Black Holes of Constant Curvature

M´aximoBa˜nados

7991 vNo 4 2v2500179/cq-r:gviXraDepartamentodeF´ısica,UniversidaddeSantiagodeChile,Casilla307,Santiago2,Chile,and

CentrodeEstudiosCient´ıficosdeSantiago,Casilla143,Santiago,Chile

Blackholesofconstantcurvatureareconstructedbyidentifyingpointsinanti-deSitterspace.Inndimensionstheresultingtopologyisℜn−1×S1,asopposedtotheusualℜ2×Sn−2Schwarzschildblackhole.

Thegoalofthistalkistoreporttheexistenceofafamilyofblackholeswithconstantcurvaturehavingthetopologyℜn−1×S1,asopposedtotheusualℜ2×Sn−2.TheseblackholeswerefirstdiscussedinRef.[1]infourdimensions,althoughthehigherdimensionalnatureofthecausalstructurewasnotexploitedinthatReference.Theℜn−1×S1existsinanydimensionandcanberegardedasanaturalextensionofthe2+1blackhole[2].Hereweshallonlymakeabriefderivationofthesolution.MoredetailscanbefoundinRef.[3].SeealsoRef.[4]forrelatedwork.

Inndimensionsanti-deSitterspaceisdefinedastheuniversalcoveringofthesurface,

−x20+x21+···+x2n−2+x2n−1−x2n=−l2

.

(1)Considertheboostξ=(r+/l)(xn−1∂n+xn∂n−1)withnormξ2=(r2

+/l2)(−x2parametricallythesurface(1)intermsofthevaluesofξ.Theren−1+

x2

n).Weplot2are

twoimportantvaluesofξ2.First,forξ2=r2

+onehasthenullsurface,

x20=x21+···+x2

n−2,

(2)

whileforξ2=0onehasthehyperboloid,

x20=x21+···+x2n−2+l2.

(3)

Letusnowidentifypointsalongtheorbitofξ.Theregionbehindthehy-perboloid(ξ2<0)hastoberemovedfromthephysicalspacetimebecauseitcon-tainsclosedtimelikecurves.Thehyperboloidisthusasingularitybecausetimelike

geodesicsendthere.Ontheotherhand,thenullsurface(2)actsasahorizonbe-causeanyphysicalobserverthatcrossesitcannotgoback.Indeed,thesurface(2)coincideswiththeboundaryofthecausalpastoflightlikeinfinity.Inthissense,thesurface(1)withidentifiedpointsrepresentsablackhole.

Letusnowintroducelocalcoordinatesonanti-deSitterspace(intheregionξ2

>0)adaptedtotheKillingvectorξ.Weintroducethendimensionlesslocalcoordinates(yα,φ)by,

x2lyα

=

α

xn−1xn

==

lr

l

lr

l

󰀂

,,

󰀂

withr=r+(1+y2)/(1−y2)andy2=ηαβyαyβ[ηαβ=diag(−1,1,...,1)].Thecoordinaterangesare−∞<φ<∞and−∞TheinducedmetrichastheKruskalform,

ds=

2

l2(r+r+)2

2(dθr+

2

+sin2θdχ2).

(6)

Thehorizoninthesecoordinatesislocatedatr=r+,thepointwhereN2vanishes.

Note,however,thatthesecoordinatesaremeaningfulonlyintheexteriorregionandtheycannotbeextendedtorTheblackholejustconstructedhasanEuclideansectorwhichcanbeobtainedbysettingτ=itin(5),ory0→iy0in(4).IntheEuclideansector,the“static”coordinatesdocoverthefullEuclideanmanifold.

Justasin2+1dimensions,angularmomentumintheplanet|φcanbeaddedbyconsideringadifferentKillingvectortodotheidentifications.Thisismosteasilydonebysettingr+=lin(5),makingthereplacements,

t→βt

r+r−

2

ll,,

(7)(8)

φ→βt

(r+>r−arbitraryconstants),andidentifyingpointsalongthenewangularcoor-dinateφ:φ∼φ+2πn.Theconstantr+parametrizesthelocationoftheouterhorizon,andthenewmetrichastwoindependentconservedcharges.IntheEu-clideanformalism,thetimecoordinateτ=−itmustbeperiodicinordertoavoid

22

conicalsingularities.Thisgivesthevalueβ=(2πr+l2)/(r+−r−)[with0≤t<1]whichcanbeinterpretedastheinversetemperatureoftheblackhole.

Sincetheabovegeometriesarelocallyanti-deSitter,theyarenaturalsolutionsofEinsteinequationswithanegativecosmologicalconstant.However,duetothenon-standardasymptoticbehaviourof(5)onefindsthatallconservedchargesareinfinite.GlobalchargesassociatedtotheseblackholescanbedefinedinthecontextofaChern-SimonssupergravitytheoryinfivedimensionsproposedsometimeagobyChamseddine5.ThisactionisconstructedasaChern-SimonstheoryforthesupergroupSU(2,2|N)5.TheenergyMandangularmomentumJoftheblackholeembeddedinthissupergravitytheoryare,

M=

2r+r−

l

S=4πr−.

.

(9)

Theentropyontheotherhandisequalto,

(10)

ThisresultisrathersurprisingbecauseitdoesnotgiveanentropyproportionaltotheareaofS1(2πr+).AsimilarphenomenahasbeenreportedbyCarlipetal6.Theentropygivenin(10)satisfiesthefirstlaw,

δM=TδS+ΩδJ,

(11)

whereMandJaregivenin(9)andT=1/β.

DuringthisworkIhavebenefitedfrommanydiscussionswithAndyGomberoff,MarcHenneaux,Cristi´anMart´ınez,ClaudioTeitelboimandJorgeZanelli.IwouldalsoliketothankPeterPeld´anformanyremarksandusefulsuggestions,andDi-eterBrillforcommentsonapreviousversionofthemanuscript.Thisworkwaspartiallysupportedbythegrant#1970150fromFONDECYT(Chile),andin-stitutionalsupportbyagroupofChileancompanies(EMPRESASCMPC,CGE,COPEC,CODELCO,MINERALAESCONDIDA,NOVAGAS,ENERSIS,BUSI-NESSDESIGNASS.andXEROXChile).

1.S.Aminneborg,I.Bengtsson,S.HolstandP.Peldan,Class.Quant.Grav.13,2707(1996)2.M.Ba˜nados,M.Henneaux,C.TeitelboimandZanelli.,Phys.Rev.D48,1506(1993).3.M.Ba˜nados,gr-qc/9703040,toappearinPhysicalReviewD.

4.Forrelatedworksee,R.B.Mann,gr-qc/9709039;W.L.SmithandR.B.Mann,gr-qc/9703007;S.HolstandP.Peldan,gr-qc/9705067;D.Brill,J.LoukoandP.Peldan,Phys.Rev.D56,3600(1997);L.Vanzo,gr-qc/9705004;J.D.E.CreightonandR.B.Manngr-qc/9710042

5.A.H.Chamseddine,Nucl.Phys.B346,213(1990).

6.S.Carlip,J.Gegenberg,R.B.Mann,Phys.Rev.D51,68(1995).

3