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

2

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].Verysimilarthingscanbesaidaboutthe3-sphere.

Herewewouldliketodescribeacoordinatesystem(τ,ω,σ)[10],analo-goustotheEuleranglesthatareusedtodescribethe3-sphere.TothisendweparametrizeanarbitrarySL(2,R)matrixas

g(τ,ω,σ)=cos

sinττ2

2(8)

StraightforwardcalculationsshowthattheKillingvectorsinthe rstSO(2,1)factorare

2sinhσ sinh coshωω22 exp σ2 .J1=

coshω τ+2sinhσ ω 2tanhωcoshσ σ.(11)

ThesecondSO(2,1)factorisspannedby

1=2sinτtanhω τ 2cosτ ω+2sinτJ

0=2 τ.J

4coshω σ(13)(14)

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

0,towhichWewillfocusonthemutuallycommutingKillingvectorsJ2andJ

ourcoordinatesystemisadapted.Theyformtwonowherevanishingvector eldsinadS3.Inanyodddimensionalanti-deSitterspacewecanconstruct 0,whilethereisnoanowherevanishingtimelikevector eldanalogoustoJ

similarhigherdimensionalanalogueforJ2.Butindimension3wehavethesetwoeverywherevanishingvector eldstoplaywith.Eachofthemde nesaninterestingcongruenceinanti-deSitterspace,andtheir owlinesaretheHopf bresalongwhichwewillsquashandstretchourspacetime.Themetriconanti-deSitterspacetakestheform

ds2=1

4

=1 (dτ sinhωdσ)+dω+coshωdσ2222(16)

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

Figure1:ThispictureisdrawnusingthesausagecoordinatesfromtheAp-pendix.Itshowsanti-deSitterspaceasacylinder(withaconformalbound-ary).Thetimelikecongruenceconsistsoftimelikespiralsrulingasetofhelicoids.Totherightweshowthatthe owbecomesnullontheboundary.horizonsinadS3occurforconjugacyclasseswherethetransformationstaketheform(hyperbolic)×(hyperbolic);theyarenumerousenoughsothateveryspacelikegeodesicisthebifurcationlineofsuchaKillinghorizon.De-generateKillinghorizonsoccurfortransformationsoftheform(parabolic)×(parabolic).Theyformatwoparameterfamilyoftotallygeodesicnullsurfaces,andcanberegardedaslightconeswithverticesontheconformalboundaryJ

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

Figure2:AgainusingsausagecoordinatesweshowthenullsurfaceX=V,andhowitisruledbythespacelikecongruence.Totherightweshowthatthe owbecomesnullontheboundary,andwhereithas xedpoints.Thereisaspecialpointactingasasinkforallthosemembersofthecongruencethatbelongtothenullsurfaceshown.

thesurface

XV VY=0 σ=0(19)

is atandminimal.WedrawitinFig.1,usingthesausagecoordinatesfrom 0= t+ φ,theAppendix.InsausagecoordinatestheKillingvectorbecomesJ

thecongruenceconsistsasetofhelices,andthesurfaceσ=0isknownasthehelicoid.(Itisaminimalsurfaceincoordinatespacetoo.)NotethatthegeodesicsbecomenullontheconformalboundaryJ

,althoughinthiscasetherearetwolinesof xed

points;therearesourcesatt φ=π/2andsinksatt φ= π/2.Insideanti-deSitterspacethecongruenceiseverywherespacelike,andeveryPoincar´ediskde nedbyt=constantcontainsonememberofthecongruence.Surfacesofconstantτ,whichare atandminimal,areruledbythesegeodesicsbutareratherhardtodraw.Anothersurfacethatisruledbythesegeodesicsisthetotallygeodesicnullsurface

7

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

X=V.(20)

InfactthissurfacecontainseveryJ2geodesicthatgoestoaparticularsinkonJ

oncewedecidetosquash

ourspacetimealongthese bres(keepingalldistancesorthogonaltothe bresconstant).Ontheboundarythe bresarechangingcharacter,fromtimelike/spaceliketonull.ThereforesquashingJ

consistingoftwodisconnectedcomponents.Somewherealongtheway,somethinghastobreak.

3.Squashing,stretching,andsymmetries

ItistimetointroducethespacetimeanaloguesoftheBergersphere.WeobtainthembysquashingadS3alongoneofthetwocongruencesdescribedintheprevioussection.Theresultingspacetimeswillbehomogeneousbutanisotropic,andwewillstudytheirsymmetriesinsomedetail.

Letusconsiderthespacelikecase rst;ithassomespecialfeaturesthat,intheend,makethiscasetheeasiesttounderstand—especiallyifFig.2iskept rmlyinmind.ThemetriconsquashedadS3is

ds2λ=1

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

howeverthat—unlikeitsanalogueforthe3-sphere—thisparticularresultdoesnothaveanystraightforwardhigherdimensionalanalogue.

Becauseofthesquashingtheisometrygroupisnowfourdimensional.TheLiealgebrachangesfromSO(2,1)×SO(2,1)fortheunsquashedspacetimetoR×SO(2,1)forthesquashedone;theleftfactorheregivestransforma-tionsbelongingtoahyperbolicconjugacyclassofSO(2,1).ThequestionweaskiswhetheranyKillinghorizonssurvive.Theanswerisyes.TherewillbebifurcateKillinghorizonscomingfromtransformationsofthetype(hyper-bolic)×(hyperbolic),althoughtheywillbelessnumerousthantheywereinanti-deSitterspace.ThedegenerateKillinghorizonsthatwerepresentintheunsquashedcasearenolongerwithus,sincetheycamefromtransforma-tionsofthetype(parabolic)×(parabolic).ButtherewerealsototallynullKillingvector eldsinanti-deSitterspace,comingfromtransformationsofthetype(identity)×(parabolic).OncewehavedonethesquashingthiswillgiveusasupplyofdegenerateKillinghorizons,asareplacementforthosethatwerelost.

Butwedonothavetorelyonanypreviousresultshere.Ashortcalcu-lationveri esthatthemostgeneralKillingvector eldthathasaspacelikecurveof xedpointsis(uptoscale)

1+bJ 2+cJ 0,ξ=J2+aJ

wheretherealnumbersa,b,cobey

a2+b2 c2=1.(24)(23)

ThisisatimelikesurfaceinthegroupmanifoldofSO(2,1).The xedpointsoccurat

ba.(25)sinτ= 2222a+ba+b

Thesecurvesarepreciselythe bresalongwhichwearesquashing.TheyarealsobifurcationcurvesforbifurcateKillinghorizons.Hencesquashedanti-deSitterspacecontainsatwoparameterfamilyofbifurcateKillinghorizons.Theunsquashedspacetimehasmore:inanti-deSitterspaceitselfeveryspacelikegeodesicisabifurcationlineforsomeKillinghorizon.

Ifwepickanexampleinthisclass,we ndthatsinhω=c,cosτ=

9

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

2||=(1 cosτcoshω)λ+1 (λ 1)cosτcoshω||J2+J222

Thesurfacegravityκisgivenby|κ|=2.Afeaturethatarisesonlyinthesquashedcaseisthatthereareactuallytwosurfaceswherethenormvanishes,butonlyoneofthemisaKillinghorizon—theotherisatimelikesurface.

AoneparameterfamilyofdegenerateKillinghorizonsarisefromtheKillingvectors

(α)=cosαJ 1+sinαJ 2+J 0.ξ

Thistimethenormis

(α)||2=(λ2 1)(sinhω+sin(τ α)coshω)2.||ξ(28)(27) .(26)

InadS3theseKillingvectorsareeverywherenull.Inthesquashedcase(λ2<1)theyaretimelikeexceptforadegenerateKillinghorizonwherethenormvanishes,andinthestretchedcase(λ2>1)theyarespacelikeagainexceptforadegenerateKillinghorizon.Inanti-deSitterspaceitselfthisfamilyofnullsurfacesisidenticaltothefamilygivenineq.(21),ifwesetα=2β+π.Intheanti-deSittercasethereareadditionaldegenerateKillinghorizonsthatdisappearwhenwesquashorstretch.

SinceweareprimarilyinterestedinKillinghorizonsbecausetheyaretotallygeodesicnullsurfaces,itisenoughtoconsiderdegeneratehorizons—thebifurcateonesdonotcontributeanythingnewinthisway.

Nextwesquashorstretchalongthetimelikecongruence.Thenthemetricis

ds2λ=1

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

whichiseverywherenullinanti-deSitterspace.After(timelike)squashingweobtain

||ξ||2= (λ2 1)(coshωcoshσ sinhω)2.(31)

Thisistimelikeorspacelike,dependingonwhetherwesquashorstretch.

4.Nullgeodesics

Ourspacetimeshaveenoughsymmetriestoensurethatthegeodesicequationcanbeseparated.Itisparticularlyinterestingtotakealookattheequationsfornullgeodesics,becausethereisasurprisewaiting.Wewilldiscussthecaseofspacelikesquashinginsomedetail,andcommentbrie yontimelikesquashingattheend.Forarelateddiscussion,includingsomeinterestingobservationsontimelikegeodesics,seeBardeenandHorowitz[6].Webeginbyintroducingtheconvenientcoordinate

w=sinhω.(32)

Usingit,itisstraightforwardtobringtheequationsforanullgeodesicwithrespecttothemetric(22)totheform

τ˙=

1+λwsinφw

λ

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

tothisdirection.Notealsothatwehavecomeacrossnullgeodesicsorthog-onaltothesquashingdirectiononcebefore—theyruletheKillinghorizonsdepictedinFig.2.

Letusnowassumethatsinφ=0.Asymptotically,thatistosayforlargevaluesofw,weobtain

τ˙～λsinφ

λ

w˙2～(λ2 1)w2sin2φ.(37)(38)

Evidentlyitispossibletoreacharbitrarilylargevaluesofwonlyifλ2≥1,thatistosayonlyforstretching,notforsquashing.Thisisthesurprisethatwewerereferringto.

Theexplicitsolutionforw(s)canbewrittendown,butisnotveryilluminating—wegettheexpectedoscillatorybehaviourforsquashing,whilestretchinggivesanexponentiallygrowingfunction.Togoon,whenλ2>1weseethat

w→∞ σ→±∞.(39)

Thisisaverydi erentkindofbehaviourfromthatoccurringintheun-stretchedanti-deSittercase.Ine ect,asymptoticallythenullgeodesicsareliningupwiththenullgeodesicsthatruletheKillinghorizonsdescribedintheprevioussection.Theimplicationsofthiswillbediscussedinsection6.Forthecaseoftimelikesquashing bresdetailedresultsareavailableintheliteraturealready.Thisisbecause,byaddinganextra atdirection,andspecializingthestretchingparametertoλ2=2[9],theresulting3+1dimensionalspacetimeisthefamousG¨odelsolution.AnelegantreviewofitsnullgeodesicshasbeengivenbyOstv´athandSchucking[13];tofollowthemweusethecoordinatesystemgivenjustbeforeeq.(91)intheAppendix,andperformthefurthercoordinatechanges

1

1 R22RT=,sinhθ=

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

Thisbringsthemetrictotheform

ds2=1

sin(φ+φ0).(42)λ

Settingφ0=0andtradingRandφforCartesiancoordinatesonthePoincar´ediskgives

λyx2+y2 x+y 2 1

2λ 2.(43)

Thisisacircle.Thefamilyofnullgeodesicsthroughtheorigin,projecteddowntothePoincar´edisk,arecircleswhoseenvelopeisacirclewithradius1/λ.

Thuswhenλ2>1allnullgeodesicsarecon nedtotheinteriorofstretchedanti-deSitterspace.WhenR=1theconformalfactorinfrontofthemetricdiverges,sothatwhathappenswhenλ=1isthatthenullgeodesicstouchJ

(1 R2)2(1 λ2R2).(44)

Henceλ2>1impliesthatthereareclosedtimelikecurvesbeyondtheenve-lopeofthenullgeodesics;this,ofcourse,wasoneofG¨odel’smainpoints.Inthesquashedcasenosuchthinghappens.Indeed

13

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

dT=02 ds=21

(48)2

′′GivenaWeylspace,thepair(gab,ωc)=(e gab,ωc+ c )de nesaWeyl

spacetoo.

TheWeylconnectionhasacurvaturetensorde nedby

[Da,Db]Vc=WabcdVd.

14(49)bb(δaωc+δcωa gacgbdωd).

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

Wealsode ne

Wab=WacbcandW=gabWab.(50)

NotethatWabisnotsymmetricingeneral.Acalculationshowsthat

W(ab)=Rab+

W[ab]=3

214Fab,ωaωb+gab 14ω2 (51)(52)

wheresquareandroundbracketsdenoteanti-symmetrizationandsymmetriza-tion,respectively.Noticethede nitionofFab.Itiseasytoseethat

Wabcd=Wab

[cd]+Fabgcd,(53)

andmoreover—becausewearein3dimensions—

Wabcd= abe cdf1

Wgab.3

FortheordinaryRiccitensorthisimpliesthat

Rab+1

4ωaωb=gab (55)12 cω+c1

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

wheretheone-formξaistheKillingvector eldthatde nesthesquashing,

ξa= aσ+sinhω aτ (aξb)=0.(58)Rab+λ2(λ2 1)ξaξb=2(λ2 2)gab,(57)

ThereforeweobtainasolutionoftheEinstein–Weylequationifweperformtherescaling

1λ2(λ2 1)ξa.(59)

Curiouslyarealsolutionisobtainedonlyforλ2≥1,thatistosayifwestretchanti-deSitterspace,butnotifwesquashit.Fortimelikesquashing,weobtainarealsolutionwhenwesquashbutnotwhenwestretch;thisisalsotruefortheRiemannianBergersphere[2].Wedonotfullyunderstandwhythisshouldbeso.Weobservethat,inanti-deSitterspace,spacelikegeodesicstendtodiverge,andtimelikegeodesicstendtoconverge.Geodesicsonthe3-spheretendtoconvergeaswell.Perhapsmoretothepoint,intheprevioussectionwenotedthatnullgeodesicsbehaveverydi erentlydependingonwhetherthespacetimeissquashedorstreched.

WiththeWeylconnectioninhandwecande neanewnotionofgeodesiccurves.Wewillcontinuetorefertogeodesicswithrespecttoourchosenmetricsas“geodesics”,whilegeodesicswithrespecttotheWeylconnec-tionwillbecalled“Weylgeodesics”.Cartanprovedthat—atleastaftercomplexi cation—athreedimensionalEinstein–Weylspaceadmitsatwopa-rameterfamilyofnullhypersurfacesthataretotallygeodesicwithrespecttotheWeylconnection.Itisthistwodimensionalspacethatisusedasamini-twistorspacebyJonesandTod[1].Intheanti-deSittercasethemini-twistorspacecanbeidenti edwithJ

can

beregardedasthevertexofapastlightconewhichistotallygeodesic—thisistrueforthedeSittercaseaswell).Itwouldbeinterestingtoseeexplicitlywhatthesenullsurfacesareinthesquashedcases.Wedonotknow,butwewillshowthatthedegenerateKillinghorizonsthatwefoundforspacelikesquashing,eq.(28),dobelongtothisset.

FollowingPedersenandTod[2],letusanalyzetheWeylgeodesics.Fromeqs.(47–48)itisseenthattheyobey

16

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

x˙bDbx˙a=x˙b bx˙a Ex˙a+1

2x˙2ωa=1

√

ω2.Asymptotically,theseWeylgeodesicslineupwiththesquashingdirection.

ItisbynowevidentthatthedegenerateKillinghorizonsthatwefoundforspacelikesquashingaretotallygeodesicwithrespecttotheWeylconnection.BeingKillinghorizonstheyaretotallygeodesicwithrespecttothemetricconnection.Nullgeodesicscoincideforbothconnections,andthespacelikeWeylgeodesicdeviatefromthespacelikemetricgeodesicsinthedirectionofthesquashing eld—whichasweknowistangentialtotheKillinghorizons.(Thisargumentdependscriticallyonthefactthatthesquashing eldis

17

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

tangentialtotheKillinghorizon.Wewouldnotbesurprisedtolearnthatthesearetheonlynullsurfacesinourspacetimesthataretotallygeodesicintheordinarysense.)

ForanyEinstein–WeylspacewithaspacelikeωaweobservethateverynullgeodesicbelongstosomenullsurfacethatistotallygeodesicwithrespecttotheWeylconnection[2].SinceallspacelikeWeylgeodesicseventuallylineupwithωa,thishasconsequencesforthebehaviourofthenullgeodesics“closetoin nity”—aphrasethatwewillexamineinmoredetailinthenextsection.

6.Conformalcompacti cation?

Wewillnowpointourmoral.ItconcernsthefragilityofJ

.

Thea neparametersonnullgeodesicswillbe nitewhentheyreachJ

isa

nullhypersurfaceifthecosmologicalconstantvanishes,whileitistimelike(spacelike)fornegative(positive)cosmologicalconstant.Buttheargumentthatleadstothisconclusion[7]reliesontheEinsteinequations,andbecomesvoidforthecaseswestudy.

ForourpurposeswewillinsistthatJ

canberegardedasthevertexofapastdirectedlightcone,with

anon-zerofractionofitsgeneratorsbelongingtotheoriginalspacetime.Letus rstrecalltheconformalcompacti cationofordinaryanti-deSitterspace,usingourunusualcoordinates.Astandardchoiceofconformalfactoris[8]

18

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

2=1/(U2+V2)=2/(coshωcoshσ+1).

Usingit,theconformallyrelatedmetricbecomes

ds 2= 2ds2=1(66)

.Actuallythiswill

giveus“onehalf”ofJ

isfoundtobe

ds 2=

coordinatesareuandv,where

tanu= sinhσ

Weseethatσ=±∞isanulllineonJ

is

ds 2= dudv.

ThespacetimeKillingvectors,restrictedtoJ

isatimelikesurface

ina2+1dimensionalEinsteinuniverse.

Squashedorstretchedanti-deSitterspacecannotworkquitelikethis.Thisactuallyfollowsfromthediscussioninsection4.Forspacelikestretchingthe(ideal)endpointsofthesetofnullgeodesicsformaonedimensionalset,andthereforetheycannotformaJ(70)v=τ.(69)dτdσ.Amoreconvenientchoiceof

atfuturein nity,weexpect

theendpointstoformazerodimensionalset.Fortimelikesquashingthesituationisagainlessclear,butitseemslikelythatthiscaseissimilartothatofspacelikestretching.SoweconcludethattherecanbenoJ

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

Letusnowproceedinadirectmannertoseeifwearriveatthesameconclusion.Alookatthemetricineq.(22)showsthat,assoonasλ=1,theasymptoticdependenceonωchangesdramatically.Togeta niteexpressionwemustchoosesomethinglike

2=4/sinh2ω

(uptosomefactorthatremains niteinthelimit).Then

22ds 2λ= dsλ=(71)1

2

Butweknowfromeq.(57)that

R=2(λ2 4). R 4g a bln 2g aln bln .abab (74) willdivergewhenω→∞,unlesstheasymptoticItisthenclearthatR

behaviourof iscarefullyadjusted.Thechoice ～exp( ω/2)thatwe ,butforallλ>0themadeforanti-deSitterspaceleadstoa niteR

choice ～exp( ω)givesacurvaturesingularityinsteadofawellde nedconformalboundaryatin nity.

Thisargumenthasits aws.Althoughds2isascalar,itdoesnotre-allyhaveaninvariantmeaning.Itissimplythelengthsquaredofavectorthatinaparticularcoordinatesystemhasthe nitecomponentsdτ,dω,dσ.However,sinceourunderstandingofthenullgeodesicsledustothesameconclusion,wedaretoclaimthattheresimplycannotbeanyconfor-malcompacti cationofsquashedorstretchedanti-deSitterspace,inany

20(75)

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

conventionalsense.Initselfthisisnotaverysurprisingconclusionsincethereismoretoconformalcompacti cationthanjustLorentziangeometry.ThemoralisthatifwedeviatefromEinstein’sequations,wecourtdis-aster.IntheEinstein–WeylcasesonemightthinkthattheexistenceofatwoparameterfamilyoftotallyWeylgeodesicnullsurfacesshouldsomehowguaranteetheexistenceofJ

+ρdθ+22 sin2θ

( r2+a2)2 a2sinθ2, ≡2Mr ae2ν

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