Volume 121,number3 PHYSICSLETTERSA 13 April 1987

THE ARROW OF TIME AND THE EXPANSION OF THE UNIVERSE

Asghar QADIR’DepartmentofPhysics,Universityof Texasat Austin,Austin,TX 78712, USA

Received7 January1987;acceptedfor publication6 February1987

It is pointed out that a model usedto testany suggestionregardingthearrowof time in a cosmologicalcontect,must havesufficient complexity.The clam of Zeh thatthearrowof time definedby auniversalwavefunctiondoesnotpermita reversalofcosmicexpansionto beobserved,is refutedon thesegrounds.

Micro-physics is time symmetric while the real was a time asymmetrybetweenthe expandingandworld is not.This differenceis accountedfor by the contractingphases,becauseofwhich thatconclusionintroductionof statistics,becauseof which processes wasnotnecessary.In a noteaddedto this paper[6],do notremainreversible.Thisis expressed,through Hawkingagreedwith Page’spoint of view. Zeh [81thesecondlawofthermodynamics,astheincreaseof claims to havefound certain flaws in Page’sargu-entropy.Thus,for most “local” physical considera- ment.Thepurposeofthispaperis to drawattentiontionswe havea well-specified“thermodynamicarrow tosomerequirementsthatanymodelconstrictedtoof time”. Is therea large-scale,“cosmologicalarrow testsuggestionsregardingthedefinition oftime in aof time”?If theuniverseexpandsforeverit couldbe cosmologicalcontext,mustsatisfy. On the basisofprovidedby the expansionfactor,a(1). An alterna- this requirementit is pointedoutthatZeh’scritiquetive suggestiondue to Penrose[1], that the Weyl ofPage’sargumentis notvalid.curvaturemay provide a measureof cosmological I shall start by giving those points madeby Zehentropy(analogoustotheblackholeentropy),could withwhich I amnotin agreement,andthenproceedprovidean arrowof time evenif the universerecol- to discussthem.Usinga model universe(originallylapses.On the otherhandGold [2] (for example) constructedby Hawking [9]) describingby auniver-hassuggestedthat the cosmologicalarrow of time, sal wavefunctionw~dependingon a massivescalardefinedby the scalefactor,shouldbeidentifiedwith field0 andthescalefactor,a (bothfunctionsoftime),thetherodynamicarrowoftime. In thiscaseentropy Zeh:shouldstartdecreasingwhenthe universestartsto (1) arguesthat “it would not be valid to extendrecollapse. classicalpathsbeyondthe WKB region”;

An entirelydifferentsuggestionarisesfrom Hawk- (2) objectsto “the conceptofcontinuedexistenceing’s quantumcosmology [31,wherein the entire of observers(referredto by Page) in a universaluniversemaybedescribedby a wavefunction.Such quantumdescription”;a theory dependson the Everett—Wheeler“many (3) claimsthat “conditionedprobabilities(givenworlds” interpretation[4] of the quantumformal- by Laplacetransformsof w(a, 0)) do not defineaism. Using the formalism set up by Hartle and directionofclassicalpaths,becausethereis noexter-Hawking [5], Hawkingdefineda cosmologicalarrow nal time”;of time [61, which heconcludedbehavedaccording (4) concludesthatthereforethescalefactor,a( t)toGold’s suggestion.Page[7] pointedoutthatthere servesasa timeparameterandhenceno expanssion

couldeverbeobserved.OnleavefromtheDepartmentof Mathematics,Quaid-i-Azam Zehis quitecorrectthat therecanbeno guaranteeUniversity, Islamabad,Pakistan.

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Volume 121, number3 PHYSICSLETTERSA 13 April 1987

that resultsobtainedusingtheWKB approximation otherquantities,providinga meaningto theLaplacewill be valid in a regime wherethe approximation transformof ~(a, 0, ...). ThisenablesPageto pro-schemedoesnothold. However,that is not in itself vide anarrowof time for the cosmologywhich doesaproofthattheresultsarewrongoutsidethatdomain, notreversewith therecollapseof theuniversemodel.Anothermethod,valid in thedomainwheretheWKB This explainsmy disagreementwith points (3) andapproximationdid nothold,maygivethesameresult. (4), statedabove.SincePageonly appealsto theWKB approximation I concludewith an asideconcerningthe specula-for the purposeof making the argumentmorecon- tion of how black holes may erode the conceptofcrete,thereseemsto beno adequatereasonto doubt “objectivisation”, with which Zeh concludeshisthe robustnessof the point of principle which he is paper.In a W-model universe(which is compact,trying to illustratethere. possessesa big banganda big crunch) it hasbeen

The secondobjection addressesa much deeper pointedout [12] that the blackholeandbig crunchproblem:the definition of an observerin quantum singularitiesamalgamate.As such, in thesemodelsmechanics.I will nottry to providesucha definition there is no distinction of principle between thehere.However, it is possibleto statesomerequire- “inside” and“outside”of ablackhole.As suchtherementsit mustsatisfy.While an elementaryquantum is no reasonto expect the conceptof “objectivisa-processis reversible,the processof measurement tion” to beerodedin thesemodels.cannotbe, as it leadsto an irreversibleincreaseofinformation.Thus a measuringdevice, in quantum I would like to thank Don Pagefor enlighteningmechanics,musthave“sufficient complexity”notto discussions.I amalsoindebtedto theCouncilfor thebe consideredasan “elementaryquantumprocess”, InternationalExchangeof Scholarsfor theFulbrightor even a finite sequenceof such processeswhich Fellowship,whichenabledmeto visit, andto Profes-would be reversible.An observermustat leastbea sor J.A. Wheeler for hospitality at the Physicsmeasuringdevice.Thus it too mustbe “sufficiently Department,Universityof Texasat Austin.complex”. Theobserver,thoughalteredabit by eachquantum interaction should largely maintain anidentity throughthevariousinteractions,onaccount R fof its complexity. It couldbemodelledon Penrose’s e erencesconstruction [101 of the measurementof angularmomentaby objectsessentiallycomposedof unitsof [1] R. Penrose,Physicsandcontemporaryneeds,Vol. 1, ed.

Raizuddin(Plenum,NewYork, 1977);Generalrelativity:angularmomentum.As such Pages observerdoes an Einstein centenary,eds. SW. Hawking and W. Israelhavea perfectright to“continuedexistence”. (CambridgeUniv. Press,Cambridge,1979).

Now I cometo the main point. Time dealswith a [2] T. Gold, La structureet l’evolution de l’universe: Proc.

chronologyof events.A universein which nothing EleventhIntern. Solvay Congress(Edition Stoops,Brus-happenshasno conceptof time [11]. A model uni- sels, 1958).

[3] S.W.Hawking,Astrophysicalcosmology:Proc.of theStudyverseconstructedto testa definition of the arrowof Weekon Cosmologyandfundamentalphysics, eds.H.A.

time musthave“sufficient complexity” to allow for BrUck, G.V. CoyneandM.S. Longair(PontificaeAcademic

eventsto occur sothat therecanbea chronology.It ScriptaVaria48 (1982) 563; Relativity,groupsandtopol-is in this sensethat I claim the modelis over-simpli- ogy II, eds. B.S. Dc Witt and R. Stora (North-Holland,

fled. While quite adequatefor its original purpose Amsterdam,1984).[4] H. Everett, Rev.Mod. Phys.29 (1957)454;

[9], it is notappropnateto discussthearrowof time. J.A. Wheeler,Rev.Mod. Phys.29 (1957)463.

Sincethe only eventsbeing consideredaregivenby [5] J.B.HartleandSW.Hawking,Phys.Rev.D 28 (1983)2960.

the scalefactor,a(t) canonly be singlevalued.This [6] SW.Hawking,Phys.Rev.D 32 (1985) 2489.

is not a consequenceof the universalwavefunction [7] D.N.Page,Phys.Rev. D 32 (1985) 2496.butof the lackof anyothereventstoprovidea chro- [8] H.D.Zeh,Phys.Lett. A 116 (1986)9.

[9] S.W.Hawking,Nucl. Phys.B 239 (1984) 257.nology. It is the lack of somethingbesidesa(t) and [10] R. Penrose,in: Combinatorialmathematicsandits appli-0(t) in w which leadsZeh to identify a(t) with t. cations, ed. D.J.A. Welsh (AcademicPress,New York,

However, Pageimplicity assumesthe existenceof 1971).

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[11] J.A. Wheeler,Frontiersof time(North-Holland,for Societa (CambridgeUniv. Press,Cambridge,1985); Black holeItalianadi Fisica, 1979). singularityas part of big crunch singularity, preprint of

[12] A. Qadir andiA. Wheeler,From SU(3) to gravity: Yual Centrefor Theoretical Physics, University of Texas atNeémanFestschrift,eds. ES. Gotsmanand G. Tauber Austin.

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