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Flat wormholes from cosmic stringsGrard Clment Citation: Journal of Mathematical Physics 38, 5807 (1997); doi: 10.1063/1.532167 View online: http://dx.doi.org/10.1063/1.532167 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/38/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Cosmic strings in Brane World models AIP Conf. Proc. 1514, 39 (2013); 10.1063/1.4791721 Fitting The Luminosity Data From Type Ia Supernovae By Means Of The Cosmic Defect Theory AIP Conf. Proc. 1059, 39 (2008); 10.1063/1.3012280 Visualizing spacetimes via embedding diagrams AIP Conf. Proc. 861, 883 (2006); 10.1063/1.2399673 The relativistic theory of gravity and Machs principle Phys. Part. Nucl. 29, 1 (1998); 10.1134/1.953058 Gravitational energy of conical defects J. Math. Phys. 38, 458 (1997); 10.1063/1.531827

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Flat wormholes from cosmic stringsGerard Clementa)Laboratoire de Gravitation et Cosmologie Relativistes, Universite Pierre et Marie Curie,CNRS/URA769, Tour 22-12, Bote 142, 4 place Jussieu, 75252 Paris cedex 05,France

~Received 5 February 1997; accepted for publication 9 June 1997!

We describe the analytical extension of certain cylindrical multi-cosmic string met-rics to wormhole spacetimes with only one region at spatial infinity, and investigatein detail the geometry of asymptotically Minkowskian wormhole spacetimes gen-erated by one or two cosmic strings. We find that such wormholes tend to lengthenrather than shorten space travel. Possible signatures of these wormholes are brieflydiscussed. 1997 American Institute of Physics.@S0022-2488~97!02110-5#

I. INTRODUCTION

It has long been recognized that the equations of general relativity carry information not onlyabout the local geometry of spacetime, but also about its possible global topologies. Early work onwormholes14 was motivated by the hope that they might provide a way to evade the problemsassociated with point singularities in particle physics. A quite different motivation was providedby the analysis of Morris and Thorne, who first investigated in some detail the possibility of usingtraversable wormholes to travel in space,5 as well as in time.6

Traversable wormholes may occur as solutions to the Einstein field equations with suitablesources violating the weak energy condition. When explicit solutions are discussed in the litera-ture, these are usually static spherically symmetric EinsteinRosen wormholes connecting tworegions at space-like infinity.7 More relevant for the purpose of interstellar travel are WheelerMisner wormholes,2 with only one region at spatial infinity. Exact solutions of the time-symmetricinitial-value problem of general relativity with such a topology have been constructed,3,4 but thesenonstatic wormholes are not traversable.5 In a recent paper,8 Schein and Aichelburg have con-structed a static WheelerMisner wormhole by matching, along two spherical shellsS1 andS2 , anouter MajumdarPapapetrou spacetime to an inner ReissnerNordstrom spacetime; this is travers-able only one way, fromS1 to S2 .

Static WheelerMisner wormholes may be obtained by suitably extending a procedure, de-scribed by Visser, to construct models of flat EinsteinRosen wormholes.9 Remove from Euclid-ean spaceR3 a volumeV. Take a second, identical copy ofR32V, and identify these two excisedspaces along the boundaries]V. The spacetime obtained by factoring the resulting space with thetime axisR is a geodesically complete EinsteinRosen wormhole~or multi-wormhole if V hasseveral connected components!, flat everywhere except on]V, where the stress-energy is concen-trated. To similarly construct a WheelerMisner wormhole, remove fromR3 two nonoverlappingvolumes V and V8 which are the image of each other under the involution (x,y,z)(2x,y,z), and identify the boundaries]V, ]V8 ~Fig. 1! ~the diffeomorphismVV8 mustreverse orientation if the resulting manifold is to be orientable10!. In a further extension of thisprocedure, the boundaries]V, ]V8 are not identified, but connected by a cylindrical tube carryingequal energy per unit length and longitudinal tension~if the surface]V is compact and simplyconnected, it follows from the GaussBonnet theorem that the energy per unit tube length is1/2G!. The geometry, as viewed by an external observer~in R32V2V8!, does not depend onthe internal distance~through the tube! between the two mouths]V, ]V8, which may bearbitrarily large, so that the advantage for space travel is not so obvious.

a!Electronic mail: gecl@ccr.jussieu.fr

0022-2488/97/38(11)/5807/13/$10.005807J. Math. Phys. 38 (11), November 1997 1997 American Institute of Physics

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In the case where]V is a cylinder, the internal tube as well as the external space, is flat. Thecurvature is then concentrated on the two wormhole mouths, each of which carries~again by theGaussBonnet theorem! a mass per unit length and a longitudinal tension both equal tom521/4G. For instance, the static conical EinsteinRosen wormholes generated by a circularcylindrical source11 may thus, in the case of a vanishing deficit angle, be extended to WheelerMisner wormholes with zero tube length, which may easily be generalized to the case of anarbitrary tube length.

Let us now discuss the case whereV is a polyhedron. Visser showed4 that in this case thecurvature of the boundary]V is concentrated on the edges, which each carry an equal energy perunit length and tension. A particular case of Vissers polyhedral wormholes is obtained in the limitof cylindrical polyhedra, i.e., configurations ofp parallel cosmic strings of tensionmi , with( imi521/2G. Following the procedure described above, these EinsteinRosen wormholes canbe straigthforwardly extended to WheelerMisner wormholes generated byp straight cosmicstrings in the case of zero tube length, or 2p cosmic strings for an arbitrary tube length.

The purpose of this paper is to investigate in more detail the construction and geometry of flatcylindrical wormholes generated by straight cosmic strings, following an analytical methodcomplementary to the geometrical method outlined above. In the second section we show howspecial multi-cosmic string metrics may be analytically extended12,13 to EinsteinRosen orWheelerMisner multi-wormhole, multi-cosmic string metrics. Because our spacetime is locallyapproximately Minkowskian, we shall be specially interested in asymptotically Minkowskianspacetimes. In the third section, we study in detail the topology and geometry of asymptoticallyMinkowskian, flat WheelerMisner wormholes generated by one or two straight cosmic strings.Geodesic paths through such wormholes are discussed in the fourth section, with applications tospace travel and geometrical optics. Our results are summarized and discussed in the last section.14

II. WORMHOLES FROM COSMIC STRINGS

We start from the well-known multi-cosmic string metric1517

ds25dt22ds22dz2, ~2.1!

where the 2-metric

ds25)i

uz2ai u28Gmi dz dz* ~2.2!

(z[x1 iy) may locally be transformed to the Cartesian form

ds25dw dw* 5du21dv2 ~2.3!

FIG. 1. Construction of almost everywhere flat wormholes:~a! EinsteinRosen wormholes;~b! WheelerMisner worm-holes.

5808 Gerard Clement: Flat wormholes from cosmic strings

J. Math. Phys., Vol. 38, No. 11, November 1997

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(w[u1 iv) with

dw5)i

~z2ai !24Gmi dz. ~2.4!

The spacetime of metric~2.1! is therefore flat outside the worldsheets of the cosmic stringsz5ai @the conical singularities, with deficit angle 8pGmi , of the surface~2.2!#, which carry anenergy per unit length and a tension both equal tomi,1/4G ~for mi>1/4G the singularityz5ai is at spatial infinity!. The metric~2.2! is also generically singular at the point at infinity in thecomplexz-plane, with deficit angle 8pG( imi . This last singularity is at infinite geodesic distanceif

(i

mi

ds25uz2cu28Gm

uz22b2udz dz* , ~2.8!

take the cut to be the geodesic segment connecting the two branch pointsz56b; analyticalcontinuation across this cut then leads to a surface with two symmetrical asymptotically conicalsheets smoothly connected along a cylindrical throat, and two conical singularitiesone in eachsheetatz5c. The corresponding spacetime