Thenatureandoriginof
time-asymmetricspacetimestructures*
H.D.Zeh(UniversityofHeidelberg)
www.zeh-hd.de
Abstract:Time-asymmetricspacetimestructures,inparticularthoserepresentingblack
holesandtheexpansionoftheuniverse,areintimatelyrelatedtootherarrowsoftime,
suchasthesecondlawandtheretardationofradiation.Thenatureofthequantumarrow,oftenattributedtoacollapseofthewavefunction,isessential,inparticular,for
understandingthemuchdiscussed"blackholeinformationlossparadox".However,this
paradoxassumesanewformandmightnotevenoccurinaconsistentcausaltreatment
thatwouldpreventtheformationofhorizonsandsingularities.
A“masterarrow”,whichcombinesallarrowsoftime,doesnothavetobeidentifiedwith
thedirectionofaformaltimeparameterthatservestodefinethedynamicsasasuccessionofglobalstates(atrajectoryinconfigurationorHilbertspace).Itmayevenchange
directionwithrespecttoafundamentalphysicalclock,suchasthecosmicexpansion
parameterifthiswasformallyextendedeitherintoafuturecontractioneraortonegative"pre-big-bang"values.
1Introduction
Sincegravityisattractive,mostgravitationalphenomenaareasymmetricintime:objectsfalldownorcontractundertheinfluenceofgravity.InGeneralRelativity,this
asymmetryleadstodrasticallyasymmetricspacetimestructures,suchasfuturehorizonsandfuturesingularitiesaspropertiesofblackholes.However,sincetherelativistic
andnonrelativisticlawsofgravitationaresymmetricundertimereversal,alltime
asymmetriesmustariseasconsequencesofspecific(onlyseemingly"normal")initial
conditions,forexampleasituationofrestthatcanbepreparedbymeansofotherar
*arXiv:1012.4708v8+.V5waspublishedintheSpringerHandbookofSpacetimePhysics(A.Ashtekarand
V.Petkov,edts.–Springer2014).
1
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,localpastsingularities),but
alsomanysolutionswithmixedorundefinedarrowsoftime–includingclosedtime-like
curvesandnon-orientablespacetimes.Therefore,themerepossibilityofposingan"initial"conditionisexceptionalingeneralrelativityfromageneralpointofview.Iwillhere
notdiscusssuchmathematicallyconceivablesolutionsthatdonotseemtoberealizedin
Nature,butinsteadconcentrateonmodelsthatcomeclosetoouruniverse–inparticularthosewhicharegloballyofFriedmanntype.AspecificarrowcharacterizingaFriedmannuniverseisgivenbyitsexpansion(unlessthiswouldbereversedatsometimeof
maximumextension–seeSect.4).
Inmanycases,non-gravitationalarrowsoftimeremainrelevantfortheevolutionof
gravitatingbodiesevenafterthelatterhavebeenpreparedinanappropriateinitial
state.Thisapplies,inparticular,tostronglygravitatingobjects,suchasstars,whoseevolutionisessentiallycontrolledbythermodynamics(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,
2
.
(1)
Whenlosingthermalenergybyradiation,thesesystemsmustgaintwiceasmuchfrom
gravitationalcontractioninordertomaintainaquasi-stablestate.Nonrelativistically,
thisnegativeheatcapacitycouldbeboundedbymeansofother(repulsive)forcesthat
becomerelevantathighdensities,orbythePauliprinciple,whichcontrolsthedensityof
electronsinwhitedwarfstarsorsolidplanets,forexample.Relativistically,eventhese
limitswillbreakdownatacertainmass,since(1)relativisticdegeneracymustultimatelyleadtothecreationofotherparticles,while(2)thepotentialenergyofrepulsiveforceswillitselfgravitate,andforasufficientlylargemassovercompensateanyrepulsion.
Therefore,itisthethermodynamicarrowunderlyingthermalradiation(andtheaccretionofmatter)thatrequiresevolutionofgravitatingsystemstowardstheformationof
blackholes.Classically,blackholeswouldthusdefinethefinalstatesintheobservable
evolutionofgravitatingsystems.
2BlackHoleSpacetimes
Themetricofasphericallysymmetricvacuumsolutionfornon-zeromassisshownin
Fig.1inKruskalcoordinatesuandv.ThisdiagramrepresentsthecompletedSchwarzschildmetricintheform
ds 2 =
32M 2 − r / 2M
e
(−dv 2 + du2 ) + r 2 (dθ 2 + sin2 θdφ 2 ) ,
r
(2)
wherethenewcoordinatesuandvareintheexternalregion(r>2M)relatedtoconven€
tionalScharzschildcoordinatesrandtby
€
u = er / 4 M
v = er / 4 M
# t &
r
−1cosh%
(
$ 4M '
2M
# t &
r
−1sinh%
(
$ 4 M ' .
2M
(3a)
(3b)
Eachpointinthediagramrepresentsaspherewithsurface4πr2.Notethatrandtinter€
changetheirrolesasspaceandtimecoordinatesforr<2M,where2MistheSchwarzschildradius.AllparametersaregiveninPlanckunitsh/2π=G=c=1.
3
AsNatureseemstoprovidespecificinitialconditionsinouruniverse,itmaythereby
excludeallpastsingularities,andhenceallpasteventhorizons.Thisinitialcondition
wouldimmediatelyeliminatetheSchwarzschild-Kruskalvacuumsolutionthatisshown
intheFigure,butwemayinsteadconsiderthefutureevolutionofasphericallysymmetricmassdistributioninitiallyatrest,suchasadustcloud.Itwouldclassicallycollapse
freelyintoablackhole,asquantitativelydescribedbytheOppenheimer-Snyderscenario3(seeleftpartofFig.2).Thevacuumsolution(2)isthenvalidonlyoutsidethesurface
ofthedustcloud,butthissurfacemustaccordingtoaclassicaldescriptionfallthrough
thearisinghorizonatsomefinitepropertime,andabitlaterhitthefuturesingularity.
Fig.1:CompleteformalcontinuationoftheSchwarzschildsolutionbymeansofunique
Kruskalcoordinates.QuadrantsIandIIrepresentexternalandinternalparts,respectively,ofaclassicalblackhole.IIIisanotherasymptoticallyflatregion,whileIVwould
describetheinteriorofa"whitehole".Inthisdiagram,fixedSchwarzschildcoordinatesr
andtarerepresentedbyhyperbolaandstraightlinesthroughtheorigin,respectively.
Propertimesoflocalobjectscouldstartatt=-¥inIoratt=+¥inIII,oratr=0onthe
pastsingularityinIV,whiletheymustendatt=+¥or-¥inIorIII,respectively,orata
secondsingularitywithcoordinatevaluer=0inII.Ontime-likeorlight-likecurvesintersectingoneofthehorizonsattheSchwarzschildradiusr=2M,thevalueofthecoordinatetjumpsfrom+¥to-¥attherimofquadrantI,orfrom-¥to+¥attherimof
quadrantIII,wheretdecreasesinthephysicaltimedirection.
4
Foracloudofinteractinggasmolecules,thisgravitationalcollapsewouldbethermodynamicallydelayedbythearisingpressure,asindicatedintheIntroduction.Gravitational
radiationwouldleadtothelossofanykindofmacroscopicstructure,whilewhatever
remainswouldbecomeunobservabletoanexternalobserver.Althoughthermodynamic
phenomenacontrolthelossofenergybyradiationduringmostofthetime,theasymmetricabsenceofpastsingularitiesrepresentsafundamentalcosmologicalinitialcondition.However,aconceivablewhiteholeinitiatedbyapastsingularitythatcompletely
representedatime-reversedblackholewouldevenrequireanti-thermodymicsandcoherentlyincomingadvancedradiation.Onemaysuspectthatallthesevariousarrows
arerelatedtooneanother,thusdefiningacommon"masterarrow".
Fig.2:Oppenheimer-Snydertypespacetimesofablackanda"white"hole.
SinceitwouldrequireinfiniteSchwarzschildcoordinatetimeforanobjecttoreachthe
horizon,anymessageitmaysendtotheexternalworldshortlybeforeitdoessowould
notonlybeextremelyredshifted,butalsodramaticallydelayed.Themessagecould
reachadistantobserveronlyatincreasinglylaterstagesoftheuniverse.(Anapparatus
fallingintoagalacticsizeblackholecouldevensendmessagesforaconsiderablelength
ofpropertimebeforeitwouldapproachthehorizon.)Soallobjectsfallingintotheblack
holemusteffectivelydisappearfromtheviewofmortalexternalobserversandtheir
descendents,eventhoughtheseobjectsneverseemtoreachthehorizonaccordingto
theirrapidlyweakening,butinprinciplestillarrivingsignals.Theonlyasymptotically
observablepropertiesoftheblackholeareconservedonesthathaveearlyenough
causedeffectsontheasymptoticmetricorotherasymptoticfields,namelyangularmomentumandelectriccharge.Thistime-asymmetricconsequenceisknownasthe"nohairtheorem"forblackholes.Duringcosmologicaltimes,ablackholeaccumulatingion
5
izedinterstellarmattermayevenloseitschargeandangularmomentum,too,forstatisticalanddynamicalreasons.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.Whilethewholeexteriorregionr>2Mcanbecompletely
foliatedbymeansof“verynice”space-likeslicesaccordingtoincreasingSchwarzschild
orsimilartimecoordinateswith-¥<t<+¥,theinteriorcanthenberegardedasits
globalfuturecontinuationbeyondtheeventhorizon,whereincreasingtimecanbelabeledbytheSchwarzschildcoordinaterdecreasingfromr=2Mtor=0.Thisstructure
mustbeessentialforallcausalconsiderationsthatincludeblackholes–notleastfor
theirownfate.Intheclassicalscenario,theinternalstateofablackholewouldbecompletelydeterminedbytheinfallingmatter,whichcouldevendependonour"free"decisionsaboutwhattodropintoablackhole.Nonetheless,propertiesofthisinfallingmatterwouldthenirreversiblybecome"irrelevant"toallexternalobservers–atermthatis
alsousedtodefineageneralizedconceptofcoarsegrainingrequiredfortheconceptof
physicalentropyinstatisticalthermodynamics.5
3ThermodynamicsandFateofBlackHoles
Intheclassicalpicturedescribedabove,ablackholewouldrepresentaperfectabsorber
atzerotemperature.ThispicturehadtobecorrectedwhenBekensteinandHawking
demonstrated,6thelatterbyexplicitlytakingintoaccountquantumfieldsotherthan
gravity,thatablackholesmustpossessfinitetemperatureandentropyproportionalto
itssurfacegravitykandsurfaceareaA,respectively:
, (4a)
6
. (4b)
Here,kandAareknownfunctionsofM,QandJ,whiletheexplicitexpressionsgivenon
therighthandsideofthearrowholdforSchwarzschildblackholes(Q=J=0)andwith
respecttospatialinfinity(thatis,bytakingintoaccountthegravitationalredshift).This
means,inparticular,thatablackholemustemitthermalradiation(Hawkingradiation)
proportionaltoT4AaccordingtoStefan-Boltzmann'slaw,andtherefore,thatitlivesonly
foralimitedtimeoftheorder1065(M/Msun)3years.Forstarsorgalaxiesthisisvery
manyordersofmagnitudemorethanthepresentageoftheuniverseofabout1010years,
butfarlessthananyPoincarérecurrencetimesforsuchmacroscopicsystems.Soone
hastobecarefulaboutwhatismeantbytheterm“asymptotic”indifferentcontexts.
Eventheselargeevaporationtimeswillbeginto“count”onlyaftertheblackholehasfor
averylongtimetocomegrowninmassbyaccretingmatter7(includinganti-matterifit
becomesavailableduringtheblackhole´slongjourneythroughtheuniverse)–andat
leastuntilthecosmicbackgroundtemperaturehasdroppedbelowtheverysmallblack
holetemperature.Althoughevaporationtimesarethusextremelylong,allradiation
wouldcausallyhavetoprecedethehorizon.Schwarzschildtimesrepresentpropertimes
ofdistantobserversintherestframeoftheblackhole,buttheircorrespondingsimultaneitiesmaybeconsistentlycontinuedinwardswhileremainingoutsidethehorizonin
ordertoformcompletetimecoordinatesforthewholeexternalregionI.Bydefinition,
theywouldthenallhavetoincludethecenterofthecollapsingmatteratapre-horizon
stage.However,ahorizonanditsinteriorregionIIcouldneverformiftheblackhole’s
energywasinfactradiatedawaybeforeanymatterhasarrivedattheclassicallypredictedhorizoninthesenseofthisglobaldynamicalfoliation.Formostofthetimethissemiclassicalscenariodoesnotrequireextremespacetimecurvature;itisamereconsequenceoftheextremetimedilationonsuchdynamicallyconsistentsimultaneities.So
whathappenstomatterthatseemstofallintotheblackholeand,inparticular,tononlocalquantumstatesinthiscausalscenario?
Schwarzschildsimultaneitiesarecounterintuitive.Forexample,onemayusetimetranslationinvarianceoftheexternalregionoftheKruskaltypediagram(Figs.1or2a)in
ordertodefinethetimecoordinatev=t=0tocoincidewithanexternaltimeclosetothe
peakoftheHawkingradiation(intheverydistantfuturefromourpointofview).As
7
sumingthatonecanneglectanyquantumuncertaintyofthemetric(whichmustinprincipleariseinquantumgravity)forthispurpose,allinfallingmatterthathadsurvivedthe
radiationprocesssofarwouldatthiscoordinatetimev=0beintheveryclosevicinity
ofthecenterbecauseoftheextremeLorentzcontractiononthissimultaneitywithrespecttotherestsystemoftheinfallingmatter.Therefore,thissimultaneityrepresents
quitedifferentpropertimesforthevariouspartsofinfallingmatterevenforacollapsing
homogeneousdustcloud–andevenmoresoforlaterinfallingthings;theseproper
timesareirrelevantfortheglobalgeometrodynamics.Mostoftheblackhole’sinitial
energymustalreadyexistintheformofoutgoingHawkingradiationatthiscoordinate
time,andmayevenhavepassedanyrealisticexternalobserver.Ifsomethinghappens
thatcanbecomerelevanttoexternalobservers(suchasthecreationofHawkingradiation),itmusthappenoutsidethehorizontobecompatiblewithrelativisticcausality.
Blackholeradiationisagainbasedontheradiationarrowofretardation,butitsconventionalformulationalsodependsonaquantumarrowthatisresponsibleforthestatisticalinterpretationofquantummechanics.Apurequantumstategravitationallycollapsingtowardsablackholewouldthendecayintomanydecayfragments(mainlyphotons),
describedbyastatisticalensembleofdifferentemissiontimesforeachofthem–similar
toaseriesofunreadmeasurementswithdifferentoutcomes,ortothecoolingofahighly
excitedquantumstateofacomplexobjectbymeansofthermalradiation.8Anapparent
ensemblecanbedefinedevenforaresultingpurestate(accordingtoaunitarydescription)bymeansofsomephysicallyrelevantcoarsegraining.Inquantumtheory,oneusuallyneglectsinthissense(thatis,oneregardsasirrelevantforthefuture)theentanglementbetweenallpossibledecayproductsandthephaserelationsbetweenalltheirdecaytimes.Suchacoarse-grainingdoesnotonlyformallyjustifytheconceptofgrowing
"physical”entropyinspiteofapureglobalstate,5butalsothephenomenonofdecoherence.Incontrasttotheglobalensembleentropythatwouldbeconservedunderunitary
dynamics(andvanishesforapurestate),physicalentropyisdefinedasanextensive
quantityinaccordwiththelocalconceptofanentropydensitythatneglectsinformation
aboutcorrelations–justasBoltzmann’sµ-spacedistribution.Themajordifferencebetweenthedecayofhighlyexcitedstatesofnormalmatterandtheevaporationofblack
holesisthatthelatter’sunitarydynamicsisnotexplicitlyknown(andoccasionallyeven
questionedtoapply).
8
Thethusdescribedsituationisnonethelessmuchdiscussedasan"informationlossparadoxforblackholes".9Itsconsequencesareparticularlydramaticifonepresumesthe
existenceofablackholeinteriorregionthatwouldnecessarilyariseintheabsenceof
Hawkingradiation;matter(andthe“information”itmayrepresent)couldthennotcausallyescapeanymore.Thisquestionablepresumption(basedonclassicalsingularity
theorems)isoftentacitlyintroducedbyusing“niceslices”thataredefinedtoavoidthe
singularitybutmay,incontrasttoour“veryniceslices”,intersectthethusalsopresumedhorizon.Aunitarydescriptionmeans,however,thattheinformationdefiningthe
initialpurestateismostlytransformedintonon-localentanglement(formallyanalogous
tothestatisticalcorrelationsarisingindeterministicBoltzmanncollisions).Inthequantumcase,unitaritytherebyleadstoasuperpositionof"manyworlds"whichthereafter
remaindynamicallyautonomous,andwhichmayincludedifferentversionsofthe
“same”observers–thusphysicallyjustifyingtheconceptofdecoherence,forexample.
Thereplacementofthissuperpositionbyanensembleofmanypossibleworldsaccordingtoafundamentalstatisticalinterpretation(acollapseofthewavefunction)would
objectivelyandirreversiblyannihilatetheinformationcontainedintheirrelativephases,thusintroducingafundamental(law-like)dynamicaltimeasymmetry.Recallthatthe
Oppenheimer-Snydermodel,onwhichtheniceslicesarebased,preciselyneglectsthe
localenergylossduetoHawkingradiation.Althoughthe("back")reactionofthemetric
inresponsetoradiationlossmayinprinciplerequirequantumgravity,myargument
aboutthenon-formationofahorizonishereonlybasedonthelocalconservationof
momentum-energyinasituationwhereitmaynothavetobequestioned.
InsteadofassuminganinitialvacuumwhencalculatingthecreationofHawkingradiationclosetothehorizon,oneshouldtherebytakeintoaccountthepresenceofinfalling
matter,inwhichcasesomekindofinternalconversionmightleadtoitsannihilation.
(Theprevailingconservationofbaryonnumberetc.wouldhavetomodifytheHawking
radiation,andmaythusleadtoadifferentscenario.)Asimilarscenariohasrecently
beenpostulatedasanovelkindofphysicsclosetothehorizon(calleda“firewall”).10
Whilethisfirewallwasmeanttopreventanobserverfromremainingintactwhenfalling
in,itshouldaccordingtomyearlierproposal(seeearlierversionsofthispaper,availableatarxiv:1012.4708v1orv2)convertallpotentiallyinfallingmatterintoradiation.
NotethatthelocalBekenstein-Hawkingtemperaturedivergesclosetothehorizon,and
thereforemustleadtothecreationofallkindsofparticle-antiparticlepairs.Aslongas
9
someinternalconversionofthiskindcannotbeexcluded,thereisnoreasontospeculate
aboutblackholeremnants,superluminaltunneling,orafundamentalviolationofunitaritythatwouldgobeyonddecoherence(thatis,beyondameredislocalizationor“globalization”ofsuperpositions).11However,unitaritycanonlyapplytotheglobal“bird’s
perspective”thatincludesallEverettbranches,anditcannotleadtoanykindof“doubleentanglement”.12
Whatmightremainasa“remnant”accordingtothissemi-classicaldescriptionofblack
holeevolutiononveryniceslicesisamasslesspointlikecurvaturesingularity,sincethe
RiemanntensoroftheSchwarzschildmetricisproportionaltoM/r3,andhencediverges
forr=2M®0.Thissingularitysignalsabreak-downofthesemi-classicaldescriptionof
geometrodynamicsatthisfinalstageonly.Quantumgravitywouldrequireaboundary
conditionforthetimelessWheeler-DeWittwavefunction,whichcannotdistinguishbetweenpastandfuturesingularities(seeSects.4and5).Thismightleadtoaneffective
finalconditionthataffectsblackholes“frominside”inananticausalmanner.13Forexample,anyinwards-directed(hencevirtual)negativeenergyradiationcompensatingthe
emissionofHawkingradiationcould“recohere”theeffectiveblackholestateinorderto
loweritsentropyinaccordancewithboththemasslossandBekenstein’srelation(4b).
Thisretro-causalitymightevenaffectthenatureoftheoutgoingHawkingradiation.
NotethattheconceptofanS-matrixwouldbeunrealisticformacroscopicobjects,such
asblackholes.Becauseoftheirnever-endingessentialinteractionwiththeirenvironments,theycanneverbecomeasymptoticallyisolated(thereasonfortheirongoing,locallynon-unitarydecoherence).Theextremelifetimeofblackholesmeans,however,
thattheinformationlossproblemisatanyrateratheracademic:anyapparentlylost
informationwouldremainirrelevantforfarmorethan1065years,anditcouldhardly
everbeexploitedevenifitfinallycameoutintheformofentangledradiation(representingonesuperpositionof“manyworlds”).The“Pagetime”,14atwhichtheentanglementbetweentheresidualblackholeanditsemittedradiationisassumedtobemaximal,canthereforenothaveanyphysicalconsequencesfortheremainingblackhole.
Severalphysicists(includingmyself)usedtoseeaproblemintheequivalenceprinciple,
whichrequiresthatobserversordetectorsfreelyfallingintotheblackholedonotregisteranyblackholeradiation.Someevenconcludedthatthemass-lossofblackholes,too,
mustthenbeobserver-dependent(notveryappropriatelycalled“blackholecomple
10
mentarity”).However,thisconclusionappearstobewrong.Whiletheequivalencebetweenablackholeandauniformlyaccelerateddetector(asregardstheirradiation)
mustindeedapplytothelocallaws,itcaningeneralnotdosofortheirboundaryconditions.Anobserverordetectorfixedatsomedistancefromtheblackholewouldnotbe
immersedinisotropicheatradiation,sincethisradiationcomesfromtheblackholesurface(oraregionclosetoit),whichwouldcovermostofthewholeskyonlyforanobserververyclosetothehorizon.Eventhoughtheinfallingdetectormaynotregisterany
radiation,thelatter’seffectonfixeddetectors,oritsfluxthroughafixedspherearound
theblackhole,mustexistobjectively,justastheclicksofanaccelerateddetectorinan
inertialvacuum(attributedtoUnruhradiation)canbeheardbyaninertialobserver,
too.Therefore,bothobserverswouldagreethattheenergyabsorbedbytheaccelerated
detectormustbeprovidedbytherocketengineand,analogously,thattheHawkingnet
fluxofenergyrequiresanobserver-independentmasslossoftheblackhole.Therefore,
theresultingspacetimereality(includingmeasurementresults)isalsoobjectivelydefined.Theinfallingobserverwouldfurthermoreheartheclicksoffixeddetectorsoccurringataveryfastrate,andthereforeasbeingcausedbyanextremelystrongoutward
fluxaccordingtohispropertime.Forthesamereason,matterattheouterrimofacollapsingdustcloudcanatlateSchwarzschildtimesnotexperienceanygravitationalfield
asthereispracticallynogravitatingenergyleftinsideitsactualradiusanymore.Soit
cannevercrossahorizon.Inthisway,thephenomenonofblackholesfromthepointof
viewofexternalobserversisconsistentwiththefateofaninfallingobserver,whomay
eithersooninhispropertimehavetobeaffectedhimselfbytheinternalconversion
process,orotherwisehavetoexperiencetheblackholesurfaceveryrapidlyshrink–
finallygivingrisetoextremetidalforces–anddisappearbeforehecouldarrive.(Note,
however,thattheauxiliaryconceptofaneventhorizonchangingintimeisill-defined,
sinceahorizonisalreadyaspacetimeconcept.)
Ifthefallingobservercouldsurvivetheinternalconversionprocess,hewouldthenend
upatveryhighspeed,andhavetravelledfarintothecosmicfutureinashortproper
timebecauseofthequasi-singulartimedilation.Ontheotherhand,notheorythatis
compatiblewiththeequivalenceprinciplecandescribebaryonnumbernon-conservationintheabsenceofasingularity,butbecauseofthehugelifetimeofblackholesthis
problemmayperhapsbesolvedonlyinconnectionwiththatofthematter-antimatter
asymmetryinouruniverse.Allsymmetriesmayinprinciplebebrokenbytheeffective
11
non-unitaritycharacterizingthedynamicsofindividualEverettbranches.Thislastremarkmightalsoberelevantfortheabovementionedpossibilityofanti-causality(recoherence)requiredbyanapparentfutureconditionthatisinaccordwithatimeless
Wheeler-DeWittequation(seeSect.5);recoherencewouldhavetoincludeare-combinationofdifferentEverettworldsbyrelocalizationofglobalsuperpositions.
RogerPenrosecomparedblackholeentropynumericallywiththatofmatterintheuniverseundernormalconditions.15Sincetheformerisaccordingto(4b)proportionalto
thesquareoftheblackholemass,macroscopicblackholeformationleadstoatremendousincreaseofentropy.Asthermodynamicentropyisproportionaltotheparticle
number,itisdominatedintheuniversebyphotonsfromtheprimordialcosmicradiation(whosenumberexceedsbaryonnumberbyafactor109).Ifourobservablepartof
theuniverseofabout1079baryonsconsistedcompletelyofsolarmassblackholes,it
wouldpossessanentropyoforder1098(inunitsofkB-1),thatis,1010timesasmuchas
thepresentmatterentropyrepresentedby1088photons.Combiningallblackholesinto
onehugeonewouldevenraisethisnumberto10121,thehighestconceivableentropyfor
this(perhapspartial)universeunlessitsvolumeincreasedtremendously.4,7,16Ifentropy
isindeedameasureofprobability,anyapproximatelyhomogenousmatterdistribution
wouldbeextremelyimprobableexceptfordensitiesmuchlowerthanatpresent(ata
verylatestageofaneternallyexpandinguniverse).Therefore,thehomogeneityofthe
initialuniverseisusuallyregardedasthe“fundamentalimprobableinitialcondition"
thatexplainstheglobalmasterarrowoftimeifstatisticalreasoningisapplicabletothe
future(seeSect.4).However,itsrelationshiptothethermodynamicallyimportantconditionofabsentor"dynamicallyirrelevant"non-localinitialcorrelations(orentanglementinthequantumcase)seemstobenotyetfullyunderstood.Ifthetwoentropyconcepts(blackholeandthermodynamic)aretobecompatible,theentropyofthefinal
(thermal)radiationmustbegreaterthanthatoftheblackhole,whilethelatterhasto
exceedthatofanykindofcollapsingandinfallingmatter.
4ExpansionoftheUniverse
Theexpansionoftheuniverseisatime-asymmetricprocess,butincontrasttomostotherarrowsitformsanindividualphenomenonratherthanawholeclassofsimilarones,
suchasblackholes,radiationemitters,orsteamengines.Itmayevenchangeitsdirec
12
tionatsometimeofmaximumextension,althoughpresentastronomicalobservations
mayindicatethattheexpansionwilllastforever.AhomogeneousandisotropicFriedmannuniverseisinclassicalGRdescribedbythedynamicsoftheexpansionparameter
a(t)accordingtothetime-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.Penrose'sentropyestimatesthendemonstratethatthehomogeneityassumedinEq.(5)is
extremelyimprobablefromastatisticalpointofview.Therefore,itmustbeunstable
undertheinfluenceofgravity(inspiteofbeingdynamicallyconsistent).
Inaccordancewithahomogeneousinitialmatterdistribution,Penrosepostulatedthat
freegravitationalfieldsvanishedexactlyattheBigBang.Thesefreefieldsaredescribed
bytheWeyltensor,thatis,thetrace-freepartofthecurvaturetensor.Thetraceitself
(theRiccitensor)islocallyfixedbythestress-energytensorofmatteraccordingtothe
Einsteinfieldequations.TheWeyltensor,ontheotherhand,isanalogoustothedivergence-freepartoftheelectrodynamicfieldtensorFµn,sincethedivergence∂µFµn(the
traceofthetensorofitsderivatives)issimilarlyfixedbythechargecurrentjn.Therefore,theWeyltensorhypothesisisanalogoustotherequirementofanabsenceofany
initialelectromagneticradiation,aconditionthatwouldallowonlytheretardedelectromagneticfieldsofallsourcesonthebackwardlightconeofadetectortoexist.This
universalretardationofradiationhadindeedbeenproposedasalawbyPlanck(ina
disputewithBoltzmann),17andlaterbyRitz(inadisputewithEinstein),18inanattempt
toderivethethermodynamicarrow.However,BoltzmannandEinsteinturnedouttobe
right,sincetheretardationcanbeunderstoodinturnasacausalconsequenceofthe
presenceofthermodynamicabsorbers1–cosmologicallyincludingtheabsorberformed
bytheradiationera,whichwouldnotallowustodiscoveranyconceivableearlierelectromagneticradiation.Incontrast,theearlyuniverseseemstobetransparenttogravitationalradiation,possiblyincludingthatwhichcouldconceivablyhavebeencreatedin
theBigBang.
13
Notethatthelowentropyandthecorrespondinghomogeneityoftheuniversecannot
beexplainedbyanearlycosmicinflationera(ashasoccasionallybeenclaimed)ifthis
inflationwasdeterministicandwouldthushaveconservedensembleentropy.
Althoughouruniversemayexpandforever,theideaofitslaterrecontractionisatleast
conceptuallyinteresting.ThomasGoldfirstarguedthatthelowentropyconditionat
highdensityshouldnotbebasedonanabsolutedirectionoftime,andhencebevalidat
aconceivableBigCrunchaswell.19ThelatterwouldthenbeobservedasanotherBig
Bangbyobserverslivingduringtheformalcontractionera,whilelocalfuturesingularitieswouldbeexcludedjustaspastones.Gold’sscenariowouldnotonlyrequireathermodynamictransitionerawithoutanywell-definedarrowinourdistantfuture–it
wouldalsoposeseriousconsistencyproblems,sincetheextremelysmallinitialprobabilityforthestateoftheuniversewouldhavetobesquaredifthetwoconditionsare
statisticallyindependentofoneanother.20Ifnonethelesstrue,itwouldhaveimportant
consequencesforthefateofmatterfallingintomassiveblackholes.Ifsuchblackholes
survivedthementionedthermodynamictransitioneraatthetimeofmaximumextensionbecauseoftheirlongevaporationtimes(cf.Sect.3),theywouldaccordingtothe
globaldynamicsenteranerawithreversedarrowsoftime.However,becauseofthe
transparencyofthelateuniversetolight,theywould“receive”coherentadvancedradiationfromtheirformalfutureevenbeforethathappens.Thisadvancedradiationmust
then"retro-cause"suchmassiveblackholestoexpandagaininordertoapproachastate
ofhomogeneityinaccordancewiththefinalcondition.21Inmathematicalterms,their
horizonisnot“absolute”inthiscaseevenintheabsenceofanyblackholeevaporation.
Areversalofthearrowoftimemaynotonlyoccurinthedistantfuture,butalsohave
occurredinthepast.Severalpre-big-bangscenarioshavebeendiscussedinnovelandas
yetspeculativetheories.Usually,onetherebyidentifiesthedirectionoftheformaltime
parameterwiththedirectionofthephysicalarrowoftime.Forexample,accordingto
argumentsfirstusedinloopquantumcosmology,22theconfigurationspaceforFriedmanntypeuniversesmaybedoubledbyinterpretingformallynegativevaluesofthe
cosmicexpansionparameteraasrepresentingnegativevolumemeasures.Thecosmic
dynamicscanthenbecontinuedbackwardsintimebeyondtheBigBangintoitsmirror
imageby"turningspaceinsideout"(turningright-handedtriadsintoleft-handedones)
whilegoingthrougha=0eveninaclassicalpicture.Forthispurpose,theclassicaldynamicaldescription(5)wouldhavetobemodifiedclosetotheotherwisearisingsingu
14
larityata=0–asitisindeedsuggestedbyloopquantumgravity.However,ifthe"initial"conditionsresponsibleforthearrowoftimeareassumedtoapplyatthesituation
ofvanishingspatialvolume,thearrowwouldformallychangedirection,and|a|rather
thanawouldrepresentaphysicalcosmicclock.Observersonbothtemporalsidesofthe
BigBangcouldonlyremembereventsinthedirectiontowardsa=0.Anotherpossibility
toavoidthesingularityisarepulsiveforceactingatsmallvaluesofa,23whichwould
leadtoaBigBouncewithsimilarconceivableconsequencesforthearrowoftimeasthe
abovemodelthatinvolvesspaceinversion.
Incosmology,quantumaspectsofthearrowoftimemustagainplayanimportantrole.
AccordingtotheCopenhageninterpretationthereisnoquantumworld–sonocomplete
andconsistentcosmichistorywouldbedefinedanymorewhenquantumproperties
becomeessential.Inotherorthodoxinterpretations,theunitaryevolutionofthequantumstateisrepeatedlyinterruptedbymeasurementsandsimilartime-asymmetric
events,whenthewavefunctionisassumedto"collapse"indeterministically.Theconsequencesofsuchstochasticeventsonquantumcosmologywouldbeenormous,butas
longasnospecificcollapsemechanismforthewavefunctionhasbeenconfirmed,one
hasagainarrivedatanimpasse.Goingforwardintimemaybeconceptuallysimplein
suchasymmetrictheories,sinceonejusthasto"throwaway"allcomponentsofthe
wavefunctionwhichrepresentthenot“actualized”potentialoutcomes,whilegoing
backwardswouldrequiretheselostcomponentstorecombineanddynamicallyform
localsuperpositionsagain.Soonehasatleasttokeeptheminthecosmicbookkeeping–
regardlessofwhethertheyarecalled"real"(asintheEverettinterpretation)ornot.GoingbacktotheBigBangwouldrequireallthosemany“worlds”thathaveeverbeen
thrownawayintheorthodoxdescriptionduringthepastofouruniverse,whileone
wouldhavetothrowawayotherswhenformallygoingbackwardsbeyondtheBigBang
inordertoobtainanindividualquasi-classical"pre-big-banghistory".Inotherwords,a
unitarycontinuationbeyondtheBigBangcanonlydescribethecompleteEverettsuperpositionofworldsonbothsidesoftheBigBang,buthardlyanyindividuallyobserved
quasi-classicalworlds.Thecorrespondingmasterarrowoftimewouldthusnotonly
affectallrealmsofphysics–itmustbetrulyuniversalinamuchdeepersense:itcan
onlyhave"multiversal"meaning.Thesamemultiversalitywasrequiredinaunitary
blackholeevolutionofSect.3,anditdoes,infact,applytotheunitaryquantumdescrip-
15
tionofallmacroscopicobjects,whenirreversibledecoherencemimicsacollapseofthe
wavefunctionandtherebyexplainsclassicality.
ThetimedirectionofEverett’sbranchingofthewavefunctionthatisbasedondecoherencerequiresahomogeneousinitialquantumstate(presumablyata=0),whichdoes
notcontainanynonlocalentanglementthatmightlaterhavelocaleffects.Quantumdynamicswillthenleadtodecoherence(theinpracticeirreversibledislocalizationofsuperpositions),andthereby"intrinsically"breakvariousglobalsymmetries–possibly
evenintheformofmanydifferentquasi-classical"landscapes",whichcanonlyrepresentdifferentbranchesofonesymmetricsuperposition.
5QuantumGravity
GeneralRelativityhastraditionallybeenconsideredinablockuniversepicture,butbecauseofthehyperbolictypeofEinstein'sfieldequationsitisadynamicaltheoryjustas
anyotherfieldtheory.Itsexplicitdynamicaldescription,whichrequiresanon-Lorentzinvariantform,wascompletedbyArnowitt,DeserandMisner(ADM).24ThisHamiltonianformulationisaprerequisiteforthecanonicalquantizationofthetheory.Ishallhere
regardtheresultofthisquantizationprocedureasaneffectivequantumtheory,without
discussinganyattemptsoftheirpossiblejustificationintermsoftheoriesthatmaybe
exactbuthavenoempiricalsupportyet(suchasstringtheoryorloopquantumgravity).
TheADMformalismisbasedonanarbitrarytime-likefoliationofspacetimethathasto
bechosen"onthefly",thatis,whilesolvinganinitialvalueproblem.(Asimilarfreedom
wasusedinSect.3forthechoiceofveryniceslices.)Ifthedynamicsofmatterisalso
defined,thisconstructionleadstoaunique(construction-independent)spacetimegeometry,whilethespatialmetriconthechosenspace-likeslicesrepresentsthecorrespondingdynamicalvariables.Thelattercanbedescribedbyasymmetricmatrixhkl(xm)
–withk,l,mrunningfrom1to3.Threeofitssixindependentmatrixelementsrepresent
thechoiceofphysicallymeaninglesscoordinates,twowouldinthelinearapproximation
correspondtothespincomponentsofagravitationalwave(±2withrespecttothedirectionofpropagationforaplanewave),whiletheremainingonecanberegardedasa
measureof"many-fingered"physicaltime(metricdistancebetweenspace-likeslices).
Thecorrespondingcanonicalmomentapkldefinetheembeddingofthespatialmetric
16
intospacetimeandthearbitrarypropagationofspatialcoordinates.Thedynamicscan
thenbeformulatedbymeansoftheHamiltonianequationswithrespecttoanarbitrary
timeparametertthatformallydistinguishesdifferentslicesinagivenfoliation.Theyare
equivalenttoEinstein'sfieldequations.Incontrasttometrictime,theparametertis
geometricallyorphysicallymeaningless,andcanthereforebereplacedbyanymonotonicfunctiont'=f(t)–includingitsinversion.
NotethatwhenSpecialRelativityissaidtoabandontheconceptofabsolutetime,this
statementrefersonlytotheconceptofabsolutesimultaneity,whilepropertimes,which
controlallmotionaccordingtotheprincipleofrelativity,arestillassumedtobegiven
“absolutely”bythefixedLorentzmetric.Thisremainingabsolutenessisabandonedonly
inGeneralRelativity,wherethemetricitselfbecomesadynamicalobjectlikematter,as
indeeddescribedbytheADMformalism.Theabsenceofanabsolutetimeparameter
(hererepresentedbyitsreparametrizability)wasalreadyrequiredbyErnstMach.Julian
Barbour,whostudieditsconsequencesinmuchhistoricaldetail,25calledit"timelessness".However,acompleteabsenceoftimewouldremoveanypossibilitytodefinean
arrow,whileaone-dimensional(dynamical)successionofstates,characterizedbyan
arbitraryparameter,stillallowsonetodefineatimedirectionasymmetry.
Theinvarianceofthetheoryunderspatialcoordinatetransformationsandtimereparametrizationiswarrantedbyfourconstraintsforthematrixhkl(t),calledmomentumand
Hamiltonianconstraints,respectively.Theymayberegardedasinitialconditions,but
theyareconservedintime.Inparticular,theHamiltonianconstraintassumestheform
H(hkl,πkl)=0.
(6)
Whenquantized,26andwhenalsotakingintoaccountmattervariables,thisconstraint
translatesintotheWheeler-DeWittequation,
HY(hkl,matter)=0,
(7)
whichmeansthatthetime-dependentSchrödingerequationbecomestrivial,
∂Y/∂t=0. (8)
Eventhetimeparameterthasnowdisappeared,becausetherearenoparametrizable
trajectoriesrepresentingcosmichistoriesanymoreinquantumgravity.Onlythisdras-
17
ticproperty,whichisaquantumconsequenceofmerereparametrizability,canberegardedasaformal“timelessness”.
ThetimelessnessoftheWheeler-DeWittwavefunctionhasbeenknownatleastsince
1967,butitseemstohaveoriginallybeenregardedas“justformal”.Atimeparameter
wasoftensmuggledinagaininvariousways–forexampleintermsofparametrizable
Feynmanpaths,bymeansofsemiclassicalapproximations,orbyattemptstoreintroduceaHeisenbergpictureinspiteoftheHamiltonianconstraint.27Theproblembecame
pressing,though,inconnectionwiththeassumptionofanonticandkinematicallycompletewavefunctioninquantumcosmology.28
ThegeneralwavefunctionalY(hkl,matter)describesentanglementofgeometryandmatter.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?ThesurprisingansweristhatthisequationisgloballyhyperbolicforFriedmanntypeuniversesonitsinfinite-dimensionalgauge-freeconfigurationspace(which
hasthereforealsobeencalled“superspace”)ratherthanonspacetime.Theexpansion
parameteraoritslogarithmappearsasatime-likevariableinthissensebecauseofthe
unusualnegativesignofitsformalkineticenergycomponent.30Therefore,theWheelerDeWittequationdefinesan“initial”valueproblem,forexampleatasmallvalueofa.For
amodifiedWheeler-DeWittequation,thispossibilitymightevenbeextendedtoa=0.
Thereisnoconceptualdifferencebetweena(multiversal)BigBangandaBigCrunch
anymore,sinceintheabsenceofatimeparameterthewavefunctioncanonlybea
standingwaveonconfigurationspace(inspiteofitsintrinsicdynamics).
18
ThemetrictensorandotherfieldsdefinedonaFriedmannsphere,a=const,mayberepresentedbyafour-dimensionalmultipoleexpansion,whichisparticularlyusefulfordescribingtheveryearly,approximatelyhomogeneousandisotropicuniverse.31Inthis
case,onemayconvenientlymodelmatterquantummechanicallybyamassivescalar
fieldF(xk).ThewavefunctionaloftheuniversethenassumestheformY(a,F0,{xn}),
whereF0isthehomogeneouspartofthescalarfield,while{xn}areallhighermultipoles
ofgeometryandmatter.Forthemetric,onlythetensormodesaregeometricallymeaningful,whiletherestrepresentsgaugedegrees(heredescribingthepropagationofspatialcoordinates).Theglobalhyperbolicnaturewithrespecttoallphysicaldegreesof
freedombecomesmanifestinthisrepresentation.
Inasimpletoymodelonemayneglectallhighermultipolesinordertosolvethe
Wheeler-DeWittequationontheremainingtwo-dimensional"mini-superspace"formed
bythetwomonopolesonly.TheremainingHamiltonianrepresentsana-dependent
harmonicoscillatorforthevariableF0,whichallowsonetoconstructadiabaticallystableGaussianwavepackets("coherentstates").32Figure3depictsthepropagationof
suchawavepacketwithrespecttothe"time"variablea=lna.Thisstandingwaveon
mini-superspacemimicsatimelessclassicaltrajectory.However,thecompletewave
functionalhastobeexpectedtoformabroadsuperpositionofmanysuchdynamically
separatedwavepackets(acosmologicallyearlyrealizationof"manyworlds").Notethat
these“worlds”arepropagatingwavepacketsratherthantrajectories(asinDeWitt’sor
DavidDeutsch’sunderstandingof“ManyWorlds”).Ifthehighermultipolesarealsotakenintoaccount,theWheeler-DeWittequationmaydescribedecoherenceprogressing
witha–atfirstthatofthemonopoleF0andofaitself,althoughthisapproachrequires
effectiverenormalizationproceduresinthisdescription.33
19
Fig.3:WavepacketforahomogeneousmassivescalarfieldamplitudeF0(plottedalong
thehorizontalaxis)dynamicallyevolvingasafunctionofthetime-likeparametera=lna
thatispartofthemetric(secondaxisinthistwo-dimensionalmini-superspace).The
classicaltrajectorypossessesaturningpointabovetheplotregion50≤a≤150–namely
atabouta=240inthisnumericalexamplethatrepresentsanexpandingandrecontractingmini-universe.Wavemechanically,thiscorrespondstoareflectionofthewavepacketbyarepulsivepotentialin(5)atthisvalueofa(withthereflectedwavebeingomitted
intheplot).Thisreflectionleadstoconsiderablespreadingofthe"initial"wavepacket.
Thecausalorderofthesetwolegsofthetrajectoryisarbitrary,however,andthephase
relationsdefiningcoherentwavepacketscouldalternativelybechosentogiverisetoa
narrowwavepacketforthesecondleginstead.Therefore,this(herenotshown)formal
spreadingdoesnotrepresentaphysicalarrowoftime(FromRef.1,Sect.6.2.1.)
This“intrinsicdynamics”withrespecttothetime-likeexpansionparameterahasnothingasyettodowiththelocaldynamicsinspacetime(controlledbypropertimesalong
time-likecurves)thatmustberelevantformatterassoonasthemetricbecomesquasiclassical.Inordertounderstandtherelationbetweenthesetwokindsofdynamics,one
mayapplyaBorn-OppenheimerexpansionintermsoftheinversePlanckmass,whichis
largecomparedtoallparticlemasses,inordertostudytheWheeler-DeWittwavefunction.34ThePlanckmassappearsinthekineticenergytermsofallgeometricdegreesof
freedomthatappearintheHamiltonianconstraint.Theformalexpansionintermsof
powersofmPlanck-1/4thendefinesan"adiabaticapproximation"inanalogytothetheory
ofmolecularmotion(withelectronwavefunctionsintheelectrostaticfieldsofslowly
movingnuclei).Inmostregionsofconfigurationspace(dependingontheboundary
20
conditions)onemayfurtherapplyaWKBapproximationtothe"heavy"degreesoffreedomQ.Inthiswayoneobtainsanapproximatesolutionofthetype
Y(hkl,matter)=Y(Q,q)=eiS(Q)c(Q,q),
(9)
whereS(Q)isasolutionoftheHamilton-JacobiequationsforQ.Theremainingwave
functionc(Q,q)dependsonlyslowlyonQ,whileqdescribesall"light"(matter)variables.
UndertheseapproximationsonemayderivefromtheWheeler-DeWittequationtheadiabaticdependenceofc(Q,q)onQintheform
.
(10)
TheoperatorhQistheweaklyQ-dependentHamiltonianforthemattervariablesq.This
equationdefinesanewtimeparametertWKBseparatelyalongallWKBtrajectories
(whichdefineclassicalspacetimes)bythedirectionalderivative
. (11)
Inthisway,oneobtainsfrom(10)atime-dependentglobalSchrödingerequationfor
matterwithrespecttothederivedWKBtimetWKB.26,28Thisparameterdefinesatimecoordinateinspacetime,sincetheclassicaltrajectoriesQ(t)inthesuperspaceofspatial
geometriesQdefinespacetimegeometries.Eq.(10)alsodecribesthedecoherenceof
superpositionsofdifferentWKBtrajectories.Decocherenceisalsorequiredtoeliminate
superpositionsthatareneededtodefinearealwavefunctioneiSc+e-iSc*,whichhasto
beexpectedfromtherealWheeler-DeWittequationunderphysicallymeaningful
boundaryconditions.
InordertosolvethisderivedtimedependentSchrödingerequationalongagivenWKB
trajectory,thatis,intermsofafoliationofaclassicalspacetimethatdoesinturnadiabaticallydependontheevolvingmatter,oneneedsa(lowentropy)initialconditionin
theregionwheretheWKBapproximationbeginstoapply.Forthispurposeonewould
firsthavetosolvetheexactWheeler-DeWittequation(oritsgeneralizedversionthat
mayapplytosomeasyetelusiveunifiedtheory)asafunctionofabyusingitsfundamentalcosmicinitialconditionata=0.Thismightbedone,forexample,byusingthe
multipoleexpansionontheFriedmannsphere,untiloneenterstheWKBregion(atsome
distancefroma=0),wherethissolutionwouldthenprovideinitialconditionsforthe
21
partialwavefunctionscforallarisingWKBtrajectories.Thederivedtime-dependent
SchrödingerequationwithrespecttotWKBthenshoulddescribefurtherdecoherenceof
matter(theemergenceofotherquasi-classicalproperties),andtherebyexplainthe
originofallotherarrowsoftime.Inparticular,itmustenforcedecoherenceofsuperpositionsofanyarisingmacroscopicallydifferentspacetimes,whichwouldformseparate
quasi-classical"worlds".26ItwouldalsodecohereconceivableCPTsymmetricsuperpositionsofblackandwhiteholes,whichareanalogoustoparityeigenstatesofchiralmolecules,ifthesehadevercomeintoexistence.16
Acknowledgement:IwishtothankClausKieferforhiscommentsonanearlydraftof
thismanuscript.
Noteaddedafterpublication:The“causaltreatment”ofblackholes,usedinSect.3as
anargumentagainsttheformationofeventhorizonsand,therefore,theexistenceofan
informationlossparadox,hasrecentlybeencorroboratedbyexplicitmodelcalculations.35Asimilarconclusionwasdrawnalreadyin1976(usingadifferentmodel)by
UlrichGerlach.36Heassumed,however,thattheblackholefinallysettlesdowninaspecificgroundstatethatisnotflatspacetimebuta“remnant”.Thebasicassumptionfor
theseconclusionsisjustthevalidityofrelativisticcausalityeveninthepresenceof
Hawkingradiation.Soonemayconjecturequitegenerallythateventhorizonscannever
formindynamicallylocalQFT.ThisargumentappearsmorerealisticfordescribingmacroscopicblackholesthanaquantumgravitationalcollapsethatneglectsHawkingradiation,althoughthismayalsoavoidasingularity.37Bothmechanismsmayberelevantin
theend,though.
22
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24