Mergers and Typical Black Hole Microstates

Mergers and Typical Black Hole Microstates

hep-th/0608217

NSF-KITP-06-26

CERN-PH-TH/2006-151

arXiv:hep-th/0608217v2 8 Sep 2006MergersandTypicalBlackHoleMicrostatesIosifBena(1),Chih-WeiWang(2)andNicholasP.Warner(2,3)(1)SchoolofNaturalSciences,InstituteforAdvancedStudyEinsteinDr.,Princeton,NJ08540,USA(2)DepartmentofPhysicsandAstronomy,UniversityofSouthernCaliforniaLosAngeles,CA90089-0484,USA(3)DepartmentofPhysics,TheoryDivisionCERN,Geneva,SwitzerlandWeusemergersofmicrostatestoobtainthe rstsmoothhorizonlessmicrostatesolu-tionscorrespondingtoaBPSthree-chargeblackholewithaclassicallylargehorizonarea.Thesemicrostateshaveverylongthroats,thatbecomein niteintheclassicallimit;nev-

ertheless,theircurvatureiseverywheresmall.Havingaclassically-in nitethroatmakesthesemicrostatesverysimilartothetypicalmicrostatesofthisblackhole.AroughCFTanalysiscon rmsthisintuition,andindicatesapossibleclassofdualCFTmicrostates.

Wealsoanalyzethepropertiesandthemergingofmicrostatescorrespondingtozero-entropyBPSblackholesandblackrings.We ndthatthesesolutionshavethesamesizeasthehorizonsizeoftheirclassicalcounterparts,andweexaminethechangesofinternalstructureofthesemicrostatesduringmergers.

August2006

Mergers and Typical Black Hole Microstates

1.Introduction

In[1]wefoundaverylargenumberofgeometriesthathaveexactlythesamesuper-symmetries,size,chargesandangularmomentaasmaximally-rotating,three-chargeBPSblackholesandblackringsin vedimensions.Thesesmooth,horizonlesssolutionshavenobranesources,andbelongtotheclassconstructedandanalyzedin[2,3],basedonearlierworkin[4-7].

ThesesolutionsareverysimilartoBPSblackholesandBPSblackrings[8,9,6,10,7]andonlydi erfromthosenearthewould-be“classical”horizon.Inthenew,horizonlesssolutionsthisregionissmoothandcompact,andcontainsalargenumberoftopologicallynon-trivialtwo-cycles.Hencethesesolutionsarecalled“bubbledblackholes”and“bubbledblackrings.”AllthesesolutionscanbedualizedtoaframeinwhichtheyareasymptotictoAdS3

holesand×S3black×T4,andthereforearedualtostatesintheD1-D5-PCFTthatdescribesblackrings.UnderstandingwhetherthiskindofsupergravitysolutionsaredualtotypicalCFTstatesisperhapsamongthemostimportantquestionsinunderstandingblackholesinstringtheory1.

Forpracticalreasons,allthe ve-dimensional,three-chargeBPSmicrostatesthathavebeenconstructed[13-17,1,2,3]useasabaseafour-dimensional“generalizedhyper-K¨ahler”metric2thathasatri-holomorphicU(1)symmetry,andisthusaGibbons-Hawking(GH)metric.Nevertheless,asrecentlyshownin[1],usingaGibbons-Hawkingbasegenericallyappearstoyieldbubbledsolutionscorrespondingtoblackringsandblackholesofzerohorizonarea.Indeed,itappearsthatnoneofthemicrostatesolutionscurrentlyintheliteraturehasthesamechargesandangularmomentaasablackholewithaclassicallylargehorizonarea.

ItisveryimportanttounderstandwhetherthislimitationarisesfromusingaGibbons-Hawkingbase.Ifthiswereso,thenwewouldneedto ndmoregeneralclassesofhyper-K¨ahlerbasemetrics,andtheseareveryhardtoconstructandanalyzeexplicitly.Beforeembarkingonthisdi culttask,oneshouldtherefore rstattempttoconstructmicrostates

Mergers and Typical Black Hole Microstates

ofpositive-entropyblackholesstartingfromaGibbons-Hawkingbase.Againthisispri-marilybecauseofcomputationalconvenience:Suchsolutionswouldbemucheasiertoobtainexplicitly,analyze,andrelatetothecorrespondingCFTmicrostates.

Anobviousplacetostartwouldbetoexaminethemergeroftwoblackholes,whichisalwaysirreversible.Nevertheless,suchacon gurationcanpreserveatmostandSO(3)thathasnotri-holomorphicU(1)subgroups;thereforethemergeroftwoblackholemicrostatescannotbedescribedusingaGHbasemetric.Ontheotherhand,themergerofablackholewithablackringinitsequatorialplanepreservesaU(1)×U(1)symmetry.Hence,usingaresultof[18],weexpectthemergerofthecorrespondingmicrostatestobedescribedusingaGHbasemetric.

ThemergerofaBPSblackholeandaBPSblackringcanbereversibleorirreversible,dependingonthechargesofthetwoobjects[19].Onethereforeexpectsthemergerofmi-crostatestoresultinanzero-entropyBHmicrostateoranpositive-entropyBHmicrostate3,dependingonthechargesofthemergingmicrostates.Moreover,sincethemergercanbeachievedbykeepingthebaseGibbons-Hawking,oneexpectstheresultingpositive-entropyBHmicrostatetohaveaGibbons-Hawkingbase.

Themainpurposeofthispaperistostudythemergerofzero-entropyBRmicrostatesandzero-entropyBHmicrostates.Weconstructsolutionsthatdescribealarge,bubbledblackringwithabubblingblackholeinthecenter,andreducesome uxparametersinordertobringthemtogether.We ndthat,whenseenfromfaraway,themergerofmicrostatescloselyparallelsthemergerofthecorrespondingclassicalcounterparts.Inparticular,themergerhappensatthesamevaluesofthechargesandangularmomenta,andtheresultingmicrostateisalwaysthatofaBMPVblackhole.

Ontheotherhand,we ndthatthereisahugequalitativedi erencebetweenthebehavioroftheinternalstructureofmicrostatesin“reversible”and“irreversible”mergers4.A“reversible”mergerofanzero-entropyBHmicrostateandanzero-entropyBRmicrostateproducesanotherzero-entropyBHmicrostate.We ndthatthebubblescorrespondingto

Mergers and Typical Black Hole Microstates

theringsimplyjointhebubblescorrespondingtotheblackhole,andformabiggerbubbledstructure.

Inan“irreversible”merger,astheringbubblesandtheblackholebubblesgetcloserandcloser,we ndthatthedistancesbetweentheGHpointsthatformtheblackholefoamandtheblackringfoamalsodecrease.Asoneapproachesthemergerpoint,allthesizesintheGHbasescaledowntozerowhilepreservingtheirrelativeproportions.Inthelimitin√whichthemergeroccurs,thesolutionshaveJ1=J2<

Mergers and Typical Black Hole Microstates

intheclassicallimit,whilethethroatofthepositive-entropyBHmicrostatesbecomesin nite.Intherestofthispaperwewillrefertothemicrostatesofpositive-entropyBH’sas“deep”microstates5,andtothezero-entropyBHmicrostatesas“shallow”microstates.

Aswehavementionedabove,allthesolutionsweconstructaredu …… 此处隐藏：16927字，全部文档内容请下载后查看。喜欢就下载吧 ……