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SU-4240-590

December,1994

EdgeStatesinGravityandBlackHolePhysics

A.P.Balachandran,L.Chandar,ArshadMomen

DepartmentofPhysics,SyracuseUniversity,

Syracuse,NY13244-1130,U.S.A.

Abstract

WeshowinthecontextofEinsteingravitythattheremovalofaspatial

regionleadstotheappearanceofaninfinitesetofobservablesandtheir

associatededgestateslocalizedatitsboundary.Suchaboundaryoccursin

certainapproachestothephysicsofblackholesliketheonebasedonthe

membraneparadigm.Theedgestatescancontributetoblackholeentropy

inthesemodels.A“complementarityprinciple”isalsoshowntoemerge

wherebycertain“edge”observablesareaccessibleonlytocertainobservers.

Thephysicalsignificanceofedgeobservablesandtheirstatesisdiscussedusing

theirsimilaritiestothecorrespondingquantitiesinthequantumHalleffect.

Thecouplingoftheedgestatestothebulkgravitationalfieldisdemonstrated

inthecontextof(2+1)dimensionalgravity.

1

I.INTRODUCTION

SincethediscoveryofblackholeradiationbyHawking[1,2],theeventualfateofinfor-

mationfallingintoablackholehasremainedanunsolvedpuzzle.Thedifferentscenarios

thathavebeenproposedcanbesummarizedinthefollowingtwoclasses:

(1)Informationgoingintoablackholeisirretrievablylostsothatunitaryquantum

theorybreaksdown[3].

(2)Informationthatwasthoughttobe“lost”reappearsinsomeformthussaving

conventionalquantumphysics[4].

Quantumblackholephysicsisusuallystudiedinasemiclassicalapproximation.Typi-

callyitisshownthatprocesseslikeHawkingradiationandthescatteringofmatterfields

byblackholesaregovernedbyphysicsoccurringclosetotheeventhorizon[5].

Therehavebeenseveralinterestingproposals[5–7]totreattheblackholehorizonas

amembranewithdynamicaldegreesoffreedomattachedtoit.Itisfoundthatforany

externalstationaryobserver,theblackholecanbewellapproximatedbyhypothesizing

astretchedmembranehavingcertainclassicalproperties,surroundingtheholeatasmall

distanceoutsideitseventhorizon.Aphysicaljustificationforthisprocedureisthatparticles

classicallycanneverleavetheinterioroftheblackholeorreachitfromoutsideinafinite

timeaccordingtoanysuchobserver.Thusitseemsthatanysuchpersoncanstudythe

quantizationofthesystemafterremovingtheinterioroftheblackholeandreplacingitby

amembrane.Themanifoldinthisapproachthushasaboundary.

Inthispaper,westudythequantumphysicsofblackholesforsuchobservers,ormore

generallyfeaturesofquantumgravityonmanifoldswithspatialboundaries.Weshowthat

thepresenceoftheboundarynecessarilyleadstoaninfinitesetofobservableswhichare

completelylocalizedatthisboundaryintheabsenceofanomalies.[Theimportantissueof

anomaliesandtheirsignificanceisalsoaddressedinthispaper.]Theseareobtainedhere

inanalogyto“edge”observablesingaugetheoriesdefinedonmanifoldswithboundaries

[8,9].Suchobservables,definedinthecontextofgaugetheories,haveenjoyedgoodphysical

2

interpretationinmanyexamplesofcondensedmatterphysics.Forinstance,ithasbeen

knownforsometimethatmanyoftheessentialfeaturesofthefractionalquantumHall

effect(FQHE)arecapturedpurelybytheexistenceoftheseedgeobservables[10].We

willdiscussmoreaboutthisanalogywiththequantumHalleffect(QHE)lateroninthis

paper.Frompreviousworkdealingwiththeconstructionoftheseobservablesforpuregauge

theories[9]andalsofromtheconstructionexhibitedinthispaper,onerealizesthattheedge

observablesareindependentofobservablesdefinedinthebulkandthattheycommute

withtheHamiltonian.Thisisquiteinterestingbecauseitmeansthat,nomatterwhatthe

quantumtheoryofthebulkobservablesis,thereisalwaysaHilbertspacelivingentirelyon

theedge.Onecanthentalkofexcitationswhichonlyinvolvetheedgeandleavethebulk

unaffectedintheabsenceofanycouplingbetweenthem.

Hence,inspiteofthedeepproblemsinquantizinggravity,onecanstillexaminetherel-

ativelysimplertaskofquantizationofthegravitationaledgeobservablesandtheassociated

edgestatesattheboundary(which,forablackhole,isS×lR).Inthiswaywecanalso

2

hopetomakemeaningfulphysicalpredictionsbecauseoftheabovementionedclaimthat

thequantumaspectsofablackholearelinkedtotheexistenceofanimaginaryboundary

neartheeventhorizon.

Itmeritsemphasisthatthestretchedmembrane[6]istobethoughtofassurrounding

theblackholeatasmallbutfinitedistanceoutsideitseventhorizon.Forthisreason,

ourboundarywillalsoassumedtobesituatedslightlyawayfromtheactualeventhorizon.

Thereisalsoagoodtechnicalreasonforthisassumptionsincetheinducedmetricbecomes

degenerateontheeventhorizon.

Itisworthpointingoutoneotherimportantobservationemergingfromtheanalysisof

Section3.Wenoticethefactthateventhoughthereexistsaninfinitesetofobservables

attheedgeoftheblackhole,notallofthesearerelevantforalltheobservers.Infact,a

sortofcomplementarityprinciplecanbeshownaccordingtowhichtheexistenceofcertain

observablesas“good”observableswillprecludesomeothersfrombeingso.Thechoice

ofthe“good”observablesisdictatedbyphysicalconsiderations.Thus,forexample,we

3

shallseethatforanasymptoticobserver,physicswilldictatethatthereexistsaninfinite

sub-familyof“good”edgeobservableswhereasforanobserverstationedclosetotheevent

horizon,onlyafinitenumberoutofthisinfinitesetsurviveas“good”observables.This

differencebetweenthenotionof“good”observablesforthesetwoobserverscertainlyneeds

tobeunderstoodmorecarefullysincethismaybeattheheartofmanyoftheconceptual

difficultiesconcerningthequantumphysicsofblackholes.[Inthisconnection,see[5],where

adifferentkindofcomplementarityprinciplehasbeendiscussed.]

Theorganizationofthepaperisasfollows.InSection2,webrieflyreviewthecanonical

formulationofpuregravityin(3+1)dimensionsfortheconvenienceofthereader,follow-

ingtheapproachof[11]closely.InSection3,weusethisformalismin(3+1)dimensions

foramanifoldobtainedbyremovingaspatialball.Inblackholephysics,thisballmay

beregardedasenclosingtheholeincludingitseventhorizoninitsinterior,itsboundary

beingthemembrane.Weshowthat,insuchaspacetime,wearenaturallyledtocertain

observablesthatareconfinedpurelytotheedge.InSection4,wediscusstheanalogyof

theaforementionededgeobservableswithsimilarobservablesarisingincondensedmatter

physics,notablyintheQHE.Inparticular,wearguethateventhoughformallytheedge

observablesdonotmixwiththeotherobservableslivinginthebulk,thisceasestobethe

caseespeciallywhenwehaveananomalouscouplingbetweentheedgeandthebulk.Such

acouplingisnotamatterofchoice,itbeingrequiredfromverygeneralargumentsofgauge

invarianceinthecaseoftheQHE[10].Thesurpriseisthatthisanalogyextendsevento

thecaseofgravity,whereitturnsoutthatananomalouscouplingbetweentheedgeandthe

exteriorisforcedonusifwerequiregeneralcoordinateinvariance(diffeomorphisminvari-

ance).InSection5,weexplicitlydemonstratethatsuchacouplingindeedappearswhenwe

aredealingwith(2+1)gravity.ThefinalSection6outlinesanargumentsuggestingthatthe

edgeandbulkstatescouplein(3+1)dimensionsaswell.Thisargumentisunfortunately

incompleteaswelackasatisfactorydescriptionofedgedynamics.

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II.CANONICALFORMULATIONOFTHEEINSTEIN-HILBERTACTION

Considerafour-manifoldMwhichistopologicallyΣ×lRandletitstime-slicesΣbe

t

parametrizedbyt.HereafterweassumethatΣisdiffeomorphictotheexteriorofaball

t

BinlR.Lettbethevectorfieldwhoseaffineparameteristandletndenotetheunit

3a

3

a

normaltothesurfacesΣinthedirectionofincreasingt.AmetricgonMinducesa

tab

metricqonΣ.SinceΣisthespatialsliceattimet,weneedqtohave+++signature.

abttab

Thenn,beingnormaltoΣ,willbetimelikeand(say)future-directed.Thuswehavethe

at

relation

q=g+nn.(2.1)

ababab

Itgives

gt=N+Γn,(2.2)

abaa

b

NbeingtangenttoΣ.HereΓandNarecommonlyreferredasthe‘lapse’functionand

aa

t

the‘shift’vectorfieldrespectively[11].

SincewewanttointerprettasthevectorfieldalongwhichthevariablesdefinedonΣ

a

t

evolve,ithastobetimelikeandfuture-directed.Thus

Γ−NN>0,

2a

a

Γ>0.(2.3)

Furthermore,sinceweareinterestedinasymptoticallyflatspacetimes,theappropriate

asymptoticconditionstoimposeonthelapseandshift,inorderthattreducestoatimelike

a

Killingvectorfieldnormaltothespatialsliceatspatialinfinity,are

Γ→1,N→0(2.4)

TheEinstein-Hilbertactionis

S=(−g)