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−∞ ds= 2 l2(r+r+)2 2(dθr+ 2 +sin2θdχ2). (6) Thehorizoninthesecoordinatesislocatedatr=r+,thepointwhereN2vanishes. Note,however,thatthesecoordinatesaremeaningfulonlyintheexteriorregionandtheycannotbeextendedtor 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 因篇幅问题不能全部显示,请点此查看更多更全内容
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