Anti-de Sitter space, squashed and stretched

We study the Lorentzian analogues of the squashed 3-sphere, namely 2+1 dimensional anti-de Sitter space, squashed or stretched along fibres that are either spacelike or timelike. The causal structure, and the property of being an Einstein--Weyl space, depe

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aStockholmUSITP05-4September2005RevisedNovember2005ANTIDESITTERSPACE,SQUASHEDANDSTRETCHEDIngemarBengtsson1PatrikSandinStockholmUniversity,AlbaNovaFysikumS-10691Stockholm,SwedenAbstractWestudytheLorentziananaloguesofthesquashed3-sphere,namely2+1

dimensionalanti-deSitterspacesquashedorstretchedalong bresthatareeitherspacelikeortimelike.Thecausalstructure,andthepropertyofbeinganEinstein–Weylspace,dependcriticallyonwhetherwesquashorstretch.Wearguethatsquashing,andstretching,completelydestroystheconformalboundaryoftheunsquashedspacetime.AsaphysicalapplicationweobservethatthenearhorizongeometryoftheextremalKerrblackhole,atconstantBoyer–Lindquistlatitude,isanti-deSitterspacesquashedalongcompacti edspacelike bres.

We study the Lorentzian analogues of the squashed 3-sphere, namely 2+1 dimensional anti-de Sitter space, squashed or stretched along fibres that are either spacelike or timelike. The causal structure, and the property of being an Einstein--Weyl space, depe

1.Introduction

TheHopf brationofthe3-sphereappearsthroughoutmathematicalphysicsinmanyguises;itisusedtodescribequbits,magneticmonopoles,Taub-NUTuniverses,andwhatnot.Thereisabeautifulpicturebehindit:theHopf bresformaspace- llingcongruenceoflinkedgeodesiccirclesinthe3-sphere.IntheTaub-NUTcosmologiesthe3-sphereissquashedalongtheHopf bres.SuchspheresareknownasBergerspheresbymathematicians.TheyaresolutionstotheconformallyinvariantEinstein–Weylequations.Thesquashed3-spherehasaLorentziananalogue.InfactithastwoLorentziananalogues,since3dimensionalanti-deSitterspaceadS3canbesquashed(orstretched)alongHopf bresthatareeitherspacelikeortime-like.Thisconstructionwasbrie ydiscussedbyJones,TodandPedersen

[1][2],becausesuchspacetimesadmitatwistorialdescription(withatwodimensionalfamilyoftotallygeodesicnullhypersurfacesservingastwistorspace[3]).Fromthispointofviewsquashedanti-deSitterspacebecomesinterestingasasimplebutnon-trivialexampleintwistortheory.Ithasalsobeenstudiedasanasymmetricdeformationoftheconformal eldtheorythatdescribesthepropagationofstringsonthegroupmanifoldofSL(2,R)—alsoknownasadS3[4,5].Butthereareotherusesofsuchanaturalconstruc-tion,inparticularthenearhorizongeometryoftheextremalKerrblackhole

[6]canbeunderstoodusingit.Forthisreasonwehavestudiedsquashedanti-deSitterspaceinsomedetail.Wealsouseittopointamoral:wewillarguethatthesquashingcompletelydestroystheconformalboundaryoftheunsquashedspacetime.Thistellsusthatconformalcompacti cation[7]de-pendsmuchmoreonthedetailedstructureofEinstein’sequationsthanonemightperhapsthinkitwould.

Thecontentsofthispaper:Wedescribesomerelevantfeaturesof2+1dimensionalanti-deSitterspaceinsection2,butsincethishasbeendescribedatlengthelsewhere—werecommendref.[8]andreferencestherein—somedetailsarerelegatedtoanAppendix.Insection2weconcentrateonthetwogeodeticcongruences,onetimelikeandonespacelike,thatwillplaytherolethattheHopfcirclesplayforthe3-sphere.Insection3wesquashandstretchourspacetimealongthese bres,discussthesymmetriesoftheresultingspacetimes,and ndtheKillinghorizonsthattheycontain.Section4makessomeobservationsonnullgeodesics;thedistinctionbetweensquashingandstretchingnowbeginstobecomeapparent.Fortimelikestretchingdetailed

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We study the Lorentzian analogues of the squashed 3-sphere, namely 2+1 dimensional anti-de Sitter space, squashed or stretched along fibres that are either spacelike or timelike. The causal structure, and the property of being an Einstein--Weyl space, depe

resultsareavailablealready—weareine ectstudyingtheG¨odelspacetime

[9].Insection5weestablishwhenourspacetimessolvetheconformallyinvariantEinstein–Weylequations.Insection6weattempttoconformallycompactifyourspacetimes,andarguethattheboundaryisdestroyedbysquashing(andstretching).Section7applieswhatwehavelearnedtoadiscussionoftheextremalKerrblackhole.Conclusionsandopenquestionsarebrie ylistedinsection8.

2.Geodeticcongruencesinanti-deSitterspace

Anti-deSitterspaceisde nedasaquadricsurfaceembeddedina atspaceofsignature(+...+ ).Thus2+1dimensionalanti-deSitterspaceisde nedasthehypersurface

X2+Y2 U2 V2= 1(1)

embeddedina4dimensional atspacewiththemetric

ds2=dX2+dY2 dU2 dV2.(2)

TheKillingvectorsaredenotedJXY=X Y Y X,JXU=X U+U X,andsoon.ThetopologyisnowR2×S1,andonemaywishtogotothecoveringspaceinordertoremovetheclosedtimelikecurves.Ourargumentswillmostlynotdependonwhetherthis nalstepistaken.

Forthe2+1dimensionalcasethede nitioncanbereformulatedinaninterestingway.Anti-deSitterspacecanberegardedasthegroupmanifoldofSL(2,R),thatisasthesetofmatrices

g=V+XY+U

Y UV X ,detg=U2+V2 X2 Y2=1.(3)Thegroupmanifoldisequippedwithitsnaturalmetric,whichisinvariant 1undertransformationsg→g1gg2,g1,g2∈SL(2,R).TheKillingvectorscannowbeorganizedintotwoorthonormalandmutuallycommutingsets,

J1= JXU JYV

3 1= JXU+JYVJ(4)

We study the Lorentzian analogues of the squashed 3-sphere, namely 2+1 dimensional anti-de Sitter space, squashed or stretched along fibres that are either spacelike or timelike. The causal structure, and the property of being an Einstein--Weyl space, depe

J2= JXV+JYU

J0= JXY JUV

Theyobey 2= JXV JYUJ 0=JXY JUV.J(5)(6)

1||2=||J 2||2= ||J 0||2=1.||J1||2=||J2||2= ||J0||2=1,||J(7)

LocallySL(2,R)isisomorphicwiththeLorentzgroupSO(2,1).Theisom-etrygroupSO(2,2)isthereforelocallyisomorphictoSO(2,1)×SO(2,1).Thesemattersarediscussedmorefullyinref.[8].Verysimilarthingscan …… 此处隐藏：17972字，全部文档内容请下载后查看。喜欢就下载吧 ……