1510.02182v1

BaochengZhang∗

SchoolofMathematicsandPhysics,

ChinaUniversityofGeosciences,Wuhan430074,China

Abstract

Basedonarecentproposalforthevolumeinsideablackhole,wecalculatetheentropyassociatedwiththisvolumeandshowthatsuchentropyisproportionaltothesurfaceareaoftheblackhole.Togetherwiththeconsiderationofblackholeradiation,wefindthatthethermodynamicsassociatedwiththeentropyishighlypossibletobecausedbythevacuumpolarizationnearthehorizon.

I.INTRODUCTION

Howbigisablackhole?Thisisnotaneasyquestion,sincethedefinitionofthevolumeofthespaceinsideablackholedependsonhowthespacetimeisslicedintospaceandtime,unlikethesurfaceareaoftheblackholethatisthesameforallobservers.Intuitively,anicedescriptionforthevolumeshouldbeslicinginvariant,whichwasmadefirstlybyParikh[1]andalsodiscussedbyothers[2–6].Recently,adifferentmethodwassuggestedbyChristodoulouandRovelli[7]basedonasimpleobservationthattheinteriorofSchwarzschildblackholesisnotstatic,whichleadedtoanothersensibledescriptionforthevolumeasthelargestvolumeboundedbytheeventhorizonofablackhole.ForaSchwarzschildblackhole,theyshowedthatatlatetimethevolumetooksuchanexpression,

VCR∼3

√

Thestructureofthepaperisasfollows.Firstly,wewillrevisitthedefinitionofthevolumeinsideablackholebyChristodoulouandRovelli,andpresentexplicitlythechoiceofthemaximalhypersurfaces[14]bymaximalslicing[15,16]inthesecondsection.Then,wecalculatetheentropyinthevolumeusingthestandardstatisticalmethodinthethirdsection.Intheforthsection,wediscussthefirstlawofblackholethermodynamics,inparticularforthenewlyobtainedentropy,whichisrelatedtothevacuumpolarization[17]nearthehorizonoftheblackhole.Finally,wewillgiveaconclusioninthefivesection.

II.BLACKHOLEVOLUMEANDMAXIMALSLICING

StartwiththegeometryofacollapsedobjectasinRef.[7],whichcanbedescribedwiththeEddington-Finkelsteincoordinates,

ds2=−f(r)dv2+2dvdr+r2dθ2+r2sin2θdφ2

wheref(r)=1−

2M

f(r)

(2)

=t+r+2Mln|r−2M|.In

particular,wehavetakentheunitsG=c=󰀁=kB=1.Theadvantageofthecoordinatesoverthestaticoneisthatthereisnocoordinatesingularityattheeventhorizon.Thusitcanbeanalyticallycontinuedtoallr>0,whichisrequiredforthedescriptionofthegeometryofthecollapsedmatter.

Inordertocalculatethevolume,aproperhypersurfacehastobechosen.

With

antransformationv→v(T,λ),r→r(T,λ),thecoordinates(2)becomesds2=󰀃󰀄󰀁∂v󰀂2

∂r∂v

−f+2∂λdλ2+r2dθ2+r2sin2θdφ2wherewehave∂T∂λ

assumedthecrosstermvanishesbytakingthetransformationproperly.Inparticular,ifthe

󰀁∂v

∂r√condition−f∂TthehypersurfaceΣ:T=constant,isthatchoseninRef.[7]wherethesphericallysymmetric

−fdTwhichalsoremovesthecrosstermsimultaneously)isenforced,onewillfindthat

hypersurfaceistakenasthedirectproductofa2-sphereandanarbitrarycurveparameter-izedbyλinv-rplane.Inparticular,itisnotedthatthehypersurfaceT=constantisable

√

tobegottenbyr=constantaccordingtothetransformationdr=

wherethedotrepresentsthepartialderivativewithregardtotheparameterλ,and−f(r)v˙2+2v˙r˙>0forthespacelikehypersurface.Themaximalvolumecanbeobtainedbytheintegral,

VΣ=4π󰀊dλ

󰀋

2

M,(5)

which,togetherwithEq.(4),givestheCRvolumeexpressedinEq.(1).

Ontheotherhand,themaximizationcanalsobecalculatedinmathematicalrelativity[18]whereamethodcalledasmaximalslicingcanleadtohypersurfacesofmaximalvolumewhichhavevanishingmeanextrinsiccurvature,K=0,whereKisthetraceoftheextrinsiccurvatureofthehypersurface.HereweshowthevanishingKisequivalenttothecondition(5).AccordingtoRef.[16],weusethecoordinates(2)andtakethespacelikehypersurfacesbyr=constant,sincethetimeandspaceareregardedasbeinginterchangedacrossthehorizonofaSchwarzschildblackhole.TakenasthefuturepointingtimelikeunitnormaltothehypersurfacesΣr,

n=

easytoconfirmthatg󰀋

∂r

+1∂v󰀆.

Itisµνnµnυ=−1forthecoordinates(2).ItisnotedthatthereisasimplerelationbetweenthedivergenceofthevectornandthetraceoftheextrinsiccurvaturetensorKµν,K=−▽·nwhere▽isthecovariantderivativewithrespecttothespacetimemetric.Withthegivenmetric(2),wehave

K=

1

󰀅∂f−fr󰀆=0,whichgivestheequationr=

3

hastobecountedforonecertainphase-spacewhichcanbelabeledherebyλ,θ,φ,pλ,pθ,pφ.Fromtheuncertaintyrelationofquantummechanics,∆xi∆pi∼2π,onequantumstatecor-respondstoa“cell”ofvolume(2π)3inthephase-space.Therefore,thenumberofquantumstatesisgivenby

dλdθdφdpλdpθdpφ

2

M.Forthemaximalslicing,

theslicesaccumulateonalimitinghypersurfacer=

3

√

−ggµν∂νΦ)=0,weobtain

2

E−

∂λ

1

r

∂φ

p22θ

−

1

,pθ=

∂I

.Thus,followingtheEq.(6),thenumberofquantumstates

5

withtheenergylessthanEisobtainedas

g(E)=

1

󰀊󰀋󰀇

󰀊

−f(r)v˙2+2v˙rdλdθdφ˙dλ󰀋

(9)

E2−

a2

1

=

(2π)

1

3

󰀅

E2−2π

1

r2sinθ

2

p2φdpθdpφ

12π212π2

4π

VCR,

󰀋

p2

r2sin2θφ

wheretherelationpλ=

󰀉󰀉󰀌

theintegralformula

isusedinthesecondline,

finallinewehaveusedthecondition(5).Asexpected,thenumberofquantumstatesisproportionaltotheCRvolume,butthisisstilldifferentfromanormalsituation,becausetheCRvolumeistheresultofthecurvedspacetime.

TemporarilyignoringtheexoticfeatureofCRvolume,wecancontinuetocalculatethefreeenergyatsomeinversetemperatureβ,

F(β)=

1

−

y2

3

abisusedinthethirdline,andinthe

=−=−

VCR

eβE−1

eβE−1

π2VCR

∂β

=

π2VCR

dv

=−

1

whereγisaconstantwhosevaluedoesnotinfluencethediscussioninthepaper.InRef.[10],itisdiscussedthatthelargevolumeisremaineduntilthefinalstageofblackholeevaporation.Hereweareconcernedaboutthetimethattheradiationcanlast.Thus,forablackholewiththemassM,wehave

v∼γM3,

(13)

whichalsosatisfiestherequirementinRef.[7],v>>M.ThenusingtheEqs.(1)and(11),andconsideringthetemperatureforaSchwarzschildblackhole,β=T−1=8πM,onefindtheentropy

SCR∼󰀁3√(45×83)=(90󰀁3

√×84)π

A,(14)

whereA=16πM2isthesurfaceareaoftheSchwarzschildblackhole.ThisisasurprisingandintriguingresultthattheentropyfromquantumtheoryintheCRvolumeisproportionaltothesurfaceareaoftheblackholehorizonthatboundsthevolume.Notethattheresultisdependentonthevalidityoftherelation(12),whichwasshown[21]toholdsolongasthemassofaSchwarzschildblackholeisgreaterthanthePlanckmass.Moreover,theexactdescriptionoffinalstageofblackholeevaporationhasnotexisted,butsomespecificconsiderationssuchasGeneralUncertaintyRelation(i.e.seethereview[22])impliedthatatthefinalstage,thelossmassratebecomes

dM

4

.Thenanatural

problem:whatistheentropySCR?Whetheritwillbeincludedinthefirstlawofblackholethermodynamics.Inwhatfollows,wewilldiscussit.

IV.FIRSTLAWANDVACUUMPOLARIZATION

SincethederivationofCRvolumeisinthecaseofv>>M,whichmeanstheblackholehasformedbythecollapseofthematterandisstaticfortheexternalobservers.Thus,duetotheHawkingradiation,thefirstlawofblackholethermodynamicsreadsas

dM=TdSH,

(15)

wheretheentropySH=

1

processesarecompleted.ThechangeoftheentropySHisequivalenttothechangeofthesurfacearea,sotheentropySCRintheCRvolumewillbechanged.AlthoughourearlieranalysisshowsthattheentropySCRisproportionaltothesurfacearea,itsoriginalforminEq.(11)iscloselyrelatedtotheCRvolume.Soinequilibrium,thethermalprocessrequiresatermrelevanttothechangeofCRvolume.Accordingtothegeneralthermodynamics,suchtermshouldbewrittenasPdVCRwherePisthepressure.Thus,weexpectthatthefirstlawcanbewrittenas

dM=TdS−PdVCR,

(16)

whereS=SH+SCR.Inordertoremainthevalidityofthefirstlaw,arelationisrequired,TdSCR∼PdVCR.Now,thevitalproblemisthatfromwhereandhowmuchthepressureis,somestudiespointedoutthatthenonlocaleffectwillpreventtheblackholefromcollapsingintoaphysicalsingularity,i.e.seeRef.[24]).

IntheoriginalcalculationofHawking[11],theconceptofparticlesareusedintheasymp-toticallyflatregionfarfromtheblackholewheretheycanbeunambiguouslydefined.Whiletheparticlefluxcarriesawaythepositiveenergy,anaccompanyingfluxofnegativeenergygoesintotheholeacrossthehorizon,whichcanonlybeunderstoodbythezeropointfluc-tuationsoflocalenergydensityinthequantumtheory.Thisphenomena,calledalsoasvacuumpolarization,willplayanimportantroleintheneighborhoodoftheblackhole.Thevacuumpolarizationisusuallyconsideredinthesemi-classicalEinsteingravity,inwhichthefluctuationsofgravitationalfieldaresmallandsoistheexpectationvalueoftheenergy-momentumtensoroftherelevantquantizedfieldsinthechosenvacuumstate.Bycausedbythevacuumpolarizationisgivenby[25–28],

P=

1

M4

solvingthemodifiedEinsteinequation,Gµν=8π󰀉Tµν󰀊,thequantumpressureatthehorizonafterthecollapsedmatterhadbeenconcentratedintothesingularity(itisalsonotedthat

.

3

(17)

ItisnotedthattheCRvolumeiscalculatedforthehypersurfacer=

fromthecosmologicalconstant,i.e.seeRefs.[29–31],buttheyaredifferentfromourssincethecosmologicalconstantisnotinvolvedhere.ThetermPdVCRiscalculatedas

PdVCR

󰀁√53=dM,(90×84)π

(18)

whichisonthelevelof10−5fortheparameterbeforedM,andisverysmall,asexpected.ThetermTdSCRisalsoeasytobeestimatedas

󰀁√43−5

TdSCR=dM∼10dM.

(90×84)π

(19)

ItisseenthatthetermTdSCRisnotequalexactlytothetermPdVCR,butfundamentallytheycanbecanceledonthelevelof10−5.ThereasonisthatthevaluesofthepressurePandthetemperatureTisnotsoexactinthepresentconsideration.Actually,adirectevidencethatthethermodynamicsintheCRvolumeiscausedbyvacuumpolarizationisthatthetermPdVCRisnotdependentontheblackholeparameterM,asobtainedforthetermTdSCR.

Finally,itisnotedthatiftakingsuchatermVCRdP,asmadeinRefs.[29–31],itcanbalanceexactlythechangeofthermodynamicscausedbyTdSCR,butnoevidenceshowsthatwhythepressurehereshouldbechanged,sotheexactcancelationmightbeoccasional.Infact,vacuumpolarizationcausesthequantumcorrectiontotheusualblackholethermo-dynamics,sothecorrespondingthermodynamicvariableswouldbecorrected,whichisthereasonwhythetermsPdVCRandTdSCRcanceleachotheroutapproximately.

V.CONCLUSION

Inthepaper,wehaveinvestigatedtherelationbetweenthederivationofthevolumeofthespaceinsideaSchwarzschildblackholedefinedbyChristodoulouandRovelliandthemaximalslicing,andfoundexplicitlythattheCRvolumewasjustobtainedforthehypersurfacewhosemeanextrinsiccurvatureiszero.WehavealsocalculatedtheentropyintheCRvolumethroughcountingthenumberofquantumstatesinthevolumewithastandardstatisticalmethod.Differentlyfromthenormalsituation,theentropyassociatedwiththeCRvolumeisproportionaltothesurfaceareaoftheblackhole,buttheparameterismuchsmallerthanthatrequiredfortheBekenstein-Hawkingentropy.Thesmallparameter

9

hasalsobeeninterpretedinthepaper,fromtheperspectiveofblackholethermodynamics,forwhichasuggestiveresultisgiventhatthethermodynamicsassociatedwiththeentropyintheCRvolumeiscausedbythevacuumpolarizationnearthehorizon,sincethematterhascollapsedintothesingularitywhenweinvestigatethisphenomena.Thus,ourresultverifiesfurthertherelationbetweenblackholephysicsandquantumtheoryagain.

VI.ACKNOWLEDGE

TheauthorwouldliketothankProf.C.Rovelliforreadingthispaperandhispositivecomments.ThisworkissupportedbyGrantNo.11374330oftheNationalNaturalScienceFoundationofChinaandbyOpenResearchFundProgramoftheStateKeyLaboratoryofLow-DimensionalQuantumPhysicsandbytheFundamentalResearchFundsfortheCentralUniversities,ChinaUniversityofGeosciences(Wuhan)(CUG150630).

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