An introduction to Lorentzian Geometry and its applicationsgigda.ugr.es/sanchezm/pdf/B_Libro Apuntes Lorentz.pdf · An introduction to Lorentzian Geometry and its applications Miguel

An introduction to Lorentzian Geometryand its applications

Miguel Angel Javaloyes VictoriaMiguel Sanchez Caja

XVI Escola de Geometria Diferencial,Sao Paulo, 12-16 julio 2010

Sao Carlos: Rima 2010, ISBN: 978-85-7656-180-4

Note for the reader

The aim of the present mini-course is to give an introduction on some basic el-ements about Lorentzian Geometry. The present notes, which complement theminicourse, have been prepared for the XVI Escola de Geometria Diferencial (SaoPaulo, 2010), even though many parts are based in several courses of differentlevels, explained by the second named author at University of Granada.

The purpose of the notes is not to provide a complete self-contained introduc-tion to the subject. Rather, we want to orientate the reader in the elementary toolsof Lorentzian Geometry. To this aim, we collect some of the basic results and tech-niques on the topic, study some of them in detail, and point out references where acomplete treatment is developed. We cite and discuss very recent results in somecases; the reason is twofold. On one hand, some classical concepts in LorentzianGeometry have been revisited recently, and they affect fundamental parts of thetheory. Therefore, even a novel researcher must acquire some knowledge on them.On the other hand, these questions may be an excellent stimulus for a deeper study.Nevertheless, our aim is not to make a full study of any recent topic of research.

The reader is expected to have some knowledge of Riemannian Geometry, say,as in an elementary course in the degree of Mathematics. In fact, a main aimof the course is to make clear which Riemannian elements can be extended tothe Lorentzian (or, in general, indefinite) case. Then, we study both, the specificLorentzian difficulties and new Lorentzian tools. In general, we do not developat the same time Riemannian and Lorentzian Geometry. The excellent book byO’Neill [43] is recommended for this aim. However, we summarize frequentlywhat can be obtained in the Riemannian case, so that the differences and difficultieswith the Lorentzian one become apparent.

The level and the style remain elementary. Most exercises are trivial assertionsabout the text immediate above. We refer to the cited O’Neill book [43], as well asBeem, Ehrlich and Easley [2] or Penrose and Rindler [44, 45], for deeper studieswhich include most of the contents of our mini-course. At the level of research,the reviews [11, 38] (or, in a more interdisciplinary way, [30, 56]) provide updatedprogress and references on many of the topics developed here. However, somerelatively simple open questions are stated in the mini-course: the purpose is toencourage young readers to think and research by themselves.

The mini-course also develops some simple interpretations and applications toRelativity. These interpretations are not only useful for their physical interest, butalso because they suggest new purely mathematical concepts. For detailed inter-pretations, the mathematical reader is encouraged to look at Sachs and Wu’s [52].With a language closer to physicists, Wald’s book [66] is strongly recommendedfor readers interested in physical applications.

These notes are divided into three chapters:

Chapter 1. Lorentzian vector spaces are studied. We collect elements whichare very well-known, even though they are very spread in the literature. Theseinclude the elementary properties of vector spaces with a Lorentzian (or, in general,indefinite) scalar product, Dajczer et al. criteria on bounds of symmetric bilinearforms, the classification of Lorentz transformations, the properties of the spinormap, or the canonical forms for self-adjoint endomorphisms with respect to an

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indefinite scalar product. The proofs are self-contained and, as an appendix, wecollect some interpretations for Special Relativity.

Chapter 2. Its aim is the study of semi-Riemannian manifolds (not necessarilydefinite positive) in general. We review briefly the typical elements of RiemannianGeometry (Levi-Civita connection, geodesics, curvature) with the following aim:to point out which properties remain essentially equal, must be modified or plainlydo not hold, in the indefinite case. Special care is taken on lightlike pregeodesics, asthey constitute a conformally invariant element, with no analog in the Riemanniancase. We finish with a tour on properties which may seem patological from theRiemannian viewpoint, but become natural in the indefinite one. Some examplesare constructed here with elementary tools. They may orientate the intuition of theindefinite case, and suggest nowadays open questions.

Chapter 3. It is devoted specifically to Lorentzian manifolds. In the first threesections, we develop basic elements that, essentially, come from a combination ofthe properties of Lorentzian vector spaces and Riemannian manifolds. In the fourthone we explain Causality theory. This is a powerful tool specific of the Lorentziancase, with no analog in the Riemannian one. Causality is a vast subject, and itsfoundations have been revised recently. We make just a brief summary whichmay serve either as a first contact for a future analysis, or as a reasonable guideto be used in another Lorentzian topics. At the last part, we introduce O’Neill’squasi-limits and prove some properties of the Lorentzian distance, in order to makereasonably self-contained the next results. In the fifth section, a classical theoremby Hawking on singularities is proven. This theorem is chosen because (apartfrom the fact that its well-known conclusion is very appealing) its similarities anddifferences with the Riemannian case become especially illustrative. In fact, aparallel Riemannian result, which can be proven by using Myers theorem, is alsodeveloped. We end the chapter with an appendix on interpretations in GeneralRelativity.

The authors would be very glad if these notes helped the reader to enjoy study-ing Lorentzian Geometry, a topic which is fascinating for us.

MS acknowledges warmly conversations with Alfonso Romero on the top-ics of the course, which they have taught along the last years. Both authors ac-knowledge the partial support by the Regional Junta de Andalucıa Grant P09-FQM-4496. MAJ is also partially supported by MICINN project MTM2009-10418and Fundacion Seneca project 04540/GERM/06 and MS by MEC-FEDER GrantMTM2007-60731. MAJ finished the writing of these notes during a research stayin the University of Sao Paulo supported by FAPESP grant 2010/00082-9.

Miguel Angel Javaloyes and Miguel Sanchez.

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Contents

Note for the reader 3

Chapter 1. Lorentzian vector spaces 71. Symmetric bilinear forms and scalar products 72. Lorentzian vector spaces 163. The Lorentz group 194. The spin covering of the restricted Lorentz group SO↑1(4) 295. Jordan canonical forms for self-adjoint endomorphisms 406. Appendix: Special Relativity 48

Chapter 2. General theory of semi-Riemannian manifolds 591. Similarities between Riemannian and semi-Riemannian geometries 592. Bounds for the sectional and Ricci curvatures 673. Conformal properties and lightlike pregeodesics 694. A tour on the differences with explicit counterexamples 73

Chapter 3. Lorentzian manifolds and spacetimes 811. Existence of Lorentzian metrics and time orientability 812. Local Lorentzian geometry 833. Variations of the length and focal points 864. Elements of the theory of Causality. 945. Hawking and Myers theorems 1136. Appendix: General Relativity 115

Bibliography 123

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CHAPTER 1

Lorentzian vector spaces

Throughout this chapter, V = V (R) will denote a real vector space of dimen-sion n unless something else is pointed out.

1. Symmetric bilinear forms and scalar products

The current section introduces elementary algebraic properties of symmetricbilinear forms (2-covariant symmetric tensors). As main references one can choose[2, Chapter 2] and [43, Section 2.3].

1.1. Symmetric bilinear forms and scalar products.

DEFINITION 1.1. Let b : V ×V → R be a symmetric bilinear form. We willsay that b is

(i) positive definite (resp. negative definite) if b(v,v) > 0 (resp. b(v,v) < 0) forevery v ∈V \0,

(ii) positive semidefinite (resp. negative semidefinite) if b(v,v)≥ 0 (resp. b(v,v)≤0) for every v ∈V \0,

(iii) indefinite if it is neither positive semidefinite nor negative semidefinite,(iv) nondegenerate if the condition b(v,w) = 0 for every w∈V implies that v = 0.

Otherwise, we will say that b (and possibly V ) is degenerate, and we will callthe subset N = v ∈V : b(v,w) = 0,∀w ∈V the radical of b.

Let qb : V →R be the quadratic form associated to b defined as qb(x) = b(x,x).It is easy to see that b can be reconstructed from qb using that

b(u,v) =12(qb(u+ v)−qb(u)−qb(v)).

DEFINITION 1.2. Let (V,b) be as above, we will say that v ∈V is(i) timelike if qb(v)< 0,

(ii) lightlike if qb(v) = 0 and v 6= 0,(iii) spacelike if qb(v)> 0,(iv) causal if v is timelike or lightlike.

We also define|v|=

√|qb(v)|.

NOTATION 1.3. There is no universally accepted criterion for the character ofthe zero vector. Sometimes it is considered as lightlike and other times as spacelike.We will use:

(i) null vector to refer to lightlike vectors together with the zero vector,(ii) non spacelike vectors to refer to causal vectors together with the zero vector,

(iii) and non causal vectors to refer to spacelike vectors together with the zerovector.

7

8 1. LORENTZIAN VECTOR SPACES

EXERCISE 1.4. Show that if b is indefinite, then there exists a basis made upof timelike vectors (resp. lightlike, spacelike).

Generalizing the Euclidean nomenclature, we give the following definitions.

DEFINITION 1.5. A scalar product g on V is a nondegenerate symmetric bi-linear form.

DEFINITION 1.6. If f is a (vector) endomorphism of (V,g), we will say thatf is an isometry (resp. self-adjoint) if g(u,v) = g( f (u), f (v)) (resp. g(u, f (v)) =g( f (u),v)) for every u,v ∈V .

DEFINITION 1.7. Let (V,g) be a vector space endowed with a scalar productg. If v,w∈V , we say that v is orthogonal to w, v⊥w, if g(v,w) = 0. Consequently,for A,B⊆V , we say that A is orthogonal to B, A⊥ B, if v⊥ w for every v ∈ A andevery w ∈ B. We will denote

A⊥ = w ∈V : g(v,w) = 0,∀v ∈ A.Moreover, a basis e1,e2, . . . ,en of V is said orthonormal if its elements are unit, thatis, |ei| =

√|g(ei,ei)| = 1, i = 1, . . . ,n and orthogonal to each other, g(ei,e j) = 0,

i, j = 1, . . . ,n.

NOTATION 1.8. We will assume that all the bases are ordered and the order isthat of appearance when written.

LEMMA 1.9. Given an orthonormal basis B of (V,g), the number ν of timelikevectors in the basis does not depend on the chosen basis, but only on (V,g).

PROOF. Assume that there exists another orthonormal basis B′ with, ν ′ < ν .Then the intersection of the subspace U ′ generated by the n−ν ′ spacelike vectorsof B′ and the subspace W generated by the ν timelike vectors of B would havedimension greater than 0, that is, there would exist a vector u ∈U ′∩W \0 suchthat b(u,u)> 0 and b(u,u)< 0.

NOTATION 1.10. The span generated by a subset of vectors v1, . . . ,vm willbe denoted as 〈v1, . . . ,vm〉R. The subscriptRwill indicate that the span is computedtaking into account real numbers as scalars (later, we will use C as subscript whenusing complex numbers). This will allow us to distinguish between the span andthe usual scalar product of Rn.

EXAMPLE 1.11. Consider R2 endowed with the scalar product

〈(x, t),(x′, t ′)〉1 = xx′− tt ′.

Observe, for example, the following relations of orthogonality (see also Fig-ure 1), which strongly contrast with the Euclidean ones:

(y,1)⊥±(1,y),〈(1,1)〉R ⊥ 〈(1,1)〉R.

DEFINITION 1.12. Let (V,g) be a vector space endowed with a scalar product.A vector subspace W <V is said nondegenerate if W ∩W⊥= 0 (or, equivalently,if the restriction gW = g|W×W is nondegenerate).

PROPOSITION 1.13. If W <V , then

1. SYMMETRIC BILINEAR FORMS AND SCALAR PRODUCTS 9

timelike

spacelike

lightlike

(1,y)

(y,1)

FIGURE 1. R2 endowed with the usual Lorentzian product. Twovector straight lines are orthogonal if and only if they are obtainedone from the other as a reflection along a diagonal line.

(i) dimW +dimW⊥ = dimV ,(ii) (W⊥)⊥ =W,

(iii) V =W +W⊥⇔W is nondegenerate (⇔W⊥ is nondegenerate).

PROOF. (i) Let e1, . . . ,eρ ,eρ+1, . . . ,en be a basis of V such that e1, . . . ,eρ isa basis of W . If v = ∑

ni=1 aiei, then

v ∈W⊥⇔ g(v,ei) = 0 ∀i ∈ 1, . . . ,ρ⇔n

∑j=1

gi ja j = 0 ∀i ∈ 1, . . . ,ρ,

where gi j = g(ei,e j). As g is nondegenerate, the subspace W⊥ is given by thesolutions of a linear system of ρ independent equations with n variables, andtherefore dimW⊥ = n−ρ .

(ii) The inclusion (W⊥)⊥ ⊃W is trivial. By (i), dim(W⊥)⊥ = dimW , and theequality holds.

(iii) It is straightforward from (i) (the sentence enclosed in brackets follows from(ii)).

As a consequence of the previous result, we will give a proof of the known theoremof Sylvester. Along the proof we will focus on the nondegenerate case so that theconstructive procedure of orthonormal bases (eventually extending orthonormalsubsets) will be apparent.

THEOREM 1.14. Let (V,g) be a vector space endowed with a scalar product.Then (V,g) admits an orthonormal basis.

PROOF. Let us prove it by induction. Assume first that dimV = 1. As g isnondegenerate, there exists v ∈ V such that g(v,v) 6= 0. Therefore u = 1

|v|v is anorthonormal basis. Assume now that n = dimV and that the result is true fork < n. Again as g is nondegenerate, there exists v ∈ V such that g(v,v) 6= 0, and


by part (i) and (iii) of Proposition 1.13, 〈v〉⊥R has dimension n− 1 and it is non-degenerate. Hence by induction we deduce the existence of an orthonormal basise1,e2, . . . ,en−1 of 〈v〉⊥R. Finally e1,e2, . . . ,en−1,

1|v|v is an orthonormal basis for

V .

COROLLARY 1.15. Every orthonormal basis of a nondegenerate subspace Uof V can be extended to an orthonormal basis of (V,g).

PROOF. As U⊥ is nondegenerate, we can complete the orthonormal basis ofU with an orthonormal basis of U⊥.

DEFINITION 1.16. The number of timelike vectors in an orthonormal basis,which depends only on (V,g) (see Lemma 1.9), is called the index of (V,g).

NOTATION 1.17. All the orthonormal bases e1,e2, . . . ,en are supposed to beordered, in such a way that if εi = g(ei,ei) for i = 1, . . . ,n, all of them have thesame value of (ε1, . . . ,εn), which is called the signature of g. We will assume thatthe signature is of the form (+, . . . ,+,−, . . . ,−).

DEFINITION 1.18. We will say that a scalar product g is(i) Euclidean if ν = 0,

(ii) Lorentzian if ν = 1 and n≥ 2.We will say that the scalar product is indefinite if it is as symmetric bilinear form.

EXAMPLE 1.19. In Rn we will define the usual scalar product of index ν ,〈·, ·〉ν , as

〈(a1, . . . ,an),(b1, . . . ,bn)〉ν =n−ν

∑i=1

aibi−n

∑i=n−ν+1

aibi.

THEOREM 1.20 (Sylvester’s law of inertia). If b is any symmetric bilinearform, then there exists a Sylverster basis, i. e., an orthogonal basis such that all itsvectors are unit or lightlike. The number of spacelike, timelike and lightlike vectorsin the basis does not depend on the chosen basis and they are called, respectively,coindex, index and nullity of b. We will denote the index by ν and the nullity by µ .

PROOF. Observe that from b, we can define a symmetric bilinear nondegen-erate form b on the quotient space V/N, as b([u], [v]) = b(u,v) ∀u,v ∈ V , where[u], [v] denote, resp., the classes of u and v in V/N. Now apply Theorem 1.14to b to obtain an orthonormal basis [e1], [e2], . . . , [ep] of V/N and choose a basisep+1,ep+2, . . . ,ep+µ of N. Then e1, . . . ,ep,ep+1, . . . ,ep+µ is the required basis ofV . The independence of the index, coindex and nullity of such a basis follows fromLemma 1.9 and the fact that the lightlike vectors in the basis always generate theradical N.

REMARKS 1.21. After Sylvester theorem we can conclude that(1) the subspace generated by the n− µ lightlike vectors of any Sylvester

basis coincides with the radical of b and, therefore, it does not dependon the chosen basis. However, the subspaces generated respectively bythe timelike and spacelike vectors in the Sylvester basis do depend on thebasis.

(2) If we consider a subspace W < V , the restriction bW := b|W×W of b is anew bilinear form on W which can be degenerate even when b is nonde-generate. It is easy to see that an orthogonal basis of W with all its vectors


unit or lightlike can be extended to a basis of V with the same propertiesif and only if the radical of bW is included in the radical of b.

(3) Let V and V ′ be two vector spaces and b and b′ symmetric bilinear formsin V and V ′ respectively. Then there will exist an isomorphism f : V →V ′

that preserves b and b′ (i.e. (b(x,y) = b′( f (x), f (y)) for every x,y ∈V ) ifand only if n = n′, µ = µ ′ and ν = ν ′.

NOTATION 1.22. If f : V →V ′ is a linear map and B,B′ are (ordered) bases ofV,V ′ resp., being B given by v1, . . . ,vn, then M( f ,B′← B) will denote the matrixwhose columns are made up orderly by the coordinates of f (v1), . . . , f (vn) in B′.Thus, given a vector v∈V , the product of the matrix M( f ,B′← B) and the column-vector of the coordinates of v in B is equal to the column-vector of the coordinatesof f (v) in B′. In particular, when f is the identity in V , that is IV , we obtain thetransformation matrix from B to B′. According to our notation, given a third basis,B′′, we have that: M(IV ,B′′← B′)M(IV ,B′← B) = M(IV ,B′′← B). In the case inthat V = V ′ and B = B′ we will write M( f ,B) rather than M( f ,B′ ← B). For abilinear form the matrix of the elements b(vi,v j) is denoted by MB(b).

REMARK 1.23. Using Sylvester theorem and basic computations of linear al-gebra we conclude the following:

(i) Let B and B′ be two bases, and P = M(IV ,B← B′), then:

MB′(b) = PtMB(b)P,

det(MB′(b)) = (detP)2 det(MB(b)).

In particular, the rank of the matrix MB(b) is independent of B and equal tothat of b. Furthermore, the symmetric bilinear form b is a scalar product ifand only if det(MB(b)) 6= 0 (for a basis B and, then, for everyone), and forevery scalar product g, we have that

(−1)ν det(MB(g))> 0.

(ii) If g is a scalar product and e1, . . . ,en−ν ,en−ν+1, . . . ,en any orthonormal basis,then

v =n

∑i=1

εig(ei,v)ei, εi = g(ei,ei)(=±1)

(the proof is trivial taking into account that v−∑ni=1 εig(ei,v)ei is orthogonal

to every vector in the basis and, therefore, it belongs to the radical of b).

1.2. Bounds of a symmetric bilinear form by a scalar product. Let g bea Euclidean scalar product. It is known that for every symmetric bilinear form b,there exists an orthonormal basis e1, . . . ,en of g such that b(ei,e j) = λiδi j. Thus,if λi ≥ 1 for every i, necessarily b(v,v) ≥ g(v,v) for every v ∈ V . The situationchanges dramatically if the scalar product is indefinite. We will need the followingsimple technique.

LEMMA 1.24. Let g be a scalar product admitting a lightlike vector v. Then,then there exist both sequences ukk and wkk converging to v such that uk istimelike, and vk is spacelike for every k ∈N.

PROOF. Let u be a timelike vector such that g(u,v) ≤ 0, and w a spacelikevector such that g(v,w)≥ 0 (obviously, such vectors can be obtained changing u,w


by −u,−w if necessary). The required sequences are

uk = v+1k

u, wk = v+1k

w.

The following result can be seen as a key point to understand the differencesof meaning of the bounds for different elements in Riemannian and Lorentziancurvature as the Ricci tensor or the sectional curvature, which will be studied inthe next chapter. Most of our proof follows the one in [2, Lemma 2.1].

THEOREM 1.25 (Dajczer-Nomizu [15], Kulkarni [33]). Let (V,g) be a vectorspace endowed with an indefinite scalar product, n = dimV , ν = the index of g,and b, a symmetric bilinear form. Then the following conditions are equivalent:

(i) b = c ·g for any c ∈R,(ii) qb = 0 on lightlike vectors of g,

(iii) ∃a > 0 : |qb/qg| ≤ a on timelike vectors of g,(iv) ∃a′ > 0 : |qb/qg| ≤ a′ on spacelike vectors of g distinct from zero,(v) ∃a ∈R : qb/qg ≤ a on non-null vectors of g,

(vi) ∃a′ ∈R : qb/qg ≥ a′ on non-null vectors of g.

PROOF. (i)⇒ (the others). It is obvious.(ii)⇒ (i). We distinguish four cases:Case I: n = 2, ν = 1.

Let B be an orthonormal basis e1,e2 for g, then v1 =1√2(e1+e2) and v2 =

1√2(e1−

e2) are lightlike vectors; let B be the basis v1,v2 and c = b(v1,v2). Then

MB(b) = c(

0 11 0

)= cMB(g),

which clearly implies that b = c ·g.Case II: n > 2, ν = 1.

Let B be an orthonormal basis e1, . . . ,en for g with g(en,en) = −1, and bi j =b(ei,e j). Applying Case I to each plane 〈ei,en〉R, i ∈ 1, . . . ,n− 1, we get thatbin = 0, bii =−bnn, ∀i ∈ 1, . . . ,n−1. Thus we have:

MB(b) =

c ?

. . . 0? c

0 −c

with c = −bnn. If we define the lightlike vector v(θ) = cos(θ)ei + sin(θ)e j + en,for i, j > 1, i 6= j, it follows that

0 = b(v(θ),v(θ)) = cos2(θ)bii + sin2(θ)b j j

+2sin(θ)cos(θ)bi j +bnn = sin(2θ)bi j.

Therefore bi j = 0, and hence

MB(b) =

c 0

. . . 00 c

0 −c

.


Case III: n≥ 2, ν = n−1.This case is solved as the previous one by considering the scalar product −g ratherthan g.

Case IV: n≥ 3, 1 < ν < n−1.Let B be an orthonormal basis e1, . . . ,en−ν ,en−ν+1, . . . ,en for g, and bi j = b(ei,e j).Reasoning as in the Case II and restricting ourselves to the Lorentzian subspace〈e1, . . . ,en−ν ,ek〉R for every k ∈ n−ν +1, . . . ,n, we get

MB(b) =

c In−ν 0

−c ?

0. . .

? −c

.

The proof is concluded reasoning as in the Case III by restricting ourselves to thesubspace 〈ek,en−ν+1, . . . ,en〉R for a value of k ∈ 1, . . . ,n−ν:

MB(b) =(

cIν 00 −c In−ν

).

For the remainder of the proof, we will need to use Lemma 1.24.(iii) ⇒ (ii). Let u be a lightlike vector and uk a sequence of timelike vectors

converging to u; from our hypothesis (iii) it follows that

0≤ |qb(uk)| ≤ a|qg(uk)|.Taking limits we obtain

0≤ |qb(u)| ≤ a|qg(u)|(= 0).

Thus qb(u) = 0 for every lightlike vector u.(iv)⇒ (ii). It is shown analogously to (iii)⇒ (ii), using the previous lemma.(v)⇒ (ii). Let u be a lightlike vector, uk and vk both sequences of timelike

and spacelike vectors, respectively, converging to u. By hypothesis,

qb(uk)≥ aqg(uk), qb(vk)≤ aqg(vk).

Taking limitsqb(u)≥ aqg(u) = 0, qb(u)≤ aqg(u) = 0.

Hence, qb(u) = 0, for every lightlike vector u.(vi)⇒ (ii). It is proven analogously to (v)⇒ (ii).

As an immediate consequence of this theorem we obtain the following charac-terization of the homothetic scalar products.

COROLLARY 1.26. Let g, g′ be two indefinite scalar products on V with thesame index ν 6= n/2. Then

∃c > 0 : g′ = cg⇔ g,g′ have the same lightlike vectors.

The condition ν 6= n/2 is imposed just to ensure that the constant c is positive.On the other hand, we will see in the following example that if we consider slightvariations of the inequalities in Theorem 1.25, the conclusions of this theorem donot hold.

EXAMPLE 1.27. Consider in R2 the natural basis e1,e2, the standard Lorentzproduct 〈 ·, · 〉1 and a symmetric bilinear form b such that b(e1,e2)= 0 and b(ei,ei)=λi for i = 1,2. Easily:


(1) b(v,v)≥ 〈v,v〉1 for all v⇔ λ1 ≥ 1, λ2 ≥−1(2) b(v,v)≥ 〈v,v〉1 for all v timelike⇔ λ2≥−1, λ1≥−λ2. Such inequalities

do not lie under Theorem 1.25.

1.3. Index raising and lowering.1.3.1. The dual space. Recall that given a vector space of finite dimension

V (R), its dual space is defined as the vector space V ∗(R) made up of the linearforms on V , that is, the linear maps ϕ : V → R. Necessarily, the dimensions ofV and V ∗ coincide; in fact, if we fix any (ordered) basis B given by the vectorsv1, . . . ,vn of V , the dual basis B∗ is defined by ϕ1, . . . ,ϕn, where ϕ i, i = 1, . . . ,n, isdetermined by ϕ i(v j) = δ i

j with j = 1, . . . ,n, where δ ij is the Kronecker delta. As a

consequence, we have

v =n

∑i=1

ϕi(v)vi, ∀v ∈V, ϕ =

n

∑j=1

ϕ(v j)ϕj, ∀ϕ ∈V ∗.

Let V , V be two vector spaces. It is well-known that if f : V → V is a linearmap, the transpose map of f , f t : V ∗ → V ∗, defined by f t(φ) = φ f , is linear.Given two bases B, B of V and V , resp., we have that M( f t ,B∗← B∗) = M( f , B←B)t . Moreover, there exists a canonical isomorphism between V and its bidualspace V ∗∗= (V ∗)∗ defined by Φ : V →V ∗∗, v 7→Φv, where Φv(ϕ) =ϕ(v),∀ϕ ∈V ∗.In any basis B and the corresponding bidual B∗∗ = (B∗)∗, we have that M(Φ,B∗∗←B) is the identity matrix In.

1.3.2. Flat and sharp isomorphisms.

DEFINITION 1.28. Let g be a scalar product of V , the isomorphism flat, [ :V −→V ∗, is defined as

[(v)≡ v[ : V −→R

v[(w) = g(v,w), ∀v,w ∈V.

The inverse isomorphism of [ is the so-called sharp isomorphism, ] : V ∗ −→V ,

](ϕ)≡ ϕ],

characterized by

g(ϕ],w) = ϕ(w), ∀w ∈V, ∀ϕ ∈V ∗.

It is not difficult to prove that, in fact, both maps are vector isomorphisms1.The flat and sharp isomorphisms are written in coordinates as follows. Let B be abasis v1, . . . ,vn of V , and B∗ its corresponding dual basis ϕ1, . . . ,ϕn. Denote

gi j = g(vi,v j), gi j = (MB(g)−1)i j.

If v = ∑ni=1 aivi, then

v[ =n

∑j=1

a jϕj, where a j =

n

∑i=1

gi jai;

this way of looking at the flat isomorphism is called “index lowering”. Moreover,we call “index raising” to the analogous process for the sharp isomorphism: if

1The flat map makes sense as a linear map for any bilinear form, even if it is degenerate, but itis not one-to-one in general, because the kernel of [ coincide with the radical of b.


ϕ = ∑nj=1 b jϕ

j, then

ϕ] =

n

∑i=1

b jv j, where b j =n

∑i=1

gi jbi.

REMARK 1.29. Let e1, . . . ,en−ν ,en−ν+1, . . . ,en be an orthonormal basis,

gi j = δi jεi, εi =

1 if i≤ n−ν ,−1 if i > n−ν +1.

Then ai = εiai. So, if g is Euclidean, ai = ai, and if g is Lorentzian, ai = ai, fori < n−1 and an =−an.

1.3.3. Extension to other types of tensors. The maps [ and ] allow us to estab-lish isomorphisms between the space of tensors (r,s) (r contravariant, s covariant),and (r′,s′) on V , whenever that r+ s = r′+ s′. For example, consider a 2-covarianttensor

T : V ×V −→R.

We can construct the 2-contravariant tensor

T : V ∗×V ∗ −→R

as T (v[,w[) = T (v,w), ∀v,w ∈V . In coordinates we say that T is obtained raisingthe indices of T . Explicitly, if

T =n

∑i, j=1

ti jϕi⊗ϕ

j, T =n

∑k,l=1

tklvk⊗ vl,

then

tkl =n

∑i, j=1

ti jgikg jl.

EXERCISE 1.30. Obtain analogous expressions for the tensors (1,1) obtainedby raising only the first index and only the second index of T . When do thesetensors coincide?

1.3.4. Metric contraction. Recall that, given a tensor (r,s) with r,s ≥ 1, wecan define a new tensor by choosing a covariant index, a contravariant index andcontracting in them. Thus, for example, if T is a tensor (1,1), its contraction (in theonly possible indices) is just ∑

ni=1 t i

i = ∑ni=1 T (ϕ i,vi), where v1, . . . ,vn is any basis

of V and ϕ1, . . . ,ϕn, its dual basis.As in the Euclidean case, any scalar product allows one to do the metric con-

traction of any pair of covariant (or contravariant) indices of a tensor. So, forexample, if T is a 2-covariant tensor, its metric contraction is obtained raising oneof the two indices and contracting. In coordinates:

n

∑i, j=1

gi jT (vi,v j).

1.3.5. (1,1)-Tensors and endomorphisms. Recall that there exists a naturalisomorphism (independent of g) between the space of the (1,1) tensors and theone of endomorphisms on V . To be precise, we can assign to each endomorphismf the tensor Tf defined as Tf (φ ,v) = φ( f (v)),∀v ∈V,φ ∈V ∗.

EXERCISE 1.31. Show that for any basis B, the matrices of coordinates asso-ciated to f and Tf in B coincide.


Fixed the scalar product g, we can also assign an endomorphism fT to each2-covariant tensor T : it is enough to raise the first index, and to consider the endo-morphism associated to the corresponding (1,1)-tensor . Explicitly, fT is charac-terized by the relation

g(u, fT (v)) = T (u,v),∀u,v ∈V.

EXERCISE 1.32. Show that the metric contraction of T coincides with the traceof fT .

It is interesting to remark that, in the particular case in that T is a 2 covariantsymmetric tensor, the endomorphism fT is self-adjoint for g.

EXERCISE 1.33. Let V be a vector space, g a scalar product on V , b a sym-metric bilinear form and fb the associated self-adjoint endomorphism.(a) Show that, for any basis B:

M( fb,B) = MB(g)−1MB(b).

(b) Show that the following conditions are equivalent:(i) The endomorphism fb is diagonalizable.

(ii) There exists an orthonormal basis B of g such that MB(b) is diagonal.(iii) The matrix MB(g)−1MB(b) is diagonalizable (by a similarity transforma-

tion).(c) Justify that if the scalar product g is Euclidean, then every self-adjoint endo-

morphism is diagonalizable, but it is not true if g is an indefinite scalar product.

2. Lorentzian vector spaces

2.1. Timelike cones. In this subsection, we will denote by (V,g) a vectorspace endowed with a Lorentzian scalar product, that is, a Lorentzian vector spaceof dimension n≥ 2.

2.1.1. Existence.

PROPOSITION 1.34. The subset of the timelike vectors (resp., causal; lightlikeif n > 2) has two connected parts.

Each one of these parts will be called timelike cone, (resp. causal cone; light-like cone).

PROOF. Let e1, . . . ,en be an orthonormal basis of V , and v ∈ V such that v =

∑ni=1 aiei. Obviously,

v is lightlike ⇔|an|=

√(a1)2 + . . .+(an−1)2

an 6= 0

v is timelike ⇔ |an|>√

(a1)2 + . . .+(an−1)2,

v is causal ⇔|an| ≥

√(a1)2 + . . .+(an−1)2

an 6= 0Therefore in each case there exist two connected parts, the corresponding to an < 0and the corresponding to an > 0.

DEFINITION 1.35. A time orientation of a Lorentzian vector space is a choiceof one of the two timelike cones (or, equivalently, of one of the causal or lightlikecones). The chosen cone will be called future, and the other one, past.

2. LORENTZIAN VECTOR SPACES 17

From now on, the vectors in the future (resp. past) cone will be called future-directed or future-pointing (resp. past-directed or past-pointing).

PROPOSITION 1.36. Two timelike vectors v and w lie in the same timelike coneif and only if g(v,w)< 0.

PROOF. We can assume without loss of generality that |v| = 1, and v can becompleted to an orthonormal basis e1,e2, . . . ,en−1,v. Observing that

w = g(e1,w)e1 +g(e2,w)e2 + . . .+g(en−1,w)en−1−g(v,w)v,

it is clear that v and w are in the same cone if and only if −g(v,w) > 0 (see proofof Proposition 1.34).

PROPOSITION 1.37. If v,w are timelike vectors in the same cone, then so isav+ bw for any a,b > 0. In particular, each timelike cone is convex (the segmentthat joins two given points in the timelike cone is also contained in the cone).

PROOF. Proposition 1.36 implies that g(v,w)< 0, and then

g(v,av+bw) = ag(v,v)+bg(v,w)< 0,

g(av+bw,av+bw) = a2g(v,v)+b2g(w,w)+2abg(v,w)< 0.

From the last inequality, we deduce that av+bw is timelike, and from the first one,it follows that av+bw belongs to the same cone as v.

2.2. Reverse inequalities.

THEOREM 1.38 (Reverse Cauchy-Schwarz inequality). If v,w∈V are timelikevectors, then

(i) |g(v,w)| ≥ |v||w|, moreover the equality holds if and only if v,w are colinear.(ii) If v and w lie in the same cone, then there exists a unique ϕ ≥ 0, called the

hyperbolic angle between v and w such that

g(v,w) =−|v||w|cosh(ϕ).

PROOF. (i) Let a be a real number and w a vector such that w = av+w withw⊥ v (see part (iii) of Proposition 1.13). Then

g(w,w) = a2g(v,v)+g(w,w),

and hence a2g(v,v) = g(w,w)−g(w,w). Using last equation and g(w,w)≥ 0,we deduce that

g(v,w)2 = a2g(v,v)2 = g(v,v)(g(w,w)−g(w,w))≥ g(v,v)g(w,w) = |v|2|w|2.Moreover, equality holds if and only if g(w,w) = 0, that is, if and only if v,ware colinear.

(ii) If v,w lie in the same cone, then

−g(v,w)|v||w|

≥ 1.

Therefore there exists a unique ϕ ≥ 0 such that

cosh(ϕ) =−g(v,w)|v||w|

.


THEOREM 1.39 (Reverse triangular inequality). If v,w∈V are timelike vectorsin the same cone, then

|v|+ |w| ≤ |v+w|,and the equality holds if and only if v,w are colinear.

PROOF. As v,w lie in the same cone, v+w is timelike and g(v,w) < 0 (seePropositions 1.36 and 1.37). Therefore

|v+w|2 =−g(v+w,v+w)

= |v|2 + |w|2 +2|g(v,w)| ≥ |v|2 + |w|2 +2|v||w|= (|v|+ |w|)2.

Moreover, equality holds if and only if |g(v,w)|= |v||w|, that is, by Theorem 1.38,if and only if v,w are colinear.

2.3. Lightlike and causal cones. In the following, we will consider someanalogous properties of lightlike and causal cones.

PROPOSITION 1.40. If u,v ∈V are lightlike vectors, then

u,v is linearly dependent⇔ g(u,v) = 0.

PROOF. (⇒) Obvious.(⇐) Without loss of generality and up to an irrelevant non-zero constant of pro-porcionality for u and another for v, we can assume that the coordinates of u and vin some orthonormal basis e1, . . . ,en are

u =n−1

∑i=1

aiei + en, v =n−1

∑i=1

biei + en.

Therefore, as g(u,v) = g(u,u) = g(v,v) = 0,

1 = a1b1 + . . .+an−1bn−1 = (a1)2 + . . .+(an−1)2 = (b1)2 + . . .+(bn−1)2.

The Cauchy-Schwarz inequality implies then that

a1e1 + . . .+an−1en−1 and b1e1 + . . .+bn−1en−1

are colinear vectors, and that the constant of proporcionality is positive. Takinginto account that |a1e1 + . . .+ an−1en−1| = |b1e1 + . . .+ bn−1en−1|, it follows theequality u = v.

EXERCISE 1.41. Show the following:(1) If u,v ∈V are two independent causal vectors, then

u,v are in the same causal cone ⇔ g(u,v)< 0.

(2) The causal cones are convex.

2.4. Subspaces.

DEFINITION 1.42. Let (V,g) be a Lorentzian vector space. We will say that asubspace of V , W <V is

(i) spacelike, if g|W is Euclidean,(ii) timelike, if g|W is nondegenerate with index 1 (that is, Lorentzian whenever

dimW ≥ 2),(iii) lightlike, if g|W is degenerate, (W ∩W⊥ 6= 0).

3. THE LORENTZ GROUP 19

PROPOSITION 1.43. A subspace W <V is timelike if and only if W⊥ is space-like.

PROOF. Assuming that g|W is nondegenerate, we deduce that g|W⊥ is non-degenerate and vice versa (see Proposition 1.13), that is W ∩W⊥ = 0. Thus,V =W ⊕W⊥ and therefore

index(g) = index(g|W )+ index(g|W⊥).Hence, for the implication to the right, index(g|W⊥) = 0, and for the converse,index(g|W ) = 1.

PROPOSITION 1.44. If W <V , with dim(W )≥ 2, the following conditions areequivalent:

(i) W is timelike,(ii) W contains two linearly independent lightlike vectors,

(iii) W contains one timelike vector.

PROOF. (i)⇒ (ii). As W is timelike, given an orthonormal basis e1,e2, . . . ,ekof W , e1 is spacelike and ek timelike. Then e1 + ek and e1− ek are two linearlyindependent lightlike vectors.

(ii)⇒ (iii). Let v,w be two linearly independent lightlike vectors of W , theneither v+w or v−w is timelike. This follows from

g(v±w,v±w) = g(v,v)±2g(v,w)+g(w,w) =±2g(v,w).

Observe that g(v,w) is not zero by Proposition 1.40.(iii)⇒ (i). Let u be a timelike vector of W . Assume by contradiction that g|W

is degenerate. Then there exists a vector z 6= 0 in the radical of g|W (in particular, zis lightlike). As u,z are linearly independent, we know by Exercise 1.41 that

if u,z are in the same causal cone ⇒ g(u,z)< 0if u,z are in different causal cones ⇒ g(u,z)> 0.

Both cases contradict the fact that z lies in the radical of W . Therefore, g|W mustbe nondegenerate.

EXERCISE 1.45. If W <V , the following conditions are equivalent:(i) W is lightlike.

(ii) W contains a lightlike vector, but not a timelike one.(iii) The intersection of W with the subset of null vectors (lightlike or zero) con-

stitutes a vector subspace of dimension 1.

3. The Lorentz group

3.1. Generalities.

DEFINITION 1.46. The Lorentzian vector space Ln = (Rn,〈·, ·〉), with n ≥ 2,will be called Lorentz-Minkowski spacetime, where 〈·, ·〉 will denote from now onthe scalar product 〈·, ·〉1 of Example 1.19.

Let B0 be the canonical basis of Rn and Mn(R) the set of n× n real matrices.Denote

η = MB0(〈·, ·〉) =(

In−1 00 −1

)


DEFINITION 1.47. We define the Lorentz transformation group as

Iso(Ln) = f : Ln −→ Ln | f is a vector isometry,and the Lorentz group as

O1(n) = A ∈Mn(R)/AtηA = η.

If we consider Ln as a Lorentzian affine space (defined analogously to an affineEuclidean space, but endowed with a Lorentzian scalar product), the group ofaffinities that preserve the Lorentzian scalar product is called Poincare transfor-mation group, and the corresponding matricial group, Poincare group.

Fixed any orthonormal basis B of Ln, we can construct the group isomorphismIso(Ln)−→O1(n) that maps every Lorentz transformation f to the matrix M( f ,B).In particular, when fixed the usual basis B0, we have

Φ : Iso(Ln)−→ O1(n)

Φ( f ) = A f = M( f ,B0).

Denote fA = Φ−1(A). Everything is analogous to the Euclidean case betweenIso(Rn) and O(n).

REMARK 1.48. As AtηA = η , it follows that

(detA)2 = 1⇒ det fA =±1.

On the other hand, the automorphisms of a vector space with positive determinantare in correspondence with those that preserve the orientation of the basis; thus, iff ∈AutRV , the following conditions are equivalent:(1) det f > 0,(2) there exists an ordered basis B such that f∗(B) has the same orientation as B,

(i. e. det(M(IV ,B← f∗(B))> 0), and(3) for every ordered basis B, the orientation of f∗(B) if the same as the one of B.

DEFINITION 1.49. We say that a Lorentz transformation, f , is proper if det f (=detA f ) = 1, and improper otherwise.

The subgroup of proper Lorentz transformations will be denoted by Iso+(Ln)(resp. the subset Iso−(Ln)), and the isomorphic subgroup Φ(Iso+(Ln)) by O+

1 (n)(analogously for the subset of improper ones Iso−(Ln),O−1 (n)).

From the usual basis e1, . . . ,en of Ln, we can fix the standard time orientation:Future causal cone C↑ : the one that contains en,Past causal cone C↓ : the one that contains − en.

REMARK 1.50. Let f be a Lorentz transformation,

A f =

a11 · · · a1n...

. . ....

an1 · · · ann

.

Observe that

−1 = 〈 f (en), f (en)〉= a21n +a2

2n + . . .+a2(n−1)n−a2

nn,

and then |ann| ≥ 1. As 〈en, f (en)〉 = −ann and the causal cones are connected, weconclude

ann ≥ 1⇔ f (en) ∈C↑⇔ f (C↑) =C↑⇔ f (C↓) =C↓,


andann ≤−1⇔ f (en) ∈C↓⇔ f (C↑) =C↓⇔ f (C↓) =C↑.

DEFINITION 1.51. Let f be a Lorentz transformation, we will say that f isorthocronous if f (C↑) = C↑. We will denote by Iso↑(Ln) the subgroup of ortho-cronous transformations and O↑1(n) = Φ(Iso↑(n)). Analogously, we will denote thesubset of nonorthocronous transformations as Iso↓(Ln), and O↓1(n) = Φ(Iso↓(Ln)).

NOTATION 1.52. We will combine the notation in an obvious way for the or-thocronous proper transformations (O+↓

1 (n), etc.). Nevertheless, we will use thespecial notation SO↑1(n) for the restricted Lorentz group, that is, the subgroup ofproper orthocronus transformations. On the other hand, we will write A rather thanA f or fA whenever there is no possibility of confusion.

PROPOSITION 1.53. If f ∈ Iso(Ln), then the following conditions are equiva-lent:

(i) f ∈ Iso↑(Ln),(ii) there exists a causal vector v ∈ Ln such that 〈v, f (v)〉< 0,

(iii) for every timelike vector v ∈ Ln, it holds 〈v, f (v)〉< 0,(iv) in every orthonormal basis, the element (n,n) of the matrix of f is greater

than zero (and, actually, greater than or equal to 1).

PROOF. It is a consequence of Exercise 1.41 and Remark 1.50.

EXAMPLE 1.54. In−2 0

01 00 1

∈ SO↑1(n)

In−2 0

01 00 −1

∈ O−↓1 (n)

In−2 0

0−1 0

0 −1

∈ O+↓1 (n)

In−2 0

0−1 0

0 1

∈ O−↑1 (n)

After these four examples, it is clear that O1(n) has at least four connectedparts. Actually there are only four (see Exercise 1.70 below).

3.2. The Lorentz group in dimension 2. The group O1(2) can be studiedin the same way as the isometry group of the Euclidean space. Namely, computingdirectly the 2×2 matrices which satisfy

At(

1 00 −1

)A =

(1 00 −1

).

However, to appreciate better the peculiarities of the Lorentzian case, we will usea basis of lightlike vectors.

Let B0 be the canonical basis e1,e2 of R2 and consider the basis B of lightlikevectors u,v, where u = (1/

√2)(e1 + e2) and v = (1/

√2)(−e1 + e2). Clearly they

satisfy〈u,v〉=−1, 〈u,u〉= 〈v,v〉= 0,

that is,

MB(〈·, ·〉) =(

0 −1−1 0

).


In particular, u and v lie in the same causal cone. Denote

P = M(IR2 ,B0← B) =1√2

(1 −11 1

),

P−1 = M(IR2 ,B← B0) =1√2

(1 1−1 1

).

To determine explicitly O1(2), we will obtain the matrix of every f ∈ Iso(L2)first in the basis B and then in the basis B0. Recall that with our notation M( f ,B0)=P ·M( f ,B) ·P−1 (see Notation 1.22).

If f (u) = au+bv and f (v) = au+ bv, then

0 = 〈u,u〉= 〈 f (u), f (u)〉=−2ab,

0 = 〈v,v〉= 〈 f (v), f (v)〉=−2ab.

Thus, there are two possibilities:Case I: f (u) = λu (λ 6= 0). Taking into account that f ∈ Iso(L2) and, in

particular,

−1 = 〈u,v〉= 〈 f (u), f (v)〉=−λ b,

we conclude that b = 1λ

and necessarily a = 0. Therefore, f (v) = 1λ

v, andthen

M( f ,B) =(

λ 00 1/λ

).

Hence it follows that f ∈ Iso+(L2), (det f = 1), and as f (u) = λu, f ∈Iso↑(L2) if and only if λ > 0.

Case II: f (u) = λv. As in the previous case, we deduce that f (v) = 1λ

u,and then

M( f ,B) =(

0 1/λ

λ 0

).

Thus f ∈ Iso−(L2), and as v lies in the same causal cone as u, f ∈Iso↑(L2) if and only if λ > 0. Anyway, the matrix is diagonalizable andit admits an orthonormal basis of eigenvectors.

Therefore, it is immediate that, if f ∈ Aut(L2), then

(1) A f ∈ O+↑1 (2)⇔M( f ,B) =

(λ 00 1/λ

)for some λ > 0⇔

⇔M( f ,B0) = PM( f ,B)P−1 =12

(λ +1/λ λ −1/λ

λ −1/λ λ +1/λ

)for λ > 0⇔

⇔M( f ,B0) =

(coshθ sinhθ

sinhθ coshθ

),

where θ = lnλ ∈ R. We can proceed analogously with the other three connectedparts of O1(2) (or composing with the matrices in Example 1.54); as a matter offact, we get

Isometries with determinant equal to 1: all of them admit a basis of lightlikeeigenvectors and

SO↑1(2) =(

coshθ sinh θ

sinh θ coshθ

): θ ∈R

; O+↓

1 (2) =−A : A ∈ SO↑1(2)

.


Isometries with determinant equal to −1: all of them admit an orthonormalbasis of eigenvectors and

O−↓1 (2) =(

coshθ sinh θ

−sinh θ −coshθ

): θ ∈R

; O−↑1 (2) =

−A : A ∈ O−↑1 (2)

.

Observe that, unlike the Euclidean case, all these matrices are diagonalizable. Theorthocronous proper Lorentz transformations in dimension 2 are usually calledpure or, in Relativity jargon, boosts. As we will see below, any orthogonal Lorentztransformation in dimension 4 can be essentially described by the compositionof a pure Lorentz transformation and a Euclidean bidimensional isometry. As aconsequence, the boosts turn out to be paradigmatic for the interpretations of therelativistic effects.

REMARK 1.55. (Interpretation of θ ) Let v,w be two timelike vectors in thesame cone. In the Lorentzian case we have already defined their hyperbolic angleϕ ≥ 0 from the reverse Cauchy-Schwarz inequality. The situation is completelyanalogous to that of the Euclidean case, where the angle ϕ ∈ [0,π] is defined forevery pair of non null vectors.

So in the Lorentzian case as in the Euclidean one, if the vector space is oriented(and it is bidimensional), we can give a sign to the angle ϕ . In fact, ϕ is considerednegative if and only if the pair (v,w) forms a negatively oriented ordered basis. Weobtain in this way an oriented hyperbolic angle ϕ(v,w) ∈R (in the Euclidean casewe have ϕ(v,w) ∈ (−π,π], even if some authors consider obvious modifications insuch a way that ϕ(v,w) ∈ [0,2π)). In the following exercise, two characterizationsof ϕ(v,w), which are analogous to the Euclidean ones, are proposed.

EXERCISE 1.56. Let (V,g) be an oriented bidimensional Lorentzian vectorspace. Show:(A) There exists a unique 2-covariant antisymmetric tensor det satisfying: det(u,u′)is the determinant of the matrix whose columns are, orderly, the coordinates of uand u′ in any positively oriented orthonormal basis (for every u,u′ ∈V ).(B) Let v,w be two unit timelike vectors in the same cone. Then

(B1) sinh(ϕ(v,w)) = det(v,w).(B2) The unique positively oriented orthonormal bases Bv, given by v,v2, and

Bw, given by w,w2, obtained respectively from v and w satisfy: ϕ(v,w) coincideswith the unique value of θ determined by the matrix M(IV ,Bv← Bw) (according tothe explicit form of SO↑1(2) exhibited above).(C) Let v,w be two timelike vectors in the same cone. Then ϕ(v,w) is the uniquereal number that satisfies

det(v,w) = |v||w| sinh(ϕ(v,w)), −g(v,w) = |v||w| cosh(ϕ(v,w)).

3.3. Some properties of the Lorentz group in greater dimensions. The firstproperty has simple analogies and differences with the Euclidean case.

PROPOSITION 1.57. If A ∈ O1(n), then:(i) non-lightlike eigenvectors of A, if any, have +1 or −1 as eigenvalues,

(ii) the product of the eigenvalues of two linearly independent causal eigenvectorsis 1,

(iii) if U is an eigenspace of A with eigenvalue ε = ±1 (in particular, if theeigenspace U contains a non-null vector), then any other eigenspace is or-thogonal to U,


(iv) if U is an A-invariant subspace, then U⊥ is also A-invariant.

PROOF. (i) Let v be a non-lightlike eigenvector of A, Av = av. Then

〈v,v〉= 〈Av,Av〉= a2〈v,v〉 ⇒ a =±1.

(ii) Let v,w be two linearly independent lightlike eigenvectors, Av= av, Aw= bw.By Proposition 1.40 and Exercise 1.41,

0 6= 〈v,w〉= 〈Av,Aw〉= ab〈v,w〉 ⇒ ab = 1.

(iii) Let the eigenvector z ∈U be with eigenvalue ε (in particular, this happens ifz is non-lightlike by item (i)). Let w be an eigenvector of an eigenvalue λ

distinct from ε . For every u ∈U ,

〈u,w〉= 〈Au,Aw〉= λε〈u,w〉.

Thus, either 〈u,w〉= 0, which concludes the proof, or λε = 1⇒ λ = ε , whichcontradicts our hypothesis on w.

(iv) As A is an isometry, A(U) =U . Moreover, A−1(U) =U . Consider w ∈U⊥,then

〈Aw,u〉= 〈w,A−1u〉= 0, ∀u ∈U,

and therefore, Aw ∈U⊥, which concludes.

The following result summarizes the possibilities when A admits a non-spacelikeinvariant plane.

PROPOSITION 1.58. Let π ⊂ Ln be an A-invariant plane that is not spacelike.Then,

(i) If π is timelike, exactly one of the following three possibilities occur:(a) A|π is plus or minus the identity,(b) A|π admits two independent lightlike eigenvectors, with eigenvalues

λ 6=±1 and 1/λ ,(c) A|π admits two independent non-lightlike eigenvectors, one of eigen-

value +1 and the other one −1.(ii) If π is lightlike, then A|π admits a lightlike eigenvector v, Av = λv. If

λ 6=±1, then it also admits a spacelike eigenvector.At any case, A admits a causal eigenvector contained in π and the two roots of thecharacteristic polynomial of A|π are real.

PROOF. The case when π is timelike follows from Subsection 3.2. In fact, thecases (a), (b) can occur if and only if A|π preserves the orientation.

For the case when π is lightlike, notice that, by Exercise 1.45, the radical of〈·, ·〉π has dimension 1 and the remainder of the vectors are spacelike. Thus, ifv ∈ π is lightlike, necessarily it belongs to the radical and A(v) = λv. Finally, ifz ∈ π does not belong to the radical, necessarily z is spacelike and Az = ±z+ bvfor some b ∈ R. So, if λ 6= ±1 then the characteristic polynomial of A|π has twodistinct real eigenvalues, and A|π is diagonalizable.

LEMMA 1.59. If the characteristic polynomial of A ∈O1(n) admits a complexnon-real root a ∈C\R, then A admits an invariant plane π which is spacelike andsatisfies that the roots of A|π are both, a and its conjugate a.


PROOF. Considering A as a complex matrix, let z ∈ Cn \ 0 be an eigenvec-tor with Az = az. Clearly, Az = az, thus, z, z is C-linearly independent and itgenerates a complex plane 〈z, z〉C. Taking real and imaginary parts x = (z+ z)/2,y = (z− z)/(2i) we have that x,y ∈ Rn ⊂ Cn are independent when regarded ascomplex (and thus also as real) n−uples, indeed, 〈z, z〉C = 〈x,y〉C. Moreover, thereal plane π = 〈x,y〉R is also A-invariant (use that A is a real matrix) and, then, theroots of the characteristic polynomial of A|π are a, a. From Proposition 1.58, sucha plane π must be spacelike.

LEMMA 1.60. If A ∈ O1(n), with n≥ 2, then A admits a causal eigenvector.

PROOF. It will be proven by induction on n. For n = 2, the result follows fromProposition 1.58. So, assume by hypothesis that the result is true for the naturalnumbers less than n, and let us prove it for n. Let a be a root of the characteristicpolynomial of A.

If a ∈R, then it admits an eigenvector v. If v is causal we are done, otherwise,v is spacelike and the subspace v⊥ is both, Lorentzian and A-invariant. So, theinduction hypothesis can be applied to A|v⊥ .

If a ∈C\R, then consider the invariant spacelike plane π provided by Lemma1.59 and apply the induction hypothesis to the restriction of A to2

π⊥.

LEMMA 1.61. Assume that A admits a lightlike eigenvector v with eigenvalueλ 6= ±1. Then A admits a second lightlike eigenvector u of eigenvalue 1/λ and,thus, leaves invariant a timelike plane Π, such that v ∈Π.

PROOF. This will be proven by induction on n. For n= 2 (and, as a limit trivialcase, for n = 1) the result holds, recall Proposition 1.58. So, assume that it is truefor all the natural numbers less than n, and let us prove it for n. Let λ1, . . . ,λn ∈ Cbe the roots of the characteristic polynomial of A, and choose λ1 = λ .

If all the roots are real then, as detA f =±1, we have that λ1 · · ·λn =±1. Thus,there exists an eigenvalue λi 6= ±1,λi 6= λ1, and the result follows directly fromProposition 1.57 and Proposition 1.44.

If an eigenvalue is complex, take the invariant spacelike plane π provided byLemma 1.59. The induction hypothesis can be applied on A|π⊥ as v ∈ π⊥. Thisis because Ln = π⊕π⊥ and then v = v1 + v2 with v1 ∈ π and v2 ∈ π⊥. MoreoverA(v1)+A(v2) = λv1 +λv2. As A(v1) ∈ π and A(v2) ∈ π⊥, it follows that A(v1) =λv1. But then v1 = 0, because A|π does not have real eigenvalues.

THEOREM 1.62. If A∈O1(n) and fA ∈ Iso(Ln) is its associated Lorentz trans-formation, then one of the following three mutually exclusive cases holds:

(i) A admits a timelike eigenvector. In such a case, there exists an orthonormalbasis B such that M( fA,B) is of the form(

Rn−1 00 ±1

),

where Rn−1 ∈ O(n− 1). If A ∈ O+↑1 (n), we say that fA is a pure spacelike

rotation (in the hyperplane which is orthogonal to the eigenvector).

2As the dimension of π⊥ is n− 2, before applying the induction hypothesis the result must bechecked for the two lower natural numbers where it applies. We can get this either admitting as alimit case that the result is true for n = 1, or checking that for n = 3 previous arguments yield theresult with no problem. Analogous considerations also apply for the proof of Lemma 1.61.


(ii) A admits a lightlike eigenvector with eigenvalue λ 6= ±1. Then there existsan orthonormal basis B such that M( fA,B) is of the form(

Rn−2 00 R

),

where Rn−2 ∈ O(n− 2) and R ∈ O1(2). In this case fA is a bidimensionalLorentz transformation in a timelike plane π composed with a Euclideanisometry in its orthogonal π⊥.

(iii) A admits a unique independent lightlike eigenvector of eigenvalue +1 or −1.

PROOF. Reasoning by induction, for n = 2 the conclusion follows from thestudy of O1(2) done in Subsection 3.2. Actually, in dimension 2, case (iii) cannotoccur.

Assume that the theorem is true for n−1 and prove it for n. By Lemma 1.60,there exists a causal eigenvector v, Av = λv. Then,1) If v is timelike, we obtain (i) (see also part (iv) of Proposition 1.57).2) If v is lightlike, it can happen that

a) λ 6∈ ±1, in this case we obtain (ii) by Lemma 1.61.b) λ ∈ ±1, and assume that (iii) does not hold. Let us see that the case (i)

occurs: if w is another lightlike eigenvector linearly independent of v, by (ii)of Proposition 1.57, the eigenvalues of w and v are equal, so either u+w oru−w is a timelike eigenvector.

Finally, let us see that the cases are exclusive. Clearly, the possibilities (ii) and(iii) are exclusive. If we assume that (i) occurs, let v be a timelike eigenvectorAv = av, a = ±1, and let us show that the cases (ii) and (iii) cannot happen. If wis a lightlike eigenvector Aw = bw, necessarily a = b = ±1 (from the part (ii) ofProposition 1.57), which excludes the possibility (ii). Moreover the timelike plane〈v,w〉R is made up of eigenvectors with eigenvalue a, excluding (iii).

The exercise below shows explicitly that possibility (iii) can happen. However,in dimension 4 it also holds that every Lorentz transformation can be decomposedas the composition of an isometry for the timelike plane π1 (which leaves invari-ant π⊥1 ) and an isometry in a spacelike plane π2 (which leaves invariant π⊥2 ) notnecessarily orthogonal to π1. In Relativity, they usually refer to this result say-ing that the (orthocronous, proper) transformations are compositions of boosts androtations (see Theorem 1.92).

EXERCISE 1.63. Construct all the isometries of L3 which admit a unique in-dependent lightlike eigenvector u with eigenvalues +1 or −1. If f is one of theseisometries, you can proceed as follows:

(1) choose a basis B1 with elements u,e,v of L3 in such a way that u,v arelightlike and orthogonal to e, g(e,e) = 1=−g(u,v), u is a lightlike eigen-vector of eigenvalue ε (ε2 = 1) and 〈u〉⊥R = 〈u,e〉R.

(2) show that in the basis B1, f is represented by the matrixε λ ελ 2

20 ν ενλ

0 0 ε

,

where ν2 = 1 and λ 6= 0.


(3) check that in the orthonormal basis B2 with elements 1√2(u−v),e,1

√2(u+

v), the associated matrix to f is

M( f ,B2) =

ε(1− 14 λ 2) 1√

2λ ε

λ 2

4

− 1√2ενλ ν

1√2ενλ

−14 ελ 2 1√

2λ ε(1+ λ 2

4 )

.

3.4. Further properties of SO↑1(n) as a Lie group. Recall that a Lie group Gis a group with a structure of differentiable manifold such that the operation of thegroup G×G→ G and the inverse map G→ G, g 7→ g−1 become differentiable3.Any closed subgroup of the real general linear group Gl(n,R) is naturally a Liegroup (and both, the differentiable structure and the operation, become analytic).Moreover, any Lie group is isomorphic to some closed group of matrices. The Liealgebra g of G can be regarded as the tangent space of the group at the identity. Formatrix groups, g is also a vector space of matrices, which becomes closed undercommutation (i.e. X ,Y ∈ g⇒ [X ,Y ] ∈ g). The algebra g determines locally the Liegroup G, and also globally, if G is simply connected [67].

Here, some properties derived of the Lie group character of SO↑1(n) are stressed.The properties of the special orthogonal group, i.e., the group of rotations SO(n),will be developed at the same time. So we will write Gε , which will be equal toSO↑1(n) for ε =−1 and SO(n) for ε = 1.

3.4.1. Generalities. Let ηε be the diagonal matrix equal to the identity up tothe last diagonal element, which is equal to ε = ±1, and Mn(R) the n× n realmatrices. Put,

F : Mn(R)→Mn(R), A 7→ AtηεA.

Clearly, Gε is the connected part of the identity of F−1(ηε). So, Gε is a closedmatrix subgroup of Gl(n,R) and, then, a Lie group. Its algebra gε can be obtainedby taking any smooth curve s 7→ A(s) ∈ Gε , with A(0) = In, and taking the usualderivative at s = 0 in the equality

At(s)ηεA(s) = ηε .

Explicitly:

PROPOSITION 1.64. The Lie algebra of Gε is:

gε = X ∈Mn(R) : X t =−ηεXηε.That is, the Lie algebra associated to SO(n) is composed of the anti-symmetricmatrices, and the one associated to SO↑1(n) by matrices of the type(

An−1 aat 0

),

where a = (a1, . . . ,an−1) ∈ Rn−1 and An−1 is anti-symmetric. In particular, thedimension of both, SO↑1(n) and SO(n), is n(n−1)/2.

3The theory of Lie groups is very precise and, so, this definition is somewhat redundant. Forexample, the differentiability of the inverse map can be deduced from the differentiability of theoperation, by using the Implicit function Theorem [60]. Moreover, any Lie group will admit ananalytic atlas –and any second countable locally Euclidean topological group admits a Lie groupstructure, which is the content of Hilbert’s fifth problem [40]. We will not care on these subtleties,and will regard all the elements as (C∞) differentiable, with no further mention.


The Lie group SO(n) is compact, as it can be regarded as a closed boundedsubset of Mn(R) (for boundedness, recall that taking the trace in AtA = In one has∑i, j(Ai j)

2 = n). However, SO↑1(n) is not compact for n ≥ 2, as SO↑1(2) can beregarded as a closed non-compact subgroup of SO↑1(n). Recall that SO(n−1) canbe also regarded as a subgroup of SO↑1(n). More precisely,

(2) SO(n−1) = (

A 00 1

): A ∈ SO(n−1)

is the intersection of both groups SO(n) and SO↑1(n) (and, moreover, SO(n−1) isa maximal compact subgroup of SO↑1(n)).

3.4.2. Hyperbolic space as a homogeneous manifold. First, recall some no-tions from homogeneous spaces. Let G be a Lie group and M a manifold. A leftaction of G on M is a smooth map G×M→M, (g,x) 7→ gx, which satisfies,

(i) (g1g2)x = g1(g2x),(ii) ep = p,

for the identity e of G and all g1,g2 ∈ G,x ∈M. The action is transitive if, for anyx1,x2 ∈M there exists some g ∈ G such that gx1 = x2. The isotropy group at x isthe closed subgroup Gx := g ∈ G : gx = x.

EXERCISE 1.65. For transitive actions, the isotropy groups of any two pointsare conjugate.

The following properties are well-known in Lie group theory (see for example[67]):

THEOREM 1.66. (i) If G is a Lie group and H a closed subgroup of G, then themap

G×G/H→ G/H, (g,g′H) 7→ (gg′)His a transitive left action, and the natural projection Π : G→ G/H a (continu-ous) open map. Moreover, G/H admits a unique differentiable structure so that Π

becomes a (differentiable) submersion.(ii) If G×M→M is a transitive left action, then G/Gx is naturally diffeomor-

phic to M for all x ∈M.

When M lies under the hypotheses of the case (ii), M is called homogeneousspace.

EXERCISE 1.67. If G/H and H are connected then G is connected.

After this general preliminaries, let us come back to the groups Gε . Let the(upper) hyperbolic spaceHn−1 in Ln be

Hn−1 = p ∈ Ln : 〈p, p〉=−1, pn ≥ 1.In order to study at the same time the Euclidean case, consider also the usual roundsphere Sn−1 in Rn, and put Mε =H

n−1 for ε =−1 and Mε = Sn−1 for ε = 1. The

next results follow from the definitions above.

PROPOSITION 1.68. The map

Gε ×Mε →Mε , (A, p) 7→ Ap

is a (well-defined), smooth, transitive left action of Gε on Mε .The isotropy group of the action at en = (0, . . . ,0,1)t is SO(n−1).

4. THE SPIN COVERING OF THE RESTRICTED LORENTZ GROUP SO↑1(4) 29

Thus, taking into account Theorem 1.66, one has:

THEOREM 1.69. Considering H = SO(n−1) as a subgroup of Gε , the quotientGε/H is naturally diffeomorphic to Mε , i.e.:

SO↑1(n)/SO(n−1)∼=Hn−1 SO(n)/SO(n−1)∼= Sn−1

These results allow us to prove easily that Gε is connected, as the followingexercise suggests (for an alternative proof, written directly in any signature, see[43, Lemma 9.6]).

EXERCISE 1.70. From Theorem 1.69 and Exercise 1.67, prove inductively thatSO(n) is connected. Then, prove that SO↑1(n) is connected.

4. The spin covering of the restricted Lorentz group SO↑1(4)

Amongst the groups SO↑1(n), the case n = 4 becomes especially interesting forboth, its mathematical structure and its physical interpretations.

Our aim is to show that the universal covering of SO↑1(4) is the group Sl(2,C),constructing explicitly the universal covering homomorphism Sl(2,C)→ SO↑1(4)or spin map. In particular, this map also yields the universal covering SU(2)→SO(3). Among the consequences, a decomposition of any (proper, ortochronous)Lorentz transformation as a composition of a boost and a rotation is derived. Theinterested reader can learn more from the references [8] and [44].

4.1. Construction of the spin covering. In order to construct the spin mapSl(2,C)→ SO↑1(4), our plan of work is:

(1) A brief study of the topology of Sl(2,C) (and SU(2)), showing in partic-ular that it is 1-connected.

(2) The Hermitian matrices H(2,C) constitute naturally a (real) Lorentz vec-tor space, canonically isomorphic to L4.

(3) The natural action Sl(2,C)×H(2,C) → H(2,C) induces the requiredspinor map

Sl(2,C)→ Iso+↑(H(2,C),gL)≡ SO↑1(4).

4.1.1. The groups Sl(2,C) and SU(2). Let M2(C) be the space of complex2× 2 matrices, and Sl(2,C) ⊂M2(C) be the group of complex matrices with de-terminant 1. Obviously, Sl(2,C) can be regarded as a closed subgroup of M4(R)and, thus as a (real) Lie group4. Put:

A =

(a cb d

)∈M2(C).

LEMMA 1.71. Assume that |a|2 + |b|2 6= 0. Then A ∈ Sl(2,C) if and only ifthere exists λ ∈ C such that

(3)(

cd

)=

1|a|2 + |b|2

(−b

a

)+λ

(ab

).

In this case, λ is unique.

4In fact, it can be also regarded as a complex Lie group, but we will not take into account thisstructure.


PROOF. Recall that∣∣∣∣ a cb d

∣∣∣∣=∣∣∣∣∣ a − b

|a|2+|b|2b a

|a|2+|b|2

∣∣∣∣∣+∣∣∣∣∣ a c+ b

|a|2+|b|2b d− a

|a|2+|b|2

∣∣∣∣∣ ,the second determinant equal to 1. So, the first determinant is 1 iff the last one is0, that is, iff (3) holds for some (necessarily unique) λ .

PROPOSITION 1.72. The map

F :(C2 \0

)×C→ Sl(2,C), (

(ab

),λ ) 7→

(a − b

|a|2+|b|2 +λab a

|a|2+|b|2 +λb

)

is a diffeomorphism.

PROOF. The differentiability of F is obvious, and the existence of F−1 is en-sured by Lemma 1.71. The differentiability of F−1 is ensured by the explicit formof λ (namely, λ = (c+ b(|a|2 + |b|2)−1)/a when a 6= 0 and λ = (d− a/(|a|2 +|b|2)−1)/b when b 6= 0).

COROLLARY 1.73. Sl(2,C) is diffeomorphic to R3×S3.

PROOF. Apply Proposition 1.72 noticing that C2 \ 0, as a real manifold, isequivalent to R4 \0 and, therefore, diffeomorphic to R×S3.

REMARK 1.74. The six-dimensionality of Sl(2,C) can be also deduced bylooking at its Lie algebra. Take any smooth curve s 7→ A(s)∈ Sl(2,C), with A(0) =I2 and derive at 0 the constant function s 7→ det(A(s)) ≡ 1 (which characterizesSl(2,C)). Then, one obtains the algebra:

sl(2,C) = X ∈M2(C) : trace(X) = 0.

As a consequence, Sl(2,C) is as a 6-dim. real (3-dim. complex) Lie group.

Similarly, one can study the unitary and special unitary groups. In any dimen-sion, they are defined, respectively, as

U(n) = A ∈Mn(C) : A†A = I2, SU(n) =U(n)∩Sl(n,C),

where A† is the conjugate tranponse matrix A† = At .

EXERCISE 1.75. (1) Prove that A ∈M2(C) belongs to SU(2) if and only if:

A =

(a −bb a

)for some a,b ∈ C with |a|2 + |b|2 = 1.

Deduce that SU(2) is diffeomorphic to S3. Prove that the Lie algebra of SU(2) isthe 3-dimensional real algebra of the traceless anti-hermitian 2×2 matrices, i.e.:

su(2) = X ∈M2(C) : X +X† = 0, trace(X) = 0.

(2) Obtain analogous results for U(2).


4.1.2. The Lorentz space H(2,C). Consider the vector space H(2,C) of the2×2 Hermitian matrices (i.e. A = A†). Explicitly,

H(2,C) = (

a zz d

)∈M2(C) : a,d ∈R,z = x+ iy ∈ C.

This is naturally a real vector space of dimension four. In this space, the (minus)determinant

−∣∣∣∣ a z

z d

∣∣∣∣= x2 + y2−ad

is a quadratic form of Lorentzian signature. Recall that the subspace H(2,C)∗ ofthe traceless matrices (d =−a) constitutes a natural spacelike hyperplane. So, wewill consider the Lorentzian vector space (H(2,C),gL) and the Euclidean vector3-space (H(2,C)∗,gE).

The Pauli matrices

σ1 =

(0 11 0

),σ2 =

(0 −ii 0

),σ3 =

(1 00 −1

),σ4 =

(1 00 1

)constitute a natural orthonormal basis, which we wil call BP, for (H(2,C),gL);moreover, the first three ones (the classical Pauli matrices) constitute an orthonor-mal basis for (H(2,C)∗,gE).

Then, two canonical isometries between L4 and (H(2,C),gL) can be defined:(4)

L4 → H(2,C)x = (x1, . . . ,x4)t 7→ x := ∑

4µ=1 xµσµ

x =(

x3 + x4 x1− ix2

x1 + ix2 −x3 + x4

)L4 → H(2,C)

x = (x1, . . . ,x4)t 7→ x := x4σ4−∑3j=1 x jσ j

x =(−x3 + x4 −x1 + ix2

−x1− ix2 x3 + x4

)Notice that the first of the two isometries above maps the canonical orthonormalbasis of L4 in the Pauli basis BP. In what follows, we will choose this first isom-etry as the preferred one, and (H(2,C),gL) is then identified canonically to L4.Therefore, we also have a canonical isomorphism between the groups of isome-tries, concretely:

(5) Iso(H(2,C),gL)→ O1(4), f 7→M( f ,BP),

explicitly, for all x ∈ L4:

(6) M( f ,BP)x = f (x).

So, from now on, both Lie groups will be also regarded as canonically identified.

EXERCISE 1.76. (1) By construction: 〈x,x〉=−det(x) =−det(x).(2) By direct computation: xx = xx =−〈x,x〉1I2.(3) Determine the matrix of gL in the canonical basis Bc of H(2,C) constitutes by

the vectors: (1 00 0

),

(0 00 1

),

(0 11 0

),

(0 −ii 0

).

(4) Make a similar study of H(2,C)∗, and obtain similar expressions for this case.


4.1.3. Action of Sl(2,C) on H(2,C) and L4. Consider the following map:

(7)Sl(2,C)×H(2,C) → H(2,C)

(A,X) 7→ A∗X := AXA†.

Straightforward relevant properties are:(1) It is well-defined: (AXA†)† = AXA†.(2) It is an action: (A1 ·A2)∗X = A1 ∗ (A2 ∗X).(3) It gives a linear isometry for (H(2,C),gL):

(a) Linearity in the second variable: A∗ (aX1+bX2) = a(A∗X1)+b(A∗X2),

(b) Preserves gL: det(A∗X)=det(X),for all A,A1,A2 ∈ Sl(2,C),X ,X1,X2 ∈ H(2,C),a,b ∈R.

The third property implies that the well-defined map

A∗ : H(2,C)→ H(2,C), X 7→ A∗X ,

is an isometry of (H(2,C),gL). So, we also have a well defined map:

∗ : Sl(2,C)→ Iso(H(2,C),gL), A 7→ A∗.

Moreover, as (7) defines an action, we have (A1 ·A2)∗ = (A1)∗ (A2)∗. That is, themap ∗ is a Lie group homomorphism. Notice also that Sl(2,C) is connected (Cor.1.73) and, so, the image of the homomorphism ∗ must lie in Iso+↑(H(2,C),gL).Then, taking into account the identification of the isometry group Iso(H(2,C),gL)with O1(4) stated in (5), we also have the Lie group homomorphism:

(8) Λ : Sl(2,C)→ SO↑1(4), A 7→M(A∗,BP).

Explicitly, for all x ∈ L4 (recall (4), (5), (6))

(Λ(A))x = A∗ x = AxA†.

THEOREM 1.77. The spin map defined in (8) is a double covering map, whichyields the universal covering group of SO↑1(4). Thus,

Sl(2,C)/±I2 ∼= SO↑1(4).

PROOF. We have already justified that Λ is a Lie group homomorphism. Letus prove that its kernel is just ±I2. Notice that if A∗(X) = X for all X ∈H(2,C),then, taking X = I2, the matrix A must be unitary (A† = A−1). So, AX = XA for allX , and A =±I2 follows easily (say, apply that equality for X in some natural basisof H(2,C)).

As the kernel is a discrete subgroup, the homomorphism Λ is locally injective.Moreover, as the dimensions of Sl(2,C) and SO↑1(4) coincide (Remark 1.74, Prop.1.64), the differential of Λ at I2 must be bijective. So, it is well-known from Liegroup theory that Λ must be a covering map [67, Prop. 3.26] –and a universalcovering map, as Sl(2,C) is 1-connected by Cor. 1.73.

EXERCISE 1.78. Prove: (a) the restriction of the action (7) to

SU(2)×H(2,C)∗→ H(2,C)∗

is well defined, (b) this yields the (universal) double covering SU(2)→ SO(3), and(c) SO(3) is diffeomorphic to the projective space RP3 (recall also Exercise 1.75).


4.2. Lorentz transformations in Sl(2,C) and polar decomposition. Thespin covering Λ yields a further insight of the Lorentz group. We have alreadyseen that SU(2) appears as a privileged subgroup of Sl(2,C). We will find anotherprivileged subset SH+(2,C) of Sl(2,C), which will correspond to some boosts ofSO↑1(4), and we will see how such two subsets, SH+(2,C) and SU(2), allow us toreconstruct all SO↑1(4). Our plan of work is:

(1) As a previous technical step, to recall the polar decomposition A = PR ofany regular complex matrix A ∈ Gl(n,C) by means of a positive Hermit-ian matrix (P ∈ H+(n,C)) and a unitary matrix (R ∈U(n)).

(2) To check that, through the spin map Λ : Sl(2,C)→ SO↑1(4), the unitarymatrix R ∈ SU(2) corresponds to a rotation which fixes the timelike axisof L4. The positive Hermitian matrix P ∈ SH+(2,C) corresponds to aboost in a timelike plane which contains the t axis.

(3) As a consequence any matrix on SO↑1(4) admits several decompositionsand, in particular, it can be written as the composition of a rotation and aboost.

As a bit of terminology, we consider a complex (finite-dimensional) vectorspace V (C). EndCV denotes the vector space of all the endomorphisms of V (C)and AutCV the group of all its automorphisms. A sesquilinear form is a mapV (C)×V (C)→ C, which is conjugate-linear in the first variable and linear inthe second one. A Hermitian form h is a sesquilinear one which is also sym-metric, i.e. h(u,v) = h(v,u). An inner product is a Hermitian form G which ispositive definite, i.e., G(v,v)≥ 0, with equality only at 0. As before, Iso(V,G) de-notes the subgroup of AutCV containing the isometries of (V,G). The adjoint f † ∈EndCV of an endomorphism f ∈ EndCV with respect to G is defined by the equal-ity G( f †(u),v) = G(u, f (u)); clearly, ( f †)† = f . One says that f is self-adjointif f = f †. If B is any orthonormal basis then M( f †,B) = M( f ,B)†, the endomor-phism f is self-adjoint if and only if M( f ,B) is Hermitian; in this case, we canchoose a basis B of eigenvectors. Moreover, f is an isometry if and only f−1 = f †,that is, if and only if M( f ,B) is unitary for any orthonormal B.

4.2.1. Polar decomposition. Let us begin with the following classical result.

LEMMA 1.79. Let (V,G) be a complex vector space endowed with an innerproduct and f ∈ AutCV . Then:

(i) f f † is self-adjoint, and all its eigenvalues are positive. So, there exists aG-orthonormal basis B such that M( f f †,B) is diagonal, real and definitepositive.

(ii) ∃!h ∈ AutCV self-adjoint and with all its eigenvalues positive, such that h h = f f †.

(iii) h−1 f ∈ Iso (V,G).

PROOF. (i) Self-adjointness follows from G(u, f f †(v)) = G( f †(u), f †(v)) =G( f f †(u),v), and positive definiteness from G( f †(u), f †(u)) > 0 for all u 6= 0,as f is injective.

(ii) Notice that M(h,B) is necessarily the positive root of M( f f †,B) above.


(iii) As h−1 is also self-adjoint, we have:

G(h−1 f (u),h−1 f (v)) = G(h−1 h−1 f (u), f (v))

= G(( f †)−1(u), f (v)) = G( f † ( f †)−1(u),v) = G(u,v).

Let H+(n,C) be the set of all the Hermitian matrices with positive eigenvalues.

THEOREM 1.80. For all A ∈Gl(n,C) there exist a unique P ∈H+(n,C) and aunique R ∈U(n) so that: A = PR.

Moreover, the map

(9) H+(n,C)×U(n)→ Gl(n,C), (P,R) 7→ PR,

is a homeomorphism.

PROOF. The existence follows easily by writting the automorphism fA asso-ciated to A as fA = h r, where r := h−1 fA and h is as in the previous lemma.For the uniqueness, notice that for a second decomposition fA = h r, necessarily hmust be a root of f f † with positive eigenvalues and, thus, it is necessarily unique.

The continuity and bijectivity of (9) is then obvious. The continuity of theinverse might be proven by analyzing the uniqueness of the process of decompo-sition. However, it is a trivial consequence of the classical Brouwer’s invarianceof domain theorem, which states that any continuous bijective map between openssubsets of Rn or, with more generality, manifolds, is an open map (and, so, hascontinuous inverse).

EXERCISE 1.81. Prove directly Theorem 1.80 by means of the following steps:(i) AA† is a Hermitian matrix with positive eigenvalues.(ii) Assuming the existence of the decomposition A = PR, the uniqueness

would follow from AA† = P2.(iii) To prove existence, take P as the unique Hermitian root of AA† with posi-

tive eigenvalues and prove that R := AP−1 is unitary.

EXERCISE 1.82. Prove an analogous version of Theorem 1.80 but, now, writ-ting A = RP instead of A = PR. Give an example where the matrices P,R obtainedin this alternative decomposition are not equal to those obtained in the original one.

Now, defineSH+(2,C) := H+(2,C)∩Sl(2,C).

COROLLARY 1.83. If A ∈ Sl(2,C) and P,R are the matrices obtained in itspolar decomposition, then P ∈ SH+(2,C) and R ∈ SU(2).

Therefore, the restricted map

(10) SH+(2,C)×SU(2)→ Sl(2,C), (P,R) 7→ PR,

is a homeomorphism.

PROOF. Notice that det(P) must be real and positive, det(P) · det(R) = 1 and|det(R)|= 1. The remainder is obvious.


REMARK 1.84. Clearly, SH+(2,C) is homeomorphic to R3, as

SH+(2,C) = (

a x+ iyx− iy d

)∈M2(C) : x,y ∈R, a,d > 0, ad− x2− y2 = 1

and one can remove the last restriction substituting a = (1+ x2 + y2)/d. So, themap (10) recovers that Sl(2,C) is homeomorphic to R3×S3.

Notice also that, under the polar homeomorphism

SH+(2,C)×SU(2)∼= Sl(2,C),

the quotient Sl(2,C)/±I2 corresponds to SH+(2,C)× (SU(2)/±I2), which ishomeomorphic to R3×RP3 (recall Exercises 1.75 and 1.78). Therefore, we havealso:

COROLLARY 1.85. SO↑1(4) is homeomorphic to R3×RP3.

4.2.2. Rotations and boosts in Sl(2,C). Let us come back to the spin mapΛ : Sl(2,C)→ SO↑1(4), and consider SU(2) and SH+(2,C) as subsets of Sl(2,C).Recall that A ∈ SO↑1(4) is called a rotation (resp. boost) on the plane π ⊂ L4 if π

is spacelike (resp. timelike), A restricted to π⊥ is the identity and A restricted to π

is a classical two-dimensional rotation (resp. restricted Lorentz transformation). IfA is a rotation, any non-null direction in π⊥ is called an axis of rotation.

PROPOSITION 1.86. Let R ∈ Sl(2,C). R ∈ SU(2) if and only if Λ(R) ∈ SO(3)(included in SO↑1(4), according to (2)), that is, Λ(R) is a rotation, being x4 an axisof rotation.

PROOF. To the right, notice that the Pauli matrix σ4 = I2 represents a timelikedirection of (H(2,C),gL), and it is an eigenvector of R∗ of eigenvalue 1 (R∗(σ4) =RI2R† = I2 = σ4). So, the restriction of R∗ to σ⊥4 = (H(2,C)∗,gE) is an isometrywhich preserves the orientation and, thus, it also has a spacelike invariant eigen-vector.

The converse is straightforward.

EXERCISE 1.87. Let A(θ) =(

e−iθ 00 eiθ

)(∈ SU(2)) for θ ∈R. Prove that

Λ(A(θ)) is a rotation of axis x3 and angle 2θ .

To study Λ(SH+(2,C)), consider first the following simple case.

LEMMA 1.88. For α > 0,(α 6= 1), let Pα =

(α 00 α−1

)(∈ SH+(2,C)).

Then Λ(Pα) is a boost on the timelike plane 〈x3,x4〉R with eigenvalues α2,α−2.

PROOF. Easily, (Pα)∗(σ3±σ4) = Pα(σ3±σ4)Pα = α±2(

1±1 00 1∓1

)=

α±2(σ3±σ4).The computation (Pα)∗(σi) = σi for i = 1,2 is straightforward.

EXERCISE 1.89. (a) In the previous lemma, consider the matrices Pα withα ∈ C\0. Prove that Λ(Pα) is the composition of a boost on the timelike plane〈x3,x4〉R with eigenvalues |α|2, |α|−2 and the rotation on the plane 〈x1,x2〉R ofangle 2θ where eiθ = α/|α| (in particular, Λ(Pα) = Λ(P−α)).


(b) Show that both, the set of real and complex matrices Pα ,α 6= 0 are sub-groups of Sl(2,C). Is SH+(2,C) another subgroup?

PROPOSITION 1.90. If P ∈ SH+(2,C), there exists a timelike plane π whichcontains the x4-axis such that Λ(P) is a boost on π .

PROOF. To the right, as P is Hermitian, there exists R ∈ SU(2) such that P =R−1PαR for some positive α(6= 1). Let σ ′i so that R∗σ ′i = σi for i = 1,2,3. Clearly,P∗σ ′i = σ ′i for i = 1,2. So P∗ is a boost on the orthogonal plane π = 〈σ ′3,σ4〉R.

EXERCISE 1.91. (a) Check that the previous result holds for P ∈ H(2,C)∩Sl(2,C).

(b) Conversely, let P ∈ Sl(2,C) and prove: if there exists a timelike planeπ ⊂ L4 which contains the x4-axis such that Λ(P) is a boost on π and the identityon π⊥, then P ∈ H(2,C)∩Sl(2,C) (that is, either P or −P belong to SH+(2,C)).

4.2.3. Decomposition of SO↑1(4) in boosts and rotations. Previous results canbe summarized in the following theorem, whose proof is detailed for the conve-nience of the reader.

THEOREM 1.92. Let L ∈ SO↑1(4). Then there exists a boost B on a timelikeplane π1 which contains the x4 axis, and a rotation S on a spacelike plane π2orthogonal to the x4 axis (but not necessarily orthogonal to π1) such that L = BS.Moreover, B and S are univocally determined5.

PROOF. Let A ∈ Sl(2,C) such that Λ(A) = L (Th. 1.77), and take its polardecomposition A = PR (Th. 1.80, Cor. 1.83). Now, L = Λ(P)Λ(R), where Λ(P) isa boost for some plane π1 (Prop. 1.90) and Λ(R) is a rotation for some spacelikeplane π2 (Prop. 1.86), as required.

For uniqueness, notice that if L = B′ S′ then there are P′ ∈ SH+(2,C) andR′ ∈ SU(2) such that B′ = Λ(P′),S′ = Λ(R′) (again by Props. 1.86, 1.90, seealso Exer. 1.91) and P′R′ = ±A(= Λ−1(L)) (Th. 1.77). By the uniqueness ofthe polar decomposition (Th. 1.80) necessarily P′ = P and R′ =±R, which impliesB′ = B,S′ = S.

Previous results and techniques allow one to obtain other decompositions, arelevant one suggested in the following exercise.

EXERCISE 1.93. (1) Let A ∈ Sl(2,C). Prove that there exists some Pα , α > 0and R1,R2 ∈ SU(2) such that A = R−1

1 PαR2.(2) Deduce that each L ∈ SO↑1(4) can be written as the composition of a first

rotation on a plane orthogonal to the x4 axis, a boost on the x3,x4 plane and asecond rotation on a new plane orthogonal to x4.

4.3. The Mobius group in the starred nights. A nice consequence of thespin covering is to show the existence of a natural isomorphism between SO↑1(4)and the Mobius group, with further suggestive relations.

4.3.1. The Riemann sphere and Mobius group. The Mobius group Moeb(SR)is the set of the linear fractional transformations of the complex plane, i.e., the setof meromorphic maps of the type:

C → C

z 7→ w = az+bcz+d

a,b,c,d ∈ C with ad−bc = 1,

5Notice that then π1,π2 are also univocally determined except if either B or S are the identity.


endowed with the composition as group operation.Such maps are also naturally identifiable as the (orientation-preserving) con-

formal transformations of the Riemann sphere SR, i.e, the usual sphere of R3 re-garded as a smooth manifold endowed with its natural orientation and conformalstructure. Recall that there are several natural ways of looking at this space. Thefirst one is as the extended complex plane C∗ = C∪ ∞. A second one is thecomplex projective line CP1, i.e., the quotient space:

CP1 :=(C2\0)/∼(

uv

)∼(

u′

v′

)⇔∃λ ∈C\0 :

(uv

)= λ

(u′

v′

).

Each class [

(uv

)] can be denoted then as z = u/v, which is naturally identified

with an element of C∗. Moreover, when z 6= 0,∞, there are two natural represen-

tatives of the class, namely(

z1

)and

(1z−1

). These two representatives can

be regarded as the images of a single point in the Riemann sphere SR, via stereo-graphic projection from the north and the south poles.

Summing up, the Mobius transformations can be also written as

SR → SR

z = uv 7→ z′ = u′

v′ =au+bvcu+dv

,

being(

u′

v′

)=

(a bc d

)(uv

).

4.3.2. Spin covering of the Mobius group. Notice that the natural action

Sl(2,C)×C2 → C2

(A,(

uv

)) 7→ A

(uv

)induces naturally and action

Sl(2,C)×CP1→ CP1,

and, thus, an onto group homomorphism,

Λ : Sl(2,C) → Moeb(SR)

A =

(a bc d

)7→ z 7→ z′ = az+b

cz+d .

Clearly, Ker(Λ) = ±I2 and we have a canonical group isomorphism6

Sl(2,C)/±I2 ∼= Moeb(SR).

So, as a straightforward consequence of Theorem 1.77,

COROLLARY 1.94. The Lie groups SO↑1(n) and Moeb(SR) are canonically iso-morphic.

6In fact, this is a Lie group isomorphism, if Moeb(SR) has already been regarded as a Lie group.Otherwise, the isomorphism can be also used to induce a topology and, so, a Lie group structure inMoeb(SR).


4.3.3. The Riemann sphere and the lightcone. In order to understand betterthe relation between SO↑1(n) and Moeb(SR), recall first that the future lightcone ofL4 is naturally diffeomorphic to the unit sphere S2 multiplied by the interval (0,∞).Moreover, it inherits the degenerate metric

t2 (dθ2 + sin2

θdφ2) ,

at any (p, t) ∈ S2× (0,∞) with spherical coordinates (θ ,φ) on the S2 part. The setof all the future lightlike directions can be identified with S2×1 or, with moregenerality, with the image of any section

(11) σ : S2→ S2× (0,∞), σ(p) = (p, t(p)).

Regarding σ(S2) as an embedded surface, the induced metric can be written as

gσ = t2(p)(dθ

2 + sin2θdφ

2) ,at each (p, t(p)). So, the pullback metric σ∗gσ is conformal to the canonical oneon S2. Moreover, the canonical orientation and time-orientation of L4 inducesa natural orientation on σ(S2) –namely, by declaring that a basis v1,v2 tangentto σ(S2) at some p is positively oriented when the basis v1,v2,N,T (where T istimelike future pointing and N spacelike and pointing outwards the causal cone) ispositively oriented in L4. Summing up, we have justified:

Claim 1. Any surface σ(S2) as in (11) represents the space ofall the (future-pointing) lightlike directions from the origin inL4. All such surfaces are naturally diffeomorphic to the sphereS2 and inherit a Riemannian metric which is conformal to theusual one, as well as an orientation.

Therefore, the space of all the future-directed lightlike di-rections is naturally represented by the Riemann sphere SR.

Now, recall that any restricted Lorentz transformation L will map future-directedlightlike directions in future-directed lightlike directions and, so, tangent planes tothe future lightcone in tangent planes to the future lightcone. Moreover, as L is anisometry, if S1 = σ1(S

2) is the image of a section as in (11), then S2 = L(S1) isboth isometric to S1 and the image of a second section7 S2 = σ2(S

2). As S1 and S2are conformal to S2, L induces a (orientation-preserving) conformal map from S2

in itself, namely,σ−12 |S2 Lσ1.

Summing up:

Claim 2. Any L∈ SO↑1(4) induces a unique orientation-preservingdiffeomorphism of the future lightcone in itself. Moreover, Lmaps isometrically the image S1 of any section as in (11) in theimage S2 of some other section.

Therefore, L induces naturally a (orientation-preserving) con-formal transformation of the Riemann sphere SR.

7Notice that, if L is not the identity, then the map L σ1 is not a section, that is, L must mapsome lightlike direction in a different lightlike direction. Otherwise, all the lightlike vectors would beeigenvectors –but the existence of three independent lightlike eigenvectors implies that all the eigen-values must be equal to 1, as the product of the eigenvalues of two independent lightlike eigenvectorsmust be 1.


The converse of this last assertion is also true, i.e., any orientation-preserving con-formal transformation of SR yields a unique L ∈ SO↑1(4) (which consistently gener-ates the original conformal transformation). In order to check this and to analyzequantitatively the claims, recall again the model (H(2,C),gL) of L4.

PROPOSITION 1.95. The smooth map

j : C2 \0→ H(2,C),(

ξ

η

)7→(

ξ

η

)(ξ , η) =

(|ξ |2 ξ η

ξ η |η |2)

satisfies:(i) The image of j is the set of all the future-directed lightlike vectors of the

Lorentzian vector space (H(2,C),gL).(ii) j induces a bijection j between CP1 and the set of all the future-pointing

lightlike directions8

j : CP1 (≡ SR)→ S2×1 (≡ future lightlike directions of (H(2,C),gL)≡ SR).

(iii) For any A ∈ Sl(2,C) and the corresponding restricted isometry A∗ of(H(2,C),gL):

j(A(

ξ

η

)) = (A∗ j)

(ξ

η

)j([A

(ξ

η

)]) = [(A∗ j)

(ξ

η

)].

PROOF. (i) The images has 0 determinant, so, they represent lightlike vectors.Moreover, recalling the explicit form of the canonical isometry with L4 (mappingx 7→ x in (4)), clearly: (a) the images are future-directed (their traces are greater than0), and (b) all the future-directed lightlike vectors lie in j(C2 \ 0) (a preimagecan be obtained even with ξ ∈R,ξ ≥ 0).

(ii) From j(λ (ξ ,η)) = |λ |2 j((ξ ,η)), the map j is well-defined (and onto by(i)). For injectivity, if j((ξ ,η)) = t2 j((ξ ′,η ′)) for some t > 0, then the explicitform of j yields, on one hand, |ξ |= t|ξ ′|, |η |= t|η ′| and, on the other,

ξ

ξ ′= t2 η ′

η= t

η

tη ′=

η

η ′,

the second equality taking into account that tη ′/η is unit.(iii) Just by direct computation:

jA(

ξ

η

)= A j(

(ξ

η

))A† = A∗ j

(ξ

η

).

As a final summary:(1) As we have already justified (Corollary 1.94), any restricted Lorentz trans-

formation L=Λ(±A) defines univocally a Mobius transformation Λ(±A)and vice versa.

(2) In Claim 2, we argued:a) All the future-directed lightlike directions from the origin in L4

admits naturally the structure of Riemann sphere (oriented conformalsphere), and

8In fact, a conformal and orientation preserving diffeomorphism.


b) any L ∈ SO↑1(4) yields naturally also an (orientation-preserving)conformal transformation of this Riemann sphere.

(3) Part (ii) of Proposition 1.95 yields a canonical bijection between thestandard Riemann sphere CP1 and the Riemann sphere of all the future-directed lightlike directions.

(4) Part (iii) of Proposition 1.95 shows the consistency between the Mobiustransformation Λ(A) and the conformal transformation induced by Λ(A)on the set of all the future-directed lightlike directions.

REMARK 1.96. R. Penrose gave a nice interpretation of the equivalence be-tween SO↑1(4) and Moeb(SR). When we look at the sky in a starred night, thelight of each star comes from a lightlike direction and, so, the stars we see corre-spond naturally with points in the Riemann sphere9 Now, consider two observers,who measure spacetime through oriented, inertial frames. Their transformation ofcoordinates belongs to SO↑1(4), and induce a conformal mapping of the Riemannsphere. So, if they take spherical coordinates on the sky, their change of coordi-nates will be a Mobius transformation. If they do not move with respect to eachother (i.e., they only rotate their axis), then the change of coordinates will be just arotation. But if they move at a (high) relative speed, then the change of coordinateswill be a conformal non-isometric map.

REMARK 1.97. A nice connection appears between Relativity (plus Newto-nian Mechanics) and Quantum Mechanics through the spin map. In Quantum Me-chanics, a complex vector space V (C) must be used in order to describe internalproperties of the particles, such as the spin. For example, if the particle spin isbeing measured (say, along some spacelike direction, for example, x3 = z for someobserver), and the spin may take the values ±1/2 (as in the case of a single elec-tron) then V (C) has complex dimension 2. If we have two inertial observers Oand O (see the Appendix 6), each one will assign a pair of complex coordinates(z1,z2)t , (z1, z2)t to represent the spin of the particle. Which is the relation betweenthese coordinates? As the spacetime coordinates of the observers O and O willbe related by some restricted Lorentz transformation L ∈ SO↑1(4), and L = Λ(±A)for some A ∈ Sl(2,C), the transformation of the complex coordinates will be ofthe type (z1, z2)t = ±A(z1,z2)t . In general, the transformation of the complex co-ordinates are related to the representations of Sl(2,C) on a complex vector spaceV (C). The non always trivial role of the choice of sign (i.e. the existence ofphysically meaningful representations of Sl(2,C) which are not inducible in rep-resentations of SO↑1(4)) has been stressed by some spectacular physical effects, asthe Ahanorov-Bohm one.

5. Jordan canonical forms for self-adjoint endomorphisms

5.1. Canonical form of a linear endomorphism. Even though we will obtainthe canonical forms in an essentially self-contained way, for the convenience of thereader, we recall throughout this subsection some well-known facts of the Jordanforms.

9Notice that, as the light arrives at our eye, we have to consider the Riemann sphere of the past-directed lightlike directions, which is completely analogous to the future-directed one developed inthis section.

5. JORDAN CANONICAL FORMS FOR SELF-ADJOINT ENDOMORPHISMS 41

Let us introduce our terminology and fix our notations by recalling a few ele-mentary facts concerning the Jordan canonical form for matrices representing lin-ear endomorphisms of Rn. Let A : Rn → Rn be a linear endomorphism; whenneeded, we will consider the C-linear extension of A to an endomorphism of Cn,defined by cA(x+ iy) = Ax+ iAy, where x,y ∈Rn. Given a complex number z, wewill denote by Im(z) its imaginary part.

By s(A) we will mean the spectrum (set of the complex eigenvalues) of cA; forλ ∈ s(A), let Hλ (A) denote the complex generalized eigenspace of A:

Hλ (A) = Ker( cA−λ )n.

As A is real, if λ ∈ s(A) then obviously λ ∈ s(A); we set

Fλ (A) =

Hλ (A), if λ ∈ s(A)∩R;

Hλ (A)⊕Hλ(A), if λ ∈ s(A)\R,

so that we can obtain the Jordan decomposition of Cn as:

(12) Cn =⊕

λ∈s(A)Im(λ )≥0

Fλ (A),

(see for example [26]). Finally, let F oλ(A) denote the real generalized eigenspace

of A:F o

λ(A) = Fλ (A)∩Rn;

Fλ (A) is the complexification of F oλ(A), i.e., Fλ (A) =F o

λ(A)+ iF o

λ(A), and thus

Rn =⊕

λ∈s(A)Im(λ )≥0

F oλ(A).

Clearly, if λ ∈ R, then dimC(Ker( cA− λ )

)= dimR

(Ker(A− λ )

)and F o

λ(A) =

Ker(A−λ )n; the dimension of Ker(A−λ ) will be called the geometric multiplicityof the eigenvalue λ , while the dimension of Ker(A−λ )n, the algebraic multiplicityof λ . Moreover, for any λ ∈ C, F o

λ(A) is an A-invariant real subspace.

The spaces Hλ (A) (and Fλ (A)) are cA-invariant, and the restriction cA|Hλ (A)of cA to Hλ (A) is represented in a suitable basis by a matrix which is the directsum of λ -Jordan blocks, i.e., matrices of the form:

(13)

λ 1 0 0 . . . 00 λ 1 0 . . . 0

...0 0 . . . λ 1 00 0 . . . 0 λ 10 0 . . . 0 0 λ

.

By direct sum of the k1 × k1 matrix α and the k2 × k2 matrix β , we mean the(k1 + k2)× (k1 + k2) matrix given by:

α⊕β =

(α 00 β

).

We will denote by Jk(λ ) a Jordan block of the form (13) having size k× k whenk > 1; J1(λ ) is defined to be the 1×1 matrix (λ ).


When λ ∈C\R, the restriction of A|F oλ(A) is represented in a suitable basis by

a matrix which is the direct sum of λ -Jordan blocks of the form

(14)

a b 1 0−b a 0 1 0

a b 1 0−b a 0 1

a b−b a

. . . 1 00 1

0 a b−b a

,

where λ = a+bi, a,b ∈R.The decomposition of cA|Hλ (A) and A|F o

λ(A) into direct sum of λ -Jordan blocks

is not unique, but the number of blocks (and their dimensions) appearing in thisdecomposition is fixed, and it is equal to the complex dimension of Ker( cA−λ ).

5.2. Canonical form of a self-adjoint endomorphism for arbitrary scalarproducts. Let us now consider a scalar product g(·, ·) on a vector space V . Wewill denote by cV the complexification of V , that is, the complex vector space de-fined formally as cV = v+ iw : v,w ∈V. Moreover cg will denote the Hermitianform obtained as the sesquilinear extension of g determined by the linearity on thesecond entry and the property cg(v,w) = cg(w,v) for every v,w ∈ cV . In particularthis implies that cg(αv,w) = α cg(v,w) for every v,w ∈ cV and every α ∈ C andthat cg(v,w) = cg(w, v).

We will assume that A : V → V is a self-adjoint endomorphism with respectto the scalar product g, meaning that g(Av,w) = g(v,Aw) for all v,w ∈ V ; it isimmediate that A is self-adjoint with respect to g if and only if cA is self-adjointwith respect to cg.

LEMMA 1.98. If λ ,µ ∈ s(A) are such that λ 6= µ , then the generalized ei-genspaces Hλ (A) and Hµ(A) are cg-orthogonal. In particular, if λ ∈ C\R, thenHλ (A) is isotropic, i.e. cg|Hλ (A) ≡ 0. Furthermore, if λ ∈ s(A), the restriction ofthe Hermitian form cg to Fλ (A) is nondegenerate, and so is the restriction of g toF o

λ(A).

PROOF. We show by induction on k = k1+k2 that Ker( cA−λ )k1 and Ker( cA−µ)k2 are cg-orthogonal spaces (in particular, this will hold when k1 and k2 reach thealgebraic multiplicities of λ and µ , yielding the result). This is trivial when k = 0,as well as if either k1 or k2 is zero.

Assume now that Ker( cA−λ )k1 and Ker( cA−µ)k2 are cg-orthogonal spaces forall pairs k1 and k2 such that k1 + k2 < k; let s1,s2 ≥ 1 be such that s1 + s2 = k, andlet v ∈ Ker( cA− λ )s1 and w ∈ Ker( cA− µ)s2 . Since ( cA− λ )v ∈ Ker( cA− λ )s1−1

and ( cA−µ)w ∈ Ker( cA−µ)s2−1, by the induction hypothesis, we have:

0 = cg(( cA−λ )v,w

)= cg

( cAv,w)− λ

cg(v,w),

0 = cg(v,( cA−µ)w

)= cg

(v, cAw

)−µ

cg(v,w).

Thus, the equality between the right-hand sides and the self-adjointness of cA yieldthe result.


The orthogonality of the generalized eigenspaces shows that (12) is in fact acg-orthogonal direct decomposition of cV , from which it follows that the restrictionof cg to each Fλ (A) is nondegenerate, since cg is nondegenerate on cV . Finally, thenondegeneracy of the restriction of cg on Fλ (A) is equivalent to the nondegeneracyof the restriction of g to F o

λ(A) (if z belongs to the kernel of Fλ (A), then so does

z, and z+ z belongs to the kernel of F oλ(A)).

In order to study the restriction of g to the generalized eigenspaces of A, wewill now determine the form of the matrix representing g in a suitable Jordan basisfor A. Lemma 1.98 tells us that it is not restrictive to consider the case that A hasonly two complex conjugate eigenvalues or one real eigenvalue: once the matrixrepresentation gλ of g|F o

λ(A) has been determined for each λ ∈ s(A) with Im(λ )≥

0, the matrix representation of g will be given by the direct sum of all such gλ ’s.Using the terminology of [26], a sip matrix will be an n×n matrix Sipn of the form:

(15) Sipn =

0 0 . . . 0 0 10 0 . . . 0 1 0

...0 0 1 . . . 0 00 1 0 . . . 0 01 0 0 . . . 0 0

.

In the next two results, we will distinguish the cases in that the eigenvalue λ is realand non real. Let us give some previous technical lemmas.

LEMMA 1.99. If W <V , then A(W )⊂W implies that A(W⊥)⊆W⊥.

PROOF. Let v ∈W⊥ and w ∈W . Then using the self-adjointness of A and thefact that Aw ∈W , we deduce that g(Av,w) = g(v,Aw) = 0, as required.

LEMMA 1.100. Let a j = (A−λ ) j−1a1 for j = 1, . . . ,s such that (A−λ )as = 0.Then the a j’s are linearly independent.

PROOF. Assume on the contrary that there exists a non-trivial linear combina-tion

α1a1 +α2a2 + . . .+αsas = 0,

with α1, . . . ,αs ∈R and let j ∈ 1, . . . ,s be the smallest number such that α j 6= 0.Then

0 = (A−λ )s− j(0) = (A−λ )s− j(α1a1 +α2a2 + . . .+αsas) = α jas 6= 0,

and they must be independent.

Adapting the proof of [26, Theorem 3.3], we get the following:

PROPOSITION 1.101. Let λ be a real eigenvalue of A, with r = dim(Ker(A−λ )) and n = dimV . Then, the real generalized eigenspace can be written as ag-orthogonal direct sum:

Ker(A−λ )n =r⊕

i=1

Vλ ,i,

for which the following property holds:


(i) for all i, there exists a basis vi1, . . . ,v

ini

of Vλ ,i and a number εi ∈ −1,1such that in this basis the matrix representation of A|Vλ ,i is as in (13), and thematrix representation of g|Vλ ,i is given by εi ·Sipni

.In particular,

(ii) g|Vλ ,i is nondegenerate for all i = 1, . . . ,r;(iii) each Vλ ,i is A-invariant;

PROOF. It will suffice to show the existence of a number ε = −1,1, of asubspace V ⊂ Ker(A−λ )n and of a basis w1, . . . ,ws of V with the properties:

• Aw1 = λw1 and Aw j = w j−1 +λw j for j = 2, . . . ,s;• g(w j,wk) = εδ j+k,s+1 for all j,k = 1, . . . ,s.

The two properties above imply that V is A-invariant and that the restriction g|V isnondegenerate. The matrix representation of A|V in the basis w1,. . . ,ws is as in (13)and the matrix representation of g|V is ε ·Sips; the conclusion will follow easilyfrom an induction argument by considering the g-orthogonal complement of V inF o

λ(A) (see Lemma 1.99).To infer the existence of such a subspace V with the desired basis, let us argue

as follows. Let s≥ 1 such that (A−λ )s|Ker(A−λ )n = 0, but (A−λ )s−1|Ker(A−λ )n 6= 0.Since B = g

((A− λ )s−1·, ·

)is a non zero symmetric bilinear form on Ker(A−

λ )n, there must exists a vector a1 ∈ Ker(A−λ )n such that B(a1,a1) 6= 0. We cannormalize a1 in such a way that g

((A−λ )s−1a1,a1

)= ε , for some ε ∈ −1,1;

the case s = 1 is concluded by setting w1 = a1, and we will now assume s > 1.For j = 1, . . . ,s, let us define a j = (A−λ ) j−1a1 and let V be the space spanned

by the a j’s; the a j’s are linearly independent (see Lemma 1.100). For j+k = s+1,we have:

g(a j,ak) = g((A−λ ) j−1a1,(A−λ )k−1a1

)= g((A−λ ) j+k−2a1,a1

)= g((A−λ )s−1a1,a1

)= ε,

(16)

while if j+ k > s+1 we have:

(17) g(a j,ak) = g((A−λ ) j+k−2a1,a1

)= 0.

Now, set b1 = a1 +α2a2 + . . .+αsas, with α1, . . . ,αs ∈ R and b j = (A−λ ) j−1b1for j = 1, . . . ,s. Observe that g(b j,bk) = 0 if j + k > s+ 1, and g(b j,bk) = ε ifj+ k = s+ 1 for every choice of the real coefficients α2,α3, . . . ,αs. Now we willchoose these real coefficients in such a way that g(b1,b j)= 0 for all j = 1, . . . ,s−1,which will imply that g(bk,b j) = 0 whenever k+ j < s+ 1. Such a choice of theαi’s is indeed possible (and unique); namely, the equality g(b1,b j) = 0 is given, inview of (16) and (17), by:

0 = g(a1 +∑

sk=2 αkak, a j +∑

s− j+1k=2 αka j+k−1

)= g(a1,a j)+2εαs− j+1 + terms in α2, . . . ,αs− j,

so that the αi’s can be determined recursively by taking j = s−1,s−2, . . . ,1 in theabove equality. Applying Lemma 1.100, we deduce that the b j’s form a basis ofV . Finally, set w j = bs− j+1 for all j = 1, . . . ,s; an immediate computation showsthat the w j’s satisfy the required properties.

PROPOSITION 1.102. Let λ ∈ C \R, with λ = a+ bi, a complex eigenvalueof A, such that a,b ∈ R and r = dim(Ker( cA− λ )). Then, the real generalized


eigenspace can be written as a g-orthogonal direct sum:

F oλ(A) =

r⊕i=1

Vλ ,i⊕Vλ ,i,

for which the following property holds:(i) for all i, there exists a basis vi

1,wi1,v

i2,w

i2, . . . ,v

ini,wi

niof Vλ ,i⊕V

λ ,i with allscalar products zero except g(vi

j,vik) = 1 =−g(wi

j,wik) if j+ k = ni +1, such

that in this basis the matrix representation of A|Vλ ,i⊕Vλ ,i

is as in (14) with aand b such that λ = a+bi.

In particular,(ii) g|Vλ ,i⊕V

λ ,iis nondegenerate for all i = 1, . . . ,r;

(iii) each Vλ ,i⊕Vλ ,i is A-invariant;

PROOF. Let us first show that the statement of the theorem can be reduced tothe following complex canonical form.

Claim 1: it will suffice to show the existence of a subspace V ⊂ Ker( cA−λ )n

and of a basis z1, , . . . ,zs of V with the properties:• cAz1 = λ z1 and cAz j = z j−1 +λ z j for j = 2, . . . ,s;• cg(z j, zk) = 2δ j+k,s+1 for all j,k = 1, . . . ,s.

Indeed, consider V ⊕ V , where V = v : v ∈ V , and let vi and wi be respec-tively the real and imaginary part of zi, so that zi = vi+ iwi for i = 1, . . . ,s. The firstproperty above implies that

Av1 =cA(

12(z1 + z1)

)=

12(λ z1 + λ z1) = av1−bw1,

Aw1 =cA(−i

12(z1− z1)

)=−i

12(λ z1− λ z1) = bv1 +aw1

and

Av j =cA(

12(z j + z j)

)=

12(z j−1 + z j−1)+

12(λ z j + λ z j) = v j−1 +av j−bw j,

Aw j =cA(−i

12(z j− z j)

)=−i

12(z j−1− z j−1)− i

12(λ z j− λ z j)

= w j−1 +bv j +aw j,

for j = 2, . . . ,s in agreement with (14). Moreover, the second property above andthe fact that Ker( cA−λ )n is isotropic (see Lemma 1.98) imply that

g(v j,vk) =14

cg(z j + z j,zk + zk) = δ j+k,s+1,

g(w j,wk) =14

cg(z j− z j,zk− zk) =−δ j+k,s+1,

g(v j,wk) = i14

cg(z j + z j,zk− zk) = 0.

Therefore, V ⊕ V is A-invariant and the restriction g|V ⊕V is nondegenerate. Thematrix representation of A|V ⊕V in the basis v1,w1,. . . ,vs,ws is as in (13) and theg-products between the elements of the basis are as required; the conclusion willfollow easily from an induction argument by considering the cg-orthogonal com-plement of V in Ker(A−λ )n.


To infer the existence of such a subspace V with the desired basis, let us argueas follows. There exists s ≥ 1 with the property that ( cA−λ )s|Ker( cA−λ )n = 0 but( cA−λ )s−1|Ker( cA−λ )n 6= 0. We will need the following property:

Claim 2: there exists a vector a1 ∈ Ker( cA−λ )n such that

cg(( cA−λ )s−1a1, a1

)= 2.

Let us define B : Fλ (A)×Fλ (A)→ C as

B(v,w) = cg((A−λ )s−1v,w)

for every v,w ∈Fλ (A). Clearly this map is sesquilinear and, by the choice of s, itis not identically zero. Then there exists z such that B(z,z) 6= 0. It is easy to seethat there exists v,w ∈Hλ (A) such that z = v+ w. Thus

0 6= B(z,z) = B(v,v)+B(w, w)+2Re(B(v, w)).

As, by Lemma 1.98, Hλ (A) and Hλ(A) are cg-isotropic, B(v,v) = B(w, w) = 0,

and from the above equation follows that Re(B(v, w)) 6= 0. Finally compute

B(v+w, v+ w) = B(v, v)+B(w, w)+2Re(B(v, w)).

Then one of the vectors v, w or v+w must satisfy the condition required in Claim2.

For j = 1, . . . ,s, let us define a j =( cA−λ ) j−1a1 and let V be the space spannedby the a j’s; by Lemma 1.100, the a j’s are linearly independent. Analogously to thereal case, for j+ k = s+1, we have:

cg(a j, ak) = cg(( cA−λ ) j−1a1,(

cA− λ )k−1a1)

= cg(( cA−λ ) j+k−2a1, a1

)= cg

(( cA−λ )s−1a1, a1

)= 2,

(18)

while if j+ k > s+1 we have:

(19) cg(a j, ak) =cg(( cA−λ ) j+k−2a1, a1

)= 0.

Now, set b1 = a1 +α2a2 + . . .+αsas and b j = ( cA−λ ) j−1b1 for j = 1, . . . ,s. Ob-serve that cg(b j, bk) = 0 if j + k > s+ 1, and cg(b j, bk) = 2 if j + k = s+ 1 forevery choice of the complex coefficients α2,α3, . . . ,αs. Now we will choose thesecomplex coefficients in such a way that cg(b1, b j) = 0 for all j = 1, . . . ,s−1, whichwill imply that cg(bk, b j) = 0 whenever k+ j < s+1. Such a choice of the αi’s isindeed possible (and unique), namely, the equality cg(b1, b j) = 0 is given, in viewof (18) and (19), by:

0 = cg(a1 +∑

sk=2 αkak, a j +∑

s− j+1k=2 αka j+k−1

)= cg(a1, a j)+4αs− j+1 + terms in α2, . . . ,αs− j,

so that the αi’s can be determined recursively by taking j = s−1,s−2, . . . ,1 in theabove equality. Applying Lemma 1.100 we deduce that the b j’s form a basis of V .Finally, set z j = bs− j+1 for all j = 1, . . . ,s; an immediate computation shows thatthe z j’s have the required properties of Claim 1.


5.3. Jordan canonical form of a self-adjoint endomorphism with respectto a Lorentzian scalar product. Let us now consider the case in that the scalarproduct is Lorentzian. The number of canonical forms are drastically reduced. Wewill denote by Dk a k× k diagonal matrix.

PROPOSITION 1.103. Let (V,g) be a vector space V endowed with a Lorentzianscalar product g and A : V → V a self-adjoint endomorphism with respect to g.Then any of the following possibilities happens

(i) there exists an orthonormal basis in that the matrix of A is diagonal or of theform Dn−2 0

0a b−b a

(ii) there exists a basis e1,e2, . . . ,en−2,u,v with g(ei,ei) = 1 for i = 1, . . . ,n−

2, g(u,v) = 1 and all the other products equal to zero, in that the matrixrepresentation of A is either Dn−2 0

0λ ε

0 λ

with ε =±1, or

Dn−2 0

0λ 0 11 λ 00 0 λ

.

PROOF. Consider a subspace Vλ ,i of Proposition 1.101 and ni = dimVλ ,i. Thenthe basis in Proposition 1.101 has index equal to

ni/2 if ni is even,(ni−1)/2 or (ni +1)/2 if ni is odd;

this is because every pair vij,v

ik such that j+k = s+1, j 6= k generates a subspace

of dimension 2, which is orthonormal to the other elements in the basis and withindex 1.

Consider now a subspace Vλ ,i⊕Vλ ,i of dimension 2s as in Proposition 1.102.

The the index of this subspace is equal to s. This is because if j+ k = ni + 1 andj 6= k, then the subspace generated by vi

j,vik,w

ij,w

ik is orthogonal to the other

elements of the basis and its matrix representation of g is0 1 0 01 0 0 00 0 0 −10 0 −1 0

,

which clearly has index 2. If j+ k = ni +1 and j = k, then the subspace generatedby vi

j,wij has index 1.

Summing up, in the Lorentzian case we can find one (and only one) Jordanbox corresponding either to a real eigenvalue of order at most 3 or to a complexeigenvalue of order at most 2. Actually, if there is a real Jordan box of order 2, theassociated basis u,v of Proposition 1.101 yield a matrix for the metric g(

0 ε

ε 0

).

Considering the basis u,εv we obtain easily the first case in (ii).


If there is a real Jordan box of order 3 with a basis u,e,v such that g(u,v) =g(e,e) = ε and all the other products zero, then ε = 1, otherwise 〈u,e,v〉R wouldhave index 2. Reordering the basis as e,u,v we obtain immediately the second casein (ii).

If there is a complex Jordan box of order 2, then we obtain easily the case (i).

6. Appendix: Special Relativity

6.1. Physics foundations. The fundamental fact because Relativity Theorywas introduced is the following: light speed has to be independent on the inertialframe of reference considered to measure it. We will denote this constant speed asc (namely, approximately equal to 300.000 Km/sec). One can always assume, aswe will do in the following, that in our system of units c = 1.Theoretical argument on the constancy of the light speed. It is supported on thefollowing facts, well-known by the physicists at the end of 19th century:

• The speed of propagation of a wave depends on the medium in which thewave propagates.• This property should be satisfied for the light, which is an electromagnetic

wave. Actually, in electromagnetism, Maxwell equations allow one todeduce the light speed propagation with respect to the medium in whichit propagates.• But the medium in which the light propagates is the vacuum, which is

assumed to be “the same” for all the “inertial frames of reference”.

Experimental argument on the constancy of the light speed. Michelson-Morleyexperiment. When the speed of the light which come from the Sun is measuredin the Earth, one obtains the same value in different periods of the same year.However:

• If the light propagated along some material medium (say, the aether), thenthe Earth would move in different directions inside this medium duringone year. So, the relative velocity of the Earth and the medium wouldaffect the measured values of the speed of light. To measure this variationwas the original motivation for the experiment.• Even if the medium of propagation is the vacuum, the motion of the Earth

around the Sun is an elliptic trajectory and, thus, both bodies go away andapproach during each year. From the intuitions of Newtonian mechanics,this relative motion would be reflected in variations of the measured speedof light, but such variations do not occur in the experiment.

Implications. The constancy of the light speed implies that the “usual rule of addi-tion of velocities” is not satisfied. This entails surprising consequences for peoplewho is familiarized with the classical Newtonian ideas of “time” and “length”, asthe following experiment suggests.

The experiment consists in observing a light ray which departs from the floorand rebound off a mirror A, from the viewpoint of two “inertial frames” O and O′.These frames move at a constant speed v 6= 0; the mirror moves together with theobserver O′ at a speed v with respect to O (see Figure 2).

6. APPENDIX: SPECIAL RELATIVITY 49

a′ a b

O′ O L

FIGURE 2. Light ray rebounded in a mirror, as observed fromtwo inertial frames O,O′.

We can express the elapsed time in each frame as

T ′ =2a′

c, T =

2bc,

where b =

√(L2

)2+a2, L is the distance that O measures between the departing

and the arrival points, and c is the light speed. Because of the natural symmetriesbetween left and right sides (in the direction orthogonal to the motion), we willassume that a′ = a. Then:

T =2bc

=2c

√(L2

)2

+a2 =2c

√(vT2

)2

+

(cT ′

2

)2

since L = vT . Therefore, T 2 = (v2T 2/c2)+T ′2 and, finally,

T =T ′√1− v2

c2

.

Let us abstract the situation, O′ observes a first event and, after a time T ′, a secondevent, which happens at the same point of the space for O′ as the first event. Then,any other inertial frame of reference O which moves at constant speed v with re-spect to O′ must measure that, between the two events, the elapsed time is greater,concretely T = (1−v2/c2)−1/2T ′. This is called the “time dilation” (due to relativevelocity), and it is observed from the inertial frame O, that is, a frame where theevents happen in two different points of the space.

Moreover, we can wonder about the distance L′ measured from O′ between thepoints in which O observes the departure and arrival of the ray. These two pointscan be regarded fixed at a distance L for O. However, for O′, these points are inmotion, and therefore, O′ observes how the first point passes with the speed −vand, a time T ′ afterwards, the second one passes with the same speed. Actually,you can think that there exists a fixed rigid stick between the two points which isat rest for O. This stick moves at speed −v for O′ (notice that we assume, again bysymmetry, that if the speed of O′ with respect to O is v, the one of O with respectto O′ is −v). We will have then

L′ = vT ′ = vT

√1− v2

c2 = L

√1− v2

c2 .


Abstracting again the situation, we assume that the inertial frame O measuresthe distance L between two points at rest with respect to O (or the length of abody at rest for it). Then, any inertial frame O′ which moves at a constant speedv in the direction determined by these two points, measures a smaller distanceL′ = (1− v2/c2)1/2L between them. This is called “contraction of length”, and itis measured by the inertial frame which moves with respect to the distance that isbeing measured.

Finally, let (x1, t1) and (x′1, t′1) be the coordinates of the second event in O

and O′ respectively. We can assume that the first event has coordinates (x0, t0) =(x1, t1) = (0,0), and as for the frame O′ the two events happen in the same point ofthe space, then x′1 = 0. Therefore (x1−0)2−c2(t1−0)2 = L2−c2T 2 =−c2(T ′)2 =(0− 0)2− c2(t ′1− 0)2. This suggests that the quantity (x1− x0)

2− c2(t1− t0)2 or,more generally, (x1− x0)

2 +(y1− y0)2 +(z1− z0)

2− c2(t1− t0)2, is preserved inthe two inertial frames. This key point leads to consider Lorentzian metrics.

6.2. Mathematical model. Next, our aim is to introduce a mathematical modelfor previous physical intuitions. We will check a posteriori that the physical pre-dictions above are obtained. About the essential uniqueness of the model a priori(in both, Special and General Relativity), see [3].

6.2.1. Physical postulates, or mathematical definitions.

DEFINITION 1.104. A spacetime in absence of gravity or spacetime in SpecialRelativity is a time-oriented affine Lorentzian space, (A,V,g), of dimension 4.

A is the set of points or ‘events’ of the affine space, V is the associated vectorspace, and g, a Lorentzian metric on V , with a time-orientation implicitly assumed.

In the remainder of the current appendix, a spacetime will always be a space-time in Special Relativity (A,V,g).

DEFINITION 1.105. An instantaneous observer is a pair (P,e), where P ∈ A,and e ∈V is a future-directed timelike unit vector.

The trajectory of a non-accelerated observer (or in free fall) is the line P+se :s ∈R, generated by any instantaneous observer (P,e).

An inertial frame of reference O is the pair formed by an instantaneous ob-server (P,e) and an orthonormal basis10 e1,e2,e3 of e⊥.

For short, “the observer O” will mean the trajectory of the corresponding non-accelerated observer generated by (P,e) in the inertial frame of reference O, whenthere is no possibility of confusion. (after Definition 1.110, the observer can beassumed with unit future-directed parametrization).

DEFINITION 1.106. A lightlike trajectory is a line of the type L = P+ su :s ∈R, where u is a (future-directed) lightlike vector, and P ∈ A.

As in any affine space, if a point P0 ∈ A is chosen, then there is a naturalidentification between A and V , namely:

A −→ VQ 7→ −−→

P0Q.

For simplicity, we will assume that such a point P0 has been chosen from now on.

10That is, O is an orthonormal affine basis of (A,V,g)) such that its timelike vector is future-pointing.


Let O and O′ be two inertial frames of reference and assume for the sake ofsimplicity that the origin point of their corresponding instantaneous observer isequal to P0, and then both observers identify in the same way A and V . Given thevector v of V , v = x1e1 +x2e2 +x3e3 + te (or the point P0 +v ∈ A), we will say that(x1,x2,x3, t) are the coordinates with respect to O of v (or indistinctly of the pointP0 + v). Analogously, ((x′)1,(x′)2,(x′)3, t ′) denote the coordinates with respect toO′ of the vector (x′)1e′1+(x′)2e′2+(x′)3e′3+ t ′e′ ∈V (or the corresponding point inA). With this convention, the following definition has an unambiguous meaning.

DEFINITION 1.107. The rest space of the inertial frame of reference O in theinstant t0 is the affine hyperplane of A with equation t ≡ t0 in the coordinates intro-duced by O.

We define now the trajectory that O measures of O′. For this, we assumethat physically O reparametrizes with its timelike coordinate the non-acceleratedobserver se′ |s ∈R. More precisely, by taking the coordinates of e′ in O we have

e′ = X1e1 +X2e2 +X3e3 +Te = T(

X1

Te1 +

X2

Te2 +

X3

Te3 + e

).

The trajectory that O measures of the observer O′ is the curve in 〈e1,e2,e3〉R =e⊥(⊂V ):

t 7→ t(

X1

Te1 +

X2

Te2 +

X3

Te3

),

and the three-velocity that O measures of the observer O′ is its derivative

→v=

X1

Te1 +

X2

Te2 +

X3

Te3.

As e′ is future timelike and unit, it follows that −1 = ∑3i=1(X

i)2− T 2, and thenT ≥ 1. Therefore

| →v |2 =3

∑i=1

(X i

T

)2

= 1− 1T 2 < 1.

Furthermore, observe that by writing→v= vxe1 + vye2 + vze3 and v2 = | →v |2, we

obtain in the coordinates of O,

e′ ≡ (vx√

1− v2,

vy√1− v2

,vz√

1− v2,

1√1− v2

).

EXERCISE 1.108. Give analogous definitions for the trajectory and the veloc-ity that the initial frame of reference O measures of a lightray, checking that thenorm of the velocity is equal to 1.

6.2.2. Classical Lorentz transformations. Observe that two inertial frames Oand O′ at a constant non-zero speed generate a timelike plane 〈e,e′〉R. We willassume for simplicity, that e1 and e′1 belong to this plane and, moreover e2 = e′2and e3 = e′3. The coordinates in e2 and in e3 will be omitted, as they are the samefor both observers, and we can describe the velocity

→v by means of a single real

number v. Let us see the relation between the coordinates of O and the ones of O′,assuming in addition that the induced bases B given by e1,e and B′ given by e′1,ehave the same orientation. Graphically,

O −→ BO′ −→ B′


and the transition matrix for B and B′ belongs to O+↑1 (2). So, there exists a θ ∈R

such that

M(Id,B← B′) =(

cosh(θ) sinh(θ)sinh(θ) cosh(θ)

),

and then→v=

sinh(θ)cosh(θ)

e1, v = tanh(θ) ∈]−1,1[.

Using that cosh2(θ)− sinh2(θ) = 1, 1− tanh2(θ) = 1/cosh2(θ), it follows

cosh2(θ) =1

1− v2 ⇒ cosh(θ) =1√

1− v2⇒ sinh(θ) =

v√1− v2

,

and therefore

M(Id,B← B′) =1√

1− v2

(1 vv 1

).

Finally, if (x, t) are the coordinates in O and (x′, t ′) are the coordinates in O′, wehave the relations known as the bidimensional Lorentz transformations:

x = 1√1−v2 (x

′+ vt ′)≡ 1√1−v2/c2

(x′+ vt ′)

t = 1√1−v2 (vx′+ t ′)≡ 1√

1−v2/c2( v

c2 x′+ t ′)

(in the last member of the equalities, the speed of light c is recovered). These trans-formations constrast drastically with the classical Galilean ones (in an analogousphysical situation, t = t ′,x = x′+ vt ′), which can be regarded as an approximationto the Lorentz ones when v c.

6.3. Reinterpretation of the physical phenomena.6.3.1. Time dilation. Consider two inertial frames of reference O and O′ with

the previous conventions, let S be the straight line described by the non-acceleratedobserver associated to O′, and let P be and ‘event’ along this line (see Figure 3).

O′O

L

P

S

FIGURE 3. Time dilation


The coordinates of P for O are (L,T ) and for O′ are (0,T ′). Using the Lorentztransformations they are related by

L =1√

1− v2vT ′, T =

1√1− v2

T ′.

The first equality is the time dilation that we had already announced by physicalgrounds. Observe that, for O′, both events, P and P0 (P0 was the initial crossingof the observers, described by the coordinates (0,0) of O and O′), occur in thesame point of the space. This is the reason why physicists call T ′ the proper timebetween those two events. For O, the two events happen in different places ofthe space. From the relation between T and T ′, the inertial frame O measures adilation of the proper time (measured by O′).

6.3.2. Length contraction. Consider again two inertial frames O and O′ andthe associated bases e1,e and e′1,e

′, respectively. Assume that, for the frame O,there is a rigid stick of length L at rest, such that its endpoints at the rest space t = 0,have coordinates (0,0) and (L,0) (and therefore at any other rest space t = t0, theyhave coordinates (0, t0) and (L, t0)). For the frame O′, the stick moves at a constantspeed−v. So, for O′ the extremes of the stick in its space at rest t ′ = 0 are the eventP0 (≡ (0,0)) and a new event P, in the intersection of t ′ = 0 and the straight linet 7→ (L, t) in the coordinates of O (see Figure 4).

O′O

L

P

S

e

e′

FIGURE 4. Length contraction

Thus, if the event P has coordinates (L,T ) for O, and (L′,0) for O′, they willbe related by

L =1√

1− v2L′, T =

1√1− v2

vL′.

Therefore L′ =√

1− v2 L, that is, O′ will measure a length contraction in the di-rection of the motion (with respect to the lengths “at rest” measured by O).

6.3.3. Twin paradox. This phenomenon is a direct consequence of the Lo-rentzian triangle inequality. Consider three inertial frames O, O′ and O′′ with futuretimelike vectors e,e′,e′′ respectively, which are not colinear. Assume that the affinestraight lines of the three observers meet at the events P,Q,R ∈ A (see Figure 5).


Firs

t Tw

in B

roth

er

Seco

nd Tw

in

ee′

P

R

Qe′′

FIGURE 5. Twin paradox

We know by Proposition 1.39 that

|PR|> |PQ|+ |QR|.Now, let us discuss the following physical situation. Two twin brothers meet

first at the event P, and then their trajectories split up until they meet again at theevent R. The first one do it following an affine straight line which is identifiedwith the non-accelerated observer associated to O. The second one passes by Q,being identified until that event with the non-accelerated observer O′ and, fromthat moment, with O′′. The first twin measures the proper time |PR| between thetwo meeting points. Nevertheless, the second twin measures the addition of propertimes |PQ|+ |QR|, which is strictly smaller.

Summing up, comparing the measures taken by each one of the twins, the timeelapsed between the two events P,R for the first twin is bigger than for the secondone. This still holds for their “biological clocks”: the first twin has aged more thanthe second one.

From this viewpoint, the phenomenon is not paradoxical but perfectly legiti-mate (actually, there are a lot of experimental evidences of this fact). The apparentparadox comes from thinking that the physical situation of the two twins is essen-tially interchangeable (that is, in the “frame of reference” of the second twin, thefirst twin “goes away and comes back” so, this twin should age less).

As we will see later (Remark 3.10), the extreme idealization that means thenon-differentiable trajectory at the point Q does not play any relevant role.

6.3.4. The Relativistic law for velocity addition. Let O1,O2,O3 be three iner-tial frames of reference, and B1,B2,B3 their corresponding orthonormal bases (all


of them with the same orientation). We denote by vi j the velocity that Oi measuresof O j, for i, j = 1,2,3. Now, we wonder: known v12 and v23, which is the value ofv13?.

We know that vi j = tanh(θi j), where

M(Id,Bi← B j) =

(cosh(θi j) sinh(θi j)sinh(θi j) cosh(θi j)

).

Therefore,

M(Id,B1← B3) = M(Id,B1← B2) ·M(Id,B2← B3)

=

(cosh(θ12 +θ23) sinh(θ12 +θ23)sinh(θ12 +θ23) cosh(θ12 +θ23)

)(the last equality is obtained by a straightforward computation). Then θ13 = θ12 +θ23 and

v13 = tanh(θ13) = tanh(θ12 +θ23) =tanh(θ12)+ tanh(θ23)

1+ tanh(θ12) tanh(θ13)=

v12 + v23

1+ v12v23,

which solves the searched relation between relative velocities.

EXERCISE 1.109. (1) Show using the previous equality that |v13|< 1.(2) ∗ : ]−1,1[× ]−1,1[−→ ]−1,1[ defined as v∗v′ = v+v′

1+vv′ endowed ]−1,1[ witha structure of commutative group.

(3) tanh :R−→]−1,1[ is an isomorphism between (R,+) and ( ]−1,1[,∗).

6.3.5. Particles and Relativistic Dynamics. In Newtonian mechanics, inertialobservers assign a kinetic energy E = mv2/2 and a linear momentum ~p = m

→v to

every particle of mass m. However, such a linear momentum of three componentsdoes not seem to make any intrinsic sense in a spacetime of four dimensions. Onthe other hand, consider two inertial frames O and O′, and recall that the coordi-nates of e′ for O are:(

vx√1− v2

,vy√

1− v2,

vz√1− v2

,1√

1− v2

),

where→v= vxe1+vye2+vze3 is the three-velocity. If one thinks in e′ as the velocity

of the trajectory of a particle of mass m, it seems clear that the three first com-ponents of me′ must be related with the classical linear momentum of the particlewith respect to O. What is more, for the last component, the Taylor expansion withrespect to v is:

1√1− v2

m = (1+12

v2 + . . .)m(≡ mc2 +

12

mv2 + . . .

).

This suggests that, up to a constant mc2, the first component of me′ must play arole analogous to the classical kinetic energy measured by O. This suggests thefollowing postulates:

(1) We assign (experimentally) a constant m ≥ 0 to every particle, which iscalled mass at rest.

(2) If m > 0, the physical particle is called material. In absence of any otherforce, its trajectory in the physical spacetime is identified with a non-accelerated observer p+ se′ : s ∈ R and the future-directed timelikevector me′ is defined as its energy–momentum.


(3) If m = 0, the physical particle is called a lightray, and its trajectory in thespacetime is identified with a lightlike trajectory P+ su : s ∈R.

Summing up, we introduce the following mathematical definitions:

DEFINITION 1.110. A particle of mass m > 0 is a future-directed timelikecurve ρ : I ⊆ R → A, parametrized with the normalization g(ρ ′,ρ ′) = −m2(≡−mc2).

The velocity ρ ′(s) will be called energy-momentum in the instant s ∈ I.When the particle ρ is an affine straight line, we say that it is non-accelerated

(or in free-fall), and its energy-momentum is a (constant) vector of V .

As in the case of General Relativity, an observer can be defined as a materialparticle with unit normalization. So, a material particle can be also regarded asan observer to whom a mass m has been assigned, in such a way that its energymomentum is the product of m times the velocity.

DEFINITION 1.111. A light ray is any affinely parametrized straight line ρ :I ⊆R→ A with future-directed lightlike velocity ρ ′.

This velocity (identifiable at every instant s ∈ I with a unique lightlike vectoru of V ) is its energy–momentum.

Recall that any inertial frame produces a split of the associated vector space:

(20) V = 〈e〉R⊕ e⊥.

Thus, if we have a particle ρ , its energy-momentum at every instant s0 ∈ I can bewritten as

(21) ρ′(s0) = Ee+~P, where E =−g(e,ρ ′(s0))> 0, ~P ∈ e⊥.

We call E the energy of ρ measured by O, ~P the momentum and que quotient~p = ~P/E ∈ e⊥ the three-momentum. If ~v is the three-velocity that O measures ofρ ′(s0)/m, we have

(22) E = m/√

1− v2/c2(≡ mc2/√

1− v2/c2), ~P = E~v, ~p = m~v.

It is worth pointing out that, in order to obtain the decompositions (20), (21),it is enough to consider an instantaneous observer (P0 = ρ(s0),e). Therefore, itmakes sense to define the energy, momentum and three-momentum as in (22) withrespect to an instantaneous observer, without making any reference to an inertialobserver. Physically, the decomposition can be made by any instantaneous ob-server (P0,e) just in the “tangent space” to P0, and therefore it has a physical mean-ing only under the restriction ρ(s0) = P0. Actually, this is what happens in thewider setup of General Relativity. Nevertheless, in Special Relativity the affinestructure of the spacetime allows one to make the mathematical decomposition inthe associated vector space, for all the events.

Let us digress about the existence of physical particles that are able to travelfaster than light, and as a consequence about the properness of our definition ofparticle. The Relativistic kinematics, and especially the Relativistic law for ve-locity addition, suggest that speed of light cannot be reached. However, there isno contradiction in assuming the existence of particles that have always travelledfaster than light moving through spacelike affine straight lines (or more generalspacelike curves). Nevertheless, such particles (called tachyons) have never been


detected. However, we can glimpse the following contradiction for the RelativisticDynamics (which studies the motion in relation with its causes), if one assumesthat a particle is initially at rest for an inertial frame, and we can accelerate it untilreaching the speed of light. We have seen that, given a material particle of mass mmoving with a constant velocity v with respect to an inertial frame O, the energythat O measures of the particle is E = mc2/

√1− v2/c2. This suggests that if we

tried to accelerate the particle until reaching the speed of light, we would need aninfinite amount of energy.

Finally, we remark that the definition of energy provided above implies thateven a material particle at rest must have a minimum of energy E = mc2; it is verywell-known that this relation has been checked experimentally.

CHAPTER 2

General theory of semi-Riemannian manifolds

Throughout the current chapter, M will denote a differentiable manifold of fi-nite dimension n. Unless the contrary is explicitly said, the defined objects will besmooth, and this term will mean (both for the manifold M and for any function,tensor field, etc.) C∞-differentiable. We will assume that M is Hausdorff and para-compact (recall that, if M is connected, the last property is equivalent to the secondaxiom of countability).

1. Similarities between Riemannian and semi-Riemannian geometries

1.1. Concept.

DEFINITION 2.1. A semi-Riemannian metric g on a manifold M is a tensorfield of scalar products (2-covariant symmetric tensors nondegenerate at everypoint) of constant index ν ∈ 0,1, . . . ,n.

The pair (M,g) will be called a semi-Riemannian manifold. If ν = 0 we willsay that (M,g) is Riemannian, and if ν 6= 0,n, that is indefinite. In particular, ifν = 1,n≥ 2, (M,g) is Lorentzian.

In other words, g assigns to every point p ∈M, a scalar product gp defined onthe tangent space TpM, with constant index νp = ν .

REMARK 2.2. We would like to point out several facts:

(1) The requirement that the index ν is chosen constant must be taken intoaccount only when M is not connected (it is not difficult to show that, asg is nondegenerate at every point, the index must be locally constant).

(2) It is well-known that, on a differentiable manifold, the topological con-dition of paracompactness is equivalent to the existence of a Riemannianmetric, and this property is equivalent to metrizability. It can be shownthat the existence of a semi-Riemannian metric implies paracompactness(see [61, pg. 8-52] and [34]). However, there exist topological restrictionson the indices of semi-Riemannian metrics that a differentiable manifoldadmits, as we will see explicitly in the Lorentzian case.

All the properties that we have studied on scalar products and tensors can beapplied to every tangent space (TpM,gp). Let X(M) denote, respectively, the setof vector fields and Λ1(M) the set of differential (one-)forms on M, both with theirnatural operations. Applying the flat [ and sharp ] isomorphisms to each tangentspace, any X ∈X(M) yields a unique differential form X [ ∈ Λ1(M) which satisfiesX [(Y ) = g(X ,Y ) for every Y ∈ X(M). Conversely, any ω ∈ Λ1(M) determinesunivocally the vector field ω] ∈ X(M) characterized by ω](Y ) = g(X ,Y ), for allY ∈ X(M).

59

60 2. SEMI-RIEMANNIAN GEOMETRY

We will denote by Rnν , the set Rn endowed with the usual semi-Riemannian

metric gν of index ν , that is:

gν =n

∑i=1

εi(dxi)2,

being εi = 1 for i = 1, . . . ,n−ν and εi =−1 for i = n−ν +1, . . . ,n.

1.2. The Levi-Civita connection. As in the Riemannian case, every semi-Riemannian metric has an affine connection canonically associated. Therefore, inevery semi-Riemannian manifold, we can derive vector fields and from them, anytype of tensor fields.

THEOREM 2.3. Let (M,g) be a semi-Riemannian manifold. Then there existsa unique affine connection ∇(≡ ∇g) that satisfies:

(i) ∇ is symmetric,

∇XY −∇Y X = [X ,Y ], ∀X ,Y ∈ X(M).

(ii) ∇ parallelizes g, that is,

X(g(Y,Z)) = g(∇XY,Z)+g(Y,∇X Z), ∀X ,Y,Z ∈ X(M).

The so-defined affine connection ∇ is called the Levi-Civita connection of the met-ric g.

Idea of the proof. The proof of this result is completely analogous to the Funda-mental Lemma of Riemannian Geometry, that is:

• Check that, if the affine connection exists, then it must satisfy the Koszulformula:

(23) 2g(∇VW,X) =V (g(W,X))+W (g(X ,V ))−X(g(V,W ))

−g(V, [W,X ])+g(W, [X ,V ])+g(X , [V,W ]),

for every X ,V,W ∈ X(M). This formula relates the differential form(∇VW )[ (first member) with an expression that depends on g and on theLie bracket (second member). As a consequence, assuming the existenceof ∇, it is univocally determined by (23).• Check that the Koszul formula allows one to define an affine connection

and this affine connetion satisfies the conditions (i) and (ii) of the state-ment of the theorem.

In particular, if (U,ϕ =(x1, . . . ,xn)) is a coordinate neighborhood and ∂i(≡ ∂/(∂xi))denotes the i−th coordinate vector field, Koszul formula allows one to computeimmediately the Christoffel symbols ∇∂i∂ j of ∇. That is, if

∇∂i∂ j =n

∑k=1

Γki j∂k,

then

Γki j =

12

n

∑l=1

gkl(∂ig jl +∂ jgli−∂lgi j).

Of course for Rnν in the usual coordinates, ∇∂i∂ j ≡ 0.

1. RIEMANNIAN AND SEMI-RIEMANNIAN GEOMETRIES 61

1.3. Elements associated to the affine connection. As every affine connec-tion, the Levi-Civita one has the following associated elements:

• Covariant derivative of arbitrary tensor vector fields, also denoted by us-ing ∇.• Covariant derivative D/dt of vector and tensor fields on piecewise smooth

curves on M, γ : I ⊂R→M (I interval).• Parallel transport of tensors fields along such a curve γ . The parallel trans-

port of vector fields (and, then, for all tensor fields) becomes an isometryfor the Levi-Civita connection.• Geodesics, that is, curves whose velocity vector field is parallel. For any

affine connection, if the curve γ is an inextendible geodesic, then it isunivocally determined by its velocity at a point γ ′(t0) ∈ Tγ(t0)M. For theLevi-Civita one, additionally we have that g(γ ′(t),γ ′(t)) is a constant.This allows one to speak about spacelike, timelike or lightlike geodesicsdepending on the causal character of γ ′(t0).

As we will see later, none of the conclusions of Hopf-Rinow theorem for Rie-mannian manifolds holds for Levi-Civita connections in the indefinite case. Forexample, there are compact Lorentzian manifolds which are incomplete and non-geodesically connected. However, one can define the exponential map exp forevery Levi-Civita connection, as for every affine connection. Namely, the expo-nential at p ∈M is:

expp : Wp ⊆ TpM→M, expp(v) = γv(1),

where γv is the unique inextendible geodesic with velocity equal to v at 0 (γ ′v(0) =v), and Wp is a starshaped neighborhood of the origin 0 ∈ TpM such that γv isdefined in 1 for any v ∈Wp; we will assume that the neighborhood Wp is maximaland, therefore, unique. Then, the exponential map on M is:

exp : W ⊆ T M→M, exp(v) = expp(v), ∀v ∈ TpM∩W,∀p ∈M,

where W = ∪p∈MWp. In the remainder, we will drop the subindex p of Wp, as thethe confusion between W and Wp =W ∩TpM is harmless.

The following remarks about this map are in order:(1) As for every affine connection, the differential map

(d expp)0 : T0(TpM)→ TpM

turns out to be equal, with the usual identifications, to the identity inTpM. Therefore, one can find a starshaped neighborhood of the originU ⊂ TpM such that the restriction of expp to U is a diffeomorphism onits image U := expp(U ). Chosen a basis Bp of TpM, we obtain a chart(U ,ψ) which maps any q ∈ U into the coordinates of exp−p 1(q) in thebasis Bp. Such (U ,ψ) is called a normal neighborhood1 of p. Then, innormal coordinates at p,

(24) Γki j(p) = 0, ∀i, j,k ∈ 1, . . . ,n.

For a semi-Riemannian metric, the basis Bp is always assumed orthonor-mal for gp and, so,

(25) gi j(p) = εiδi j

1In some books, U is not assumed starshaped, as in [17].


with εi = ±1. Of course, the identities (24) and (25) hold in principleonly at p (the curvature of ∇ will prevent its extendibility).

(2) For every p ∈M, it is possible to show the existence of a convex neigh-borhood C (see [43, pg. 29]), that is, a normal neighborhood of everyoneof its points. For every two points p,q ∈ C there is a unique geodesicγpq : [0,1]→ C that joins them and it is contained in C . Moreover, themap

C ×C → T M, (p,q)→ γ′pq(0),

is smooth.(3) For any affine connection, it makes sense to speak of critical points of

the map expp and therefore, to define conjugate points along a geodesic.However, the properties of the conjugate points in the semi-Riemannianindefinite case can be dramatically distinct from the Riemannian one; forexample, the conjugate points along a geodesic do not have to be isolated(see [29, 46, 47]). Of course, it does not make sense in general to speakabout a cut point, that is, the point where the geodesic in a Riemannianmanifold does not minimize anymore for the associated distance. As wewill see later, essentially only in the Lorentzian case and for timelike orlightlike geodesics, similar properties to those of Riemannian geodesicshold (something similar to the cut locus can be seen in [2, Chapter 9]).There even exists a Morse theory developed initially by K. Uhlenbeck(see [64] and also [2, Chapter 10]). The peculiarities of the lightlike casecan be studied for example in [38, Section 2].

(4) Gauss Lemma holds, and its proof can be checked in a similar way to theRiemannian case [43, p. 117, Lemma 1]: if p ∈ M, 0 6= x ∈ TpM andvx,wx ∈ Tx(TpM)) with vx parallel to x, then

gp((d expp)x(vx),(d expp)x(wx)) = 〈vx,wx〉,where 〈·, ·〉 is the scalar product in Tx(TpM) naturally determined by gp.

1.4. The curvature tensor. Any affine connection has associated a curvature,which is a 1-contravariant 3-covariant tensor field defined on vector fields

R(X ,Y )Z = ∇X ∇Y Z−∇Y ∇X Z−∇[X ,Y ]Z, ∀X ,Y,Z ∈ X(M),

and, then, inducible on tangent vectors (and linear forms) on each tangent space.

REMARK 2.4. Recall that there is no universally accepted convention for thecurvature tensor, and some authors define it with the opposite sign to ours. How-ever, our convention is universally accepted for both, the sectional curvature andthe Ricci tensor.

If, as we will assume from now on, ∇ is the Levi-Civita connection of the semi-Riemannian metric g, from this tensor we can construct the 4-covariant curvaturetensor:

R(X ,Y,Z,W ) := g(R(X ,Y )Z,W ).

These tensors present the following classical symmetries, which hold not only inthe Riemmanian case but in the general one:

(i) Symmetry in the two first variables: R(X ,Y,Z,W ) = −R(Y,X ,Z,W ) (or, forshort: R(X ,Y ) =−R(Y,X)).

(ii) Symmetry in the two last ones: R(X ,Y,Z,W ) =−R(X ,Y,W,Z).


(iii) Symmetry of the first pair of variables with the two second one:

R(X ,Y,Z,W ) = R(Z,W,X ,Y ).

(iv) First Bianchi identity: R(X ,Y )Z +R(Z,X)Y +R(Y,Z)X = 0.(v) Second Bianchi identity: (∇ZR)(X ,Y )+(∇Y R)(Z,X)+(∇X R)(Y,Z) = 0.

As in the Riemannian case, the unique non-necessarily null contractions of thecurvature tensor are, up to a sign the Ricci tensor, Ric, and the scalar curvature S.The first one is the 2-covariant symmetric tensor that is obtained as the contractionwith respect to the last indices of the (1,3)-curvature tensor. This contraction doesnot require to use the metric, although it is useful to compute it:

Ric(Y,Z)(p) := traceR(·,Yp)Zp =n

∑i, j=1

gi jg(R(∂i|p,Yp)Zp,∂ j|p)

=n

∑i, j=1

gi jR(∂i|p,Yp,Zp,∂ j|p).

The scalar curvature S is the metric contraction of the Ricci tensor. In coordinates:

S(p) =n

∑i, j=1

gi j(p)Ric(∂i |p,∂ j |p).

1.5. Sectional Curvature. If g is a Riemannian metric and Π = 〈u,v〉R ⊆TpM is a plane of the tangent space to p, the sectional curvature of Π is defined bythe relation

(26) KS(Π) =R(u,v,v,u)

g(u,u)g(v,v)−g(u,v)2 ,

which is independent of the chosen basis u,v of Π (see (27) below). In the casethat g is any semi-Riemannian metric, the definition (26) remains the same, but

the sectional curvature is defined if and only if the tangent planeΠ is nondegenerate.

Obviously, this is due to the fact that if g |Π is degenerate, then the denominator of(26),

Q(u,v) = g(u,u)g(v,v)−g(u,v)2,

is zero. Actually,

Q(u,v)

> 0 ⇔ g|Π is definite (positive or negative)= 0 ⇔ g|Π is degenerate< 0 ⇔ g|Π is indefinite

For a nondegenerate plane Π, the expression (26) can be simplified if we takeu,v to be an orthonormal basis of TpQ, since then the denominator Q(u,v) turns tobe ±1. When g |Π is Euclidean, Q(u,v) coincides with the square of the area ofthe parallelogram generated by u,v and, in general, whenever that g |Π is nonde-generate, we can keep this interpretation (every semi-Riemannian metric generatesnaturally both a volume element up to a sign and a measure on M, as the Riemann-ian ones do).

As we have already observed, if Π is a degenerate plane, then Q(u,v) is zeroand the definition of sectional curvature in (26) does not make sense. However,the sign of the numerator of (26) is independent of the chosen basis, as we deduce


from the following identity, which follows immediately from the symmetries of thecurvature tensor:

(27)u′ = au+bvv′ = cu+dv

=⇒ R(u′,v′,v′,u′) = (ad−bc)2R(u,v,v,u),

for all u,v,u′,v′ ∈ TpM, p ∈M. Thus, if Π = 〈u,v〉R ⊆ TpM is a degenerate plane,we define the signature of the curvature

(28) N (Π) =

+1 if R(u,v,v,u)> 0,0 if R(u,v,v,u) = 0,−1 if R(u,v,v,u)< 0.

Summing up:

DEFINITION 2.5. Let Π = 〈u,v〉R ⊆ TpM be a tangent plane:(i) If Π is nondegenerate, we define its sectional curvature KS(Π) using (26).

(iii) If Π is degenerate, we define the signature of its curvature N (Π) using(28).

In some particular cases, however, there exists some privileged lightlike vectorfield U ∈ X(M). Then, one defines the null sectional curvature [27] with respectto a lightlike vector u ∈ TpM as

(29) Ku(π) =R(u,v,v,u)

g(v,v),

where π = 〈u,v〉R is any tangent plane in TpM which contains u such that gp|πis degenerate but not identically equal to 0. This is particularly interesting inLorentzian signature, as the plane π generated by u and any (necessarily space-like) orthogonal vector v satisfies the required properties.

As in the Riemannian case, the sectional curvature KS(Π) for every nondegen-erate plane Π tangent to p determines the value of the curvature tensor R in p in thegeneral case. It is easy to see that the proof of this fact in the Riemannian case canbe extended to the indefinite case using continuity, because every degenerate planeis limit of nondegenerate ones, both definite (positive or negative) and indefiniteones. To be more precise:

LEMMA 2.6. Let (V,g) be a vector space endowed with an indefinite scalarproduct, and let Π⊂V be a degenerate plane. Then there exist two sequences

ukk→ u, vkk→ v

with Π = 〈u,v〉R, such that the planes Πk = 〈uk,vk〉R are nondegenerate for everyk.

Moreover, the sequences can be chosen in such a way that all the planes Πkare indefinite or all of them are definite.

PROOF. Observe first that the radical of g restricted to Π must have dimension1 or 2 (in the last case, g|Π is identically null). Distinguish these two cases.

Case 1: g|Π is not identically null. As g|Π is degenerate, we can assume that vgenerates the radical in Π and g(u,u) 6= 0. Thus, for any x ∈V and δ ∈R:

Q(u,v+δx) = Q(u,v)+2δ (g(u,u)g(v,x)−g(u,v)g(u,x))+δ2Q(u,x)

(30) = 2δa[x]+δ2Q(u,x),


witha[x] = g(u,u)g(v,x)−g(u,v)g(u,x) = g(u,u)g(v,x).

As g is nondegenerate in all of V , there exists any x ∈V such that a[x]> 0. Consid-ering then a k0 ∈N big enough, the sign of (30) for δ =±1/(k+k0) is independentof the term in δ 2, for every k ∈N, and the required sequences are

(31) uk = u, vk = v± 1k+ k0

x.

Case 2: g|Π is identically null. Observe first that Π ⊂ Π⊥, and hence, thedimension n and the index ν of V satisfy (recall Proposition 1.13) 4≤ n, 2≤ ν ≤n−2. If Π = 〈u,v〉R, taking a vector y non-lightlike in v⊥ ⊂V , the plane

Π j = 〈u+1jy,v〉R

lies in the previous case (up to just at most one j ∈ N such that |u+ 1j y| = 0).

Therefore, we obtain two sequences (31) for every j:

uk j = u+1jy, vk j = v± 1

k+ k0( j)x j,

where we can even assume that k0( j) and x j are chosen in such a way that x j/k0( j)belongs to a fixed neighborhood W of 0 ∈V . The required sequences are then thediagonals

uk = ukk, vk = vkk.

PROPOSITION 2.7. The sectional curvature KS(Π) on all the nondegenerateplanes Π ⊂ TpM tangent to a fixed point p, together with the value of the semi-Riemannian metric g in this point p, determines univocally the curvature tensor Rat p.

PROOF. Let us see first that the conclusion of the theorem holds for R(u,v,v,u).It is obvious that R(u,v,v,u) is zero if u,v is linearly dependent and that

(32) R(u,v,v,u) = KS(〈u,v〉R) ·Q(u,v)

if 〈u,v〉R is a nondegenerate plane. If the plane 〈u,v〉R is degenerate, then theprevious lemma allows one to obtain R(u,v,v,u) by continuity as the limit of thesubsequence R(uk,vk,vk,uk)k, where in every plane 〈uk,vk〉R we can use (32).From this point on, the proof follows essentially the same lines as in the Riemann-ian case, that is, using the symmetries of R:

(i) The value of R(u,v)v is computable from R(u,v,v,x) for every x, and the latterfrom the known expressions of the type (32), using

R(u+ x,v,v,u+ x) = R(u,v,v,u)+R(x,v,v,x)+2R(u,v,v,x).

(ii) The values of R(v,w)u is computable from the known expresion of the previ-ous case (i) by substracting the identities

R(u+w,v)(u+w) = R(u,v)u+R(w,v)w+R(u,v)w+R(w,v)u

R(u+ v,w)(u+ v) = R(u,w)u+R(v,w)v+R(u,w)v+R(v,w)uand applying the first Bianchi identity.


As in the Riemannian case:

PROPOSITION 2.8. If the sectional curvature on every nondegenerate plane atp is a constant c(p), then

R(u,v)w = c(p)(g(w,v)u−g(w,u)v) .

Moreover, if this happens in a connected neighborhood U and the dimension n is≥ 3, then c(p) is a constant independent of p at every2 U.

In general, algebraic expressions in terms of the curvature tensor R, whichare valid both in the Riemannian case and in the general one, come into play in theresults of the type “the curvature determines locally the metric”. Thus, for example:if g and g′ are two semi-Riemannian metrics on the same connected manifold Mwhich coincide in a point p (gp = g′p) and they have the same curvature tensors inall of M (Rg = Rg′), then g = g′.

In other interpretations of the curvature, some slight algebraic details must betaken into account to go from the Riemmanian case to the general one:

(1) Assume that dim(M) = 2 and let Π be a (nondegenerate) tangent plane,c = KS(Π) 6= 0 and BΠ an orthonormal basis e1,e2 of Π. In the Riemann-ian case or, in general, if gΠ is definite, the linear map

R(e1,e2) : Π→Π, v→ R(e1,e2)v,

is the composition of a rotation and a homothety of ratio c. In fact, in theRiemannian case:

R(e1,e2)e1 =−c · e2, R(e1,e2)e2 = c · e1.

However, if Π is an indefinite plane, this linear map turns out to be ananti-isometry (isometry of (Π,g|π) in (Π,−g|π)) composed with the ho-mothety of ratio c:

R(e1,e2)e1 = c · e2, R(e1,e2)e2 = c · e1,

(this interpretation can be extended to any nondegenerate plane of a tan-gent space of arbitrary dimension by taking the exponential of this plane).

(2) In the semi-Riemannian case, as in the Riemannian one, both the Riccitensor and the scalar curvature can be expressed algebraically from thesectional curvatures. But we must take into account some precautions forthe interpretations of Ric and S as a mean of sectional curvatures. Thus,for example, if v is non-lightlike, fixing an orthonormal basis e2, . . .en ofits orthogonal space v⊥, we can write:

Ric(v,v)g(v,v)

=n

∑i=2

KS(〈ei,v〉R).

EXERCISE 2.9. If u ∈ TpM is lightlike, prove

Ric(u,u) =n

∑i=3

Ku(πi),

where u,e3, . . . ,en denotes a Sylvester basis of the degenerate hyperplane u⊥ andKu(πi) denotes the null sectional curvature of the plane πi = 〈ei,u〉R according to(29).

2The last conclusion is an immediate consequence of the first one and Theorem 2.11.

2. BOUNDS FOR THE SECTIONAL AND RICCI CURVATURES 67

A more remarkable difference between the sectional curvature in the Riemanniancase and in the indefinite one, is that the sectional curvature is hardly bounded, asshown next.

2. Bounds for the sectional and Ricci curvatures

One of the striking differences between the Riemannian and the indefinite caseis the apparent triviality related with some bounds for the sectional and Ricci cur-vatures in the latter case. For the Ricci tensor, this follows immediately from thealgebraic criterion on the bounds of bilinear forms (see Theorem 1.25). In fact,the following result is an immediate consequence of that criterion just by choosingRicp as the symmetric bilinear form:

COROLLARY 2.10. Let (M,g) be an indefinite semi-Riemannian manifold andp ∈M. The following conditions are equivalent:

(i) Ricp = c ·gp for any c ∈R,(ii) Ricp(u,u) = 0, on any lightlike vector u ∈ TpM,

(iii) ∃a > 0 : |Ric(z,z)| ≤ a · |gp(z,z)| on any timelike vector z ∈ TpM,(iv) ∃a > 0 : |Ric(z,z)| ≤ a · |gp(z,z)| on any spacelike vector z ∈ TpM,(v) ∃b ∈R : Ric(z,z)

gp(z,z)≤ b on any non-lightlike vector z ∈ TpM\0,

(vi) ∃b ∈R : Ric(z,z)gp(z,z)

≥ b on any non-lightlike vector z ∈ TpM\0.

As a consequence, if these equivalent conditions hold on all of M, then theRicci is equal to a certain function f multiplied by the metric. But, then, the fol-lowing Schur-type result implies that this function is (locally) independent of thepoint, up to the trivial case in dimension 2.

THEOREM 2.11 (Schur). Let (M,g) be a connected semi-Riemannian manifoldof dimension n≥ 3. If the Ricci tensor satisfies

Ric = f ·g for some function f ∈C∞(M),

then f is constant, that is, the manifold is Einstein:

Ric = c ·g for some constant c ∈R.

PROOF. The classical proof of the Riemannian case (see for example [31, Note3, page 292]) is still valid for the indefinite one.

The bounds for the sectional curvature come from the combination of severalfacts that we have already commented:(a) the sectional curvature is not defined for degenerate planes, but the sign of the

numerator of (26) is well-defined for degenerate planes,(b) every degenerate plane is the limit of definite and indefinite planes (Lemma

2.6), and(c) the sign of the denominator of (26) is distinct in the definite and indefinite

planes.Concretely, we have the following result:

THEOREM 2.12 (Kulkarni [33]). Let (M,g) be an indefinite semi-Riemannianmanifold and p ∈M. The following conditions are equivalent:

(i) The sectional curvature KS is constant on all the nondegenerate planes inTpM.


(ii) N (Π) is null on every degenerate plane in TpM.(iii) ∃a > 0 : |KS(Π)| ≤ a on any indefinite plane Π⊂ TpM.(iv) ∃a > 0 : |KS(Π)| ≤ a on any definite plane Π⊂ TpM.(v) ∃b ∈R : b≤ KS(Π) on any nondegenerate plane Π⊂ TpM.

(vi) ∃b ∈R : KS(Π)≤ b on any nondegenerate plane Π⊂ TpM.

PROOF. It is immediate that (i) implies all the other equivalences (for (i) ⇒(ii) use either the continuity of the numerator of (26) or Proposition 2.8).(ii)⇒ (i). It is an algebraic result that can be deduced in three steps:

(a) If x,u,z are three orthonormal vectors, the first two of opposite character,then the curvatures of 〈x,z〉R and 〈u,z〉R coincide. (This follows from

R(x+u,z,z,x+u) = 0, R(x−u,z,z,x−u) = 0,

summing the two expressions and using Q(x,z) =−Q(u,z)).(b) If g is Lorentzian (or of index ν = n− 1) and x,u,z is orthonormal with u

timelike, then KS(〈x,u〉R) = KS(〈x,z〉R) = k(u) (depends only on u). But if v isother timelike vector independent of u, then KS(〈u,v〉R) = k(u) = k(v) (hence weconclude the result for the Lorentzian case).

(c) For arbitrary index, take an orthonormal basis and reason analogously tothe Case IV in the proof of Theorem 1.25.It is easy to show that (ii) is implied by any between (iii) and (vi), using that anydegenerate plane is the limit of definite and indefinite planes (Lemma 2.6). Thus,(iii) ⇒ (ii) (and analogously (iv) ⇒ (ii)). If Π = 〈u,v〉R is a degenerate plane,take the sequences

ukk→ u, vkk→ v,

such that the planes Πk = 〈uk,vk〉R are indefinite for every k. Then we have

aQ(uk,vk)≤ R(uk,vk,vk,uk)≤−aQ(uk,vk).

Taking limits, it follows that 0≤ R(u,v,v,u)≤ 0, as desired.(v) ⇒ (ii) (and analogously (vi) ⇒ (ii)). Reasoning as in the previous case wehave that

R(uk,vk,vk,uk)≤ bQ(uk,vk),

if the planes Πk are chosen definite, and taking limits R(u,v,v,u)≤ 0. Then, choos-ing indefinite planes Π′ks, the opposite inequality is obtained.

Of course, if the sectional curvature is constant at a point, then necessarily theRicci tensor at this point is a multiple of g (use Proposition 2.8). Therefore, bycombining the results of Kulkarni and Schur, one can obtain severe restrictions tofind non-trivial bounds of the curvature. This situation constrasts dramatically withthe Riemannian case, where necessarily all the sectional curvatures for tangentplanes at any p are bounded from above and from below. In fact, the sectionalcurvature is a continuous function which is defined on all the Grassmannian oftangent planes at p in the Riemannian case, and only in a open dense subset inthe indefinite one; the former domain is compact, but the latter is not. Recallthat a classical problem in Riemannian geometry is relative to the comparison ofproperties between two manifolds whose curvatures satisfy some inequality. Theextension to the indefinite case of these types of problems presents then some nontrivial subtleties (see for example [1, 18] and references therein).

3. CONFORMAL PROPERTIES AND LIGHTLIKE PREGEODESICS 69

EXERCISE 2.13. Let (M,g) be a semi-Riemannian manifold with n ≥ 3. As-sume that it satisfies the vacuum Einstein equation

Ric− 12

S ·g+Λ ·g≡ 0,

for some cosmological constant Λ ∈R.Show: (i) (M,g) is Einstein, (ii) if Λ = 0, then Ric≡ 0.

Finally, we emphasize some simple but striking algebraic differences in the indef-inite case. In the Riemannian case, the vanishing of the scalar invariants whichcan be constructed from the curvature tensor R (that is, from functions constructedby using contractions of R and its covariant derivatives of any order) implies thevanishing of R. For example, the vanishing of the scalar ∑i jkl Ri jklRi jkl is enough.This property does not hold in the indefinite case. A (local) classification of all theLorentzian manifolds having zero all the scalar invariants of the curvature has beenobtained recently in [49]. More subtly, in the Riemannian case, a celebrated result[42, 63] says that the vanishing of second (or higher) order covariant derivative ofR implies the vanishing of R, i.e.:

∇(∇R)≡ 0⇒ R≡ 0.

However, such a result does not hold in the indefinite case. In the Lorentzian case,it is easy to find counterexamples in the class of plane waves which, essentially,turn out to be the unique non-trivial local ones [58, 7].

EXERCISE 2.14. Consider R4 endowed with the metric in Cartesian coordi-nates (x1,x2,u,v):

g = dx21 +dx2

2 +2 du dv+H(x1,x2,u) du2, ∀(x1,x2,v,u) ∈R4,

where H :R3→R is a function (pp-wave spacetime). Show:(1) Ric= 0 (gravitational pp-wave) if and only if the ordinary Laplacian of H with

respect to the variables x1,x2 vanishes (i.e., ∆xH(x,u)≡ 0),(2) in the particular case H(x1,x2,u) = f (u)(x2

1 − x22) + 2g(u) x1x2, f 2 + g2 6≡ 0

(gravitational plane wave) prove that R 6≡ 0 and ∑i jkl Ri jklRi jkl = 0 (as well asany other scalar invariant of R vanishes),

(3) particularize further choosing f ,g as two polynomials of degree 1, and checkthat ∇R 6= 0 but ∇(∇R) = 0.

3. Conformal properties and lightlike pregeodesics

3.1. Isometries, homotheties, conformal transformations.

DEFINITION 2.15. Let (M,g), (N,h) be two semi-Riemannian manifolds andφ : M→ N a diffeomorphism (resp. local diffeomorphism). We will say that φ is aconformal transformation (resp. local conformal transformation) if there exists afunction Ω ∈C∞(M) such that

φ∗h = Ωg, that is, h(dφp(vp),dφp(wp)) = Ω(p)g(vp,wp), ∀p ∈M.

In particular, for a conformal transformation φ :• if Ω is a constant (Ω≡ λ ), then φ is a homothety.• if Ω ≡ 1, then φ is an isometry, and if Ω ≡ −1 we say that φ is an anti-

isometry.


We say that (M,g) and (N,h) are conformal (homothetic, isometric, anti-iso-metric) if there exists a transformation of the corresponding type between bothmanifolds. If g,g∗ are two semi-Riemannian metrics on M, we say that they areconformal (resp. homothetic, opposite) if the identity is a conformal (resp. ho-mothetic, anti-isometric) transformation between (M,g) and (M,g∗). As in theRiemannian case, general results on transformation groups imply that the set ofall the isometries (resp. homotheties, conformal transformations) of (M,g) in it-self constitutes a (finite dimensional) Lie group, with the exception of conformaltransformations in dimension < 3. The vector fields such that their local flows areisometries (resp. homotheties, conformal transformations) are called Killing (resp.homothetic-Killing, conformal-Killing).

Essentially, two semi-Riemannian isometric manifolds have the same proper-ties. From Koszul formula we deduce that two homothetic metrics have the sameLevi-Civita connection (see (33)) and, therefore, any homothetic transformation isalso affine. Thus, if we consider two homothetic metrics g and λg (λ ∈R,λ 6= 0):

• the Ricci tensor of g and λg coincide.• the sectional (in each nondegenerate tangent plane) and scalar (in each

point) curvatures for λg is equal to the corresponding one for g multipliedby the factor 1/λ .• the signature of the curvature N of degenerate planes coincides for g and

λg if λ > 0, and it has the opposite sign if λ < 0.In particular, the effect of changing the metric g by its opposite −g can be

summed up in that the spacelike vectors turn out to be timelike (and vice versa),and that the signs of N and the sectional and scalar curvatures change.

3.2. Conformal changes of metric. Theorem 1.25 produces a fundamentaldifference between the properties of the conformal metrics in the definite and in-definite cases:

COROLLARY 2.16. Let g and g∗ be two semi-Riemannian metrics on a mani-fold M, and assume that g is indefinite. Then g∗ is conformal to g if and only if thelightlike vectors for g are also lightlike for g∗.

PROOF. Apply Theorem 1.25 in each point p ∈M taking each gp as an indefi-nite scalar product and each g∗p as a symmetric bilinear form.

This result produces a surprising simplification of the conformal properties ofan indefinite metric, in strongly contrast with the situation for Riemannian met-rics. Moreover, these simplications have an outstanding physical interpretation inLorentzian signature because, as we will see in the next chapter, the lightlike vec-tors determine the causal structure of spacetimes.

On the other hand, the relation between the Levi-Civita connections for twoconformal metrics is formally equal in the Riemannian case and in the general one.Thus, if g is a semi-Riemannian metric and g∗ a conformal one,

g∗ = Ω ·g, Ω = e2u(> 0), u ∈C∞(M),

the relations between their Levi-Civita connections ∇, ∇∗ follow directly from theKoszul formula,

(33) ∇∗XY = ∇XY +X(u)Y +Y (u)X−g(X ,Y )∇u, ∀X ,Y ∈ X(M),

where ∇u = (du)[ is the g-gradient of u.

3. CONFORMAL PROPERTIES AND LIGHTLIKE PREGEODESICS 71

It is also interesting to observe that the definition of the Weyl tensor (for n≥ 4,as well as Schouten and Cotton tensors for n ≥ 3) can be done in the general casein the same way as in the Riemannian one. Moreover, Riemannian interpretationsremain true in the general case. Thus, when n ≥ 4, the Weyl tensor is identicallyequal to 0 if and only if the metric is locally conformally flat (i.e., locally conformalto a semi-Euclidean space).

3.3. Invariance of lightlike pregeodesics. We have seen that if two indefinitemetrics have the same lightlike vectors, then they are conformal. The converse istrivially true, but we will prove a stronger implication: if two metrics are confor-mal, then their lightlike geodesics coincide up to reparametrization. To this end,we will prove before that if the acceleration of a curve γ is pointwise proportionalto its velocity, then it is a pregeodesic, that is, it admits a reparametrization as ge-odesic (from now on, we will always assume, without loss of generality, that thereparametrizations are increasing).

LEMMA 2.17. Let (M,g) be a semi-Riemannian manifold, and let γ : I → M(I ⊆R interval) be a curve whose velocity is not zero in any point. If γ satisfies

Dγ ′

dt= f (t)γ ′(t), ∀t ∈ I,

for some function f : I→R, we define the reparametrization

(34) γ(s) = γ(t(s)), γ : J→M,

(J ⊆R interval), where t : J→ I,s→ t(s), is any function whose inverse s(t) sat-isfies

(35) s′(t) = s′(t0)e∫ t

t0f (t)dt

,

for some t0 ∈ I,s′0 > 0. Then γ is a geodesic for g.

PROOF. From (34), it follows immediately (we denote with a dot · and a prime‘ natural derivatives, namely d/ds and d/dt, resp.):

γ′(s) = t(s)γ ′(t(s))

Dγ ′

ds= t(s)γ ′(t(s))+ t(s)2 Dγ ′

dt(t(s)) =

(t(s)+ t(s)2 f (t(s))

)γ′(t(s)).

Therefore, γ is a geodesic if and only if the reparametrization satisfies the differen-tial equation

t(s)+ t(s)2 f (t(s)) = 0.

It is immediate to check that (35) corresponds to the solutions of this equation.

THEOREM 2.18. Let g and g∗ = Ωg,Ω = e2u be two conformal semi-Riemann-ian metrics on the manifold M. Then the lightlike pregeodesics for g and g∗ coin-cide.

PROOF. It is enough to prove that, if γ is a lightlike geodesic for g, then it is alightlike pregeodesic for g∗. Using (33), we deduce the following general relationbetween the accelerations of the same curve for the two conformal metrics:

(36)D∗γ ′

dt=

Dγ ′

dt+2du(γ ′(t)) · γ ′−g(γ ′,γ ′)∇u,


where D∗/dt denotes the covariant derivative for g∗, and the acceleration Dγ ′/dt iszero since γ is a geodesic for g. As γ is lightlike, the last term of (36) is also zero,and hence we can write

D∗γ ′

dt= 2

d(u γ)

dt· γ ′.

As a consequence, it suffices to apply Lemma 2.17 with

(37) f = 2d(u γ)

dt.

Conformal changes preserve not only lightlike pregeodesic but also conjugatepoints with their multiplicity along lightlike pregeodesics (see [38, Theorem 2.36]).

This result will have a natural interpretation in Lorentzian signature for rela-tivistic spacetimes, since the trajectory of a light ray in a spacetime is precisely alightlike pregeodesic. Thus, the conformal structure (local Causality) of a space-time is characterized by the trajectories of the light rays spreading through it.

Finally, studying the maximal intervals where γ and γ are defined (that is,the reparametrization t : J → I of Lemma 2.17), we can obtain straightforwardconsequences of these results about the completeness of lightlike geodesics. Fromnow on, all the geodesics will be assumed to be inextendible (therefore, they willbe complete, by definition, if their domain is all of R).

COROLLARY 2.19. Let g and g∗ = Ωg,Ω = e2u, be two conformal indefi-nite metrics on a manifold M. Let γ be a lightlike geodesic for g and γ thereparametrization of γ as a geodesic of g∗.

(i) If Inf(Ω)> 0 and γ is complete (that is, its interval of definition is R) then γ

is complete.(ii) If Sup(Ω)<+∞ and γ is incomplete, then γ is incomplete.

In particular, if M is compact, the lightlike geodesics of g will be complete if andonly if the lightlike geodesics of g∗ are complete.

PROOF. Observe first that by (37) and (35),

dsdt

=C e2u(γ(t)) ,

for some constant C.(i) As u is also bounded from below, ds/dt ≥ ε > 0, for some ε > 0. As a

consequence, if the domain I of γ(t) is the whole R, the image J of s(t) is also thewhole R.

(ii) Analogously, we will have that 0< ds/dt <P, for some P> 0. Thus, if I =(a,b) with b<∞, then the image of s(t) cannot take the value s(t0)+P(b−t0).

More precise results in this direction can be found in [11], [51]. As far aswe know, no compact example in the literature shows that geodesic completeness(for non-lightlike geodesics) can be lost by conformal changes; in the next sectionwe will see that this question is related to the independence of the causal type ofcompleteness.

4. A TOUR ON THE DIFFERENCES WITH EXPLICIT COUNTEREXAMPLES 73

4. A tour on the differences with explicit counterexamples

Corollary 2.19 suggests possibilities which make no sense or seem strangefrom the Riemannian viewpoint, namely, the existence of semi-Riemannian mani-folds with all the geodesics of some causal character complete, but some geodesicsof a different causal character incomplete, as well as the existence of indefinitecompact geodesically incomplete manifolds. Our goal in this section is to pro-vide some explicit examples which show that these and other possibilities, far tobe strange or exceptional, become a natural ingredient of semi-Riemannian Ge-ometry. So, even though in the previous sections many algebraic definitions andelementary properties of the Riemannian case were naturally extended to the semi-Riemannian one, the global geometry for indefinite manifolds differs drasticallyfrom that of Riemannian manifolds. Such examples will be as simple as possi-ble and, for our purposes, to remain in dimension 2 (Lorentzian surfaces) will beenough.

Recall that a connected Riemannian manifold (M,g) has a distance d canoni-cally associated to g. The well-known Hopf-Rinow theorem establishes the equiv-alence between the following conditions:

(1) M is complete as a metric space endowed with the distance d (that is,each Cauchy sequence of elements of M converges in M).

(2) There exists a point p ∈M such that M is geodesically complete from p(expp is defined on all the tangent space TpM).

(3) (M,g) is geodesically complete (inextendible geodesics are defined on all(−∞,∞)).

(4) Every closed subset of M which is bounded for d is compact.Moreover, in this case (M,g) is geodesically connected. In particular, if M is com-pact, then (4) holds and, therefore, all the other conditions hold.

Some of the examples in the remainder, will show that none of the conclusionsof the Hopf-Rinow theorem (for example, the completeness or geodesic connectiv-ity for compact manifolds) stands in the indefinite case.

4.1. Independence of completeness with the causal character. In general,for an indefinite semi-Riemannian manifold, the term “completeness” refers to ge-odesic completeness exclusively, because there is no any distance canonically as-sociated to the metric. Nevertheless, as there exist three causal types of geodesics,we can distinguish between spacelike, timelike or lightlike completeness depend-ing on the causal type of geodesics required to be complete. There exist exampleswhich show a total independence between the three types of completeness, that is,they are complete in one or two arbitrary causal characters, and incomplete in theothers. Detailed references about such examples (constructed by Kundt, Gerochand Beem) can be found in [2, page 203]. It is especially simple to construct thefollowing spacelike incomplete and timelike and lightlike complete example.

EXAMPLE 2.20. Consider the Lorentzian metric g∗ on R2 which is conformalto the usual one g0 = dx2−dy2:

g∗ = e2u(x,y)(dx2−dy2),

where u :R2→R satisfies the following properties:(1) ∂yu(x,0) = 0 (for example u is symmetric with respect to the axis x:

u(x,y) = u(x,−y)).


(2) It vanishes away from a “horizontal strip”: u(x,y) = 0 if |y|> 1.(3)

∫∞

−∞eu(x,0)dx < ∞.

It is easy to prove then:

(a) The parametrization of the axis x, γ(t) = (t,0), which is a spacelike ge-odesic for g0, is also a pregeodesic for g∗. In fact, by property (1), wehave: ∇u(γ(t)) = ∂xu(t,0)γ ′(t). Then it is enough to use (36) and theLemma 2.17.

(b) The reparametrization γ(s) = γ(t(s)) is necessarily an incomplete geo-desic. To show this, it is enough to take into account that this reparametri-zation satisfies (35) with f (t) = d(uγ)

dt (t), and to use (3).(c) Any causal geodesic α(t) = (x(t),y(t)) is complete. In order to show

this, observe that y′(t) 6= 0 at every point, as α is causal. Then, thereparametrization α(y) = (x(y),y) is well defined, and |dx/dy| ≤ 1. Thus,if the reparametrization was not defined on all R, but only in an interval(y−,y+) with y+ < ∞, then the map y→ x(y) would be continuously ex-tendible to y = y+. Taking into account that if a geodesic has a limitpoint, then it is extendible through it (see for example [43, Lemma 5.8]),the domain of x(y) must be allR, that is, the function y(t) must be strictlymonotone and its image will cover all R. As a consequence, up to somevalues of t in a compact interval of R, the curve α(t) is away from thehorizontal strip |y| < 1. The result follows by observing that away fromthis horizontal strip, α(t) is also a geodesic of the complete metric g0because of condition (2).

4.2. Incomplete closed (lightlike) geodesics. ConsiderR2 endowed with themetric

g0 = du⊗dv+dv⊗du

(that is, L2 in lightlike coordinates (u,v)). Consider the isometry

φ(u,v) = (2u,v/2)

and the group that it generates G = φ k : k ∈ Z. Our final aim is to check theexistence of incomplete closed geodesics in the quotient M/G of M = R+×Rendowed with the restricted metric g0. But, first, the two next exercises alreadyshow other relevant differences between the Riemannian and the Lorentzian case.

Recall that if · : G×X→ X is a (left) action (recall Subsection 3.4.2 of Chapter1) of the group G on a set X , we say that it is:

(a) free if the equality g · x = x for some x implies that g is the identity of G,(b) discontinuous if for every sequence of distinct elements gmm ⊂ G, the

sequence of points gm · xm is not convergent for any x ∈ X ,(c) and properly discontinuous if:

(i) for any x,x′ that do not belong to the same orbit (x′ 6= g · x,∀g ∈ G\e), there exist neighborhoods U 3 x,U ′ 3 x′ such that gU ∩U ′ =/0,∀g ∈ G,

(ii) the isotropy group Gx of each point x (Gx = g ∈ G : g · x = x) isfinite, and

(iii) ∀x ∈ X ,∃U 3 x open: gU ∩U = /0,∀g ∈ G\Gx.


EXERCISE 2.21. Clearly, the action of G on all R2 is not free, since the origin(0,0) is a fixed point. Let R2

∗ = R2 \ (0,0) and consider the natural action by

isometries of G on R2∗ obtained by restriction. Prove:

(1) The action of G on R2∗ is free and discontinuous.

(2) The quotient R2∗/G is a non-Hausdorff manifold admitting a Lorentzian

metric g0.(3) If a group acts by isometries on a Riemannian manifold and the action is

free and discontinuous, then the action is properly discontinuous. As aconsequence, the quotient is a Riemannian (Hausdorff) manifold.

(See also [31], especially Proposition 4.4).

REMARK 2.22. Let Mn be the group of rigid motions of Rn (semidirect prod-uct of translations and rotations) and An the one of affinities (semidirect product oftraslations and linear automorphisms). As a consequence of the point (3), we havethat if G is a subgroup of Mn that acts free and discontinuously on Rn, then thequotient space Rn/G is a Hausdorff manifold. However, up to our knowledge, itis an open question to determine if the same statement is true when replacing Mnby An (see [13, page 4]). The previous points (1) and (2) show that the conclusionwould be false if the action of An acted on the subset obtained by removing a pointfrom R2.

EXERCISE 2.23. A semi-Riemannian homogeneous space is a semi-Riemann-ian manifold (M,g) such that its group of isometries acts transitively on M (recallalso Subsection 3.4.2 of Chapter 1). Prove:

(1) (M = R+×R,g0) is an incomplete homogeneous Lorentzian manifold.(Hint: both the v-translations and maps similar to φ are isometries).

(2) Every homogeneous Riemannian manifold is complete. (Hint: a Rie-mannian manifold is complete if there exists an ε > 0 such that the closedball of center any point and radius ε is compact. This property is clearlysatisfied in the homogeneous Riemannian manifolds.)

REMARK 2.24. By a theorem of J.E. Marsden (see [35]) every homogeneoussemi-Riemannian compact manifold is geodesically complete. Observe that bothhypotheses (homogeneity and compactness) are necessary in the indefinite case,whereas in the Riemannian case any of them is enough.

Naturally, G acts by isometries on M, and the quotient C = M/G is topolog-ically a cylinder, the Misner cylinder (see [39]), which is a simplification of therelativistic Taub-NUT spacetime. It can be visualized as the strip

(x,y) ∈R2 : x ∈ [1,2]with the identification

(1,y)∼ (2,y/2), ∀y ∈R.Clearly, the incomplete lightlike geodesic γ(t) = (t,0), t > 0 of M projects on acurve γ of C, which is a closed incomplete lightlike geodesic.

REMARK 2.25. Timelike or spacelike geodesics of the type t → (t,at), t > 0also project on incomplete geodesics for the cylinder. The half-geodesic obtainedby restricting the domain to t ∈ (0,1] is incomplete, but it is contained in a compactregion of the cylinder.


γ(t)(1,0) (2,0)

R+×R

a

b

c

d

a′

b′

c′

d′

C

FIGURE 1. Misner Cylinder

Taking into account this example, let us revisit the concept of closed geodesic.

DEFINITION 2.26. Let (M,g) be a semi-Riemannian manifold and let γ : I→M be a non constant (inextendible) geodesic. We will say that γ is closed if thereexists λ > 0 and a,b ∈ I, a < b such that

(38) γ′(b) = λγ

′(a).

In this case, if λ = 1, we will say that γ is periodic.

REMARK 2.27. It is easy to check that, if γ is periodic, then there exists aminimum T > 0 (the period of γ) such that γ(t) = γ(t+T ), for every t. In particularγ is complete.

PROPOSITION 2.28. Let γ be a closed geodesic of (M,g).(i) If γ is not lightlike, then it is periodic.

(ii) γ is complete if and only if it is periodic.

PROOF. (i) Observe that, as the geodesic is closed,

g(γ ′(b),γ ′(b)) = λ2g(γ ′(a),γ ′(a)),

and as it is not lightlike, the constant g(γ ′,γ ′) is distinct from 0.(ii) As we have noted, if γ is periodic (λ = 1), then it is also complete. Next,

let us see that if λ > 1, then γ is incomplete toward +∞ (an analogous reasoningshows that if λ < 1, then γ is incomplete toward −∞).

Define b1 := b and T := b−a. Since γ ′(b1) = λγ ′(a), from b1 and until b2 :=b1+T/λ , the geodesic γ is a reparametrization of γ|[a,b], in such a way that γ ′(b2)=

λγ ′(b1) = λ 2γ ′(a). Reasoning inductively, the sequence bi+1 = bi+T/λ i satisfies:

γ′(bi) = λ

iγ′(a).

So, γ cannot be extended until

limi→∞bi= b+

Tλ −1

<+∞.


4.3. Incomplete semi-Riemannian compact manifolds. Starting at Misnercylinder, it is easy to understand intuitively that an incomplete compact Lorentziansurface can be constructed. Simply, deform conveniently Misner metric outside aneighborhood of the incomplete closed geodesic, so that outside a compact region,the metric behaves as the usual Lorentzian one on a cylinder obtained as a quotientof L2 by a translation. Then, cut the ends of the cylinder and glue to obtain a torus,which retains an incomplete lightlike geodesic (and also incomplete timelike andspacelike geodesics, see Remark 2.25), see Figure 2.

Cut and Paste

Misner Cylinder TransitionRegion

TransitionRegion

UsualCylinder

UsualCylinder

by TranslationsQuotiens

by TranslationsQuotiens

γ(t)

γ(t)

FIGURE 2.

This intuitive idea can be completely formalized, producing heuristically fam-ilies of incomplete Lorentzian tori. However, we will simply check the existenceof such a family explicitly. The first example of an incomplete compact Lorentzianmanifold was the Clifton-Pohl torus, which was not published by its authors but itis already contained in [59].


Consider on R2 the metric written in usual coordinates as

g = dx⊗dy+dy⊗dx−2τ(x)dy2,

where τ :R→R is a function such that:(1) it is periodic of period 1.(2) τ(0) = 0.(3) τ ′(0) 6= 0.

Recall the the y−translations are isometries (∂y is a Killing vector field) and, bycondition (1), the unit translations in the x−coordinate are also isometries. There-fore, the metric g is induced in the quotient torus R2/Z2. By (2), the coordinateaxis y is the image of a lightlike geodesic. Computing the Christoffel symbol,

Γyyy(x,y) =

12

gyx (2∂ygxy−2∂xgyy) = τ′(x),

the reparametrization of the y-axis as a geodesic γ(t) = (0,y(t)) must satisfy

y′′(t)+ τ′(0)y′(t)2 = 0.

From (3), the solutions of this equation (essencially, y(t) = ln(t)/τ ′(0)) cannot bedefined on all R and, so, γ is incomplete.

REMARK 2.29. Integrating explicitly the equation of geodesics, it is not dif-ficult to show that the previous torus is also spacelike and timelike incomplete.Actually, this situation is general for Lorentzian tori admitting a Killing vectorfield K 6≡ 0 (as in the previous example). These tori are either complete in thethree causal senses or they are incomplete in the three causal senses. The firstpossibility occurs when the causal character of K is constant (and, then, globallyconformally flat), and the second one when it changes (for this and other prop-erties of Lorentzian tori, see [54]). Up to our knowledge, there are no explicitexamples which show the independence of completeness for compact Lorentzianmanifolds. It is appealing the conjecture that incompleteness must imply lightlikeincompleteness for compact Lorentzian surfaces (see a discussion in [50]). Carriereand Rozoy [12] have proven that this is generically true, and have also suggestedthe possibility of a counterexample.

4.4. Geodesic connectivity. As a consequence of Hopf-Rinow theorem wehave that a complete Riemannian manifold, (in particular, a compact one), is geo-desically connected. The situation is drastically different in the indefinite case. Infact, there exist Lorentzian manifolds

(a) complete and non geodesically connected,(b) complete and starshaped from a point, but non geodesically connected,(c) compact and non geodesically connected.

The frequency of such situations may be surprising. Actually, a Lorentzian space-form (that is, a complete Lorentzian manifold of constant curvature) of dimensionn ≥ 3 and positive curvature is geodesically connected if and only if it is not timeorientable (see [43, Proposition 9.19]). The classification of when (a) (and thesubtler property (b)) occurs in spaceforms was done by E. Calabi and L. Markusin [9].

EXERCISE 2.30. Prove that the simply connected spaceform of dimension 2and curvature -1 (universal anti de Sitter spacetime) is not geodesically connected.


Recall that this space can be seen as any of the following two Lorentzian sur-faces: (i) in usual coordinates (t,x):

R×]−π/2,π/2[, g1 =1

cos2(x)(−dt2 +dx2)

R2, g2 = dx2− cosh2(x)dt2,

and (ii) the universal covering of the surface in (R3,〈·, ·〉2) defined by 〈x,x〉2 =−(x1)2− (x2)2 +(x3)2 =−1, with the induced metric.

Dropping the examples of constant curvature, it is also very easy to constructan example of (c). Consider the metric on R2 of Exercise 3.4. Because of the turnof the timelike cones, no timelike or spacelike curve (geodesic or not) departingfrom (0,0) can reach the straight lines x = ±2. As a consequence, R2/4Z2 is notgeodesically connected.

4.5. Flat tori with non compact isometry groups. As we have already com-mented (Subsection 3.1), the isometry group Iso(M,g) of a Riemannian or, ingeneral, a semi-Riemannian manifold (M,g) is a Lie group. Assume that M isconnected, and consider the fiber bundle of orthonormal frames OM, that is, themanifold naturally constructed taking at every p ∈M all the orthonormal bases ofTpM for gp (OM is a subbundle of the fiber bundle of linear frames LM). If we fixan orthonormal basis Bp ⊂ TpM, then Iso(M,g) can be naturally identified with thesubmanifold

C = φ∗Bp : φ ∈ Iso(M,g) ⊂ OM

obtained as the orbit of Bp in OM. Moreover, C is always a closed subset of LM.Assume now that M is compact. If g is Riemannian, the compactness of the

group O(n) implies the compactness of OM and, so, the compactness of the closedsubset C. That is:

the isometry group of a compact Riemannian manifold3 (M,g)is compact.

Clearly, the proof of this result depends strongly on the compactness of O(n). AsO1(n) is not compact the result cannot be extended to the Lorentzian case and,again, it is very easy to find a Lorentzian counterexample. In fact, we are going toconstruct a flat Lorentzian torus with non-compact isometry group (see [16]).

The idea is the following. Let

A =

(a bc d

)be a real matrix 2×2 such that;

(i) A has integer coefficients a,b,c,d ∈R.(ii) A is diagonalizable (for example, b = c).(iii) The eigenvalues λ1,λ2 of A satisfy λ1 ·λ2 = 1,λ1 > 1.

Let e1,e2 be a basis of eigenvectors, and define a Lorentzian metric g in R2 byimposing g(ei,ei) = 0, g(e1,e2) = 1. This metric satisfies:

(1) It induces a metric g on the torus T 2 = R2/Z2, because the metric coeffi-cients in the usual coordinates are constant.

3The result holds even if M is non-connected because the compactness of M implies that thenumber of connected parts is finite.


(2) The matrix A corresponds to an isometry of (R2, g) (by (iii)), which can beinduced on (T 2,g) (by (i)).

(3) The group G of isometries generated by A has non compact closure inIso(T 2,g), and it cannot be identified with any compact subset of the fiber bundleof linear frames LT 2. In fact, Ak admits λ k

1 → ∞ as an eigenvalue.

EXAMPLE 2.31. Consider the flat Lorentzian metric on R2 given in the usualcoordinates as

g = dx21 +dx1⊗dx2 +dx2⊗dx1−dx2

2,

and induce it naturally in a metric g on the torus T 2 =R2/Z2. The matrix

A =

(1 11 2

)generates an isometry for g and (T 2,g). But the closed subgroup An : n ∈ Z ⊂Iso(T 2,h) is not compact, as desired.

Anyway, one can obtain general results about the compactness of Iso(M,g) (if(M,g) is Lorentzian and compact). In fact, the following result is proven in thepioneering paper by D’Ambra [16]:

if a compact semi-Riemannian manifold (M,g) is simply con-nected (or with finite fundamental group), analytic and Lorentzian,then Iso(M,g) is compact.

The tori constructed above show that simply connectedness is necessary. The ne-cessity of being Lorentzian can be checked by means of a counterexample, asS3×S3×S3 admits an analytic metric of index 2 with non compact isometry group.The analicity has been recently removed in the simply connected case when thereexists a timelike Killing vector field by P. Piccione and A. Zeghib (see [48]).

CHAPTER 3

Lorentzian manifolds and spacetimes

1. Existence of Lorentzian metrics and time orientability

In this section, (M,g) will denote a Lorentzian manifold.

1.1. Time orientability. Any tangent space (TpM,gp), p∈M, is a Lorentzianvector space with two timelike cones, and one can choose one of them as a timeorientation, as we did in the first chapter. However, now the question of when thischoice can be carried out in a continuous way arises naturally. The problem issimilar to that of the orientability of a differentiable manifold.

DEFINITION 3.1. A time orientation in a Lorentzian manifold (M,g) is a mapτ which assigns, to each p ∈M, a timelike cone τp ⊂ TpM and satisfies: for everyp ∈M, there exists an open neighborhood Up of p and a (differentiable) timelikevector field X on Up such that

Xq ∈ τq, ∀q ∈Up.

In this case, the assigned cone τp will be called the future cone at p, and the non-assigned one is the past cone). A Lorentzian manifold which admits a time orien-tation will be called time orientable.

In the time orientable case, if M is connected then it admits exactly two timeorientations (say, τ and−τ), as the points where the two time orientations coincideare always open and closed. The following result characterizes the time orientabil-ity in a simple way.

PROPOSITION 3.2. A Lorentzian manifold (M,g) is time orientable if and onlyif if admits a globally defined timelike vector field X ∈ X(M).

PROOF. (⇐) It is enough to choose as τp the cone that contains Xp.(⇒) Fixed a time orientation τ , consider for every p ∈M, the open neighbor-

hood Up and the timelike vector field X ∈X(Up) of Definition 3.1. Let µi : i∈Nbe a partition of unity subordinate to the open covering Up : p ∈ M of M, thatis, 0 ≤ µi ≤ 1,∑i µi ≡ 1, and the support of each function µi is contained in someopen subset Ui (= Upi) of the covering. Calling Xi to the corresponding timelikevector field on Ui, and using the convexity of the timelike cones, it is immediate tocheck that

X := ∑i

µiXi

is a globally defined timelike vector field on M.

REMARK 3.3. (1) Clearly, to change timelike cones by causal cones in thedefinition of time orientability becomes irrelevant. It is also irrelevant to assumethat X is just continuous rather than differentiable (as for ordinary orientability,continuous time orientations are also differentiable).

81

82 3. LORENTZIAN MANIFOLDS AND SPACETIMES

(2) Some other properties similar to those for orientability happen. So, if (M,g)is timelike orientable, then: (i) it admits exactly 2k time orientations, being k thenumber of connected parts of M, and (ii) any Lorentzian covering (M, g) of (M,g)will be time orientable. If (M,g) is not time orientable, then it admits a time ori-entable (Lorentzian) covering of two sheets (M, g). There exists also some simi-larity between Proposition 3.2 and a known property for the ordinary orientability,namely: M is orientable if and only if it admits a volume element (i.e., a differentialn-form which does not vanish in any point).

(3) Orientability and time orientability are completely independent, that is,there exist examples of Lorentzian manifolds of the four possible types (orientableand time orientable, orientable and non time orientable, non orientable and timeorientable, non orientable and non time orientable). Actually, observe that theconcept of orientability depends purely on the differentiable structure, and the oneof time orientability depends also on the metric. As we will see below (Theorem3.5), if an (orientable or not) manifold M admits a non time orientable Lorentzianmetric, then M admits a different metric which is time orientable (the converseis not true: if M is simply connected, then it cannot admit a non time-orientableLorentzian metric).

Identify

X2 X1 X2

X1 X2

X1

X2 X1

X2

X1

FIGURE 1. Construction of an orientable and non time orientableLorentzian surface.

EXERCISE 3.4. We consider the vector fields on R2

X1 = cos(πx)∂x + sen(πx)∂y

X2 =−sen(πx)∂x + cos(πx)∂y

and the Lorentzian metric g in that the both vectors are lightlike and g(X1,X2)≡−1(see Figure 1). Check:(1) In the usual basis of coordinate fields (∂x,∂y), the matrix of g is:(

−sen(2πx) cos(2πx)cos(2πx) sen(2πx)

).

(2) The translation T (x,y) = (x+1,y) is an isometry that reverses X1 and X2, thatis, T∗Xi =−Xi, i = 1,2. The translation T ′(x,y) = (x,y+1) is an isometry thatpreserves X1 and X2.

(3) The Lorentzian torus quotient T 2 =R2/Z2 is not time orientable.

2. LOCAL LORENTZIAN GEOMETRY 83

(4) Construct explicitly time orientable and non time orientable Lorentzian metricson the Mobius strip and the Klein bottle.

1.2. Existence of Lorentzian metrics. A very well-known classical conse-quence of the existence of partitions of unity is that every (paracompact) differ-entiable manifold M admits a Riemannian metric. Now, we ask ourselves whathappens in the Lorentzian case. If M admits a vector field X without zeros, and gRis a Riemannian metric on M, then

(39) gL = gR−2

gR(X ,X)X [⊗X [

is a Lorentzian metric. This is a consequence of: (i) gL(X ,X) =−gR(X ,X), (ii) theorthogonal distribution of X coincides for gR and gL, and (iii) if Y,Z are orthogonalto X , then gL(Y,Z) = gR(Y,Z). Moreover, as the vector field X is timelike for gL,this metric turns out to be time orientable. Moreover, we know from Proposition3.2 that if M admits a time orientable Lorentzian metric, then it admits a timelikevector field (without zeros). Such considerations plus elementary properties ofalgebraic topology, answer completely the question on the existence of Lorentzianmetrics:

THEOREM 3.5. Let M be a connected differentiable manifold. Then the fol-lowing conditions are equivalent:

(i) M admits a Lorentzian metric.(ii) M admits a time orientable Lorentzian metric.

(iii) M admits a vector field X without zeros.(iv) Either M is non compact or its Euler characteristic χ(M) is 0.

PROOF. (ii)⇒ (i). Trivial(iii)⇒ (ii). Recall the metric (39).(iv)⇔ (iii). The equivalence between these two properties is classical in Al-

gebraic Topology (see for example [65]).(i)⇒ (iv) Assume that M admits a Lorentzian metric g and it is compact. If it

is time orientable, then it admits, by Proposition 3.2, a vector field without zeros,and therefore its Euler characteristic is null.

Otherwise, its time orientable two-sheet covering (M, g) will have χ(M) = 0.But, as the covering has two sheets, χ(M) = 2χ(M), and we are done.

EXERCISE 3.6. By using elementary algebraic topology, prove:(1) The spheres of even dimension do not admit Lorentzian metrics.(2) Compact manifolds of odd dimension admit Lorentzian metrics.(3) If a compact manifold M of dimension 4 admits a Lorentzian metric, then it is

non simply connected. (Hint: By Poincare duality, the first cohomology groupof M cannot be zero).

(4) There exist compact simply connected Lorentzian manifolds of any dimensionn > 1 except n = 2,4.

2. Local Lorentzian geometry

In Chapter 2, we studied some local similarities between the geodesics in Rie-mannian and semi-Riemannian manifolds. In the Lorentzian case there are someadditional similarities which concern the local minimization properties of Rie-mannian geodesics and the local maximization ones for causal geodesics. In both


cases, the result comes from the triangle inequality (the ordinary one in the Rie-mannian case, and the reverse one in the Lorentzian case). In the present sectionwe focus on timelike geodesics; the lightlike case will be postponed to the end ofthe next section (after the study of some technicalities by means of the index form).

DEFINITION 3.7. A continuous curve α : [a,b]→ M in a manifold will becalled piecewise smooth if there exists a partition

P = a = t0 < t1 < .. . < tk−1 < tk = bof the interval [a,b] in such a way that the restriction of α to each subinterval[ti−1, ti] is smooth (C∞). In particular, α admits the left and right derivatives at eachbreak ti, i = 1, . . . ,k−1 (denoted α ′

(t−i),α ′(t+i)

as usual).A piecewise smooth curve in a Lorentzian manifold (M,g) is called timelike

(resp. causal, lightlike) if its restriction to each subinterval [ti−1, ti] is a timelike(resp., causal, lightlike) curve and each vector α ′

(t−i)

lies in the same causal coneas α ′

(t+i). In this case, if (M,g) is also time-oriented then the curve will be future

or past directed depending on the causal character of its derivative at all the points.

Piecewise smooth timelike curves are related with curves in the tangent spaceby means of the exponential map, as the following result states.

LEMMA 3.8. Let α : [0,L]→ TpM be a piecewise smooth curve included inthe domain of expp such that α (0) = 0 and the composition α = expp α : [0,L]→U ⊆M is a piecewise smooth timelike curve. Then α (]0,L]) lies in a single timelikecone of TpM.

PROOF. Consider first the smooth piece α|[0,t1]. A Taylor expansion of α tothe right of 0 shows that there exists some ε > 0 such that α|(0,ε] lies in a singletimelike cone C, namely, the timelike cone selected by α ′(0) (notice that α ′(0) istimelike and the timelike cones are open subsets). Now, let

ε0 = Supε ∈ (0,L] : α((0,ε])⊂C.We have already proven that ε0 > 0, and it is enough to check that α(ε0) ∈ C.Equivalently, consider the (piecewise smooth) function f (s) := gp(α(s), α(s)), s∈[0,L]; we know that f < 0 on (0,ε0) and it is enough to prove that f (ε0) < 0.Consider the radial position vector field P on α

(40) P(s) = γ′s(0) with γs(t) = expα(0)((1+ t)α(s)), ∀s ∈ [0,L].

By Gauss Lemma (see Chapter 2, Subsection 1.3 part (4)) in each interval [ti−1, ti]:

f ′(s) = 2gp(α′(s), α(s)) = 2g(α ′(s),P(s)).

Notice that, if s < ε0, then the last expression is negative. In fact, it varies smoothlyon each [ti−1, ti] and cannot vanish (by Gauss lemma, P(s) is timelike when s < ε0and α ′(s) is always timelike by hypothesis). It must be negative for small s > 0,because α ′(s) and α(s) lie in the same cone, and its sign cannot change at anybreak, as the causal cone of α ′(t+i ) and α ′(t−i ) are equal.

For some δ ∈ (0,ε0), the restriction of α to [ε0−δ ,ε0] is smooth. So, the resultfollows from f (ε0) = f (ε0−δ )+

∫ε0ε0−δ

f ′(s)ds < 0.

The “infinitesimal” application of the triangle inequality implies the followingresult.

2. LOCAL LORENTZIAN GEOMETRY 85

PROPOSITION 3.9. Let U be a normal neighborhood of a point p in a Lorentzmanifold (M,g) and let p′ ∈ U be any other point. Assume that there exists atimelike curve α : [0,L]→U that joins α (0) = p with α (L) = p′. Then the radialgeodesic segment σ : [0,1]→ U that joins σ (0) = p with σ (1) = p′ is, up to a(monotonic) reparametrization, the timelike curve from p to p′ contained in Uwith strictly longest length amongst all the piecewise smooth timelike curves thatjoin p with p′ and which are contained in U .

PROOF. Consider α = exp−1p α and the vector field P on α defined in (40),

both (necessarily) timelike as in previous lemma. Let us define the timelike unitvector field on α , U(s) = 1

r(s)P(s), where r(s) = |P(s)|, s ∈ (0,L]. Then,

(41) α′(s) =−g(α ′(s),U(s))U(s)+N(s),

for some spacelike N(s) ∈ Tα(s)M in the orthogonal subspace to U(s) and, so,

−g(α ′(s),α ′(s))≤ g(α ′(s),U(s))2.

By Gauss Lemma,

r(s) =√−g(P(s),P(s)) =

√−gp(α(s), α(s)),

and, so

r′(s) =− 1r(s)

gp(α(s), α ′(s)) =− 1r(s)

g(P(s),α ′(s)) =−g(U(s),α ′(s)).

By Lemma 3.8, g(U(s),α ′(s))< 0 and, using (41),

L(α) =∫ L

0|α ′|ds≤ r(L) = L(σ).

Finally, L(α) = L(σ) if and only if the equality holds in (41), that is, N = 0 andtherefore α is a reparametrization of σ .

This means that inside of a normal neighborhood, timelike geodesics maximizethe length between all timelike curves that connect its endpoints. This property iscompletely analogous to the Riemannian case. Indeed, when varying a timelikecurve with fixed endpoints, the curves that are close to the timelike one are alsotimelike, from which we can claim that “timelike geodesics locally maximize thelength”.

REMARK 3.10. The previous proposition gives a deeper understanding of thetwin paradox, which was derived in Special Relativity (see the appendix to Chap-ter 1) as a consequence of the Lorentzian reverse triangle inequality. Namely, the“infinite acceleration” which appeared from the physical viewpoint in the vertex ofthe Lorentzian triangle, does not play any role now. In fact, all Lorentz-Minkowskispace lies in a normal neighborhood and, so, in the framework of Special Relativity,timelike geodesics maximize the length among all timelike curves which connectits endpoints. Therefore, non-accelerated observers measure a bigger proper timebetween two events on its trajectory than accelerated ones. Moreover, such aneffect will happen not only in Lorentz-Minkowski spacetime, but also in normalneighborhoods of arbitrarily general Lorentzian manifolds which may model gen-eral physical spacetimes. So, the twin paradox cannot be regarded as a consequenceof the special symmetries of L4.


3. Variations of the length and focal points

Even though this section is self-contained, the reader is assumed to have someacquaintance with the calculus of variations in the Riemannian case. Our studywill be restricted to variations of length, and variations of the energy will not beconsidered. We refer to [17, 53] for a more detailed study of this case, and [2,43] for Lorentzian and semi-Riemannian cases, respectively. As in Riemanniangeometry, a basic problem in Lorentzian geometry is to study the variations of thelength functional L = L(g):

L(γ) =∫ b

a|γ ′|ds,

when considering small perturbations of a curve γ . This functional is naturallydefined on any piecewise smooth curve α : [a,b]→M (according to Definition 3.7),with smooth intervals denoted [ti−1, ti], i = 1, . . . ,k. At each break ti, the differencebetween the right and left limits of α ′ will be denoted

∆α′(ti) = α

′(t+i )−α′(t−i ).

Accordingly, any variation x : (−δ ,δ )× [a,b]→ M of α (i.e., x(0, t) = α(t) forall t ∈ [a,b]) will be piecewise smooth, in the sense that x is continuous and therestrictions x : (−δ ,δ )× [ti−1, ti]→M are differentiable for all i = 1, . . . ,k (even-tually, one can assume that some new values ti ∈ [a,b] are regarded as breaks of α

in order to regard x as piecewise smooth). We will denote by w the first variableof x, and then curves at constant w are longitudinal, and by t the second variable,so constant t yields transversal curves. The partial derivatives of x will be denotedxw and xt . In particular, the variation vector field V (t) = xw(0, t) is also piece-wise smooth. Recall that, given a piecewise smooth vector field V along a curveα , there always exists a variation associated to V , for example, we can choosex(w, t) = expα(t)(wV (t)) (for w ∈ (δ ,δ ) and δ > 0 small enough).

3.1. First Variation. In the following, V (t) = xw(0, t) will denote the varia-tion vector field associated to a variation x(w, t) of some α as above. The prime ′

will denote the natural derivative, either usual or covariant. For example V ′(t0) =DVdt (t0) and, as V is not necessarily differentiable at the breaks t1, . . . , tk−1,

∆V ′(ti) :=V ′(t+i )−V ′(t−i ).

Analogously xtt ,xwt ,xtw,xww denote natural second derivatives such as xtw =Dxtdw . It is easy to check the Schwartz rule xtw = xwt , which comes from the sym-metry of the Levi-Civita connection. Nevertheless, if a new covariant derivativeis done, the curvature appears as a measure of the loss of conmutativity of thecovariant derivatives and, so, it is not hard to check:

(42) xwwt = xwtw +R(xt ,xw)xw.

We also denote by L′x(0) and L′′x (0), respectively, the derivatives ∂

∂w L(x(w, ·))|w=0

and ∂ 2

∂w2 L(x(w, ·))|w=0. We will consider a curve α of constant speed c = |α ′|defining its sign ε as the sign of g(α ′,α ′).

THEOREM 3.11. Let α : [a,b]→M be a piecewise smooth curve of constantspeed c = |α ′| > 0 and sign ε . For any variation x of variational vector field V

3. VARIATIONS OF THE LENGTH AND FOCAL POINTS 87

along α:

(43) L′x(0) =ε

c

∫ b

ag(α ′,V ′)ds

=−ε

c

(∫ b

ag(α ′′,V )ds+

k−1

∑i=1

g(∆α′(ti),V (ti))−g(α ′,V )|ba

).

PROOF. As c > 0, if δ is small enough, g(xt ,xt) 6= 0. Then

L′x(w) =∫ b

a

∂

∂w

√εg(xt ,xt)dt = ε

∫ b

a

g(xt ,xtw)

|xt |dt.

Using xtw = xwt and evaluating last equation at w = 0, the first equality in (43) isobtained. For the second one, it is enough to use g(α ′,V ′) = d

dt g(α ′,V )−g(α ′′,V )in the first equality and to integrate.

This formula will be applied to both cases, when the variation has fixed end-points and when one of the endpoints is allowed to vary in a submanifold. Fromnow on we will say that a piecewise smooth curve is a broken geodesic if its re-striction to each interval [ti−1, ti] is a geodesic.

COROLLARY 3.12. A piecewise smooth curve α of constant speed c > 0 is an(unbroken) geodesic if and only if the first variation of the arclength functional iszero for every variation x of α with fixed endpoints.

PROOF. The implication to the right follows immediately from (43). For theconverse, if tk−1 < s < tk, assume that α ′′(s) 6= 0 and choose y ∈ Tα(s)M such thatg(α ′′(s),y)> 0 (this is possible because g is nondegenerate). Let Y be the paralleltranslation of y along α and choose η > 0 such that g(α ′′(t),Y (t)) > 0 for everyt ∈ (s−η ,s+η). Finally consider the variation vector field V (t) = f (t)Y (t), wheref : [a,b]→R is a nonnegative function equal to zero out of (s−η ,s+η) and withf (s)> 0. Then using (43) we deduce that

L′x(0) =−ε

c

∫ s+η

s−η

f (t)g(α ′′(t),Y (t))dt 6= 0,

which is a contradiction. Therefore, α is a broken geodesic. To check that, infact, it is smooth, assume that ∆α ′(ti) 6= 0 and repeat the previous process to obtaina variation vector field V (t) such that g(∆α ′(ti),V (ti)) 6= 0 and it is zero out of(ti−1, ti+1). Then (43) gives a contradiction: 0 = L′x(0) = g(∆α ′(ti),V (ti)) 6= 0.

Now, assume that P is a nondegenerate (embedded) submanifold in M. Fixthe interval [a,b] and consider the space Ω(P,q) of all piecewise smooth curvesα : [a,b]→M from P to a point q ∈M.

COROLLARY 3.13. A piecewise smooth curve α of constant speed c is an (un-broken) geodesic orthogonal to P if and only if the first variation of the arclengthfunctional is zero for every variation x of α by means of curves in Ω(P,q).

PROOF. The implication to the right follows immediately from (43), takinginto account that V (a) must be tangent to P. For the converse, the proof of Corol-lary 3.12 also implies that α must be a geodesic. To see that it is orthogonal to P,choose y ∈ Tα(a)P and a variation vector field V such that V (a) = y. Then formula(43) implies 0 = L′x(0) =

ε

c g(α ′(a),y).


3.2. Second Variation. Recall that if σ is a geodesic and ( ·)⊥ denotes theorthogonal part to σ ′, it is easy to see that (V ′)⊥ = (V⊥)′.

THEOREM 3.14. If σ is a geodesic segment of speed c > 0 and sign ε and x avariation of σ , then

(44) L′′x (0) =ε

c

(∫ b

a

(g((V ′)⊥,(V ′)⊥)−g(R(σ ′,V )V,σ ′)

)ds+g(σ ′,A)|ba

),

where A(t) = xww(0, t)

PROOF. Let h(w, t) = |xt(w, t)|. As each curve of the variation has length equalto∫ b

a h(w, t)dt and

hw ≡∂h∂w

=εg(xtw,xt)

h,

a direct computation shows that

L′′x (0) =∫ b

ahww(0, t)dt =

ε

h

∫ b

a

[g(xtw,xtw)+g(xt ,xtww)−

ε

h2 g(xt ,xtw)2]

dt.

Now, using the symmetries xwt = xtw, xtww = xwtw plus (42), we obtain

L′′x (0) =ε

c

∫ b

a

[g(V ′,V ′)−g(R(σ ′,V )V,σ ′)+g(σ ′,A′)− ε

c2 g(σ ′,V ′)2]

dt.

Observing finally that g(σ ′,A′) = ddt g(σ

′,A) (because σ is a geodesic) and V ′ =ε

c2 g(V ′,σ ′)σ ′+(V ′)⊥, substituting in the previous equation, integrating and sim-plifying, we obtain (44).

3.3. The Index Form. Although we will not consider the space Ω(P,q) as amanifold, we will need a sort of tangent space to introduce the index form.

DEFINITION 3.15. Let σ : [a,b]→ M be a geodesic. We define the tangentspace to Ω(P,q) as the vector space Tσ Ω that consists of piecewise smooth vectorfields along σ such that V (a) ∈ Tσ(a)P and V (b) = 0.

Each V ∈ Tσ Ω will be regarded as the variation vector field for some variationx of σ , regarding eventually some values in [a,b] as breaks of σ . Recall that σ hasno breaks, and V is allowed to have breaks just for consistency with the previousapproach.

DEFINITION 3.16. The index form of a non-lightlike geodesic σ ∈ Ω(P,q) isdefined as the unique bilinear symmetric form

IPσ : Tσ (Ω)×Tσ (Ω)→R

such that for every V ∈ Tσ (Ω), it holds

IPσ (V,V ) = L′′x (0).

In the trivial case P = p we just write Iσ .

DEFINITION 3.17. The second fundamental form of P is the tensor II : X(P)×X(P)→ X(P)⊥ defined as II(V,W ) = nor(∇VW ) for every V,W ∈ X(P), where ∇

is the Levi-Civita connection of M, nor denotes the normal part to P, and X(P)⊥

the normal vector fields to P. Moreover, we define the tensor II : X(P)×X(P)⊥→X(P) as II(V,Z) = tan(∇V Z) for every V ∈X(P) and Z ∈X(P)⊥, where tan denotesthe tangent part to P, so that: g(II(V,Z),W ) =−g(Z, II(V,W )) for all W ∈ X(P).


COROLLARY 3.18. Given a geodesic σ ∈Ω(P,q) of speed c > 0 and sign ε , itholds

IPσ (V,W ) =

ε

c

∫ b

a

(g((V ′)⊥,(W ′)⊥)−g(R(σ ′,V )W,σ ′)

)ds

− ε

cg(σ ′(a), II(V (a),W (a)))

=−ε

c

∫ b

a

(g((V ′′)⊥+R(V⊥,σ ′)σ ′,W⊥)

)ds−

k−1

∑i=1

ε

cg(∆(V ′)⊥,W⊥)(ti)

− ε

cg(V ′(a)⊥,W⊥(a))− ε

cg(σ ′(a), II(V (a),W (a)))

for every V,W ∈ Tσ (Ω(P,q)) with breaks at t1 < t2 < .. . < tk−1.

PROOF. First equality follows from (44), taking into account that a symmetricbilinear form is determined by its quadratic form and that if α(w) = x(w,a), then

g(σ ′(a),A(a)) = g(σ ′(a),nor(α ′′(a)))

= g(σ ′(a), II(α ′(a),α ′(a))) = g(σ ′(a), II(V (a),V (a))).

For the second equality, integrate by parts g((V ′)⊥,(W ′)⊥) = g((V ′)⊥,W⊥)′ −g((V ′′)⊥,W⊥) and recall that g((V ′)⊥,W⊥)|ba =−g(V ′(a)⊥,W (a)⊥).

3.4. Conjugate and focal points. Next, Jacobi equation is introduced andfocal points (in particular, conjugate ones) are studied.

DEFINITION 3.19. A (smooth) vector field J along a geodesic σ : [a,b]→Mis called a Jacobi field if it satisfies the equation

J′′+R(J,σ ′)σ ′ = 0.

Moreover, if P is a submanifold orthogonal to σ in σ(a), we say that J is a P-Jacobivector field if

(i) J(a) is tangent to P,(ii) tanJ′(a) = II(J(a),σ ′(a)), where tan denotes the tangent part to Tσ(a)P.

This definition is suggested by the last formula in Corollary 3.18 replacingV,W by J. In fact, as J is smooth, ∆J′(ti) = 0. In the P-Jacobi case J(a)⊥ = J(a)and so:

g(J′(a)⊥,J(a)⊥) = g(J′(a),J(a)) = g(tanJ′(a),J(a)) = g(II(J(a),σ ′(a)),J(a)).

Therefore, (recall Definition 3.16 and 3.17),

g(J′(a)⊥,J⊥(a))+g(σ ′(a), II(J(a),J(a))) = 0.

As a consequence, if there exists a P-Jacobi J which also vanishes on σ(b) (so thatJ ∈ Tσ (Ω)) then IP

σ (J,J) = 0.It is not hard to prove that Jacobi fields are the vector fields along the geo-

desic σ generated by variations in which every longitudinal curve is a geodesic.From previous considerations, P-Jacobi fields are those Jacobi fields generated byvariations by means of longitudinal geodesics departing orthogonally from P.


DEFINITION 3.20. Given a geodesic σ : [a,b]→M, we say that r0 ∈ (a,b] is aconjugate value if there exists a Jacobi field J 6= 0 along σ such that J(a) = J(r0) =0. Moreover, given a submanifold P which is orthogonal to σ through σ(a) we saythat r0 is a P-focal value if there exists a P-Jacobi field J such that J(r0) = 0.

Roughly speaking, conjugate (resp. focal points) are associated to variationsof geodesics departing from p (resp. orthogonally from P) and arriving to q up toan infinitesimal of first order. However, the corresponding Jacobi field belongs toTσ (Ω) and, so, it can be generated also by means of a variation with longitudinalcurves arriving at p. As discussed above, if the focal value is r0 = b, then the indexform vanishes on such a Jacobi field.

From now on, we will study conjugate points as a particular case of P-focalpoints. Following [43], the similar Lorentzian and Riemannian cases will be stud-ied at the same time by means of the following notion.

DEFINITION 3.21. Let (M,g) be a semi-Riemannian manifold. A geodesic σ

is cospacelike if the orthogonal subspace to the geodesic is positive definite, thatis, either if g is Riemannian or if g is Lorentzian and σ is timelike.

LEMMA 3.22. Let σ be a cospacelike geodesic of sign ε and constant speedc in a Lorentz or Riemannian manifold. Then there exists V ∈ Tσ (Ω), such thatεIP

σ (V,V )> 0.

PROOF. Consider a unit vector y∈ σ ′(a)⊥, and let Y be the parallel translationof y along σ . Let Vk = δk sin[(t−a)/δk]Y with δk = (b−a)/(πk), k ∈N. Then

εIPσ (Vk,Vk) =

1c

∫ b

a[g(V ′k ,V

′k)− εc2KS(〈Vk,σ

′〉R)g(Vk,Vk)]d t

=1c

∫ b

a

(cos2[(t−a)/δk)]− εc2KS(〈Y,σ ′〉R)δ 2

k sin2[(t−a)/δk])

d t.

As the sectional curvature KS(〈Y,σ ′〉R) is bounded on [a,b], if δk is small enough(that is, k is big enough), then εIP

σ (Vk,Vk) is positive.

REMARK 3.23. If one considers a spacelike geodesic in a Lorentzian manifold,the same method of the previous proof allows one to find infinite-dimensional sub-spaces where IP

σ (or Iσ ) is both, negative or positive definite: simply take y∈σ ′(a)⊥

timelike or spacelike, respectively. So, the following results are stated just bycospacelike geodesics. (This also suggests that the celebrated Morse index the-orem cannot hold for spacelike geodesics in Lorentzian manifolds).

THEOREM 3.24. Consider a cospacelike geodesic σ ∈Ω(P,q) of sign ε . Then,(i) if there are no P-focal points of p=σ(a) along σ , the index form IP

σ restrictedto the orthogonal vector fields to σ is definite,

(ii) if there is a P-focal point σ(r) along σ with a < r < b, then IPσ is not semi-

definite.

PROOF. Let us prove (i). It is enough to show that IPσ is definite for V ∈ Tσ Ω

orthogonal to σ . First of all, observe that if V and W are P-Jacobi fields, then

(45) g(V ′,W ) = g(V,W ′).

This is becauseg(V ′,W )−g(V,W ′) = constant


along σ , whenever V and W are Jacobi fields ((g(V ′,W )−g(V,W ′))′ vanishes fromJacobi equation), and g(V ′(a),W (a)) = g(V (a),W ′(a)) (use the definition of P-Jacobi fields and the properties of II and II). Let k be the dimension of P. Nowchoose a basis of P-Jacobi fields J1,J2, . . . ,Jn−1 in such a way that J1(a), . . .Jk(a)constitute a basis of TpP, Jk+1(a) = · · · = Jn−1(a) = 0 and J′k+1(a), . . . ,J

′n−1(a)

are linearly independent. We claim that every vector V ∈ Tσ Ω can be expressed asthe sum

(46)n−1

∑i=1

fiJi,

with fi smooth functions in [a,b]. As there are no P-focal points along σ , thevectors

J1(t),J2(t), . . . ,Jn−1(t)

form a basis of σ ′(t)⊥ for every t ∈ (a,b]. This gives (46) for t ∈ (a,b] and re-duces the claim to the smooth extension of the functions to t = a. Recall that,g(Ji(a),J′k+ j(a)) = g(J′i(a),Jk+ j(a)) = 0 for any allowed i, j, in fact, the first equal-ity follows from (45) and the second one from Jk+ j(a) = 0. Therefore, the vectors

J1(a), . . . ,Jk(a),J′k+1(a), . . . ,J′n−1(a)

constitute a basis of σ ′(a)⊥. Moreover, as Jk+ j(a) = 0 we can write Jk+ j(t) =(t−a)Jk+ j(t), where Jk+ j(t) is a smooth vector field along σ and, thus, J′k+ j(a) =Jk+ j(a). It follows that

J1(t), . . . ,Jk(t), Jk+1(t), . . . , Jn−1(t)

is a basis of σ ′(t)⊥ for all t ∈ [a,b]. Then, we can put V (t) = ∑ki=1 hi(t)Ji +

∑n−1j=k+1 h j(t)J j(t), where hi : [a,b]→ R are smooth functions for i = 1, . . . ,n− 1.

Then, choose fi = hi for i = 1, . . . ,k and fi =hi

t−a for i = k+1, . . . ,n−1. Formula(46) follows because if i = k+1, . . . ,n−1, then hi(a) = 0, and fi can be extendedsmoothly to a as fi(a) = h′i(a).

Now, let us show that

(47) g(V ′,V ′)−g(R(σ ′,V )V,σ ′) = g(A1,A1)+ddt

g(V,A2),

where A1 = ∑n−1i=1 f ′i Ji and A2 = ∑

n−1i=1 fiJ′i . Observe that V ′ = A1 +A2, and thus,

(47) is easily equivalent to

g(V,A′2) = g(A1,A2)−g(R(V,σ ′)σ ′,V ).

This is just the computation

g(V,A′2) = g(V,∑i

f ′i J′i)+g(V,∑i

fiJ′′i ) =∑i, j

f j f ′i g(J j,J′i)−∑i

fig(V,R(Ji,σ′)σ ′)

= ∑i, j

f j f ′i g(J′j,Ji)−R(V,σ ′,σ ′,V ) = g(A2,A1)−R(V,σ ′,σ ′,V ),


where both, the Jacobi equation and (45), has been used. Then, recalling the firstformula for the index form in Corollary 3.18, the equality (47) yields

εcIPσ (V,V ) =

∫ b

a

(g(V ′,V ′)−g(R(V,σ ′)σ ′,V )

)dt−g(σ ′(a), II(V (a),V (a)))

=∫ b

ag(A1,A1)dt +g(V,A2)|ba−g(σ ′(a), II(V (a),V (a))).

However, the last two terms cancel, as:

g(V,A2)|ba =−g(V (a),A2(a)) =−∑i

fi(a)g(V (a), tanJ′i(a))

=−g(V (a), II(V (a),σ ′(a)) = g(σ ′(a), II(V (a),V (a))).

Thus, we have the inequality

εcIPσ (V,V ) =

∫ b

ag(A1,A1)dt ≥ 0.

Moreover, if IPσ (V,V ) = 0, then g(A1,A1) = 0, and therefore V = 0, which finishes

(i).Let us show (ii). There must exist a P-Jacobi field J(6≡ 0) that is zero in r. Let

Y be the vector field that is equal to J on [a,r] and zero in [r,b]. From the discussionon Jacobi equation below Definition 3.19 and Corollary 3.18, IP

σ (Y,Y ) = 0. Nowconsider any vector field W ∈ Tσ (Ω) such that W (r) = (∆Y )′(r) =−J′(r)(6= 0) andW (a) = 0. Given δ > 0, we have IP

σ (Y +δW,Y +δW ) = IPσ (Y,Y )+2δ IP

σ (Y,W )+δ 2IP

σ (W,W ). As IPσ (Y,Y ) = 0 and εcIP

σ (Y,W ) =−g(J′(r),J′(r))< 0 (g(σ ′,J′) = 0implies that J′ is spacelike), one has

εcIPσ (Y +δW,Y +δW ) =−2δg(J′(r),J′(r))+ εcδ

2IPσ (W,W ).

Thus, if δ is small enough, εIPσ (Y +δW,Y +δW ) is negative. Moreover, by Lemma

3.22 there always exists Z such that εIPσ (Z,Z)> 0.

REMARK 3.25. The interpretation of focal points is now clear from Theorem3.24 and the first and second formulas of variation. Namely, when a cospacelikegeodesic σ : [a,b] → M of sign ε = 1 (resp. ε = −1) starts orthogonally to aspacelike submanifold P, then:(1) if there is no focal value r ∈ [a,b], then σ is (up to a reparametrization), the

curve of minimum (resp. maximum) length which connects P and σ(b), amongall the curves close to it (that is, among the longitudinal curves obtained fromany variation of σ and sufficiently small values of the transverse parameter),and

(2) if there exists a focal value r ∈ (a,b) then there are arbitrarily close connectingcurves with both, bigger and lower length.

EXERCISE 3.26. Extend Theorem 3.24 to the case that r = b is a focal valueof σ .

3.5. Applications. We will present in this subsection some elementary appli-cations of focal and conjugate points, to be used later.

In order to expect the existence of conjugate or focal points, the natural condi-tions involve the curvature of the manifold and the initial focusing of the submani-fold. Classical Myers theorem focus on the role of (Ricci) curvature for conjugatepoints. We pay attention here to the focusing of the submanifold.


DEFINITION 3.27. The mean curvature vector ~H of of a nondegenerate sub-manifold P is the trace of the second fundamental form II divided by dim P.

THEOREM 3.28. Let P be a spacelike hypersurface (dimension m = n−1) in aRiemannian or Lorentzian manifold M, and σ : [0,b]→M a cospacelike geodesicwhich is normal to P at p = σ(0). The conditions

(i) H(σ ′(0)) = g(σ ′(0), ~Hp)> 0,(ii) Ric(σ ′,σ ′)≥ 0,

imply that there exists a focal point q=σ(r) along σ such that 0< r≤ 1/H(σ ′(0))whenever σ is defined on the interval [0,1/H(σ ′(0))].

PROOF. Let e1, . . . ,em be an orthonormal basis of Tσ(0)P and E1, . . . ,Em thevector fields along σ obtained by parallel translation of e1, . . . ,em. If k =H(σ ′(0)),define f (t) = 1−kt on [0,1/k]. Clearly f Ei ∈ Tσ |[0,1/k]

Ω. If ε and c are the sign andspeed, then:

εcIPσ ( f Ei, f Ei) = k−

∫ 1/k

0f 2g(R(σ ′,Ei)Ei,σ

′)d t−g(σ ′(0), II(ei,ei)).

Adding the expressions above from i = 1 to m we get

εc∑i

IPσ ( f Ei, f Ei) =mk−

∫ 1/k

0f 2Ric(σ ′,σ ′)d t−g(σ ′(0),m~Hp)

=−∫ 1/k

0f 2Ric(σ ′,σ ′)d t ≤ 0.

This means that εIPσ is not positive definite, and then by the proof of part (i) of

Theorem 3.24, there must exist a P-focal point in (0,1/H(σ ′(0))].

The second application is an extension of Proposition 3.9 to lightlike geodesics.(For more precise properties on the maximization properties, see [37, Section 2]).

PROPOSITION 3.29. If (M,g) be a Lorentzian manifold, and p, q two pointswhich can be joined by a piecewise smooth causal curve α which is not a causalpregeodesic. Then there exists a variation x of α with fixed endpoints such that allthe longitudinal curves but α are timelike.

Moreover, if the each smooth piece of α is timelike or lightlike, the formervariation can be chosen in such a way that the lengths of the longitudinal curvesdecrease strictly with the transverse parameter w.

PROOF. First, observe that it is enough to find a variation x of α with associ-ated vector field V satisfying g(V ′,α ′)< 0. In fact, by the compactness of the do-main of the curves in the variation, there exists δ > 0 such that g(V ′(w, t),xt(w, t))<0 if 0≤ w < δ . Then, all the longitudinal curves [a,b] 3 t→ x(w, t) with w ∈ (0,δ )are timelike, because

∂

∂wg(xt ,xt) = 2g(V ′,xt)< 0

and g(xt ,xt) decreases strictly on any transverse curve. Moreover, the length of thelongitudinal curves would increase with w. Clearly the strict absolute minimumis reached for w = 0 (at α) and, for any longitudinal curve with 0 < w < δ theformula of the first variation yields

∂Lx

∂w=−

∫ b

a

g(V ′,xt)

|xt |dt > 0.


Assume first that there exists t0 ∈ (a,b) such that α is smooth there and α ′(t0)is timelike. Let W be the parallel translation of α ′(t0) along α . As α ′ and W arealways in the same timelike cone, g(W,α ′) < 0. Now, choose δ > 0 such that α

is timelike in (t0−δ , t0 +δ ), and a smooth function f : [a,b]→R vanishing at theendpoints and satisying that f ′ > 0 on [a, t0−δ )∪ (t0 +δ ,b]. Then the variationalvector field V = fW gives the required variation, because g(V ′,α ′) = f ′g(W,α ′)<0 on [a, t0− δ )∪ (t0 + δ ,b] and the longitudinal curves remain timelike on [t0−δ , t0 + δ ] for w small enough (notice that the longitudinal curves of any variationof a timelike curve remain timelike for small values of w).

For the remainder case and the last assertion, assume that every smooth pieceis timelike or lightlike. Then we can assume with no loss of generality that α

is parametrized affinely on every smooth piece. As a consequence g(α ′,α ′′) = 0(except at most at the breaks) and g(α ′′,α ′′) ≥ 0 (α ′′ is orthogonal to the causalvector α ′).

Consider first the case that g(α ′′,α ′′) is identically equal to zero. Then, α ′′

must be proportional to α ′, (as α ′ generates the radical of (α ′)⊥). Then, apply-ing Lemma 2.17, we deduce that α is the reparametrization of a geodesic. Now,consider each (non-smooth) break ti, i = 1, . . .k. Let Wi be the parallel translationof ∆α ′(ti) = α ′(t+i )−α ′(t−i ). As these two velocities lie in the same causal cone,g(Wi,α

′) is negative in [t+i−1, t−i ] and positive in [t+i , t−i+1]. Now choose a piecewise

smooth function fi on [a,b] such that f ′i is positive in [t+i−1, t−i ], negative in [t+i , t−i+1]

and fi is zero outside [ti−1, ti+1]. Putting V = ∑i fiWi, necessarily g(V ′,α ′)< 0 andwe are done.

Assume now that g(α ′′,α ′′) is not identically zero. Let W be a parallel timelikevector field along α in the same causal cone as α ′, therefore g(W,α ′) < 0, andV = fW +hα ′′, where f and h are to be determined in order to have g(V ′,α ′)< 0.As g(α ′,α ′′) = 0, we deduce that g(α ′′,α ′′)+g(α ′,α ′′′) = 0, and hence,

g(V ′,α ′) = f ′g(W,α ′)−hg(α ′′,α ′′).

As u = g(α ′′,α ′′)/g(W,α ′)≤ 0 is not identically zero, there exists a real functionh≥ 0 in [a,b] that vanishes at its endpoints and such that∫ b

auhdt =−1.

Let f (t) =∫ t

a(uh+1)ds. Then f is zero at the endpoints and

f ′ = uh+1 > uh = hg(α ′′,α ′′)/g(W,α ′).

Hence g(V ′,α ′)< 0 as required.

4. Elements of the theory of Causality.

Here, we introduce some elementary tools in Causality Theory. We refer to therecent reviews [38] or [23] for a detailed account. Extending the notion of space-time for Special Relativity, we will use the following one, natural in the Lorentziancase for curved spaces (see the Appendix to this chapter for motivations in theframework of General Relativity).

DEFINITION 3.30. A spacetime is a connected time-orientable Lorentzianmanifold (M,g) endowed with a time-orientation (which will be implicitly as-sumed).

4. ELEMENTS OF THE THEORY OF CAUSALITY. 95

4.1. Primary definitions. The claim that observers (as well as material par-ticles) must move by using future-directed timelike curves and light rays alonglightlike geodesics makes natural the following definitions.

DEFINITION 3.31. Let (M,g) be a spacetime and p,q ∈M. We will say that:• p lies in the chronological past of q (or that q lies in the chronological

future of p), denoted by p q, if there exists a piecewise smooth future-directed timelike curve that departs from p and arrives to q;• p lies in the strict causal past of q (or that q lies in the strict causal future

of p), denoted by p < q, if there exists a piecewise smooth future-directedcausal curve that departs from p and arrives to q;• p lies in the causal past of q (or that q lies in the causal future of p),

denoted by p≤ q, if, either p = q, or p < q.

These definitions extend in a natural way to any subset A ⊆ M. Thus, wewill call chronological future (resp., past) and it will be denoted by I+ (A) (resp.,I− (A)), the subset of points of M which lie in the chronological future (resp., past)of some point of A, namely:

I+ (A) = q ∈M / ∃ p ∈ A such that p q,I− (A) = q ∈M / ∃ p ∈ A such that q p .

Analogously, the causal future (resp., past) of A, denoted by J+ (A) (resp., J− (A)),is the subset of points of M which lie in the causal future (resp., past) of some pointof A:

J+ (A) = q ∈M / ∃ p ∈ A such that p≤ q,J− (A) = q ∈M / ∃ p ∈ A such that q≤ p .

REMARK 3.32. With these relations of causality in M we can deduce somesimple properties (by means of the simple concatenation of piecewise differentiablecurves and, eventually, the use of Proposition 3.29). For example,

(a) p q, q r⇒ p r(b) p q, q≤ r⇒ p r(c) p≤ q, q r⇒ p r

As a consequence, B⊂ I+(A)⇒ I+(B)⊂ I+(A), etc.

The notation will be simplified for points and, so, we will write, I+(p) insteadof I+(p). Moreover, the chronological and causal relations can be regarded asbinary relations on M and, then, one can define:

I+ ⊂M×M, (p,q) ∈ I+⇔ p q,J+ ⊂M×M, (p,q) ∈ J+⇔ p≤ q.

4.2. Local vs Global Causality. As an extension of the previous definitions,consider an open connected subset U of a spacetime (M,g). We know that U , withthe restriction of the metric and the time orientation, is another spacetime. Given asubset A ⊆U , we will denote by I+ (A,U), I− (A,U), J+ (A,U) and J− (A,U) thechronological and causal future and past of A with the causality of the Lorentzianmanifold U .

EXERCISE 3.33. Show that

I+ (A,U)⊆ I+ (A)∩U

and give an example where the strict inequality holds.


NOTATION 3.34. If C is a convex open subset (see Chapter 2, Section 1.3)of a Lorentzian manifold (M,g) and p,q ∈ C are two points of such a subset, wewill denote by −→pq the unique geodesic of M which joins p to q and is completelycontained in C , that is:

−→pq : [0,1]→ C , −→pq(0) = p, −→pq(1) = q

The following lemma shows that Causality is well behaved in convex subsets.

LEMMA 3.35. Let C be a convex open subset of a spacetime (M,g). Then(i) Let p and q be two distinct points in C . The inclusion q ∈ J+ (p,C ) holds if

and only if the geodesic −→pq is future-directed causal.(ii) The subset I+ (p,C ) is open in C (and also in M).

(iii) The subset J+ (p,C ) is the closure in C of the open subset I+ (p,C ).(iv) The relation ≤ is closed in C , that is, if p,q, pn,qn ∈ C satisfy pn→ p andqn→ q with qn ∈ J+ (pn,C ) , ∀n ∈N, then q ∈ J+ (p,C ).

(v) Any causal curve γ : [0,L)→ C ,L≤∞ contained in a compact subset K of Ccan be extended in a continuous way to the whole interval [0,L].

PROOF. (i) If q ∈ I+ (p,C ), apply Lemma 3.8 to realize that −→pq is timelike.If q ∈ J+ (p,C )\ I+ (p,C ), Proposition 3.29 ensures that the causal curve inC which joins p and q is a lightlike pregeodesic and, so, this curve is −→pq, upto a reparametrization.

(ii) For the second property, observe that I+ (p,C ) is the image by the exponentialmap at p of the timelike future cone in TpM (Lemma 3.8), and then it is openbecause the restriction of the exponential map to C is a diffeomorphism.

(iii) It follows from two facts: the same property holds in the tangent space and,using the exponential map at p, the sequences in I+ (p,C ) are lift to the time-like future cone of TpM.

(iv) Consider p,q, pn,qn ∈ C such that pn→ p, qn→ q and qn ∈ J+ (pn,C ),forall n ∈N. As pn ≤ qn, the first part states that the geodesic −−→pnqn is causalfuture-directed. Using the continuity of the map

C ×C → T M, (p,q) 7→ −→pq′ (0) ,

we deduce that as −−→pnqn′ (0)n∈N is a sequence of non-spacelike (timelike or

null) vectors, then its limit is −→pq′ (0) and is non-spacelike. Applying againpart (i) we conclude that q ∈ J+ (p,C ).

(v) If γ is inextendible to L, then there will exist two sequences tn and sn ofpoints of [0,L[ such that tn → L and sn → L, and their images convergeto different points

γ (tn)→ p and γ (sn)→ q being p 6= q.

We can assume that tn < sn < tn+1,∀n ∈N. As γ is a causal curve that joinsγ (tn) and γ (sn) (and also γ (sn) and γ (tn+1)), we have

tn < sn < tn+1⇒ γ (tn)≤ γ (sn)≤ γ (tn+1) .

Taking limits, it follows that p ≤ q ≤ p, and therefore, by part (i), p = q, incontradiction with p 6= q. Thus, γ must be continuously extendible to L.

Globally, the only property amongst all of the previous lemma that is preservedin the whole manifold M is the second one (see Proposition 3.39 below). None of


the others holds in general, and it is not difficult to give counterexamples of everysituation. This is obvious for the first one (i), and we will check explicitly theothers.

EXAMPLE 3.36 (Counterexample of (iii) ). In general, the subset J+ (p) is notthe closure in C of the open I+ (p). Consider the points p = (0,0) and q = (1,1) inL2, and define the spacetime M = L2 \q. Clearly, the subset J+ (p) is not closed(observe that the discontinuous line of Figure 2 is not contained in J+ (p)).

REMOVE(1,1)

(2,2) ∈ Cl(J+(p))\ J+(p)

qn

p = (0,0)

J+(p)

M = L2 \(1,1)

FIGURE 2. Counterexample of (iii)

EXAMPLE 3.37 (Counterexample of (iv) ). In general, the relation ≤ is notclosed in M. Consider in the previous example a sequence that converges to (2,2)horizontally with respect to Figure 2. We have that (0,0)< qn, but (0,0) (2,2).

EXAMPLE 3.38 (Counterexample of (v) ). Any inextendible causal curve in acompact spacetime yields a counterexample. In particular, an inextendible causalgeodesic suffices (recall that a geodesic is continuously extendible if and only ifit is extendible as a geodesic). The incomplete ones (as, for example the incom-plete closed lightlike geodesics in a time-oriented Lorentzian tori, in Chapter 2,Subsections 4.2 and 4.3) show that the reparametrization or the value of L≤ ∞ areirrelevant.

The following result shows, in particular, that part (ii) holds in general. Giventwo curves γi : [ai,bi]→M, i = 1,2, we denote the concatenation of them as γ2 ?γ1;this curve maps each t ∈ [a1,b1 +(b2−a2)] on γ1(t) if t ∈ [a1,b1] and on γ2(a2 +t−b1) otherwise.

PROPOSITION 3.39. If (M,g) is a spacetime, then the relation is open in M,that is, if p,q ∈M are points such that p q, then there exist two neighborhoodsU of p and V of q in M such that

p′ q′, ∀p′ ∈U, ∀q′ ∈V.

In particular, I+ (p) is an open subset of M, for all p ∈M.


PROOF. As p q, let γ : [0,1]→M be any future-directed timelike curve thatjoins γ (0) = p with γ (1) = q. Let Cp and Cq be open convex subsets respectivelyof p and q in M; we can assume that Cp∩Cq = /0. Take two arbitrary points

p0 ∈ I+ (p,Cp)∩ γ ([0,1]) and q0 ∈ I− (q,Cq)∩ γ ([0,1])

From Lemma 3.35 part (ii), the subsets U = I− (p0,Cp) and V = I+ (q0,Cq) areopen in M.

V = I+(q0,Cq)

V ⊂ I+(U)

U = I+(p0,Cp)

q0

q

q′

γ2

V

Cq

Cp

γ

γ1

p′p

p0

U

FIGURE 3.

Given p′ ∈U , there exists a future-directed timelike curve γ1 that joins p′ withp0 (by definition) and, analogously, given q′ ∈ V , there exists a future-directedtimelike curve γ2 that joins q0 with q′. Thus, the curve γ2∗ γ|[p0,q0]

∗γ1 is a piecewisefuture-directed timelike differentiable curve that joins p′ with q′, and p′ q′, asdesired (see Figure 3).

For the last claim, if we consider q ∈ I+ (p), we have found an open V of q inM in such a way that V ⊆ I+ (p), and therefore I+ (p) is an open subset of M.

EXERCISE 3.40. For any spacetime (M,g), check:(i) I+ is an open subset of M×M(ii) When a convex subset C ⊂ M is regarded as a spacetime, J+ is a closed

subset of C ×C .

REMARK 3.41. The binary relations I+,J+ are the basic elements of CausalityTheory, and they depend exclusively on the conformal class of (M,g) (plus thechoice of orientation). Moreover, the chronological relation in each convex setdetermines the lightcones of the metric and, so the conformal class of the metricin C by Corollary 2.16. Roughly speaking, local Causality is the same thing aslocal conformal geometry in Lorentzian signature, and sometimes Causality and


conformal geometry are regarded as equivalent. Then, the choice of a spacetimewhich represents the conformal class is considered as convenient from a technicalviewpoint, because one can work with its geodesics (and metric notions such as theLorentzian distance, to be studied next). Notice, however, that the global relationsI+,J+ may not characterize the conformal class (for example, in the totally viciousspacetimes below). For more subtleties on the meaning on Causality, it is usefulthe notion of isocausality introduced in [22] (see also [21]).

4.3. Time separation or Lorentzian distance. Let (M,g) be an arbitraryspacetime. Given two points p,q ∈ M, we will denote by C c

pq the family of allthe piecewise smooth future-directed causal curves that join p with q.

DEFINITION 3.42. We will call time separation (or Lorentzian distance) in(M,g) the following map d : M×M→ [0,+∞] defined by

d(p,q) =

0 if C c

pq = /0,sup

L(α) , α ∈ C cpq

if C cpq 6= /0.

Actually, d is not a distance properly speaking, but there are some properties(coming from the inverted triangle inequality) which suggest that d will play a rolesimilar to the associated distance to the metric in a Riemannian manifold.

PROPOSITION 3.43. Let p,q,r∈M be three points of a spacetime (M,g). Then(i) d(p,q)> 0⇔ p ∈ I− (q)⇔ q ∈ I+ (p);

(ii) d(p, p) is equal to +∞ if there exists a piecewise smooth timelike curve thatjoins p with itself, and it is equal to 0 otherwise;

(iii) if 0 < d(p,q)<+∞, then d(q, p) = 0; in particular, d is not symmetric;(iv) if p≤ q≤ r, then d(p,q)+d(q,r)≤ d(p,r).

PROOF. (i) It is straightforward from the definition of I± (p).(ii) Assume that there exists a piecewise smooth future-directed timelike curve

γ that joins p to p. As γ is timelike, its length L(γ) is strictly positive, and thenwe can use it to construct new timelike curves γk,k ∈N, that connect p to p, justconcatenating γ with itself a number k of times. Then d(p, p)≥ L(γk) = kL(γ)→+∞.

If there does not exist any future-directed timelike curve that connects p toitself, there are two possibilities with the same conclusion. If there is no future-directed causal curve that joins p with itself, by definition, d(p, p) = 0. Otherwise,this causal curve cannot be timelike at some point (see Proposition 3.29), and itslength must be 0. Therefore, again d(p, p) = 0.

(iii) If 0 < d(p,q)<+∞, there does not exist any future-directed causal curvethat joins q with p. Otherwise, we could concatenate this curve with a future-directed timelike curve from p to q, concluding d(p,q) =+∞ by the previous part.

(iv) The proof is completely analogous to the triangle inequality for the dis-tance associated to a Riemannian metric taking into account that: (1) as supremesare involved in the proof, the orientation of such inequality changes, and (2) inprinciple, one must do a simple distinction of cases, since some time separationcan be +∞.

The time separation can be a map constantly equal to +∞ (d ≡ +∞). Whenthis happens, the spacetime is called totally vicious.


EXERCISE 3.44. Let (M,g) be a spacetime.

(1) Assume that there is some point p ∈ M such that d(p, p) = +∞. Then,for any q ∈M, the only possible values of d(p,q) are either 0 or+∞.

(2) (M,g) is totally vicious if and only if for each p ∈M there exists a closedtimelike curve through p.

(3) Construct examples of:(a) a non-compact spacetime which is totally vicious.(b) a compact spacetime which is non-totally vicious.[Hint for (b): recall the example of a non-geodesically connected

torus in Chapter 2, Subsection 4.4.]

PROPOSITION 3.45. In a spacetime (M,g), the time separation map is lowersemicontinuous, that is, if p,q, pm,qm ∈M with pm→ p and qm→ q, then

liminfm→+∞

d(pm,qm)≥ d(p,q)

PROOF. As d ≥ 0, if d(p,q) = 0, there is nothing to prove. So, we have justto prove that, given any future-directed causal curve γ : [0,L]→M which joins pwith q then liminf

m→+∞d(pm,qm)≥ L(γ).

pε

pm p

I−(pε)

qε

I+(qε)

qqm

FIGURE 4.

With no loss of generality, we can assume |γ ′| ≤ 1. Given some 0 < ε < L/2take pε = γ(ε/2), qε = γ(L− ε/2) (see Figure 4). As pm → p ∈ I− (pε) andqm → q ∈ I+ (qε), there exists a natural number mε ∈ N such that, if m ≥ mε ,then pm ∈ I− (pε) and qm ∈ I+ (qε) (the chronological futures and pasts are openin M). Then

L(γ)≤ d(pε ,qε)+ ε ≤ d(pm, pε)+d(pε ,qε)+d(qε ,qm)+ ε,

and from the triangle inequality:

L(γ)≤ ε +d(pm,qm)


if m≥ mε . Taking the inferior limit,

L(γ)− ε ≤ liminfm→∞

d(pm,qm)

and, as ε > 0 is arbitrary, we conclude the inequality with ε = 0, as required.

EXAMPLE 3.46. There exist examples where the time separation is not uppersemi-continuous (and then it is discontinuous). In fact, the time separation can gosuddenly from +∞ to 0, as suggested in Exercise 3.44. Other interesting exampleis M =L2 \SL, where SL = (x,0) : x ∈ [−L,0] is a closed segment removed fromthe negative x semiaxis. Take p, q and the sequences pm → p and qm → q asin the Figure 5, so that d(pm,qm)≥ 1.

p←p∞

γ

length(γ)< 1/2

length = 1

M = L2 \S

S

π/4

q←q∞

FIGURE 5.

If L is big enough (including the case that the full negative semiaxis is re-moved) then q 6∈ J+ (p) and so, d (p,q) = 0. We can choose L also so that 0 <d (p,q)< 1/2. In both cases, d is not upper semi-continuous.

4.4. Global conditions of Causality. We have seen that, in a totally viciousspacetime, the chronological relation is trivial, that is, I+ = M×M (Exercise 3.44).In order to get both, mathematically fruitful techniques and physically applicablespacetimes, some restrictions must be imposed on the Causality of the spacetimeThis yields the so called causal hierarchy or causal ladder of spacetimes: a set ofstrictly more restrictive Causality conditions to be satisfied by spacetimes.

An account of this ladder is carried out in the book [2], and a detailed summarywith updated results and references is carried out in [38]. Here, we give just a briefidea of this ladder, in order to estimulate the intuition, pointing out also somesubtleties in the definitions, which have been solved recently (and, thus, do not


agree exactly with some classical references). The top class in the hierarchy, theglobally hyperbolic spacetimes, is emphasized, as these spacetimes constitute themost important class from both, the mathematical and physical viewpoint.

First, we include the list of the conditions in the main levels of the ladder.Many of these conditions can be formulated in alternative ways, and we choose asimple one.

DEFINITION 3.47. Let (M,g) be a spacetime. We will say that (M,g) is

(i) chronological, if there does not exist any closed timelike curve in M;(ii) causal, if there is no closed causal curve in M;

(iii) distinguishing, if each point is characterized by its chronological future andalso by its chronological past, that is: if p,q∈M satisfy either I+ (p) = I+ (q)or I− (p) = I− (q) then p = q;

(iv) strongly causal, if the topology of M is generated by the chronological futuresand pasts, that is, the set I+(p)∩ I−(q) : p,q∈M is a basis for the topologyof M;

(v) stably causal, if the spacetime admits a time function, that is, a continuousfunction which is strictly increasing on any future-directed causal curve;

(vi) causally continuous, if (M,g) is distinguishing and it is reflecting, that is,it satisfies that for any p,q ∈ M: x ∈ Cl(I−(y))⇔ y ∈ Cl(I+(x)) (here, Cldenotes closure);

(vii) causally simple, if it is causal and the subsets J+ (p) and J− (p) are closed inM, for every point p ∈M;

(viii) globally hyperbolic, if it is causal and each intersection J+ (p)∩ J− (q) is acompact subset of M, for any p,q ∈M.

The relation between the previous properties is given in the following diagram.We emphasize that none of the converse implications holds. Perhaps the mostusual conditions are the chronology one, strong causality and global hyperbolicity,however, each one of the other conditions plays an important role in its own right.We will comment these levels very briefly, taking into account previous definitions.

Globally hyperbolic⇓

Causally simple⇓

Causally continuous⇓

Stably causal⇓

Strongly causal⇓

Distinguishing⇓

Causal⇓

Chronological


We also stress that all these properties depend only on the conformal class ofthe metric g. So, even though we will go on speaking on metrics, one can consideralways the study with conformal classes (as remarked, even notationally, in [38,Remark 2.9]).

4.4.1. Chronological spacetimes. The simplest condition on Causality is toforbid the existence of timelike closed curves. This has a clear interpretation: anobserver cannot travel to its past and, so, typical paradoxes about the use of timemachines cannot appear. As a matter of fact, the following result explains the re-luctancy of many relativists to admit compact spacetimes as physically reasonable.

THEOREM 3.48. No compact spacetime can be chronological.

PROOF. Consider the covering of M by open subsets I+ (p) , p ∈M. AsM is compact, there exists a finite subcovering I+ (p1) , I+ (p2) , . . . , I+ (pm). Ifpi ∈ I+ (p j), then I+ (pi) ⊆ I+ (p j), and we can remove I+ (pi) from the previoussubcovering. Therefore, we can assume that if i 6= j, then pi /∈ I+ (p j). Obviously,

p1 ∈M =m⋃

j=1I+ (p j)

p1 /∈m⋃

j=2I+ (p j)

⇒ p1 ∈ I+ (p1) .

Therefore, there exists a future-directed timelike curve that joins p1 to itself, andthe spacetime is not chronological.

Technically, the condition of chronology is strictly more restrictive that thecondition of being non-totally vicious, see Exercise 3.44.

4.4.2. Causality. Just a bit more restrictive than the inexistence of closed time-like curves it is the inexistence of causal ones. The interpretation is also clear: notonly observers, but also light rays (and, say, energy, information) can travel to thepast. Obviously, every causal spacetime is chronological, but the converse is nottrue. In fact, it is interesting to recall the following property.

PROPOSITION 3.49. A chronological spacetime which is not causal, admits aclosed lightlike geodesic.

PROOF. As it is not causal, there exists a closed (piecewise) differentiablecausal curve γ . Let p be any point along γ . If this curve is not a lightlike pre-geodesic, Proposition 3.29 guarantees that we can find a future-directed timelikecurve that joins p to itself, but this is impossible because M is chronological. Asa consequence, γ must be a lightlike pregeodesic. Moreover, (up to reparametriza-tion) the pregeodesic must be closed in the sense studied in Chapter 2, that is, thevelocities at the endpoints must coincide or, at least, be proportional (otherwise,concatenating the curve γ with itself, a closed causal curve that is not a pregeodesicwould be obtained).

EXAMPLE 3.50. The previous result suggests how to construct an example ofchronological manifold that is not causal: a cylinder obtained as the quotient of L2

by a non trivial lightlike translation. More precisely, let

M =(R2,−2dudv

)/G,

where G is the group of translations generated by (u,v) 7→ (u+1,v) (see Figure6). Every horizontal line “v = constant” is a closed lightlike pregeodesic, hence


the quotient spacetime is non causal. But the component v(s) of any timelike curvemust have non vanishing derivative, which forbids the existence of a closed time-like curve; summing up, the spacetime is chronological.

Identify

timelike timelike

Closed lightlikegeodesic

FIGURE 6. Non causal chronological cylinder

4.4.3. Distinguishing. It is natural to expect that, if Causality contains relevantinformation on the spacetime, the first elements which must be characterized arethe points of the spacetime themselves. Exactly, this means to be distinguishing:the chronological future (and also the chronological past) of any p ∈ M singlesout p. If only the future (resp. past) singles out the point, then the spacetime isfuture (resp. past) distinguishing. By using the transitivity of the causal relation,it is straightforward to check that any distinguishing spacetime is causal. In orderto understand at what extent distinguishing is a more restrictive hypothesis, thefollowing charaterization is illuminating (see [38, Lemma 3.10] for a more generalresult and a proof).

PROPOSITION 3.51. A spacetime (M,g) is distinguishing if and only if forany p ∈M the following property holds: for any neighborhood U 3 p there existsa neighborhood V ⊂U, p ∈ V, such that any causal curve which starts at p andleaves V , cannot enter again at V .

Now, it is easy to construct a causal but non-distinguishing spacetime.

EXAMPLE 3.52. First, modify Example 3.50 in order to obtain a spacetime thatis chronological and non-causal, but it only admits one closed lightlike geodesic γ

(see Figure 7). If we remove a point p along the geodesic γ of this spacetime, weobtain a new spacetime that is causal (it does not admit any closed geodesic), but itis not distinguishing: all the points on the image of γ (except the removed p) havethe same chronological pasts and futures.

4.4.4. Strong causality. Going further than for distinguishing spacetimes, onecan impose that Causality not only must characterize the points of M but also itsmanifold topology. This is the meaning of strong causality. To understand well thishypothesis, recall first.

DEFINITION 3.53. Let (M,g) be a spacetime. The topology on M which ad-mits as a basis the subsets

(48)

I+ (p)∩ I− (q) : p,q ∈M

is called the Alexandrov topology of (M,g).


Identify Non causalCausal

non distinguishing

C C \p

p

FIGURE 7. Construction of a causal spacetime that is not distinguishing.

EXERCISE 3.54. Show that (48) always defines a basis for a topology.

As every subset of the type I+ (p)∩ I− (q) is open in M (with its manifoldtopology), the Alexandrov topology is finer than the one of M. Obviously, it maybe strictly finer: in a totally vicious spacetime, the Alexandrov topology is thetrivial one (M and /0 are the unique open sets).

Thus, the equivalence between Alexandrov topology and the one of the mani-fold is our definition of strong causality. However, in order to understand the roleof this assumption, we have to check its meaning in the sequence of conditions onthe inexistence of certain causal curves of the previous levels in the causal hierar-chy. The following result becomes essential (a proof can be found in [38, Theorem3.27, Lemma 3.22], compare with [2, pg. 60]).

THEOREM 3.55. (Characterizations of strong causality). For any spacetime(M,g), the following conditions are equivalent:

(i) at any p∈M, the spacetime is strongly causal at p, i.e., the following propertyholds: given any neighborhood U 3 p there exists a neighborhood V ⊂U, p∈V, such that any causal curve with endpoints at V is entirely contained in1 V .

(ii) the Alexandrov topology on M coincides with the canonical topology of M,(iii) the Alexandrov topology on M is Hausdorff.

EXAMPLE 3.56. An example of a distinguishing spacetime which is not strong-ly causal can be constructed as follows (see Figure 8). Consider L2 with its usualmetric dx2−dy2, where we have removed two closed horizontal half-lines as in theFigure 8 (its vertexes are joined under an angle of π/4 radians over the horizontalline). Then, identify the upper and lower lines, generating in this way a spacetimeM which admits an open neighborhood U of a point p with the following property:for every neighborhood V ⊂ U of p there exists a causal curve not contained inV with endpoints in V . Thus, the spacetime is not strongly causal, and it is notdifficult to check that it is distinguishing.

1It is an equivalent condition to assume here that it is entirely contained in U .


Identify

V

Remove

Remove

Causal curvewith endpoints in V

FIGURE 8. Distinguishing spacetime that is not strongly causal.

Intuitively, if a spacetime is not strongly causal at p then there exist causalcurves which are “almost closed at” p, that is, with endpoints as close to p as onelikes (but leaving a fixed neighborhood of p).

These spacetimes have interesting properties which concern the space of causalcurves. Essentially, this can be also topologized and natural properties of conver-gence appear, see the study of limit curves and the C0 topology of curves in [2] orthe summary in [38, Section 3.6.3] (see also Subsection 4.4.8 below).

It is also worth mentioning that, in strongly causal spacetimes, any causal curvecontained in a compact subset can be continuously extended (recall the last item inLemma 3.35). More precisely:

DEFINITION 3.57. A causal curve γ : [0,b[→ M is called imprisoned in acompact subset K⊂M toward b if there exists δ ∈ ]0,b[ such that γ ([b−δ ,b[)⊆K.We will say that γ is partially imprisoned in K toward b if there exists a sequencetmm∈N ⊆ [0,b[ such that tm→ b and γ (tm) ∈ K, ∀m ∈N.

Taking into account that, if γ is imprisoned in a compact subset then a subse-quence of γ (tm)m must converge to a point p, the condition of strong causalitywould yield a contradiction if limtb γ(t) 6= p (and, so an analogous sequence con-verging to a different point p′ would appear, see also [43, Lemma 14.13]). That is,one has:

PROPOSITION 3.58. No inextendible causal curve is partially imprisoned (orimprisoned) in a compact subset K if strong causality holds for all p ∈ K.

4.4.5. Stable causality. The appearance of a time function t : M → R in thedefinition of stable causality seems to represent an important jump with respectto the previous levels. Say, this recovers the primary physical intuition that, in aphysically reasonable spacetime, a “globally defined time” must exist -even thoughthis time is not canonically determined. Because of the topological character ofprevious levels, the time function t which appears here is only continuous, and it isnot easy to find then a smooth one (in fact, this was an open folk question until therecent result [5]). The following result yields four alternative definitions of stablecausality, and shows the role of this condition in the causal ladder.


THEOREM 3.59. (Characterizations of stable causality). For a spacetime(M,g), the following conditions are equivalent:

(i) There exists a time function.(ii) There exists a temporal function, that is, a smooth function such that its

gradient at each point is timelike and past-directed.(iii) There exists another Lorentzian metric g′ which is both, causal and with

causal cones more open than the ones of g, that is, every causal vector v for g istimelike for g′ (g(v,v)≤ 0⇒ g′(v,v)< 0).

(iv) In the space of Lorentzian metrics on M endowed with the natural C0

topology, there exists a neighborhood U of the metric g such that all the metrics inU are causal.

For the proof of this result and explanations on the C0 topology, see the detailedstudy in [55] and the summary [38, Sect. 3.8.2, 3.8.3] (a fifth alternative definitionof stable causality follows from the result in [36]). Then, it is not hard to prove thatany stably causal spacetime is strongly causal (see for example [38, Proposition3.57]), but the converse does not hold. Now, we comment briefly this theorem.

First, notice that the items (iii) and (iv) strengthen naturally the conditionsabout the inexistence of closed or almost closed causal curves in the previous lev-els. It is easy to check the equivalence between both items, and the name “stablecausality” becomes obvious from (iv).

It is more surprising that such conditions (say, (iii)) imply the appearance ofa time function. The idea goes as follows. Consider an admissible measure onM. Say, the measure m obtained from any representative g′ of the conformal classof g, or even the measure associated to any auxiliary Riemannian metric serves,whenever m(M) < ∞ is fulfilled. Now, consider the future t− and past t+ volumefunctions associated to m, defined as:

(49) t−(p) = m(I−(p)), t+(p) =−m(I+(p)), ∀p ∈M.

Clearly, the function t− is non-decreasing on any future-directed causal curve γ ,as I−(γ(s)) only can make bigger with s. Analogously, t+ is also non-decreasing(the sign - is introduced for t+ in order to ensure this property). Moreover, if g isdistinguishing, then t± are also strictly increasing on γ .

However, t± may be non continuous. In order to ensure that they are continu-ous, the chronological futures and pasts I±(p) must vary continuously with p; thiswill be ensured in the next level, causal continuity. Nevertheless, a nice result byHawking [28] ensures that, if the spacetime is stably causal, a time function can beobtained as follows. Consider a parametric family of metrics gλ ,λ ∈ [0,ε],ε > 0,obtained by varying g, such that each gλ is causal with wider causal cones than g.A time function t can be obtained from the volume function t+

λassociated to each

gλ , namely, t(p) is the integral of t+λ(p) on λ .

Previous discussion justifies (iv) ⇔ (iii) ⇒ (i). The problem (i) ⇒ (ii), eventhough it seems harmless, has remained open until recently. The difficulty is thatboth, the involved tools for the construction of the time function t (say, volumefunctions) or to the approximation of t by means of smooth ones by using standardtechniques (say, convolution ones), fail to construct the desired temporal function.So, a new procedure to construct causally-related functions had to be developed [5].This procedure has some resemblances with the construction of some partition of


the unity by means of functions which behave as a temporal one in an appropriatepart of their support.

Finally, it is easy to prove that (ii)⇒ (i) (any temporal function is a time one)and (ii)⇒ (iii) (the gradient of the temporal function t allows widening the causalcones in such a way that t remains as a temporal function for these metrics).

4.4.6. Continuous causality and causal simplicity. Let us comment briefly thenext two classical levels in the hierarchy, previous to global hyperbolicity.

The intuitive idea for a causally continuous spacetime is that the set-valuedmaps, I± : p→ I±(p), vary continuously with p. One must impose additionallythat the spacetime is distinguishing (i.e., the set valued functions are one to one),because otherwise such a variation may be trivial. From the discussion above onvolume functions, one also has:

PROPOSITION 3.60. A spacetime is causally continuous if and only if it isdistinguishing, and its volume functions (49) are continuous, that is, if and only ifthe volume functions t± are time functions.

We refer to [38, Sections 3.9 and 3.7.3] for the proof of this result and exhaus-tive discussions.

Finally, about causal simplicity, notice that we have imposed two conditions:(1) the spacetime is causal, and (2) the sets J±(p) are closed. In fact, in the clas-sical references, one impose that the spacetime is distinguishing, instead of casual.However, as emphasized in [6], conditions (1) and (2) imply the condition of beingdistinguishing. Then, one can prove that causally simple spacetimes are causallycontinuous too (see for example [38, Section 3.10]).

Notice that, even in Lorentz-Minkowski spacetime L2, if one removes a point(say, (1,1)) the causal futures and pasts of some points may be non-closed (in ourcase, J+(0,0) is not closed). Of course, such a non-causally simple spacetimebelongs trivially to the previous levels of causality. Any open convex subset of Ln

is causally simple.

4.4.7. Global hyperbolicity. The final strengthening of Causality conditionsis global hyperbolicity. Namely, one starts at the definition of causal simplicity,and the condition that the sets J±(p) are closed is replaced by the following morerestrictive one: (M,g) does not contain any naked singularity, that is, the intersec-tions

J(p,q) := J+(p)∩ J−(p)

are compact. The bizarre name of this condition is explained below (Remark 3.70).

REMARK 3.61. (i) As in the case of causal simplicity, we have imposed twoconditions for global hyperbolicity: (1) the spacetime is causal, and (2) the absenceof naked singularities. In the classical references, the spacetime is assumed to bestrongly casual. However, the simplification of the definition was also emphasizedin [6]. Therefore, a globally hyperbolic spacetime is also causally simple and satis-fies all the properties of the other levels. However, a semiplane x > 0 of L2 showsthat the converse is not true.

(ii) Global hyperbolicity can be interpreted as a sort of completeness from theCausality viewpoint. In fact, we will see explicitly in Section 5 that, in some


Lorentzian results with a Riemannian analog, the role of this hypothesis is similarto Riemannian metric completeness2.

The definition of global hyperbolicity yields a much more powerful globalgeometric structure. The key is the behavior of the volume functions t± definedin (49). Concretely, recall the following result, which lies in the core of Geroch’scelebrated theorem in [25] (see also [38, Lemma 3.76] or [55] for a detailed study).

LEMMA 3.62. Let (M,g) be a globally hyperbolic spacetime, and let γ : (a,b)→M be any inextendible future-directed causal curve. The (necessarily continuous)function

t(p) = log(− t−(p)

t+(p)

)satisfies:

lims→a

t(γ(s)) =−∞, lims→b

t(γ(s)) = ∞.

The importance of this result is that, as a consequence, each level hypersurfaceSt0 = t−1(t0) must be crossed by γ , that is, St0 is an (acausal) Cauchy hypersurface,in the sense described next.

THEOREM 3.63. (Characterization of global hyperbolicity). For a spacetime(M,g), the following conditions are equivalent:

(i) (M,g) is globally hyperbolic.(ii) (M,g) admits a Cauchy hypersurface, that is, a subset S which is crossed

exactly once by any inextendible timelike curve(iii) (M,g) admits a Cauchy time function, i.e., an onto time function t : M→R

such that all its levels St0 = t−1(t0), t0 ∈R, are (acausal) Cauchy hypersurfaces.(iv) (M,g) admits a spacelike Cauchy hypersurface (a smooth hypersurface

which is spacelike and Cauchy).(v) (M,g) admits a Cauchy temporal function, i.e., an onto temporal function

t : M → R such that all its levels St0 , t0 ∈ R, are (necessarily spacelike) Cauchyhypersurfaces. Therefore, the full spacetime is isometric to an orthogonal Cauchyproduct

S×R, g =−βdt2 + g,where S = t−1(0), β : M→ R is some (smooth) positive function and g is a pos-itive semi-definite 2-covariant tensor field with radical spanned by ∇t (so that g,restricted to each Cauchy hypersurface St0 , is a Riemannian metric which dependson t0).

The first three items were proven by Geroch [25], and the two last ones corre-spond to the recent solution of the folk problems on smoothability in [4, 5]. Thefollowing comments are in order (a complete discussion can be found in [38, Sec-tion 3.11]). Lemma 3.62 proves the implication (i)⇒ (iii), and the implication(iii)⇒ (ii) is trivial. The definition of Cauchy hypersurface for S provided aboveimplies that S is an (embedded) topological hypersurface. Moreover, S must bealso crossed by any inextendible causal curve ρ , but ρ may intersect S in more

2This analogy goes further when one studies causal boundaries. Globally hyperbolic spacetimesare characterized because their causal boundaries do not contain timelike points. These points corre-spond to the Cauchy boundary in the Riemannian case. Non-timelike boundary points may appear,but these correspond with directions at infinity in the Riemannian case, [19].


than a point (say, in a segment). Nevertheless, as t is a time function, item (iii)ensures that this possibility does not happen, that is, the Cauchy hypersurfaces areacausal (this is automatically satisfied by spacelike Cauchy hypersurfaces). Theproof of (ii)⇒ (i) can be found in several books, for example, in [43, Theorem14.38, Corollary 14.39].

The proof of (iii) ⇒ (iv) is carried out in [4] by means of a relatively simplesmoothing argument. Spacelike Cauchy hypersurfaces are fundamental for Ein-stein equation (see the appendix to this chapter). In fact, the initial conditionsfor this equation are posed typically on a spacelike hypersurface, which must beCauchy for the spacetime which solves the equation. As commented for the sta-bly causal case (Theorem 3.59), the proof of (iii) ⇒ (v) is carried out by meansof a more complicated smoothing procedure. A simplification of this procedure(specific for globally hyperbolic spacetimes) is carried out in3 [41].

4.4.8. A note on the techniques, limits of the curves and time separations.Throughout the previous sections, quite different Causality techniques have ap-peared. For example, the deformations of causal curves have been used in Propo-sition 3.29 and also in other different parts. The rough idea for the applicabilityof admissible measures in the causal ladder has been explained in the Subsections4.4.5 and 4.4.7. Also in these subsections, the techniques for the smoothabilityproblems have been at least cited.

However, the study of Causality requires some notion of limit for sequences ofcausal curves. In general, such limits may be complicated. Namely, if one needsgood properties of convergence in the family C c

pq of all the piecewise differentiablefuture-directed causal curves from p to q, this family must be enlarged to includecontinuous causal curves (see [10, Appendix A]). As commented in Subsection4.4.4, an attractive feature of strongly causal spacetimes is the existence of verygood properties for the convergence of curves in the sense of C0 topology and limitcurves.

Here, we are going to explain briefly the notion of quasi-limit, which has beendeveloped in full detail in O’Neill’s book [43]. Even though this notion is quitetechnical, it has some clear advantages. On one hand, it is applicable to arbitraryspacetimes. Therefore, it can be used for the low levels of the causal ladder, orfor the proof of properties which make it consistent (for example, to check thatthe existence of a Cauchy hypersurface implies strong causality and, then, globalhyperbolicity). On the other hand, a quasi-limit is, by definition, a broken geodesic(say, which can be adapted to the sequence of curves with the accuracy one likes).This is an especially simple object, and so, easy to deal with.

Next, our aim is to explain and comment the quasi-limit framework, referringto [43] for a detailed study. We pay special attention to the application of quasi-limits to the time-separation d in globally hyperbolic spacetimes. Recall that this isa metric (non-conformally invariant) object, but the general properties to be studiedremain true in each conformal class. In particular, we will prove Avez-Seiferttheorem, which will be used for the proof of Hawking’s theorem in the next section.

To start with quasi-limits, one must fix a convex covering R of the spacetime(M,g). That is, R is a covering of M such that all U,V ∈R are convex and satisfy

3The aim of this last reference is to solve a classical problem studied by Clarke [14] in the spiritof Nash’ theorem: to show that any globally hyperbolic spacetime can be isometrically embedded inLN , for some big N.


that U ∩V is convex too. By using the paracompactness of M, it is not hard toprove that any open covering of M admits a refinement which is a convex covering[43, Lemma 5.10]. It is worth pointing out that only properties of causal curveswill be used. So, the conformal character of the approach can be stressed by takingglobally hyperbolic neighborhoods instead of normal ones (recall [38, Theorem2.14]).

DEFINITION 3.64. Let p∈M, αnn∈N be a sequence of future-directed causalcurves starting at p and R be a convex covering of M. We say that a (finite orinfinite) sequence p = p0 < p1 < p2 < .. . in M is a limit sequence for αn startingat p if:

(L1) For every pi there is a subsequence α imm, m ∈ N, of αn such that

there exist numbers sim0 < si

m1 < .. . < simi satisfying:

(a) limm→∞ α im(s

im j) = p j for every j ≤ i, and

(b) for each j < i the points p j, p j+1 and the segments α im([s

im j,s

im( j+1)])

lie in a convex set C j of R.(L2) If the sequence pi is infinite, then it is nonconvergent. Otherwise, the

limit sequence p0, p1, . . . , pr is not trivial (that is, r ≥ 1) and it is max-imal, i.e., it cannot be increased with a point pr+1 in such a way that theproperty (L1) still holds.

In this case, the broken future-directed causal geodesic γ obtained by connectingeach p j with p j+1 by means of a geodesic segment −→p j p j+1 : [0,1]→ C j is called aquasi-limit of the sequence αn.

Recall that, by definition, if the limit sequence is infinite then the quasi-limit γ

is inextendible. In this case, γ is not exactly a piecewise smooth curve accordingto our definitions, as its domain is non-compact and its number of breaks may beinfinite. However, its inextendibility prevents that the breaks may accumulate atsome point. If a limit sequence exists, there will be also infinitely many distinctlimit sequences even though, in some cases, all the quasi-limits may coincide, upto a reparametrization.

Before going on, the reader can gain some intuition by thinking in a sim-ple case. For example, consider a spacetime (M,g) such as L2 and L2 \ (0,0)and choose some simple convex covering. Consider a sequence of future-directedcausal curves αn : [0,1]→M, n ∈N, such that the endpoints sequences convergein M:

αn (0)→ p and αn (1)→ q.Now, think in the different possibilities for a limit sequence. We anticipate that, ifM is globally hyperbolic, the limit sequence must be finite, start at p and end at q.However, the limit sequence may be infinite in L2 \(0,0) (say, when the naturalquasi-limit in L2 would cross the removed origen).

The essential result on the existence of quasi-limits is the following one (see[43, Proposition 14.8]).

THEOREM 3.65. Let R be any convex covering of M, and αn : [0,bn)→ M,n ∈N, a sequence of causal curves which satisfy:

(i) the sequence αn(0) ∈M converges to a point p ∈M,(ii) there exists a neighborhood U of p such that the curve αn is not entirely

contained in U for an infinite number of values of n.


Then, αnn∈N admits a limit sequence relative to R starting at p.Moreover, if the curves αn are all future inextendible, then the quasi-limit γ

is also future-inextendible.

With this result at hand, we can prove interesting properties on the time-sepa-ration in globally hyperbolic spacetimes. A pair of technical lemmas are requiredfirst.

LEMMA 3.66. Let αn : [0,1]→ M, n ∈ N, be a sequence of future-directedcausal curves such that αn(0) → p for some p ∈M and αn(1) does not con-verge to p. Assume that αn has a finite limit sequence pir

i=0. Then, a subse-quence of αn(1) converges to the last point pr(6= p).

PROOF. Consider the subsequence αrmm, and the corresponding sr

mim suchthat, in the particular case i = r,

αrm(s

rmr)m→ pr.

Now, restrict the curves of the subsequence to the interval [srmr,1]. This subse-

quence of restricted curves must also admit a limit sequence p0 < p1..., and nec-essarily p0 = pr. Then, the point p1 violates the maximality of the original limitsequence.

LEMMA 3.67. Let K be a compact subset of a spacetime (M,g) such that thestrong causality condition holds at all the points of K, and let p,q ∈ K. Let αn :[0,1]→M, n ∈N, be a sequence of future-directed timelike curves in K such thatαn(0)→ p ∈ K and αn(1)→ q ∈ K. Then there exists a quasi-limit γ startingat p and finishing at q and a subsequence αnk of αn satisfying liminfk L(αnk)≤L(γ).

PROOF. Consider a limit sequence p0 < p1 < ... and the quasi-limit γ . Noticethat the limit sequence cannot be infinite because (as γ cannot be partially impris-oned in K, Proposition 3.58) some vertex pi of γ would lie outside K. Moreover,the last point pr of the limit sequence must be equal to q by Lemma 3.66.

So, the limit sequence satisfies p = p0 < p1 < · · ·< pr = q and we are going tocheck that the lengths of the curves αr

mm satisfy the required inequality. This canbe checked because, for each j = 0, . . . ,r, the restricted curves αr

m|[srm j,s

rm( j+1)]

haveendpoints pm j, pm( j+1) which lie in a single convex neighborhood C j and, thus

L(αrm|[sr

m j,srm( j+1)]

)≤ L(−−−→pm j, pm( j+1))

(recall Proposition 3.9). The result follows taking the sum in j = 0, . . . ,r− 1in this inequality and realizing that the right hand term converges to L(γ) (asL(−−−→pm j, pm( j+1)) varies continuously with pm j and pm( j+1)); so, the result follows.

LEMMA 3.68. If p < q, the subset J(p,q) = J+(p)∩J−(q) is compact and thestrong causality conditions holds in it, then there exists a future-directed causalgeodesic from p to q of maximal length.

PROOF. If d(p,q) = 0, then, by Proposition 3.29, every causal curve joining pand q must be a lightlike pregeodesic. Assume now that d(p,q) > 0. Then usingagain Proposition 3.29 we can assume that there exists a sequence αn : [0,1]→M

5. HAWKING AND MYERS THEOREMS 113

of future-directed timelike curves from p to q such that their lengths converge tod(p,q). Applying Lemma 3.67 with K = J(p,q), we obtain a quasi-limit γ withlength equal to d(p,q).

Recall that γ has a finite number of breaks and, in principle, it is a piecewisesmooth geodesic. However, the last assertion of Proposition 3.29 forbids the ex-istence of any break because, otherwise, γ could be varied into a strictly longertimelike curve with the same endpoints, in contradiction with L(γ) = d(p,q).

THEOREM 3.69 (Avez-Seifert). Let (M,g) be a globally hyperbolic spacetime:(i) If p,q ∈M are such that p < q, then there exists a causal geodesic joining p

to q with length d(p,q).(ii) The time separation d is finite and continuous on all M.

PROOF. Assertion (i), and then the finiteness of d follows from Lemma 3.68.For the continuity of d, recall that only upper semi-continuity has to been proved(Proposition 3.45).

Reasoning by contradiction, there exist two convergent sequences pn→ p ∈M and qn → q ∈M and some ε > 0 such that d(pn,qn) > d(p,q)+ ε for everyn ∈N. Choose p− ∈ I−(p) and q+ ∈ I+(q). Then, p,q ∈ I+(p−)∩ I−(q+) and, upto a finite number of values of n, pn and qn lie also in I+(p−)∩ I−(q+).

Now, take each causal geodesic αn (ensured by part (i)) of length d(pn,qn)which connect pn and qn. Obviously all αn belongs to the compact set J(p−,q+).Therefore, Lemma 3.67 ensures the existence of a causal curve γ from p to q suchthat L(γ)≥ d(p,q)+ ε , which is a contradiction.

REMARK 3.70. Theorem 3.65 also provides the promised physical interpre-tation for the condition of absence of naked singularities in globally hyperbolicspacetimes (see also the appendix to this chapter). Assume that J(p,q) is notcompact and, so, there exists a sequence rn ⊂ J(p,q) with no converging sub-sequence. Consider a sequence of curves αn, each one starting at p, passingthrough rn and ending at q. Choosing any convex covering of J(p,q), we will havea limit sequence p = p0 < p1 < .... This sequence can neither reach q nor be finite.Now, the quasi-limit γ presents the following behavior: γ starts at p and can be “ob-served” always from q (as γ lies in J−(q)), but it “disappears suddenly” from thespacetime. That is, in principle, γ (as well as the curves αn) could describe sometype of particle or travelling information, but such a particle or information woulddisappear from the spacetime. For a physically reasonable spacetime, this seemsto mean that γ finishes at some sort of singularity. This singularity is naked, thatis, the singularity (or at least, the disappearing particles as γ) can be observed fromthe event q. This has a different nature from the singularities of black holes, whichare hidden beyond an event horizon, or the primordial Big-Bang/ Big-Crunch ones.

5. Hawking and Myers theorems

Our aim in this section is to provide a self-contained proof of a classical singu-larity theorem by Hawking, by using the results and tools studied in this chapter.We want also to stress the similarities and differences with the Riemannian case byproving, at the same time an analogous Riemannian result.

LEMMA 3.71. Let S be a Cauchy hypersurface and p ∈ I−(S). Then J+(p)∩Sis compact.


PROOF. Let xn be a sequence in J+(p)∩ S and choose causal curves αn :[0,1]→M from p to xn. By Lemma 3.65, there exists a limit sequence p < p1 <p2 < ... of the sequence αn. If pi is finite, then, by Lemma 3.66, a subsequenceof xn converges and we are done. If pi is infinite, then the quasilimit γ isinextendible to the future, and must meet S at some point. As a consequence,there exists a point pi0 of the sequence which lies in I+(S). Then, there existsubsequences α i0

mm, si0i0mm such that α i0

m (si0i0m)m→ pi0 , and all α i0

m (si0i0m) lies

in the chronological future of S for large m. But in this case the restriction ofthe curve α i0

m to [0,si0i0m] must also cross S (its endpoints are, respectively, in the

chronological past and future of S). This is a contradiction with the fact that S isa Cauchy hypersurface, because the full curve α i0

m crosses also S at its endpoints = 1.

THEOREM 3.72. Let (M,g) be a spacetime satisfying the following conditions:

(i) (M,g) is globally hyperbolic,(ii) there exists some spacelike Cauchy hypersurface S with an infimum C > 0 of

its expansion, that is, such that its mean curvature vector ~H = H~n, where~n isthe future-directed unit normal, satisfies H ≥C > 0,

(iii) the timelike convergence condition holds: Ric(v,v) ≥ 0 for every timelikevector v.

Then, any past-directed timelike curve starting at S has length at most 1/C.

PROOF. If q ∈ I−(S), then previous Lemma 3.71 and the continuity of thetime separation ( part (ii) of Theorem 3.69), imply the existence of a point p ∈ Ssuch that d(q,S) = d(q, p) (d(p,S) denotes the supremum of lengths of causalcurves from q to points of S). By Avez-Seifert theorem (part (i) of Theorem 3.69)there exists a timelike geodesic γ : [0,b]→M from q to p that maximizes the timeseparation between q and S. By Corollary 3.12, this geodesic is normal to S and,even more, the index form is positive semi-definite by the formula of the secondvariation. This implies, by part (ii) of Theorem 3.24, that there is no focal pointalong γ|[0,b). However, the hypotheses (i) and (ii) allows one to apply Theorem3.28 and, then, a focal point exists whenever b > 1/C. As a consequence of thesetwo facts

I−(S)⊂ q ∈M : d(q,S)≤ 1/C,which clearly implies the statement of the theorem.

The analog Riemannian result, which proof relies in Riemannian focal pointsand, at the end, in classical Myers theorem, is the following.

THEOREM 3.73. Let (M,g) be a Riemannian manifold satisfying the followingconditions:

(i) g is complete,(ii) there exists some embedded S which separates M as a disjoint union M =

M− ∪ S∪M+, with an infimum C > 0 of its expansion towards M+, that is,such that its mean curvature vector ~H = H~n, where~n is the unit normal whichpoints out M−, satisfies H ≥C > 0,

(iii) Ric(v,v)≥ 0 for every v.

Then, dist(p,S)≤ 1/C for every p ∈M−.

6. APPENDIX: GENERAL RELATIVITY 115

PROOF. Observe that the proof of this theorem is isomorphic to the one ofTheorem 3.73, where each hypothesis (i), (ii) and (iii) plays a role similar to itspartner, and I±(S) corresponds with M±. The role of Avez-Seifert theorem there isplayed by Hopf-Rinow theorem here.

REMARK 3.74. In spite of the similarities between the proofs of Theorems3.72 and 3.73, the interpretation of the conclusion is very different. In the lattercase, the conclusion is just that, in the complete Riemannian manifold, certaindistances must be bounded by 1/C. In the former one, the conclusion is not onlythat all timelike geodesics must be past incomplete, but also that any inextensibletimelike curve γ , which necessarily must cross S, has a length bounded by 1/C inI−(S).

One can interpret physically the curve γ as an observer or material particle (seealso the Appendix for the interpretations here). Then, the conclusion of Theorem3.72 suggests strongly the existence of some sort of initial common singularity,namely a Big-Bang. In fact, this theorem shows that the past timelike incomplete-ness, which appears as a consequence of a initial singularity in the simplest modelsof the Universe (say, the Friedman-Lemaitre-Robertson-Walker ones), is not a con-sequence of the idealizations or symmetries of these models, but a general fact. Ofcourse, this physical conclusion must be accepted if the hypotheses of the theoremare realistic. And, fairly, this is the case: (i) the existence of a Cauchy hypersurfaceis expected, because this is equivalent to the classical predictability of the full Uni-verse from initial conditions and differential equations (the so-called strong cosmiccensorship hypothesis also supports this viewpoint), (ii) the measured expansionof the Universe supports that H > 0, say, for the spacelike level hypersurface S ofsome natural temporal function T where we live now say (namely, T would be theinverse of the absolute temperature at a big scale; the big scale homogeneity andisotropy observed in the space becomes relevant at this point), and (iii) the timelikeconvergence condition can be interpreted as the fact that gravity must be attractive,at least on average –a fact that has been classically accepted as true, even thoughthe accelerated expansion of the Universe measured along the last decade, maymake to reconsider it.

Because of these interpretations, Hawking’s result is one of the celebrated sin-gularity theorems. Other singularity theorems combine subtler Causality propertiesand other tools. They conclude not only singularities of Big Bang type, but also ofblack holes type. We refer to [56] for a easily readable overview on these topicsand [20] and [57] for a detailed review and some recent advances, respectively.

6. Appendix: General Relativity

Special Relativity considered space and time as aspects of a unique entity,mathematically modeled by means of a Lorentzian affine space. General Rela-tivity goes a step further, considering that space, time and gravity are aspects ofa unique entity, say, a relativistic spacetime, modelled by means of a (possiblycurved) Lorentzian manifold. There are different reasons for this step. On onehand, the impossibility to implement satisfactorily Newtonian gravitation in theframework of Special Relativity. On the other hand, some gedanken experimentssuggest:

(i) A freely falling observer perceives at a first order of approximation thephysics of Special Relativity (with no gravity). The observer must be freely falling


because it must not experiment any forces (except, at most, gravity). Recall, forexample, that a person sat on the floor experiments an electromagnetic force by thefloor.

(ii) The presence of gravity affects at a second order. For example, if twoastronauts start to fall freely toward the Earth, at a first order of approximationtheir relative positions would be the same as if they were not freely falling. Butat a closer look one finds possible “second order effects”. For example, assumethat the two astronauts start to fall exactly at the same radial distance to the centerC of the Earth. Then, as their accelerations would not be exactly parallel (theypoint out C), the relative acceleration among them would yield some progressiveapproximation of the astronauts. Or assume that the two astronauts start to fallin the same radial half-line from C. Then, one of the astronauts would start tofall at a smaller distance from C than the other one. So, the first astronaut wouldexperiment a somewhat bigger initial acceleration than the second one, and bothastronauts would experiment a progressive separation due to this effect.Such reasons (prior to experimental evidences, which came historically quite later)support the ideas: (a) physical space and time must be modelled by a Lorentzianmanifold, (b) freely falling observers are especially well adpated to make mea-sures, they will follow future-directed timlike geodesics, and they may try to findphysical coordinates well-adapted to the geometry (say, normal coordinates), and(c) the curvature or Levi-Civita connection of this manifold (which in general, actsa second order effect on the trajectories of the geodesics) will collect the role ofthe gravity.

In our mathematical definition (Subsection 4) a spacetime was a connectedtime-oriented Lorentzian manifold (M,g). The role of the additional conditionsfor the Lorentzian manifold is clear. On one hand, we are supposed to distinguishat each event p∈M between the future and past time directions. On the other hand,if there were other connected parts of the Universe (which could not influence us byany known procedure) we are not especially worried. One could require additionalnatural conditions. For example, the dimension n of M could be assumed to beequal to 4. Or M could be assumed to be orientable (an even oriented): not onlybecause this can be regarded as an inoffensive mathematical simplification, butalso because the physicists find feasible to fix continuously an orientation (thanksto the violation of parity produced by the weak force). However, these additionalhypotheses (except the value of n for some particular topics) will not play a relevantrole here.

Summing up, we will accept that our mathematical definition of spacetime(assuming, when necessary, n=4) provides a general ambient to describe physicalspace, time and gravity4, and we will discuss some features of this description.

The concepts of observer and light ray would extend naturally to the ones inSpecial Relativity.

DEFINITION 3.75. An observer (respectively particle) inside of a spacetime(M,g) is a unit future-directed timelike curve γ : I → M, |γ ′| = 1 (respectively,normalized to its rest mass m = |γ ′|). If this curve is a geodesic, we say that theobserver (respectively particle) is in free fall (or non accelerated).

4The implicit hypothesis of differentiability (and continuity) of M deserves a separate attention.It is essential for the construction of the relativistic model, and can be admitted as a valid macroscopicapproach. However, the model would be unappropriate at a microscopic (and quantum) scale.


A light ray is a future-directed lightlike geodesic.An instantaneous observer is any unit future-directed timelike vector.

For the observers, the parameter t of the curve γ(t) is interpreted as the propertime, measured by this observer; the normalization for particles is chosen for con-venience.

Of course, not every spacetime, according to our general definition, is physi-cally realistic. The following elements play a role in the construction of realisticones. They also suggest geometric ideas with interest in its own right, some ofthem studied in the text.

(i) Causality conditions. We discussed them in Section 4 of Chapter 3. From aphilosophical viewpoint, the inexistence of closed timelike curves (chronol-ogy condition), the existence of a time function (stable causality), and thecondition of possible predictability (global hyperbolicity), are probably themost discussed.

The first one means that an observer cannot travel to events in its ownpast, and it is related with the inexistence of a “time machine” –that philo-sophically is implied by the free will. However, spacetimes such as the Godelone, or the interior region of the “slow” Kerr spacetime, present such curves.These spacetimes are physically realistic in some parts, but their analityc pro-longations imply the existence of closed timelike curves. Remarkably, themanifold of Godel spacetime is R4, which shows that these curves are nota consequence of a complicated topology. The possible existence of closedtimelike curves, and even their creation from the so-called wormholes hasbeen speculated; mathematical arguments about the impossibility of its de-terministic creation have been also provided (see [32]).

Once the existence of a Newtonian absolute time has been discarded,the existence of, at least, a time function is both, mathematically useful andphysically reassuring. However, as no canonically determined time functionseems to appear in the physical spacetime, the phylosophical problems asso-ciated to the time are persistent. There exist some good candidates as physi-cally acceptable time functions; the inverse of the temperature (measured onaverage at cosmic scale) would be one. Notice that this would not mean thatthe inverse of the temperature grows with the time, but that it defines the time(or one possible time).

Finally, if one would like to predict the spacetime from initial data, Cauchyhypersurfaces (the element which characterizes global hyperbolicity) yieldthe minimum structure to pose the problem.

(ii) Attractive gravity and timelike convergence condition. That gravitation at-tracts becomes the most classical assumption on it. Due to the observationsof accelerated expansion of the universe during the last decade, some authorsrenounce this property. At any case, its implementation in General Relativityis quite subtle.

In principle, the geometrical translation of attractive gravity would beto claim that, when the spacetime is curved, timelike geodesics tend to getcloser (or to move away slower) than in the Lorentz-Minkowski spacetime.The meaning of this claim can be formalized in a precise way in terms of theJacobi equation. In a clear analogous to the Riemannian case, one can say


that the effects of curvature “attract” (on timelike geodesics) if and only if thesectional curvature of timelike planes is non-positive5.

However, a careful analysis of the Newtonian analogs, shows that thegravitational “attraction” cannot be reduced to this condition. Indeed, in theexample of the astronauts at the beginning, we saw that, when they are sit-uated in a position orthogonal to the radial one, then effectively gravitationtends to join them. However, when they are in a radial position, it tends toseparate. A careful analysis (see [52]) would show that, in this example, thegravity neither attracts nor repels. Nonetheless it is true that, at the end, thetwo astronauts are attracted to the Earth, and this must be reflected in someway.

The way to model this in General Relativity is the following. One imposethe so called timelike convergence condition Ric(v,v) ≥ 0 for every timelikevector v. This means that, “gravitation attracts (or, at least, does not repel)on average” 6. Now, through the Einstein equation (see below), the time-like convergence condition is related with the acceptable hypotheses on thestress-energy tensor. The Hawking’s theorem proved above suggests how the“local” attractive effects of curvature may yield a “global” effect of attraction.

(iii) Inextendibility, singularities. According to our definition of spacetime, themetric of a spacetime is well-defined at all the points. Therefore, if at some“event” of the physical spacetime the Lorentzian metric could not be defined,such an event would be automatically excluded from our model of spacetime.However, such excluded part of the physical reality may be of big importance.It would be desirable to describe it (preferably in terms of the included part).Let us comment this question in more detail.

It seems reasonable to think that, if a spacetime describes the physicalUniverse as a whole, it should be inextendible, that is, it should not be con-sider as a proper open subset of a greater spacetime. (If it describes justa region of the Universe, then one should specify carefully the “boundary”conditions that connect it with the remainder of the Universe). This leads toseveral interesting problems. For example, in the Riemannian case, a naturalcondition that implies inextendibility is the completeness. In the Lorentziancase, the (geodesic) completeness also implies inextendibility (even if it isassumed just for geodesics of one causal character). Nevertheless, the con-dition of completeness is very restrictive for realistic spacetimes, as shownabove in Hawking’s theorem. So, (geodesically) incomplete spacetimes canbe accepted, in principle, as models of our physical Universe, whenever theyare inextendible. However, one can wonder then which is the physical causethat produces incompleteness.

From a physical viewpoint, one expects that the possible incompletenessof a realistic spacetime must be attributed to some kind of “singularity”. How-ever, it is not easy to formalize what this may mean (the delicious dialogueby R. Geroch in [24] is recommended). It seems desirable to define thatthere exists a singularity when, along an incomplete causal geodesic, some

5This condition does not imply constant curvature in spite of Theorem 2.12 in Chapter 2, be-cause no assumption is imposed on the spacelike planes.

6Notice also that the curvature is a “second order effect” difficult to measure locally. So, it isalso preferred to impose conditions just on average.


geometric invariant (say, associated to the curvature tensor R) diverges. Nev-ertheless, some characteristic properties of the indefinite case make difficult atotally satisfactory formalization of this idea. For example, geometric invari-ants such as, e.g., ∑

ni, j,k,l=1 Ri jklRi jkl , can be identically null when R is not,

(see Chapter 2). However, if such an invariant diverges along an incompletegeodesic γ , it is obvious that the spacetime cannot be extended so that γ hasan endpoint.

Summing up, it is commonly accepted as sufficient requirements for theexistence of a singularity the following:(a) there exist incomplete geodesics (at least one timelike or lightlike), and(b) there exist some invariant associated to the curvature tensor that diverges

along some incomplete geodesics.(iv) Einstein equation and the energy conditions. The Einstein equation relates

geometrical concepts associated to g, as the Ricci tensor and the scalar cur-vature, with physical concepts as the energy and momentum. For the de-scription of the energy and momentum (in a wide sense), a 2-covariant tensorfied, the stress-energy-momentum tensor, or simply stress tensor T , is used.To explain this a bit, we consider dimension n = 4. Essentially, for everyorthonormal basis e1,e2,e3,e4 in TpM, p ∈ M, the term T (e4,e4) measuresthe density of energy at this point for the instantaneous observer e4, the term−T (e4,ei) measures the linear momentum density for e4 in the direction ei,and T (ei,e j) represents the force F per unit of area in the ei direction exertedon the matter on one side of the spacelike plane orthogonal to e j (and e4), bythe matter on the other side (this interpretation is analog to the one for theclassical stress tensor of a body in equilibrium). Write this in a 4×4 matrix:

T (ei,e j) T (ei,e4)

T (e4,e j) T (e4,e4)

,

where T (ei,e j), i, j,k ∈ 1,2,3 is a 3×3 matrix. As the tensor T is symmet-ric, the associated field of endomorphisms T (namely, T (v,w) = g(v, T (w)),for all v,w ∈ TpM, p ∈ M) is self-adjoint for g. From Chapter 1 we knowthe possible canonical forms of T . Not all of the T ’s admit an orthonormalbasis of eigenvectors, but for almost all the forms of known matter this istrue. In this case, if v1,v2,v3,v4 is such a basis, the eigenvalue ρ = T (v4,v4)is the density of energy (or density of mass at rest) and every eigenvaluepi = T (vi,vi) is a principal pressure. (A detailed study of the algebraic prop-erties of T and Ric, including the so-called Segre types can be found in [62],especially Chapter 5.)

In principle, the tensor T that is constructed from the distribution of en-ergy of the spacetime should determine its geometry (with suitable boundaryconditions). The expected equality between the elements associated to g andT is the Einstein equation. In the case of null cosmological constant and unitsG = c = 1, this is written:

(50) Ric− 12

Sg = 8πT


(see also the Exercise 2.13). In principle, one should pose initial conditions(including eventually asymptotic behaviors) on a smooth 3-dimensional man-ifold S. These conditions must be compatible with the fact that, in the space-time which solves the equation, S must satisfy the Gauss and Codazzi equa-tions. In the non-empty case (T 6≡ 0), equations on state, which describe thestructure of the matter, must be added on T . Under quite general properties,there exists a unique solution (M,g) such that S is a Cauchy hypersurface init (see [66]).

There are several physical arguments of plausibility in favor of (50), e.g.: (a) taking a suitable limit, one reobtains the Poisson equation of Newto-nian dynamics, (b) from Einstein equation, it is possible to deduce the law ofconservation divT = 0, where

divT (v) =4

∑i, j=1

gi j(∇eiT )(v,e j),

with no necessity of imposing additional conditions (as an advantage in re-spect to Newtonian Mechanics), or (c) the (Hilbert) equation (50) correspondswith extremal properties for compactly supported variations of the scalar cur-vature S. However, this equation must be accepted as a postulate. Admittingthat (50) must be satisfied, the restrictions on the stress tensor also implyrestriction on the geometry. In particular cases, as perfect fluids, T has aconcrete expression. But in general, one can consider an arbitrary T whichsatisfies some minimal restrictions which make it physically acceptable. Thecommonly used ones are:(a) Weak energy condition. It holds if and only if the density of energy is

always non negative, that is, for every v timelike, T (v,v)≥ 0. Taking intoaccount Einstein equation, this is equivalent to the following restrictionfor the geometry of the spacetime:

Ric(v,v)≥ 12

Sg(v,v).

(b) Strong energy condition. It holds if and only if for every timelike v:

T (v,v)≥ 12

trace(T )g(v,v),

or, equivalently, if the timelike convergence condition explained aboveis satisfied.

(c) Dominant energy condition. It holds if and only if for every future-directed timelike vector v, the energy-momentum density −T (·,v)] is afuture-directed causal vector (this means that the flow of energy is prop-agated toward the future at a speed not bigger than the speed of light).

EXERCISE 3.76. Solve the following algebraic questions:(A) The dominant energy condition always implies the weak energy condi-

tion. Is there any other implication between the different conditions ofenergy?

(B) Assume that T is diagonalizable. Check that its eigenspaces are neces-sarily orthogonal and:

(1) the weak energy condition holds if and only if

ρ ≥ 0, and ρ + pi ≥ 0 (i = 1,2,3)


(2) the strong energy condition holds if and only if

ρ +3

∑i=1

pi ≥ 0, and ρ + pi ≥ 0 (i = 1,2,3)

(3) the dominant energy condition holds if and only if

ρ ≥ |pi| (i = 1,2,3).

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