Two Topics concerning Black Holes Extremality of the Energy, Fractality of the Horizon

TwoTopicsconcerningBlackHoles:Extremalityofthe

Energy,FractalityoftheHorizon*

RafaelD.Sorkin

DepartmentofPhysics,SyracuseUniversity,Syracuse,NY13244-1130and

InstitutodeCienciasNucleares,UNAM,A.Postal70-543,D.F.04510,Mexico.

internet:rdsorkin@mailbox.syr.edu

Abstract

Wetreattwoaspectsofthephysicsofstationaryblackholes.Firstweprovethattheproportionality,d(energy)∝d(area)forarbitrarypertur-bations(“extendedfirstlaw”),followsdirectlyfromanextremalitytheo-remdrawnfromearlierwork[1].Secondweconsiderquantumfluctuationsintheshapeofthehorizon,concludingheuristicallythattheyexhibitafractalcharacter,withorderλfluctuationsoccurringonallscalesλbelowM1/3innaturalunits.

Thetheoryofblackholethermodynamicsisincomplete.Ononehandtheidentifi-cationofblackholeentropywithhorizonareaseemsestablishedbyapreponderanceofdirectandindirectevidence.Ontheotherhandwearestillinthedarkaboutthephysicalvariableswhose“states”thisentropycounts.(Forarecentreviewsee[2].)

Thetwomainsectionsofthispaperbelongwiththetwo“hands”justmentioned.Thefirstprovidesanewproofofoneofthemainpiecesofevidenceforthethermodynamicalinterpretationofblackholeproperties,namelythe“firstlaw”(initsextendedformwhichdealswitharbitraryvariations,notjuststationaryones).Itisessentiallythetextofmytalkattheconference.Thesecondmainsectionpresentssomeevidenceforafractalstructureofthehorizoninthecontextofcontributionstotheentropyfromfluctuationsinambientquantumfieldsandfluctuationsintheshapeofthehorizonitself.ItreportsonsomeideasIdiscussedinformallywithparticipantsintheconference,especiallyAndreiZelnikovandValeriFrolov.

I.Derivationofthe“FirstLaw”

TheworkIwilldescribeinthissection,donetogetherwithMadhavanVaradarajan[3],grewoutofourwishtounderstandwhathappenstothetheoremthatstationarityimpliesextremality,whenspacetimehasaboundary.Ithasbeenknownforalongtimethatforgravity,oranyotherLagrangianfieldtheory,everysolutionofthefieldequationswhichhasaKillingvectoralsohasacorrespondingextremalityproperty:theconservedquantityassociatedtotheKillingvectorisunchangedbyinfinitesimalperturbationsofthefields.BernardSchutzandIhadfoundaproofofthiswhichweliked[1]andwewonderedatthetimewhatwouldhappenifweappliedittoaspacetimecontainingablackhole.ThemainmessageIwanttoleaveyouwithtoday,isthatwhathappensisthattheso-calledfirstlawofblackholethermodynamicsemergesinaverydirectmanner.

Thederivationwhichresultsinthiswayisofinterestmainlybecauseofitsconceptualsimplicity,butitalsoshowsonenewthing.Itshowsthatthe3-surfaceΣonwhichtheenergyisevaluatedcanmeettheblackholehorizonanywhere;itdoesn’thavetogothroughanyspecialplacelikeabifurcationsubmanifold1.Ibelievethisisimportant,becausetheabilitytopushΣforwardalongthehorizoniscrucialtounderstandingwherethesecondlawofblackholethermodynamicscomesfrom[5].

Theproofalsomakesclearwhythefirstvariationoftheenergygetscontributionsonlyfromthehorizonitself,anditprovidesanexplanation(viatheRaychaudhuriequation)ofwhyitisspecificallythechangeinhorizonareawhichgovernsthechangeintheenergy.

Thederivationalsoillustrateshowintegralformulationsofconservationlawscanoftenbemoreconvenientthandifferentialones.Ittakesplacein4DforEinsteingravity(withapossibleelectro-magneticfield),butthereisnoreasonitcouldnotbeextendedtohigherdimensions,ortootherlagrangians.Theproofisalsoinsuchaformthatitmighthelpinunderstandingthebehaviorofthesecondvariationoftheenergy.Thisvariationisimportantinconnectionwithstability,butIdon’thaveanynewresultstoreportonit.

Sinceadetailedaccountwillsoonbeavailable[3],thereisnoreasontotryforcom-pletenesshere.InsteadIwillpresentthemainstepsoftheanalysisassimplyasIcan,inamannerwhichIhopewillbecomplementarytothatofreference[3].InthesameminimalistspiritIwillmainlyrestrictmyselftothecaseofpureEinsteingravityandwillsettheelectromagneticfieldandblackholerotationratetozero.[TheNoetheroperatorandthetotalenergy]

Beforewecangettotheproofproper,weneedthenotionofNoetheroperatorandatechniqueIwillcallasymptoticpatching.TheNoetheroperatorformalizesherexpla-nationofhowcontinuoussymmetriesimplyconservationlaws.Forafirst-orderActionS

dependingondynamicalfieldsQandbackgroundfieldsB,andforageometricalsymmetrylikeenergyorangularmomentum,theNoetheroperatorisdefinedthroughtheidentity,

󰀇󰀂

δL

δ(f,ξ)S=fTba·ξbdσa−

∂X

X

2

󰀅

RdV.Insteadwedosomething

elsewhichcanbedescribedindifferentways.Perhapsthebestdescriptionisjustthatwereplacethemetricgbyonewhichisstrictlyflatnearspatialinfinity.Inrealitytheverylongrangepartofthemetricrepresentinganisolatedsystemlikeablackholeismeaninglessinanycase,sinceitcannotbeisolatedfromthefieldsofotherobjectswhichareinvariablypresent.Hence,thereshouldbenodistinction,inaphysicalsense,betweengabandametricwhichhasbeen“cutoff”atlargeradiiby“patching”ittoaflatmetric.TheSwhosevariationyieldstheenergyoftheisolatedsystemisbestviewed,Ibelieve,asnothingbutthecovariantAction3ofthiscutofffield󰀁gab;andthetechniqueforobtainingg󰀁abbygraduallydeformingtheoriginalmetrictotheflatoneassomeradialparameterrincreasesfromRto2RiswhatImeanby“asymptoticpatching”[1][3].

AsymptoticpatchingcanalsobeunderstoodinapurelytechnicalwayinrelationtoanintegrationbypartsperformedtorendertheActionfinite.ForgenericO(1/r)falloffinthemetric,theRicciscalarRwilldecayonlylike1/r3,whichleadstoalogarithmicallydivergentintegralforS.ByaddingasuitabledivergencetotheintegrandweeliminatefromRthetermsoftheformg∂∂g,therebyimprovingitsfalloffto1/r4,whileatthesametimemakingSfirst-ordersothattheabovedefinitionoftheNoetheroperatorapplieswithoutmodfication.TheimprovedfalloffsufficestorenderbothSandEwell-defined.4HavingmodifiedSinthisway,wecanthenperformthesamepatchingtoaflatmetricatinfinitywithoutproducinganyfurtherchangeintheActionortheenergy(inthelimitinwhichthepatchingradiusRrecedestoinfinity)[1][3].Thissecondviewpointisperhapssomewhatmoreadvantageoustechnically,butitrequirestheintroductionofextrabackground:agloballydefinedconnectionwithrespecttowhichonecanperformtheintegrationbyparts.

Theupshotfromeitherpointofviewisthatweenduphavingtodealonlywithmetricswhicharestrictlyflatnearinfinity.Thiswillfreeusfromhavingtoworryabouttheeffectsofvariationsatinfinity,leavingonlyboundarytermsatthehorizontocontribute.Italsomeans,ofcourse,thatwecannolongerexpresstheenergyasthefluxintegral(4)takenattrueinfinity,but(4)stillholdsifevaluatedjustinsidethepatchingradius,andtheexpression(2)intermsofaspatialintegralremainsgenerallyvalid,undertheassumption(whichwillalwaysbeinforce)thatξaremainsanexactKillingvectoroftheflatasymptoticmetricthroughoutthepatchingregion.

Henceforth,wewillconsideronlymetricswhichhavebeenpatchedtobecomestrictlyflatnear∞,and”solution”willalwaysmeansolutionpatchedtoflatmetricatlarger.Inaddition,wewillconsideronlyvectorfieldsξwhichpreserveanybackgroundwhichhasbeenintroduced,andwhichinparticulararestrictKillingvectors(oftheflatmetric)nearinfinity.

[Theextremumproofwithoutreferencetoahorizon]

Settingf=1inthedefiningidentity(1)oftheNoetheroperator,andrecallingthatthelefthandsidethenvanishesautomatically,weobtainthebasicidentity

󰀇󰀂

δL

T·ξ=

foranarbitraryperturbationδgandanarbitraryregionX.Butforourδgthisexpressionitselfisby(2)noneotherthanthedifferenceE(g)|Σ0−E(g′)|Σ,whichaccordinglymustvanish.InotherwordsE′=EorδE=0,wherenowδEjustmeansthevariationinEonΣingoingfromgtog′.Thisisourfirstmainresult:thetotalenergyisanextremumforanyasymptoticallyflatstationarysolutiontothefieldequations.[Applicationtoaspacetimewithinternalboundary]

ThusfarIhavebeentacitlyassumingthatthe3-surfaceΣisacompleteRieman-nianmanifoldpossessingasoleasymptoticregion.Whenspacetimehasmorethanoneasymptoticregion,ormoreimportantlyforus,whenithasaninternalboundary,thefor-mula(2)forEmustbeappliedwithcare.Inorderthatitcorrectlyfurnishtheenergyassociatedwithagiven∞,ξmustbeatimetranslationthere,butitmustvanishatalltheotherboundaries(includingtheactualinternalonesandtheidealonesatinfinity).ThisrulefollowsfromtheprescriptionthatErepresentsthechangeinSwhichresultsfromaperturbationthatrigidlydisplacestheentirespacetimerelativetotheinfinityinquestion.Alternatively,itcanbederivedbyrevertingtotheformula(4)fortheenergyofa“non-patched”solution,andconverting(4)toavolumeintegralviaStokestheorem.

Inthecaseofinteresttheboundarywillbethehorizonofablackhole(orholes).Thissurfacedoesnotrepresentaphysicallyreal“edge”ofspacetime,ofcourse,butratheraboundaryweimposeonthesubmanifoldweworkwith,inordertomakeeffectiveuseoftheidentity(5).Beingafuturehorizon,thisboundingsurface(whichIwillcallH)willbenullwithitsfuturesidefacingawayfromtheouterworld.5

LetusnowtrytogeneralizethereasoningoftheprevioussubsectiontoaregionXformedasbefore(withfutureboundaryΣandpastboundaryΣ0)butwithanextrainternalboundaryHrepresentingtheportionofthehorizonbetweenΣ0andΣ.Inreferringtothis

setupIwilldenotethe2-surfaceΣ∩HbyS,andthecorresponding,butearlier,2-surfaceΣ0∩HbyO.(SeeFigure2.)

SH.Σ.OΣΟFigure2.AregionXanalogoustothatofFigure1,buttruncatedatthehorizon.ThenullsurfaceHisthatportionofthehorizonbetweenΣ0andΣ;itsfutureboundaryisthe2-surfaceSanditspastboundaryO.

Nowinordertousetheidentity(6)aswedidintheprevioussubsection,weneedξtobeaKillingvectoroftheunperturbedsolution,whichcontradictstherequirementthatitvanishonHinorderthat(2)bethetotalenergy.HoweverξisaKillingvectoratlargeradii,sothereisnothingtostopusfrommakingitKillingeverywherebyuseoftherelation(3).ApplyingthisrelationinconjunctionwithStokes’TheoremtotheregionΞ⊆ΣwhereξdeviatesfrombeingKilling,weimmediatelyobtain6

󰀂󰀇−E=T·ξ+W·ξ,(7)

Σ

S

wherenowξisKillingeverywhereandtheintegraloverSappearsbecauseSistheinner

boundaryoftheregionΞ.Expressedinthismanner,theenergyappearsasavolumeintegralaugmentedbyahorizoncontributionwhichitwouldbenaturaltodesribeasthe“energyoftheblackhole”.

Nowletusapplyto(7)avariationleadingfromthestationarysolutiongtoanearbysolutiong′,andletustemporarilyassumethatg′=ginaneighborhoodofS.Sincethevariationofthesecondintegralin(7)thenvanishestrivially,exactlythesameproofasearliershowsthatδE=0.FromthisitfollowsimmediatelythatδEforageneralperturbationcandependonlyonthevalueoftheperturbation(anditsderivatives)atthehorizonitself,i.e.atthe2-surfaceS.Noticethatessentiallynocomputationwasinvolvedinreachingthisconclusion.

Consider,then,aperturbedsolutiong′forwhichg′−gdoesnotnecessarilyvanishonthehorizon.Wecanstudyitsenergybyintroducingthesamekindof“interpolatingperturbation”δgasweusedearlierintheabsenceofaboundary;howeverbeforedoingthis,itwillbeconvenienttoprepareourselvesbyextendingtheΣ-integralin(7)allthewaybacktoΣ0withtheaidoftheidentity(3).Theresultis

󰀂󰀂󰀂−E=T·ξ+Gξ+W·ξ.(8)

Σ∪H

H

O

Nowwhenweapplythevariationδ,thefirstintegralin(8)isunchangedforexactlythe

samereasonasearlierandweareleftwith

󰀂

−δE=δGξ(9)

H

(thethirdintegralin(8)beingtriviallyunchangedbecauseδgvanishesinaneighborhood

ofΣ0).

Thisisoursecondmainresult.ItexpressesδEasthechangeinthefluxofthefictitious

b

(conserved)energycurrentGabξacrossthehorizonHingoingfromthestationarytothevariedsolution.7Noticethatallreferencetoauxiliarybackgroundfieldshasnowdroppedout.

[ReductionofδEtoanintegralonS]

WehavealreadyseenongeneralgroundsthatδEmustbeexpressibleintermsofquantitiesdefinedonlyonthe2-surfaceSinwhichour3-surfaceΣmeetsthehorizon.Todiscovertheexplicitformofthisexpressionrequiresustoconvert(9)fromanintegraloverHtooneoverSalone.Clearly,itsufficestore-expressitastheintegralofatotaldivergenceofsome“potential”.8

Itturnsoutthatthereisasystematicmethod[8]forconstructingsuchapotential(andthepotentialisuniquelydeterminedbytheconstruction);itsapplicabilityisguaranteed

√

bythefactthatδ(

7

Forthecaseofarotatingblackhole,therelevantKillingvectorwouldbeξ=t+Ωφwheret,now,denotestheKillingvectorwhichisatime-translationatinfinity,whileφdenotestherotationalone(Ωbeingtheangularvelocityofthehorizon).HencethevariationδEin(9)wouldbereplacedbyδE−ΩδJ,Jbeingtheangularmomentum.

AnotherapproachwouldbetoshrinkHtoa2-surfacebybringingtogetherthesurfacesΣ0andΣ.

Thisidentity(cf.ref.[7])istheanalogofeq.(6)forthecovariantAction

1

8

9

wouldrequireonlystraightforwardcalculation,butinsteadoffollowingthisroute,wecaninvoketheRaychaudhuriequationtoevaluatetheintegralin(9)directly,anapproachwhich—thoughitislesssystematicthanthemethodofreference[8]—affordsasimpleexplanationofhowthehorizonareaemergesasthemeasureofδE.

Inessence,allthatisinvolvedisusingtheRaychaudhuriequationtoconverttheintegrandof(9)intoanexpressioninvolvingtheexpansionofthehorizon,andthenper-forminganobviousintegrationbyparts.Inpreparation,however,weneedtorecallafewdefinitionsandmakeaconvenientchoiceofgaugeinwhichtoexpresstheperturbationδg.

Letusbeginbynotingthatforanon-rotating,stationaryblackhole(Schwarzschildmetric),thetimelikeKillingvectorξisautomaticallynullonthehorizon,10whencepro-portionaltothenullgeodesicgeneratorsofH.Accordingly,ifweparameterizethelatterwithanaffineparameterλ,thenwehave

ξ=α

a

dxa

dλ

=κ,(10b)

wheretheblackhole’s“surfacegravity”κisdefinedbytheequationξb∇bξa=κξa.

Nowincomparingthestationarysolutiongwiththeinterpolatingmetricg+δg,wearefreetochoosethediffeomorphism-gaugesothatthehorizonisthesamesurfaceHforbothmetrics.InfactweclearlycanarrangethatξremainsanullgeneratorofHwithrespecttog+δgandalsothatλremainsanaffineparameteralongeverysuchgenerator.(ForgivenchoicesofΣ0andδg,thisalsodeterminestofirstorderinδgwhereΣisembeddedinthevariedspacetime.)Inthisgauge,equations(10a,b)willremaintrueevenafterthevariation(withκdenotingthesurfacegravityoftheunvariedmetricg,asalways.)

Finallywewillusethefactthattheextensor11dSarepresentinganelementofthesurfaceHcanbewrittenas

dSa=−d2Adxa(11)

foraportionofHwithcross-sectionalaread2AandextensionalongthenulldirectioninHgivenbythe(futurepointing)nullvectordxa.

Nowwearereadytosubstituteinto(9)theRaychaudhuriequationforthegeneratorsofH,namely

dxadθσ2Rab=−−

dλ2

dxbdλ

dλ

.(13)

NotingfurtherthatGabalsovanishesfortheunvariedmetric,wecannowtransform(9)

asfollows:

󰀂

b

−δE=δdSaGabξ

󰀂H

b

=dSaGabξ

H󰀂

b

=−d2A(dx)aGa(byeq.(11))bξ

󰀂b

2adx=−dA(dx)aGbα

dxbdλ

dxbdλ

dλ󰀂

dα

=−d2Adλ

H

(byeq.(13))

=−

󰀂

κ

󰀂

dλ

d(d2A)dλ

κ+

󰀂

d2Aαθ

S

Hereinthesixthequalityweusedthatdx/dλisnull;intheninthweusedeq.(10b)andthatθcanbeexpressedas

1

;θ=

dλ

andinthelastequalityweusedthatκisconstantonH.

ThuswehavereducedtoδEtoanexpressionpertainingsolelytothecrosssectionSofthehorizon,andthisisourthirdmainresult:

󰀂

−δE=−κδA+d2Aαδθ(14)

S

[Locatingthehorizon]

Withequation(14)ourworkisessentiallycomplete,exceptthat,inadditiontothe

desired(first)termitcontainsanintegraldependingonδθ.Inordertorealizewhythisunwantedtermispresent,weonlyhavetoaskourselveswherewehaveusedthefactthatHisactuallythehorizonoftheperturbedsolutiong′,andnotjustsomerandomnullsurfacetherein.Thepointis,ofcourse,thatwehaven’tusedityet,meaningthattheareaofSmighthavechangedjustbecauseitwasdisplacedinlocationwithoutevenleavingtheunvariedspacetime!InordertodistinguishsuchabogusδAfromthetrueone,weneedacriteriontolocatethehorizonwithrespecttothemetricg′.Suchacriterion,Iclaim,ispreciselytherequirementthatδθ=0everywhereonH(wherehereandhenceforth‘δg’justmeansg′−g,notthemorecomplicatedinterpolatingperturbationofearliersubsections).

Inprinciplethisclaim,iftrue,shouldbederivablefromtheEinsteinequation,andsuchaderivationdoesnotlooktooimpractical,atleastinconnectionwiththeSchwarzschildmetric,whoseperturbationsarefairlywellunderstood.Fornowhowever,wewillderiveδθ=0fromanassumptionwhichisaspecialcaseoftheso-calledcosmiccensorshipcon-jecture.Wewillassumethatthehorizonofthestationarysolutiongcannotbedestroyedbyarbitrarilysmallperturbationsofthemetric.

Ifthisisso(andifitisnot,thenblackholesdonotexistinrealityanyway!)thennoinfinitesimalperturbationofthemetricgcanmaketheexpansionθnegativeanywhere,becauseifitdid,thentherewouldbearbitrarilynearbysolutionsg′withnegativeexpansionsomewhereontheirhorizons,butitiswell-knownthatnegativeexpansionimpliesthatthehorizonencountersasingularityinafinite“time”(reallyaffineparameter).12Butifθ=δθcanneverbenegative,thenitcanneverbepositiveeither,becauseasimplechangeinthesignofδgwillsimilarlychangethesignofδθ(andofcourse,−δgwillalsobeasolutionof

thelinearizedEinsteinequation).Henceθonthetruehorizonmustremainzerotofirstorderinanyperturbationaboutastationaryblackholemetric.[Summary:thefirstlaw]

Ouranalysisisnowcomplete;letussummarizethehighlights.Usingtheidentity(6)wefirstfoundthatδE=0foranyvariationδgsupportedawayfromacross-sectionSofthehorizonH.Thisimpliedthatforgeneralperturbations,δEcandependonlyonthebehaviorofδgintheneighborhoodofS.ToevaluateδEexplicitly,wehadtore-expressthe3-dimensionalintegral(9)astheintegralofadivergence.Asystematicmethodfordoingsoexists,butweusedtheRaychaudhuriequationinstead,leadingtoequation(14).Byinvokingthe“stability”ofthehorizonwe“situated”Hwithintheperturbedspacetime,showingtherebythatthesecondtermin(14)isinfactzerowhenevaluatedonthecorrectlyidentifiedperturbedhorizon.Theremainingtermyields13theso-called“firstlaw”

δE=κδA.

[Possiblefurtherwork]

WithAidentifiedasentropy,thefactthatδA=0wheneverδE=0canbeinter-pretedasthefirst-orderexpressionofthermodynamicstability(intheh¯→0limit),athermodynamicallystablesolutionbeingonewhichmaximizesentropyatfixedenergy.Atsecondorder,thismaximizationisgenericallyequivalentto

A′′−κ−1E′′≥0onkerE′,

(16)(15)

where(·)′denotesFr´echetderivative.Aninterestingproblemwouldbetotrytoprove(16)forSchwarzschild(say)byextendingtheforegoinganalysistosecondorder.14

Anotherworthwhileextensionoftheanalysiswouldbetogeneralizedgravitytheories,includinginaKaluza-Kleinsetting(cf.[9]).Thereour“Raychaudhuritrick”wouldprobablyfail,andonewouldhavetofindanothertrickorfallbackonthegeneralmethodreferredtoearlier.Indeedthisgeneralmethod[8]meritsfollowingupeveninordinarygravity,bothasa“warmup”formorecomplicatedLagrangians,andfortheadditionalinsightitmightofferintotheoriginsofthefirstlawitself.

II.FractalityoftheHorizon

Onepossiblesourcefortheentropyofablackholeisinthefluctuationsofaquantumfieldpropagatingnearthehorizon.Whenthefieldinquestionisthelinearizedmetric(“graviton”),theassociatedentropyisgeometricalincharacter,buttherearemanyotherquantumfieldswhichareabletocontributeaswell.Themechanisminallcasesisthesame:fluctuationsinthefieldoccuronallscales,andwhenafluctuationwithcharacteristicsizeλisastridethehorizonitsetsupacorrelation(“entanglement”)betweeninsideandoutsidewhichmetamorphosesintoentropywhenone“tracesout”thefieldmodesinsidetheblackholeinordertoobtaintheeffectivedensity-operatordescribingthefieldoutsidetheblackhole[10].

Whenonetriestocomputethevalueofthisentropyforafreefield,oneobtains,atfirst,aninfiniteresultderivingfromthefactthatfreefieldsarescale-invariantintheultravioletregime,whenceaninfinitenumberofmodescontributewithconstantentropypermode.However,ifoneintroducesacutoffatsomescalel,theentropytakesonthefinitevalueS=cA/l2,whereAistheareaofthehorizon,andcisadimensionlessconstantoforderunity.Sincethisgivestherightareadependence,andalsotherightgeneralmagnitudeifonechoosesl=lPlanck,oneistemptedtoconcludeontheonehandthatonehasexplainedblackholeentropy,andontheotherhandthatonehasobtainedpersuasiveevidencefortheexistenceofspatio-temporaldiscretenessinnature.

AnotherthingwhichspeaksinfavorofidentifyingSwithsomesortofentanglemententropyisthattheprospectofanaturalproofoftheSecondLawthenarisesnaturally.Indeed,onecanarguethat,iffullquantumgravityfurnishesus(atsomelevelofcoarse-graining)withawell-defined,autonomouslyevolvingdensity-operatorρdescribingtheoutsideworld,then−trρlnρnecessarilyincreasesasthesurfaceΣwithwhichitisasso-ciatedmovesforwardintime.Theargument[5]restsonthefactthatthetotalenergyisconservedanddeterminablefromthegravitationalfieldoutsidetheblackhole(s),nomatterwhatmaybeoccuringinsideofthehorizon(i.e.itrestsonequation(4)or(7)above.Noticethattheentropydoesnotchangeifthecodimension-twosurfaceSinwhichΣmeetsthehorizondoesnotmoveforwardalongH;hencethesignificance,referredtoearlier,ofbeingabletochooseSfreely.)

Althoughtheargumentjustalludedtodoesnotcarewhatdegreesoffreedomitdealsin(aslongasthenumberofeffectiveexternalstatesisfiniteatfinitetotalenergy),ourinteresthereisinthosevariablesassociatedwiththefluctuationsofquantumfields.Totakeseriouslytheircontributiontotheentropyleadstotheseemingdifficultythat—forfixeddiscretenessscalel—themagnitudeofSwoulddependonthetotalnumberoffieldspresentinnature,seeminglyatoddswiththesimplegeometricalcharacteroftheformulaS=2πA,whichjustequatestheentropytothecircumferenceoftheunitcircletimesthe

√

areaofthehorizonmeasuredinPlanckunits.(WetakelPlanck=

the“rationalizedgravitationalconstant”,andh¯=c=1.)Thissimpleformulaseemsmoreinharmonywithadirectly“geometrical”characterfortherelevantdegreesoffreedom,perhapstheshapeofthehorizonitself[11],ortheconfigurationofsomeunderlyingdiscretestructurecomposingthehorizon,suchas(theappropriateportionof)acausalset.

The“fractal”pictureofthehorizonIwilldescribeinamomentgrewoutofmywonderingwhetheronecouldavoidtheabove“speciesdependenceproblem”bysomehowwritingthequantumfieldsoutofthescriptinfavorofmoresuitablygeometricaldegreesoffreedom.Inthemeantimeithasbecomemuchlessclearthatthereisinfactanydifficultytobeavoided,inviewoftheobservation[12]thatachangeinthenumberoffieldswouldaffectnotonlySbutalsotherenormalizedvalueofκ≡8πG,andindeedwouldalterκinjustthemannerneededtocompensateforthechangeintheentanglemententropy,leavingtheformulaS=2πA/κstillvalid.Althoughthedetailsoftheirargumentcanbecriticized,itsoverallstructureis“tooprettytobewrong”,andsoisprobablycorrectatsomelevel.Atthesametime,itmanifestlyignorestheinfluenceofthefluctuationsonthehorizonitself(“backreaction”),andtothatextentislimitedtoasemiclassicalregime.

InthepictureIamproposing,thenumberofspeciesisirrelevantforanentirelydifferentreason,namelyforthereasonthat—duepreciselytotheback-reaction—theconstantcisnotconstantatall,butratherdependsonthesizeoftheblackholeinsuchamannerastobecomenegligiblysmallforallbutPlancksizedblackholes.MoreaccuratelyIwilltrytoshowthattheapproximationoffixedhorizonlocationandshapebecomesinvalidatalengthscalemuchgreaterthanPlanckian,namelyatascaleofthemagnitudeM1/3,Mbeingthemassoftheblackhole.15Belowthatscale,thefieldfluctuationsbecomestronglycoupledtothehorizonshape,andasemi-classicalanalysisbecomesunreliable.Atthesametime,theshapeofthehorizonitselfbecomes“fractal”duetotheeffectsofthefluctuations,perhapsprovidingtheanticipatedgeometricaldegreesoffreedomto“absorb”thefieldones.

Thepointisthat,atleastforfreeorasymptoticallyfreefields,fluctuationsoccurwithequalintensityatallsufficientlysmallscalesλ.Givenafluctuationofsizeλ,onewouldexpecttheassociatedenergyofmagnitude∼1/λtoinduceaconcomitantdistortionofthehorizon.Heuristically,wemayperhapspicturethesituationasfollows.Asonedescendsinscale,onewillreachathresholdsizeλ0,atwhichthe“virtualenergy”ofatypicalfluctuationwillbebigenoughtodistortthehorizonshapebyanamountcomparabletothesizeofthefluctuationitself.Then,likeasleeperwhoisuncomfortableinbedandeitherburieshim/herselfundertheblanketsorpushesthemallonthefloor,thefluctuationwilleitherpullthehorizonupoveritselfor(inthecaseofnegativeenergy-density)drivethehorizonentirelyaway.Ineithercasethefluctuationwillnolongeroverlapthehorizon,and

itthereforewillnolongercontributetotheentanglemententropy.Moreover,thiseffectevidentlyentailsastrongcouplingbetweenthehorizonshapeandthefieldfluctuationsofsizeλ<∼λ0;whencesuchafluctuationsshouldnotcountasindependently“entangled”degreesoffreedomeveniftheydohappentomeetthehorizon.Weconcludethen,thatthescaleλ0setsalimittoourunderstandingofentanglemententropy,andthattheonlyreliableestimateswecanmakeforthelatterpertaintofluctuationswithcharacteristicsizesgreaterthanλ0.

Butisn’titobviousthatλ0willjustturnouttobeofPlanckiansizeinanycase?Tobegintoanswerthisquestionreliably,onewouldhavetoanalyzetheeffectonthehorizonofaspacetime“energyfluxloop”ofcharacteristicsizeλ,situatedonornearthehorizonof,say,aSchwarzschildblackhole.Here,Iwilldosomethinglessaccuratebutmucheasier:IwillcomputeforNewtoniangravity,thedisturbanceinthe“horizon”inducedbyasmalladditionalmassm∼1/λdistributedthroughoutaspatialregionofsizeλlocatedinthevicinityofthehorizon.

SolettherebepresentattheoriginasphericalmassM,anddefineitshorizonasthelocusofpointswheretheescapevelocityequalsunity(i.e.c2),thatis,where

V=−1/2,

(17)

Vbeingthegravitationalpotential,−GM/r,ofthemass.16Itwillbeconvenienttowork,notwithM,butwiththecorresponding“geometrizedmass”or“Schwarzschildradius”R:=2GM.IntermsofRwehaveforapointmass,V(r)=−R/2r,sothatthehorizonoccurspreciselyatr=R,awell-knowncoincidence.

Nowletusaddinthegravitationalpotentialofthefluctuation,towhichforanalyticalconvenience,wewillassigntheeffectivemass-densityρ=aλ/r1(r1+λ)3,resultinginthepotential

−a/2V1=

r

16

+

f/λ

Oneshouldpresumablyconceiveoftheperturbationasenduringonlyforatimeoforderλ,buttheassociatedretardationeffectswouldbehardtoincorporateintheNewtonianframework,andinanycase,theywouldnotseemlikelytoalterthequalitativepicturederivedfromtreatingthehorizonasdeterminedbytheinstantaneousNewtonianpotential.

15

Forsimplicity,letusnowplacethecenterofthefluctuationwheretheunperturbedhorizonmeetsthey-axis,andletusalsomovetheoriginofourcoordinatesystemtothatpoint.Thenifwerestrictourselvestothepositivey-axis,thepotentialassumestheparticularlysimpleform

R

.(18)−2V=

λ+yIntheapproximationthaty,λ≪Rthisreducesto

−2V≈1−

y

λ+y

.

(19)

ItisnoweasytolocatetheperturbedhorizonbysolvingtheequationV=−1/2or−2V=1.Workingwiththeapproximation(19),wehaveonthehorizon,

yλ+1)=

fR

λ

󰀃󰀄1+

λ

)2+(

y

λ3

.

Fromthis,oneseesthatthecharacteristicparameterfR/λ3governstheshapeofthebulgeaswellasitsheight,andthatthewidthofthebulgeiscomparabletoλwhenλ<∼λ0.

Tosummarize:Thesizeandshapeofthebulgeinthehorizonraisedbythefluctuationdependsontheratioλ/λ0.Forλ≪λ0thefluctuationraisesabulgemuchsmallerthanitself,whereasforλ≫λ0itis(inourNewtonianpicture)muchlarger.Inparticular,thebulgebecomescomparabletothesizeofthefluctuationpreciselywhenλ∼λ0.Thisconclusiondoesnotdependonthespecificprofilechosenfortheeffectivemassdensityofthefluctuation.Adelta-functionwouldleadtothesameconclusion,aswouldadipolarsourcewithvanishingtotalenergy(perhapsmoreappropriateasamodelofavirtualfluctuationofaquantumfield).Anditappearsthatfullgeneralrelativityagainyields

16

asimilarrelationshipbetweenscaleλanddistortionheighthifonemakesthedrasticapproximationofsphericalsymmetry.

Theformula(20),iftakenliterally,impliesthatafluctuationonscaleλ≪λ0inducesadistortionofthehorizonmuchgreaterthanitsownsize.Howeveritseemsimplausiblethatsuchaneffectwouldbepresentinafullyrelativisticsetting,whereretardationeffectswouldmakesuchextreme“actionatadistance”bythefluctuationappearveryunrealistic,andonewouldnotexpecttheinfluenceofafluctuationtoextendmuchbeyonditsimmediatevicinity.Ifthisiscorrect,thenitbecomesplausiblethattheactualperturbationsinthe

1/3

horizonduetofluctuationsofsizeλwouldthemselvesbeofsizeλforallλ<λ∼M.0∼

Theresultingstructureofthehorizoncouldthenbedescribedasfractalonscalesbetween1andM1/3(itbeingdoubtfulwhetherspacetimeitselfexistsasacontinuousmanifoldonscalesbelowunity).Inprinciplethereisnolimittohowlargethisscale-invariantwrinklingcouldgrowifsufficientlymassiveblackholeswereavailable,butunfortunatelytheprospectofhuman-sizedfluctuationsinthehorizondisappearswhenoneplugsinthenumbers.Thewrinklesonasolarmassblackhole,forexample,wouldonlyreachascaleofaround10−20cm,andforthefluctuationstoattainasizeof1cm,ablackholeoftheabsurdmassof1091gramswouldbecalledfor.

Inconclusion,IwouldliketothankR.Salgadoforhisindispensableaidinpreparingthefigures.ThisresearchwaspartlysupportedbyNSFgrantPHY-9307570.

References

[1]Schutz,B.F.andR.D.Sorkin,“VariationalAspectsofRelativisticFieldTheories,withApplicationtoPerfectFluids”,AnnalsofPhys.(NewYork)107:1-43(1977)[2]J.Bekenstein,“Doweunderstandblackholeentropy?”,gr-qc/9409015

[3]R.D.SorkinandM.Varadarajan,“EnergyExtremalityinthePresenceofaBlackHole”,(inpreparation)

[4]D.SudarskyandR.Wald,Phys.Rev.D46,1453(1990)

[5]R.D.Sorkin,“TowardanExplanationofEntropyIncreaseinthePresenceofQuantumBlackHoles”,Phys.Rev.Lett.56,1885-1888(1986);seealsoR.D.Sorkin“ForksintheRoad,ontheWaytoQuantumGravity”,talkgivenattheconferenceentitled“DirectionsinGeneralRelativity”,heldatCollegePark,Maryland,May,1993,(SyracuseUniversitypreprintSU-GP-93-12-2).

[6]R.D.Sorkin,“ConservedQuantitiesasActionVariations”,inIsenberg,J.W.,(ed.),MathematicsandGeneralRelativity,pp.23-37(Volume71intheAMS’sContemporary

17

Mathematicsseries)(Proceedingsofaconference,heldJune1986inSantaCruz,California)(Providence,AmericanMathematicalSociety,1988)

[7]R.D.Sorkin,“TheGravitational-ElectromagneticNoether-OperatorandtheSecond-OrderEnergyFlux”,ProceedingsoftheRoyalSocietyLondonA435,635-644,(1991)[8]R.D.Sorkin,“OnStress-EnergyTensors”,Gen.Rel.Grav.8:437-449(1977)

[9]Lee,J.andR.D.Sorkin,“ADerivationofaBogomol’nyInequalityinFive-dimensionalKaluza-KleinTheory”,Comm.Math.Phys.116,353-364(1988);Bombelli,L.,Koul,R.K.,Kunstatter,G.,Lee,J.andR.D.Sorkin,“OnEnergyinFive-dimensionalGravity”,Nuc.Phys.B289,735-756(1987).

[10]R.D.Sorkin,“OntheEntropyoftheVacuumOutsideaHorizon”,inB.Bertotti,F.deFelice,Pascolini,A.,(eds.),GeneralRelativityandGravitation,vol.II,734-736(Roma,ConsiglioNazionaleDelleRicerche,1983);Bombelli,L.,Koul,R.K.,Lee,J.andR.D.Sorkin,“AQuantumSourceofEntropyforBlackHoles”,Phys.Rev.D34,373-383(1986).

[11]Seethesecondreferencein[10],p.374.

[12]L.SusskindandJ.Uglum,Phys.Rev.D50:2700(1994)

18