Thermodynamics of Large AdS Black Holes

arXiv:0709.3738v2[hep-th]10Oct2007PreprinttypesetinJHEPstyle.-PAPERVERSIONRH-05-2007ThermodynamicsofLargeAdSBlackHolesSamuliHemmingAcademyofFinlandVilhonvuorenkatu6P.O.Box99,00501Helsinki,FinlandE-mail:[e-mail]´arusThorlaciusUniversityofIceland,ScienceInstituteDunhaga3,107Reykjavik,IcelandE-mail:[e-mail]:WeconsiderleadingorderquantumcorrectionstothegeometryoflargeAdSblackholesinasphericalreductionoffour-dimensionalEinsteingravitywithnegativecosmologicalconstant.TheHawkingtemperaturegrowswithoutboundwithincreasingblackholemass,yetthesemiclassicalback-reactiononthegeometryisrelativelymild,indicatingthatobserversinfreefalloutsidealargeAdSblackholeneverseethermalradiationattheHawkingtemperature.ThepositivespecificheatoflargeAdSblackholesisastatementaboutthedualgaugetheoryratherthananobservablepropertyonthegravityside.ImplicationsforstringthermodynamicswithanAdSinfraredregulatorarebrieflydiscussed.Keywords:blackholethermodynamics,semiclassicalgravity,AdS/CFTcorrespondence.Contents1.Introduction12.Classicalsolutions42.1AdS–Schwarzschildblackholes42.2Self-gravitatingradiation63.Semiclassicalgeometry83.1One-loopaction83.2Staticsolutions94.Energyconsiderations124.1Classicaltotalenergy124.2Semi-classicalenergy135.Discussion141.IntroductionBlackholesplayanimportantroleintheAdS/CFTcorrespondence[1,2,3].Inparticular,theso-calledlargeAdS–Schwarzschildblackholes,whosehorizonareaislargeinunitsofthecharacteristicAdSlengthscale,correspondtohigh-temperaturethermalstatesinthedualgaugetheory[4,5].AnothermotivationforstudyingAdSblackholescomesfromstringthermodynamics.Theinfraredinstabilityofaself-gravitatinggasofstringscanberegulatedbyembeddingthesysteminanasymptoticallyAdSbackgroundandinthiscontextlargeAdS–Schwarzschildblackholesarethedominantconfigurationsatsufficientlyhightotalenergy[6].LargeAdS–SchwarzschildblackholesdifferfromordinarySchwarzschildblackholesinasymptoticallyflatspacetimeinanumberofways,butperhapsthemoststrikingdifferenceisthattheHawkingtemperatureoflargeAdS–Schwarzschildblackholesgrowsasafunctionoftheblackholemass[5].Inthissensetheseblackholeshavepositivespecificheat.DuetothecurvatureintheasymptoticAdSregion,HawkingradiationfromablackholegetsreflectedbackandatthequantumleveloneexpectstofindequilibriumconfigurationswhereablackholeemitsHawkingradiationandabsorbsparticlesfromtheenvironmentatthesamerate.InSec-tion3weconsidersemiclassicalcorrectionstotheAdS–Schwarzschildblackhole1geometry,whichareobtainedviasphericalreductionoffour-dimensionalgravity.WelookforstaticsolutionstodescribeequilibriumconfigurationsinvolvinglargeAdS–Schwarzschildblackholesandstudysomeoftheirphysicalproperties.FormallythereisnoupperlimitonthemassofalargeAdS–SchwarzschildblackholeandtheHawkingtemperaturecanbearbitrarilyhigh.Doesthismeanthatalargesemiclassicalblackholeisinequilibriumwithahigh-temperatureenvironment?Ouranswerisno,butbeforewegettothatletusconsiderthequestionmorecare-fully.Firstofall,inanasymptoticallyAdSgeometrythereisinfinitegravitationalredshiftasonegoestospatialinfinityandthereforethelocaltemperaturegoestozeroasymptoticallyfarfromaAdS–Schwarzschildblackhole,regardlessofhowhightheHawkingtemperatureis.Thequestionisratherwhetherthereisaregionneartheblackhole,i.e.withinaproperdistanceofordertheAdSlengthscaleorsooftheeventhorizon,wherethelocaltemperaturebecomesofordertheHawkingtemperature.Theanswertothisquestiondependsonwhatkindofobserverwehaveinmind.Letusfirstconsiderafiducialobserver,whoiskeptatrestwithrespecttothestaticAdS–Schwarzschildcoordinates.SuchanobserverwillindeedmeasurehightemperaturesnearalargeAdS–SchwarzschildblackholebutthiscanbeattributedtotheaccelerationrequiredtokeeptheobserverfromfallingintotheblackholeandisthesameeffectasfoundoutsideanordinarySchwarzschildblackhole[7,8].Inparticular,thisfiducialtemperaturegoestoinfinityasweapproachtheblackholeeventhorizonbutwedonotexpectthistoleadtostrongbackreactioneffectsforAdS–SchwarzschildblackholesanymorethanitdoesforSchwarzschildblackholes.Itismoreinterestingtoaskwhattemperaturewouldberecordedbyanobserverinfreefallneartheblackhole.Thisistherelevanttemperaturewhenweconsiderquantumcorrectionstotheblackholegeometry.Iftheblackholeisinequilibriumwithalayerofhigh-temperatureradiation,thentheproperenergydensityishighinthatlayerandonewouldexpectlargecorrectionstothegeometry.ThiswouldhavetobereconciledwiththefactthatinthelimitoflargeblackholemassthespacetimecurvatureinthehorizonregiondoesnotgrowbeyondtheAdSscale,aswewillseeinsection2.1,andthusapriorithesizeofquantumcorrectionsisnotexpectedtogrowwithincreasingblackholemass.ThequestionofthelocaltemperatureoutsideablackholealsocomesupinthecontextofstringthermodynamicsinanasymptoticallyAdSbackground.TheJeansinstabilityundermineseffortstodefineathermodynamiclimitinagravitationaltheorysuchasstringtheorybutthetroublesomeinfraredbehaviorcanberegulatedbyintroducinganegativecosmologicalconstant.Atweakcouplingandlowenergiesthereexistsphericallysymmetricequilibriumconfigurationsconsistingofagasofstringsconfinedtoalocalregioninanasymptotica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