Flat wormholes from cosmic strings

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<ul><li><p>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 </p><p> This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:</p><p> On: Mon, 22 Dec 2014 10:26:25</p><p>http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcovhttp://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/578858469/x01/AIP-PT/CiSE_JMPArticleDL_121714/Awareness_LibraryF.jpg/47344656396c504a5a37344142416b75?xhttp://scitation.aip.org/search?value1=Grard+Clment&amp;option1=authorhttp://scitation.aip.org/content/aip/journal/jmp?ver=pdfcovhttp://dx.doi.org/10.1063/1.532167http://scitation.aip.org/content/aip/journal/jmp/38/11?ver=pdfcovhttp://scitation.aip.org/content/aip?ver=pdfcovhttp://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4791721?ver=pdfcovhttp://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.3012280?ver=pdfcovhttp://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.2399673?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/ppn/29/1/10.1134/1.953058?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/jmp/38/1/10.1063/1.531827?ver=pdfcov</p></li><li><p>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</p><p>~Received 5 February 1997; accepted for publication 9 June 1997!</p><p>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#</p><p>I. INTRODUCTION</p><p>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</p><p>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 .</p><p>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.</p><p>a!Electronic mail: gecl@ccr.jussieu.fr</p><p>0022-2488/97/38(11)/5807/13/$10.005807J. Math. Phys. 38 (11), November 1997 1997 American Institute of Physics</p><p> This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:</p><p> On: Mon, 22 Dec 2014 10:26:25</p></li><li><p>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.</p><p>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.</p><p>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</p><p>II. WORMHOLES FROM COSMIC STRINGS</p><p>We start from the well-known multi-cosmic string metric1517</p><p>ds25dt22ds22dz2, ~2.1!</p><p>where the 2-metric</p><p>ds25)i</p><p>uz2ai u28Gmi dz dz* ~2.2!</p><p>(z[x1 iy) may locally be transformed to the Cartesian form</p><p>ds25dw dw* 5du21dv2 ~2.3!</p><p>FIG. 1. Construction of almost everywhere flat wormholes:~a! EinsteinRosen wormholes;~b! WheelerMisner worm-holes.</p><p>5808 Gerard Clement: Flat wormholes from cosmic strings</p><p>J. Math. Phys., Vol. 38, No. 11, November 1997</p><p> This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:</p><p> On: Mon, 22 Dec 2014 10:26:25</p></li><li><p>(w[u1 iv) with</p><p>dw5)i</p><p>~z2ai !24Gmi dz. ~2.4!</p><p>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&gt;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</p><p>(i</p><p>mi</p></li><li><p>ds25uz2cu28Gm</p><p>uz22b2udz dz* , ~2.8!</p><p>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~2.1! is therefore a two-cosmic string EinsteinRosen wormhole. Note that the masses per unit lengthm of the sources are different from thetotal masses per unit lengthM determined from the asymptotic behavior of the metric at eitherregion at spatial infinity,18,13</p><p>M51</p><p>4G1m, ~2.9!</p><p>the difference 1/4G being the topological contribution of the wormhole. For the spatial sections tobe open (M</p></li><li>is the union of the left and right regions discussed above, and has a closed boundarygeodesicG of length 2l @the segment~u50, 2 l</li><li><p>However, the metric~2.15! admits a more economical analytical extension to a topologicallynontrivial Riemann surface with only one sheet. Such a possibility derives from the observation13</p><p>that the torus may be pinched only once into a single tetracone joined to itself by an identificationof the two edges. This identification corresponds to an identification of the two cuts, leading to theidentification of the two sheets, of the Riemann surface for the complex variablez(w), the pointz in the first sheet of the symmetrical extension being identified with the point2z in the secondsheet~other possible identificationsz6z* between the two sheets would lead to the nonorient-able manifoldU2!. A large circle geodesicv5const. is thus mapped into a line connecting the twocuts either in the upper or in the lower half-plane of thez-plane, so that a particle going around thetorus along this geodesic falls into, e.g., the left-hand cut to come out again from the right-handcut ~Fig. 4!.</p><p>Such a one-sheeted extension is possible whenever the distribution of both then cuts and thep conical singularities of the flat metric~2.12! is invariant under the isometryz2z, so that thetwo sheets of the symmetrical extension may be identified together. In the casen52 the resultingsurfacea topological torus with a point at infinity~provided M,1/4G! and p conicalsingularitiesis a WheelerMisner wormhole. In fact, our construction is the three-dimensionalcounterpart of Lindquists4 reinterpretation of a four-dimensional EinsteinRosen manifold withtwo identical spherical bridges as a single WheelerMisner wormhole by identifying correspond-ing points on the two sheets of the EinsteinRosen manifold. In the next section we investigate thevarious possible geometries for asymptotically Minkowskian (M50) WheelerMisner worm-holes withp51 or 2.</p><p>III. TWO-STRING AND ONE-STRING ASYMPTOTICALLY MINKOWSKIANWHEELERMISNER WORMHOLES</p><p>The WheelerMisner wormhole generated by two cosmic strings is the one-sheeted extensionof a symmetricaln5p52 metric ~2.12!. In the asymptotically Minkowskian case, this metric</p><p>ds25uz22c2u2</p><p>u~z22a2!22b4udz dz* ~3.1!</p><p>depends on three complex parametersa, b, c. According to the relative values of these param-eters, the nonextended geometrical configuration may belong to one of three possible generic typesDD, AA, or Q.</p><p>~1! Dipoledipole~DD!. The sequences of closed u-geodesics surrounding each of the twosymmetrical geodesic cuts terminate in two disjoint geodesic segments, each connecting one of thesingularities with itself. A single v-geodesic segment, bissecting the angles formed by the con-tinuations of these critical u-geodesics, connects the two singularities together. An instance of thiscase isa,b,c real, a2.b21c2 @Fig. 5~a!#. The analytical extension of this geometry to theRiemann surface obtained by identification of the two cuts, as described in the previous section,leads to the DD wormhole geometry. The geometrical construction of this wormhole@Fig. 5~b!#follows closely that of the dipole EinsteinRosen wormhole, except that the two copies of the</p><p>FIG. 4. A torus is pinched into a tetracone, mapped to a one-sheeted Riemann surface with two cuts identified under theinvolution P8P.</p><p>5812 Gerard Clement: Flat wormholes from cosmic strings</p><p>J. Math. Phys., Vol. 38, No. 11, November 1997</p><p> This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:</p><p> On: Mon, 22 Dec 2014 10:26:25</p></li><li><p>(u,v) plane deprived of a semi-infinite strip are replaced by a single Euclidean plane deprived ofa rectangular strip. Two opposite edges, of length 2d, of this rectangle are glued together, whilethe other two edges, of length 2l , are glued to the two ends of a truncated cylinder of circumfer-ence 2l and length 2L.</p><p>~2! Antidipoleantidipole~AA !. In this type again, two disjoint critical u-geodesic segmentsconnecting each of the two singularities with itself enclose concentric closed u-geodesics sur-rounding a cut. However, the two singularities are now connected by two symmetrical v-geodesicsegments, bissecting the two angles formed by a critical closed u-geodesic segment and its con-tinuation to infinity @Fig. 6~a!, drawn for a, b, c real, b2,a2,c22b2#. The correspondingWheelerMisner wormhole geometry turns out to be equivalent to that of the Q wormhole, as weshall presently explain.</p><p>~3! Quadrupole~Q!. In this case the sequences of closed u-geodesics surrounding the two cutsterminate in two contiguous geodesic contours made from three u-segments connecting the twosingularities together@Fig. 6~b!, drawn for a and b real with b2,a2, and c imaginary#. Thecritical v-geodesics bissecting the two angles formed by this se...</p></li></ul>