodesics. not without con-

jugate points, but without focal points: If N is a submanifold of a manifold

M , then a point q f M wi!l be said to be a focal poznt o j N from p (p 6 N ) if

q is the image of a critical point of exp in the normal bundle of 1V at p. For

v a vector in the tangent bundle TM. -W(v) will denote the submanifold of -14

which is the image under exp of a small enough neighborhood of the origin in

the perpendicular space of v such that exp is an embedding on it. Thus N ( v )

is an (n - 1)-dimensional submanifold orthogonal to u.

In the following statement of the second Rauch Theorem, and elsewhere in

this appendix, A A B will denote the 2-plane spanned by the vectors A and B.

Theorem A.1 (Timelike &uch II). Let t/, be a Jacob] field along a unit

speed timelike geodesic y, in a space-time M,, z = 1.2, with y, : 10, L] -+ hf,;

let T, = 4%'. Suppose that (VI, VI)O = {VZ, %)o, (VI,TI)O = (V2, T2)0, and

(VT,V,)(0) = 0. Further suppose that for any vectors X, a t y,(t)

K(Xi A Ti) 2 K(X2 A 7'2)

and that 7 2 has no f0ca-I point of N(yzl(0)) from yz(0). Then for all t Jn [0, L],

An important corollary of this theorem shows how curvature can affect the

lengths of timelike curves.

Corollary A.2. Let y, : :O. L] --+ M1, i = 1,2 , be two timelike (or two

null or two spacelike) geodesics, and let E, be parallel unit timelike vector

fields along 7, with (E1,Tl) = (Ez,T2) (Te = %I). Let f . [O,L] -. R be any

smooth red-valued function. Suppose that for all t in [O, L], exp,,(,)(f (t)E,(t))

is defined; call this c,(t). Suppose further that for all t in [O, L], the geodesic

11 : s w exp,2(t)(~Ez(t)) has no focal point of N(vl(0)) from ~ ( 0 ) for s < f ( t ) ,

Finally, suppose that for all tirnelike 2-planes o, in M,,

APPENDIX A JACOBI FIELDS AND TOPONOGOV'S THEOREM 569

Then, for all t in [0, L],

(cll, cI')~ > (02/, c2')t

Thus if el is a nonspaceljke curve, then c2 is also nonsprtcclikc, and

Thls corollary makes possible a triangle comparison theorem for "thin"

triangles. The model spaces used for comparison are the two-dimensional

de Sitter and antl-de Sltter spaces of constant curvature [cf. Hawking and

Ellis (1973, pp. 124-134). Wolf (1961, pp 114-118)j. A triangle of timelike

geodesics (7lry2,?3) IS "th~n" In this context ~f the following holds: For a

given H. let yl and be limelike geodesics In the simply connected two-

dimensional Lorentzian man~fold of constant curvature H (denoted by h I H )

with ~ ~ ( 0 ) = ~ ~ ( 0 ) . L(Tl) = L1, L(y2) = Lz, and 0 3 = as. Flrst, suppose

there is a timelike geodesic T3 between the endpoints of 7, and T2. Let E be

the parallel translate of TI1(0) along Tiiz For each t m [O, Lz], there 1s a smallest

positive number f ( t ) such that exp(f(t)E(t)) lies on Y3. Second, suppose that

for ail such t , the geodesic s c-t exp(sE(t)) has no focal point of N ( E ( ~ ) ) from

T2(t) for s 5 f (t). If these two suppositions hold. and if y3 is maximal and ail

t~rnelike planes 0 in M satisfy K(u) < H. then L(y3) > L(7,).

The problem is to start with a more ger~eral tirr~elike geodesic triangle,

slice it up into "thin" triangles, apply the result jusi, mentioned to each sllce,

and then put them back together. In the Riemannian theorern. ~onipleteness

is used to ensare that In any triang!~ (yl. ~ 2 . 1 ~ ) . mlnirnal geodesics can be

extended from y3(L3) to yl. slicing up the original trisngle. In the Lorentman

context, global hjrperbolicity succeeds just as well as long as H 2 0. For

N = -q2, however. a problem arises: Not even the model spaces blH arc

globally hyperbolic; indeed, by Proposition 11.8, no tinlelike geodesic of length

greater than ~ / q tan be maximal in a Lorentzian manifold whose timelike

planes a satisfy K(o) 5 -q2. A sol~ltinn to this problem lies In a new concept.

a sort of global hyperbo1ic:ty in the small: For x and y in a l,orenizim manifold

M , let C(x,y) denote the space of nonspacelike curves in A4 from T to y ,

modulo reparametrizat~on. wlth the compact-open topology.

570 APPENDIX A JAGOBI FIELDS AND TOPONOGOV'S THEOREM

Definition A.3. A Lorentzian manifold M is globably hyperbolzc of order q

(q > Of if M is strongly causal and for ail points x and y in M with sup(L(y)

y E C(s, y ) ) < x/q, this space C(x , y) is compact.

The Lorentzian analogue of Toponogov's Theorem can now be stated (all

geodesics parametrized with un:t speed).

Theorem A.4 (Lorentzizen Triangle Comparison Theorem). Let M

be a space-time whose timelike planes a satisfy K(a ) 5 W for some constant

H; M is to be globally hyperbolic or, in case W = -q2, globally hyperbolic of

order q. Let yZ, -y3) be a triangle of timelike geodesics with the future

directed side becween the pastmost and futuremost of the three endpoints,

the other future directed side froxn yz(0), and y3 the remaining future direcsed

side. Suppose that 7 2 m d y3 are maximal and, if H = -qa, that L, < r / q ,

i = 1,2,3. Then

(1) There is in MH a timelike geodes~c triangle (TI, 72, T3) with L(7,) = L,,

i = 1,2,3, and wjthB2 2 a2 andB3 2 C Y ~

(2) For timel~ke geodesics 7, end constructed in MH with T1 (0) = 72(0),

L(T1) = L1, L(T2) = L2, and 3 3 = C Y ~ , if there is a tinlelike geodesic

T3 between the endpoints of Tl and %, then L(y3) 5 L(y3).

The Toponogov Triangle Comparison Theorem can be used to show ho

a bound on sectional curvature can rigidly determine a complete Rieman '

manifold lf the l~mits of the curvature-imposed constraints are attained.

such result is the maximal diameter theorem [cf. Cheeger and Ebin (19

p. 110)]: If a complete Riernannian manifold Mn satisfies K(u) >_ q2 > 0

all planes a , and if the diameter of M attains the maximum thus allowable

x / q , then M is isometric to the sphere Sn of curvature q2.

Theorem A.4 can be used similarly.

Theorem A.5. Let Mn be a space-time which is globally hyperboli

order q and satisfies K(IT) 5 -q2 for dl timelike planes a. Suppose t

possesses a. complete tirnelike geodesic y which is maximal on a2I inter

length nlq. Then M 1s isometnc to the s~mply connected geodesically co

n-dimensional Lorentz~an manifold of constant curvature -q2.

APPENDIX A JACOB1 FIELDS AND TOPOKOGOV'S THEOREM 571

In addition to tirnelike geodesics, it is possible to study the effects of curva-

ture on Jacobi fields along null geodesics. Sectional curvature cannot be used

for this since it is undefined for null (singular) planes, bu t a similar concept,

introduced here, will work.

Definition A.6. If o is a null plane and IV is any nonzero clement of the

one-dimensional space of null vectors in o, then the null aectzonal curvature of

u wzth respect to N, KN (IT), is defined by

where A is any nonnufl vector in a.

This expression for null sectional curvature is independent of the vector A and depends in a quadratic fashion on the vector N. Thls curvature quan-

tity has certain interesting relations with ordinary sectional curvature. For instance, see Proposition 2.3 in Harris (1982a).

Proposit ion A.7. If at d single point in a Lorenczian manlfold the null

sectjonal curvature5 are all positive (respectivel~ negative), then timelike sec-

tional curvature at that point is unbounded below (respectively. above): and

if the null sectional curvatures all vanish, thexi the tirnelike and spacelike sec-

tional curvatures are all equal. Thus, a Lorentzian rllanifold of clirner~sion a t

least three has constant curvature iff it has null sectional curvature everywhere

Null sectional curvature can be used much like timelike sectional curvature

ith regard to Jacobi fields.

Proposit ion A.8. Let ,8 : [0, L] --, iMn be a null (&neIy parametrized)

geodesic with T = 13' If for all nonnull vectors V perpendicular to T, KT(V A

< 0, then p has no conjzlgate points. If for some q, KT(V A T) 2 q2 for all

V or, more generallj: if Elic(T. Tj 2 (n - z ) ~ ~ , chen L 2 n/q impiies 0 oes have a conjugate point.

Nuli sectional curvature also lends itself to a Rauch-type theorem

572 .4PPENDIX A JACOBI FIELDS AND TOPONOGOV'S THEORZM

Theorem A.9 (Rauch Comparison Theorem for Nu11 Geodesics).

Let p, . [O. L] - M,, z = 1,2, be nuli geodesics; T, = Pa' Let V, z = 1.2 he

perpendicular Jacobi fields along P,, not everywhere parallel to T,, (and thus

nowhere parallel to T,), with V(0) = 0 and &'I' Vl')o = (Vz'. V2')o

Suppose hat for any t in 10, L] and for any nonnuli vectors X, at &(tj.

perpendicular to T,

K T ~ (Xi A Ti) < K ~ ~ ( x 2 A T2)

and that p2 has no points conjugate to &(O). Then for all t in (0, L],

{vi,pi)t L {v~,Vz)t.

The necessity of using perpendicular vector fields makes this theory less

tractable than that for timelike geodesics.

Details for Proposition A.8 and Theorem A.9 can be found in Harris (1982d).

Flnally in Harris (1985), consideration is gmen to the case where the rlull

curvature function of all null planes associated to a null congruence IS point

function. It is shoun that this condlt~on locally characterizes Robertson-

Walker metrics and that ~f further completeness and causality assuinptions

are added, then a global characterlzation may be obtained

APPENDIX B

FROM THE JACOBI, TO A mCGATI. TO THE

RAYCNAUDHURI EQUATION: JAGOBI TENSOR

FIELDS AND THE EXPONEKTIAL MAP REVISITED

The use of Jacobi field techniques in the so-called "comparison theory" in

Riemannian geometry stems from the close relationship between Jacobi fields

and the differential of the exponential map. One very b a i c example is provided

by Proposition 10.16 in Section 10.1, in which the Jacobi field J along a unit

timelike geodesic c with J(0) = 0 and J1(0) = ui is gi\ien by

In this appendix we give a second illustration of the relationship between the

differential of the experiential map and Jacobi fields which has come to promi-

nence in the 1980's version of cornparison theory in Riemannian geometry. In

this comparison theory, the use of the Jacobi equation has been partly s u p

planted by the use of a Riccati equation for the ae~ond fundamental tensor of

a local foliation of a Riemannian manifold by hypersurfaceb arising a, the level

sets of a differentiable function with gradient of constant length [cf. Karcher

(1989)) Eschenburg (1975, 1987, 1989)j. A recent applicat~on of the Rccati

inequality method to submanifolds of semi-Riemannian manifolds is given in

Andersson and Howard (1994).

B.1 Generalities on Semi-Riemannian Manifolds

Let (M, g) be a semi-Eemannian manifold with semi-Remannlan metrlc g

of arbitrary signature ( 5 , ~ ) but which is nondegenerate. Let U be an open

subset of &I. This section is cor,cerned with the basic differential geometry of

the foliation induced on U by the level sets of a siriooth fidx~ction f : U -, R wlth

574 APPENDIX B JACOB1 TENSCR FIELDS

.e constant nowhere vanishing gradient vector field V f = grad f of everywhe*

length. Upon resealing, it may be supposed that

(B.1) gfgrad f , grad f ) = X

for some choice of X in (-1. 0, +I). Such functions have been termed solutions

to the eikonal equasion.

This formalism has been useful in a number of different geometric con-

texts. The first and perhaps most well-known example occurs in the case

where (M, g) is Riemannian, U is a deleted neighborhood of p which does not

intersect the cut locus of p, and f = d(p, .), where d denotes the Riemann-

ian distance function of (M,g). In this case: X = 1 in (B.l). The analogous

situation may be considered on a strongly causal space-time where U could

be taken to be a subset of It(p) which does not meet the future timelike cut

locus of p, and f = d(p, .), where d this time denotes the Lorentzian distance

function of (M,g). In this case, X = -1. The case X = -1 also arises (a pos-

teriori) in the course of the proof of the Lorentzian splitting theorem, as in

Beem, Ehrlich, hlarkvorsen, and Galloway (1985, p. 391, in which U could

be taken to be a tubular neighborhood of a complete timelike geodesic line

c : R -+ (M.9) in the globally hyperbolic space-time ( M , g ) , and f = b+

(or f = b-) 1s the Busemann function associated to the future timelike ray

(respectively, the past timelike ray) determined by the given timelike line.

The third case, X = 0, arises in the context of semi-time functions for space-

times, i s . , g(grad f , grad f ) < 0 IS required for a semi-time function The case

that grad f is identically null 1s encountered for the spare-time M = R x S1

with Lorentzian metric ds2 = d @ d t and f ( t , @ ) = -t. A second larger class

of examples is furnished by the gravitational plane wave space-times, Here

the global coordinate function f ( 2 . y,u. v) = u hm gradient grad f = d/Dv,

which is a global null parallel field. The geometric properties of this so-called

"quasi-time fi~nction" have been systematically studied for these space-times

in Ehrlich and Emch (1992a,b, 1993) [cf. Chapter 131.

The following basic remit explains the significance of constant gradient

length in (B.1).

B.l GENEIRALITIES ON SEMI-RIEMANNIAN MANIFOLDS 575

Lemma B.1. Let X = -1 (respectively, -1, 0). Suppose f : U - R is a

smooth function with g(V f. Vf) = A. Let c : [a, b ] -t U be an integral curve

of grad f . Then c is a tirnelike (respectively. spacelike, riullj geodesic.

Proof. Fix any p in U and 2. in T,M. Denoting the Hessian D ( D j ) by ~ f ,

we have

Since v was arbitrary and the semi-Riemannian metric ~vas assumed to be

nondegenerate, we have

on U . Thus the integral curves of grad f are geodesics. O

In the case that (M. g) is Lorentzian and X = -1, or in the case that (,2.f. g)

is Riemannian and X = +1, it is standard that (B.l) has the even stronger

implication that any such integral curve of grad f is a maximal geodesic in

the space-time (U, i , ] ~ ) (respectively, a minimal geodesic in the Riemannian

manifold (U,glu)) by virtue of the reverse (or wrong way) Cauchy Schwarz

inequality for timelike vectors [cf. Sachs and Wu, (1977a)J (respectively, the

Cauchy-Schxvarz inequality for tangent vectors).

We present here the argument for the space-time case, under the assumption

that the vector field grad f is future directed. Thus let y . fa b] + U be any

future directed nonspacelike curve in U with y(n) = r( f 1 ) and y (6) = c(tz) ,

where c is a maximal integral curve of grad f lying in U . Then we first have

that 4.f 07) , g(grad f , 7') = -$---

since both vectors are future directed and grad f is timelike. Secondly. by the

wrong-way Cauchy-Schwarz inequality in the case that yi ( t ) is timelikc, and

trivially if y l ( t ) is null, we have

576

so that

APPENDIX B JACOB1 TENSOR FIELDS

as required. The argument for the Lorentzian case with grad f null is deferred

until after Lemma B.3.

Let us now record an elementary identity in the context of a differentiable

map f : U -+ B for U an open subset of the (nondegenerate) semi-ftiernannian

manifold (M, g) that will be useful in the sequel:

for any p in U and v in TpM. This identity follows immediately from the

The identity (B.3) then has the following consequence.

Lemma B.2. For X in (-1,0, fl), let f : U -+ IR have constant gradie

length X as in (B.1). Then im(f,p) = Tj(@ for each p in U. Hence, f

a submersion and dim( f -l ( T ) ) = dim(M) - 1 for each r in f (U) in a11 t

cases simultaneously.

Proof. This is an immediate consequence of the assumed nondegeneracy

the given semi-Riemannian metric and (B.3). For tl

- he only way that f,, (

B.l GENERALITIES ON SEMI-RIEMANNIAN MANIFOLDS 577

fail to be surjective is that Imf fa,) = (01, since d i ~ n ( T ~ ( ~ ) R ) = 1. But In this

case, we then have from (B.3) that g(v, grad f (p)) = 0 for all v in T,M. Bus

this then implies that grad f(p) = 0, in contradiction to the assumption that

grad f is nowhere vanishing. O

The following is the translation to the semi-Riemannian context of a basic

result in elementary differential topology.

Lemma B.3. Let c : [0, a) - - U be an integral curve of grad f with c(0) in

f - ' ( r ) Then

(1) I f X = 0, then c(t) E f -l(r) for all t ;

(2) I f X = +l, then e(t) E f - l ( r + t ) for all t , and

(3) I f X = -1, then c(t) E f -l(r - t ) for all t .

Proof. Since c(t) is an integral curve of grad f , wc have from identity (B.3)

that

( f 0 c)'(t) = f*c'(t) = g(c1(t), grad f (c ( t ) ) )

= !?( Vf (clt)) , Vf (c(t)j 1 = A.

Hence if X = 0, then (f 0 c)'(t) = 0 for ali t and (1) follows. The other cases

are similar. (if X = -1, then (f 0 c)'(t) = -1 so that (f o c](t) = T - t.)

Thus in the case of a null gradient, instead of mapplng a level surface to a

different level surface, the gradient flow o f f maps each level surface into itself.

This phenomenon is encountered for the gravitational plane wave space-times

where f (3. Z,U,V) = 2 is the "natural" qu~ i - t ime function for this class of

space-times and has everywhere nonvanishing but null gradient (cf Chaptcr

13).

Also as mentioned above, we may derlve the following consequence from

Lemma B.3. Let (M,g) be Lorentzian, and let c . ( a , b) --, U be an :ntegral

curve of f : U -+ IW with gradf = 0. Then c is globally maximal with respect

to arc length among future directed nonspacelike curves lying in U For by

considering -grad f if necessary, we may assume that c is future directed. If c

fails to be maxima1 in the above sense. then there exist s, t with a < s < t < b

518 APPENDIX B JACOB1 TENSOR FIELDS

and a future dirccted timellke curve 5 : [O, 11 -4 U with ~ ( 0 ) = C ( B ) and

a(1) = c(t ) (cf. Section 9.2). But then ( f o a)'(u) = (V f ( o (u ) ) , o l (u ) ) > 0 ,

so that f (c(s)) = f (n(0)) > f (a(1)) = f (c(t f) , in contradiction to part (1) of

Lemma B.3.

Related to Lemmas B 2 and B.3, and also a consequence of equat~on (3.31,

is the following description of the tangent spaces and normal vertor field to

the local foliation ( f -"u) : :L E W) induced on U by the level submanifolds of

f .

Lemma B.4. For any X in (-1: 0, i-11, we have simultaneousiy

(1) The tangent space at any p in f -' (r) is given by

(B.4) T ~ ( f ( r ) ) = ker( f e p ) = (2. E TpM : g(v, grad f (P)) = 0) ;

(2) N = grad f is a normal field of constant length for each level subman-

ifold f - ' (r); and

(3) For any v in T p ( , f - l ( r ) ) we have D, grad f E T p ( f ( T ) ) .

Proof. (3) In view of (B.3) it suffices to note that (as before)

In the case that (M,g) is Lorentzian and grad f is timelike (respectively.

spacelike), the induced metric on the level submanifolds 1s nondegenerate and

the submanifold is thus spacelike (respectively, timelike) in the sense of Definl-

tion 3.47. In the case that grad f is null, the induced metric on the level sub-

manifold 1s degenerate in view of (B.4) slnce grad f (p) is both in Tp( f - ' ( f ( p ) ) )

and orthogonal to all vectors in Tp( f - ' ( f Hence, in the case that grad f

is null, the level submanifolds are not "semi-Riemannian submanifolds" i11 the

sense of OWeill (1983), so that some care needs to be taken in considering the

concepts of the second fundamental form and of a totally geodesic submanifold.

To be consistent with Section 3.5 in notation, given grad f on U we define

an operator L 011 Tp( f - ' ( ~ ) ) by

B.l GENERALITIES ON SEMI-RIEMANNIAN MANIFOLDS 579

In the case that A = +1 or X = -1 and ( M , g ) is Lorentzian, since the level

submanifolds inherit nondegenerate induced metrics, i t is well known that

is the shape operator whose %an~shing determ~nes whether the level submani-

fold is totally geodesic. In the case that A = 0 (or ( M , g) is not Lorentzian),

it is a consequence of part (3) of Lemma B.4 rather than familiar abstract

nonsense that (B.6) is still \slid, so we may contlnue to regard L as the shape

operator. This ib further onf firmed by the following lemma which is standard

when the induced metric is nondegenerace.

Lemma B.5. Let (M. g) be an arbitrary (nondegenerate) semi-R~eman-

njan manifold, and let X E (-1.0, $1). Suppose f : U -+ R satisfies the

further condition

03.7) Ct, grad f = 0

for all v in T,M and for all p IR U. Then every Jet-el subrnanifold f - ' (r)

in U is totally geodesic in the mowing sense: Let v E T,( f - ' ( r ) ) , and let

c . (a. b) -, U be (the restriction of) the geodesic In ( M , g ) with cl(0) = V.

Then c((u, b)) is contained in ~+-'((r).

Proof. Consider the function F : (a, 6) --+ R given h>r ~ ( t ) = f o c ( t ) .

Then we have F(0) = r , and we need to show that F1(t ) = O for all t to

obtain the desired result. Now F1( t ) = ( f o c ) ' ( t ) = g(V f (c j t ) ) , cJ( t ) ) and thus

F1(Q) = g ( V f ( p ) , v ) = O by (B.4). Further,

since c is a geodesic and (3.7) is assumed. Thus F1( t ) = FJ(0) = 0 m r e

quired.

580 APPENDIX B JACOBI TEh'SOR FIELDS

B.2 Jacobi Tensors and t h e Normal Exponential Map in the

Timelike Cease

In this section. we will assume that ( M , g ) is Lorentzian and that U is equipped with a smooth function f : U -+ R with everywhere unit t~melike

gradient. i.e., g(grad f , grad f ) = -1. Since grad f is normal to the level

surfaces of f, we will denote N = g a d f . Also, since we are only interested

in local calculations in this seetlon, we will not bother to carefully specify

domains of geodesics or of maps In the t or s-parameters.

Fix r in R with r In f ( U ) . With the above proviso in mind, recall that it

is traditional to study the geometry of the submanifold f-'(r) by considering

the s ~ c a l l e d normal exponential map E ( . , t ) on f -l(.r), which may be defined

Since the integral curves of N = grad f are geodesics, it follows that in this

particular geometric situation (B.8) is equivalent to followrng the gradient flow

of f starting at points of f -'(T), and thus Lemma B.3 implies that E( . $ t ) is

a map

(B.9) E ( - . t ) : f -"r) -+ ff-'(r - t).

The normal exponential map may be placed in a variational context with

notation ;I? in Chapter 10 for the purpose of studying Jacobi fields along

timelike geodesics initially perpendicular to f -l(r) as in Karcher (1989) for

Riemannian manifolds.

Fix p in fe l ( r ) , and let c : [O,a) -+ M denote the integral curve of N with

c(O) = p; by the above comments, c(t) = exp(tN(p)). Now let v E T,(f-'(r))

be glven, and choose any smooth curve b : (-E, E ) -+ f -'(r) with b(0) = p and

bt(0) = v for purposes of computation. Then define a variation a(t , s ) of c by

(B. 10) 3) = E(b(s), t ) = ex~b(s)(tN(b(s)))

so that

~ ( t , . ) : f -l(r) 4 f -'(r - t )

B.2 THE NORMAL EXPONENTIAL MAP 58 1

and

a, 8/& = grad f o a.

Since s H a ( . , s ) is a variation of c through geodesics, the variation vector

field

is a Jacobi field. By using the chain rule or by introducing some fancy notation,

one may remite the third term of equation (B.11) in a form closer to that of

Proposition 10.16 as recalled in the introduction to this Appendix. Explicitly.

if we let

i : ~ x I R i f-'(r) xR

be the map given by d(s,t) = (b(s),t), then a = E o i and i, =

(b' (0); 0) = (v, 0). Hence, we have

for all t in [O, a). Hence, the set of Jacobi fields J ( t ) as v varies over all vectors

tangent to f-'(r) at p is identified with the differential E, of the normal

exponential map E by means of equation (B.12).

Lemma B.6. The variation vector field J ( t ) of the variation a(t . s) given

by (B. 10) satisfies the differential equation

(B. 13) J I M = -L t (J ( t ) )

for all t in [O,a) with initid condi~ions J ( 0 ) = 1: and J'(0) = -L(v), where L

is the second fundarr~ental tensor for the level surfaces { f - ' (T - t ) ) defined as

in (B.5) by Lt(w) = -D,N.

Proof. Using (B. 11). one calculates

J'it) = D~iat (a* g) I(,,,, = (a*&) / ( t , O )

= Dalaa grad f o a

= Dn.(~/ds)(t,o) grad f = Dq9N = -L,(J(t)) . n

582 APPENDIX B JACOB1 TENSOR FIELDS

Thus in our particular geometric situation, the usual submanifold initial

condition for Jacobi fields, Ji(0) = -L(v) = -L(J(O)) , is propagated for all t

along the level submanifolds f - l ( r - t ) and the transversal timelike geodesic c.

Further, we may reinterpret equation (B.13) in the language of Jacobi tensor

fields as utilized in Section 10.3 and Sections 12.2-12.3. Let

be the Jacobi tensor along c : [0, a) + M satisfying the initial conditions

(B.15) A ( O ) = E and A1(0 )=-L .

Then as me remarked In Lemma 12.21, A is a Lagrange tensor field which

intuitively represents the vector space of Jacobi fields J along c which satisfy

the initial condition J1(0) = -L(J(O)) . In view of equation (B.13), we have

in the present context the further identification for the associated tensor field

B = AIA-I occurring in the Raychaudhuri equation that

for all t in [O, a) for which A( t ) is non-singular, i.e., which are not focal points

top. Also, one may interpret equation (B.12) as identifying the differential E,

of the normal exponential map with the Jacobi tensor A. Equation (B 16) then

has the implication that the Rqchaudhuri equation for A (or equivalently, E,)

is related to the trace of L and its derivative L', hence to the mean curvature

of the ievel surfaces ( f - ' ( r - t ) ) (cf. equation (12.2) in Section 12.1).

B.3 The Rieeati Equdion for L

There is a venerable technique in the disconjugacy theory of ordinary dif-

ferential equations in which a Jacobi equation such as u" + q(t)u = 0 is linked

to an associated Riccatr equation r' + r2 + q( t ) = 0 by the change of variables

r = ul/u [cf. Bartman (1964)]. The n x n matrlx version of this theory involves

forming B = AIA-' exactly as is done In general relativity in passlng from

the Jacobi equation A" + RA = 0 to the associated hychaudh>~ri equation

(cf. Sectioris 10.3 and 12.2-12.3).

B.3 THE RICCATi EQUATION FOR L 583

In this section, we shall follow the route suggested in Karcher (1989) and

obtain an intermediate Rtccatr equatlon satisfied by the second fundamental

forrn operator in the geometric context under consideration. Thus. let (M, g)

be a Lorentzian manifold and f : U -+ R be a smooth function with everywhere

timelih gradient of constant length -1, as in Sectron B.2. Let p E f - ' ( r ) be

fixed, and let c : [O, a ) --, U denote the integral curve o f f starting at p. Choose

any v E Tp(f - ' ( T ) ) , and let J = J, denote the Jacobi field along c with initial

conditions J(O) = v and J1(0) = -L(v) . We established in Section B.2 the

identity

(B.17) J1 ( t ) = - L o J ( t )

for all t in [0, a). Recalling here that X1(S) = DaIatX for vector fields and

tensors along the geodesic c, we have

or in more compact notation,

Thus differentiating (B.17) covariantly with respect to t yields

J" = - (LO J)' = -L1 (J ) - L(J1 )

= -L'(J) - L( -L(J ) )

again using (B.17). Combining this last result with the Jacobi equation J" + R(J, c1)c' = 0 then yields the desired Riccati type equation for J,:

(£3.13) L1(J,) - ( L o L)(J,) = R(J,. d)cl for a11 t in [O. a )

if we denote the composition L o L by L? then prior to the first focal point

to t = 0 along c to the submanifold f - ' ( r ) , we h ~ v e from (B 18) the operator

equatlon

(B. 19) L' - L~ = R( - , c')cl.

584 APPENDIX B JACOB1 TEXSOR FIELDS

Recall from equation (B.16) that B = A'A-"may be identified with -L prior

to the &st focal point to p along c, and recall also the earlier notation of

Definition 12.2 of Section 12.1 of 6 = tr(B). We rnay then rewrite equation

(B.19) as

(B. 20) B' = -B o B - R(,, ct)c'

in exact accord with equation (12.1) of Section 12.1. Recalling further that

tr(I3') = [tr(B)I1, we may obtain either from (B.19) or (B.20) the equation

(B.21) 8' = -tr(B o B) - Ric(cl~ c') = -tr(L o L) - Ric(cl, c')

which corresponds (with change in sign conventions) to equation (1.5.5) in

Karcher (1989): d/ds[trace (S ) ] = - t race(R~) - trace(S2), which is as far

as Karcher proceeds in decomposing this particular quation. In general rel-

ativity, the Raychaudhuri equation for 6' is derived from (B.21) by further

decomposing B in terms of the expansion, shear, and vorticity tensors and

calculating B2 in terms of this decomposition (cf. the derivation of equation

(12.2) in Section 12.1).

Note also that equation (B.21) occurs in another related form in connection

with the "Boehner trick" type identities derived by manipulations of the curva-

ture tensor and by utilizing the fact that the normal field used to calculate the

second fundamental tensor is a gradient, i.e., N = grad f. In this formalism,

taking a parallel orthonormal basis {El, E2,. . . , En) along c with En = c', one

is then led to the following version of (B.21) (U denotes the d2Alembertian of

f 1:

and hence the inequality

1 (B.23) -Ric(cl. c') b (m(f) 0 c)' + n_i(I(f) 0 el2,

which plays a role in the proof of the so-called splitting theorems in both

Riemannian and Lorentzian geometry. In Beern, Ehrlich, Markvorsen, and

B.3 THE XCCAT! EQUATION FOR L 585

Gdloway (1985, Lemma 4.3 and Corollary 4 4 , for instance. the untraced

version of equation (B.23) corresponding to equation (B.20) is employed.

Finally, let us conclude this section by deriv~ng as in Karcher (1989) a

we!l-known second link between E, and the mean curv~ture tr(L(t)) in the

present context by using equations (£3.12) and (B.16). Put a( t ) = det(E,),

or more formally, a(t) = det(A(t)). Tnen using the identity tr(AIA-') =

(detA)-"det A)' of Section 12.1, Definition 12.2, we obtain

Thus the expansion tensor B(A(t)) is, up to sign, precisely the mean curvature

at c ( t ) of the hypersurface in the family of f-level surfaces passing through the

point c(t), Karcher's viewpoint is that the Jacobi equation should be regarded

as giving geometric control by considering the Jacobi equation as equivalent

to the so-called Riccati inequality (B.23) and to equation (B.24). Of course,

equation (B.24) also demonstrates rather explicitly the well-known fact that

in the hychaudhuri framework, the expansion tensor @(A) is the quantity

that measures the infinitesimal change in hypersurface volume; indeed, in Es-

chenburg (1975, p. 32) the tensor 6 is even called the "volumen-expansion''

tensor.

In the context of equation (B.241, the following elementary calculation has

been noted. Suppose that a( t ) and b(t) are differentiable functions which

satisfy the ordinary differential equations

and

respectively, Then

Hence if a(t) and b(t) are positive valued and f 2 g , then a/b is monotonic

nondecreasing. This remark provides, after integrating along radial geodesics

586 APPEXDIX B JACOB1 TENSOR FIELDS

emanating from p, a proof of the Bjshop-Gromov Volume comparison The-

orem for Riemannian manifolds, choosing $ = d(p, . ) and taking g to be the

correspo~lding distance function on the appropriately chosen model space of

constant curvature [cf. Karcher (2989, pp. 186-I87)j.

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LIST OF SYMBOLS

BALLS

B+(P, €1, B- (PI 6) inner balls: future and past. 142, 143

O+(PIE), O-(P,E) outer balls: future and past, 144

BOUNDARIES

aa -44 dbM BCM d i f , i- io

abstract boundary, 198, 232-238 Schmidt boundary, 214 causal boundary, 215-218 topological boundary, 495 timelike infinity: future and past, 178 spacelike infinity, 178

BUSEMANN FUNCTIONS

b Busemann function, 512 b+, b- Busemann functions: future and past, 550 b ~ , bp,, various Busernann type functions, 512, 520

CAUSALITY

D+(s), D - ( s ) Cauchy development: future and past, 287 E + ( s ) , E- (S) horismos: future and past, 287

H + ( s ) , f f - (S) Cauchy horizon: future and past, 287

I+ ( P I , I - (P) Chronological future and past, 5. 6, 54

J+ { P I , J - (P) Causal future and past, 5, 6, 54

P K q p in chronological past of q, 54 P Q ~ p in causal past of q. 54

CUT LOCUS

rz", Iff

AS V

[X, Yl grad f = Vf

DIAMETER

GROUPS

INDEX

LIST O F SYMBOLS

nonspacelike cut locus, 311-322

nompacelike cut locus: future and past. 311-322 null cut locus: future and past. 296-318 timelike cut locus: future and past. 296-305 cut locus in T,M, 302

Christoffel symbols, 18 Hessian of f, 24 Laplacian of f , 24, 120

(Levi-Civita) connection, 15, 22 Lie bracket. 18 gradient of f , 24. 307

timelike diameter, 399. 401, 402

Riemannian distance, 2-4 Lorentzian distance, 8, 137

fundamental group, 399, 419 isometry group, 188 isotropy group at p, 186 left and right translation, 190

index of vector fields X, Y, 325 quotient index, 326 index for hypersurface, 458 extended index, 326, 342, 395 index, 326, 342, 395

LIST OF SYMBOLS

INNER PRODUCTS

g(V, W ) = (V, W ) inner product on tangent space, 327

(w , Xr) = u(X) on one forms and vector fields. 20 << , >> inner product on T,(T,:W). 338

J Jacobi field, 31

Jt (4 Jacobi fields vanishing at a and t , 326

Jt (P) quotient Jacobi tensors vanishing at a and t . 326

A @ ) , a t ) Jacobi tensors, 426, 431 B (= A'A-I), fS used with Jacobi tensors, 428, 431

LENGTH

L(Y length of curve y. 8, 137

MANIFOLDS AND SPACE-TIMES

MAPS

ring of smooth, real-valued functions on M, 16 class of srnooth vector fields on hl, 16 Euler characteristic, 50 warped product, 95

semi-Euclidean space. 181 (universal) anti-de Sltter. 183, 199, 306 hlinkowski, 25, 116. 177 de Sitter, 183, 286 real projective 3-space, 189

exponential map, 53 canonical isomorphism, 337

METRICS

c ( M . 9 ) set of all metrics conformal t o g. 6, 142 Con(iM) conformal equivalence classes, 239

i sww) quotient metric. 370-398

g1 < g2

partial order on metncs, 64

91 < 92 partial order on metrics, 64 4 Lor(M) i

'iet of all Lorentzian rnetrics on M , 50, 63, 89

620 LIST OF SYMBOLS LIST OF SYMBOLS

SECOND FUNDAMENTAL FORMS

second fundamental form operator, 93, 165 second fundamental form, 92 in direction of n. 93

TANGENT

T ~ N normal space to N, 444

TM tangent bundle of .nA, 16

T,M tangent space to M a t point p, 16

T-lM unit observer bundle, 298-304

TENSORS AND CURVATURES

sectional curvature, 29-32 curvature, 19, 20 Riemann-Christoffel, 22 scalar curvature, 23 Ricci curvature, 23, 30 energy-momentum, 44-48 torsion, 13 Wronskian, 385 expansion, 428,431 shear, 429, 431 vorticity, 428, 431

TOPOLOGIES

C0 topology on curves, 49. 72, 79

CO topology on metrics, 63, 89, 239, 247

C' topology on metrics, 63, 247

VECTOR FIELDS, CURVES, AND BUNDLES

C ( ~ + 7 ) piecewise smooth timelike curves from Conn(P0, u) in P(u) and connected to Po by

a geodesic, 485 p to g, 354-365

G(P) quotient bundle, 326, 368 space orthogonal to c'(t), 327

orthogonal to /?'(t). 368 null conjugate locus, 489 null tail. 433 null hyperplane in R4 481

piecewise smooth and orthogonal to c, 327, 369 piecewise smooth, orthogonal to c, vanishing at endpoints, 327 piecewise smooth and orthogonal to a, 366 piecewise smooth, orthogonal to P, vanishing at endpoints, 367 change in X'. 329 path space from p to q, 326

a-boundary (abstract boundary 3,). 198, 232. 233-238

Achronal. 57, 142, ~dentity. 538

Adapted coordmates. 251 normal neighborhoods, 251

Adm~ssl ble chain, 252, 253-255 deformation, 389, 390 measure. 67 variation, 376. 388

Affine f~mctlon, 332. 333 parameter, 12, 17. 112 parameter, generalized, 208

Alexandroff topology. 7, 8, 39-60 Almost maxrrnal curve. 146,272-275 Angular momentum for Kerr space-

time, 181 Anti-de Sitter space-tlrne, 183, 199.

306, 569 Arc length

functional, 356 Lorentzim, 135 upper semicontinuity, 83.273, 278,

51 1 Arzela's theorem, 75, 76 Astigmatic conjugate. 485, 486-4'39

~ s y m p t o t e. 518

b. a. complete (bounded aceelerati 197, 207

b-bcundary (bnndle boundary at), b-complete (bundle c~rnplet~e), 15

208 Basis

orthonormal. 21, 30, 404 natural, 1 G pseudo-orthanormal, 36, 460

Big bang cosmological model, 10 174, 185, 190, 477

see also Friedmann cosmologic, models

see also Robertson-Walker spac t!me

Bi-invariant Lorentziari merrlr, 19( 191-195

Birkhoff's Theorem, 132 Black hole. 480

Schwarzschild. 173, 179-182 Kerr, 174, 179-181

Bonnet-Myers Theorem. 399-400,4 Bore1 Measure, 67 Boundary pomnts

a-boundary (d,), 198, 233-238 approachable, 234 b-boundary ( d b ) . 214 causal boundary (8,). 21 5-218

425, 474 C"-rep~llar 219

624 INDEX

from enveloped manifold: 230 quafi-regular, 225 re,@%, 225 smooth, 472478

Bounded wceleration, 197, 207 Bounded parameter property, 233 Boyer-Lindquist coordinates, 181 Busemann functions, 503-566

Lorentzian, 512, 550 Riemannian, 507 see also horoball see also splitting theorems

C(M,S), 6, 142 c-topology

on c ~ w e s , 49,72, 79,81-83,272, 273, 278,288, 297

on Lor(M), 50, 63, 89, 239 Q-stable, 242 Cr-topology, 63, 247 Canonical

isomorphism (r,) , 337 variation, 331, 388

Cauchy complete, 4. 209, 425 development (D+, D-), 287,466 horizon (Nf, H-) , 287 partial surface, 539 reverse Cauchy-Schwarz, 575 surface, 65, 362 time function. 65 timelike complete, 211 see also horismos

Causal boundary (a,), 215-218,425,472-

478 future ( J f ) , 5, 6 past (J-), 5, 6 relation (I), 54 space-time, 7 stably, 63, 89 vector (= nompaeelike), 24 see also globally hyperbolic

Causally continuous, 59-60, 158-160,479 convex, 59, disconnected, 272,228-293 related, 54 simple, 479

Chistoffel symbols (r,k,), 18, 248 Chronologicd

future ( I f ) , 5, 6, 54, 55 past (I-), 5, 6, 54, 55 relation (a), 54 space-time, 7

Closed geodesics, 145-146 trapped surface, 148-151, 463

Cluster curve, 73 Compact

space-time, 58, 418 Complete

b, 197 b. a., 197 geodesic, 12, 17, 202 geodesicdly, 4, 425 nonspacelike, 12 m111, 12 pregeodesic. 1 7 Riemannian manifold, 4 spacelike, 12 timelike, 12 timelike Cauchy, 21 1 see also Hopf-Know Theorem

Con(M), 239 Cone

null, 15, 25-29 Conformal

changes and ndl cnt point,s, 306 changes and null geodesics. 308 factor, 91 global transformations, 28, 307 metrics, 6. 28

Conformally stable property of Lor(M), 249

Conjugacy index, 485

INDEX 625

Conjugate points, 295, 298, 302-322, null sect~ond cllrvature. 571 413 Riemann curvature (R or RiJkm),

definition of, 314, 328, 374 19, 20, 30 locus of first need not be closed Kernam-CIzristoEel (RJk,), 22,

in Riemannian, 315 30 Conn(P0, u), 485, 489-496

Kcci (Ric or K J ) , 23, 30, 45 Connection (V), 16 coeficients (I':), 18, 22 R(X,Y)Z, 19

scalar (R or r ) , 23 curvature of ( R ( X , Y)Z), 19, 22 mduced on submanifold (VO), 92 sectional(K(E,p)), 29-32, 399.

Levi-Civita, 15, 22 403

symmetric, 19 singularity, 225 torsion free, 19. 22 tensor, 20

torsion of ( T ( X , Y)), 19 Curve Conservation laws, 45 dmost maximal, 146, 272-275 Convex 6'-topology on space of, 79, 81-

hull, 268, 270 83, 272, 273, 278, 288, 297 neighborhood, 53-54 endless, 61 normal neighborhood. 53-54,292 endpoint of, 61, 233-234

Co-ray, 518 future d~rected, 50 asymptote, 518 geodesic, 17, 18 generalized, 510, 518 impr~soned, 61 , 221, 264 (generalized) timelike co-ray con- inextendible (= endless), 61

dition, 519, 522, 531 lirmt. 72, 81, 297 Cosmological constant (A), 44, 189 limit point, 233-234

see Einstein Equations maximal. 11, 66, 146, 166-171, Cosmologicd models

272, 275. 278, 296. 298, 356 see F'riedmann cosmologicd mod- null (= lightlike), 25 els

partially imprisoned, 62, 264.426 see Robertson-Walker space-tlmw past directed, 50 Cosmologieaf space-time, 563

Covariant pregeodesle, 86, 295, 286, 307-

curvature tensor, 22 309

derivative (;), 18, 370 spacehke, 65 Covering boundary sets, 231 timelike, 25

Covering space, 52, 86, 148, 419, 473 Gut points Critical point, 355, 356, 361 comparison to conjugate po~nts, Curl, 120 302-322 Curvature

bounded, 31 components ( R 3 k m ) , 20 constant, 31, 181-185 covariant tensor, 22 identities. 22-23

nonspacellke, 311-323 null, 296-298, 305-318. 395 Riemannian, 295. 31 7 time!ike. 296, 302-305 see also conjugate point

CW-complex, 355. 363. 354

626 INDEX

d'Alembertian (a), 114.534,535,545 Degenerate plane section. 29 de Sitter space-time, 163, 184, 286,

549, 569 Diameter, 399

timelike (diam(M, g)), 399, 401, 402

Disprisorment, 401, 415, 416, 418, 420, 422

Distance function, Lorentzim, 8, 137 continuity of for globally hyper-

bolic, 140 distinpshing space-tlmes, 158 finite, 142, 162 finite compactness. 211 globally hyperbolic spacetimes,

11, 65. 140, 142 inner (metric) balls (B', B-), 142,

143 local distance function. 160 lower semicontinuity of, 140. 277,

290-291, 508 nonsymmetric, 138 outer (metric) balls (Of, 0-), 144 preserving maps, 151 reverse triangle inequality, 140.

274. 279, 508 totally vicious space-t~mes, 68.

137 Distance function, Riemanman, 2-4

complete, 4 finite compactness, 4 see also Hopf-Rinow Theorem

Distance homothetic map, 151 Distingilishng

space-time, 58. 70, 158 Distribution, 121, 123 Diverge to infinity, 271, 272. 283.

288 Domain of dependence, 287

see Cauchy developrneilt

Edge, 538 Eikonal equation. 541, 574 Eimtein

equations, 44-48 manifold, 133 space-time, 117 static universe. 189,286, 290, 306,

307: 310. 313, 318, 323,477 silmmation convention, 23

Einstein-de Sitter universe, 115 Embeddings in high-dimens~onal Min-

kowski spacetime, 354 Empty spacetimes. 180

see also Ricci flat spacetimes Ends, of manifolds, 272, 282 Endpoint, 61, 234 Energy

conditions, 434 density, 47, 189 function, 375

Energy-momentum tensor (T, ) ,44- 48, 180

see also Einstein equations Enveloped mmifold, 2.30 Equivalent boundary sets. 231 Euler characteristic ( x ) , 50 Expansion tensor (8). 428, 585

quotient expansion (8), 431 Exponential map (exp,), 53. 314-

322, 399, 410 normal exponential map, 580 see also conjugate points see abo normal coordinates

Extended index for null geodesics; 395 for timelike geodesics, 342, 407-

408 Extemion, 198, 21 9. 219-225

C1-extemion, 232 local, 198, 220, 221-225

INDEX

given a pseudo-orthonormal sis, 37

Finite related to Ricci curvature, 39 compactness, 197, 21 1 related to sectional curvature distance condition, 142, 162 nonnull, 34

First variation formula, 360, 450 77.

Geodesic r lat f i n e parameter, 17, 202

metrics, 45, 141 closed, 149, 295 Ricci flat, 45, 124-127, 180 complete, Lorentzian, 91, 20, see also semi-Euchdean spaces

n, . I complete, Riemamm, 4 r m a

see perfect fluid Focal point, 444, 445, 446-449, 462

to a spacelike hypersurface, 447 Free, action of a group, 419 Freely falling, 31, 43, 425 Ekiedmann cosmological models, 286,

306, 311. 402

comwtedness, 400,417,418, equations of, 17, I8 incomplete, 108. 202, 309 instability of completeness, 2 instability of incompleteness, line, 272, 519 line, nonspacelike, 273, 28.3: 2

29.1 -- - see also Robertson-Walker space- line. null. 286

times Fundamental group (T I ) , 399,419 Future

Cauchy development (D'), 287 causal ( J + ) , 5 chronological ( I+) , 5 directed vector field, 5 horismos (E'), 175-177,287,288-

293, 317, 322, 468 horizon (H'), 287 ~mprisoned, 221 one-connected, 351,352,353,400,

414. 415 set, 55 trapped set, 287. 288, 465, 468 unit observer bundle (T- lM), 298-

304, 523

1 - - - line, timelike, 273, 290, 519 loop, 295. 312 maximal. 66, 146, 166-171. 2

275, 278, 296, 298, 356 minimal, 3, 4, 272 pregeodesic, 86, 295, 307-309 ray, 271, 279, 281,289

Geometric realization for quotient dle, 368, 369, 372, 373

Geroch splitting theorem, 65, 102 Global

conformal transformations, 28 time function, 64, 665

Globally hyperbolic, 11, 65 embedding in high-dimensions

kowski space-time. 354 of order q, 570

Gradient, (grad f ) , 24,307, 363,4

N Gallss' Lemma, 338, 379 General position, 40 Hadamard-Cartan theorems, 399- Generalized affine parameter, 208 411-423 Generic condition, 33-44, 309-3 11, Hausdorff

433 closed limit, 74 given an orthonormal basis, 33 distance, 258

lower limit (Lirn i d {A,)). 74 upper limit (Lim sup (A,)), 74

Hessian (Hf), 24 see also Laplacian see also d'Alembertian

Homogeneous completeness of (pos. def. case).

185 implied by isotropic, 186, 187 spatially, 155-188,261 two point. 185-187

Nomothetic, 97, 99, 151, 155, 188 Womotopy, 149, 400

groups, 399. 419 type, 325,326 see also fundamental group

Wopf-Rinow Theorem, 4 , 75, 197, 205, 271, 276, 399, 417

see also Cauchy complete Horismos (E+, E-1, 175-177, 287,

288-293, 317, 322, 468 Horizon (H', H-), 257 Noroball, 540, 541

see also Busemann function Norosphere, 538, 541 Hypersurface

aehronal, 57 spacelike, 64, 539

Ideal boundaries, 198,214-218,232- 238

Imprison&, 221,264 Incomplete

geodesic, 108, 202, 309 nonspac&ke, 425 null, 108 space-time, 108 timelike, 108

Inequality reverse Cauchy-Sehwarz, 575 reverse triangle, 10,140,274,279

Indecomposible future (past) set, 215- 218

Index, 323-398 comparison theorem, Lorentzian,

400. 406-410 ramparison theorem. Riemannian,

400. 406 extended index (Indo), 342 forms, 325, 328, 375, 377, 406,

458 geodesic, 324 timelike (Ind), 342, 407-408 to a spacelike hypersurface, 458

Inextendible space-time, 219, 220

Infinity diverge to, 271. 272, 283, 288 future timelike (if) , 178 spacelike (io), 178 past timelike (i-), 178

Inner continuo^^, 59 Inner products

metric (g and < , >), 21 on T,, (T,(M)) (< . >>) . 338 on vectors and co-vectors ( w ( X f ) ,

20 signature ( ( 8 , r ) ) , 15, 21

Inner (metric) ball (B+, B-) , 142. 163

Interval topology, 241 Irrotational, 120 Isometry, 97, 99, 354

local, 181 group ( I ( M ) ) . 188

Isotropic, 185, 186, 261 see also two point homogeneous

Isotropy group (I ,(M)), 186

Jacobi classes, 372, 373-398 classes, mslxirndity of, 988 equation, 31, 41, 328 fie&, 51 , 328, 332-398 fields, maximality of, 342 tensor, 383, 426

INDEX

see also second variation see also tidal accelerations

Kernel of a (1,1) tensor field along a null geodesic, 382

Kerr space-time, 179-181 Killing field, 120, 121, 482 Koszul formnla, 109 Kruskal diagram. 182

Lagrange tensor field. 385,427,428, 432

Laplacian (A), 24, 120, 545 see also Hessian see also dlAlembertian

Length Lorentzian, 8, 136 see also energy function

Levi-Civita connection (V), 15, 22 metric compatible and torsion free,

22 Lie bracket ([X, Y)), 18 Lie group, 190-195

translations of, 190 see also bi-invariant metric

Light cone. 15, 25-29 used to partial order metrics, 64 see also null cone

Lightlike (= null), 24 Lirni t

curve lemma, 5.7 2 curve of a sequence, 49, 12, 297 in Co topology, 79, 81-83, 272.

273, 278, 288. 297 Hamdorff, 74 maximizing sequence of causal curves,

515 point of a curve, 233-234

Line, maximal geodesic, 273-293 Lipschitz, 57, 75, 136 Local extension, 198. 220, 221-225 Loop space, 363, 364

Lor(M), 50, 63,64,89; 101. 23s Lorentzian

arc length. 135 distance function, 8, 137 manifold, 5, 25 metric, 5 product, metric, 95, 176, 31 quotient metric (g) , 370 signature (-, +, . . . , +), 15

51, 400 triangle comparison theorem warped product, 49-50, 95,

133 180-189

Madfold Lorentzlan. 25 Riemannian (= pos. def.), 1 seml-Riemannian, 15, 264 tangent bundle of ( T M ) , 16

RIaxlmal curve, 146, 296, 298 geodesic, 66. 166171, 356 space-time, 198

Maxlmal~ty of Jacob: classes. 388 of Jaeobi fields. 342

Mean clirvature. 213, 544, 585 Measure zero, 355 Metric

compatible with conneetion, , complete Lorentzian, 2/33 complete %emaman, 4 inner ball, 142. 163 Levi-Civita connection of, 22 Lorentzian. 25, 51 nondegenerate. 21, 92 outer ball. 144-145 quotient, 370 Rimannian (= pos. def.), I

25 Schwarzschild. 45, 131. 132, 17

179-182 semi-Rlernannian, 20 21 264

630 INDEX

semi-Eudidean, 181, 245 signatufe of, 21 tensor (g or g,), 21

Minimal geodesic segment, 4. 272 hilinkowski space-time (Ln = Ry),

25, 116, 159, 177, 173-184, 270, 286, 311. 354, 410: 480

Morse function, homotopic, 355,356, 363

Morse Index Theorem null, 365, 398 Riemannian, 324 timelike, 346, 405

Natural basis, 16 Nconj (Po), 489-497 Nondegenerate

metric tensor, 21 plane section, 29, 403 submanifold, 92

Nongeneric, 33 Noiiimprisoment, 263 Nonspacelike (= causal)

curve, 54 cut locas, 311-323 geodesic line, 273, 283, 285-293 geodesic ray, 279, 281 geodesically complete, 202 geodesically incomplete, 202,425 i i d t curve, 73, 297 vector, 24

Normal coordinate neighborhood, 53- 54

see also convex normal neighborhood

Null boundary point, 4 74 cone, 15, 25-29 conjugate point, 306, 395 convergence condition, 434 curve, 25 cut locus, 296-298,306309,395 index forms, 325,375,377,395

geodesic completeness, 91, 202 geodesic incompletenms, 202,306.

309 geodesic ray, 289 geodesics and conformal changes,

308 line, 286, 290 Morse Index Theorem, 324, 365,

398, 405 pregeodesics, 110, 307-309 tail (NT), 493, 494497 tangent vector, 24

Orientation by timelike vector field, 5, 25, 50

Orthogonal vectors, 21 Orthonormal basis, 21, 30, 404 Outer continuous, 59 Outer (metric) ball, 144

P

P(UO)I 481 Parallel translation, 1 7 Partial

imprisonment, 264, 265, 416 order on metrics, 64

Past Cauchy development (D- ) , 287 causal ( J - ) , 5 chronological ( I - ) , 5 directed vector field, 5 set, 55

Path space from p to q, 326 timelike (C(,,q)), 146, 354-365, 400

Penrose diagram, 179, 284 Perfect fluid, 47, 189, 311 Piecewise smooth

variation, 329, 389-391, 450 vector field, 327

Plane section degenerate, 29 nondegenerate: 29: 92, 403

sectional curvature of, 29-32,403 spaceiike. 29 timelike. 29, 403

Plane wave solution, 479-499 definition of gravitational plane

wave, 483 definition of plane fronted waves,

480 polarized, 483 sandwich wave, 483 unimodal gravitational plane wave,

497 Positive definite, 15 Pregeodesic, 17, 86, 110, 111. 295,

307-309 Pressure, 47, 180. 189 Proper variation, ,329. 389-391 Properly discoi~ti~mously. 419 Pseudoeonvex, 265, 268, 269, 270.

401, 415, 416. 418, 420, 422, 488

nonspacelike, 269 Pseudo-orthonormal bais, 36, 460

INDEX 6

Quasi-regular singularity, 225 Qliotient

bundle (G(P)), 326, ,368 bundle index form, 377 covariant derivative. 370, 371 c~irvatme tensor, 371 expansion tensor (8 ), 431 metric, 370, 371-398 shear tensor ( 8 ), 431 vorticity tensor ( G ), 431

Quotient topology on Con(hf). 241

Railch Comparison Theorems. 400, 406. 408, 572

Ray. 271 nonsparelike geodesic, 279 null geodesic, 289, 290

Flaycha~idh~lr~ eqnatiun, 402

along mdl geodesics, 432 along timelike geodesics. 430 v~rt~icity free, 430, 432

Real projective space, 189 Reflect,lng, 69 Regular boundary points. 225 Reissner-Nordstrcm space-time, 15

179 Residual set, 355 Retraction. 357 Reverse triangle inequality, 10, 14t

274, 279 Riccati equation. 573. 582 Rieci clirvatlire (&c or a3), 23, 30

45, 180 Riemann-ChristofFel tensor ( a 3 k s n )

22 Riemannian

distance function, 2-4 manifold, 25, 30 nletric (= pos. def.), 15, 24 t,wo point homogeneous, 185-188

Robe1 tso~i-Walker space-tlme, 103, 174. 185-190. 250-263, 311, 477

having closed trapped surfaces, 472

Root warping ftinction (S = a), 119

see also warping function Rotation group (SO(3)), 180

Scalar c~~rvature ( R or T ) , 29 Schmidt &boundary ( d b ) , 197. 214 Schwarzsch~ld space-time. 45. 131,

132, 173, 179-182 Second fundamental forms (S. S, L,)

165, 445 1x1 general (S), 92 in the direction n (5,). 93 opelator (L,), 93, 456, 461 see also closed trapped surface see abo totally geodesic

632 INDEX

Second variation formula, 324, 329 Sectio~lal curvature (K(E.p)) , 29-32,

399. 403 bounded, 31,403 null, 5.iI timelike, 29-32, 403

Self-abjoint, 430 Semicontinuity. 140, 273, 277, 278.

290-291.299-301, 511 Semi-E~iclidean (R:), 181, 24 5

area, 29 classification of planes, 29

Semi-Riemannian metric, 15, 264 Semi-time function, 68, 490 Shear tensor (a), 428

quotient ( 5 ), 431 Signature

degenerate, 29 Lorentzian, metric, 21, 25, 51.

400 Riemannian (pos. def.), 15, 25 semi-Eudidean. 29, 181, 245 semi-Riemannian, 21 type (ti, r), 21, 181, 245

Simply connected. 352 Singalar

C'-singular point. 237 spacetime. 12,203,225-238.426

Singtilarity theorems, 468, 469, 472, 478

Smooth boundary point, 474, 472- 478

Spacelike botindary point, 474 curve, 25 hypersurface, 539 infinity (to), 178-179 stibmanifold, 92

Space-time, 25, 51 a-boimndary, 198, 232-238 anti-de Sitter, 183, 199, 306, 569 big bang models, 103-104 b-boundary, 214 b complete, 208

b. a. complete, 207 causal, 7. 58, 73 causal bo~mdary, 215-218, 425.

472-478 causally continuous, 59, 71, 73,

158-160,479 causally simple, 65, 73, 479 chronological, 7, 58, 73 cosmological, 563 c~trvat,u~re sin~ilarity, 215 definition of, 5. 51 de Sitter, 183, 184, 286, 549, 569 di~t~i~lguishing, 58, 79, 73, 158 Einstein static, 189, 286,289, 306,

307. 310, 313, 318, 323, 477 flat, 141 Riedmann cosmological models,

286 306, 311, 402 f~iture one-connected, 351. 352.

353, 400, 414, 415 geodesically complete, 91. 203 globally hyperbolic, 11, 65, 71,

73, 354 incomplete, 203, 309, 425 Kerr, 174. 179-181 locally inextendible, 198,220-238 Minkowski (Ln = R;), 25, 116.

159, 177, 173-184, 270, 286. 311, 354, 410. 480

nonspacelike geodesirally complete, 203

null geodesically complete, 253 plane wave solution, 479-499 qi~nsi-reglllar singularity, 225 refiecting, 69 Reissner-Nordstrom, 139, 179 Ricci flat, 45, 124, 180 Robertson-Walker, 103, 174, 185-

190, 250-263 311. 477 Schwarzxhild, 45, 131, 132, 173,

179-182 singular, 203, 425, 468 stably catlsd, 63, 73, 89, 242 static, 121, 180

INDEX 6:

strongiy causal, 7, 59, 73 timeiike geodesically complete, 2&? totally vlcious, 68. 137, 168

Sphere Theorem, topological. 502 Sph~tmg theorems

Cheeger-Gromoll. 503 Geroch, 49, 65, 102 Lorentzian. 506, 550, 562

causality, 63. 89, 242 conformally, 242 in the C' t?opology. 239, 242 properties in Lor(M). 239, 247

Static. 121, 180 Strong callsalits 7, 59 St,rong curvature singularity, 225 Strong energy condit~on, 434

In the first edztzon of thzs book, we used strong energy condition to mean what zs now termed the timelike conver- germ condition.

Submanifold indticed connection (VO), 92 nondegenerate, 92 second ftindamental form. 92-94 spacelike, 92 timelike. 92. 164 tottally geodesic, 99 94. 98, 165

summation con.i.entio11. 23 Synge world filnct~on. 491

Tangent b~lndle ( T M ) . 16 Tangent space (T,M), 16. 25. 336

null (= lightlike). 24 spacelike, 24

cnrvatiire (R(.Y. Y ) Z ) , 19 energy-momentum (T or T,,), 4

48 expansion (8) . 428. 585 expansion, quotient ( i? ), 4 31 Jacobi. 383. 426 Lagrange. 385. 427, 428. 432 metric (g or g,J). 21 Rcci (Ric or a?), 23, 30, 180 Bemann-Christoffel (R,J,,), 2; scalar ctirvatiire (R or T ) , 23, 21 shear (c) 428 shear. qt~ot~erit ( 8 ), 431 torsion ( T ( X , Y ) or T , J k ) . 19 vortit~ty (w) , 428 vorticity, quotient (Z j ), 431 Wrorlskian (W(A. B)). 385

Term~nsl indecomposible futtire set (TIF), 215-218 past set (TIP), 215-218

Theoreins. selected Arzela's Theorem, 75, 76 Birkhoff's Theorem, 132 Bonnet-Myers Theorem, 339-40(

405 comparison theorems. 400, 408 Geroch Sphtt~ng Theorem, 65,

102 Hadamard-Cartan 'Theorems. 39'

401, 411-423 Hopf-Rinow Theorem, 4.75.197

205. 271 276, 399, 417 Maximal Diameter Theorem. 570 blorse Index Theorems. 324. 346,

365, 398, 405 Ranch Comparison Theorems, 40(

406, 408. 572 singi~larity theorems, 468, 469,

472, 478 splitting theorems 49. 506, 550,

562 Timelike Index Comparison The-

orem. 406 Topolog~cal Sphere Theorem, 502

634 INDEX

Toponogov's Dlameter Theorem, 406

Toponogov Triangle Comparison Theorem, 567

Tkiangle Comparison Thecrems, 567, 570

Tidal acceleration, 31-32. 42 ~inboilndedness of, 32, 42-44 see also Jacobi fields

TIF's and TIP'S, 215-218 Time

function, 64, 70 generalized time function, 70 oriented. 5, 25. 50, 52 q11asl-time fi~nction, 490, 577 semi-time function, 68, 490

Tlrnelike convergence condition, 46. 112

see comment afier strong energy conditton

Timelike boundary point, 4 74 Caiichy complete, 21 1 chain, 356, 357 conjugate point, 328. 413 convergence condition. 46, 1 12 curve, 25 curve space jiWb,,)). 357 cut loc~~s , 298-305 diameter, 399, 401, 402 geodesic completeness. 209 geodesic incompleteness, 203 geodesic line, 273, 290 index comparison theorem, 400,

4 08 index form, 325. 328, 407-408 infinity (zi, z-), 178 Morse Index Theorem, 324, 346.

405 path space (C( , , ) ) , 146, 354-365 sectional curvature ( K ( E , p ) ) , 29-

32, 403 spares, 209 siibma~fold, 92, 164

two plane, 29. 403 vector field, 50

Topology Alexandroff, 7, 8, 59-60 Co on cilrves, 79, 81-63. 272,

273, 278, 288. 297 C' on Con( M), 239 C' on Eor(M), 239, 247 interval on Con(A4), 241 quotient on Con(M), 241

Toponogov Triangle Comparison The- orem, 567

Torston tensor ( T ( X , Y) or T , J k ) , 19 Totally

geodesic, 93, 94, 98. 165 viciotis. 68, 137. 165

?l-apped set, 287, 465, 468 s~irface, closed, 463

Triangle cornparison theorems, 567, 570

Two point homogeneo~is, 185-155

u Unit

observer biindle (T-I hf), 298-304 vector. 21

Upper slipport fiinction, 520. 530

Vactium space-times. 482 see also Ricri flat and empty

Variation, 329 admissible. 376, 388-391 canonical. 331, 388 first, 360, 450 plecewise srnooth. 329, 389-391 proper. 329, 389-391 second. 324 vector field. 323. 331, 389

Vector field along a cirrve. 17 Killing, 120. 121 nonsparehke, 24

INDEX

nilli, 22 , 24 spacelike. 24 timellke. 24 ~ m i t , 21

Volnme form, 67 Vorticity tensor (w). 428

quotient (IJ ). 431

Warped producr (M x f H ) . 95, 94- 133. 180 163. 250-263

fibers. 119 horizontal vectors, 119 leaves. 119 local warped products, 117 133 vert~cal vectors, 119

Tarping fl~nctlon, 112. 189 see also root warptng fiinct~on

Weak energy condition, 434 Wheeler universes, 399, 401 Wlder light cones. 64 !Vroiiskidn (IV'(A. B)), 385

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