How Fast Does Information Leak out from a Black Hole

Howfastdoesinformationleakoutfromablackhole?

JacobD.Bekenstein*

DepartmentofPhysics,UniversityofCaliforniaatSantaBarbara,

SantaBarbara,CA93106

and

TheRacahInstituteofPhysics,HebrewUniversityofJerusalem,

GivatRam,Jerusalem91904,Israel**

PACS:97.60.Lf,95.30.Tg,04.60.+n,05.90.+m

ABSTRACT

Hawking’sradiance,evenascomputedwithoutaccountofbackreaction,departsfromblackbodyformduetothemodedependenceofthebarrierpenetrationfactor.Thustheradiationisnotthemaximalentropyradiationforgivenenergy.Bycomparingestimatesoftheactualentropyemissionratewiththemaximalentropyrateforthegivenpower,andusingstandardideasfromcommunicationtheory,wesetanupperboundonthepermittedinformationoutflowrate.Thisisseveraltimestheratesofblackholeentropydecreaseorradiationentropyproduction.Thus,ifsubtlequantumeffectsnotheretoforeaccountedforcodeinformationintheradiance,theinformationthatwasthoughttobeirreparablylostdowntheblackholemaygraduallyleakbackoutfromtheblackholeenvironsoverthefulldurationofthehole’sevaporation.

Followinghistheoreticaldiscoveryoftheblackholeradiancethatbearshisname,Hawkingnoted[1]thatsuchradiationseemstocontradictacceptedquantumphysics.Ifablackholeformsfrommatterpreparedinapurestate,andthenradiatesawayitsmassinostensiblythermalradiation,oneisleftwithahighentropymixedstateofradiation.ThiscontradictsthequantumdogmathatapurestatewillalwaysremainpureunderHamiltonianevolution.Arelatedcontradictionfollowsfromtheinterpretationofblackholeentropyasthemeasureoftheinformationhiddenintheblackholeaboutthewaysitmighthavebeenformed[2].Sincefullythermalradiationisincapableofconveyingdetailedinformationaboutitssource,thatinformationremainssequesteredastheblackholeradiates,andwhenitfinallyevaporatesaway,theinformationislostforever.Thesetwocontradictionsarefacetsoftheblackholeinformationlossparadox.

Threereactionstotheparadoxarepossible(forreviewsseeRefs.3and4).Thefirstistoacceptthelossofinformationandthetrasmutationofpureintomixedstateasaninevitableconsequenceofthemergingofgravitywithquantumphysics[1].Specificschemesforaccomplishingthishavebeenfoundtobeincompatiblewithlocalityorconservationofenergy[5].Asecondpointofview[1,6]holdsthatblackholeevaporationleavesamassiveremnantofPlanckdimensionswhichretainsalltheinformationinquestion.Thispossibilityisnotasconservativeasitsounds.Accordingtotheboundonspecificentropyorinformation[7],orconsiderationsfromquantumgravity[8],anobjectofPlanckmassanddimensioncanholdonlyafewbitsofinformation,sothatthepositedmassiveremnantscannotfittheinformationbillofalargeevaporatingblackhole.Variationsoftheremnantidea,theirmeritsandproblemshavebeendiscussedinRefs.8,9and4,amongmany.Yetathirdview[10,11]isthatexploitingsubtlecorrelationsintheradiation,theinformationmanagestoleakbackoutfromtheincipientblackholeinthecourseoftheevaporation.Theleakcannotbepostponedtothelatestagesofevaporationwithoutincurringtheproblemsaccompanyingremnants[6,4].Forinformationleakthroughouttheevaporationtobeareasonableresolutionoftheparadox,itmustbeshownthataninformationflowoftheappropriatemagnitudecancomeoutoftheblackhole’snearenvirons.Astepinthisdirectionistakeninthepresentpaper.

Latelythesethreeviewpointshavebeenwidelyexaminedbymeansofthe1+1dimensionsdilaton-gravitymodelofanevaporatingblackholeproposedbyCallan,Giddings,HarveyandStro-minger[12].Thismodelallowsexplicittreatmentofthequantumradianceanditsbackreactiononthehole(forreviewsseeRef.3).Whateverthefinaloutcomeofthistypeofinvestigations,itwillbenontrivialtoprojecttheconclusionsfromthismodeltotherealisticcaseof1+3blackholes.Therefore,anynewmodelindependentaproachwhichcanaddressthe1+3dimensionalcasewouldbeofgreatconceptualhelp.Wehereemployathermodynamicargument(whichfactmakesitvirtuallymodelindependent)toshowthatforthe1+3dimensionalSchwarzschildblackhole,anoutflowofinformationoftherequiredmagnitudetoresolvetheinformationproblemispermittedinprinciple.Wedonotexploreherespecificmechanismsforinformationextraction,butnotethatthissubjectshasalreadyreceivedattention[13].

AlthoughtheHawkingradiancehasthermalfeatures,ascertifiedbytheexponentialdistribution

2

ofthenumberofquantaemittedineachmode,andthelackofcorrelationsbetweenmodes[14],itisnotpreciselyofblackbodyform.ThewouldbeblackbodyspectrumisdistortedbythemodedependenceofthebarrierpenetrationfactorΓsjmp(ω),wheresstandsfortheparticlespecies,jandmfortheangularmomentumquantumnumbersandpforthepolarization,withΓ<1ingeneral[1].ForaSchwarzschildblackholeofmassM,inversetemperatureβbh=8πGM/¯handentropySbh,theaverageenergyinamodeis(henceforthwesetc=1)

εsjmp(βbh,ω)=

hωΓsjmp(ω)¯

holetospontaneouslyemitnquantainmode{i,ω}isgivenby[14]

psp(n)=(1±e−γi)∓1e−γin

whereγi(βbh,ω)isdefinedby

1

hωeβbh¯

(2)

±1

γi

(3)

Fromthisfollowstheentropyinthegivenmode:

σi(βbh,ω)=±ln(1±e−γi)+

eγi±1

Theentropyoutfluxrateandthepowermaynowbeexpressedas

󰀂

dω·󰀁∞

S=σi(βbh,ω)

i

0

(5)

2π

wheredω/2πistherateatwhichmodesoftypeiemanatefromthehole.

(7)

··

Page[17,18]hascalculatednumericallythecontributionsofvariousparticlespeciestoEandS,

···

andstatestheresultsintermsofthedimensionlessratiosµ≡E(GM)2¯h−1andν≡S/(βbhE).Foreachspeciesoflightneutrinosorantineutrinoshefindsµ=4.090×10−5andν=1.639withmodeshavingj=12,5

·

Atinversetemperatureβeff,ablackbodyofareaAphotemitspowerE′=Nπ2Aphot/(60βeff4¯h3),whereNistheeffectivenumberofparticlespeciesemitted.Photonsandgravitonscontribute1eachtoN;eachspeciesoffermionscontributes7/16.Therefore,wemustsetN=37/8iftherearethree

·

lightneutrinospeciesandN=15/4iftherearetwo.ComparingtheseresultsforE′withPage’s

·

resultsforEweobtainβeff=1.230ξ1/4βbhforthreelightneutrinosandβeff=1.271ξ1/4βbhfor

·

two.Forblackbodyradiationflowinginthreespacedimensions,S′=4

ex±1

=

π2(3∓1)

2βeff

2

·=E′=

π

forabosonorfermionmode,respectively.Summingovertheparticlespecies

󰀃󰀃

andmodeswhichPageconsidered,weobtainigi=90forthreelightneutrinosandigi=78for

····

two.EquatingE′withPage’sEgivesβeff=11.48βbhandS′=22.97βbhEiftherearethreelight

··

neutrinoswhileβeff=12.68βbhandS′=25.36βbhEiftherearetwo.Usingthecitedvaluesofνweconcludethat

····Imax=S′−S=21.35|Sbh|

5

(11)

forthreelightneutrinos.Ifthereareonlytwo,thenumericalfactoris23.75.

·

AlthoughtheabovefigureforImaxisanoverestimate,itissolargeastosuggestthataninformationleakofsuficientmagnitudetoresolvetheinformationproblemisallowed.Forexample,··

ifI,theactualinformationoutflowrate,amountsto1.619|Sbh|throughoutthecourseofevaporationofamassiveblackholedowntoM≈1×1014g(whentheemissionofmassiveparticlesbecomesimportantandmostoftheinitialblackholeentropyhasdisappeared[17]),theoutgoinginformationequalsthetotalHawkingradianceentropy.Hence,givenanappropriatemechanism,theradiationcanendupinapurestate.

·

ButhowexageratedistheaboveboundonImax?Weshallnotattempttoexcludethelowfrequencymodesbyhandfromourcalculation;suchataskwouldbefraughtwithambiguities.Ratherweask,ifitwerepossibletomodifythecurvaturebarriersurroundingtheblackhole,andconsequentlytomodifytheΓi(ω),whatwouldbethemostentropicspectrumthatcouldcomeoutoftheblackhole?Asweshallseepresently,theanswerisnotblackbody:theΓi(ω)cannotallbeunity.However,thenewspectrumismorerelevantforcomparisonthanthepureblackbodyonebecause,forgivenangularmomentum,itdoessupresslowfrequencymodes.

·

IfwecouldmanipulatetheΓi(ω),thelargestentropyflowS′wouldbeobtainedwiththeΓi(ω)aslargeaspossible.Thisisseenbydifferentiatingσi[Eq.(4)]withrespecttoγi,andtransformingthederivativetoonewithrespecttothecorrespondingΓi(ω)withhelpofEq.(3);theresultispositivedefinite.WearethusinterestedinthehypotheticalsituationwhenalltheΓi(ω)areaslargeasphysicallypossible.ForfermionmodesnoreasonisknowntopreventΓi(ω)fromapproachingunity.However,forbosonmodesthevalueofΓi(ω)issubjecttoabound.

Thisboundstemsfromtheformula

hω)Γ(ω)Γi(ω)=(1−e−βbh¯i0

(12)

where1−Γi0istheprobabilitythatasingleincidentquantumisscatteredbackfromtheblackhole[20,21].Formula(12)followsbycombinatoricsfromtheinterpretationintermsofacombinationofscattering,andspontaneousandstimulatedemissionoftheconditionalprobabilityp(m|n)thattheSchwarzschildblackholereturnsoutwardmquantainamodewhichhadnincidentones.Thep(m|n)hasbeenobtainedindependentlybyinformationtheoretic[20]andfieldtheoretic[22]methods.ItturnsouttobeimpossibletounderstanditsformasduetoacombinationofHawkingemissionandscattering[20];theinclusionofstimulatedemissionsuppliesthemissingelement.ThestimulatedemissiondepressesthevalueofΓi(ω)underthenaiveabsorptionprobabilityΓi0(ω).Infact,because

hω.(ThecaseΓ=Γ=1isnotactuallyexcludedbytheconsiderationsΓi0≤1,Γi(ω)≤1−e−βbh¯ii0

ofRef.20,butaΓiclosetounityisnotallowed).

··

Inviewoftheabove,letuscomparetheactualSwiththeS′ofaspectrumoftheformofEq.(1)

withinversetemperatureβeffandhavingΓi(ω)=1(perfectblackbody)forallfermionmodes,but

hωforallbosonmodes.ThisistheclosestablackholeemissionspectrumcouldΓi(ω)=1−e−βeff¯

·

cometoblackbody,andthusgivesthelargestS′forgivenpower.Notethatthenewcomparison

6

spectrumispoorinlowfrequencybosonsascomparedwiththeblackbodyspectrum.Thuswehavegonepartofthewaytowardsrepairingtheproblemnotedearlier.Sincethefermionsareblackbody

··

asinourpreviouscalculation,weshalljustconcentrateonthebosoncontributionstoSandS′.Inwhatfollowsthesubscript“b”standsforbosons.

·′

.FromthechosenΓi(ω)itfollowsthatWeshallfirstcomputethebosoncontributionEb

·2)−1󰀃g.Thesumoverthebosonhω.ThusEq.(7)givesE′=(2πh¯βεi(βeff,ω)=h¯ωe−βeff¯effbib

·

modescalculatedbyPageis54.EquatingtheresulttohisEbdeterminesthatβeff=19.04βbh.We

hω.ItthenfollowsfromEq.(4)thatnowcompareεiwithEq.(5)todeterminethateγi=1+eβeff¯

hω)+e−βeff¯hωln(1+eβeff¯hω)forbosonmodes.Afterintegrationbyparts,σi(βeff,ω)=ln(1+e−βeff¯

Eq.(6)gives

·′

Sb=(π/24+ln2/π)

󰀃

bgi

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