1

Thenatureandoriginof

time-asymmetricspacetimestructures*

H.D.Zeh(UniversityofHeidelberg)

www.zeh-hd.de

Abstract:Time-asymmetricspacetimestructures,inparticularthoserepresentingblack

holesandtheexpansionoftheuniverse,areintimatelyrelatedtootherarrowsoftime,

suchasthesecondlawandtheretardationofradiation.Thenatureofthequantumar-

row,oftenattributedtoacollapseofthewavefunction,isessential,inparticular,for

understandingthemuchdiscussed"blackholeinformationlossparadox".However,this

paradoxassumesanewformandmightnotevenoccurinaconsistentcausaltreatment

thatwouldpreventtheformationofhorizonsandtime-likesingularities.

A“masterarrow”,whichcombinesallarrowsoftime,doesnothavetobeidentifiedwith

thedirectionofaformaltimeparameterthatservestodefinethedynamicsasasucces-

sionofglobalstates(atrajectoryinconfigurationorHilbertspace).Itmayevenchange

directionwithrespecttoafundamentalphysicalclock,suchasthecosmicexpansion

parameterifthiswasformallyextendedeitherintoafuturecontractioneraortonega-

tive"pre-big-bang"values.

1Introduction

Sincegravityisattractive,mostgravitationalphenomenaareasymmetricintime:ob-

jectsfalldownorcontractundertheinfluenceofgravity.InGeneralRelativity,this

asymmetryleadstodrasticallyasymmetricspacetimestructures,suchasfuturehori-

zonsandfuturesingularitiesaspropertiesofblackholes.However,sincetherelativistic

andnonrelativisticlawsofgravitationaresymmetricundertimereversal,alltime

asymmetriesmustariseasconsequencesofspecific(onlyseemingly"normal")initial

conditions,forexampleasituationofrestthatcanbepreparedbymeansofotherar-

*arXiv:1012.4708v12+.V5waspublishedintheSpringerHandbookofSpacetimePhysics(A.AshtekarandV.Petkov,edts.–Springer2014);seethe“Noteaddedafterpublication”attheendofthistext!

2

rowsoftime,suchasfriction.Otherwise,conclusionslikegravitationalcontraction

wouldhavetoapplyinbothdirectionsoftime.Indeed,thesymmetryofthegravitational

lawsdoesallowobjectstobethrownup,wheretheirfreemotioncouldinprincipleend

byanotherexternalintervention,ortheconceivableexistenceof"whiteholes",which

wouldhavetocontainpastsingularitiesandpasthorizons.

Theabsenceofsuchpasthorizonsandsingularitiesfromourobserveduniverse(except,

perhaps,foraveryspecificbigbangsingularity)mustberegardedasatimeasymmetry

characterizingourglobalspacetime(seeSects.2and4),whileEinstein'sfieldequations

wouldnotonlyadmittheoppositesituation(forexample,inhomogeneouspastsingular-

ities),butalsomanysolutionswithmixedorundefinedarrowsoftime–including

closedtime-likecurvesandnon-orientablespacetimes.Therefore,themerepossibility

ofposingan"initial"conditionisexceptionalingeneralrelativityfromageneralpointof

view.Iwillherenotdiscusssuchmathematicallyconceivablesolutionsthatdonotseem

toberealizedinNature,butinsteadconcentrateonmodelsthatcomeclosetoouruni-

verse–inparticularthosewhicharegloballyofFriedmanntype.Aspecificarrowchar-

acterizingaFriedmannuniverseisgivenbyitsexpansion(unlessthiswouldbereversed

atsometimeofmaximumextension–seeSect.4).

Inmanycases,non-gravitationalarrowsoftimeremainrelevantfortheevolutionof

gravitatingbodiesevenafterthelatterhavebeenpreparedinanappropriateinitial

state.Thisapplies,inparticular,tostronglygravitatingobjects,suchasstars,whoseevo-

lutionisessentiallycontrolledbythermodynamics(emissionofheatradiationintothe

colduniverse).Therelationbetweentheelectrodynamicandthermodynamicarrows

(retardationandthesecondlaw,respectively)1isquiteobviousinthiscase.

Gravitatingsystemsarenonethelessthermodynamicallyunusualinpossessingnegative

specificheat.2Thismeans,forexample,thatstarsbecomehotterwhenlosingenergyby

emittingheat,andthatsatellitesaccelerateasaconsequenceoffrictionintheearth's

atmosphere.Itcanbestbeunderstoodbymeansofthevirialtheorem,whichstatesinits

nonrelativisticform,andforforcesthatdecreasewithdistanceaccordingtotheinverse

squarelaw(thatis,gravitationalandCoulombforces),thatallboundstateshavetoobey

therelation ,wheretheoverbarmeansaveragingover(quasi)periodsof

time.Therefore,

3

.

(1)

Whenlosingthermalenergybyradiation,thesesystemsmustgaintwiceasmuchfrom

gravitationalcontractioninordertomaintainaquasi-stablestate.Nonrelativistically,

thisnegativeheatcapacitycouldbeboundedbymeansofother(repulsive)forcesthat

becomerelevantathighdensities,orbythePauliprinciple,whichcontrolsthedensityof

electronsinwhitedwarfstarsorsolidplanets,forexample.Relativistically,eventhese

limitswillbreakdownatacertainmass,since(1)relativisticdegeneracymustultimate-

lyleadtothecreationofotherparticles,while(2)thepotentialenergyofrepulsiveforc-

eswillitselfgravitate,andforasufficientlylargemassovercompensateanyrepulsion.

Therefore,itisthethermodynamicarrowunderlyingthermalradiationthatrequires

evolutionofgravitatingsystemstowardstheformationofblackholes.Classically,black

holeswouldthusdefinethefinalstatesintheevolutionofgravitatingsystems.

2BlackHoleSpacetimes

Themetricofasphericallysymmetricvacuumsolutionfornon-zeromassisshownin

Fig.1inKruskalcoordinatesuandv.ThisdiagramrepresentsthecompletedSchwarz-

schildmetricintheform

€

ds2 =32M 2

re−r / 2M −dv 2 + du2( ) + r2 dθ 2 + sin2θdφ 2( ) , (2)

wherethenewcoordinatesuandvareintheexternalregion(r>2M)relatedtoconven-

tionalScharzschildcoordinatesrandtby

€

u = er / 4M r2M

−1cosht4M#

$ %

&

' ( (3a)

€

v = er / 4M r2M

−1sinh t4M#

$ %

&

' ( . (3b)

Eachpointinthediagramrepresentsaspherewithsurface4πr2.Notethatrandtinter-

changetheirrolesasspaceandtimecoordinatesforr<2M,where2MistheSchwarz-

schildradius.AllparametersaregiveninPlanckunitsh/2π=G=c=1.

4

AsNatureseemstoprovidespecificinitialconditionsinouruniverse,itmaythereby

excludeallpastsingularities,andhenceallpasteventhorizons.Thisinitialcondition

wouldimmediatelyeliminatetheSchwarzschild-Kruskalvacuumsolutionthatisshown

intheFigure,butwemayinsteadconsiderthefutureevolutionofasphericallysymmet-

ricmassdistributioninitiallyatrest,suchasadustcloud.Itwouldclassicallycollapse

freelyintoablackhole,asquantitativelydescribedbytheOppenheimer-Snyderscenar-

io3(seeleftpartofFig.2).Thevacuumsolution(2)isthenvalidonlyoutsidethesurface

ofthedustcloud,butthissurfacemustaccordingtoaclassicaldescriptionfallthrough

thearisinghorizonatsomefinitepropertime,andabitlaterhitthefuturesingularity.

Fig.1:CompleteformalcontinuationoftheSchwarzschildsolutionbymeansofunique

Kruskalcoordinates.QuadrantsIandIIrepresentexternalandinternalparts,respec-

tively,ofaclassicalblackhole.IIIisanotherasymptoticallyflatregion,whileIVwould

describetheinteriorofa"whitehole".Inthisdiagram,fixedSchwarzschildcoordinatesr

andtarerepresentedbyhyperbolaandstraightlinesthroughtheorigin,respectively.

Worldlinesoflocalobjectscouldstartatt=-¥inIoratt=+¥inIII,oratr=0onthepastsingularityinIV,whiletheymustendatt=+¥or-¥inIorIII,respectively,orata

secondsingularitywithcoordinatevaluer=0inII.Ontime-likeorlight-likecurvesin-

tersectingoneofthehorizonsattheSchwarzschildradiusr=2M,thevalueofthecoor-

dinatetjumpsfrom+¥to-¥attherimofquadrantI,orfrom-¥to+¥attherimof

quadrantIII,wheretdecreasesintheglobaltimedirection.

5

Foracloudofinteractinggasmolecules,thisgravitationalcollapsewouldbethermody-

namicallydelayedbythearisingpressure,asindicatedintheIntroduction.Gravitational

radiationwouldleadtothelossofanykindofmacroscopicstructure,whilewhatever

remainswouldbecomeunobservabletoanexternalobserver.Althoughthermodynamic

phenomenacontrolthelossofenergybyradiationduringmostofthetime,theasym-

metricabsenceofpastsingularitiesrepresentsafundamentalcosmologicalinitialcondi-

tion.However,aconceivablewhiteholeinitiatedbyapastsingularitythatcompletely

representedatime-reversedblackholewouldevenrequireanti-thermodymicsandco-

herentlyincomingadvancedradiation.Onemaysuspectthatallthesevariousarrows

arerelatedtooneanother,thusdefiningacommon"masterarrow".

Fig.2:Oppenheimer-Snydertypespacetimesofablackanda"white"hole.

SinceitwouldrequireinfiniteSchwarzschildcoordinatetimeforanobjecttoreachthe

horizon,anymessageitmaysendtotheexternalworldshortlybeforeitdoessowould

notonlybeextremelyredshifted,butalsodramaticallydelayed.Themessagecould

reachadistantobserveronlyatincreasinglylaterstagesoftheuniverse.(Anapparatus

fallingintoagalacticsizeblackholecouldevensendmessagesforaconsiderablelength

ofpropertimebeforeitwouldapproachthehorizon.)Soallobjectsfallingintotheblack

holemusteffectivelydisappearfromtheviewofmortalexternalobserversandtheir

descendants,eventhoughtheseobjectsneverseemtoreachthehorizonaccordingto

theirrapidlyweakening,butinprinciplestillarrivingsignals.Theonlyasymptotically

observablepropertiesoftheblackholeareconservedonesthathaveearlyenough

causedeffectsontheasymptoticmetricorotherasymptoticfields,namelyangularmo-

mentumandelectriccharge.Thistime-asymmetricconsequenceisknownasthe"no-

hairtheorem"forblackholes.Duringcosmologicaltimes,ablackholeaccumulatingion-

6

izedinterstellarmattermayevenloseitschargeandangularmomentum,too,forstatis-

ticalanddynamicalreasons.4Onlyitsmassanditscenterofmassmotionwouldthen

remainobservationallymeaningful.Ablackholeisusuallycharacterizedbyitscenterof

massmotionanditslong-lastingproperties,namelyitsmassM,chargeQ,andangular

momentumJ,inwhichcaseits"Kerr-Newmanmetric"isexplicitlyknown.Theinternal

topologicalstructuresofthesemetricsforJ≠0and/orQ≠0areradicallydifferentfrom

thatoftheKruskalgeometryinFig.1,thusraisingfirstdoubtsinthevalidityofthese

classicalcontinuationsinsidethehorizon.

Itisimportant,though,tokeepinmindtheessentialcausalstructureofablackhole:its

interiorspacetimeregionIIneverentersthepastofanyexternalobserver,thatis,itwill

neverbecomea“fact"forhim.Thisremarkincludeseventsofobjectscrossingthehori-

zon.Whilethewholeexteriorregionr>2Mcanbecompletelyfoliatedbymeansof“very

nice”space-likeslicesaccordingtoincreasingSchwarzschildorsimilartimecoordinates

with-¥<t<+¥,theinteriorcanthenberegardedasitsglobalfuturecontinuationbe-

yondtheeventhorizon,whereincreasingtimecanbelabeledbytheSchwarzschildco-

ordinaterdecreasingfromr=2Mtor=0.Thisstructuremustbeessentialforallcausal

considerationsthatincludeblackholes–notleastfortheirownfate(Sect.3).Inthe

classicalscenario,theinternalstateofablackholewouldbecompletelydeterminedby

theinfallingmatter,whichcouldevendependonour"free"decisionsaboutwhatto

dropintoablackhole.Nonetheless,propertiesofthisinfallingmatterwouldthenirre-

versiblybecome"irrelevant"toallexternalobservers–atermthatisalsousedtodefine

ageneralizedconceptofcoarsegrainingrequiredfortheconceptofphysicalentropyin

statisticalthermodynamics.5

3ThermodynamicsandtheFateofBlackHoles

Intheclassicalpicturedescribedabove,ablackholewouldrepresentaperfectabsorber

atzerotemperature.ThispicturehadtobecorrectedwhenBekensteinandHawking

demonstrated,6thelatterbyexplicitlytakingintoaccountquantumfieldsotherthan

gravity,thatablackholesmustpossessfinitetemperatureTandentropySproportional

toitssurfacegravitykandsurfaceareaA,respectively:

7

, (4a)

. (4b)

Here,kandAareknownfunctionsofM,QandJ,whiletheexplicitexpressionsgivenon

therighthandsideofthearrowholdforSchwarzschildblackholes(Q=J=0)andwith

respecttospatialinfinity(thatis,bytakingintoaccountthegravitationalredshift).This

means,inparticular,thatablackholemustemitthermalradiation(Hawkingradiation)

proportionaltoT4AaccordingtoStefan-Boltzmann'slaw,andtherefore,thatitlivesfora

verylargebutlimitedtimeoftheorder1065(M/Msun)3years.Forstarsorgalaxiesthisis

verymanyordersofmagnitudemorethanthepresentageoftheuniverseofabout1010

years,butfarlessthananyPoincarérecurrencetimesforsuchmacroscopicsystems.So

onehastobecarefulaboutwhatismeantby“asymptotic”indifferentcontexts.

Eventheselargeevaporationtimeswillbeginto“count”onlyaftertheblackholehasfor

averylongtimetocomegrowninmassbyfurtheraccretingmatter7(includinganti-

matterifitbecomesavailableduringtheblackhole´sverylongjourneythroughtheuni-

verse)–atleastuntilthecosmicbackgroundtemperaturehasdroppedbelowthevery

smallblackholetemperature.Althoughevaporationtimesarethusextremelylong,all

radiationregisteredbyanexternalobservermusthavebeencausedoutsidethehorizon.

Schwarzschildtimesrepresentpropertimesofdistantobserversintherestframeofthe

blackhole,butthespacelikeslicesthattheydefinemaybeconsistentlycontinuedin-

wardswhileremainingoutsidethehorizoninordertoformacompletefoliationofthe

wholeexternalregionI.Bydefinition,theywouldthenallhavetoincludethecenterof

thecollapsingmatteratapre-horizonstage.However,ahorizonanditsinteriorregionII

couldneverformiftheblackhole’senergywasindeedradiatedawaybeforeanyinfall-

ingmatterarrivedattheclassicallypredictedhorizoninthesenseofthisglobaldynam-

icalfoliation.Althoughsuchmattermayneedonlysecondsofpropertimetoreachthe

classicallyexpectedhorizon,theremustalwaysexistsimultaneitieswhichinclude

eventsonthelatepre-horizonpartofitstrajectoriesaswellasexternalonesinourfar

future–includingthoseatt»1065yearsormorefromnow.Thissingulargravitational

timedilationdoesnotrequireanyextremespacetimecurvatureintheregionwhereit

applies.Attemptstofindforcesorstresstermsthatpreventinfallingmatterfromcross-

ingthehorizonforthispurposewouldbereminiscentofPoincaré’ssearchforforcesto

8

explaintheLorentzcontraction.Sowhathappenstomatterthatseemstofallintothe

blackhole(andthatmayevenbeentangledwithmatterthatremainsoutside)?

Schwarzschildsimultaneitiesmaythusbecounterintuitive.Onemayalsousetimetrans-

lationinvarianceoftheexternalregionoftheKruskaltypediagram(Figs.1or2a)in

ordertodefinethetimecoordinatev=t=0tocoincidewithanexternaltimeclosetothe

peakoftheHawkingradiation(intheverydistantfuturefromourpointofview)with-

outcominganyclosertothehorizonthatisdefinedbytheremainingblackholemass.

Assumingthatonecanneglectanyquantumuncertaintyofthemetric(whichmustin

principleariseinquantumgravity),allinfallingmatterthathadsurvivedtheradiation

processsofarwouldatthiscoordinatetimev=0beintheveryclosevicinityofthecen-

ter.Therefore,thissimultaneityrepresentsquitedifferentpropertimesforthevarious

partsofinfallingmatterevenforacollapsinghomogeneousdustcloud–andevenmore

soforlaterinfallingthings.Propertimesareirrelevantfortheglobalgeometrodynam-

ics.Mostoftheblackhole’soriginalmass-energymustalreadyexistintheformofout-

goingHawkingradiationonthissimultaneity,andmayevenhavepassedanyrealistic

“asymptotic”observer.Inordertobeobservedbyhim,itcanhaveitscausalrootonly

outsideanhorizon.

Blackholeradiationisagainbasedontheradiationarrowofretardation,butitsconven-

tionalformulationalsodependsonaquantumarrowthatisdefinedbythestatistical

interpretationofquantummechanics.Apurequantumstateformingablackholewould

accordingtothistraditionalpicturedecayintomanyfragments(mainlyphotons,gravi-

tonsandneutrinos),describedbyastatisticalensembleofdifferentemissiontimes–

similartotheensembleofallpotentialoutcomesinaseriesofmeasurements,ortothe

coolingofahighlyexcitedquantumstatebymeansofmanystochasticradiationevents.8

However,anapparentensembleisalreadydefinedbymeansofanappropriateconcept

ofcoarsegrainingforanoutgoingpurestatethatwouldbetheresultofaunitaryde-

scription(withoutanyeventsthatmightalsocauseingoingparticleswithnegativeener-

gy).Inquantumtheory,oneusuallyneglectsinthissense(thatis,oneregardsasirrele-

vantforthefuture)theentanglementbetweendecayfragments.Suchacoarse-graining

(neglectofinformation)doesnotonlyformallyjustifytheconceptofgrowing"physical”

entropyinspiteoftheconservationofapureglobalstate,5butalsothephenomenonof

decoherence(whichwouldhereoccurinany“particle”detectors).Incontrasttothe

globalensembleentropythatisconservedunderunitarydynamics(andvanishesfora

9

purestate),physicalentropyisdefinedasanextensivequantitythatgivesrisetothe

localconceptofanentropydensitywhichneglectsinformationaboutcorrelations–just

asBoltzmann’sµ-spacedistributiondoes.ThethermalHawkingradiationcanthusnot

representapropermixtureforthesamereasonwhydecoherencedoesnotexplaina

“real”collapseofthewavefunction.40Themajordifferencebetweenthedecayofhighly

excitedstatesofnormalmatterandtheevaporationofblackholesisthatthelatter’s

unitarydynamicsisnotexplicitlyknown(andoccasionallyevenquestionedtoapply).

Thethusdescribedsituationisnonethelessmuchdiscussedasan"informationlosspar-

adoxforblackholes".9Itsconsequencesareparticularlydramaticifonepresumesthe

existenceofablackholeinteriorregionthatwouldnecessarilyariseintheabsenceof

Hawkingradiation;matter(andthe“information”itmayrepresent)couldthennotcaus-

allyescapeanymore.Thisquestionablepresumption(oftenbasedonclassicalsingulari-

tytheorems)maybetacitlyintroducedbyusing“niceslices”thataredefinedtoavoid

thesingularitybutwould,incontrasttoour“veryniceslices”,intersectthethusalso

presumedhorizon.Unitarydescriptionmeans,however,thattheinformationwhichde-

finestheinitialpurestateismostlytransformedintonon-localentanglement.Global

unitaritythusleadstoasuperpositionof"manyworlds"whichthereafterremaindynam-

icallyautonomous,andwhichmayincludedifferentversionsofthe“same”observers–

thusphysicallyjustifyingdecoherenceasdescribinganapparentcollapse.40There-

placementofthissuperpositionbyanensembleofmanypossibleworldsaccordingtoa

fundamentalstatisticalinterpretation(arealcollapseofthewavefunction)wouldin-

steadobjectivelyannihilatetheinformationcontainedintheirrelativephases,andin

thiswayintroduceafundamental(law-like)dynamicaltimeasymmetry.Recallthatthe

Oppenheimer-Snydermodel,onwhichtheniceslicesarebased,preciselyneglectsthe

energylossoftheblackholebyHawkingradiation.Althoughthe("back")reactionofthe

metricinresponsetoradiationlossmayinprinciplerequirequantumgravity,myargu-

mentaboutthenon-formationofahorizonishereonlybasedonthelocalconservation

ofmomentum-energyinasituationwherethismaynothavetobequestioned.

InsteadofassuminganexternalvacuumwhencalculatingprobabilitiesforHawkingra-

diation,oneshouldtakeintoaccountthelocalpresenceofinfallingmatter,inwhichcase

somekindofinternalconversionmightleadtoitsannihilation.(Theconservationof

baryonnumberetc.wouldhavetomodifytheHawkingradiation,andmaythusleadto

anessentiallydifferentscenario.)Asimilarscenariohasrecentlybeenpostulatedasa

10

novelkindofphysicsclosetothehorizon(calleda“firewall”).10Whilethisfirewallwas

meanttopreventanobserverfromremainingintactwhenfallingin,itshouldaccording

tomyearlierproposal(seeearlierversionsofthispaper,availableatarxiv:1012.4708v1

orv2)convertallinfallingmatterintooutgoingradiation.NotethatthelocalBeken-

stein-Hawkingtemperaturedivergesclosetothehorizon,andwouldthereforedescribe

allkindsofparticle-antiparticlepairsinanon-inertialframe(suchasatafixeddis-

tance).Aslongassomeinternalconversionofthiskindcannotbeexcluded,thereisno

reasontospeculateaboutblackholeremnants,superluminaltunneling,orafundamen-

talviolationofunitaritythatwouldgobeyonddecoherence(thatis,beyondameredis-

localizationor“globalization”ofsuperpositionsthatjustrendersthemirrelevantfor

localobservers).11Unitaritycanonlyapplytotheglobal“bird’sperspective”thatin-

cludesallEverettbranches,anditcannotleadtoanykindof“double-entanglement”.12

Whatmightremainasa“remnant”accordingtothissemi-classicaldescriptionofblack

holeevolutiononveryniceslicesisamasslesspointlikecurvaturesingularity,sincethe

RiemanntensoroftheSchwarzschildmetricisproportionaltoM/r3,andhencediverges

forr=2M®0.Thissingularitysignalsabreak-downofthesemi-classicaldescriptionof

geometrodynamicsatthisfinalstageonly.Forexample,quantumgravitywouldrequire

aboundaryconditionforthetimelessWheeler-DeWittwavefunction,whichcannotdis-

tinguishbetweenpastandfuturesingularities(seeSects.4and5).Thismightleadtoan

effectivefinalconditionthataffectsblackholes“frominside”inananticausalmanner.13

Anyinwards-directed(hencevirtual)negativeenergyradiationcompensatingtheemis-

sionofHawkingradiationaccordingtosomepicturescouldthen“recohere”theeffective

blackholestateinordertoloweritsentropyinaccordancewithboththemasslossand

Bekenstein’srelation(4b).

NotethattheconceptofanS-matrixwouldalsobeunrealisticformacroscopicobjects,

suchasblackholes.Becauseoftheirnever-endingessentialinteractionwiththeirenvi-

ronments,theycanneverbecomeasymptoticallyisolated(thereasonfortheirongoing,

locallynon-unitarydecoherence).Theextremelifetimeofblackholesmeansthatthe

informationlossproblemisclearlyanacademicone:anyapparentlylostinformation

wouldremainirrelevantforfarmorethan1065years,anditcouldhardlyeverbeex-

ploitedevenifitfinallycameoutasentangledradiation.Itcanonlydescribeonesuper-

positionof“manyworlds”whichformanapparentensemble.The“Pagetime”,14when

11

theentanglementbetweentheresidualblackholeanditsemittedradiationisassumed

tobemaximal,canthereforenothaveanyconsequencesfortheobservedblackhole.

Severalphysicists(includingmyself)usedtoseeaproblemintheequivalenceprinciple,

whichrequiresthatobserversordetectorsfreelyfallingintotheblackholedonotregis-

teranyHawkingradiation.Someevenconcludedthatthemass-lossofblackholes,too,

mustthenbeobserver-dependent(notveryappropriatelycalled“blackholecomple-

mentarity”).However,thisconclusionappearstobewrong.Whiletheequivalencebe-

tweenablackholeandauniformlyaccelerateddetector(asregardstheirspecificradia-

tion)mustindeedapplytothelocallaws,itcaningeneralnotapplytotheirboundary

conditions.Anobserverordetectorfixedatsomedistancefromtheblackholewouldnot

beimmersedinisotropicheatradiation,sincethisradiationiscomingfromthedirection

oftheblackholesurface,whichwouldcovermostoftheskyonlyforanobserververy

closetothehorizon.Eventhoughthefreelyfallingdetectormaythennotregisterany

radiation,thelatter’seffectonfixeddetectors,oritsfluxthroughafixedspherearound

theblackhole,mustexistobjectively–justastheclicksofanaccelerateddetectorinan

inertialvacuum(attributedtoUnruhradiation)canbenoticedbyallobservers,regard-

lessoftheirownacceleration.Theyallhavetoagreethattheenergyabsorbedbythe

accelerateddetectormustbeprovidedbytherocketengineand,analogously,thatthe

Hawkingnetfluxofenergyrequiresanobserver-independentmasslossoftheblack

hole.Therefore,thedynamicallyresultingspacetimegeometry(includingconsequences

ofstochasticmeasurementoutcomes)isalsoobjectivelydefined.Thefreelyfallingob-

serverwouldfurthermoreheartheclicksoffixeddetectorsoccurringataveryfastrate,

andsoasbeingcausedbyaveryintenseoutwardfluxaccordingtohispropertime.For

thesamereason,matterattheouterrimofacollapsingdustcloudcanatlateSchwarz-

schildtimesnotexperienceanygravitationalfield,asthereispracticallynogravitating

energyleftinsideitspresentpositionanymore.Hence,itcannevercrossahorizon.

Inthisway,thephenomenonofblackholesfromthepointofviewofexternalobservers

isconsistentwiththefateofafreelyfallingobserver,whomayeithersooninhisproper

timehavetobeaffectedhimselfbytheinternalconversionprocess,orotherwisehaveto

experiencetheblackholesurfaceveryrapidlyshrinking–finallygivingrisetoextreme

tidalforces–anddisappearingbeforetheobserver’sremainsarrive.Notethattheauxil-

iaryconceptofaneventhorizonchangingintimeisinprincipleill-defined,sinceahori-

zonisalreadyaspacetimeconcept.Theapparentblackholesurfacer=2M(u),where

12

M(u)»M(t)characterizesthecorrespondingVaidyametric,whileuisheretheoutgoing

Eddington-Finkelsteincoordinate,maynonethelessshrinkadiabaticallyinordertodis-

appearbeforeanyinfallingmatterhasgotachancetoentertheregionr≤2M(t)forany

finitecoordinatetimet.

Ifthefreelyfallingobservercouldsurvivetheinternalconversionprocess,hewould

havetravelledfarintothecosmicfutureinashortpropertimebecauseofthequasi-

singulartimedilation.Ontheotherhand,notheorythatiscompatiblewiththeequiva-

lenceprinciplecandescribebaryonnumbernon-conservationintheabsenceofasingu-

larity.Becauseofthehugelifetimeofblackholesthisproblemmayperhapsbesolvedin

connectionwiththatofthematter-antimatterasymmetryinouruniverse.Allsymme-

triesmayinprinciplebebrokenbytheeffectivenon-unitaritycharacterizingthedynam-

icsofindividualEverettbranches.Thislastremarkmightalsoberelevantfortheabove

mentionedpossibilityofanti-causality(recoherence)requiredbyanapparentfuture

conditionthatisinaccordwithatimelessWheeler-DeWittequation(seeSect.5);reco-

herencewouldrequireare-combinationofdifferentEverettworlds.

RogerPenrosehadcomparedblackholeentropynumericallywiththatofmatterinthe

universeundernormalconditions.15Sincetheformerisaccordingto(4b)proportional

tothesquareoftheblackholemass,macroscopicblackholeformationleadstoatre-

mendousincreaseofphysicalentropy.Asthermodynamicentropyisproportionaltothe

particlenumber,itisdominatedintheuniversebyphotonsfromtheprimordialcosmic

radiation(whosenumberexceedsbaryonnumberbyafactor109).Ifourobservable

partoftheuniverseofabout1079baryonsconsistedcompletelyofsolarmassblack

holes,itwouldpossessanentropyoforder1098(inunitsofkB-1),thatis,1010timesas

muchasthepresentmatterentropythatisrepresentedby1088photons.Combiningall

blackholesintoasingleonewouldevenraisethisnumberto10121,thehighestconceiv-

ableentropyforthis(perhapspartial)universeunlessitsvolumeincreasedtremen-

dously.4,7,16Ifentropyisindeedameasureofprobability,anyapproximatelyhomoge-

nousmatterdistributionwouldbeextremelyimprobableexceptfordensitiesmuchlow-

erthanatpresent(ataverylatestageofaneternallyexpandinguniverse).Therefore,

thehomogeneityoftheinitialuniverseisusuallyregardedasthe“fundamentalimprob-

ableinitialcondition"thatexplainstheglobalmasterarrowoftimeifstatisticalreason-

ingisapplicabletothefuture(seeSect.4).However,itsrelationshiptothethermody-

namicallyimportantconditionofabsentor"dynamicallyirrelevant"non-localinitial

13

correlations(orentanglementinthequantumcase)seemstobenotyetfullyunder-

stood.Ifthetwoentropyconcepts(blackholeandthermodynamic)aretobecompati-

ble,theentropyofthefinal(thermal)radiationmustbegreaterthanthatoftheblack

hole,whilethelatterhastoexceedthatofanykindofcollapsingandinfallingmatter.

4ExpansionoftheUniverse

Theexpansionoftheuniverseisatime-asymmetricprocess,butincontrasttomostoth-

erarrowsitformsanindividualphenomenonratherthanawholeclassofsimilarob-

servableones,suchasblackholes,radiationemitters,orsteamengines.Itmayeven

changeitsdirectionatsometimeofmaximumextension,althoughpresentastronomical

observationsmayindicatethattheexpansionwilllastforever.Ahomogeneousandiso-

tropicFriedmannuniverseisinclassicalGRdescribedbythedynamicsoftheexpansion

parametera(t)inaccordancewiththetime-symmetric“energytheorem"forln[a(t)],

(da/adt)2/2=(4π/3)r(a)+L/6–k/2a2, (5)

whereristheenergydensityofmatter,Lthecosmologicalconstant,andk=0,±1thesign

ofthespatialcurvature.Thevalueoftheformal"totalenergy"(thedifferenceofboth

sidesoftheequation)isthusfixedandvanishesingeneral-relativisticcosmology.Pen-

rose'sentropyestimatesthendemonstratethatthehomogeneityassumedinEq.(5)is

extremelyimprobablefromastatisticalpointofview.Therefore,itmustbeunstable

undertheinfluenceofgravity(inspiteofbeingdynamicallyconsistent).

Inaccordancewithahomogeneousinitialmatterdistribution,Penrosepostulatedthat

freegravitationalfieldsvanishedexactlyattheBigBang.Thesefreefieldsaredescribed

bytheWeyltensor,thatis,thetrace-freepartofthecurvaturetensor.Thetraceitself

(theRiccitensor)islocallyfixedbythestress-energytensorofmatteraccordingtothe

Einsteinfieldequations.TheWeyltensor,ontheotherhand,isanalogoustothediver-

gence-freepartoftheelectrodynamicfieldtensorFµn,sincethedivergence∂µFµn(the

traceofthetensorofitsderivatives)issimilarlyfixedbythechargecurrentjn.There-

fore,theWeyltensorhypothesisisanalogoustotherequirementofanabsenceofany

initialelectromagneticradiation,aconditionthatwouldallowonlytheretardedelec-

tromagneticfieldsofallsourcesintheuniversetoexist.Thisuniversalretardationof

radiationhadindeedbeenproposedasalawbyPlanck(inadisputewithBoltzmann),17

14

andlaterbyRitz(inadisputewithEinstein),18inanattempttoderivethethermody-

namicarrow.However,BoltzmannandEinsteinturnedouttoberight,sincetheretarda-

tioncaninturnbeunderstoodasaconsequenceofthepresenceofthermodynamicab-

sorbers.1Incosmology,thisincludestheabsorberformedbytheradiationera,which

wouldnotallowustodiscoveranyconceivableearlierelectromagneticradiation.Incon-

trast,theearlyuniverseseemstobetransparenttogravitationalradiation,including

thatwhichmighthavebeencreatedintheBigBang.

Notethatthelowentropyandthecorrespondinghomogeneityoftheuniversecannot

beexplainedbyanearlycosmicinflationera(ashasoccasionallybeenclaimed)ifthis

inflationwasdeterministicandwouldthushaveconservedensembleentropy.

Althoughouruniversemayexpandforever,theideaofitslaterrecontractionisatleast

conceptuallyinteresting.ThomasGoldfirstarguedthatthelowentropyconditionat

highdensityshouldnotbebasedonanabsolutedirectionoftime,andhencebevalidat

aconceivableBigCrunchaswell.19ThelatterwouldthenbeobservedasanotherBig

BangbyobserverslivingduringtheformalcontractioneraiftheWeyltensorwasre-

quiredtovanishthereaswell.Gold’sscenariowouldnotonlyrequireathermodynamic

transitionerawithoutanywell-definedarrowinourdistantfuture–itwouldalsopose

seriousconsistencyproblems(similartoWheelerandFeynman’sabsorbertheory1),

sincetheextremelysmallinitialprobabilityforthestateoftheuniversewouldhaveto

besquaredifthetwoconditionsarestatisticallyindependentofoneanother.20Ifnone-

thelesstrue,itwouldhaveimportantconsequencesforthefateofmatterfallinginto

massiveblackholes.Ifsuchblackholessurvivedthementionedthermodynamictransi-

tioneraatthetimeofmaximumextensionbecauseoftheirlongevaporationtimes(cf.

Sect.3),theywouldaccordingtotheglobaldynamicsenteranerawithreversedarrows

oftime.However,becauseofthetransparenceofthelateuniversetolight,theywould

“receive”coherentadvancedradiationfromtheirformalfutureevenbeforethathap-

pens.Thisadvancedradiationmustthen"retro-cause"suchmassiveblackholestoex-

pandagaininordertoapproachastateofhomogeneityinaccordancewiththefinal

condition.21Inmathematicalterms,theirhorizonisnot“absolute”inthiscaseevenin

theabsenceofanyblackholeevaporation.

Areversalofthearrowoftimemaynotonlyoccurinthedistantfuture,butmayalso

haveoccurredinthepast.Severalpre-big-bangscenarioshavebeendiscussedinnovel

15

andasyetspeculativetheories.Usually,onetherebyidentifiesthedirectionofthefor-

maltimeparameterwiththedirectionofthephysicalarrowoftime.Forexample,ac-

cordingtoargumentsfirstusedinloopquantumcosmology,22theconfigurationspace

forFriedmanntypeuniversesmaybedoubledbyinterpretingformallynegativevalues

ofthecosmicexpansionparameteraasrepresentingnegativevolumemeasures.The

cosmicdynamicscanthenbecontinuedbackwardsintimebeyondtheBigBangintoits

mirrorimageby"turningspaceinsideout"(turningright-handedtriadsintoleft-handed

ones)whilegoingthrougha=0eveninaclassicalpicture.Forthispurpose,theclassical

dynamicaldescription(5)wouldhavetobemodifiedclosetotheotherwisearisingsin-

gularityata=0–asitisindeedsuggestedbyloopquantumgravity.However,ifthe"ini-

tial"conditionsresponsibleforthearrowoftimeareassumedtoapplyatthesituation

ofvanishingspatialvolume,thearrowwouldformallychangedirection,and|a|rather

thanawouldrepresentaphysicalcosmicclock.Observersonbothtemporalsidesofthe

BigBangcouldonlyremembereventsinthedirectiontowardsa=0.Anotherpossibility

toavoidthesingularityisarepulsiveforceactingatsmallvaluesofa,23whichwould

leadtoaBigBouncewithsimilarconceivableconsequencesforthearrowoftimeasthe

abovemodelthatinvolvesspaceinversion.

Incosmology,quantumaspectsofthearrowoftimemustagainplayanimportantrole.

AccordingtotheCopenhageninterpretation,thereisnoquantumworld–sonocom-

pleteandconsistentcosmichistorywouldbedefinedanymorewhenquantumproper-

tiesbecomeessential.Inotherorthodoxinterpretations,theunitaryevolutionofthe

quantumstateisrepeatedlyinterruptedbymeasurementsandsimilartime-asymmetric

events,whenthewavefunctionisassumedto"collapse"indeterministically.Theconse-

quencesofsuchstochasticeventsonquantumcosmologywouldbeenormous,butas

longasnocollapsemechanismforthewavefunctionhasbeenconfirmed,onehasagain

arrivedatanimpasse.Goingforwardintimemaybeconceptuallysimpleinsuchasym-

metrictheories,sinceonejusthasto"throwaway"allcomponentsofthewavefunction

whichrepresentthenot“actualized”potentialoutcomes,whilegoingbackwardswould

requirealltheselostcomponentstorecombineanddynamicallyformlocalsuperposi-

tionsagain.Soonehasatleasttokeeptheminthecosmicbookkeeping–regardlessof

whethertheyarecalled"real"(asintheEverettinterpretation)ornot.Goingbacktothe

BigBangbymeansoftheunitarydynamicswouldrequireallthosemany“worlds”that

haveeverbeenthrownawayintheorthodoxdescriptionduringthepastofouruni-

16

verse,whileonewouldhavetothrowawayotherswhenformallygoingbackwardsbe-

yondtheBigBanginordertoobtainanindividualquasi-classical"pre-big-banghistory".

Inotherwords,aunitarycontinuationbeyondtheBigBangcanonlydescribethecom-

pleteEverettsuperpositionofworldsonbothsidesoftheBigBang,buthardlyanyindi-

viduallyobservedquasi-classicalworlds.Thecorrespondingmasterarrowoftime

wouldthusnotonlyaffectallrealmsofphysics–itmustbetrulyuniversalinamuch

deepersense:itcanonlyhave"multiversal"meaning.Thesamemultiversalitywasre-

quiredinaunitaryblackholeevolutionofSect.3,anditdoes,infact,applytotheunitary

quantumdescriptionofallmacroscopicobjects,whenirreversibledecoherencemimics

acollapseofthewavefunctionandtherebyexplainsclassicality.

ThetimedirectionofEverett’sbranchingofthewavefunctionthatisbasedondecoher-

encerequiresahomogeneousinitialquantumstate(presumablyata=0),whichdoes

notcontainanynonlocalentanglementthatmightlaterhavelocaleffects.Quantumdy-

namicswillthenleadtodecoherence(theinpracticeirreversibledislocalizationofsu-

perpositions),andthereby"intrinsically"breakvariousglobalsymmetries–possibly

evenintheformofmanydifferentquasi-classical"landscapes",whichcanonlyrepre-

sentdifferentbranchesofonesymmetricsuperposition.

5QuantumGravity

GeneralRelativityhastraditionallybeenconsideredinablockuniversepicture,butbe-

causeofthehyperbolictypeofEinstein'sfieldequationsitisadynamicaltheoryjustas

anyotherfieldtheory.Itsexplicitdynamicaldescription,whichrequiresanon-Lorentz-

invariantform,wascompletedbyArnowitt,DeserandMisner(ADM).24ThisHamiltoni-

anformulationisaprerequisiteforthecanonicalquantizationofthetheory.Ishallhere

regardtheresultofthisquantizationprocedureasaneffectivequantumtheory,without

discussinganyattemptsofajustificationintermsoftheoriesthatmaypossiblybeexact

buthavenoempiricalsupportasyet(suchasstringtheoryorloopquantumgravity).

TheADMformalismisbasedonanarbitrarytime-likefoliationofspacetimethathasto

bechosen"ontheflight",thatis,whilesolvinganinitialvalueproblemnumerically.(A

similarfreedomwasusedinSect.3forthechoiceofveryniceslices.)Ifthedynamicsof

matterisalsodefined,thisconstructionmustleadtoaunique(foliation-independent)

17

spacetimegeometry,whilethespatialmetriconthechosenspace-likeslicesrepresents

thecorrespondingdynamicalvariables.Thelattercanbedescribedbyasymmetricma-

trixhkl(xm)–withk,l,mrunningfrom1to3.Threeofitssixindependentmatrixelements

representthechoiceofunphysicalcoordinates,twowouldinthelinearapproximation

correspondtothespincomponentsofagravitationalwave(±2withrespecttothedirec-

tionofpropagationforaplanewave),whiletheremainingonecanberegardedasa

measureof"many-fingered"physicaltime(metricdistancebetweenadjacentspace-like

slices).Thecorrespondingcanonicalmomentapkldefinetheembeddingofthespatial

metricintospacetimeandthearbitrarypropagationofspatialcoordinates.Thedynam-

icscanthenbeformulatedbymeansoftheHamiltonianequationswithrespecttoan

arbitrarytimeparametertthatformallydistinguishesdifferentslicesinagivenfolia-

tion.TheseHamiltonianequationsareequivalenttoEinstein'sfieldequations.Incon-

trasttometrictime,theparametertisgeometricallyorphysicallymeaningless,andcan

thereforebereplacedbyanymonotonicfunctiont'=f(t)–includingitsinversion.

NotethatwhenSpecialRelativityissaidtoabandontheconceptofabsolutetime,this

statementrefersonlytotheconceptofabsolutesimultaneity,whilepropertimes,which

controlallmotionaccordingtotheprincipleofrelativity,arestillassumedtobegiven

“absolutely”bythefixedLorentzmetric.Thisremainingabsolutenessisthusabandoned

onlyinGeneralRelativity,wherethemetricitselfbecomesadynamicalobjectlikemat-

ter,asdescribedbytheADMformalism.Theabsenceofanabsolutetimeparameter

(hererepresentedbyitsreparametrizability)wasalreadyrequiredbyErnstMach.Julian

Barbour,whostudieditsconsequencesinmuchhistoricaldetail,25calledit"timeless-

ness".However,acompleteabsenceoftimewouldremoveanypossibilitytodefinean

arrow,whileaone-dimensional(dynamical)successionofstates,characterizedbyan

arbitraryparameter,stillallowsonetodefineatimedirectionasymmetry.

Theinvarianceofthetheoryunderspatialcoordinatetransformationsandtimerepara-

metrizationiswarrantedbyfourconstraintsforthematrixhkl(t),calledmomentumand

Hamiltonianconstraints,respectively.Theymayberegardedasinitialconditions,but

theyareconservedintime.Inparticular,theHamiltonianconstraintassumestheform

H(hkl,πkl)=0. (6)

Whenquantized,26andwhenalsotakingintoaccountmattervariables,thisconstraint

translatesintotheWheeler-DeWittequation,

18

HY(hkl,matter)=0, (7)

whichmeansthatthetime-dependentSchrödingerequationbecomestrivial,

∂Y/∂t=0. (8)

Eventhetimeparameterthasnowdisappeared,becausetherearenoparametrizable

trajectoriesrepresentingcosmichistoriesanymoreinquantumgravity.Onlythisdras-

ticproperty,whichisaquantumconsequenceofclassicalreparametrizability,canbe

regardedasaformal“timelessness”.

ThetimelessnessoftheWheeler-DeWittwavefunctionhasbeenknownatleastsince

1967,butitseemstohaveoriginallybeenregardedas“justformal”.Atimeparameter

wasoftensmuggledinagaininvariousways–forexampleintermsofparametrizable

Feynmanpaths,bymeansofsemiclassicalapproximations,orbyattemptstoreintro-

duceaHeisenbergpictureinspiteoftheHamiltonianconstraint.27Theproblembecame

pressing,though,inconnectionwiththeassumptionofanonticandkinematicallycom-

pletewavefunctioninquantumcosmology.28

ThegeneralwavefunctionalY(hkl,matter)describesentanglementofgeometryandmat-

ter.Ifwedidhaveasuccessionofsuchquantumstates(formingaquantumtrajectoryor

quantumhistory),averyspecial,initiallynotentangled,statecouldexplainanarrowof

growingentanglementanddecoherence–asusual.Theresultingbranchingofthewave

functionaccordingtoanappropriateparametertwouldthenincludebranchingstatesof

spacetimegeometry(thatis,branchingquasi-classicalwavepacketsintheconfiguration

spaceofthree-geometries).Althoughthereisnosuchtimeparameteranymore,the

metrichklstillcontainsameasureofmetrictime.Therefore,itdescribesaphysicaltime

dependenceintheformofanentanglementofthismeasurewithallotherdegreesof

freedom–evenforaformallytime-lesssolutionof(7).29ForFriedmannuniverses,the

expansionparametera,whichispartofthemetrichkl,issuchanappropriatemeasureof

time,buthowdoesthathelpustodefineaninitialvalueproblemforthisstaticwave

equation?Thesurprisingansweristhatthisstaticequationisgloballyhyperbolicfor

Friedmanntypeuniversesonitsinfinite-dimensionalgauge-freeconfigurationspace

(whichhasthereforealsobeencalled“superspace”)ratherthanonspacetime.Theex-

pansionparameteraoritslogarithmappearsasatime-likevariableinthissensebe-

causeoftheunusualnegativesignofitsformalkineticenergycomponent.30Therefore,

19

theWheeler-DeWittequationdefinesan“initial”valueproblem,forexampleatasmall

valueofa.ForamodifiedWheeler-DeWittequation,thispossibilitymightevenbeex-

tendedtoa=0.Thereisnoconceptualdifferencebetweena(multiversal)BigBangand

aBigCrunchanymore,sinceintheabsenceofatimeparameterthewavefunctioncan

onlybeastandingwaveonconfigurationspace(inspiteofitsintrinsicdynamics).

ThemetrictensorandotherfieldsdefinedonaFriedmannsphere,a=const,mayberep-

resentedbyafour-dimensionalmultipoleexpansion,whichisparticularlyusefulforde-

scribingtheveryearly,approximatelyhomogeneousandisotropicuniverse.31Inthis

case,onemayconvenientlymodelmatterquantummechanicallybyamassivescalar

fieldF(xk).ThewavefunctionaloftheuniversethenassumestheformY(a,F0,{xn}),

whereF0isthehomogeneouspartofthescalarfield,while{xn}areallhighermultipoles

ofgeometryandmatter.Forthemetric,onlythetensormodesaregeometricallymean-

ingful,whiletherestrepresentsgaugedegrees(heredescribingthepropagationofspa-

tialcoordinates).Theglobalhyperbolicnaturewithrespecttoallphysicaldegreesof

freedombecomesmanifestinthisrepresentation.

Fig.3:WavepacketforahomogeneousmassivescalarfieldamplitudeF0(plottedalong

thehorizontalaxis)dynamicallyevolvingasafunctionofthetime-likeparametera=lna

thatispartofthemetric(secondaxisinthistwo-dimensionalmini-superspace).The

classicaltrajectorypossessesaturningpointabovetheplotregion50≤a≤150–namely

atabouta=240inthisnumericalexamplethatrepresentsanexpandingandrecontract-

ingmini-universe.Wavemechanically,thiscorrespondstoareflectionofthewavepack-

etbyarepulsivepotentialin(5)atthisvalueofa(withthereflectedwavebeingomitted

intheplot).Thisreflectionleadstoconsiderablespreadingofthe"initial"wavepacket.

20

Thecausalorderofthesetwolegsofthetrajectoryisarbitrary,however,andthephase

relationsdefiningcoherentwavepacketscouldalternativelybechosentogiverisetoa

narrowwavepacketforthesecondleginstead.Therefore,this(herenotshown)formal

spreadingdoesnotrepresentaphysicalarrowoftime(FromRef.1,Sect.6.2.1.)

Inasimpletoymodelonemayneglectallhighermultipolesinordertosolvethe

Wheeler-DeWittequationontheremainingtwo-dimensional"mini-superspace"formed

bythetwomonopolesonly.TheremainingHamiltonianrepresentsana-dependent

harmonicoscillatorforthevariableF0,whichallowsonetoconstructadiabaticallysta-

bleGaussianwavepackets("coherentstates").32Figure3depictsthepropagationof

suchawavepacketwithrespecttothe"time"variablea=lna.Thisstandingwaveon

mini-superspacemimicsatimelessclassicaltrajectory.However,thecompletewave

functionalhastobeexpectedtoformabroadsuperpositionofmanysuchdynamically

separatedwavepackets(acosmologicallyearlyrealizationof"manyworlds").Notethat

these“worlds”arepropagatingwavepacketsratherthantrajectories(asinDeWitt’sor

DavidDeutsch’sunderstandingof“ManyWorlds”).Ifthehighermultipolesarealsotak-

enintoaccount,theWheeler-DeWittequationmaydescribedecoherenceprogressing

witha–atfirstthatofthemonopoleF0andofaitself,althoughthisapproachrequires

effectiverenormalizationproceduresinthisdescription.33

This“intrinsicdynamics”withrespecttothetime-likeexpansionparameterahasnoth-

ingasyettodowiththelocaldynamicsinspacetime(controlledbypropertimesalong

time-likecurves)thatmustberelevantformatterassoonasthemetricbecomesquasi-

classical.Inordertounderstandtherelationbetweenthesetwokindsofdynamics,one

mayapplyaBorn-OppenheimerexpansionintermsoftheinversePlanckmass,whichis

largecomparedtoallparticlemasses,inordertostudytheWheeler-DeWittwavefunc-

tion.34ThePlanckmassappearsinthekineticenergytermsofallgeometricdegreesof

freedomthatappearintheHamiltonianconstraint.Theformalexpansionintermsof

powersofmPlanck-1/4thendefinesan"adiabaticapproximation"inanalogytothetheory

ofmolecularmotion(withelectronwavefunctionsintheelectrostaticfieldsofslowly

movingnuclei).Inmostregionsofconfigurationspace(dependingontheboundary

conditions)onemayfurtherapplyaWKBapproximationtothe"heavy"degreesoffree-

domQ.Inthiswayoneobtainsanapproximatesolutionofthetype

Y(hkl,matter)=Y(Q,q)=eiS(Q)c(Q,q), (9)

21

whereS(Q)isasolutionoftheHamilton-JacobiequationsforQ.Theremainingwave

functionc(Q,q)dependsonlyweaklyonQ,whileqdescribesall"light"(matter)varia-

bles.UndertheseapproximationsonemayderivefromtheWheeler-DeWittequation

theadiabaticdependenceofc(Q,q)onQintheform

. (10)

TheoperatorhQistheweaklyQ-dependentHamiltonianforthemattervariablesq.This

equationdefinesanewtimeparametertWKBseparatelyalongallWKBtrajectories

(whichdefineclassicalspacetimes)bythedirectionalderivative

. (11)

Inthisway,oneobtainsfrom(10)atime-dependentglobalSchrödingerequationfor

matterwithrespecttothederivedWKBtimetWKB.26,28Thisparameterdefinesatimeco-

ordinateinspacetime,sincetheclassicaltrajectoriesQ(t)inthesuperspaceofspatial

geometriesQdefinespacetimegeometries.Eq.(10)mustalsodecribethedecoherence

ofsuperpositionsofdifferentWKBtrajectories.Decoherenceisalsorequiredtoelimi-

natesuperpositionsthatareneededtodefinerealwavesfunctioneiSc+e-iSc*,which

havetobeexpectedfromtherealWheeler-DeWittequationunderphysicallymeaningful

boundaryconditions,intermsofthecomplexonesin(9).

InordertosolvethisderivedtimedependentSchrödingerequationalongagivenWKB

trajectory,thatis,intermsofafoliationofaclassicalspacetimethatdoesinturnadia-

baticallydependontheevolvingmatter,oneneedsa(lowentropy)initialconditionin

theregionwheretheWKBapproximationbeginstoapply.Forthispurpose,onewould

firsthavetosolvetheexactWheeler-DeWittequation(oritsgeneralizedversionthat

mayapplytosomeasyetelusiveunifiedtheory)asafunctionofabyusingitsfunda-

mentalcosmicinitialconditionata=0.Thismightbedone,forexample,byusingthe

multipoleexpansionontheFriedmannsphere,untiloneenterstheWKBregion(atsome

distancefroma=0),wherethissolutionwouldprovideinitialconditionsforthepartial

wavefunctionscforallarisingWKBtrajectories.Thederivedtime-dependentSchrö-

dingerequationwithrespecttotWKBshouldthendescribefurtherdecoherenceofmatter

(theemergenceofotherquasi-classicalproperties),andtherebyexplaintheoriginofall

otherarrowsoftime.Inparticular,itmustenforcedecoherenceofsuperpositionsofany

22

arisingmacroscopicallydifferentspacetimes,whichwouldformseparatequasi-classical

"worlds".26ItwouldalsodecohereconceivableCPTsymmetricsuperpositionsofblack

andwhiteholes,whichareanalogoustoparityeigenstatesofchiralmolecules,ifthese

hadevercomeintoexistence.16

Acknowledgement:IwishtothankClausKieferforhiscommentsonanearlydraftof

thismanuscript,andDanielTernoforarecentdiscussion.

Noteaddedafterpublication:The“causaltreatment”ofblackholes,usedinSect.3for

anargumentagainsttheformationofeventhorizonsand,therefore,theexistenceofan

informationlossparadox,hasrecentlybeensupportedbytheexplicitmodelofacollaps-

ingthinmassshell.35Adifferentattempt36describedamodificationofthesuggestionof

asingularheatbathfrommyfirstarXivversionsofthepresentpaper(inthatform

calleda“firewall”),whileanotherscenariohadalreadybeenproposedin1976(usinga

differentmodel)byUlrichGerlach.37Heassumedthattheblackholefinallysettlesdown

inaspecificgroundstatethatisnotflatspacetimebutwouldinsteadrepresentastable

“remnant”.Theessentialassumptioninallthesemodelsisthevalidityofrelativisticcau-

salityinthepresenceofHawkingradiationandveryclosetotheexpectedhorizon.This

semiclassicalassumptionmaywellbeproblematic,butitshouldatleastbemorerealis-

ticthanclassicalGRwithitsinevitablehorizonsanditsoftenmisrepresentedprinciple

ofequivalence–seeSect.3.(Incontrasttonon-localphotonnumbereigenstates,general

quantumfieldstatespossessalocalbasisthatpermitsadefinitionofdynamicallocali-

ty.40GRisthenappliedbytakingintoaccountalocalizedmassloss,thatis,acausalout-

goingenergycurrentthatisinaccordancewiththedynamicallyarisinglightconestruc-

ture,suchasobtainedbyanappropriateADMconstructionstartingfromregularinitial

conditions.)Onemaythushavetodrawtheconclusionthateventhorizonscannever

formifmatterisdescribedbydynamicallylocalQFT–inmyopinionaveryconvenient

andevenplausibleresult,whichwouldmeanthattheveryconceptofeventhorizonsis

nomorethanamathematicalartifactfromtheformalismofclassicalGR.Observersat

fixeddistancesfromtheblackholewouldfeelaheatbathofdivergingtemperaturefor

r®2M(t),whichrepresentstheHawkingradiationclosetotheexpectedhorizon.Even

thoughthisheatbathmaynotbenoticedbyaninertial(freelyfalling)observer,thelat-

termaythenbedisruptedbytheextremetidalforcesofthe,fromhispointofview,rap-

idlyshrinkingblackhole,andmaylaterhimselfbetransformedintoHawkingradiation

bysomeunitarymechanismthatwouldhavetooccuratverystrongcurvaturecloseto

23

thecenterofthecollapsingmatterifBekensteinandHawking’spredictionofthermal

radiationremainsvalidatthislatestage.Observablephenomenacausedbyblackholes,

ontheotherhand,dependstronglyontheangularmomentaofscatteredobjects,38and

thusseemtoremainhardlyaffectedbytheabsenceofaneventhorizon.

Thissemi-classicaldescriptionofblackholesappearspresentlyalsomorerealisticthan

aquantumgravitationalcollapsethatneglectsHawkingradiation,althoughthismayalso

avoidacurvaturesingularity.39Bothaspectsmayberelevantintheend.

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