The Information Loss Problem of Black Hole and the

arXiv:hep-th/9705100v210Mar1998May97OU-HET266hep-th/9705100TheInformationLossProblemofBlackHoleandtheFirstOrderPhaseTransitioninStringTheoryKenjiHotta,1DepartmentofPhysics,GraduateSchoolofScience,OsakaUniversity,Toyonaka,Osaka560,JAPANABSTRACTInrecentyears,Susskind,ThorlaciusandUglumhaveproposedamodelforstringsnearablackholehorizoninordertorepresentthequantummechanicalentropyoftheblackholeandtoresolvetheinformationlossproblem.However,thismodelisinsufficientbecausetheydidnotconsiderthemetricmodificationduetomassivestringsanddidnotexplainhowtocarryinformationfrominsideofthehorizontotheoutsideworld.Inthispaper,wepresentapossible,intuitivemodelforthetimedevelopmentofablackholeinordertosolvetheinformationlossproblem.Inthismodel,weassumethatafirstorderphasetransitionoccursneartheHagedorntemperatureandthestringgaschangestohypotheticalmatterwithvanishingentropyandenergywhichwecall‘thePlancksolid’.Wealsostudythebackgroundgeometryofblackholesinthispictureandfindoutthatthereisnosingularitywithinthemodel.1E-mailaddress:[e-mail]11IntroductionImaginethatweareobservingthegravitationalcollapseofastartoablackholefromadistance.Accordingtogeneralrelativity,themotionoftheconstituentsofthestarshouldlookextremelyslowbecauseoftimedilationinthestronggravitationalfield.Whenwouldtheblackholebeformedforus?AccordingtoHawking’stheorem,theblackholewouldradiateoutlikeablackbodywithfinitetemperature[1].WhatisthemechanismoftheHawkingradiation?Allthematterfallsintothesingularity,andnothingisknownafterward.Then,wherehastheinformationgone?Inthispaper,wegiveonepossibleresolutiontothesenaivequestions.Theinformationlossproblemstatedaboveisoneofthemostseriousdifficultiesinpresentphysics[2].TheessenceofthisproblemisthefactthatthequantumstateofHawkingradiationdoesnotdependontheinitialstateofthecollapsingbody.Thisisbecausethestateoftheradiationisdeterminedonlybytheoutsidegeometryoftheblackholehorizon.LetussupposethattheHilbertspaceHisfactorizedintoHinandHout,whichcorrespondtotheinnerandoutersidesoftheeventhorizon,respectively.Takingthestates|ψin(t)i∈Hinand|ψout(t)i∈Hout,wecanrepresentthestate|ψ(t)ioftheentirespaceas|ψ(t)i=|ψin(t)i⊗|ψout(t)i.(1)Theinformationoftheinitialstateofthecollapsingbodyisincludedin|ψin(t)i,since|ψout(t)i,whichconsistsoftheHawkingradiationstate,doesnotdependontheinitialstate.Thisimpliesthattheinformationislostforever,becausenomattercanescapefromtheblackhole.Inotherwords,sincethereisnocorrelationbetweentheinitialstate|ψinitialiandthefinalstate|ψfinaliwhichappearswhentheHawkingradiationhasfinished,wecannottracethetimedevelopmentofthequantumstatesoftheblackholefromtheinformationofthecollapsingbodyaslongasweremainwithintheframeworkoftheusualquantumtheoryandgeneralrelativity.Thisfactimpliesthattheunitarityofquantumtheorydoesnothold.Arelatedquestionarisesinquantumstatisticalmechanics.Iftheblackholehasentropy,itmustbeabletoberepresentedbycountingthenumberofquantumstates.’tHoofthasshownthatifoneintroducesaspatialcutoffataboutaPlancklengthseparatedfromtheeventhorizon,onecanrepresenttheBekenstein-Hawkingentropybycomputingtheentropyofscalarfieldsoutsideoftheblackhole[3].Thisimpliesthattheinformationmaybestoredoutsidethehorizon.Inrecentyears,Susskind,ThorlaciusandUglumhaveconstructedamodelforstringsneartheblackholehorizoninordertoresolvetheproblemofinformationlossbyrepresentingtheBekenstein-Hawkingentropyquantummechanically[5]∼[10].Theyprovedthatforanobserverfarfromtheblackhole,stringsneartheeventhorizonarethermalizedandstoredbetweentheeventhorizonandthestretchedhorizon,thelatterofwhichispositionedastring-scalelengthawayfromtheformer.Theyalsoprovedthat,inparticularwhentheblackholemassisinfinite,wecanrepresenttheblackholeentropybycountingthenumberofstringstates.Thesefactssuggestthatinformationregardingthematterfallingintotheblackholeisstoredinthelongstringsnearthehorizon.Letuscallthismodelthe‘stretchedhorizonmodel’.However,therearetwoproblemsinthismodel.Wemustconsiderthemechanism2underwhichinformationiscarriedoutsideofthehorizonandthemetricmodificationbymassivestrings.Wewillseedetailsoftheseproblemsin§2,togetherwithareviewofthestretchedhorizonmodel.Wewishtoreconstructthismodelwhilekeepingitsadvantages.In§3,weshowthatifweassumetheexistenceofmatterwithvanishingentropyandenergy,whichwecall‘thePlancksolid’,wecanconstructascenarioinwhichtheinformationlossproblemdoesnotexistfromtheviewpointofanoutsideobserver.Inthisscenario,theinformationisexcludedfromtheinsideoftheblackholeduetothecharacterofthePlancksolid,anditisstoredinthestringgasnearthehorizon,asinthestretchedhorizonmodel.Thisstringgasprovidesalmostallthecontributiontothemassenergyoftheblackhole.Then,weproposeatoymodel,whichislaterreferredtoasthe‘stringbitmodel’,thatgivesusanintuitivepictureofthePlancksolidin§4.Itisconcludedthatasthetemperaturegrows,thestringgasintheweakcouplingregionchangestothePlancksolidinthestrongcouplingregionbyafirstorderphasetransition.AfirstorderphasetransitioninstringtheoryhasbeenproposedbySathiapalan[17],Kogan[18],AtickandWitten[19].Thisisseenin§4.Withre