the nature and origin of time-asymmetric spacetime structures*
TRANSCRIPT
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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