Black Holes and the Holographic Principle

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

BlackHolesandtheHolographicPrinciple

L´arusThorlacius

ScienceInstitute,Dunhaga3,107Reykjavik,Iceland

lth@hi.is

Abstract

arXiv:hep-th/0404098v1 14 Apr 2004Thislecturereviewstheblackholeinformationparadoxandbrie yappraisessomeproposedresolutionsinviewofdevelopmentsinstringtheory.Itgoesontogiveanelementaryintroductiontotheholographicprinciple.1IntroductionThetheoryofblackholesinvolvesasubtleinterplaybetweengravityandquantumphysics.Semiclassicalargumentsindicatethatthetimeevolutionofasystem,whereablackholeformsandthenevaporates,cannotbegovernedbythestandardpostulatesofquantummechanics.Iftheblackholeformsbygravitationalcollapsefromaninitialmattercon gu-rationthatisnonsingularanddescribedbyapurequantumstate,andifHawkingradiationistrulythermal,thentheformationandevaporationprocessevolvesapurestateintoamixedone,inviolationofquantummechanicalunitarity.Alternatively,unitaritymaybemaintainedinblackholeevolutionbutatthepriceofgivinguplocalityatafundamentallevel.Thelong-standingdebateregardingtheseissues,initiatedbyHawking[1,2],playedakeyroleinthedevelopmentoftheholographicprinciple[3,4].Thisradicalprinciplegoesbeyondblackholephysics.Itconcernsthenumberofdegreesoffreedominnatureandstatesthattheentropyofmattersystemsisdrasticallyreducedcomparedtoconventionalquantum eldtheory.Thisclaimissupportedbythecovariantentropybound[5]whichisvalidinarathergeneralclassofspacetimegeometries.Thenotionofholographyiswell

developedincertainmodelsandbackgrounds,inparticularinthecontextoftheadS/cftcorrespondence.Amoregeneralformulationislacking,however,andtheultimateroleoftheholographicprincipleinfundamentalphysicsremainstobeidenti ed.2Blackholeevolution

Letusstartbyreviewingthebasicingredientsthatgointotheblackholeinformationparadox.Weconsiderablackholeformedingravitationalcollapseandletusassumethattheinitialmattercon gurationisapproximatelysphericalandsu cientlydi usesothatspacetimecurvatureiseverywheresmallatearlytimes.Thesubsequentevolution,includingtheformationandevaporationoftheblackhole,isthenwellrepresentedby

1

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

r=0 r=∞4 3 MPl 2 4lPl M 4

r=0

Figure 1: Penrose diagram for a semiclassical black hole geometry. At early and late times the geometry approaches that of Minkowski spacetime. the Penrose diagram in Figure 1. The diagram assumes exact spherical symmetry. Only radial and time coordinates are displayed, with each point in the diagram representing a transverse two-sphere, whose area depends on the radial coordinate. Penrose diagrams are a useful means of portraying the global geometry and causal properties of a given spacetime. Without going into details we note two key properties. First of all,

the coordinates are chosen in such a way that radially directed light-rays correspond to straight lines oriented at±45 degrees to the vertical axis. The second feature is that, following a conformal mapping that brings in nity to a nite point, the entire spacetime geometry is represented by a nite region. As a result, the causal relationship between any two events is easily read o a Penrose diagram but proper distances in spacetime are not faithfully depicted. The event horizon, shown as a dotted line in Figure 1, de nes the boundary of the black hole region, inside which timelike observers cannot avoid running into the future singularity. An important feature of black holes is that local gravitational e ects are extremely weak at the event horizon of a large black hole. In fact, any curvature invariant formed from the Riemann tensor will go as an inverse power of the black hole mass. For a Schwarzschild black hole, for example, one nds Rµνλσ Rµνλσ= (1)

at the event horizon. This observation forms the basis of the semiclassical approach to black holes, which assumes that only low-energy physics is involved in the formation and evaporation of a black hole, except in the region near the singularity, and that away from this region physics can be described by a local e ective eld theory. The detailed 2

r=0

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

constructionofsuchane ective eldtheoryforblackholeevolutionpresentsaformidableandunsolvedtechnicalproblem,butletusforthemomentassumethatsuchatheorycanbefoundandsketchtheargumentforinformationlossinblackholeevolution.

2.1Semiclassicalinformationloss

The rststepistochooseappropriatespatialslicesthroughtheblackholespacetimetoprovideasetofCauchysurfacesforthequantumevolutionofoursystem.Theinitialsliceistakentolieintheasymptoticpastwherespacetimeisapproximately atandcontainsadi usedistributionofmatterthatwilllaterundergogravitationalcollapse.The nalslice,atasymptoticallylatetimes,containsalongtrainofoutgoingHawkingradiationandpossiblyalsoaPlanckmassremnantoftheblackhole.1Itturnsouttobepossibletochoosespatialslicesatanintermediatestageinsuchawaythattheyliepartlyinsidetheblackholeregionandpartlyoutside,asindicatedinFigure2.Foralargeenoughblackholethiscanbedoneinsuchawaythatthefollowingtworequirementsaremet:

1.ThereareCauchysurfacesthatintersecttheworldlinesoftheinfallingmatterinsidetheblackholebutalsotheworldlinesofmostoftheoutgoingHawkingradiationthatisemittedduringtheblackholelifetime.

2.Thesespatialslicesavoidtheregionofstrongcurvatureneartheblackholesingularityandarealsosmoothinthesensethattheirextrinsiccurvatureiseverywheresmall.Anexplicitconstructionofafamilyofniceslicesofthistypeisforexamplegivenin[6].Thesemiclassicaltheoryofblackholeevolutionrestsontheassumptionthat,givensuchafamilyofCauchysurfaces,thedynamicsofthecombinedmatterandgravitysystemisgovernedbyalow-energye ective eldtheoryandthatnoPlanckscalee ectsenterintothephysicsexceptnearthecurvaturesingularity.Theargumentforinformationlossisbasedontheexistenceofthise ective eldtheorybutnotonitsdetailedform.

Supposeaninitialcon gurationofcollapsingmatterinaweaklycurvedbackgroundisdescribedbyapurequantumstate|ψ(Σin) de nedonthesurfaceΣininFigure2.TheHamiltonianofthee ective eldtheorygeneratesalinearevolutionofthisstateintoanotherpurestate|ψ(ΣP) onthesurfaceΣP,whichispartiallyinsideandpartiallyoutsidetheblackholeregion.TheinsideandoutsideportionsofΣP,denotedbyΣbhandΣextrespectively,arespacelikeseparatedandasaresultallobservablesinthee ective eldtheorythathavesupportonΣextcommutewithobservablesthathavesupportonΣbh.ThestateonΣPisthereforeanelementofatensorproductHilbertspace,

|ψ(ΣP) ∈Hbh Hext.(2)

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

r=0 r=0Σbh r=0P

ΣoutΣext

Σin

Figure 2: Cauchy surfaces used in the argument for information loss in black hole evolution. Observers outside the black hole have no access to the part of|ψ(ΣP ) that is inside the black hole and ultimately runs into the singularity. As a result we are instructed to trace over states in Hbh giving rise to a mixed state density matrix on Hext . This mixed state will then evolve into another mixed state on the late time Cauchy surfaceΣout . We now have a paradox on our hands, for if the entire process of black hole formation and evaporation is to preserve unitarity, then the nal con guration of the system must be described by a pure quantum state|ψ(Σout ) . This nal state is obtained from the initial state by a unitary S-matrix,|ψ(Σout )= S|ψ(Σin ), (3) where S S = 1. This relation can in principle be inverted to express the initial state in terms of the nal one,|ψ(Σin )= S |ψ(Σout ), (4)

In other words, in a unitary theory the nal quantum state carries all information that is contained in the initial state. The semiclassical argument, on the other hand, resulted in a mixed nal state from which there is no way to recover the initial state. In other words, quantum information is lost in semiclassical black hole evolution.

2.2

Proposed resolutions

There are a number of ways to respond to this paradox. Let us brie y review the three main proposals

.

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

2.2.1Informationloss

Hawkingadvocatedthattheabovesemiclassicalargumentshouldbetakenatfacevalueandthatanaddedfundamentaluncertaintyistobeincorporatedintoquantumphysicswhengravitationale ectsaretakenintoaccount[2].Healsohadaconcreteproposalinvolvingamodi edsetofaxiomsforquantum eldtheorythatallowspurestatestoevolveintomixedones.InHawking’sformalismtheunitaryS-matrixofconventionalquantum eldtheory,whichmapsaninitalquantumstatetoa nalquantumstate,isreplacedbyasuperscatteringoperator$,whichmapsaninitialdensitymatrixtoa naldensitymatrix,andprocessesinvolvingblackholeformation,orevenvirtualprocessesinvolvinggravitational uctuations,giverisetosuperscatteringthatmixesquantumstates.

Thisproposalwascriticisedbyanumberofauthors[7,8].Inparticular,Banksetal.arguedthatHawking’sdensitymatrixformalismisequivalenttoconventionalquantum eldtheorycoupledtorandomly uctuatingsourceseverywhereinspacetime.Asaresultthetheorydoesnothaveemptyvacuumasit’sgroundstatebutratherathermalcon gurationatthePlancktemperature[7].Thisisclearlyphenomenologicallyunacceptablebutitshouldbenotedthattheargumentrestsoncertaintechnicalassumptionsandconceivablyaloopholemaybefoundtoavoidthethermaldisaster.TheearlycritisicmappearstohaveputastoptofurtherdevelopmentsinthisdirectionandthegeneralviewisthatHawking’sdensitymatrixformalismisnotaviableoptionforresolvingtheinformationparadox.This,ofcourse,doesnotruleoutatheoryincorporatinginformationlossbeingdevelopedinthefuture.

Atthemoment,however,ourbestcandidateforatheoryofquantumgravityissuper-stringtheoryandthistheorydoesnotfavorinformationloss.Init’soriginalformulation,stringtheoryisanS-matrixtheoryandassuchitismanifestlyunitary.Ontheotherhand,theoriginalformulationofstringtheoryreallyonlyamountstoaperturbativeprescriptionforscatteringamplitudesandisnotadequatefordescribingmacroscopicprocessessuchastheformationofalargemassblackhole.Ontheotherhand,wenowhavenon-perturbativeformulationsofstringtheorybothinanasymptotically atbackground[9]andinanti-deSitterspacetime[10],wherethegravitationaldynamicshasadualdescriptionwhichisuni-tary.Admittedly,inbothcasesthedualityisfoundedonconjecturesthatareunproven,andwillbedi culttoproveinallgeneralitybecausetheyinvolveweak-strongcouplingdualities,butsupportingevidencehasbeenpouringinforseveralyearsnow,buildingastrongcase.Wewillreturntothisinsection4.

2.2.2Blackholeremnants

Analternativeviewpoint,putforwardbyAharonovetal.[11],isthatblackholesdonotcompletelyevaporatebutratherleavebehindremnantsthatarestableorextremelylong-lived.Quantummechanicalunitarityisthenmaintainedbyhavingtheblackholeremnantcarryinformationabouttheinitialquantumstateofinfallingmatterthatformstheblackhole.

IfweassumethatHawking’ssemiclassicalcalculationofparticleemissionremainsvalid

5

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

untiltheremainingblackholemassapproachesthePlanckscaleandthatnoneoftheinitialinformationgoesoutwiththeHawkingradiationthenthemassofablackholeremnantcanbenomorethanafewtimesthePlanckmassandthereneedstobeadistinctremnantforeverypossibleinitialstate.Asaresult,thedensityoftheseremnantstatesatthePlanckenergymustbevirtuallyin niteandthisleadstoseverephenomenologicalproblemsiftheremnantsbehaveatalllikelocalizedobjects.Theire ectonlow-energyphysicscouldthenbedescribedintermsofane ective eldtheoryandcontributionsfromvirtualremnantstateswoulddominatealmostanyquantumprocessonemightconsider.EveniftheamplitudeforproducinganygivenPlanckmassremnantasanintermediatein,say,e+e scatteringatacolliderisextremelysmall,thein nitedensityofsuchstateswouldneverthelessmakeremnantsthedominantchannel.Anin nitedensityofstatesalsoleadstoapergentpairproductionrateofremnantsinweakbackground eldsandtopergentthermalsums.Sincethesee ectsarenotobservedeithertheinformationcarriedbyablackholeisnotleftbehindinaPlanckscaleremnantorsuchremnantsaredescribedbyveryunconventionallawsofphysicsatlowenergy.

Abouttenyearsagosomeremnantmodelsweresuggested,wherethesepathologiesweretobeavoidedbyaccommdatingthehighdensityofstatesinalargeinternalvolumecarriedbytheremnantandconnectedtotherestofspacetimeviaaPlanckscalethroatregion[12].Althoughthemodelshadsomesuccessandservedasawarningagainstdrawingtoo rmaconclusionfromargumentsbasedone ective eldtheory,theyhavenotbeendevelopedfurther.Amajorreasonforthiscanbetracedtosubsequentdevelopmentsinstringtheory.Withtheadventofstringdualityandbranesthebasicdegreesoffreedomofstringtheoryhavemoreorlessbeenidenti ed,andtheydonotincludeexoticblackholeremnantsatthePlanckscale.

OneofthetriumphsofstringtheoryintheninetieswasthemicrophysicalexplanationanddirectcalculationoftheBekenstein-Hawkingentropyofcertainextremalandnear-extremalblackholes[13].Theentropyisobtainedbycountingcon gurationsofstringsandbranesthatcarrythesamechargesastheblackholeinquestion.Suchcountingisonlyreliableforaweaklycoupledcollectionofstringsandbranesin atspacetimewhichbearslittleresemblancetothestronglycurvedgeometryofablackhole,butforacon gurationthatcorrespondstoanextremalblackholeoneappealstoextendedsupersymmetryandnon-renormalizationtheoremstoarguethatthecountingwill,infact,alsoholdatstrongcouplingwherethesystemismoreappropriatelydescribedasablackhole.Thenotionthat,bymovingaroundinsidetheparameterspaceofthetheory,onecan ndadualdescriptionofblackholesintermsofweaklycoupled(highlyexcited)stringsandbranes[14]leavesnoroomforapergentdensityofremnantstatesatthePlanckenergy.

2.2.3Blackholecomplementarity

Athirdpossibility,pioneeredbyPage[15]and’tHooft[16],isthatHawkingradiationisnotexactlythermalbutinfactcarriesalltheinformationabouttheinitialstateoftheinfallingmatter.Thisinformationmustthenbeencodedinsubtlecorrelationsbetweenquantaemittedatdi erenttimesduringtheevaporationprocessbecauseeveniftheformationand

6

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

evaporationprocessasawholeisgovernedbyaunitaryS-matrixtheradiationemittedatanygivenmomentwillappearthermal.Detectingtheinformationwouldrequirestatisticalanalysisofalargenumberofobservationsmadeonanensembleofblackholesformedfromidenticallypreparedinitialstates.Thisisaconservativeviewpointinthatitassumesunitarityinallquantumprocesses,alsowhengravitationale ectsaretakenintoaccount,butitleadstoanovelviewofspacetimephysicsandrequiresustogiveupthenotionoflocalityatafundamentallevel.

ThequestioniswhethertheinfallingmatterwillgiveupallinformationaboutitsquantumstatetotheoutgoingHawkingradiationorwhethertheinformationgetscarriedintotheblackhole.IftheinformationisimprintedontheHawkingradiationthenitmustalsoberemovedfromtheinfallingmatterforotherwisewewouldhaveaduplicationoftheinformationinaquantumstateinviolationofthelinearityofquantumevolution.Wecancomparethistothemoreconventionalinformationlosswhenabookisburned.Alltheinformationthatwasoriginallycontainedinthebookcaninprinciplebelearnedfrommeasurementsontheoutgoingsmokeandradiation,butatthesametimethisinformationisnolongertobefoundintheremainsofthebook.Inthiscase,however,itisawellunderstoodmicrophysicalprocessthatremovestheinformationfromthebookandtransfersittotheoutgoingradiation,whereasmatterinfreefallenteringalargeblackholedoesnotencounteranydisasterbeforepassingthroughtheeventhorizon.

Theprincipleofblackholecomplementarity[17]statesthatthereisnocontradictionbetweenoutsideobservers ndinginformationencodedinHawkingradiation,andhavingobserversinfreefallpassthroughtheeventhorizonunharmed.Thevalidityofthisprinciplerequiresmattertohaveunusualkinematicpropertiesatveryhighenergybutitdoesnotcon ictwithknownlow-energyphysics.Contradictionsonlyarisewhenweattempttodirectlycomparethephysicaldescriptioninwidelydi erentreferenceframes.Thelawsofnaturearethesameineachframeandlow-energyobserversinanysingleframecannotestablishduplicationofinformation[18].

2.2.4Thestretchedhorizon

Inordertoillustratetheconceptofblackholecomplementarityitisusefultohaveaphysicalpictureoftheevaporationprocessintheoutsideframe.Forsometimeastrophysicistshavemadeuseofthemembraneparadigmofblackholestodescribetheclassicalphysicsofaquasistationaryblackhole[19].Fromthepointofviewofoutsideobserverstheblackholeisthenreplacedbyastretchedhorizon,whichisamembraneplacedneartheeventhorizonandendowedwithcertainmechanical,electricalandthermalproperties.Thisdescriptionisdissipativeandirreversibleintime.Onedoesnothavetobespeci caboutthelocationofthestretchedhorizonaslongasitisclosetotheeventhorizoncomparedtothetypicallengthscaleoftheastrophysicalproblemunderstudy.

Inthecontextofblackholeevaporationonegoesastepfurtherandviewstheclassicalstretchedhorizonasacoarsegrainedthermodynamicdescriptionofanunderlyingmicro-physicalsystem,aquantumstretchedhorizon,locatedaPlanckdistanceoutsidetheeventhorizonandwithanumberofstatesgivenbyexp(A/4),whereAistheblackholeareain

7

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

Planckunits[17].Thenatureofthemicrophysicsinvolvedwasleftunspeci edin[17]butitwaslatersuggestedthatthedynamicsofthestretchedhorizonmightbeexplainedinthecontextofstringtheory[20].Thestickypartisthat,inordertoimplementblackholecomplementarity,wehavetostipulatethatthismembrane,orstretchedhorizonisabsentinthereferenceframeofanobserverenteringtheblackholeinfreefall.

Theevaporationofalargeblackholeisaslowprocessand,forthepurposeofourdiscussionhere,theevolvinggeometrymaybeapproximatedbyastaticSchwarzschildsolution,2M) 1dr2+r2d 2.(5)ds2= (1 r

ProvidedthemassMissu cientlylarge,anobserverinfreefallwillnotexperienceanydiscomfortuponcrossingtheeventhorizonatr=2Mbutaso-called ducialobserver,whoisatrestwithrespecttotheSchwarzschildcoordinateframe,willbebathedinthermalradiationatatemperaturethatdependsontheradialposition,

T(r)=1

r) 1/2.(6)

Thistemperaturepergesneartheblackhole,T≈(2πδ) 1,whereδistheproperdistancebetweenthe ducialobserverandtheeventhorizon.Theradiationcanbeattributedtotheaccelerationrequiredtoremainstationaryat xedr,whichpergesasδ→0.Inthespiritofthemembraneparadigmtheradiationcanalsobeviewedasthermalradiationemanatingfromahotstretchedhorizon.Theregionnearesttheeventhorizon,wherethetemperature(6)formallyperges,isthenreplacedbyaPlancktemperaturemembranelocatedaproperdistanceofonePlancklengthoutsidetheeventhorizon.

ThestretchedhorizonisthenthesourceofHawkingradiationasfarasoutsideobserversareconcerned.ItemitsPlanckenergyparticlesbutmostofthesewillfallbackintotheblackhole.ThosewhoescapeitsgravitationalpullpredominantlycarrylowangularmomentumandareredshiftedtoenergiesofordertheHawkingtemperaturebythetimetheyreachtheasymptoticregion.Inthisview,HawkingradiationistheenormouslyredshiftedglowfromaPlanckscaleinfernoatthestretchedhorizon.Itcarriesallinformationabouttheinitialstateofthematterthatformedtheblackhole,albeitinaseverelyscrambledform.Noinfallingobserversurvivestheencounterwiththehotmembrane.Infactnothingeverenterstheblackholeinthereferenceframeofoutsideobservers,includingthematterthatformedtheblackholeinthe rstplace.Itisfamiliarfromclassicalgeneralrelativitythatmatterfallingintoablackholeappearstoslowdownasitapproachesthehorizonandfadesoutofviewduetothegravitationalredshift.Thisisafterallwherethetermblackholecomesfrom.Inthequantummembranepicture,theinfallingmatterrunsintothestretchedhorizon,getsabsorbedintoitandthermalized,andisthenslowlyreturnedbackoutalongwiththerestoftheblackholeasitevaporates.

Thisscenariocon ictswiththenotionthatinfallingobserversfeelnoille ectsuponpassingthroughtheeventhorizonofalargeblackhole.Theanalysisofseveralgedankenexperiments,designedtoexposepossiblecontradictionsbetweentheexperienceofinfallingobserversandthedescriptionintheoutsidereferenceframe,ledtotheconclusionthatthe

8

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

apparentcontradictionscouldineachcasebetracedtoassumptionsaboutphysicsatorabovethePlanckscale[18].Thisobservationdoesnotbyitselfresolvetheinformationproblembutitchallengestheunderlyingassumptionofthesemiclassicalapproachthattheparadoxcanbeposedintermsoflow-energyphysicsalonewithoutanyreferencetothePlanckscale.

Ifwetaketheprincipleofblackholecomplementarityatfacevaluewehavetoacceptaradicallynewviewofspacetimephysics.Thenotionofalocalevent,thatisinvariantundercoordinatetransformations,iscentraltogeneralrelativity.Accordingtoblackholecomplimentarityobserversindi erentreferenceframescantotallydisagreeaboutthelo-cationoftherathersigni canteventwhereanobserverfallingintoablackholemeetshisorherend.Theproperdistancebetweentheeventhorizonandthe nalsingularityisproportionaltotheblackholemassandcanthereforebearbitrarilylarge.Theprinciplethereforeintroducesanewdegreeofrelativityintofundamentalphysicsbeyondthefamil-iarrelativityofmeasuringsticksandtimepieces.Itrequiresphysicstobenon-localonarbitrarilylargelengthscales,yetconventionallocalityandcausalitymustberecoveredineverydayprocessesatlowenergy.

Adetailedphysicaltheoryofblackholeevaporationthatincorporatesblackholecom-plementarityhasnotbeendeveloped.2Thereareindications,however,fromstringtheorythatsuchadescriptionshouldbepossible,especiallyinthecontextoftheadS/cftcorre-spondence.3Inthiscase,thedualgaugetheoryisunitaryanditsS-matrixinprincipleincludesprocesseswhereblackholesareformedintheadSbackgroundandsubsequentlyevaporate[22].Theproblemisthatthespacetimeinterpretationofgaugetheoryob-servablesisobscureanditisnon-trivialtoestablishlocalcausalityatlowenergyinadSspacetimeinthelanguageofthegaugetheory,evenintheabsenceofblackholes[23].3Theholographicprinciple

Localquantum eldtheoryleadstounitarityviolationinthecontextofblackholeevolu-tion.Thiscanbeavoidedbyadoptingtheviewpointofblackholecomplementaritybutthenspacetimephysicsisrequiredtobenon-localatafundamentallevel.Thisnotionofnon-localitywastakenfurtherby’tHooft[3]andSusskind[4],whoarguedthatthenumberofavailablequantumstatesinanygivenregionismuchsmallerthanonemightnaivelyexpect.Theclaimisthatthisnumberisnotanextensivequantity,i.e.onethatscalesasthevolumeoftheregioninquestion,asonewould ndinanylocalquantum eldtheorywithanultravioletcuto ,butisinsteadproportionaltoasurfaceareaassociatedwiththeregion.Thisdramaticreductioninthenumberofstatesascomparedtoconventionalquantum eldtheoryisreferredtoastheholographicprinciple.

Thisprincipleonlyariseswhengravitationale ectsaretakenintoaccount.Letus

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

4A,(10)

insteadofbeingproportionaltothevolumeV.Whydoesquantum eldtheorysogrosslyoverestimatethemaximalentropy?Theanswerhastodowithgravitationalbackreaction.

10

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

Γ

Figure 4: The Susskind process: The regionΓ is placed inside an imploding shell that forms a black hole with area A. Most of the states of the spin system are highly excited and will curve the surrounding spacetime to the extent that it collapses to a large black hole, in fact much larger than the one formed in the Susskind process.

3.1

The spacelike entropy bound

A useful way to state the holographic nature of spacetime physics is through entropy bounds that relate the number of available states to a surface area associated with a given quantum system rather than its volume.4 We start with a simple entropy bound motivated by the analysis of the Susskind process. Spacelike entropy bound: The entropy contained in any spatial region will not exceed 1/4 of the area of the region’s boundary in Planck units. This bound turns out to be too naive and is only valid under very restrictive assumptions. It is nevertheless useful as a step towards the much more universal covariant entropy bound of Bousso[5] which we state later on. It is instructive to consider some of the objections to the spacelike entropy bound. (i) Particle species: The entropy of a matter system depends on the number of species of particles in the theory, and the entropy bound will be violated if this number is su ciently large. Just how large the number has to be depends on the size of the system. A simple estimate[24] shows that the spacelike entropy bound fails for a weakly coupled gas in a volume with surface area A in Planck units if the number of massless particle species is N> A. For a surface area of 1 cm2 the required number of species is enourmous, N～ 1066 . As there is no upper bound on the possible number of species in a eld theory the entropy bound can formally always be violated. On the other hand, it is too much to expect an arbitrary eld theory coupled to gravity to give sensible results. The holographic principle4

For a detailed review of the holograph

ic principle and various entropy bounds see[24].

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

4σ Smatter(Γ)>1

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

time+ B++

+

Figure 6: The circle represents a spherical surface B (with the polar angle suppressed). The lines denote future and past directed lightrays orthogonal to B.

3.2

Lightsheets and the covariant entropy bound

The foregoing examples demonstrate the failure of the spacelike entropy bound but this does not mean that the holographic principle fails. What is needed is a more geometric entropy bound that adapts to dynamical situations like the ones in examples (ii) and (iii). This is provided by Bousso’s covariant entropy bound[5], which involves light-cones rather than spacelike volume, but before we get to that we need to introduce a few geometric concepts. We do this in the context of a simple example to avoid making the discussion too technical. Consider a spherical surface B, with area A(B), at rest in at spacetime and imagine emitting light simultaneously from the entire surface at some time t0 . The light front will propagate in two directions, radially inwards and radially outwards. One can also consider light arriving radially at B at time t0 from the outside and inside respec

tively. Thus there are four families of lightrays orthogonal to B,++ + + future directed, outgoing, future directed, ingoing, past directed, outgoing, past directed, ingoing.

There is nothing special about our spherical surface in this respect. Every surface in Lorentzian geometry has four orthogonal light-like directions, two future directed and two past directed, and the corresponding families of lightrays trace out null hypersurfaces (at least locally) in spacetime. The expansionθ of a family of lightrays, that are orthogonal to a surface, is positive (negative) if the rays are perging (converging) as one moves along them away from the surface. In our simple example the sign of the expansion is easily determined. The location at in nitesimal time t= (or t= for past directed lightrays) as measured in the rest frame of B, of photons that were at B at time t= 0 de nes a new surface B′ and we are interested in the area of B′ relative to B. If A(B′ )> A(B) the expansion of this particular family of lightrays is positive. Beyond our example, the expansion of a family of lightrays 13

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

L B

L′

Figure 7: A lightsheet L of a surface B is a lightlike hypersurface traced out by following a family ofconverging lightrays orthogonal to B.

orthogonal to any smooth surface in curved spacetime can be de ned locally in a coordinate invariant manner[26] and its sign determined by comparing areas of neighboring surfaces intersected orthogonally by the lightrays. The surface B in our example is in a normal region where outgoing lightrays (both future and past directed) have positive expansion and ingoing lightrays have negative expansion,θ++> 0,θ+ < 0,θ +> 0,θ < 0 . (14) In a future trapped region on the other hand both the future directed families of lightrays have negative expansion,θ++< 0,θ+ < 0,θ +> 0,θ > 0 . (15)

Such behavior is for example found in the collapsing region inside a black hole. There are other possibilities besides normal and future trapped but one nds in all cases that at least two out of the four families of orthogonal lightrays have non-positive expansion,θ≤ 0, locally at the surface. In degenerate cases this can be true of three or even all four families. A lightsheet L of a surface B is a lightlike hypersurface obtained by following a family of lightrays that is orthogonal to B and hasθ≤ 0. It follows from our previous comment that every surface has at least two lightsheets. The expansion will in general change as we move along L and the lightsheet ends whereθ becomes positive. For example, when converging lightrays self-intersect their expansion turns positive and they no longer form a lightsheet. In other words, lightsheets do not extend beyond focal points as indicated in Figure 7 for our spherically symmetric example. Covariant entropy bound: The entropy on any lightsheet of a surface B will not exceed 1/4 of the are

a of B in Planck units. 14

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

Thecovariantentropyboundappearstobeuniversallyapplicable.Atleasttherearenophysicallyrelevantcounterexamplesknown.Ithasevenbeenproveninthecontextofgeneralrelativitywiththeaddedassumptionthatentropycanbedescribedbyacontinuum uidandwithsomeplausibleconditionsrelatingentropydensityandenergydensity[27].Ofcourse,aswaspointedoutin[27],entropyatafundamentallevelisnota uidandtheassumedconditionsrelatingentropyandenergyarenotawayssatis ed.Itshouldthereforebestressedthatthecovariantentropyboundhasnotbeenderivedfrom rstprinciples,afterallthe rstprinciplesofquantumgravityareunknown,butisratheranobservationaboutthenatureofmatterandgravitythatshouldbeexplainedbyafundamentaltheory.UndercertainassumptionsthecovariantentropyboundimpliesthespacelikeboundofSection3.1[5]butthecovariantboundisvalidmuchmoregenerally.Itisinstructivetoseehowthecovariantbounddealswiththevariousobjectionsthatwereo eredtothespacelikebound.

(i)Particlespecies:Thespeciesproblemisthesameasinthespacelikecaseandisequallyrelevant(orirrelevant)here.

(ii)ClosedFRWuniverse:LightraysdirectedtowardsthenorthpoleofthethreespherefromBinFigure5,i.e.onesthattraverseΓ,havepositiveexpansion.Thereforetheydonottraceoutalightsheetandthecovariantentropybounddoesnotapply.BothlightsheetsofBaredirectedtowardsthesouthpoleinthiscaseandthecovariantentropyboundisvalidforthecomplementofΓ.

(iii)Spatially atFRWuniverse:Withoutgoingintodetailswenotethattheproblemherehadtodowithverylargeregionsinaspatially atuniverse.Thesurfaceofsucharegionwillhaveapastdirectedlightsheetbutthislightsheetterminatesattheinitialsingularityandhasthereforemuchlessentropyonitthaniscontainedintheoriginalspatialvolume.ItcanbeshownthattheentropythatpassesthroughthelightsheetislessthataquarterofthesurfaceareaoftheregionunderconsiderationinPlanckunits[28].4TheadS/cftcorrespondence

Theholographicprincipleisputforwardasabasicprincipleofphysicsandassuchitshouldbemanifestinanysuccessfulfundamentaltheory.Itisthereforenaturaltoasktowhatextentsuperstringtheory,theleadingcandidateforauni edquantumtheoryofmatterandgravity,isholographic.Sincestringtheoryisfarfroma nishedproduct,withmajorconceptualproblemsunsolved,itmaybeprematuretosubjectittothistest.Yet,remarkably,ithasalreadyproducedasettingwhereholographyisexplicitlyrealized.We nishthislecturewithaquicksketchoftheadS/cftcorrespondenceandanorder-of-magnitudeestimateshowinghowthenumberofdegreesoffreedominthisnonperturbativede nitionofstringtheoryisinlinewiththeholographicprinciple.

ThesettingfortheoriginaladS/cftcorrespondence[10]isthephysicsofNcoincidentDirichletthree-branesinten-dimensionalspacetime.Thisphysicsisgovernedbyanactionoftheform

S=Sbulk+Sbrane+Sint.(16)

15

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

HereSbulkistheten-dimensionalgravitationalaction,i.e.typeIIBsupergravityalongwithα′correctionsfromstringtheory,whileSbraneistheworldvolumeactionoftheNDirichlet-branes,i.e.d=4,N=4supersymmetricSU(N)Yang-Millstheoryalongwithitsownα′corrections.Sintdescribesthecouplingbetweentheten-dimensionalbulkandthebranesandthiscouplingcanbeignoredinthelimitofweakstringcoupling,Ngs 1.AstackofcoincidentDirichletthree-braneshasadualdescriptionasanextendedobjectinsupergravity.Thecorrespondingten-dimensionalmetricisgivenbythelineelement

ds=21

H(r) dt+23 i=1dxidxi+

.TheparameterRisacharacteristiclengthwhichmustbelarge√comparedtothestringlengthr4

z2+d 25, (19)

andxi=Rxwherewehaveintroduceddimensionlessvariablesthroughr=Rz,t=Rt i.

ThisisthemetricofadS5×S5usingPoincar´ecoordinatesfortheadS5part.

TheadS/cftcorrespondencefollowsfromtheobservationthattheabovetwodescrip-tionsoftheD3-branesystembothinvolvetwodecoupledfactors,andineachcaseoneofthefactorsisbulkten-dimensionalsupergravity.ByidentifyingtheotherfactorswitheachotherweareledtoadualitybetweentheworldvolumeSU(N)gaugetheory,whichisafour-dimensionalconformal eldtheory,andstringtheoryintheadS5×S5near-horizongeometry.Thetwodualdescriptionsapplyatdi erentcouplingstrength,Ngs 1vs.gs 1 Ngs.

4.1AdS/cftandtheholographicprinciple

Wenowgiveanargument,duetoSusskindandWitten[29],thatthenumberofdegreesoffreedominadS-gravityinfactsatis esaholographicbound.Forthisitismoreconvenienttoadoptso-calledcavitycoordinatesforadS5,

ds2=R2 1+u2

(1

16u2)2 2du2+u2d 23+d 5. (20)

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

Thetotalentropyofthegravitationalsystemisin nitebecausethespatialpropervolumeofadSspacetimeperges.IncavitycoordinatesthespatialboundaryoftheadSgeometryislocatedatu=1andweimposeacuto atu=1 with 1.Withthisinfraredregulatorinplacethespatialvolumeis nite.Theproperareaofthespatialboundaryisalso niteandcaneasilybeobtainedfrom(20).Thisleadstothefollowingholographicentropybound, 31R85S≤R=

3,(22)

whichisseentosaturatetheholographicbound(21)whenwetakeintoaccounttherelation

(18)betweenRandN.

5Discussion

Wereviewedtheblackholeinformationproblemandarguedthatdevelopmentsinstringtheorystronglyfavoritsresolutionintermsofunitaryevolution.Thiscomesatapriceofintroducingafundamentalnonlocalityintophysics,buthistoricallythisnonlocalityinblackholeevolutionservedtomotivatetheholographicprinciple,afarreachingnewparadigmforquantumgravity.

Manyintrestingaspectsofholographywerenottoucheduponhere.Thegoalwastoconveysomeofthebasicideasratherthangiveasurveyofthe eld,whichisbynowquitewide.Recentworkhasincludedarevivalofinterestinmatrixmodelsoftwo-dimensionalgravity[30],whichprovidearelativelysimplerealizationoftheholographicprinciple,andalsotheapplicationofholographicideastocosmology[31].

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This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

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This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

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