时间、空间与哲学 281页

\nTIME,SPACEANDPHILOSOPHYInthisbook,ChristopherRayaddressesfundamentalissuesinthephilosophyofspaceandtime,whileavoidingdauntingtechnicalitiesandjargon.Alwayscarefultoelucidatethephilosophicalproblemsassociatedwithspaceandtime,RayexaminestheworkofNewton,Einstein,Hawkingandotherscientificgiantsanddiscussesthereactionsofphilosopherstothiswork—frommetaphysicalworriesaboutthenatureandrealityofspaceandtimetoquestionsaboutthestatusofrelativityanditsrivaltheories.Heinvestigatesthepuzzlingnatureofspace—fromtheinfinitesimallysmalltotheunimaginablylarge,thedisturbingparadoxesoftimeandtimetravel,andthecuriousideasofmoderncosmology—fromthebigbangandthepossibilityofcreationexnihilotothequantumworldofblackholes.ChristopherRayisAssistantProfessorinHistoryandPhilosophyofScienceatPortlandStateUniversity,Oregon.HispublishedworksincludeTheEvolutionofRelativity(1987).\nPHILOSOPHICALISSUESINSCIENCEGeneralEditoW.H.Newton-SmithTHERATIONALANDTHESOCIALJ.R.BrownTHENATUREOFDISEASELawrieReznekINFERENCETOTHEBESTEXPLANATIONPeterLiptonTHEPHILOSOPHICALDEFENCEOFPSYCHIATRYLawrieReznekMATHEMATICSANDTHEIMAGEOFREASONMaryTilesTHELABORATORYOFTHEMINDJ.R.BrownForthcomingMETAPHYSICSOFCONSCIOUSNESSWilliamSeager\nTIME,SPACEANDPHILOSOPHYChristopherRayLondonandNewYork\nFirstpublished1991byRoutledge11NewFetterLane,LondonEC4P4EESimultaneouslypublishedintheUSAandCanadabyRoutledgeadivisionofRoutledge,ChapmanandHall,Inc.29West35thStreet,NewYork,NY10001RoutledgeisanimprintoftheTaylor&FrancisGroupThiseditionpublishedintheTaylor&Francise-Library,2003.©1991ChristopherRayAllrightsreserved.Nopartofthisbookmaybereprintedorreproducedorutilizedinanyformorbyanyelectronic,mechanical,orothermeans,nowknownorhereafterinvented,includingphotocopyingandrecording,orinanyinformationstorageorretrievalsystem,withoutpermissioninwritingfromthepublishers.BritishLibraryCataloguinginPublicationDataRay,ChristopherTime,spaceandphilosophy.—(Philosophicalissuesinscience).1.Space&time.PhilosophicalperspectivesI.TitleII.Series115LibraryofCongressCataloginginPublicationDataRay,Christopher.Time,space,andphilosophy/ChristopherRay.p.cm.—(Philosophicalissuesinscience)Includesbibliographicalreferencesandindex.1.Spaceandtime.I.Title.II.Series.BD632.R39199190–24118115–dc20ISBN0-203-01873-7Mastere-bookISBNISBN0-203-19146-3(AdobeeReaderFormat)ISBN0-415-03221-0—ISBN0-415-03222-9(pbk.)\nForCarol\n\nCONTENTSPrefacexIntroduction11ZENOANDTHELIMITSOFSPACEANDTIME5Introduction5Divisibilityversusindivisibility6Infinitesimalsandlimits11Thomson’sinfinitesuper-task14Theparalleltaskparadox15Abstractionsandthephysicalworld202CLOCKS,GEOMETRYANDRELATIVITY24Introduction24Mytimeandyourtime33Theparadoxofthetwins:foreveryoung?36Fromtwinstotriplets41Phantomsofperspective443TRAVELLINGLIGHT46Introduction46Measuringthespeedoflight49Absolutesimultaneity?53Slowclocktransport57Spaceliketravel:ataleoftwotachyons60Justthetwoofus:acrosstheuniverse?664ACONVENTIONALWORLD?69Introduction69Whenparallellinesmeet71vii\nCONTENTSWilltherealgeometrypleasestandup?74Conventionandtopology79Dimensions82Thefutureoftheuniverse84TheCosmologicalPrinciple:conventionorfact?86Theunderdeterminationoftheorybydata905NEWTONANDTHEREALITYOFSPACEANDTIME99Introduction99Absolutespaceandtime100MatterintheNewtonianworld103Leibnizandrelationism105Clarke’sdefenceofNewton108Absolutemotionwithoutabsolutespace?1136MACHANDTHEMATERIALWORLD116Introduction116Mach’srelationism118Simplicityandscience120Positivisminaction122Canweseespace?125Experimentandintervention1277EINSTEINANDABSOLUTESPACETIME131Introduction131Mach’sPrinciple133Absolutely,ProfessorEinstein?134Empty,almostempty,androtatingworlds139Relationismandrelativity:anempiricalview?143Theholeargumentandspacetimepoints1468TIMETRAVEL151Introduction151Spacetimestructure154Backtothepast156Forwardtothepast166Correlationsandbackwardscausation1719EINSTEIN’SGREATESTMISTAKE?176Introduction176Spaceandinfinity177Einstein’suniverse180viii\nCONTENTSThecosmologicalconstant:didEinsteinblunder?184Lawsandtheoreticalchange187TheAnthropicPrinciple18910COSMOLOGICALCONUNDRUMS193Introduction193Thebigbang:asingularidea196Thebeginningoftime?199Inflationarycosmology:somethingfornothing?204Blackholes209Cosmiccensorship211Determinismversusindeterminism215CONCLUSION:RELATIVITY—JUSTANOTHERBRICKINTHEWALL?217Introduction217Whatisatheory?218Thestructureandscopeofspacetimetheories221Thelastword?226NOTES229SELECTBIBLIOGRAPHY260INDEX263ix\nPREFACEThisbookpresentsmyreflectionsuponaseriesofproblemsabouttimeandspace.Muchdiscussedherehasalonganddistinguishedheritage.Ihaveeveryreasonforgratitudetobothearlierandpresentgenerationsofscientistsandphilosophersfortheirexplorationandclarificationofourideasaboutspaceandtime:fromSamuelClarke’sdefenceofNewtontoHansReichenbach’sempiricism;fromAristotle’sdiscussionofZenoonmotiontoHughMellor’sthoughtsabouttimeandtimetravel;fromAlbertEinstein’srevolutionarythoughtsaboutmatterandthesourceofinertiatoStephenHawking’sequallystartlingdiscussionofthepropertiesofblackholes.Ialsohaveeveryreasontothankthosewhohelped,invariousways,withthisbook,readingpartorallofthevariousdraftsordiscussingtheideasinvolved,andgivingsomanyvaluablesuggestions,andsteeringmeawayfromerrortoooftenformetohaveanythingelsebutamarkedsenseofmyfallibility.IamespeciallygratefultoHarveyBrownandBillNewton-SmithofOxfordUniversity,MartheChandleratDePauwUniversity,CarlHoeferofStanfordUniversity,AlexanderRuegerattheUniversityofOregonandRobertWeingardofRutgersUniversity.IamgratefultoofortheassistancegiventomebyRichardStonemanandtheeditorialstaffatRoutledge,andfortheirpatience.AndCarolRay,readingthemanuscriptasanon-specialist,didmorethananyonetohelpmetoclarifythoseideaswhichwereexpressedtooclumsilyortootechnically.Sothemeritsofthisbookderiveinpartfromtheendeavoursofothers;butthedefectsyoumustblameonme.Someofthematerialinthisbookisbasedonarticlespublishedinjournals,withrevisionswhereappropriate,andIamgratefultotheeditorsofthejournalsinvolvedforallowingmetousethismaterialhere.ThecentralpartofChapter1appearsas‘Paradoxicaltasks’inAnalysis502(1990);thelastsectionofChapter3isbasedon‘Canwetravelfasterthanlight?’inAnalysis421(1982);thefinalsectionofChapter7isbasedonpartofareview,writtentogetherwithCarlHoefer,ofJohnEarman’sWorldEnoughandSpace-Time,intheBritishJournalforthePhilosophyofScience423(1991);andmuchofChapter9appearedin‘Thex\nPREFACEcosmologicalconstant:Einstein’sgreatestmistake?’StudiesintheHistoryandPhilosophyofScience214(1990).MythanksmustgoaswelltoMrP,neverfarfromanycentreofactivity,forhisslumberingandIhopeappreciativefelinereflectionsonmyendeavours.ChristopherRayPortland,Oregon,USAxi\n\nINTRODUCTIONUnderthestartlingheadlines‘Revolutioninscience:NewtheoryoftheUniverse:Newtonianideasoverthrown’,theNewYorkTimesreported,in1919,theeffectsofSirArthurEddington’sdramaticconfirmationofEinstein’sGeneralTheoryofRelativityanditspredictionthatalightrayfromadistantstarwould‘bend’inthecurvedspaceclosetotheSun:YesterdayafternoonintheroomsoftheRoyalSociety,atajointsessionoftheRoyalandAstronomicalSocieties,theresultsobtainedbyBritishobserversofthetotalsolareclipseofMay29werediscussed.Thegreatestpossibleinteresthadbeenarousedinscientificcirclesbythehopethatrivaltheoriesofafundamentalphysicalproblemwouldbeputtothetest,andtherewasaverylargeattendanceofastronomersandphysicists.Itwasgenerallyacceptedthattheobservationsweredecisiveinverifyingthepredictionofthefamousphysicist,Einstein,statedbythePresidentoftheRoyalSocietyasthemostremarkablescientificeventsincethediscoveryoftheplanetNeptune.Buttherewasadifferenceofopinionastowhethersciencehadtofacemerelyanewandunexplainedfact,ortoreckonwithatheorythatwouldcompletelyrevolutionizetheacceptedfundamentalsofphysics.(NewYorkTimes1919)1Laterthatyear,EinsteinwasinvitedtoexplainhisideastotheBritishpublic.Inashortarticle,hepresentedtheessentialfeaturesofhistheory:hetoldthereadersofTheTimesthat‘Inthegeneralisedtheoryofrelativity,thedoctrineofspaceandtime…isnolongeroneoftheabsolutefoundationsofgeneralphysics’(Einstein:28November1919).1\nTIME,SPACEANDPHILOSOPHYOurconceptsofspaceandtime,alreadychallengedbyEinstein’sSpecialTheoryofRelativity,werenowunderfurtherattackfromhisGeneralTheory.FewunderstoodtheimplicationsofEinstein’sworkinthoseearlyyears.Manyfoundithardtobreakfreefromthewell-establishedNewtonianideas.ButmoreandmorethescientificcommunityembracedEinstein’stheories.Someoftheinitialimplicationsofboththeorieswerehardtoswallow:theideathattimeisnotanabsoluteframework;andthepossibilityofanon-Euclideanuniverseinwhichthethreeinternalanglesofatriangledonotaddupto180degrees.EvenEinsteinfoundsomeoftheimplicationshardtostomach:hisequationswereconsistentwiththepossibilityofanexpandinguniverse—apossibilitywhichheinitiallyrejectedinamovewhichhecametoregardashisgreatestmistake.Andmoresurprisesweretocomeasthetheoriesweredevelopedfurther:thebigbang,timetravel,andblackholesallseemedtobeconsistentwiththeideasofrelativitytheory.Inthisbook,weshallexploresomeofthemajorideasandproblemsbehindourviewsofspaceandtime.MostofthecentralquestionsaboutspaceandtimearisefromconsideringtheideasofscientistssuchasIsaacNewton,ErnstMach,AlbertEinstein,andStephenHawking.Sowemustconsidertheessentialfeaturesoftheworkonspaceandtimebysuchscientistsasthese:fromspeculationsabouthowmanydimensionsspacemighthavetotheproblemofinfinitesimals;fromquestionsaboutwhetherspaceandtimeareinfinitetoworriesaboutthescientificstatusofentitieswhichcannotbeseen;fromtheideasofblackholesandthebigbangtoconjecturesabouttimetravel.Weshallthenbeinabetterpositiontounderstandthephilosophicalissuesconnectedwithalltheseproblems.InChapter1,weshalllookatthefiveparadoxespresentedbytheearlyGreekphilosopherZeno.Hisworriesaboutthewayweregardspace,time,andmotionhaveaclearmessageforthewaywethinkofgeometryanditsapplicabilitytothephysicalworld.TheproblemsofgeometryarepursuedfurtherinChapters2to4.Firstweshalldiscussthecelebratedparadoxofthetwinsandintroducethelesswell-knownparadoxofthetriplets;weshalltheninvestigatetheimportanceofthespeedoflightinrelativitytheory,asking,amongstotherquestions,whathappenswhenwerelaxthegenerallyheldconventionthatnothingtravelsfasterthanlight;andthenweshallfocusonthegeneral2\nINTRODUCTIOINimplicationsofrelativity’scommitmenttonon-Euclideangeometries.InChapters5,6,and7,weshalllookcloselyatthequestionofabsoluteandrelationalspaceandtime,firstthroughtheargumentsofNewtonandLeibniz,andthenthroughtheideasofMachandEinstein.WeshallseethattheproblemsidentifiedbyNewtonmayberaisedinbothNewtonianandrelativisticcontexts.Chapter8focusesontheproblemsandpossibilitiesoftimetravel.Weshalldiscussseveralwaysinwhichtimetravelmightbepossible;butweshallfindthatsomeofthemmayinvolvelogicalcontradictionsormayrequireratherpeculiarviewsofthephysicalworld.TheproblemsofclassicalandmodernideasofcosmologyareaddressedinChapters9and10.Particularattentionisgiventothecosmologicalconstant—theideadismissedbyEinsteinasablunder.Butweshallalsoreviewproblemsconnectedwithblackholesandthebigbang.Thefinalchapterpresentsanoverallimpressionofthestatusofclaimsaboutspace,time,andmotion:howmuchshouldwebelieveofthestoriestoldtousbyphysicistswhentheyseemtochangetheirmindssooften?ThroughoutIhaveaimedtodrawabalancebetweenexplainingthephysicsandexaminingthephilosophicalassumptions,arguments,andperspectivesinvolvedinthevariousphysicalaccountsahead.Ihavetriedtokeeptechnicaldetailstoaminimum,butsometimestheproblemswhichwemeetcannotbegraspedwithoutatleastsomeappreciationofthemathematicalandgeometricalideasinvolved.Wherepossible,Ihaveuseddiagramstohelpthereadervisualisethesituationsbeingdiscussed.Inwritingthisbook,Ihavetriedtoprovideacomprehensive,up-to-date,andaccessibleintroductiontothephilosophyofspaceandtime,tohelpthosewithoutspecialistbackgroundsinthephysicsofspaceandtimebegintounderstand(andnotjustbedazzledby)someofthefundamentalissuesarisingfromclassicalandmodernideasofspaceandtime—issueswhichwillalsointroducethereadertophilosophicalproblemsinmetaphysics,thetheoryofknowledge,thephilosophyofreligion,andthephilosophyofscience.However,Ihopethatmanyreaderswillregardthisbookasastarting-pointforfurtherstudiesinthephilosophyofspaceandtime.Soaselectbibliographyreviewsthemostimportantandhelpfulliteratureinthefield.Anddetailednotestoeachchapteramplifythetext,suggestfurtherreading,andpointthosewishingtoengageinfurtherresearchintherightdirection.3\nTIME,SPACEANDPHILOSOPHYTheideasofspaceandtimeprovideuswitharichandrewardingfieldofstudy.ThechallengewhichfacedNewtonandEinsteinmaybesharedbyeveryone.Wemaynothavetheirgenius,butwecansharetheirinsights.Andtheseinsightscangiveusabetterappreciationoftheroleofphilosophyasitmeetstheproblemsofscience.4\n1ZENOANDTHELIMITSOFSPACEANDTIMEINTRODUCTIONWetypicallythinkofspaceandtimeasthreedimensionsplusone.Mathematicianstellusthateachdimensionmaybecontinuouslysub-divided.Buttheyalsotellusthatwemayconstructmodeluniverseswithratherdifferentproperties.Wemayhaveotherstructureswhichmaynotbecontinuouslysub-divided.And,tocomplicatematters,wemayconstructworldswithwhateverdimensionalityweplease.So,canwereallychop‘real’spaceandtimeupassmallaswelike?Thepre-SocraticphilosopherZenoofElea—aGreeksettlementinSouthernItaly—issaidtoberesponsibleforfive‘paradoxes’whichwrestlewiththepropertiesofspace,time,andmotion.ThemainfocusofZeno’sparadoxesisthe‘small-scale’characterofspaceandtime.Isthissmall-scalestructurereallycontinuous,orisit‘indivisiblyatomistic’or‘discrete’insomesense?Ifthreedimensionalspaceisacontinuum,thenwemaycontinuouslyandindefinitelysub-divideitsparts.Butifspaceortimearediscreteinsomeway,thenanyprocessofsub-divisionwillhaveadefinitelimit.AristotlegivesabriefandperhapsincompleteaccountofthefirstfourparadoxesinhisPhysics,andSimpliciusdiscussesthefifthinhiscommentaryonAristotle.1Zenoisthoughttohaveproducedhisideasaround460BC.WeshallreviewZeno’sdiscussion,andweshallfindthattheseparadoxesdoidentifysomerealdifficultiesforour‘continuum’viewofspaceandtime.ManymathematiciansandphilosophersbelievethatathoroughacquaintancewiththemathematicsofthecontinuumshouldbesufficienttodispelanyworriesthatmightarisefromZeno’sparadoxes.5\nTIME,SPACEANDPHILOSOPHYButtheproblemsraisedbyZenoliveonandsomewriters,includingthephilosopherWesleySalmonandthetheoreticalphysicistRogerPenrose,adviseagainstanyuncriticalandcompleteacceptanceoftheroleofthecontinuuminourphysicaltheories.2Arelatedproblem,suggestedbyJamesThomsonin1954,concernstheparadoxicalnatureofanysuper-taskconsistingofaninfinitenumberoftasks.Ishallarguethatthisproblemisgenuinelyparadoxical,onthemathematicians’ownterms.ButIshallnotjoinZenoinrejectingtherealityofacomplex,diverseworld.Ishallmerelyquestiontheextenttowhichmathematicsandgeometrymayserveasanadequatemodelforthephysicalworld.Imaginethatwehavetwotheoriesaboutthewayobjectsmoveintheworld.Onetheoryassumesthatspaceandtimemaybecontinuouslysub-divided.Theotherdeniesthis.Butalsoimaginethatboththeoriesareperfectlyconsistentwitheverymeasurementandobservationwecanpossiblymake.Ifwecanactuallyconstructsuchanempiricallyimpeccablerivaltothe‘continuum’theory,thenwemightbegintowonderaboutthestatusofthecontinuum.Wemaybewillingtoadmitthatitgivesusanextremelyusefulwayoforganisingourexperience.Butshouldwebelievethattheworldisreallylikethat?Theadvantageofmathematicsisthatithelpsustothinkclearlyaboutthosestructureswhichwebelievetobetheactualstructuresoftheworld;buttheproblemwithmathematicsisthatitallowsustogenerateallsortsofweirdandwonderfulpossiblestructuresfortheworld.Thejobofsortingoutwhich,ifany,weshouldacceptasthe‘real’pictureislefttothephysicist.Andsometimesthechoiceisfarfromstraightforward.DIVISIBILITYVERSUSINDIVISIBILITYZeno’sparadoxesofspace,time,andmotionattacktheveryideaofthedivisibilityofspaceandtime.Webeginbyimaginingadistanceoratemporaldurationwhichisdividedbytwo;andweimaginethattheprocessofdivisioniscontinued.Whymaywenotimaginethattheprocesscouldcontinueindefinitely?Zenotellsusthatanyassumptionthattheprocesscouldgoonindefinitelywillleadusintologicalcontradictions.Buthealsoarguesthatanyassumptionthattheprocesshassomedefinitelimitalsoleadsusintojustasmuchtrouble.Thefirstfourparadoxesrevealthedilemma:6\nZENOANDTHELIMITSOFSPACEANDTIME1AchillesandthetortoiseZenoasksustoimaginearacebetweenAchillesandatortoiseinwhichthetortoiseisallowedtostartfirst.Afteranagreedtime,Achillessetsoffinpursuit.Althoughitseemsentirelyobviousthattheraceisamis-matchandthatAchilleswillalltoosoonovertakethetortoise,Zenoraisesadoubtinourminds.ForinordertoovertakethetortoiseAchillesmustfirstreachthepointwherethetortoisewaswhenAchilleswasgiventhesignaltostartinpursuit.LetuscallthisfirstpointP.ButwhenhereachespointP,thetortoisewillnowbealittlefurtheronatpointQ.AchillesnowmustreachQifheistocatchthetortoise.YetwhenhearrivesatQthetortoiseisstillaheadatR.WhenAchillesgetstoR,thetortoisehasreachedS.Theracecontinuesjustlikethis:everytimeAchillesreachesthetortoise’slast‘staging-post’thetortoisehasmovedfurtherontoanewpost.Ofcourse,thedistancebetweenthetwogetsshorterandshorterallthetime.ButAchillesisalwaysbehind!SodespitefirstappearancesAchillescannotevencatchletaloneovertakethetortoise.2Theracecourse(ordichotomyparadox)HereZenonotonlyarguesthatanathletewouldneverfinish,say,a100-metrerace,italsoseemsthattheathletecouldnotevengetstarted!Toreachtheendofthetrack,theathletewouldfirsthavetoreachthe50-metrepoint.Havingrun50metres,theathletewouldnowhavetoreachthehalf-waypointbetweenthe50-metrepointandthefinishline.Thatwouldtaketheathletetothe75-metremark.Butnowtheathletewouldhavetoreachthehalf-waypointbetweenthismarkandthefinish.Nomatterhowfartheathletegetsdownthetrack,therewouldalwaysbeyetanother‘half-way’pointtoreachbetweenthepointwheretheathleteisandthefinishingline.Sotheathletewouldgetcloserandclosertotheendofthetrack,butwouldneveractuallyreachthefinish.Fortherewouldbeaninfinitenumberofhalf-waypointsaheadoftheathlete.Thismightseembad,butanassociatedargumentimpliesthattheracewouldnotevenbegin.Fortoreachthefinishinglinedemandsthattheathletewouldfirstneedtoreachthe50-metremark;andtoreachthe50-metremarkdemandsthattheathletewouldalreadyhavereachedthe25-metrepoint;andtoreachthatpointwouldrequirethatathletetohavegottothe12.5-metremark;andsoon.Aswekeepdividingthedistancebytwo,7\nTIME,SPACEANDPHILOSOPHYwegetclosertothestartingline,butweneveractuallyreachit.Andwemaydividethesedistancesaninfinitenumberoftimes.Sotoreachtheendofthetracktherewouldbeaninfinitenumberofdistancestorunthrough.Indeed,nomatterhowshortthetrack,therewouldalwaysbeaninfinitenumberofdistancesahead.Theathletewouldbestuckatthestart.Togoanydistanceatall,theathletewouldhavetorunthroughaninfinitenumberofdistances—andhowcouldthatbepossible?3ThearrowTakeahigh-speedphotographofanarrowinflightandyoumayfindithardtodisagreewithZeno’sassertionthatsuchanarrowoccupiesexactlythatspacewhichisequaltoitsownshapeandsize.Weseemtohavecapturedthearrowataninstantoftime.Atsuchaninstantthearrowismotionless.Ifitwerenotmotionless,theinstantoftimecouldbesub-divided:nowthearrowishere,nowthere.Yettheentireflightofthearrowcouldbecapturedinaseriesofinstantaneousphotographs.Ateveryinstant,thearrowismotionless.Thereisnotimebetweentheinstantsforthearrowtomoveontothenextinstant.Forsuchatimewouldbecomposedofinstantsitself.Sohowcananalwaysmotionlessobjectmove?4Themovingrows(orthestadium)Imagineastadiuminwhichacolumnofsoldierspassesacolumnofsoldiersatattentionsothateachstepbringseverysoldierinthemovingcolumnintolinewiththenextcomradeinthestationarycolumn;athirdcolumnofsoldiersisalsomoving,butintheoppositedirection,sothatwitheachstepthesoldiersherealsoarebroughtintolinewiththenextcomradealonginthestationarycolumn;seeFigure1(p.9).Witheachstep,eachsoldierineachmovingcolumnencountersonecomradeinthestationarycolumnbuttwocomradesintheoppositelymovingcolumn.Nowimaginethateachsoldierrepresentsanindivisibleminimumunitoflengthandthateachsteprepresentsanindivisibleminimumunitoftime.Surelywecanaskthequestion:atwhatinstantandinwhatpositiondidthetwomovingcolumnsalignsothateachsoldierwasalongsidethenext(ratherthanthenext-but-one)soldierintheadjacentmovingcolumn?Ifwecansub-dividethetimeforthestepandthespacebetweenstepsthereisnoproblematall.Fortheywillmeetafterhalfastep.Butwehavesupposedthatthereisnosuchthingashalfofoneofourunitsoflengthortime—sincetheyare8\nMovingrowsparadox.Tworows(XandZ)movebyastationaryrow(Y)asshown.Inthetopdiagram,X1andZlareinadjacentcolumns,X1totheleftandZ1totheright.Aninstantlater,X1andZ1haveshiftedtheirpositions,sothattheyarestillinadjacentcolumnsbutwithX1nowtotherightofZ1asshowninthelowerdiagram.Zeno’sproblemisthis:whenandwherewereX1andZ1inalignmentvertically?Giventhatthechangeofpositiontookplaceintheshortestpossibletime,wecannotsaythattheywereinlineinhalfthistime.And,becausethechangeofpositioninvolvestheshortestpossibledistance,wecannotsaythattheywereinlinewhentheyhadmovedthroughhalfthisdistance.Figure1Zeno’smovingrowsorstadiumparadoxindivisibleminima.Soeitherthequestionisunreasonable(andwhyshouldthisbe?)orwearewrongtosupposethatspaceandtimeconsistinindivisibleminima.Inthefirsttwoparadoxes,Zenotriestoillustratetheabsurdityofbelievingthatalinemaybedividedupintoprogressivelysmallerchunksadinfinitum.Andthereissomethingseductiveinhisargument.ForhowcanImovefromAtoBwhenIfirstmustmovetosomepointinbetween?AndwhateverpointIchooseandnomatterhowmanytimesIdothis,thereisalwaysgoingtobeyetanotherpointinbetween.ZenowarnsusagainstsayingthatsoonerorlaterImustreachthesmallestpossible‘indivisible’distance.Forthisdiscreteviewofspacetoowillgenerate9\nTIME,SPACEANDPHILOSOPHYproblems,asdemonstratedbythefourthparadox.SomewritersapproachZeno’sparadoxeswithconfidence,sayingthatjustalittlemoderncalculuswillbesufficienttodispelanyworrieswhichtheparadoxesmayproduce.3IanStewartidentifiesthecentralissueinZenoasthewaywethinkofinfinitesimalquantities;andsaysthatonlyinthelasthundredandfiftyyearsorsohavewebeguntoseetheprobleminawaythathelpsustoresolvetheparadoxeswithouttoomanyqualms.Stewartasks:Canalinebethoughtofasasequenceofpoints?Canaplanebeslicedupintoparallellines?Themodernviewis‘yes’,theverdictofhistoryanoverwhelming‘no’;themainreasonbeingthattheinterpretationofthequestionhaschanged.(Stewart1987:66)4Mathematiciansnowseemtohavefewworriesaboutcontinuoussub-divisions.Whathaschangedistheirattitudetowardsinfinitesimalquantities.Suchquantitiesarenotregardedasextensionlesspointsinspaceorintime.Ifweregardpointsashavingnoextension,thenwefallvictimtoZeno’sfifthparadox:thatofplurality—saidbyG.E.L.OwenandotherstobeZeno’sprimaryconcernandtounderlietheotherfourparadoxes.5IndeedOwenarguesthatweshouldregardtheparadoxesasprovidingacoordinatedattackontherealityofspace,time,andmotion.Thefirsttwoparadoxeschallengetheideathatspaceandtimecanbecontinuouslysub-dividedandthesecondtwoattackthenotionthatthereareindivisibleminimaofspaceandtime;sothatZeno’soveralljudgementmaybesummarisedthus:‘nomethodofdividinganythingintospatialortemporalpartscanbedescribedwithoutabsurdity.’6Thefifthparadoxdiscouragesusfromregardingtheendresultofsomecontinuoussub-divisionaseitheranextensionlessquantitylikeapointoraquantitywithsomedefiniteifminuteextension:5TheparadoxofpluralityZeno,accordingtoSimplicius,askshowevenaninfinitenumberofextensionlessdistancescouldadduptoafinitedistanceandhowanextendedbodycanconsistofaninfinitenumberofparts(geometricalpoints?)whichthemselveshavenoextension;suchadistanceorsuchabodymustbeinfinitelysmall—i.e.itmustbejustlikeitsconstituentparts:extensionless.7Yetifweallow10\nZENOANDTHELIMITSOFSPACEANDTIMEtheseconstituentpartstohavesomefinitesize—howeversmall—thenthebodymustbeinfiniteinsize.8Owenpointsoutthatthisparadox,takentogetherwiththefirstfour,maybeseenasprovidingreasonsforZeno’sviewoftheworldasasingleglobalentityratherthanasmadeupofparts,whethertheseareindivisiblysmallorcontinuouslydivisible.Assoonaswestarttosub-dividewerunintodifficulties.Sothesensiblethingtodoistoresistthetemptationtodividetheworldupatall!Zeno’sworldisasinglebodywhichmaynotbesub-dividedinanywaywithoutabsurdity.INFINITESIMALSANDLIMITSMustweacceptZeno’sconclusions?Theanswerseemstolieinourattitudetowardsthe‘end’resultofanunendingprocessofsub-division,totheideaofinfinitesimals.Itisamistaketoregardthemashavingsome‘constant’valuewhetherthisbethe‘zero’ofextensionlessobjectsorpoints,orwhetheritisthenon-zerovalueoftheshortestpossibledistanceortime.InbothcaseswewouldfallstraightintooneorotherofZeno’straps.Weneedadifferentapproachifwearetoavoidthetrapsaltogether.ThewayoutwasfirstsuggestedbytheFrenchmathematicianCauchyin1821:heintroducedtheideaofalimit;andthenotionoftheinfinitesimalwasabsorbedintothismorecoherentconcept.9And,somethirtyyearslater,Weierstrassshowedthatwecouldmovethedebatefromtherealmofgeometrytothatofarithmetic,fromideasofspatialandtemporaldistancestothoseoffunctions.Insteadoftalkingaboutever-decreasingdistancesalongastraightline,wecouldtalkwithalittlemorerigouraboutinfiniteseriesconvergingonlimitingvaluesintermsoffunctionsandrealnumbers.Theproblemmaybehighlightedbyconsideringhowweshouldanswerthisquestion:whatspeeddoestheathletehaveatanygiveninstant?Ifwethinkintermsofinfinitesimalswitha‘zero’value,thentheequationforthespeedofanobject(distance÷time)collapsesintononsense—thespeedofanymovingobjectconsideredinthiswaywillalwaysbezerodividedbyzero!So,insteadofsayingthatwemaydescribethemotionoftheathletebyreferencetoinfinitesimaldistancesandtimes,weshouldcalculatethespeedoftheathleteatanyinstantintermsofhowtheobjectismovingintheimmediateneighbourhood,asshownbythe11\nAlthoughAchillesstartstheraceafterthetortoise,becausehisspeedisgreaterthanthatofthetortoise,heovertakesthetortoiseatthepointshown.ThespeedofAchilles(distance÷time)ratherthanthedecreasingdistancebetweenthetwoisthekeytotheproblem.Figure2Distance-timegraphcomparingAchilleswithtortoise:ideaofvelocitymathematicalfunctiondescribingtheathlete’smotion.Byconsideringsmallerandsmallerneighbourhoods,wetypicallyreachalimitingvalueforthefunction—the‘instantaneous’speed.Wegetouranswerbyconsideringwhathappensasweapproachtheinstant,notbyaskingwhatishappeningattheinstant.Similarly,weconsiderwhetherornotAchillesovertakesthetortoiseandwhetherornottheathletemayrunfromAtoBbythinkingintermsofwhathappensasAchillesapproachesthetortoiseandastheathleteapproachestheendoftheracecourse;seeFigure2(above).So,usingtheseideas,wemaygivethefollowingprovisionalresponsestoZeno’sworriesaboutacontinuumwhichmaybecontinuouslysub-divided:1ThefunctionsdescribingAchilles’andthetortoise’smotionsshowthat,whenAchillesisintheimmediateneighbourhoodofthetortoise,Achilles’speedisgreaterthanthatofthetortoiseandhethereforeovertakesit.102WhenZeno’sathleteattemptstorunfromAtoB,theathlete12\nZENOANDTHELIMITSOFSPACEANDTIMEwillindeedneedtocoverhalfthedistance,thenafurtherquarter,andsoon;butthefunctiondescribingtheseriesofdistancesrunbytheathleteconvergesuponanaturallimit:thetotaldistanceAB.3Theideaofaninstantaneous‘snapshot’ofamovingobject(e.g.anarrow)doesnotbyitselfcarrywithitanyideaofmotion;onlywhenweconsideritsimmediatespatio-temporalneighbourhoodmaywegiveanysensetotheideaofamovingarrow—butoncewedothistheideaofmotionisquitecoherent(manyactionphotographsshowmovingobjectsagainstablurredbackgroundtoconveytheideaofspeed—suchpictureswouldcapturethearrownotataninstant,butoverashortperiodofitsflight).Butwestillfacesomeimportantquestions:1Areweentitledtosaythataconverginginfiniteserieshasasumgiventhefactthataninfinitesequencehasalimitingvalue?Foralthoughwemayagreethatthelimitingvalueofthesequenceofpartialsums:is1,wedonottherebyhavesufficientreasontosaythattheseries:has1asitssum.Wemightaccusemathematiciansoffudgingtheissuewhentheyassureusthatthelimitingvalueofsuchasequenceisalsothesumofarelatedseries.So,althoughwemaysaythatthesequenceofdistancesfromArunbytheathletehasABasitslimitingvalue,thisneednotcommitustotheviewthattheseriesmaybesummedatall!Forsuchasummationseemstoinvolveaninfinitenumberofadditions,andwemightregardsuchanadditionasatbestimplausibleoratworstlogicallyimpossible.2Towhatextentmaywesaythatthemathematicalconceptsemployedaboveapplytothephysicalworld?For,evenifweallowthataninfiniteseriesmayhaveamathematicalsum,thisisnoreasonforustoagreethatwemayapplythisproceduretothephysicalworldwithimpunity.Forexample,wemayusetheideaofasimplearithmeticalsumwhenaddingonequantityofmoneytoanother;but,whenadding13\nTIME,SPACEANDPHILOSOPHYvelocitiesintheSpecialTheoryofRelativity,adifferentprocedureisrequired.Sowemightask:towhatextentaremathematicalandgeometricalconceptsandstructuresstrictlytrueofthephysicalworld?Whyshouldabstractionsapplyliterallytothephysicalworld?ThesequestionsarenowaddressedbyThomson,whochallengesustocontemplateasuper-taskconsistingofaninfinitenumberoftasks.THOMSON’SINFINITESUPER-TASKJamesThomsonasksustoimaginealampwhichmaybeswitchedonandoffaninfinitenumberoftimesinafinitetime.Ifwesetasidethequestionofwhetherornotitisphysicallypossibleforaninfinitenumberofsuchtaskstobeperformedinafinitetime,wemaystillaskwithThomsonwhetherornotitislogicallypossible.WemayeasilyimagineThomson’sreading-lampwithitsswitchintheoffpositionandhimswitchingitonthenoffthenonandsoon.Ifitisswitchedonattimezeroandoffafteroneminuteandonagainafteranother30secondsandoffagainafterafurther15secondsandsoon,thenwemightthinkthataftertwominuteswewouldhavecompletedaninfinitenumberofswitchingoperations.ButThomsonasks:attheendofthetwominutes,isthelamponoroff?…Itcannotbeon,becauseIdidnoteverturnitonwithoutatonceturningitoff.Itcannotbeoff,becauseIdidinthefirstplaceturniton,andthereafterIneverturneditoffwithoutatonceturningiton.(Thomson1954:5)11Andthis,Thomsontellsus,iscontradictory.Heconcludesthatwecouldnotinprinciplecarryoutsuchasuper-task.SainsburysaysinParadoxesthatThomson’sconclusionisunwarranted.12Hebeginsbydistinguishingbetweenthosemomentswhentheswitchingtasksarebeingperformed(theT-series)andthatfirstmomentafterthesuper-taskhasbeencompleted(T*).FollowingBenacerraf’sargumentin‘Tasks,super-tasksandthemoderneleactics’,13Sainsburythenmaintainsthat‘foranymomentintheT-series,ifthelampisonatthattimethereisalatermomentintheseriesatwhichthelampisoff;andviceversa’.14\nZENOANDTHELIMITSOFSPACEANDTIMEHowever,nothingfollowsfromthisaboutwhetherthelampisonoroffatT*,forT*doesnotbelongtotheT-series.TheT-seriesisclosedatoneend(timezero)andopenattheother‘end’.Thismeansthatalthoughwecanidentifyafirsttaskattimezero,wecannotgiveanysensetothenotionofalasttask—theopennessoftheT-seriesguaranteesthepossibilityofthetaskscontinuinganinfinitenumberoftimes.ThetimeT*doesnotoccupyanypointatthisopenendandisthereforeindependentoftheT-series.ButsinceT*isindependentoftheT-series,Sainsburypointsoutthatour‘specificationofthetaskspeaksonlytomembersoftheT-series,andthishasnoconsequences,letalonecontradictoryconsequences,forhowthingsareatT*,whichliesoutsidetheseries’(Sainsbury1988:15).SoSainsburyconcludesthatThomsonfailstodemonstratethattheideaofasuper-taskislogicallyabsurd.TwoclearimplicationsofSainsbury’sargumentare:1thatthelampiseitheronoroffatT*;and2thatwecannotsayeitheratthebeginningofthesuper-taskoronceitisunderwayjusthowthingswillturnoutatT*.Hence,thereisnowaytopredictthestateofthelampatT*,whateveritsstateattheoutset.THEPARALLELTASKPARADOXSupposeweaskanoperatortocarryoutthesuper-tasktwiceinsuccessioninexactlythesamewayeachtime.Thenthereisnoreasontosuppose,onSainsbury’sview,thatthelampwouldbeinthesamestateateachofthemomentsT*afterthetwotasksarecompleted.Otherwisewewouldalwaysbeabletopredictthefinalstateofthelamp.Wemaysharpenthisproblemasfollows.Imaginenowtwolampsandoneoperatorforeachlamp.WeaskbothoperatorstoattemptThomson’ssuper-taskatthesametime.Bothlampsareoffandattimezerobothoperatorsswitchtheirlampson.Afteroneminutetheoperatorsswitchthelampsoff;after30secondsbothlampsareswitchedontogether;after15moresecondsthelampsareoffagain;andsoon.IfwegrantthepointthatT*liesoutsidetheseries,wemayalsograntthatatT*eachlampwillbeeitheronoroff.Butareweforcedtoconcludethatbothlampswillbeinthesamestate?Giventhattheoperatorsbegintogetherandcontinue15\nTIME,SPACEANDPHILOSOPHYtogetherwiththelampsflashingonandoffinunison,wemightexpectthemtofinishtogetherwiththelampsinpreciselythesamestate.However,whathappensduringtheT-seriesis,wearetold,independentofwhatisthecaseatT*.AsSainsburyclaims,nothingconcerningT*followsfromourspecificationofthetasktobecarriedoutbecausethisspecificationrelatesonlytotimeswithintheT-series.Thefactthatthelampsareinitiallyinthesamestateisirrelevant,sincethemomentatthestartofthesuper-tasklieswithintheT-series.Andthefactthattheoperatorscontinuetogetherdoesnothelp,becauseourinstructionstothemrelatesolelytotimeswithintheT-series.SowhyshouldweexpectthelampstobeinthesamestateatT*?Weareleftwiththeunsatisfactoryconclusionthattwopeoplealwaysinstepduringaninfinitesequenceoftasksmaybeoutofstepimmediatelyafterthesequencehas‘ended’.Suchaconclusionseemstoinvolveusinrathermorethananempiricalpuzzleaboutthewaythingswillturnoutwithsuch‘parallel’super-tasks.Thereseemstobearationalifnotalogicalinconsistency.Ifthesuper-tasksruninparallelandinstepatalltimesduringtheseries,thenwehavenoreasonatalltosupposethatthispatterncouldbebrokenwhenthesuper-tasksareover.ByacceptingthedistinctionbetweentheT-seriesandthe(independent)timeT*,thenthereisalsonowareasontosupposethatthepatternmaybebrokenatT*whenthetasksareover.ThetwolampsmaybeindifferentstatesatT*becausethereisnoconnectionbetweenthestatesofthelampsduringtheswitchingoperationsandtheirstatesatT*.Giventhislackofconnection,thechancesofthelampsbeingintheidenticalstatesatT*seemtobethesameasthechancesofthembeingindifferentstatesatT*.Andthisappearstobeanacceptablereasonforbelievingthatthepatternmaybebroken.Hence,acceptanceoftheindependenceofT*fromtheT-seriesleadsustoadirectconflictwithourapparentlyreasonableexpectationthattwo‘parallel’operationsshouldalwaysremaininstepnotjustuptotheendofataskbutalsowhenthetaskisover.Thedifficultyinvolvedinthisproblemseemstoderivefromthefactthatthereisnolimitingbehaviourofthesituationtowhichwemightappealinordertodissolvethepuzzle.WhenZeno’sathleteattemptstorunfromAtoBatauniformspeed—firstpassingthroughthehalf-waypoint,thenthethree-quarter-waypoint,andsoon—themathematicalfunctionwhichdescribestheathlete’sprogresscanbe16\nZENOANDTHELIMITSOFSPACEANDTIMErepresentedonagraphofdistanceagainsttimeasastraightline.ThenaturallimitofthelinemaybeunambiguouslydefinedaspointB.Again,thereisanaturallimitforalightlydampedsimplependulum;inthiscasethegraphofdisplacementfromequilibriumagainsttimeshowsthattheamplitudesoftheoscillationsdecreasetowardsthelimitingvalueofzerodisplacementfromequilibrium;seeFigure3(pp.18–19).Butthelampsystemhasnosuchnaturallimit.Forthereisnopreferredwayforustoextendanymathematicalfunction,whichweemploytodescribethebehaviourofthesystemacrosstheopenendoftheT-seriestoT*itself.Itmightbethoughtthatthislackoflimitingbehaviourpresentsuswitharesolutiontoanyworrywemighthaveconcerningtheparallelsuper-task.Foritseemsthatwenowhaveanexplanationofthefactthattwolampswhichstartouttogethercanendupoutofstep.ThemathematicalfunctiondescribingthesystemsimplyfailstodeterminewhichoftwopossiblestateseachlampwillbeinattimeT*.Andwhyshouldthisbesoproblematic?Therearemany,manysituationsinphysicswherewecannotuniquelydetermineoutcomes.However,theparallelsuper-taskraisesmoredifficultiesthanthatofmereuncertaintyastothefinaloutcome.Indeed,thesourceofanyworrywemighthaveistheconvictionthatbothlampsshouldendupinthesamestate,whateverthatmightbe.Ofcourse,wehaveadegreeofuncertaintyastohowthingswillturnout.Yettheuncertaintyislimitedtothatwhichwewouldhaveaboutonelamp—wejustcannotsaywhetherthefinalstatewillbeonoroff.Nevertheless,asSainsburyremindsus,wedoknowthatitwillbeinoneofthesestates.Butwealsohavecompellingreasonsforbelievingthatthelampswillbeinthesamestate.Ofcourse,itmightbesaidthatthepeculiarcircumstancesofthetaskaresuchastoraisedoubtsaboutwhetherornotthelampswillbeinthesamefinalstate.Perhapsquantumeffectsmightbreakthesymmetry.However,weneedoffernoreasonotherthanconsiderationsofsymmetrybetweenthetwolampstojustifythebeliefthatthelampswillbeinthesamestate.Butwedoneedsomereasontosupposethatthesymmetrymightbebroken.Ifweweretoprovideevenasketchexplanationforthebrokensymmetry,wewouldbesupplyingsomekindoflinkbetweenthelampsinoperationduringtheinfinitetaskandthelampsatT*.ButwehaveseenthattheargumentagainstThomsonproceedsontheassumptionthatthereisnosuchlinkbetweentheT-seriesand17\nTIME,SPACEANDPHILOSOPHYAlightlydampedsimplependulumwillswingtoandfro,withitsamplitude(maximumdistancefromthecentreoftheswing)decreasing.Intheory,theamplitudewillbecomecloserandclosertozerobutthependulumwillneveractuallystop.Hencethependulumwillmovethroughainfinitedistance—butonlyinaninfinitetime.Figure3(a)Graphofinfinitedistanceininfinitetime(pendulum)Thelampisswitchedonandoff—eachswitchingactionhappeningafterjusthalfthetimeofthepreviousaction.The‘infinitesuper-task’shouldbecompleteafter120seconds.Thefingerpushingtheswitchonandoffshouldthereforemovethroughaninfinitedistanceinafinitetime—giventhefactthattheswitchhastobepushedaninfinitenumberoftimes.Figure3(b)Graphofinfinitedistanceinfinitetime(Thomson’slamp)18\nZENOANDTHELIMITSOFSPACEANDTIMEZeno’sathletetravelsfromAtoB—afinitedistance—inafinitetime.Thenaturallimitfortheamplitudeofthesimplependulumiszero.ThenaturallimitfordistancerunbytheathleteisB.Butthereisnosuchnaturallimitfortheswitchonthelamp:itmayonlybe‘on’or‘off’.Figure3(c)Graphoffinitedistanceinfinitetime(Zeno’sathlete)T*,thatthisseriesandT*areindependent.Ifthisindependenceischallenged,thenThomson’soriginalargumentagainstsuchtasksmustbefaced.Ifwewishtodenythegenuinelyparadoxicalnatureofsuper-tasks,thenoneoftwooptionsseemstobeopentous.Wemaybeforcedtochallengeourrationalexpectationsconcerningtheoutcomesofsuchtasks.Orwemaymaintainthatonlythoseparadoxesinvolvinglogicalcontradictionsaregenuineparadoxes.Butthereseemstobenoreasonatalltotakethefirstoption.And,togetherwithSainsbury,wehaveplentyofreasonstoresistthesecondoption—inhisintroduction,aparadoxisdescribedas‘anapparentlyunacceptableconclusionderivedbyapparentlyacceptablereasoningfromapparentlyacceptablepremisses’.14And,despiteitslackofanyoutrightlogicalcontradiction,thesuper-taskcertainlyfallsintothiscategory.SoweshouldfollowThomson’sadviceandruleoutsuchtasksinprinciple.AlthoughthemathematicsofthecontinuummightseemtoprovideuswithapowerfulstrategytoattackThomson’soriginalparadox,wearegiveninsufficientresourcestoresolvealltheramificationsofsuchsuper-tasks.Perhapsthemoralofthistaleisthat,whenweconjureupempiricalfictions,weshouldnotbetoosurprisedwhenourstoriesendunhappily,eveniftheydohaveimpeccablemathematicalcredentials.19\nTIME,SPACEANDPHILOSOPHYABSTRACTIONSANDTHEPHYSICALWORLDWhatlessonsdoesthelampparadoxholdforourviewofZeno’sparadoxes?Unlikethelampoperators,athleteslikeAchillesseemtohaveonlyonetaskahead:torunfromAtoB.Hence,wecannotsaythatthelampparadoxprovidesanydirectsupportforZeno’sattacksonthecontinuum:thetaskfacingAchillesdoesnotseemtobeasuper-task.YetsomeonemightstillwanttosaythatthereisasenseinwhichAchillesiscomplyingwithinstructionssimilartothosegiventothelampoperators:firstgotothehalf-waypoint,thenafurtherquarter,andsoon.ButarewereallyaskingAchillestocarryoutaninfinitenumberoftasksinanysense?Thisjustdoesnotseemtobeso.MaxBlackdrawsourattentiontothecentralissuehere:Achillesisnotcalledupontodothelogicallyimpossible;theillusionthathemustdosoiscreatedbyourfailuretoholdseparatethefinitenumberofrealthingsthattherunnerhastoaccomplishandtheinfiniteseriesofnumbersbywhichwedescribewhatheactuallydoes.Wecreatetheillusionoftheinfinitetasksbythekindofmathematicsthatweusetodescribespace,time,andmotion.(Black1950:101)15Thelinkbetween,thelampparadoxandZeno’sparadoxeslies,notinthenatureofthetasksunderdiscussion,butratherinthesharedconcernsaboutthestatusofthecontinuum.BenacerrafandSainsburyfollowanapproachtoZeno’sworriesaboutthecontinuumwhichfocusesourattentionontheconsistencyofthemathematicsusedtodiscusstheproblem.Themathematicsallowsustothinkconsistentlyintermsoflimits.But,assoonaswetrytoapplythemathematicstothephysicalworld,theproblemsbegin—aswesawinthecaseoftheparallelsupertask.AsSalmonpointsout‘theapplicabilityofmathematicalconceptstospecifictypesofphysicalphenomenaisnotanautomaticconsequenceoftheirconsistency’.16WeimagineAchillesproceedinginregularstridesdownthetrack.Nooneaskstheathletetomovethroughaninfinitenumberofever-shorteningsteps.AnynotionthattheathletemovesthroughaninfinitenumberofdistancesderivesfromthemathematicalpossibilityofsubdividingthetotaldistanceABasmanytimesaswewish(andmoreadinfinitum).Soanyclaimthattheathleteisrunningthroughaninfinite20\nZENOANDTHELIMITSOFSPACEANDTIMEnumberofphysicaldistancesreliesonthequestionableassumptionthatwhatseemstobetrueofanabstractmathematicaldomainshouldalsoapplystrictlytothephysicalworld.Inthemathematicaldomain,wemayreadilyimagineaseriesclosedatoneendandopenattheotheroralengthdivisiblecontinuouslyadinfinitum;butinthephysicalworldlengthsaremeasuredbyrulersandothersuchinstruments—andwealwaysreadoffadiscretefiniteamount,arationalnumberforanydistancewemeasure.CouldInotmakesmallerandsmallermeasurementsatleastinprincipleifnotinpractice?Thekeyphrasehereis‘inprinciple’—usingitrunstogethermathematicalandphysicalpossibilities.WhatIcandoinpracticeisconstrainedbywhatisphysicallypossible.AlthoughIcannotalwaysdecidewhetherornotagiventaskorsituationisphysicallypossibleandIcanalsoacknowledgethatadvancesinscienceandtechnologymayopenupmorepossibilities,Igenerallydoknowwhensomethingispossibleandwhensomethingisnot.Theremaybenohard-and-fastdistinction.Butdistinctiontherecertainlyis.Ofcoursewemayplaymathematicalgameshere,andimaginewhatitwouldbelikeif…;but,inthephysicalworld,itispracticeratherthanprinciplewhichdictatespossibilities.Theproblemwithtakingtoovigorousanempiricallinehereisobvious:wedousemathematicsinourphysicaltheorieswithatremendousdegreeofsuccess.Butevenanenthusiasticacknowledgementoftheutilityofmathematicsshouldnotcarrywithitthepresumptionthatbecauseitworksitmuststrictlyapplytothephysicalworld.BothmathematicalabstractionandphysicalobservationpointustowardsthesameanswerstoZeno:Achillesovertakesthetortoise,theathletereachesthegoal,andthearrowisinflight.Mathematicsprovidesuswiththeideaofalimitingvalueinanattempttocapturewhatishappeningphysically.Here,aselsewhere,mathematicsandgeometryprovideuswiththematerialsformodelsofthephysicalworld.Butmodelsareabstractanalogiesandnottheworlditself.Thespatio-temporalcontinuumisonesuchmodel.And,aswitheveryanalogy,therearepointsofdifferenceaswellaspointsofsimilarity.17Justasthelampsupertasksshouldmakeusworryabouttheapplicationofmathematicstothephysicalworld,thelessonforusprovidedbyZeno’sparadoxesisthatweshouldnotbetooquicktostressthesimilaritiesbetweenthecontinuummodelandtheobservedworldattheexpenseofwhatmaybecrucialdifferences.Butweshouldalsobewarethetemptationtosubstituteanindivisiblespaceandtimemodelforthecontinuum.Evenifthemathematicsandgeometryusedinsuchamodelisconsistentand21\nTIME,SPACEANDPHILOSOPHYpowerfulenoughtoescapeZeno’sfourthparadox—theattackon‘atomism’—thequestionofstrictapplicabilitystillremains.Aninterestingargument,discussedbyW.H.Newton-Smith,highlightsthisissueandshouldprovideafurtherreasonforustoavoidtreatingthecontinuumwithtoomuchrespect.18TheargumentrecallstheessentialfeatureoftheDuhem-Quinethesis:that‘physicaltheoriescanbeatoddswitheachotherandyetcompatiblewithallphysicaldataeveninthebroadestsense’—i.e.thatallpossibleempiricalevidencemaynotbesufficienttocommitustoanyonetheoreticalview.19Wehavealreadynotedthatthemeasurementswhichwemakearealwaysrationalnumbers—numberswhichmaybeformedbytheratioofintegersasfractions.Nomatterwhatdistancewemeasurealong,say,ametrerule,thatdistancewillbeadefinitefractionoftheentirelength.Butanirrationalnumbersuchasv2cannotberepresentedbyanyfraction.Andnopossiblemeasurementcoulddeliveranirrationalnumber.YetthestructureofthecontinuumusedinNewtonian(andrelativistic)physicsassumesthatthedistancebetweenanytwopointsinspaceortimemayberepresentedbyanorderingofrealnumbers—rationalandirrational.Newton-SmithsuggestsanalternativetheorytoNewton’smechanics(hecallsthisNotwen’stheory)whichcouldbebasedonanorderingoftherationalnumbersalone.Hence,Notwen’stheorywouldeschewthe‘smooth’continuumstructureinvolvingbothrationalandirrationalnumbersforspaceandtime.Ofcourse,Notwen’stheorycouldnotusethedifferentialequationsofthecalculustoanalysemotionsincetheseequationsassumeasmoothcontinuum.However,itcouldusedifferenceequations—rathermorecumbersome,butnoimplicitreferencetoirrationalnumberswouldbeneeded.Notwen’stheorywouldcertainlynotbeelegantnorwoulditbeeasytouse.Butitwouldbeconsistentwithallpossibleempiricalevidence!Foreverymeasurementwemakeisintermsofrationals.ThedatagiveninobservationandexperimentarenotenoughtodetermineourchoiceoftheorydespitetheunattractivenessofNotwen’stheory.Thecomplexitiesofsuchatheory,apparentevenwhenweconsiderone-dimensionalorderings,wouldincreasedramaticallyinthecontextoftwo-,letalonethree-orfour-dimensionalstructures—indeed,itisfarfromclearhowwewouldcarrythebasicideaofarationalorderingfromonetothefamiliarcontextofthreespatialdimensions.SowemightstillchoosetouseNewtoniantheory.However,theselectionwouldbemadeonpragmaticgrounds,andnotbecausethemathematicalandgeometricalconceptsemployedindicatesomething22\nZENOANDTHELIMITSOFSPACEANDTIMEaboutthe‘true’natureofthephysicalworld.But,evenifwecommitourselvestotheideaofthecontinuumasthebasisofourphysicaltheoriesofspace,time,andmotion,wemuststilldecideontheappropriate‘global’geometry.For,asweshallseeinChapter3,wenolongerhaveastraightforward‘choice’betweenEuclidandnothing.And,onceagain,weshallfacetheproblemthattheempiricalevidenceisnotalwayssufficienttoforceanyparticulartheoreticalviewoftheworlduponus.23\n2CLOCKS,GEOMETRYANDRELATIVITYINTRODUCTIONInJune1905,Einsteinpublishedapaperentitled‘Ontheelectrodynamicsofmovingbodies’inwhichhereflectsupondifficultiesarisingfromMaxwell’selectrodynamicsandtheNewtonianideasofspace,time,andmotion.InthisshortpapertheessentialideasoftheSpecialTheoryofRelativity(STR)aresetdown.Theseideas,Einsteintellsus,havea‘peculiarconsequence’:IfatthepointsAandBof[agiveninertialframeofreference1]Ktherearelocatedclocksatrestwhich,observedinasystematrest,aresynchronized,andiftheclockatAistransportedtoBalongtheconnectinglinewithvelocityv,thenuponarrivalofthisclockatBthetwoclockswillnolongerbesynchronized;instead,theclockthathasbeentransportedfromAtoBwilllag…behindtheclockthathasbeenatBfromtheoutset.(Einstein1905:153)2Einstein’spredictionhereallowsustosaythis:ifsomeoneweretoleavetheEarthinafast-movingspaceship,thenonthatperson’sreturn,saytenyearsfromnowinEarthterms,anyclockcarriedintheshipwouldhavetickedofffewersecondsthananyclockonEarth.And,sinceapersonis,inanimportantsense,nomorethanabiologicalclock,thespacetravellertoowouldhaveagedlessthanthosewhostayedbehind.STRalsoallowsustosaythatthefasterthetravellermovesintheround-trip,thegreatertheagedifferencebetweenthetravellerandthosewhostayedbehind.Somepeoplehaveregardedsuchpredictionsas‘paradoxical’,findingitatleastcounterintuitiveifnotdownrightcontradictorythatpeople24\nCLOCKS,GEOMETRYANDRELATIVITYinhabitingthesameuniverseshouldnotallageatthesamerate.Butthis‘paradox’ofSTRseemstorundeeperthananysimpleconflictwithoureverydayintuitions.ForSTRpredictsthat,justasIwouldsaythatyourclockrunsslowwhenyouareinrelativemotionwithrespecttome,youtoowouldsaythesameofmyclock,thatitrunsslowandyourownclockistickingawayasusual—sowhichofusisright?Anystraightforwardassertionthat‘movingclocksrunslow’isclearlygoingtobeproblematic,foryouandIwillbeindisagreementaboutwhichclockismovingandwhichclockisrunningslow.SinceallinertialframeshavethesamestatusinSTR,wecannotsay:‘myframeofreferenceisamoreappropriatestandardforclocksthanyourframe,somyclockwillalwaysbehavecorrectly,andyourclock,solongasitismovingwithrespecttome,willberunningslow’.ForSTRblocksmygivinganyframesuchpreferentialstatus.Yet,despitethesymmetryofourrespectivesituations,framesanddescriptions,STRalsopredictsthataclockreturningfroma‘round-trip’willindeedhavetickedofffewersecondsthantheclockthatstaysbehind—thatinsuchasituationonlyoneofuswillberightinsayingthattheotherclockistickingmoreslowlythanourownclock.Andthisapparentclashofpredictionsdoesindeedseemtobepuzzling.InApril1922,theCollègedeFrancedebatedtheimplicationsofSTR.AndEinstein’spredictionsaboutclocksinthepaperofJune1905causedtheparticipantsgreatconcern.TheFrenchphilosopher,HenriBergson,respondingtotheCollègedebate,saidthat,ifweapplythepredictionsofSTRtotherealworld,thenweshallindeed‘endinabsurditiesofparadoxes’.3Hearguesthatwhenaspacetravelleroraclockmovesawayfromusathighspeedthemovingobjectsarenomorethanphantomsofthephysicist’simagination.ThemathematicalcalculationsofSTRapplyonlytosuchphantomimages.Whentherealtravellerandclocksreturn,weshallseenodifferencebetweenthemandthepeopleandclocksleftonEarth.ButBergsonrevealshisprejudicesabouttimewhenheassertsthattheremaybea‘multiplicityofimaginarytimes’inSTRbutthereisnevertheless‘asingle,realtime’.BergsonwantsEinstein’srelativisticcake,andyetheisdeterminedtoeatitbakedàlaNewton.ThepredictionsofSTRoffendhisNewtonianbeliefthatthereisasingleglobaltimeframeinwhichallthedifferentkindsofclocks(includingthehumanbody)tickatthesamerate.SoSTRisacceptedbyBergsonasmathematicallyinteresting,butphysicallyinappropriate.25\nTIME,SPACEANDPHILOSOPHYEvenifweweretofeelsomesympathyforBergson’sprejudices,weneedtobeawarethatSTR’spredictionsconcernfarmorethanobjectsinhigh-speedmotion.Icanmakethesamepredictionforanyobjectmovingrelativetome,evenattheslowestofspeeds.Mychancesofactuallynoticingtheeffectareverylowexceptwhenobjectsaremovingatspeedsclosetothatoflight.TheeasiestwaytoconfirmSTR’spredictionsaboutclocksistoexaminehigh-speedobjectssuchasmu-mesonstravellingclosetothespeedoflight.Mu-mesonsareunstableparticleswithareasonablydefinitelifetimeinthelaboratory,buttravellingatsome0.99ofthespeedoflighttheylastaboutninetimeslonger.4Butwecanalsomeasuretheeffectandconfirmthepredictionusingclockstravellingonordinarypassengerjets,asshownbyHalfeleandKeatingintheirexperimentofOctober1971whenclocksjoinedthejet-setters:thetimeelapsedonairborneclocksinaround-the-worldtripwerecomparedwiththetimeelapsedonclockslocatedintheUSNavallaboratory—andthemeasuredtimedifferencesagreedcloselywiththepredictionsofSTR.5SoBergsoniswrongwhenhearguesthatthosereturningafterajourneywillbeexactlythesameageasusandthisappliesjustasmuchtotheobjectsofeverydayexperienceasitdoestothehighspeedlifeofmu-mesons.Weshallseethatthereissometruthinhisbeliefthat,whenweareobservingobjectsinmotion,weareobserving‘phantoms’insomesense.However,onlyamoredetailedacquaintancewiththelanguageofSTRwillhelpustorealiseinwhatsenseBergsonmayberightaboutthis.Andthismoredetailedstudywillalsohelpustoseeinwhatsenseifanythispredictionaboutclocksisparadoxical.Abrief,preliminarysurveyoftheideasofmotioninNewtonianspaceandtimeandSTRspacetimeisprovidedinFigure4(pp.27–33).ThenextsectionwillthendeveloptheseideaswithinthecontextofSTR.Somereadersmaywishtomoveonimmediatelytop.33.26\nCLOCKS,GEOMETRYANDRELATIVITYAspacecraftAisatrestrelativetoanobserverO.SpacecraftBismovingawayfromAandOatasteadyspeedv.AnobjectPismovingtowardsAatasteadyspeedc.Thediagramhereillustratestwoframesofreference:thatinwhichAisatrest,andthatinwhichBisatrest.BecauseBismovingatasteadyspeedvtowardsPasshown,theframeofreferenceinwhichBisatrestisalsomovingatvrelativetotheframeofreferenceinwhichAisatrest.ThepeopleinspacecraftAsaythatPismovingtowardsthematasteadyspeedc.ButwhatspeeddotheysaywillbemeasuredbythoseinspacecraftB?ThecalculationiseasyinNewtonianphysics:wesimplytakeintoaccounttherelativevelocitybetweenthetwospacecraftv,sothatthoseinBaresaidtomeasureaspeedc—vforP.ThusNewtonianmechanicsgivesusastraightforwardwayofcomparingeventsandmeasurementsintwoormoreframesofreference.ButNewtonianphysicsplacesnorestrictiononthevalueobtainedforthespeedoflightbydifferentobservers.Whenwealsospecifythatdifferentobserversmovinguniformlymustallmeasurethesamespeedforlight,werunintoproblems.ImaginethatPisapulseoflighttravellingtowardsbothAandB.Asaysthatitismovingatasteadyspeedc.WemaynolongersaythatthoseonBwillmeasureadifferentspeedc—v!Fortheytoo,giventherestriction,mustalsomeasurec.ThismeansthatwecannolongerrelyonNewtonianmechanicswhenwecomparethemeasurementsmadebydifferentobservers—solongastherestrictionisjustified,asitseemstobebyalltheavailableempiricalevidence.Weneedanewwayofcomparingeventsandmeasurementsindifferentframesofreference—andtheLorentztransformationsoftheSpecialTheoryofRelativityprovideuswithjustthat.Figure4(a)FromNewtontoSTR27\nTIME,SPACEANDPHILOSOPHYInadistance-timegraph(top),thesteepertheslopeofthelinerepresentingamovingobjectorsignal,thefasterthatobjectorsignalismoving.Stationaryobjectsarerepresentedbyahorizontallinealongthetimeaxis.Inatime-distancegraph(bottom),thetimeaxisisnowverticalandastationaryobjectisrepresentedbyaverticallinealongthisaxis.Thefastertheobjectorsignal,thefurtheritmovesawayfromastationaryobjectinagiventime.Figure4(b)Time-distancegraphs28\nCLOCKS,GEOMETRYANDRELATIVITYSpacetimediagramsareusuallydrawnastime-distancegraphs,withthetimeaxisverticalandwithjustonespatialdimension(left/right)representedbythehorizontalaxis.Inuniform(straight-line)motion,thereisnoaccelerationordeceleration.Anyobjectorsignalinuniformmotionmovesataconstantvelocityrelativetoanyotherobjectorsignalinuniformmotion.Iftheconstantrelativevelocitybetweentwoobjectsiszero,thenthosetwoobjectsareatrestwithrespecttoeachother.Theymay,however,stillbemovingrelativetootherobjectsinuniformmotion.Eachlinerepresentinganobjectorsignalinaspacetimediagramiscalledtheworldlineofthatobjectorsignal.Eachpointonaworldlineidentifiesauniquespacetimelocationfortheobjectorsignalrepresentedbytheline.Figure4(c)Ideasofvelocityandacceleration29\nTIME,SPACEANDPHILOSOPHYThediagramontheleftillustratestwospacecraftAandDatrestwithrespecttoeachother.Thisisshownbythefactthatthedistancebetweentheirworldlinesneverchanges.ButtwootherspacecraftBandC,movingawayfromAandtowardsD,arealsoatrestwithrespecttoeachother—heretoothedistancebetweentheirworldlinesdoesnotchange.Whenworldlinesareparallel—asinthecaseofAandDandalsoofBandC,wesaythattheobjectsorsignalsrepresentedbytheworldlinesareinthesameframeofreference.Thespacetimediagramitselffixesaparticularframeofreference—thatinwhichallobjectswithworldlinesparalleltotheverticaltimeaxisareatrest.Butwemayredrawanyspacetimediagrambytakinganyobjectorsignalmovinguniformlyasthemainreference—withitsworldlineparalleltothetimeaxis.ThediagramontherightshowsBandCparalleltothetimeaxis.Notice,however,thatthephysicalsituationdepictedinthisnewdiagramisidenticalwiththeoriginalsituation.BandCarestillmovingsteadilyawayfromAandtowardsD.AndAandDarestillatrestwithrespecttoeachother—shownbythefactthattheirworldlinesarestillparallel.Itdoesnotmatterwhichobjectwefocusuponandtaketobestationarywithrespecttothetimeaxis—thephysicalrelationshipsbetweenalltheobjectsremainthesame.AreferenceframedefinedbyauniformlymovingobjectsuchasAorCiscalledaninertialframeofreference.TheobjectrepresentedbyBoccupiesfirstoneinertialframeandthenafteritschangeofdirectionanotherinertialframeofreference.Whenanobjectchangesfromoneinertialframeofreferencetoanotheritexperiencesinertialforces.Henceanydeviationfromstraightnessforaworldlineindicatesthattheobjectrepresentedbytheworldlinehaschangedframeandhasexperiencedinertialforcesasitacceleratesordeceleratesinchangingframe.Figure4(d)Referenceframes30\nCLOCKS,GEOMETRYANDRELATIVITYIfBistheworldlineofalightsignal,thenAistheworldlineofasignaloranobjecttravellingatlessthanthespeedoflight.But,ifCrepresentsanymovingobjectorsignal,thentheobjectorsignalmustbemovingfasterthanlight.Pisapointinspacetime:imaginethatPrepresentstheexplosionofastar.Whenthestarexplodes,thelightfromtheexplosionwilltravelinalldirectionsawayfromthepointofexplosion.Thisspacetimediagramallowsustocapturealltheinformationabouttheexplosionofthestarinonespatialdimension.Fromthispointonwards,weshalldrawtheworldlinesofalllightpulsesat45degreestothetimeanddistanceaxesashere.Ifthescaleisonecentimetretoonesecondonthetimeaxis,thescaleonthedistanceaxiswillbeonecentimetretoroughly300,000,000metresonthedistanceaxis—sincethespeedoflightisjustunder300,000,000metrespersecond.Figure4(e)Basicideasofspacetimediagram31\nTIME,SPACEANDPHILOSOPHYAspacetimediagramdrawnfromthepointofviewofS:i.e.drawninS’sframeofreferenceSrepresentsastationaryobject.ThepointPrepresentsaneventsuchasamovingobjectMpassingbyS.MmovesclosertoS,meetsSatP,andthenmovesawayfromSasshown.LandL1representtwopulsesoflight—LmovingintowardsSfromtheleft,andL1fromtheright.TheymeetSatpointPandthenmoveawayasshown.Wehaveyettodetectanyobjectorsignalwhichcantravelfasterthanlight.SoasfarasSisconcerned,asignalfrompointQjustcannotreachSuntilafterP.Forthisreason,SregardsthepastofPasbeingdefinedbytheportionofspacetimewithinLandL1asthesetwopulsesconvergeuponSatP.Similarly,SregardsthefutureofPasallthoseeventswhichmightbereachedbyasignalorobjectmovingawayfromP,i.e.thatportionofspacetimewithinLandL1asthesetwodivergefromSatP.Figure4(f)SpacetimediagramshowingpastandfutureofaneventP32\nCLOCKS,GEOMETRYANDRELATIVITYMYTIMEANDYOURTIMETheLorentztransformations(LT)giveusthemathematicaltoolswhichweneedtoanalysemotioninthespacetimeofSTR.Here,withoutgoingintothemathematicaldetailsoftheLT,Ishalltrytospellouttheirsubstanceandrationale.Ifaspaceshippassesmeatasteady(inertial)speed,thentheLTallowmetopredictwhatwouldbeobservedfromanyotherinertialperspective:forexample,fromanotherspaceshipeithertravellingalongsidethefirstorrushingbyattwicethespeedofthefirst.Inthisway,Imay‘view’thepassageofthefirstspaceshipfromanyinertialframeIchoose.SotheLTallowmetodofarmorethanstayonEarthandsimplymeasurethetimepassingspaceshipsappeartotaketotravelacrossthesolarsystem.ImayalsopredicthowlongthejourneywouldseemtobeifIweretravellinginsidethespaceshipitself.Ofcourse,thiswouldhardlybeinterestingifeveryviewweretobeidenticalandeverytimepredictionweretocorrespondto‘Earthtime’.ButSTRbreaksfreefromtheglobaltimeperspectiveofNewtonianphysicswitheverythingintheuniverserunningtogetheratthesamerate.WithSTR,theperspectivefromwhichweviewaneventorsequenceofeventsbecomesofcriticalimportancewhenwemakepredictionsabouthowaclockwilltickcomparedwithsomeotherclock.6Ifthereisonetimeofwhichwecanbesure,thenitisourowntime—thetimeweusetomeasureeventsinourownlocality.Nooneisgoingtohavemuchsuccessinpersuadingusthatwecannotrelyonclocksforsuchlocalmeasurements.Ifwecannotdependonthem,thenitwouldbedifficulttoacceptanypredictionsatallabouthowclocksarebehavingindistantplaces.Fortunately,itisthis‘personal’time-keepingwhichprovidesthetouchstoneforalltimepredictionsinSTR.WhenweusetheLTwearetryingtodiscoverjusthowsomedistantandpossiblymovingclock(asfarasweareconcerned)isbehavinginitsownlocalitywhencomparedwithourown.Andwearetryingtoavoidtheclaimthathowanydistantclockappearstobetickingisthewayitreallyisticking.Atelescopemighthelpuslookandsayhowaclockseemstoberunning,butonlytheLTcancomparethebehaviouroftheclockinitslocalitywithaclockinourvicinityinaneutralway.WiththeLTwemaydeliverthesamejudgementuponalldistantand/ormovingperspectivesaswedeliveruponourownperspective.TheLTtransformawaylocalprejudicesandindoingsohelpustoseewhatishappeningatsome33\nTIME,SPACEANDPHILOSOPHYspecificdistantlocationwhencomparedwithourownoranyotherlocation(andviceversa).IfIstandinParliamentSquarelookingupatBigBenandmeasurethetimebetweentheclockreachingtwelveandone,thenmyownclockshouldshowthatonehourhaspassed.7OnehourforBigBenisonehourformetoo:letuscallthistimemeasuredbymeT0.Butwhataboutsomeoneflyingbyinajet—willonehourforBigBenbeonehourforthemaswell,whenBigBenwillseemtobemovingwithrespecttothem?AccordingtoSTR,anyclockmovingataspeedvrelativetomewillmeasureatimeTbetweentwoeventsinmyframe;andthistimeisgivenbytheLTwhichleadtothefollowingequation:T=ßT0wherethefactorßdependsupontherelativespeedv.8IfmystationaryclockmeasuresthetimeT0(=onehour),then,usingtheaboveequation,ImaycalculatethetimeTmeasuredbythemovingclock.Wemaynotethatßhasthevalue1whenv=0,i.e.whentheobjectisatrestwithrespecttous.Forlowspeedsrelativetous,therefore,ßisapproximately1andthetimesT0andTareverynearlythesame.ThisofcourseiswhyweseelittledisagreementaboutmosttimemeasurementshereonEarth.But,iftheclockismovingatall,thenineverycasethetimeT0measuredbythestationaryclockwillbelessthanthetimeTmeasuredbythemovingclock.Soonehourforthestationaryclockisnotonehourforthemovingclock.Asyouflybyinajet,yourclockwillreadTbetweenthetwoevents:(E1)BigBenreachingtwelveand(E2)BigBenreachingone;and,ifT0isonehour,Twillbegreaterthanonehour.Asfarasyouinyourjetareconcerned,BigBenwillberunningslow.Anditwillberunningslowbecauseitismovingwithrespecttoyou,i.e.itisslowrelativetoyourclock.NowimaginearatherlargejetcarryingBigBen,yourself,andaclock.Iamstrandedonthegroundwithmyclock.Wenowconsiderexactlythesametwoeventsasbefore:E1—BigBenattwelve;andE2—BigBenatone.BecauseyouandyourclockarenowinthesameinertialframeasBigBen,i.e.becausethereisnorelativemotionbetweenyou,yourclock,andBigBen,onehourforBigBenwillnowbeonehourforyourclock.ButwecanonlyfindthetimemyclockmeasuresbetweenE1andE2byusingtheLT.Andthesameequationasbeforetellsusthat34\nCLOCKS,GEOMETRYANDRELATIVITYInowmeasureatimegreaterthanonehourwithmyclock!BigBentakesonehourfromyourpointofview,butittakesmorethanonehourfrommine—BigBenwillseemtoberunningslow.Again,itwillberunningslowbecauseitisinmotion,thistimefrommypointofview,i.e.itisrunningslowrelativetomyclock.BigBenactsasastandardclockagainstwhichwemaysynchroniseourownclocks.WemaysummarisetheBigBenstoryintermsofastandardclock.Ifastandardclockisstationarywithrespecttoaclocktobesynchronised,thenonehourforthestandardclockisonehourfortheclocktobesynchronised.But,ifthestandardclockismovingfromthepointofviewoftheclocktobesynchronised,thentheLTshowthatthestandardclockandallclocksinthesameframeasthisclockarerunningslowfromtheperspectiveoftheclocktobesynchronised.So,onlyclocksinthesameframemaysynchroniseperfectly.9Andanyclock(electronic,clockwork,orbiological)inaframemovingrelativetomeoranyotherobserverwillrunslowerthanmyclock.Itisthesymmetryoftwopossibledescriptionsoftwoclocksinrelativemotionthathasdisturbedsomepeople:twoobserversinrelativemotionwilleachsayexactlythesameabouteachother’sclock:thatitisrunningslowrelativetotheirownclock.Howcanthetwodescriptionsbothbecorrect?Solongasweremainweddedtotheideaofasingle‘realtime’,weareunlikelytofindtheresultsoftheLTcredible.Forthisideacarrieswithittheassumptionthatthereisaglobaltimestandardagainstwhichallclocksmaybejudged.ButSTRchallengespreciselythisassumption.Becausethereisnoglobalstandardtowhichwecanappeal,noinertialperspectivemaybesingledoutas‘thecorrect’view.Hence,neitherofthetwodescriptionsabovemaybedeemed‘correct’—rather,theyarecomplementarydescriptionsofmotionswithinaspacetimecharacterisedbySTRandtheLT.InSTR,distancesinspacetimeareinvariant:suchdistancesarecalledspacetimeintervals;theyaredistancesfromonelocationinspacetimetoanother.10Althoughwewillneverdisagreeaboutthespacetimeinterval,weshallingeneraldisagreeaboutthelengthofanobjectorthetimebetweentwoeventsunlessourperspectivesarethesame.Thisnotionofaninvariantintervalleadsustotheideaof‘propertime’—orlocaltimeasitissometimescalled.Whenasignaltravellingatlessthanthespeedoflightisobservedtomovebetweentwopointsinspacetime,theintervalbetweenthosetwopointsissaidtobetimelike.Ifthesignalbetweenthetwopointsinspacetimeistravellingatthespeedof35\nTIME,SPACEANDPHILOSOPHYlight,thentheintervalissaidtobelightlikeornull;and,ifasignaltravellingfasterthanlightwouldbeneededtoconnecttwopointsinspacetime,thentheintervalbetweenthepointsissaidtobespacelike;seeFigure5(p.37).THEPARADOXOFTHETWINS:FOREVERYOUNG?Thestoryoftheparadoxofthetwinsmaynowbetold.IdenticaltwinstakepartinanexperimenttotestSTR’spredictions.OnetwinremainsonEarthandtheothertakesaround-tripinanextremelyfast-movingspaceshiptosomedistantstar;whenthetripisoverandthetwinsmeettocomparenotes,theonethatstayedathomeisolderthanthetraveller,justasalltheclocksonEarthhavetickedoffmoresecondsthanaclocktakenwiththetravellingtwin.Thefasterandfurtherthejourney,themoremarkedwillbethedifferencesbetweenthetravellerandtheEarth-basedtwin.Buteachoftheirnotesincludetheclaimthattheother’stimeisstretchedoutordilated.Ifweleavethestoryatthat,thenwemightthinkthatthereisadegreeofparadoxhere.Ontheonehand,thereisaremarkablesymmetryhere:theLTalloweachtwintomakethesameclaim—itistheothertwinwhoisageingslowly.But,ontheotherhand,thetwosituationsdoseemtobeasymmetric:oneofthetwinsisquitedefinitelyolderthantheother—olderinthesensethatthattwinhaslivedthroughmoreseconds.Soonlyonetwindoesagemoreslowly.Yetthisseemstoconflictwiththeviewalreadyexpressedthatthereisno‘correct’view.SurelythetwinonEarthisrightwhenthattwinsaysthattheothertwinisageingmoreslowly;andthetravellerissimplywrongtoclaimthatthetwinonEarthisageingmoreslowly!Wenowhavetwomainchoicesavailable:1wemightquestiontheassertionthatthetravellerwouldhaveageddifferentlyfromthetwinwhostayedbehind;or2wemightlookforsomeasymmetrybetweenthetwosituationswhichwillallowustoexplainthedifferenceinageing.Buttheempiricalevidencementionedintheopeningtothischaptersuggeststhattheassertionmadeiscorrect:clockstakenonround-tripsdoseemtotickofffewerseconds,and,aswehavesaid,peoplemaybeconsideredasbiologicalclocks.Sotheevidencesuggeststhetwowouldhaveageddifferently.Thesecondchoiceseemstooffermorepromise.Fortheredoesseemtobeanasymmetrywhichhasbeenglossedoverbyourfailuretotellthe36\nCLOCKS,GEOMETRYANDRELATIVITYA:thisworldlinerepresentsanobservertakentobeatrest—thisobserverjudgesBtoEtobeinmotion;thelineshownisatimelikeinterval.B:thisworldlinerepresentsanobjecttaken(byA)tobemovingataspeedlessthanthatoflight;thelineshownisatimelikeinterval.C:thisworldlinerepresentsalightsignal;thelineshownisanullinterval.D:thisworldlinerepresentsasignaloranobjectmovingfasterthanlight(accordingtoA);thelineshownisaspacelikeinterval;itisgenerallysupposedthatnosignalorobjectcouldmovebetweenpointsconnectedbyaspacelikeinterval—whetherornotthisissoisanempiricalproblemandisnotruledoutaprioribySTR.E:thisworldlinerepresentsasignalorobjectmoving‘instantaneously’inA’sjudgement;thislinetooisaspacelikeinterval.Figure5Null,timelike,andspacelikeintervalsonaspacetimediagramwholestoryaboutthetwins.Yes,itistruethateachtwin,usingtheLT,willmakethesameclaimsabouteachother.Theirsituationsmayseemtobesymmetrical,forwemightthinkthat,justasthetwinonEarthmaysaythattheEarthisatrestandthatitisclearlythespaceshipwhichmovesfirstawayfromandthenbacktotheEarth,thetravellingtwinmayalsoconsiderhisorherspaceshiptobeatrest,andviewtheEarthassteadilymovingfirstawayfromandthenbacktothespaceship.But,unlikethetwinonEarth,thetravellingtwinwillnotbeabletoconsiderhimselforherselfatrestfortheentirejourney!Whenthespaceshipturnsaround,theframeofreferenceofthetravellingtwinwillchange.If,inordertomakearound-triphereon37\nTIME,SPACEANDPHILOSOPHYEarth,someoneattemptstojumpfromthebackofatrucktravellingat30m.p.h.inonedirectiontothebackofatrucktravellingatthesamespeedbutintheoppositedirection,thenitwillsoonbeobvioustothepersontryingtodothisthattheywillchangeframebydoingso.Forthetwotrucksareinframesofreferencetravellingat60m.p.h.withrespecttoeachother.Andwhenobjectschangeframe,forcesactuponthem—thegreaterthechange,thegreatertheaccelerationordecelerationinvolved,andthegreatertheresultingforce.Anyonewhohasjumpedfromabusmovingatjust10m.p.h.relativetothepavementmightbeabletoguessjusthowfoolhardysudden,dramaticchangesofframecanbe.Thetravellingtwinwillexperiencesuchachangewhenthespaceshipturnsaround—asitslowsdownandthenspeedsupforthereturnlegofthejourney.Giventhattheeffectsofaround-tripwillonlybeclearlyvisibleasfarasthetwinsareconcernedifthetraveller’sjourneyisatspeedsclosetothatoflight,thechangeofframewillindeedbedramatic—fromatremendouslyhighspeedawayfromtheEarthtoanequallyhighspeedtowardstheEarth—especiallyifwerestrictthetimeavailableforthechangeofdirection.Butweshallsupposethatthetravellersurvivesthechange.11ThespacetimediagramsinFigure6(p.39)illustratethetwosituations:onlyoneofthetwinsmayberepresentedbyaworldlinewhichneverbendsorchangesdirection.Foraworldlinewhichisbothstraightandunidirectionalsignifiesthattheobjectrepresentedbytheworldlineismovinginertiallyanditsframeofreferenceisunchanging.Aworldlinewhichchangesdirectionindicatesthattheobjectrepresentedbytheworldlineexperiencesachangeofframe,andthereforeitalsoexperiencesanaccelerationandinertialforces.Thetravellingtwin’sworldlinechangesdirectionbuttheworldlineofthetwinonEarthdoesnot.Andthediagramsshowthatthisisthecasewhichevertwinwetaketobeatrestatthestartofthejourney.So,wemaybreaktheapparentsymmetryinthestoryofthetwinsbypointingoutthatonlyonetwinchangesframe,onlyonetwindeceleratesandthenaccelerates,andonlyonetwinexperiencestheinertialforcesduetothechangeofframe.12Whatisresponsibleforthisdifferenceinageing?Thestraightforwardansweristhefactthatthespacetimepathsofthetwinsaredifferent.InSTR,wemaynolongertreatmotionsastakingplace38\nCLOCKS,GEOMETRYANDRELATIVITYThespacetimediagramontheleftisdrawnfromthepointofviewofthetwinAwhostaysathome.Thistwinexperiencesnoinertialforces—theworldlinerepresentingthistwinremainsstraight.ButthetravellingtwinBexperiencesinertialforcesattheturnaroundpointP.Thediagramontherightillustratesthesituationfromthetravellingtwin’spointofview.Althoughthistwinmightbeabletosaythatitistheothertwinwhoismovingasthedistancebetweenthetwotwinsincreases,onlyoneofthemexperiencesanaccelerationandonlyoneofthemexperiencesinertialforces—andthishappenstothetravellingtwinattheturnaroundpoint.Thechangeofframeexperiencedbythistravellingtwindemonstratesthatthereisonlyasuperficialsymmetrybetweenthetwoperspectives.Figure6Twinsparadoxspacetimediagram39\nTIME,SPACEANDPHILOSOPHYinavastspatialarenawithsomeindependentglobalcosmicclocktickingawayinthebackground.Afulldescriptionofanymotionwillalwaysspecifyhowanobjectmovesinspacetime,takenasacompositeentity.Theworldlineofanobjectprovidessuchadescription.Andbecausetheworldlinesofthetwinsarenotthesame,thereisnoreasonforustoexpectclockswhichweresynchronisedatthestartofthejourney,whentherespectiveworldlinesintersect,tosynchroniseattheendofthejourney,whentheworldlinesnextmeet.WhenweembracethespacetimeperspectiveofSTR,whyshouldweregardthisasanymoreproblematicthanthefactthattwopeoplewhotakedifferentspatialpaths(e.g.thehighroadandthelowroad)betweentwofixedpointsonamapwillingeneraltraveldifferentdistances,eachtakingadifferentnumberofpacestocompletetheirrespectivejourneys?Yetthe‘paradox’doesseemtoleaveuswithapuzzle.Iftwoidenticalclockssynchroniseatthestartofanexperiment,butdisagreeattheend,thenwemightbeinclinedtolookforaphysicalreasonforthedisagreement.Buttheanswergivenaboveseemstoexplainthedifferencesintermsofspacetimegeometryratherthanintermsofmatterandforces.Soitistemptingtosaythatthedifferenceinworldlines,andthereforethedifferenceinageing,isexplainedbythefactthetravellerexperiencesinertialforceswhenthepropulsionsystemoperatestoaccelerateanddeceleratethespaceship.Theseforcesarethephysicalcauseofthedifference.Thisneedtofindamaterialexplanationoftheeffectseemstoderivefromapositivistoutlook.Typicallypositivistsarguethatonlythosethingswhichwecanobserveinsomestraightforwardwayshouldbeincludedinourphysicaldescriptionsoftheuniverse.Wecan‘see’neitherspacenortimenorspacetimegeometry.Sopositivistinclinationswouldnaturallyleadustodiscountexplanationsofphysicaleffectsintermsofgeometryandtolookforsomethingmoreconcretesuchasinertialforceswiththeirorigininsomekindofmaterialinteraction.13Unfortunately,STRdoesnotcompletelysatisfysuchapositivisticdesire.Forwemayreadilyreconstructthetaleofthetwinsinsuchawayastoleaveoutallmentionofinertialforces!40\nCLOCKS,GEOMETRYANDRELATIVITYThediagramshowstwowaystogetfromPtoRinspacetime:viathespacetimepathPR(alongtheverticalaxis)andviathepathPQR.Figure7DifferentspacetimepathsfortwinsFROMTWINSTOTRIPLETSThespacetimediagraminFigure7aboveshowsthespacetimepathsofthetwins.TheworldlinesdivergefrompointPandmeetagainatR.ThechangeofframeoccursatpointQ.Sotherearetwodistinctspacetimepaths:PR;andPQ…QR.Insteadofonepersonfollowingthesecondpathinitsentirety,nowimaginethatonetraveller‘movesalong’thelinePQsuchthatatnotimedoesthatpersonexperienceinertialforces;andthatanothertraveller‘movesalong’thelineQRagainwithoutexperiencinginertialforces.14Asbefore,weimaginethatthepersononEarth,representedbyPR,alsoexperiencesnoinertialforces.Sonowtherearethreepeopleinvolvedinourstory—tripletsinsteadoftwins,eachcarryinganidenticalclock.15Thethreeclocksinvolvedmaybelabelled:C1—fortheclockonEarth;C2—fortheclockwiththetravellerrepresentedbyPQ;andC3—fortheclockwiththetravellerrepresentedbyQR.41\nTIME,SPACEANDPHILOSOPHYWemaynowretellthetaleintermsofasecondthoughtexperimentasfollows:1AttheintersectionofthelinesPRandPQ,theclocksC1andC2synchronise,and,asfaraswecansee,thetwotripletsinvolvedappeartobethesameage.2AttheintersectionofthelinesPQandQR,theclocksC2andC3synchronise,and,again,thetwotripletsinvolvedhereseemtobethesameage.3AttheintersectionofthelinesPRandQR,theclocksC3andC1donotsynchronise,nordotheagesofthetwotripletsinvolvedatthispointagree.TheclockC3lagsbehindC1;andthetravellingtripletrepresentedbythelineQRisyoungerthanthetripletontheEarthlinePR.164AteachintersectionPandRoftheworldlines,therelativespeedofthoserepresentedbytheintersectingworldlinesisthesamesothat,fromthepointofviewofthetripletonEarth,thevelocityofthetripletrepresentedbyPQisequalinsizebutoppositeindirectiontothatofthetripletrepresentedbyQR.17ThelackofsynchronisationatRisexplainedbythedifferenceinspacetimepaths;seeFigure8(p.43).However,wemaynolongerlooktoinertialforcestoexplaintheasymmetry.Fornooneinvolvedinthisthoughtexperimentexperiencessuchforces.SomewritershavearguedthattheparadoxofthetwinscannotbesatisfactorilyresolvedwithinthecontextofSTR,andthatonlybyusingtheGeneralTheoryofRelativity(GTR)mayweproperlyexplaintheasymmetricaleffects;seeBohm(1965).18WhateverreasonswemighthaveformovingtoatheorylikeGTRinordertogiveanaccountofthetwothoughtexperiments,itiswrongtosuggestthatweshoulddosobecauseSTRcannotdealwithacceleratedmotion.ThisclaimaboutSTRisfalse,asFrench,amongstothers,pointsout:BecauseEinsteindevelopedawholenewtheory…baseduponthedynamicalequivalenceofanacceleratedlaboratoryandalaboratoryinagravitationalfield,itissometimesstatedorimpliedthatspecialrelativityisnotcompetenttodealwithacceleratedmotions.Thisisamisconception.Wecanmeaningfullydiscussadisplacementandallitstimederivatives[e.g.acceleration]withinthecontextoftheLorentzTransformations.(French1968:153)1942\nCLOCKS,GEOMETRYANDRELATIVITYTheclocksC1andC2synchroniseatP;C2synchroniseswithC3atQ;but,atR,C3lagsbehindC1.Figure8TripletsparadoxspacetimediagramWhatSTRdoesnotdoistogiveanaccountofmotioningravitationalfields,i.e.inarbitrarilycurvedspacetimes.Forcurvedspacetime,GTRisindeedrequired;theessentialdetailsofcurvedspacetimewillbeexploredinChapters4and7.20Thereis,interestingly,agravitationaltimedilationeffectpredictedbyGTR:fromthepointofviewofsomeoneinlowgravity,aclockinahighgravityareawillrunslow.GiventheequivalencebetweengravitationalandinertialeffectswhichispartofthefoundationsforGTR,itmightbethoughtthatthisisreasonenoughforustoshiftfromSTRtoGTR.Yetthereisnofirmevidencethatgravitationisanissueeitherdirectlyorindirectlyinthe‘paradoxes’above.And,evenifwearetemptedtomovetoGTRtoexplaintheasymmetryinthestoryofthetwinsbecauseinertialforcesareinvolvedinthatthoughtexperiment,thetaleofthetripletsindicatesthattheasymmetryhasnothingtodowithforces—fornoforcesareinvolvedinanyway.Thisproducesadilemmaforanyonewithapositivisticbias.Thepositivistisgenerallyeagertofindsomeconcreteobservationaloriginforanyphysicaleffect.This,asweshallseeinChapters6and7,ishowtheproblemofthesourceofinertialforcesishandled—thepositivist43\nTIME,SPACEANDPHILOSOPHYclaimsthatinertialforcesarisewhenobjectsacceleraterelativetosomeaverageofthematerialcontentsoftheuniverse.But,sincewecanimaginethis‘triplet’thoughtexperimenttakingplaceinanotherwiseemptyspacetime,whatmightweciteasbeingresponsiblefortheasymmetry?Eachofthethreepathsinvolvedisinertial.Andwehavenoreasontosingleoutanyonepathforpreferentialtreatmentinanyway.Evenifwedogivepreferredstatustoonepath,wewouldclearlyhavesomenon-materialreasonfordoingso.Thepositivistmightstilltrytoarguethatonlyinthecontextofacompletedynamicaltheorywhichsetsallmotionswithinarealistic,gravitationalcontextmaywetalksensiblyaboutsucheffects.However,evenifwesetthethoughtexperimentwithintheobserveduniverse,sincenoneoftheparticipantsexperiencesanyforcesandallthreemaybemovingrelativelytothematerialcontentsoftheuniverse,wemayonlypointtothefactofrelativemotiontoexplaintheresultingasymmetry.Andwhy,fromamaterialistpointofview,shouldobjectsinrelativeuniformmotionnotbehaveinthesameway?Whatistruemateriallyforoneobjectmayalsobetruefortheotherobjectsinvolved.Theonlysignificantdifferencesseemtobethedirectionsofthethreemotions.Sothereseemstobelittleencouragementhereforthepositivist,whomightperhapsregardthethoughtexperimentasgenuinelyparadoxical—orgiveuphisorherpositivisticbias.21PHANTOMSOFPERSPECTIVEBergsonarguedthatthetravellingtwinwouldbenomorethanaphantominthephysicist’simaginationandthatonhisorherreturntoEarthagesandclockswouldallagree.WecannowseewhySTRdoesnotsupportsuchaview.Imaynotsaythatapersoninmotionrelativetomeisaphantom,justbecausewemaybothmakethesameclaimabouteachother.Allwemaysayisthatneitherclaimhasanypriorityovertheother,forneitherclaimis‘thecorrect’one.Andthereisnoreasonwhyclockstravellingbetweenthesamepointsbutalongdifferentspacetimepathsshouldagree.Weappealtogeometrytoresolvethe‘paradox’—tothegeometricalideasofspacetimepathsandworldlineswhichpresentthemotionofobjectsintheonlywaypossibleinSTR:withintheunionofspaceandtime.Throughoutthetalesofthetwinsandthetriplets,Itriedtoavoidtalkofwhateachparticipantactuallysees.Thiswouldcomplicatethestoriestremendously.For,inacertainsense,thepeopleinvolvedinsuchexperimentswouldnotseeeachother‘astheyreallyare’.Whenweuse44\nCLOCKS,GEOMETRYANDRELATIVITYtelescopestoviewdistanteventswedonotseethings‘astheyreallyare’—anditisnotatallclearwhatmightbemeantbythisphrase.Therearethreeproblemswhicharisefromanysimple-mindedrelianceontelescopes:1Differenttelescopesindifferentlocationsandmovingatdifferentspeedswillnot,ingeneral,agreeonthe‘facts’aboutsomesequenceofevents.2The‘facts’whichtheyrevealarealwaysfromtheperspectiveoftheviewerandnotfromthatoftheviewed;forexample,becausetimeelapsesbetweentheemissionofanysignalfromasourceanditscapturebythetelescope,theeventviewedalwaysliestothepastoftheactofobservation—weseethings‘astheywere’.3Othereffectsmaypreventthetelescopefromgivinganaccurateobservationaldescriptionofwhatishappeningatthedistantlocality—forexample,theDopplereffectmaycauseustoattributepropertiestoadistanteventwhichitdoesnotinfactpossess.Nevertheless,wemaymakecorrectionsforallsuchproblems,sothattheresultsobtainedfromtheLTagreewithobservation.Yettheseproblemsmaystilltemptusintoacceptingthatdistanteventsandobjectsarephantomsinasense—fortheydonothavetheimmediatetangibilityofeventsandobjectsinourownlocality,andweneverseethemexceptfromourlocalandpartialviewpoint.But,asindicatedearlierinthischapter,thisisexactlywhytheLTareneeded.TheLTenableustoexplorewithoutdistinctionalleventsandmotionswithinthespacetimeofSTR.Theyprovideamechanismbywhichwecanovercomeourlocallyboundperspectives.Theyhelpustogivetodistantobjectsandeventsthesameconcretestatusasthoseinourownlocality.Bergson’sphantomsareassolidandrealasBergson(was)himself.45\n3TRAVELLINGLIGHTINTRODUCTIONManyattemptstomeasurethespeedoflightclearlyinvolvethemeasurementoflighttravellingtoagivenpointandback;sothespeedmeasuredisthatforthe‘round-trip’.Forexample,Fizeau’smethod,astandardexperimentinthemid-nineteenthcentury,involveslighttravellingfromasourcetoadistantmirrorandbacktoanobserverclosetotheoriginalsource;again,Foucault’smethod,whichreplacedFizeau’sapproachasthestandardprocedure,alsomeasuresthespeedoflighttravellingaroundaclosedpath.1Numerousactualandthoughtexperimentshavebeensuggestedinordertohelpusmeasuretheone-wayspeedoflight.WesleySalmonarguesthatallsuchattemptsinvolveafatalflaw:anyexperimentdesignedtomeasuretheone-wayspeedactuallyinvolveseitheraround-tripmeasurementorsomeotherproblematicassumption.2Hence,wearetold,theclaimthatthespeedoflight(invacua)inagivendirectionisalwayscisessentiallyanon-trivialconvention.Thishasanimmediateconsequenceforourdefinitionofsimultaneity.TheSpecialTheoryofRelativity(STR)demandsachangeinthewaywethinkaboutsimultaneity.Theideaofafixed‘invariant’speedoflightisakeyelementinSTR’saccountofsimultaneity.Thelightcone,whichrepresentslightspreadingoutinalldirectionsfromanevent,providesthefoundationforalljudgementsofsimultaneity.3Wecannolongerrelyonsomevastcosmictime-sliceprovidingaplaneofsimultaneityforallobservers.Weuselightraystosynchroniseeventsandthejudgementthattwoeventsaresimultaneousdependsonhowlightrayspropagatebetweenthem.Ifwewanttobesurethatanearbysequenceofeventssynchroniseswitha46\nTRAVELLINGLIGHTIfthelightsignaltravelsatthesamespeedinbothdirections,thediagramontheleftapplies;but,ifnot,thenthesituationontherightmightapply.Figure9PlanesofsimultaneityinSTRusinglightbeamsforsynchronydistantsequenceinourframeofreference,thenwetypicallybouncelightraysbackandforthtocheckthatthesequencesmatch.Theshuttlinglightraystransmitthelatestinformationabouteachofthesequences.Iftheround-triptakes10seconds,thenwewouldgenerallyfeelconfidentintheassertionthatthelatestinformationreceivedalwaysrelatestoanevent5secondsold.Butweneedtobesurethatthetimethelightraytakestotravelfromustothedistantsequenceisthesameasthetimeforthereturntrip.Otherwiseourjudgementofsimultaneitybetweenpairsofeventsinthesequenceswillbeimpaired;seeFigure9above.Therefore,ifourassertionthatthelighttravelsatthesamespeedinbothdirectionsisconventional,thenourjudgementsofsynchronyandsimultaneitywillalsohaveconventionalcharacteristics.4Muchthatwesaydependsuponsomeconventionorother.Beforepublicationofanovel,itstitleisagreed.Onpublication,thattitleisacceptedastheproperwayofreferringtothenovel.Butthebookcouldquiteeasilyhavehadsomeothertitle.Theparticulartitlegivenisjustamatterofconventionalagreementeventhoughsometitlesmaybemoreappropriatethanothers.Again,Iacceptandfollowthe47\nTIME,SPACEANDPHILOSOPHYgeneralconventionthattheobjectbeforemeisaMacintoshcomputer.Butallsuchobjectscouldquiteeasilyhavebeencalledsomethingelse.Whatwecallthenovelorthemachinemakesnodifferencetothecontentofthebookortheoperationofthecomputer.Suchconventionsconcernonlytheuseofonenameratherthananother.However,someconventionsmightbefarlesstrivialthanthese.Inthefirstchapter,weobservedthatitishardtojustifythechoicebetweencontinuousandmerelydensespatio-temporalstructuresonthebasisofempiricalevidencealone.But,ifweneverthelessdecidetoagreethatspaceandtimeformacontinuum,thenouragreementonthispointseemstoimplysomethingaboutthewaytheworldis.Inthiscase,suchconventionalagreementisfarfromtrivial.Inarguingfortheconventionalcharacterofatleastsomeimportantscientificbeliefs,somephilosophersadoptananti-realist‘conventionalist’viewpoint.Manyphilosophersandscientistsarereluctanttoadmitthatanymajordecisionsaboutthenatureofthephysicalworldarebasedonconventions,agreedamongstthescientificcommunity,ratherthanonindependentempiricalfacts.Theyprefertocommitthemselvestoarealistperspectiveinwhich:1whatsciencesaysabouttheworldisatleastapproximatelytrue—evenatthetheoreticallevel;2thetruthorfalsityofallscientificclaimsmaybedeterminedbyindependentreferencetothewaytheworldisandnotbyanypragmaticorconventionalbeliefsheldbythescientificcommunity;and,asamatterofhistoricalfact,3thehistoryofsciencerevealsoverallprogress—aconvergenceontruth,withbetterandbetterapproximationstothetruth.5Conventionalistsmaybe‘realists’toalimitedextent;theymaybelievethattheworldexistsindependentlyofourminds.Theymayacceptthatourtheoriesareinsomesenseabouttheworldandthatwhatwesayaboutobservations(ifnottheory)maybetrueoftheworld.However,theyarelikelytoremindusthatwearesometimesfacedwithalternativetheoriesandhypotheseswhichseemtobeequallysupportedbyempiricalevidence.Sotheytypicallymaintainthatourchoiceamongstthealternativetheoreticaldescriptionsisessentiallyconventional.Theirpositionseemstobemotivatedbyascepticalattitudetowardsourknowledgeoftheworld:somerealistswouldliketothinkthateverythingwesayabouttheworldmaybedeterminedas48\nTRAVELLINGLIGHTtrueorfalse,atleastinprinciple;butmanyconventionalistsarguethatthescopeofourknowledgeisrestrictedinprinciplebytheneedtomakeconventionaldecisions.Somewritersgofurtherthanthis.Theyarguethatallourtheoreticalchoiceshaveaconventionalcharacter:theempiricalevidencecannevertieusdowntojustoneview;thatis,ourchoiceoftheorymustalwaysbeunderdeterminedbytheempiricaldata.6Ifempiricalevidenceturnsouttobeinsufficienttosettleadisputeaboutthephysicalworld,thentherealistpositionwouldbechallenged.Inthischapterandthenext,weshallexploreanumberofsituationswhichmightbesaidtohaveaconventionalcharacter.Weshallthenbeinapositiontoassessthestrengths,andweaknesses,oftheconventionalistchallengetorealism.Anotherissueconnectedwiththespeedoflightconcernsthepossibilityofsignalsorparticlestravellingfasterthanlight.Weshallthereforetaketheopportunityinthischaptertoexploretheproblemsassociatedwithparticlestravellingfasterthanlight.Weshallfindthatrelativitydoesnotlegislateagainstsuchparticles,butthattheydoleadtosomeverystrangephysicalconsequences,particularlythepossibilityoftravelbackwardsintime.MEASURINGTHESPEEDOFLIGHTIsSalmonrightwhenhesaysthattheone-wayspeedoflightisaconventionratherthananempiricalfact?Hereviewsawidevarietyofattemptstomeasuretheone-wayspeed;but,atthisstage,weshalldiscussjustoneofthemethodswhichheexamines.7Despiteinitialappearances,thismethodassumesimplicitlythatthespeedoflightisthesameirrespectiveofitsdirection.8Intheearlynineteenthcentury,theEnglishscientistThomasYounginvestigatedthewavenatureoflightbyobservingtheeffectsoflightpassingfromasinglesourcethroughtwonarrowslitsinhiscelebrated‘two-slitexperiment’.Theoveralleffectmaybedisplayedonascreen;andthisshowsfirmevidenceofinterferencebetweenthelightwavesspreadingoutfromthetwoslits.Aregularseriesoflightanddarkbandsorfringes,theresultsofconstructiveanddestructiveinterference,isclearlyseen.Thedistancebetweenthebandsdependsuponthewavelengthofthelightused.Onceweknowthisdistance,astraightforwardgeometricalargument49\nTIME,SPACEANDPHILOSOPHYallowsustocalculatethewavelength.Giventhatthewavelengthoflightisrelatedtothespeedoflight,wemayreadilycalculatethisspeedonthebasisoftheresultsofYoung’stwo-slitexperiment.Onthefaceofit,thismethodappearstoofferastraightforwardmeasurementoftheone-wayspeedoflight:speedisgivenbytheexpression‘distancetravelled÷timetaken’.ThedistanceisastraightlineinonedirectionfromslitstoscreeninYoung’sexperiment.Noround-tripdistanceappearstobeemployedinthecalculation.However,Salmonandotherswhosupporthisviewsarguethatinordertoarriveatthefigurefortheone-wayspeedoflight,wemustassumethatlighttravelsatthesamespeedindependentlyofthedirectioninwhichittravels!Therelationshipbetweentheseparationoftheinterferencebandsandthewavelengthoftheincidentlightinthetwo-slitexperimentdependsuponanimportantassumptionaboutthepropagationoflight.ThegeometricalargumentusedbyYoungassumesthatlightspreadsoutinthesamewayinalldirectionsfromeachofthetwoslits.Hence,ifweusethismethodtomeasurethespeedoflight,wehavealreadytacitlyassumedthatthespeedoflightisindependentofitsdirection.Wehavenotprovidedanobjectivewayofdeterminingtheone-wayspeed;wehavesimplyreconfirmedourconventionthatdirectiondoesnotmatter.Thosedebatingthisissueoftheallegedconventionalityoftheone-wayspeedoflightaskustofocusonafundamentalissuewhichseemstolieattheheartoftheproblem:thestructureofspacetimeinSTR.ThestandardmethodforsynchronisingeventsinSTR’sspacetimeinvolvesthetransmission,reflection,andreceptionoflightsignalsbetweentwoobserversatrestwithrespecttoeachother:thismethodiscalled‘standardsignalsynchrony’.ImaginetwosuchobserversAandB:AsendsasignaltoBwhoholdsamirrorwhichreflectsthesignalbacktoAwithoutdelay.Asendsthesignalatt=0andreceivesthereflectedsignalatt=2.AtwhattimeaccordingtoAdidBreflectthesignal?Clearly,withafinitetransmissionspeedforlight,AdoesnotseeBatthemomentofreflectionassuch:Amustwaitforthereflectedsignaltoarrive.ButitmightseemreasonableforAtomaketheassumptionthatthetimeforthesignaltotraveloutisexactlyequaltothetimeittakestoreturn.This,ofcourse,reliesonthefurtherassumptionthatlighttravelsatthesamespeedinbothdirections:samespeed,samedistance,sosametime!Hence,Amaysaythatthemomentof50\nTRAVELLINGLIGHTAnypointonA’sworldlinefromt=0tot=2couldbechosenasbeingsimultaneouswiththearrivalofthelightsignalonB’sworldline.Wesaythatt=1ismetricallysimultaneouswithQ—thearrivalevent.Buttheentiresetofpointsfromt=0tot=2issaidtobetopologicallysimultaneouswiththearrivalevent—forQmightbelocatedatsomeotherpositiononB’sworldline;twootherpossiblepositionsareshowninthediagram.Thechoiceoft=1astheappropriatemetricallysimultaneouseventisaconventionaldecisionaccordingtoGrunbaumandothers.Figure10Metricalandtopologicalsimultaneityreflectionisatt=1onA’sclock.Theplane‘t=1’thendefinestheplaneofsimultaneityforthe‘reflection’eventQ;seethehorizontaldottedlinefromAtoQonBinFigure10above.Usingsignalsinthiswaythenallowsalldistantobserversinthesameframetoagreeonwhicheventsareandwhicharenotsimultaneouswithanyothergivenevent.51\nTIME,SPACEANDPHILOSOPHYHowever,ifwechallengetheassumptionthatthespeedoflightisindependentofdirection,thenAmaynotsay,withoutsomeotherassumptionaboutthespeedoflight,justwhattimeonA’sclockissimultaneouswithQ.NormayweusestandardsignalsynchronytoprovideageneraldefinitionofdistantsimultaneityinSTR.Fornouniqueplaneofsimultaneitymaybespecifiedforanyeventwhatsoever.Reichenbachprovidesausefulwayofhighlightingtheproblemhere.9Ingeneral,therelationshipbetweenthetimesforthetransmission(t0),reflection(t1),andreception(t2),accordingtoA,maybespecifiedasfollows:Reichenbachpointsoutthatthecasee=1/2correspondstothatinwhichthespeedoflightisthesameinbothdirections,i.e.thetimet1,supposedtobesimultaneouswithQ,occursatprecisely1/2thetimetakenfortheroundtrip(t2—t0).So,whent0=0,wemaysupposethatt1=1/2t2.Hence,whenweassumethespeedoflightisindependentofdirection,wealsoassumethatthate=1/2.AdolfGrunbaumclarifiesthedilemmaforobserverssuchasAalongthefollowinglines:1Thereisaninfinitenumberofeventsbetweent=0andt=2onA’sworldlinewhichmightbesaidtobesimultaneouswithQ;2wearesouncertainaboutwhicheventissimultaneouswithQgiventhefactthatnosignaltravellingfasterthanlightmaybeusedtonarrowthechoice;3thereforethesetofpointsbetweent=0andt=2onA’sworldlinemayonlybeconnectedtoQbyspacelikeintervals—forexample,thehorizontaldottedlinefromAtoQonBisaspacelikeintervalrepresentingthepathofanotional‘instantaneous’signal.4thissetasawholemaybesaidtobetopologicallysimultaneouswithQ;5adecisiontostipulatethespeedoflightinanygivendirectionwillpickoutauniqueeventfromthissetasbeingmetricallysimultaneouswithQ;however,6anysuchdecisionmustbeconventional:forexample,wemaychoosee=1/2andappealtothesimplicityinvolvedinourassumptionthatthespeedoflightisnotdependentondirection,andthenfixont=1asbeingsimultaneous;but,equally,7wemayadoptadifferentconvention,suchthateisnot1/2,andselectsomeotherpointfromthetopologicallysimultaneousset;ingeneral,anyvaluemaybeselectedforebetween0and1;52\nTRAVELLINGLIGHT8hencemetricalsimultaneitydependsuponaconventionaldecisionwhichisforceduponusasamatteroffactgiventhattheroundtripspeedoflightisfinite;seeFigure10(p.51).10Theimmediateproblemwhichthisanalysissuggestsisthis:canwedevelopaversionofSTRconsistentwiththebeliefthat?maytakeanyvaluebetween0and1?JohnWinnieseemstoprovideaconvincingandconsistentaccountofatheorywhichlookslikeSTRexceptinitsassumptionofe=1/2andtherepercussionsofthisassumption.11ThekeydifferencebetweenstandardSTRandthisnewversionisthatintheformerwespeakoftheinvarianceofthevelocityoflightinvacuainanydirectionasanessential(ifconventional)ideainthetheory;whereasinthelatterwemayonlyreferingeneraltermstotheinvarianceoftheround-tripvelocity.However,DavidMalamentarguesthattheconformalstructureofspacetimeinSTRforcesustoaccepttherelationshipofsimultaneitybythestandardsignalmethod.Thereareseverallevelsofstructurewithinanyspacetime:metric,affine,conformal,andtopological;seeFigure11(p.54)forareviewofthepropertiesassociatedwitheachlevel.Theconformalstructureunderwritesthenotionofanangleinspacetimeandthereforeprovidesuswiththeabilitytoconstructplanesofsimultaneityatrightanglesinalldirectionstoanyworldline.Conformalstructureisregardedforthisreasonasmorefundamentalthanthemetricalfeaturesassociatedwithmeasurementsbetweenpoints.Malamentshowsthattheimplicationsofdroppingtheassumptionthate=1/2arefar-reaching,sincewewouldhavetoreviseourideasabouttheconformalstructureofSTR’sspacetime.Yes,itistruethatwecouldinprincipleadoptsomenon-standardview.ButthenecessaryrevisionstotheframeworkofourspacetimetheorieswhichsuchamovewouldrequirewouldbesogreatastotakeusclearlyoutsideofthecontextofSTR.12ABSOLUTESIMULTANEITY?Grunbaumsupposesthattheconventionalityofmetricalsimultaneityfollowsfromamatteroffactaboutthefinitenessoflight’sround-tripspeed.Thissuppositionrevealsaninterestingassumption,whichheshareswithWesleySalmon.BothheandSalmon53\nTIME,SPACEANDPHILOSOPHYMetricalstructureofspacetimeThemetricofaspacetimegivessensetotheideaofthedistancebetweenpointsandthelengthofaline.Oncewehavespecifiedametricinagivenspacetime,weareabletotalksensiblyabout,notjustdistancesandlengths,butstraightlinesandtheanglesbetweentwostraightlines.However,thenotionofastraightlinedoesnotdependupontheideasofdistanceandlength,andsowemayintroducetwo(moregeneral)kindsofspacetimestructuretocapturetheideasofstraightnessandofangles.AffinestructureofspacetimeTheaffinestructureofspacetimepicksoutallthestraightlinesinthatspacetime.Sincewedefinestraightlinemotionasforce-freemotion,theaffinestructureallowsustodistinguishtheclassofinertialmotionsfromthatofnon-inertialmotion.ConformalstructureofspacetimeConformalstructuregivessensetotheideaofananglebetweentwointersectingstraightlines.TopologicalstructureofspacetimeThetopologyofaspacetimecapturesthepropertiesofthepointsofspacetimethemselves:whetherornotthepointsformacontinuum,howmanydimensionsthespacetimehas,whetherornotthespacetimehasboundariessettingitslimitsorholeswithinit,whetherthetimedirectionatanypointiswelldefined,andsoon.Hence,thetopologyofaspacetimeissaidtodescribethegeneralcharacteristicsofthesetofpointsformingthespacetime‘manifold’.Thismanifoldissometimesconsideredasthefundamentalbackgroundarenaintowhichtheotherthreekindsofstructureareintroduced.Figure11Synopsisofpropertiesofspacetimestructure:topological,affine,conformal,andmetricseemtobelievethat,ifthereweresomesignalwhichcouldtraveltosomedistantpointandbackinstantaneously,thentheissueoftheconventionalityofsimultaneitywoulddissolve.LikeSalmonheacceptsthat:AccordingtoNewtonianmechanics…materialparticlescanbeacceleratedtoarbitrarilyhighvelocities,wellbeyondthespeedoflight.IfNewtonianmechanicsweretrue,suchparticlescould,inprinciple,beusedtosynchronizeclocksandtoestablishrelationsofabsolutesimultaneity.13(Salmon1980:119)And,likeSalmon,healsoacknowledgesthat‘Wearenotmakinganyassumptionabouttherelativeone-wayvelocitiesofthefastestsuper-lightsignalthatisemployed.14PresumablySalmonandGrunbaumthinkthatsuchasignalwouldtravelinfinitelyfastinonedirectionandinfinitely54\nTRAVELLINGLIGHTfastintheother.Butthisneednotbethecase!Forallwewouldknowinsuchacase,accordingtoGrunbaum’sownposition,isthattheround-triptakesplaceinstantaneously.Ifweallowthepossibilityoftravelbackwardsintime,thenthefollowingsituationseemspossible:1att=0,theoutwardlightsignaltravelsatafinitespeedforwardsintime;but,afterreflection,2thereturningsignaltravelsbackwardsintimeandarrivesatthestarting-point,sothatnotimehaselapsedatthatpoint.Alternatively,theoutwardsignalmighttravelbackwardsintime,andthereturningsignalforwardstoarriveonceagainatt=0.Ofcourse,itwouldbedifficulttosingleout‘instantaneous’lightsignalsforsuchspecialtreatment;howlighttravelsdependsuponthemetricofthespacetimeanduponmorefundamentalfeaturessuchasconformalstructure.Wewouldthereforeexpectallmovingobjectstobehaveinasimilarmanner.Sowewouldneedtochoosebetweena‘simple’Newtonianworldinwhich‘timetravel’doesnottakeplaceandamorecomplexsituationinwhichitdoesoccur.Clearly,suchastrangesituationwouldrequireradicalrevisionsnotjustintheaccountwegiveofthevariousstructuresofNewtonianspaceandtimebutalsoinourotherphysicaltheories.However,anumberofpossibilitiesfor‘timetravel’areraisedbytheGeneralTheoryofRelativityandotherphysicaltheories,andsoperhapsweshouldnotbetoohastyindismissingtheideaasabsurddespitetheadviceofsomephilosophers.Butallowingsuchapossibilitymightalsoallowustosaythatthedirectiontravelledbyanysignalintimemightbedependentonthespatialdirectionofthesignal!Figure12(p.56)illustratesthreepossiblesituations:1AninfinitelyfastsignaltravelsfromAtoBandbackalongroute1,i.e.fromMtoQandback;ifthesignaltravelsinalldirectionsinthesameway,thenwecandefineaplaneof‘transmission’orthogonaltoA,i.e.atrightanglestoAinalldirections,whichiscoincidentwiththeplaneofsimultaneityt=0;theeventsY,M,andQlieonthisplane,whichiscommontoeveryobserver.2AsignaltravelsfromAtoBandbackalongroute2,i.e.fromMtoRandback,suchthattheoutwardsignaltravelsforwardsandthereturnsignalbackwardsintime;ifweareabletoconstructa‘planeoftransmission’consistentlyinsuchaspacetime,theninthiscasetheplanewouldbeangled‘downwards’tointerceptC55\nTIME,SPACEANDPHILOSOPHYFigure12AbsolutesimultaneityatXasshown;theNewtonianplaneofsimultaneitythroughthereflectioneventR(definedbythepathsthatone-wayinstantaneoussignalswouldtake)ist=1,andtheeventsZ,N,andRlieonthisplane.3ThesignaltravelsfromAtoBandbackalongroute3suchthattheoutwardsignaltravelsbackwardsandthereturnsignaltravelsforwardsintime;theplaneoftransmissionwouldnowbetiltedupwardstointerceptCatZ;theNewtonianplaneofsimultaneityforthereflectioneventPisnowt=-1;theeventsX,L,andPlieonthisplane.Inthiscaseandincase2,theangleofthetiltfortheplaneoftransmissionisclearlyarbitrary,giventhefactthatwemayselectanyfinitespeedforthetransmissionsignal.15Assumingthatwecanindeedconstructsuchplanesoftransmissionconsistently,itisclearthatthespeedandtimedirectionofsignalswillvarycontinuouslywithspatialdirection;forexample,signalscomingdirectlyoutofthepagewillhaveaninfinitespeed,sincetiltingthe56\nTRAVELLINGLIGHTplanedownorupdoesnotaffecttheslopeoftheplaneatrightanglestoAsothatalongthislinethetransmissionplanecoincideswiththeplaneofsimultaneity.Whichdirectionischosenfortheinstantaneousone-wayspeedisnotanabsolutecharacteristicofthespacetime,sinceitwouldclearlybeaconventionalmatterwhichspatialdirectionischosenasareferenceforanytiltsapplied.Therefore,eveniflightorsomeothersignalappearstotravelinstantaneouslyonsuchround-trips,weneednotselectt=0astheplaneofsimultaneity.Allpointsontheworldlineofadistantobserver(i.e.onthepathtracedoutinspacetimebythatobserver)are‘topologicallysimultaneous’withanygivenlocalevent.Andsowemayselectanyeventonthisworldlineas‘themetricallysimultaneous’event.Hence,thedegreeofuncertaintyaboutwhicheventmaybesaidtobemetricallysimultaneousinthiscasebecomesfargreaterthanthatintheSTRcaseabove.16Itisindeedamatteroffactthatlighttravelsaroundclosedspatialpathsatafinitespeed.However,wemaynotjumptotheconclusionthatinstantaneoussignalsmustimplyabsolutesimultaneity.IfweaccepttheargumentsforconventionalityinSTR,thenthereseemslittlereasontoresistthedemandforconventionsevenin‘Newtonian’contextswheresomesignalsmaymakeround-tripsinaninstant.SLOWCLOCKTRANSPORTCouldweside-steptheproblembyfindingsomeotherstandard,freefromanyconventionalelement,whichwouldallowustodeterminetheone-wayspeedoflight,atleastinthecontextofSTR?Insteadofusingstandardsignalsynchronytodeterminesimultaneitybetweendistantevents,wemightusethemethodofslowtransportofclocks.ThismethodisrecommendedbyEllisandBowmanandbymanyothersasthemostpromisingconventionfreeapproachtothesynchronisationofdistantevents.17AlthoughslowclocktransportissetfirmlywithinthecontextofSTR,thegeneralideainvolvedmaybefoundinarenownedseventeenthcenturyattempttomeasuretheone-wayspeedoflight.Inthe1670s,theDanishastronomerOleRømerarguedthatthespeedoflightisfiniteandapproximately210,000km/sec.MostscientistsintheseventeenthcenturywerereluctanttodisagreewithDescartes’sverdict,givenin1637,that‘lightcanextenditsraysinstantaneouslyfromtheSuntous’.18However,RømerrealisedthattheperiodsbetweentheeclipsesofJupiter’smoonsoughttoexhibitthesamekindofregularity57\nTIME,SPACEANDPHILOSOPHYasotherastronomicaleventsofthiskind;butthisisnotthecase.Hebelievedthattheirregularitiescouldbeexplainedbythefactthatourobservationsoftheeclipsestakeplacefromdifferentviewpoints:sometimestheEarthisrelativelyclosetoJupiterandsometimestheSunliesbetweenthetwoplanets.Hence,lightfromtheeclipsesneedstotravelacrossthediameteroftheEarth’sorbitwhentheEarthisatitsmostdistant.ButwhentheEarthisatitsclosesttoJupiterthedistancetravelledbythelightisclearlysomuchshorter.Ifthespeedoflightwereinfinitelyfast,thentheirregularitieswouldremainapuzzle.ButRømer’scalculationsindicatedthat,inordertocompensatefortheobservedirregularities,lightshouldtakesome22minutestotravelacrossthediameteroftheEarth’sorbit.19Rømerthenusedtheavailableestimateofthediametertoestimatethespeedoflight.RømerusedclocksonEarthasthestandardforhismeasurements.ButtheseclocksarenotstationarywithrespecttoJupiteranditsmoons.AswesawinChapter2,accordingtoSTR,thetimeregisteredbyaclockasitmovesbetweentwogivenpointsinagivenframedependsuponitsvelocityrelativetotheframeandonthepathtaken.ThereforewecannotsimplyassumethatthejourneytheclocksmakefromonesideoftheEarth’sorbittotheotherhasnoeffectwhatsoeveronthemeasurementsofthetimesbetweeneclipses.Thesemeasurements,ofcourse,providethedataforthecalculationofthespeedoflight.Iftheclocks’readingsareaffectedbythejourneyaroundtheSun,wewillneedtomakeasuitablecorrectiontocompensatefortheseeffects.Thefigurecalculatedforthespeedoflightwillthenbedependentupontheassumptionssupportingthiscorrection.Butaretheseassumptionsfreefromconvention?Mayweuseclockswhicharetransportedfromonelocationtoanotherasnon-conventional,objectivestandardsforourmeasurementsofthespeedoflight?EllisandBowmanbelievethatwemayuseamovingclocktoprovideanobjectivefigureforthespeedoflightsolongastheclockistransported,stepbystepthroughtiny,equalintervals,atanextremelylowspeedrelativetotheframeinwhichthebeginningandendofitsjourneyaresituated,i.e.byusingtheslowclockmethod.AlthoughtwoinitiallysynchronisedclocksmovingfromAtoBviadifferentpathswillingeneralnotbesynchronisedatB,theywillbesynchronisediftherelativevelocityandthereforethespeedoftransportationisinfinitesimallysmall.EllisandBowmanarguethatslowclocktransportispossible;theyremindusthattheLorentz58\nTRAVELLINGLIGHTtransformations(LT)allowustocalculatewhathappenswhenclocksaremovedfromplacetoplace.Wedonotactuallyneedtoslowmovingclocksdowntoastop;wemerelyneedtobeawarethatthemathematicalcharacteristicsoftheLTallowustoinferthepossibilityofsynchronisationonthebasisofthebehaviourofactualclocksaswemovethematslowerandslowerspeeds.Wemaythencorrectanyreadingsmadebyarealmovingclockonthisbasis.However,others,followingGrunbaum,arguethatouruseofanysuchcorrectiondependsuponaconventionaldecisiontoadoptametricforSTRspacetimewhichallowsustospecifythetimelikeintervalbetweeneventsandtogivesensetotheideaof‘equalintervals’;theythenarguethatnothingcompelsustochoosethisparticularmetricexceptourconventions.20Hence,theysay,thecalculationfortheone-wayspeedoflightstilldependsuponourconventionalchoice,ratherthanuponsomeobjectivefeatureoftheworld.Figure13Pathwaysinspacetime59\nTIME,SPACEANDPHILOSOPHYThelightconeaboveillustratestherangeofpossibleinfluenceoftheeventCwithintwospatialdimensions(left/right;in/out).Figure14LightconeSPACELIKETRAVEL:ATALEOFTWOTACHYONSInthespacetimesoftheSTRandTheGeneralTheoryofRelativity(GTR),wemayconnectanytwopointsgeometricallybytimelike,null,orspacelikepaths;seeFigure13(p.59).Whenatimelikepathconnectstwopoints,asignaltravellingatlessthanthespeedoflightmaypasscontinuouslyfromonepointtotheother.Whenanullpathconnectstwopoints,onlyasignaltravellingatthespeedoflightmaypassbetweenthem.Thenotionofcausalityinrelativityistiedtotheideaoflightbeingthefastestpossiblecausalsignalbetweentwopoints.WhenaneventCissaidtobeacauseofanothereventE,weusuallysaythatthesetwoeventsmaybeconnectedbynon-spacelikecurves,i.e.byeithernullortimelikepaths,sothatasignaltravellingatorbelowthespeedoflightmaytravelbetweenCandE.Sothe‘lightcone’ofaneventCmaybethoughtofastheconeofpossiblecausalinfluence—Cmayberegardedasaphysicalcauseofanyeventwithinoronthelightcone;seeFigure14above.However,spacelikepathsarenotconfined60\nTRAVELLINGLIGHTtothelightcone,sotheyarenotregardedaspossiblephysicalpathwaysforcausalsignals:ifnothingcantravelfasterthanlight,thennothingcanfollowaspacelikepath.Hence,theideaofaspacelikepathorcurvemayberegardedasmerelyanartefactwhichallowsustoexplorethegeneralgeometricalpropertiesofspacetime.Thisisastandardviewofrelativity,whichexcludesthepossibilityofparticlesmovingalongspacelikecurves.21Whathappenswhenwerelaxthestipulationthatnophysicalsignalorparticlemaytravelfasterthanlight?Thisdoesnot‘breakthelawsofrelativity’:thelawsofSTRdonotlegislateagainstparticlestravellingfasterthanlight.ThekeyideabehindSTRistheinvarianceratherthanthelimitingcharacterofthespeedoflight.Thestandardviewofrelativityincorporatestheideaofthespeedoflightasmaximalasanadditional‘physicallyplausible’hypothesis.Asanumberofrelativistshaveshown,wecanconstructframeworksconsistentwithSTRinwhichparticlesmaytravelat‘super-luminal’velocities.22But,ifwegivesuchparticlesaplacewithinthecontextofSTR,wemustfacetheratherunsettlingconsequences.Particlestravellingfasterthanlightaretypicallycalled‘tachyons’.23Andtachyons,somewriterstellus,maybeusedtotransmitmessagesbackwardsintime!Thetachyonmaytravel,weareoftentold,outsidethelightconeoftheeventwhichisregardedasitssource.Hence,tachyonsmayfollowspacelikepaths.Tachyonsneednotchallengethepostulateoftheinvarianceofthevelocityoflight.For,althoughtheideathatlightisthefastestsignalisintimatelyconnectedwithrelativity,themetricalgeometryofspacetimesinSTRandGTRdependsuponthedistinctandfundamentalnotionoftheinvarianceofthevelocityoflight.Hence,theframeworkinwhichweshallexplorethebehaviouroftachyonsisconstructedwithinthecontextofthepostulate:lighthasthesameinvariantvelocityinallinertialreferenceframes.Thispostulateallowsustodefineplanesofsimultaneityinspacetime.Andthiswillinturnenableustonoteanycausalanomaliesarisingfromtheintroductionoftachyonsintothespacetime.Althoughweshallseethattheremaybegroundsforscepticismabouttheallegedcausalanomaliesassociatedwithtachyons,thebehaviouroftachyonsdoesraiseaninterestinggeneralpointconcerningattemptstoredescribebackwardscausationintermsofforwardscausation.Manystoriesaretoldabouttachyons,butmostsharethesamegeneralcharacteristicsasrevealedintheaccountgivenbyR.C.TolmananddevelopedbyDavidBohmandothers.24Imagine61\nTIME,SPACEANDPHILOSOPHYtwoinstrumentswhicharecapableofemittingandabsorbingtachyons.OneinstrumentiscloseathandinalaboratoryL;theotherisinaspaceshipMmovingawayfromusatasteadyspeed.Wemustnotethat,althoughwecanrepresentthissituationonasinglespacetimediagram,wecannotdrawasingleplaneofsimultaneitycommontobothLandM.And,giventheconstraintsofsuchdiagrams,theworldlineofanyobjectemittedatasteadyspeedfromasourcemustdependuponthevelocityofthatsource.Theplaneofsimultaneityforagivenobserver(e.g.ascientistinthelaboratory)representsthesetofworldlinesofthosesignalsmovinginfinitelyfastawayfromthatobserver.Hence,distanteventssimultaneouswitheventsinthelaboratoryLlieonaplaneS1orthogonaltothelaboratory’sworldline;buttheplaneofsimultaneityS2ofthespaceshipMistiltedasshowninFigure15(p.63).Thefasterasignalmovesfromagivenobserver,the‘closer’theworldlineofthatsignalwillbetotheobserver’splaneofsimultaneity.WenowimagineanextremelyfasttachyonmovingawayfromLandtowardsM,representedbytheworldlineP1(closetoS1).NoticethatS2meetstheworldlineofLatatimeearlierthantheemissionofthistachyon;thisissimplyaresultofthefactthattwoobserversinrelativemotionwillingeneraldisagreeaboutwhicheventsaresimultaneous.ThetachyoniseventuallyabsorbedbytheinstrumentonboardM.Thisinstrumentisprogrammedtoemitatachyonshortlyafterabsorbingatachyon.SoafurthertachyonisthenemittedtowardsL,atthesametremendousvelocityandthereforeclosetoitsplaneofsimultaneityS2,alongtheworldlineP2.Thissecondtachyontravelsforwardsintimewithrespecttothespaceship.But,giventhetilttothespaceship’splaneofsimultaneity,thetachyonmeetstheworldlineofLatatimeearlierthantheinitialemissionfromL!Hence,fromthelaboratorypointofview,thesecondtachyonseemstobetravellingbackwardsintime,giventhe‘facts’that:1atachyonfollowsthepathP1leavingLattimet*;2atachyonfollowsthepathP2,meetingLattimet,suchthattisearlierthant*;3thetachyonfollowingP,isabsorbedbytheinstrumentonthespaceshipS;andthetachyonfollowingP2isthenemittedbythisinstrument.Ifthisstoryiscoherent,thentachyonsseemtoallowthepossibilityofsendingmessagesbackwardsintime.Ofcourse,sincepeopleare62\nTRAVELLINGLIGHTFigure15Tachyons:travelbackwardsintime?63\nTIME,SPACEANDPHILOSOPHYclearlynotmadeupoftachyons,thereisnoquestionofanyhumantravellingbackwardsinthisway.Butthereisstillaseriouscausalanomalyhere:theinstrumentinthelaboratoryseemstoreceive,atagiventime,asignalwhichmaybetraced‘back’toaneventatalatertimeinthelaboratory.Itseemsthatinsuchacase‘backwardscausation’ispossible:thatis,aneffectmayinsomecircumstanceshappenbeforeitscause.Ifweaskwhatcausesthesecondtachyontoreachthelaboratory,thentheanswerseemstobeitsemissionfromthespaceship;andthiseventonlytakesplacebecauseoftheabsorptionofthefirsttachyon,whichitselfisaconsequenceoftheinitialemissionfromthelaboratory.Hence,fromthepointofviewofanyoneinthelaboratory,theeffectdoesseemtoprecedethecause.Particlephysicistsfrequentlyrefertopairsofparticlesandantiparticles.Theelectronispairedwithitsantiparticle,thepositron;similarly,thetachyonmaybepairedwiththeantitachyon.RichardFeynmanclaimed,inapaperpublishedin1949,thatapositrontravellingforwardsintimemaybereinterpretedasanelectronmovingbackwardsintime;seeFigure16(p.65).25Thisinterpretationhasanumberofadvantages,giventhatitallowsphysiciststoconsidertwodistantbutcorrelatedeventsastheeffectsofasingleparticle,anelectron,movingforwardsandbackwardsintime.WeshalllookatFeynman’sideaingreaterdetailinChapter8.But,atthisstage,wemaynotethegeneralpointarisingfromhisanalysis:aparticletravellinginonedirectionintimemaybetreatedas(ifnotidentifiedwith)itsantiparticletravellingintheoppositedirection.Hence,atachyonmovingbackwardsintimemaybetreatedasanantitachyonmovingforwardsintime.Thisapproachmightallowustoredescribethetaleoftwotachyonsfromthepointofviewofsomeoneinthelaboratoryasfollows:1atachyonfollowsthepathP1leavingLattimet*;2anantitachyonfollowsthepathP2leavingLattimet,wheretisearlierthant*.Thisredescriptionraisesanimmediatequestion:howmayweaccountforwhatseems(fromthelaboratorypointofview)tobethenearsimultaneousabsorptionofantitachyonandtachyonbytheinstrumentonboardthespaceship?Fortheseabsorptionsseemtodemandacorrelationbetweentheemissionsfromthelaboratorywhichmaybehardtoexplaininanystraightforwardway.Isthe64\nTRAVELLINGLIGHTFeynmanarguesthatwecantellstoriesaboutparticlesindifferentways:considertheabovesituation:A*isthepathofanelectron;B*isthepathofapositron:andC*isthepathofanotherelectron.PisthepointwheretheelectronfollowingpathC*andthepositronfollowingB*arecreated;andQisthepointwherethepositronandtheelectronfollowingA*meetandannihilateeachother.Thisstoryrequirestheideasofparticlecreationandannihilation,andmodernparticlephysicsdoesindeedgivesubstancetotheseideas.Butwecanretellthestoryintermsofasingleparticle,anelectron,whichfollowstherouteA®B®C.OfcoursewenowneedtoprovidesomeexplanationofwhytheparticleswitchesdirectionintimeatPandQ.Butwenolongerneedtoleanontheideasofcreationandannihilation.Eachstoryhasitscomplexitiesandsimplicities.Theadvantageofthesecondstoryisthatwemaytreatpositronsaselectrons(inreversetimedirectionadmittedly)—butweonlyneedonekindofparticle.Ingeneral,wemighttreatallantiparticlesasparticlestravellingbackwardsintime.Figure16Feynmandiagramlaboratoryinstrument‘forced’tosendoutthetachyonhavingalreadyemittedtheantitachyon?Ifthisisthecase,whyshoulditbeso?Thesequestionsraiseageneraldifficultyformanyredescriptionsofbackwardstravelandcausation:dowenotcommitourselvestounexplainedcorrelationsbetweenevents?Thisproblemneedsfurtherexploration,andinChapter8weshallexaminetheissuesinvolved.Thereare,however,stronggroundsforsuspicionaboutthecoherenceofthetachyonstory.Tobeconsistentwemustaccountforthefactthat65\nTIME,SPACEANDPHILOSOPHYtheinstrumentonboardSisprogrammedtoemitfollowinganabsorption.Butthisinstrumentdoesnotseemtoabsorbanythingatall;rather,asRecamishows,fromthepointofviewofanobserveronboardS,itseemstoemittwoantitachyons!Inathoroughanalysisoftheproblemoftachyons,Recamiarguesthat,fortheinstrumenttobehaveasplanned,theparticleabsorbedmustleavethelaboratorybeforetheemittedparticlereturns.Hegoesontosuggestthateventheideaofabsorptionandemissionmaybeframe-dependent,sothatwhatcountsasanemissiontooneobservermaybeanabsorptionfromadifferentpointofview.Thedetailsofthisanalysisarecomplex,butRecamidrawsthemoralthatconfusionwillalwaysarisewhenwe‘mixtogetherthedescriptionsofonephenomenonyieldedbydifferentobservers’—andthattheeventsonboardSmustnotbedescribedas‘fact’fromthelaboratorypointofview.26JUSTTHETWOOFUS:ACROSSTHEUNIVERSE?RobertElliotsuggestsaningeniouswayinwhichpeopleandnotjusttachyonsmighttravel‘fasterthanlight’.27Withjudiciousplanningandbymakinguseofadvancedtechnology,hesaysthatanyonemaytravelfasterthanlight,i.e.followaspacelikepath.The(asyetunavailable)technologicalrequirementsare:theabilitytomanufacturea‘total’biochemicalblueprintofthephysiologicalandpsychologicalcompositionofagivenindividual;andtheabilitytoreconstructanindividualfromthedetailsofsuchablueprintevenafterthetransmissionofthesedetailstodistantstarsystems.Elliot’splanforfaster-than-lighttravelmayberetoldasfollows:1OntheEarth,atotalblueprintismadeofEdith—andthisisstoredsafely.2EdithgivesinstructionsthatsheshouldbeembodiedonEarthfromthedetailsoftheblueprintafterapreciselygiventime—sayexactlytenyears.3Acopyoftheblueprintisthensentby‘conventional’meanstoaplanetinadistantstarsystematthespeedoflight—ajourneywhichtakesfouryears.4AtthemomentoftransmissionfromtheEarth,Edithisdestroyed;but,whentheblueprintreachestheplanet,Edithisreconstructedfromtheblueprintdetails.66\nTRAVELLINGLIGHT5Afterspendingsixyearsseeingthelocalsights,Edithisdestroyed.6AtthemomentthatEdithisdestroyedonthedistantplanet,EdithisreconstructedfromtheblueprintdetailsleftonEarth.7The‘journey’fromtheplanettotheEarthtakesplaceinstantaneously:atonemoment,Edithisaliveandwellonthedistantplanet,andinaninstant,sheisrelocatedtoEarth;soEdithtravelsfasterthanlight!Evenifweacceptthatcausalsignalsmaytravelnofasterthanlight,ElliotassuresusthatanyonewhocarriesoutEdith’splandoesindeedmakeaninstantaneouschangeofspatialpositionsuchthatonlyaspacelikecurvemayconnectthepointsofdepartureandarrival.Ofcoursewemustaccepttwobasicpresumptions:first,itmustbepossibleinprincipletomakesuchablueprintfromwhichanindividualcouldbegenuinelyreconstructed;and,secondly,wemustagreethataplaneofsimultaneitymaybedefinedsuchthateventsonthedistantplanetandontheEarthmaybesaidtobesimultaneous.Letusgrantbothofthese.However,thequestionofwhetherornotEdithdoestravelfasterthanlightseemstoturnontheclaimthattheEdithwholeftthestaris‘psychologicallycontinuousandcohesive’withtheEdithreconstructedontheEarth.28Ifweagreethatthefirstconventional‘journey’,fromEarthtoplanet,doesnotdisruptEdith’spsychologicalandphysiologicalcontinuityandcohesiveness,thenElliotsaysthatthereislittlereasonforustoobjectinthecaseofthesecondmoreproblematic‘journey’.Letusalsoaccept(ifonlycautiously)theassertionthatpersonalidentitymaybepreservedincaseslikethefirstjourney.29WemightjustifythisdecisionbyagreeingtoLeibniz’sprincipleofidentityofindiscernibles:iftwoobjectsareidenticalinallrespects,thentheyareoneandthesameobject.Hence,iftheEdithwhoappearsonEarthisidenticalinallrespectswiththeEdithwhodisappearsfromthedistantplanet,thenwemustbetalkingaboutjustoneEdith.SoisEdith’s‘return’totheEarthreallyacaseoftravelfasterthanlight?WhenEdithspendssixyearsintheneighbourhoodofthedistantstar,wewouldexpecthertoundergomanyphysiologicalandpsychologicalchangesduringthistime.Shemightfallinloveordevelopafearofspiders;shemightloseatoothoracquireanulcer.Consequently,thephysiologicalandpsychologicalcompositionofEdithaftersixyearsislikelytodifferinmanyrespectscomparedwiththecompositionofEdithasrecordedontheblueprintonEarth.EvenifweretellthestorysothatEdithspendsjustfractionsofasecondinherembodiment,changeswilloccurinEdith’scomposition—neuronswillfireandcellswilldie:ifshespendsanytimeatallembodiedonthe67\nTIME,SPACEANDPHILOSOPHYplanetthenthisEdithwillnotbeidenticalwiththeEdithembodiedonEarth.Butthismeansthatwelosethepsycho-physicalcontinuityessentialtoElliot’sclaimthatinsuchcircumstancesEdithwouldhavetravelledfasterthanlight.ThetwoEdithswouldonlybegenuinelycontinuousifnotimeatallelapses,fromherpointofview,betweenherembodimentanddestructiononthedistantplanet.SogenuinecontinuitycouldonlybeguaranteedifEdithisnotembodiedatallontheplanet.ButthismeansthatthereisnoquestionofanyonetravellingfromtheplanetbacktoEarthatall!NoEdithisavailabletomakeanysortofjourney.And,ifshewereavailable,thenthepossibilityofchangesinthecompositionofthisEdithdisruptstheessentialcontinuityneededfortheidentificationofherwiththeEdithreconstructedfromtheoriginalblueprintonEarth.So,evenifwemakesomerathercourageousassumptionsaboutthenatureofpersonalidentity,westillfailtoprovideaconvincingcaseforspaceliketravel.So,ifshereallywantstogoplaces,Edithoughttosticktomoreconventionalmeansoftravel.68\n4ACONVENTIONALWORLD?INTRODUCTIONTheshortestdistancebetweentwopointsisastraightline.Twoparallellineswilltendtostayparallel.Theinternalanglesofatriangleaddupto180degrees.Suchstatementsbelongtotheapparentlysecuredomainofhighschoolmathematics.TheyhaveastheirbasistheaxiomsofEuclideangeometry.Themoreuseful,butcomplextheoremsofEuclideangeometryaretypicallydeductionsfromthebasicdefinitions,axioms,andpostulatesofthesystem.1Whydoweacceptthesebasicideas:becausetheyareself-evidentlytrue,becausetheyareinherentlyrational,orbecausetheyareintuitivelycorrect?Allthesereasonshaveindeedbeenadvanced.But,insteadofcommittingourselvessowhole-heartedlytothebuilding-blocksofEuclideangeometryassomehownecessarilytrue,wemightjustacceptthembecausetheyprovideuswithaninternallyconsistentsystemwhichseemstounderwritesomuchthatwesayaboutthephysicalworld.Then,ifwearefacedwithachoicebetweengeometries,wecanmakedecisionswithoutfeelingtheneedtojustifyourchoicefromfirstprinciples.Thechoiceusedtobesimple:Euclideangeometryornothing.Andwhyshouldanyonequestiontheseaxiomswhentheyseemedtobesointunewiththeevidenceofoureyes?TodoubtEuclidwasalmostasgreataheresyasatheism.However,mathematiciansliketoplayinnovativegamesandtheysometimeshavescantrespectforeventhemostsacredofcows.AndalongwiththespiritoffreeenquiryinthenineteenthcenturycamemathematiciansboldenoughtochallengeEuclidandtalentedenoughtoconstructalternativegeometricalsystems,inwhichthethreeanglesoftrianglesmightnotaddupto180degreesandinwhichlinesparallelatonelocationmightindeedmeetatanother.Someregardedthenewgeometrieswithsuspicion.HenriPoincaré69\nTIME,SPACEANDPHILOSOPHYsuggestedinScienceandHypothesis(1902)that‘Euclideangeometry…hasnothingtofearfromfreshexperiments’—hebelievedthatEuclideanandnon-Euclideangeometriesaredistinct,incompatibletheorieswhichmaybeequallysupportedbyempiricalevidenceofobservationandexperiment.2Withafew‘trifling’ifcomplicatingassumptions,wecouldretaintheEuclideanworld-viewandstillbeconsistentwiththeempiricalevidence.Wemightthenresisttheadoptionofanynon-Euclideantheory—despiteanyadvantagessuchanewoutlookmightseemtooffer.Hence,thedecisiontoadopteithergeometricalperspectiveisessentiallyconventional;seethepreviouschapterforsomeintroductoryremarksaboutconventionalism(pp.46–9).Euclidwasdestinedtofightonelastbattlesome2200yearsafterhisdeath.WiththedevelopmentoftheGeneralTheoryofRelativity(GTR)in1915–16cametherealisationthatnon-Euclideangeometriesmightbemorethanmathematicalfantasies.ThesenewgeometriesmightbefarmoreappropriateforourphysicaldescriptionsofgravitationalandotherfieldsthantheEuclideanview.Thenon-EuclideanapproachofGTRdidnotconvinceeveryonethattheEuclideanperspectivewasaltogetherredundant.HansReichenbacharguedthattheempiricalevidenceinfavourofGTRneednotforceustoacceptanon-Euclideanview.But,unlikePoincaré,hedidnotbelievethattwoviewswhichareequallysupportedbytheevidencearegenuinelydistinct.3Reichenbachsaysthatthenon-EuclideanviewisempiricallyequivalenttoaEuclideanperspective,oncewemakeadditionalassumptionsaboutthephysicalworld.4Ofcourse,wemightneverthelesshavegood(non-empirical)reasonsfordecidingtoadoptanon-Euclideanview.ButifwemoveawayfromaEuclideanview,therangeofnon-Euclideangeometriesonoffermightseemtobebewildering:spacetimeswithmetricfields,withscalarfields,andwithvector,tensor,ornon-dynamicalfields.Indeed,sometheoriesnolongerrestrictthemselvestothefourfamiliardimensionsofspaceandtime,forexample:the‘string’theoriesproposedbyGreenandSchwartzwhichemploymultiple(andnotallstrictlyspatio-temporal)dimensions;orthe‘chaoticinflation’cosmologicaltheoryofLindewhichallowsforacompartmentalizeduniverse—brokenupintomanydistinctandisolatedregions,someofwhichmayhave,say,justtwospatialdimensions.5Inthischapter,weshallexamineavarietyofsituationswhichmightbesaidtoinvolveeitherempiricalequivalenceorconventions.OurdiscussionofEuclideanversusnon-Euclideangeometrywillbeginby70\nACONVENTIONALWORLD?focusingonthemetric.Weshalltheninvestigatetheproblemsraisedbythetopologicalstructureofspaceandtime.WeshallthentrytogetafirmergriponthegeneralphilosophicalproblemsraisedbyReichenbachandPoincaré;butsadlythisisasfaraswecantakethediscussionofconventionalisminthisbook.Theissuesarecomplex,andtheaimhereistoclarifythecentralproblemsasmuchaspossible.Butsimplytryingtoreachthisgoalwillbringmanyrewards:forweshallalsoobtainamorecomprehensivepictureofbothEuclideanandnon-Euclideangeometriesandalsoofrelativitytheory.6WHENPARALLELLINESMEETThefirst‘convention’whichweshallexamineisthechoicebetweenEuclideanandnon-Euclideanmetricalgeometry.ThegeometricalpostulatewhichcausedallthetroublefortheEuclideanpictureisthefifth,thepostulateoftheparallels,whichstatesthat:throughanygivenpointthereisoneandonlyoneparalleltoagivenstraightline(whichdoesnotgothroughthegivenpoint),i.e.onestraightlinewhichliesinthesameplanewiththefirstanddoesnotintersectit.(Reichenbach1957:2)Figure17(p.72)illustratesthesituationenvisagedbythepostulate.OnlyonelinethroughPispossibleifthatlineisnottointersectastraightlineABwhilstremaininginthesameplaneasAB.Wemaysympathisewiththosewhofeltthatthispostulateisself-evidentlytrue;but,asReichenbachremarks,perhapsweshouldbesuspiciousaboutthestatusofthepostulatesincetheprohibitionagainstintersectionseemstocontinuebeyondtheconfinesofourfiniteexperiencerightontoinfinity.CarlFriedrichGauss,JanosBolyai,andNikolaiIvanovichLobatschevskiiarebelievedtobejointlyresponsibleforthechallengetothisaxiom,thusproducingthefirstnon-Euclideangeometry;onlythelasttwomathematicianspublishedtheirresults—inthelate1820sandearly1830s—andthenewgeometrywasshowntobeinternallyconsistentsomefortyyearslaterbyEugenioBeltrami.7Theend71\nTIME,SPACEANDPHILOSOPHYAnylineotherthanLwillintersectthecontinuationoflineABaccordingtoEuclid’sparallelpostulate.LineLissaidtobeparalleltoAB,anditistheonlyparallellinewhichmaybedrawnthroughPinthesameplaneasAB.Figure17ParallelpostulateofEuclideangeometryresultoftheircombinedeffortsisageometricalmodelwhichisconsistentwithallofEuclid’saxiomsexceptthefifth.Inthis‘non-Euclidean’model,theremaybemorethanoneparallellinethroughP.Thethreeanglesofatrianglewilladduptolessthan180degrees.Althoughtheparallellinesandthesidesofthetrianglearenot‘straight’inaEuclideansense,theyareneverthelessasstraightaspossiblewithinthisnon-Euclideanmodel:suchlinesarecalled‘geodesies’.FurtherworkbyGeorgRiemannledthemathematicalcommunitytowardstheideaofgenerallycurvedspacesinwhichcurvaturemightvaryarbitrarilyfrompointtopointsothatspacemightbe‘flat’orEuclideanatonepointand‘curved’atanother.8Riemann’sworkalsosuggestedtheideaofapositivelycurved‘spherical’space,whichisperhapsthemosteasilyimaginednon-Euclideanspace:seeFigures18and19(pp.73,74);andhismathematicalgeniuscontributedtowardsthedevelopmentofthetensorcalculuswhichprovidedthemathematicaltoolsneededbyEinsteinandhisco-workersintheirconstructionofGTR.Somemathematicians,includingRiemann,didaskwhichkindofgeometryshouldbeemployedtodescribetheobservedphysicalworld.However,withoutfirmempiricalevidence,therewasverylittlechanceofnon-Euclideangeometriesbeingregardedasanythingmorethanmathematicalcuriosities,tributestothetremendousingenuityoftheirdiscoverers.Thedevelopmentofthe72\nACONVENTIONALWORLD?ThethreeinternalanglesofthetrianglePQRonthesurfaceofthesphereaboveadduptomorethan180degrees.TwoparallellinesthroughPandQheadingdirectly‘north’willmeetatthe‘pole’ofthesphere.Figure18IdeaofcurvatureonsurfaceofasphereSpecialTheoryofRelativity(STR)andthenGTRchallengedthissituation.STRinvolvesafour-dimensionalviewofthephysicalworld.Minkowskishowsthatthemostnaturalwaytodescribephysicalrealityisintermsofeventsinafour-dimensionalframework:spacetime.9AndGTRlinksthedistributionofmatterandenergythroughoutthephysicaluniversetothemetricalgeometryofspacetime.Thislinkhighlightstherelationshipbetweenmotionandcurvature;inGTR,astheeminenttheoreticalphysicistJohnArchibaldWheelersoelegantlysays:mattertellsspacetimehowtocurve,andspacetimetellsmatterhowtomove.10Indeed,evenfreelymovingmasslessparticles,suchaslight,followthegeodesiesofspacetimeinGTR.Andthesegeodesiesaredeterminedinpartbythedistributionofmatter.Amassiveobject,suchastheSun,causesmarkedcurvatureinitsimmediateneighbourhood;seeFigure20(p.75).HenceGTRpredictsthat73\nTIME,SPACEANDPHILOSOPHYThisillustrationpresentstheessentialideaofagenerallycurvedspacetime.Hereweseeatwo-dimensionalsurfacecurved.Thegridonthesurfacerepresentsthepathswhichwouldbetakenbyfreeparticlesorraysoflight.Buttheanalogybetweenthecurvatureofspacetimeandthatofthissurfaceislimited.Thetwo-dimensionalsurfacesitsinathree-dimensionalbox.However,thereisnohigherspatialdimensioninwhichthree-dimensionalspacesits.Spacetimeisallwehave,anditisamistaketothinkintermsofspacetimeasathree-dimensionalspatialstructuresittinginsomebackgroundcontainer.Figure19IdeaofagenerallycurvedspacetimelightfromanydistantstarfollowsacurvedpathasthelightpassesclosebytheSun.Physicistsarenormallyreluctanttoembraceanytheorywithoutsomehardempiricalevidence.Theconfirmationofthisprediction,bytheastronomerSirArthurEddingtonin1919,helpedtoconvincemostofthemthatweinhabitanon-Euclideanworld.11WILLTHEREALGEOMETRYPLEASESTANDUP?Theempiricalevidencepointstowardsanon-Euclideanworld.However,ifthemessageofconventionalismiscorrect,thenweoughttobeabletoconstructatheorywhichistheempirical74\nACONVENTIONALWORLD?Observersseelightcoming‘straight’atthemfromthestar.ComparisonofthepositionofthestarbeforeandwhentheSunmovesacrossthelineofsightdemonstratesthatthelightdoesnotfollowthesamepathwhenamassiveobjectapproachesthatpath.Incertaincircumstances,wecanactuallyseebehindtheSun!Figure20BendingoflightingravitationalfieldequivalentofGTR,butwhichimpliesthatthemetricalgeometryoftheworldisEuclideandespitetheevidencetothecontraryandwhichwecanadoptwithoutabsurdity.PoincaréandReichenbachsuggestthatsuchworldsareindeedpossibleandthetalewhichfollowsisbasedontheirsuggestions.12Imaginethattheinhabitantsofoneoftheouterplanets,Neptune,havealwaysusedEuclideangeometryandtheyareconvincedthattheuniverseisEuclidean:everymeasurementtheyhavemadeseemsto75\nTIME,SPACEANDPHILOSOPHYbeinlinewiththisgeometry.TheyknowofnootherpossiblegeometryandtheyhavenoreasontodoubtthestatusoftheEuclideanview.Buttheyarecuriousabouttheinnerplanets,andapartyfromNeptunesetsoffinaspaceshiptoexplorethesolarsystemclosertotheSun.Ontheirjourney,inordertokeepthespaceshiponcourse,theymakeaseriesofmeasurementswithavarietyofmeasuringinstruments;someofthesearebasicinstrumentssuchasmetrerulesandclocks,andothersaremoresophisticatedsuchastelescopes.Theseinstrumentshavecommonstandardsofspatialdistanceandtemporalduration.AllgoeswelluntilthecraftpassesinsidetheorbitofEarth.TheythennoticethateverymeasurementtheymakenolongerconformstotheEuclideanview.Forexample,thelightfromdistantstarsnowseemstobefollowingcurvedratherthanstraightpaths;andthemeasurementsmadeontriangleswithinthespaceshipshowsmallbutsignificantdeviationsfromexpectedvalues.WherevertheyexplorewithinthespheredefinedbytheEarth’sorbit,thestoryisthesame.Andtheresultstheyobtainarethesameregardlessofthekindofmeasuringinstrumentused.However,assoonastheymoveoutsidethisorbit,theirmeasurementsconformtotheEuclideanviewonceagain.Theyenterandleavethespheredefinedbytheorbitseveraltimes,checkingandrecheckingthisstrangephenomenon.AsgoodEuclideans,theyarereluctanttodoubttheapplicabilityoftheirlong-standinggeometricalperspective.Infact,itneverevenoccurstothemthattheEuclideanviewmightbewrongorinneedofsomecorrection.Insteadtheydecidethattheremustbesomegeneralandall-pervasiveinfluenceatworkwithinthespheredefinedbytheEarth’sorbitwhichdistortsalltheirmeasuringinstruments.Theyalsoconcludethatthisinfluencedistortseverything,includingthepathsoflightbeams,whichmightbeusedtomakeanymeasurementofdistance.Unlikeothernaturalphenomenawithwhichtheyarefamiliar,suchasheatormagneticforces,thisstrangeinfluenceaffectsallobjectsinexactlythesameway.Forexample,copperandothermetaltrianglesseemtobejustasdistortedasplasticorwoodtriangles.TheydecidetoflybacktoNeptunetorelatetheirstartlingdiscoveries.Ontheirreturn,theyfindthatequallystartlingadvancesingeometryhavebeenmadeintheirabsence.Agroupofmathematicianshaveshownthatitispossibletoconstructnon-Euclideangeometries.AfiercedebatethenstartsonNeptune.SomearguethattheyshouldretaintheEuclideanview.Theyexplainthecuriousmeasurementsobtainedbytheexplorersintermsof‘mysterious’universalforceswhichdistorteverythingwithintheEarth’sorbit—76\nACONVENTIONALWORLD?afterall,hadnotNeptuniansalwaysfeltthatEarthwasanoddplace?13Othersclaimthatthediscoveryofnon-Euclideangeometriessolvesthewholeproblem:outsidetheEarth’sorbit,spaceisEuclidean;butinsideitisnon-EuclideananddeviationsfromtheEuclideanviewaretobeexpected.SoNeptuniansmustnowmakeachoicebetweentwoalternativehypothesesaboutthespaceinsidetheEarth’sorbit:1thatitisEuclideanjustlikespaceelsewhere,butauniversalforceisinoperationtherewhichdistortseverythingincludingeverykindofmeasuringinstrument;and2thatthesolarsystemisEuclideanexceptforacentralnon-Euclidean‘irregularity’.ThesecondhypothesisforcestheNeptunianstoacceptthattheyliveinauniversewhichis,atleastinpart,non-Euclidean.So,iftheyadopt2,Euclideangeometryaloneisinsufficienttodescribetheirenvironment.However,thepostulationofuniversalforcesdemandedby1allowsthemtoretainaEuclideanaccountofthegeometryoftheuniverse.SoisthereanyconvincingreasonwhyNeptuniansshouldaccept2andreject1?Couldwenotsaythatthesimplerviewshouldbeadopted?Butthesituationhereisnotsosimple!Forbothhypothesesseemtorequirecomplicatingratherthansimplifyingchangestothecommunity’ssystemofbeliefs:2hasitsnon-Euclideangeometricalcomplexities;and1involvesanadditionalforcewithalocal(ifall-pervasiveinfluence)inthecentralregionofthesolarsystem.And,becausetheuniversalforce,ifpresent,affectsallobjectsinthisregioninthesameway,wecannotproveitspresenceindependently.Sowecannotoffersomeindependentempiricalevidenceinfavourof1.Theevidenceseemstosupporteachhypothesistothesamedegree.Itwouldalsobedangeroustotrytodefendhypothesis1onthegroundsthatNeptunians(andhumansforthatmatter)canonlyvisualisetheworldinaEuclideanway.Clearly,thereisnologicalreasonwhytheyshouldbesorestricted,especiallygiventhefactthatlogicallyconsistentnon-Euclideangeometrieshavebeenconstructed.Arewetosaythatsuchconstructionsdonotinvolve‘visualisation’insomesense?AsReichenbachpointsout,theydonotinvolvevisualisationifwesimplymeanbythatapeculiarly‘Euclideanvisualisation’;hesays:‘wecannotvisualisenon-EuclideangeometrybymeansofEuclideanelementsofvisualisation’.14ButwecananddouseEuclidean‘pictures‘asanalogiestohelpusunderstandnon-Euclidean77\nTIME,SPACEANDPHILOSOPHYgeometry.Despitethedifferencesbetweenthegeometries,thereare,ofcourse,manysimilaritieswhichprovidetheanalogicalbaseforustodothis—forexample,thepropertiesof‘straightlines’onthesurfaceofaspherediscussedinFigure18(p.73)provideatwo-dimensionalEuclideananalogueofathree-dimensionalnon-Euclideanconfiguration.However,acoherentvisualisationofanon-Euclideanspacetimemayonlybeachievedfromanon-Euclideanperspective.OurdifficultiesinachievingthiseasilyseemtoarisefromthefactthattheEuclideanmetricalperspectiveisdeeplyentrenchedinoursystemofbeliefs.Itishardtobreakfreefromsuchahabit.Buttheconstructionofnon-Euclideanmodelsdemonstratesthatitisindeedpossibletobreakaway.15SoNeptuniansseemtobefreetochooseeither1or2.Itmightseemoddtochooseatheoreticalmodelwhichinvolvesastrangeandhighlylocalisedforce.Butwhyshouldthisbeanymorepeculiarthanadmittingthatthegeometryoftheworldhassuchanunusualanomaly?Thechoiceseemstobebetweenacceptingeitheraphysicalorageometricalmystery.Ofcourse,thetaleabouttheNeptuniansisfictionalinmorethanoneway.IfapartyofNeptuniansweretoinvestigatethegeometryofthesolarsystem,thentheywouldnotfindanysharptransitionbetweenEuclideanspaceandapparentlynon-Euclideanspace.Instead,iftheyweretoapproachtheSun,theywouldfindthatspacebecomesincreasinglynon-Euclidean—withverymarkedcurvatureclosebytheSun.GTRgivesagoodexplanationofthis:thetheoryrelatesthenon-Euclideancharacterofspace(andtime)tothedistributionofmassandenergy.Amassive,densebodysuchastheSunproducessufficientlocalcurvaturetobedetectablequitereadilywitharangeofmeasuringinstruments.Nevertheless,theNeptunianscouldcontinuetoholdtheirEuclideanbeliefsevenwiththismoreaccurateaccountofthekindsofmeasurementswhichwouldbemade.Theymightstillbelievethatsomeuniversalforceinfluencesallmeasurements.ButGTRgivesthemasoundtheoreticalbasisforanon-Euclideanview.ThewholepointaboutGTRisthatitdemystifiesthepresenceofcurvature:themass(andenergy)inagivenlocationisrelated(byasetofequations)tothecurvatureinthatregion.Hence,thetheoreticalframeworkofGTRencouragesustoacceptthenon-Euclideanhypothesisinourownuniverse.Thechoiceisnowbetweenamysteriousforceandaratherlessmysteriousifnovelgeometricalperspective.ReichenbachandPoincarésaythatwecouldresist,complicateouraccountofphysical78\nACONVENTIONALWORLD?forces,andmaintainaEuclideanperspective.ButthecostofthisresistanceisatheoreticalconcoctionwhichseemsclumsyandartificialratherthanthemoreelegantapproachofGTR.CONVENTIONANDTOPOLOGYBeforewespecifythemetricalgeometryinaspacetime,thereisnowell-definedsensetotheideasoflengths,distances,ordurationinthatspacetime.Butsucha‘metricallyamorphous’spacetimedoeshavestructure:wemayspecifyitsaffineandconformalstructureoritstopologicalcharacteristics—itsdimensions,connectedness,compactness,andorientability—independentlyofanymetricalfeatures;seeFigure11(p.54)inChapter3forageneralreviewofthedifferentlevelsofstructureinspacetime.16Aninfinitelyextendedplanedoesnothavethesametopologyasthesurfaceofacylinder.Theplaneisconnectedinawaythatthecylinderisnot:thefundamentaltopologicaldifferencebetweenthemliesinthefactthat,ontheplane,anyclosedpathmaybeshrunkcontinuouslydowntoapointandthereforetheplaneis‘simplyconnected’;but,onthesurfaceofthecylinder,someclosedpathsmaynotbeshrunktoapointinthesameway—forexample,thosearoundthecircumference.17Thepossibilityofsuchtopologicaldifferencessuggeststhisquestion:arewealwaysabletospecifythetopologyofaspacetimeonthebasisofempiricalevidence,oristheresomeelementofconventionalchoiceinvolved?18Wehavealreadyencounteredonetopologicaldisputeaboutthestructureofspacetime,inourdiscussionofZenoandthecontinuum.19Thechoicebetweencontinuousormerelydenseorderingsofthepointsinaspacetimeseemstobeessentiallyconventional.However,theideaofamerelydensethree-dimensionalspacemaynotbetheoreticallycoherent.Sothemorestraightforwardtopologicaldifferencebetweentheplaneandthesurfaceofacylindermayhelpustoinvestigatetheproblemofconventionalitywithinthecontextoftopologywithalittlemoreclarity.Manycosmologistsbelievethattheapproximateoverallshapeoftheuniversemaybecomparedwithaninfinitelyextendedplane,sothatanystraightlinestartingatanygivenpointwillgoonandonwithoutend.Insuchaspacetime,aspacecraftmovingawayfrom79\nTIME,SPACEANDPHILOSOPHYFigure21(a)TopologyofcylinderworldtheEarthwouldcontinuetodojustthat,withthedistancebetweenthemgrowingever-greater.Contrastthisspacetimewithauniversewhichpossessesacylindricaltopology,inwhichstraightlinesmaytraversethesamegroundbytrackingaroundthecircumferenceofthecylinder;seeFigure21(a)above.Inthiscase,aspacecraftmighttravelawayfromtheEarthmovingaroundthecylinderandfinditswaybacktotheEarthwithouteverchangingdirection.Couldwedeterminewhichkindofuniverseweinhabit?Clearly,themetricalstructureofthetwoworldsmightbedifferent,sincethecylindricalworldmighthaveacurvedmetric,whereastheplaneisalwaysmetricallyflat.Butwemightinvoketheideaofuniversalforcestoaccountforthismetricaldifference;orwemightconcernourselvesonlywith‘cylinders’inatopologicalsenseonly.Butwhateverwesay,thereisstillanimportanttopologicaldifferencebetweenthetwoworlds:theplaneissimplyconnectedandthesurfaceofthecylinderisnot.Andwemightdiscoversomeempiricalevidenceassociatedwiththisdifference.Butnowimagineaninfiniteplanewhichisdividedintoaseriesofstrips,eachofwhichisqualitativelythesame;seeFigure21(b)(p.81).Whatdifferencemightwefindbetweenthissegmentedplaneandacylindricalworldwhichsharesthesamequalitativefeatureswithanyonestrip?Ifwecouldjourneyaroundthecylinder,thenwewouldsee,overandoveragain,thesamefeaturesandthesameevents.But,ifwetravelacrosstheplane,eachnewstripencounteredwouldpresentuswiththesesamefeaturesandevents.Despitethetopologicaldifferencebetweenthetwoworlds,itishardtoimaginewhatkindofempiricalevidencemightbeunearthedtoallowustodistinguishbetweentheseapparentlyidenticaldomains.20Hence,itmightseemthat,onceagain,wemaychoosefreelybetweentwodistinctworlds.However,as80\nACONVENTIONALWORLD?Figure21(b)Topologyofstrip-worldReichenbachadvises,wemustspelloutalltheconsequencesofcompetingviewsbeforemakingadecisioninsuchcases.21Inthecylindricalworld,everyeventisfixedbyitsuniquepositiononthecylinder.However,inthestrip-planeworld,anysingleeventhasaninfinitenumberofoccurrences—oneoneachstrip.Thechainofeventsleadingtoanysingleeventalsorepeatsendlessly.So,ifweresistthehypothesisthattheworldiscylindrical,wemustsay:1theworldhastheglobaltopologyofaplane;2universalforcesarethecauseofanyobserveddeviationsfromthemetricofaninfinitelyextendedplane;and3aneventwhichiscausallyfixedatonepointmustalsobecausallyfixedinthesamewayonallother‘strips’.Reichenbachsaysthatbeliefssuchasthethirdleadtocausalanomalies:‘Theinterdependenceofalleventsatcorrespondingpointscannotbeinterpretedasordinarycausality,becauseitdoesnotrequiretimeforitstransferenceanddoesnotspreadasacontinuouseffectthatmustpassconsecutivelythroughtheintermediatepoints’(Reichenbach1957:65).Ifthereisnotsomemysteriouscausalconnectionbetweeneachstrip,thentheonlyotherexplanationseemstobeacosmiccoincidenceor‘pre-establishedharmony’whichunderwritesthisapparentlymiraculous81\nTIME,SPACEANDPHILOSOPHYrepetition.Eitherwaywewouldbechoosingastrangeworldindeedifwedecidetorejectthemorecausallyrespectablecylindricalworld.Yeteventhecylindricalworldhasitscausalproblems—foritsclosedpathsmaybeclosedintime,leadingtothecontroversialpossibilitiesoftimetravelandofaneventinmyfutureinfluencinganeventinmypast.InChapter8weshalltakeacloserlookatsuchpossibilitiesandthecausalissuesinvolved.DIMENSIONSOneofthecentralfeaturesofEuclideangeometrysetinatemporalframeworkisits‘three-plus-one’-dimensionality:forward-back;left-right;up-down;before-after.Itseemsdifficulttoimaginethatwearewrongaboutthefourdimensionsofspaceandtime.Butaretheworldswithdifferentdimensionalityphysicallypossible?Riemannshowedusthatwemayconstructconsistentmulti-dimensionalworlds:thisformedthecoreofhisworkonnon-Euclideangeometries.Whenwethinkofmulti-dimensionalworlds,weregarddimension,notasa‘physicalproperty’,butasa‘degreeoffreedom’orasa‘variable’neededtodescribeatopologicalmanifold.Amulti-dimensionalmanifoldistheanalogueofthetwo-dimensionalsurfaceofasphere.Weneedtwovariablestopickoutanypointonthesurfaceofasphere:e.g.east-westandnorth-southreferencesareallweneedtopickoutpreciselocationsonthesurfaceoftheEarth.Inathree-dimensionalspace,weneedthreevariables.InanN-dimensionalspace,weneedNvariables.AfterthedevelopmentofGTR,severalphysicistsinvestigatedtherelationshipbetweendimensionalityandthephysicalworld.Ehrenfestarguedthatmanyphysicalstabilitiesaredirectconsequencesofauniversewithonlythreespatialdimensions;andWeylshowedthatconformalpropertieswouldbeinvariantonlyinsuchauniverse.22ButthesuccessofthegeometricisationofgravitybyEinsteinledmanytoattemptthegeometricisationofallmaterialproperties.Physicswouldstickwiththreespatialdimensions,but,inadditiontoafurtherdimensionfortime,therecameagrowingdemandforextradimensionstocharacterisethepropertiesofmatter.Superstringtheoryisthelatestversionofthisprogramme.Thistheory,developedbyJohnSchwarzandMichaelGreen,involvesanadditionalsixdimensionsina‘string’tightlywrappedupintoaballinaspaceonly10–35metresacross.Thesedimensionsallowustorepresentthevariouspropertiesofmatterintermsofthevariousmodeswhichthestringmaybein:justasreal82\nACONVENTIONALWORLD?stringsmayoscillateandvibrate,thesesuperstringshavevariousmodesof‘vibration’andeachmoderepresentsaparticularpropertyofmatter.23Somephysicistswelcomethishighlyspeculativetheory,butothersremaindeeplysceptical.RichardFeynman’stypicallycausticresponse,madeshortlybeforehedied,revealstheantagonismofthosewhosharehiscommitmenttotheneedforafirmempiricalbasisbeforeacceptingatheory:I’manoldmannow,andthesearenewideas,andtheylookcrazytome,andtheylookliketheyareonthewrongtrack.NowIknowotheroldmenhavebeenveryfoolishinsayingthingslikethis,andtherefore,Iwouldbeveryfoolishtosaythisisnonsense.Iamgoingtobeveryfoolish,becauseIdofeelstronglythatthisisnonsense!…SoperhapsIcouldentertainfuturehistoriansbysayingIthinkallthissuperstringstuffiscrazy…Idon’tlikethatforanythingthatdisagreeswithanexperiment,theycookupanexplanation…it’saquestionofverifyingyourideasagainstexperiment.(Feynman1988:193–4)24But,asthetheoryisdevelopedanditstheoreticalpowergrows,theempiricalbasismayalsobegintogrow.Althoughsuperstringtheoryisspeculative,itresiststhetemptationtoaddorsubtractspatialdimensions.AndreiLindeislesscautious.The‘chaoticinflation’cosmologicaltheoryofLindeallowsforacompartmentalizeduniverse—brokenupintodistinctandisolatedregions.Andsomeoftheseregionsmayhavefewerthanthreespatialdimensions.25IfwegrantLinde’sideaasaphysicalpossibility,couldthenumberofspatialdimensionsbeotherthanthreeinourregionoftheuniverse?JohnBarrowsuggeststhatwearerestrictedtothreedimensionsbythestableatomicconditionsrequiredforlife:‘Thedimensionalityoftheuniverseisareasonfortheexistenceofchemistryandtherefore,mostprobably,forchemistsalso’(Barrow1983:339).26Barrow’sclaimisunderwritten,inpart,bytheAnthropicPrinciple—aprinciplewhichmakesadecisivelinkbetweenphysicsandbiology.WeshallexplorethisideafurtherinChapter9.83\nTIME,SPACEANDPHILOSOPHYTHEFUTUREOFTHEUNIVERSEOurownuniversedoesnotseemtohaveanythinglikeacylindricalstructure—metricallyortopologically.Soitmightbethoughtthatsuchquestionsabouttheoverallstructureoftheuniversearesomewhatcontrived.Butproblemsaboutglobalstructuredofaceus.Isouruniverseclosedoropen?Ordoestheuniversefollowamiddlewaybetweenthesetwopossibilities—theso-called‘flat’universe?Oristheuniversecyclical?Whatwedecidehasimplicationsforthewaywewillseethefuture.Foreachofthesefourkindsofuniversehasadifferentfutureand,inaddition,thecyclicaluniverseinvolvesadifferentpast:1A‘closed’universeisusuallycharacterisedasonewhichexpandsfromasingularitywithextremelyhot,denseinitialconditions(theso-called‘bigbang’)andthencollapsestowardsafinalsingularity.2An‘open’universeneverstopsexpanding:theaverageoveralldensityofmatterinsuchauniverseisnotlargeenoughtostopthematerialintheuniversegettingfurtherandfurtherapart.Gravitationalattractiontendstoslowtheexpansiondown,butinanopenuniversetheoveralldensityofthematerialcontentsisinsufficienttocounteracttheglobalexpansion.3A‘flat’universeholdsthebalancebetweenaclosedandanopenuniverse;thereisanaturallimittotheglobalexpansion:theoveralldensityisjustsufficienttobringtheexpansiontoahalt‘atinfinity’.4Acyclicaluniverseisonewhichfollowsa‘closed’patternofexpansionandcollapse,butwhich‘bounces’outofthefinalsingularityjustbeforethemomentoftotalcollapsetonothingnesstostarttheprocessalloveragainallowingthecycletorepeatendlessly—thisphaseoftheuniverseisjustoneofaninfinitenumberofcycles;seeFigure22(p.85).Asinthecaseofthecylindricalandplaneworlds,theclosedandcyclicaluniverseshavedistinctmetricsandtopologies.Eventhoughthe‘local’metricalandtopologicalfeaturesofagivencycleofexpansionandcollapseinthecyclicaluniversemaybethesameasthoseinthecloseduniverse,thereareobviousglobaldifferencesbetweenthem:forexample,thecyclicaluniverseisinfiniteinawaythatthecloseduniverse,withamoreclearlydefined‘beginning’and‘end’,clearlyisnot;butseethediscussionoftheideaofa84\nACONVENTIONALWORLD?Figure22Graphshowingopen,closed,flat,andcyclicuniversesbeginningoftheuniverseinChapter10.27Whetherweliveinacloseduniverseornotisan‘open’question!Theavailableevidencesupportsneitherendlessexpansionnorsomefuturerecontraction—a‘flat’universeisthemostlikelyalternative.ButtheimportantFriedmanncosmologicalmodelsoftheuniverseprovideacogenttheoreticalbasisforeachofthethreemainoptions:open,closed,orflat.28Additionalevidencemightindeedpushustoadoptanyoneoftheseoptions:wemight,forexample,discoverthattheaveragedensityoftheuniverseismorethansufficienttobringaboutglobalgravitationalcollapse.Andeventhemostardentconventionalistsupporterofanopenuniversemightbegintoacceptthattheuniverseisclosedwhenitsmaterialcontentsbegintocollapseinwards.However,wemightnotbeabletochoosebetweenacloseduniverseandacyclicaluniverseonsoclearanempiricalbasis.Supposethattheevidenceweretosuggestthatweliveinacloseduniverse,howmightwetellwhetherornottheexpansionandcollapsesequencehappensonlyonce?Theconditionsclosetotheinitialandfinalsingularitiesmightobscuretheexistenceofanybounceandthereforethepossibilityofanothercycle,leavingusuncertainastotheeventualfateoftheuniverse.Sowemightbetemptedtoallowthepossibilityofacyclicaluniverse—perhapsonconventionalgrounds.Ofcourse,ifwefavouracyclicalview,wecouldnotsaywhethereachcyclewouldbeidenticalinallrespectstotheobservedexpansion-recollapsepatternofthecloseduniverse.Indeed,thereissomeevidencewhichsuggeststhateachcyclewould85\nTIME,SPACEANDPHILOSOPHYhavetobedifferent,forexample,atlowerandlowerentropylevels.29However,itishardtoimaginewhatkindofphysicalprinciplesmightbeinvolvedin,andwhatsortofphenomenamightbeassociatedwith,a‘bounce’.Ourknowledgeofthefutureoftheuniverseseemstobe,atleastatpresent,somewhatfragmentarywhencomparedwithourunderstandingofmorelocaleventsandphenomena.Soweareleftina(relative)stateofignorancewith(hopefullyinformed)speculationasourguide.Itseemsclearthatourinabilitytomakeachoicebetweenfuturesseemstoarisesimplybecauseofthelackofavailableevidence,ratherthanbecausedecisiveempiricalevidencecouldnotbefoundinprinciple.THECOSMOLOGICALPRINCIPLE:CONVENTIONORFACT?Whereverwelook,whateverdirectionourtelescopespoint,theuniverseseemstobemuchthesame.Clustersofgalaxiesseemtobespreadthroughouttheuniverseinagenerallyuniformwayandthereseemtobefewlarge-scaleanomalies.Hence,fromourperspective,theuniverseappearsonaglobalscaletobeapproximately:1thesameatallpoints,i.e.homogeneous;and2thesameinalldirections,i.e.isotropic.Thecentralassumptioninmostacceptedcosmologicaltheoriesisthattheobservedlarge-scaledistributionofmatterandenergyissosmooththattheuniverseisgenerallyhomogeneousandisotropiceverywhere.Henceanyobserveranywherewouldseeessentiallythesamelarge-scalepicture;seeFigure23(p.87).ThisassumptioniscalledtheCosmologicalPrincipleanditliesattheheartofthestandardFriedmanncosmologicalmodelsoftheuniverse.This‘Copernican’ideathatneithertheEarthnoritsgeneralenvironmenthasaprivilegedlocationinthecosmic‘scheme’isdevelopedtodeliveratrulyegalitarianuniverse.Neitherthesolarsystem,norourgalaxy,noranyothergalaxy,forthatmatter,isregardedasbeing‘central’orhavinganyspecialstatusintheuniverseasawhole.Theuniverseexpandsbutneednothavea‘centre’assuch.Two-dimensionalanalogiesmaybe86\nACONVENTIONALWORLD?Arepeatingpatternonthesurfaceofasphereillustratestheideaofhomogeneity—whereverwelookwefindessentiallythesamesituation,thesamelocalpicture;and,regardlessofthepointwechooseasareference,essentiallythesameviewpresentsitselfineverydirection,thusillustratingtheideaofisotropy.Asthesphereincreasesinsize,itremainshomogeneousandisotropic;andexactlythesameamountofmaterialispresentasintheearlierpicture.Figure23Ideaofhomogeneity:expandingsphereusedtoillustratethisidea:thesurfaceofanexpandingspheredoesnothaveacentre;nordoesthesurfaceofanexpandinginfiniteplane;norindeeddoesthespatiallyandtemporallyclosedsurfaceofatorus.Allpointsandalldirectionsonsuchsurfacesareequivalent.Andthemostlikelygeometryforourspacetimeseemstobethefour-dimensionalanalogueofeitheraverygentlycurvingsphereoraninfinitelyextendedflatornearlyflatplane.30Itisraretohearanyseriousandinformedchallengetotheempirical87\nTIME,SPACEANDPHILOSOPHYstatusoftheCosmologicalPrinciple.ButsuchachallengeismadebytheeminentphysicistGeorgeEllisinhisprize-winningessay:‘Istheuniverseexpanding?’31Hearguesthatitsplaceincosmologyisguaranteedmorebyphilosophicalcommitmentthanbyempiricalevidence.Forwemayconstructotheruniverseswhichareconsistentwiththeavailableevidencebutwhichareneverthelessinhomogeneousandanisotropic.Toprovehispoint,Ellisconstructsaspacetimeinwhichmattercirculatesbetweentwo‘centres’—itispumpedoutfromoneandsuckedinattheother;bothofthese‘centres’aresingularities.Theuniversedoesnotexpand,buttheimpressionofexpansionisgivenbythecontinuousmovementawayfromonesingularityandtowardstheother.Theconditionsatthefirstsingularityareidenticalwiththoseattheinitialsingularityinstandardcosmologicalmodels.Ourowngalaxyisinapreferredpositioninthesensethatonlyincertainperiodsinthelifeofagalaxydophysicalconditionspermittheexistenceofpeople.Asourgalaxymovestowardsthesecondsingularity,conditionsdeteriorateandhumanlifeisnolongerpossible.Butothergalaxiesarecontinuouslymovingtowardsthepreferredpositionforlife.Sowecansay,atleastinprinciple,thathumanlifemightcontinuealwaysingalaxieswhilsttheyoccupythispreferredposition.Ellisavoidsmakinganyover-enthusiasticclaimsforhismodeluniverse.HepreferstopointtothemodelasawarningtothosewhoareuncriticalintheiracceptanceoftheCosmologicalPrinciple.Yes,thisprinciplehasanexcellentempiricalpedigree;buteveryobservationinfavourofthestandardviewalsosupportstheEllismodel:hencetheassertionthattheuniverseishomogeneousandisotropiccertainlyseemstohaveaconventionalcharacter.Ofcourse,wemightsaythatthestandardmodelsaremorerespectableinasmuchaswehavedevelopedandtestedthemextensively.Ellisrepliesbyaskingwhythereissomuchreluctancetostepoutsidethestandardview.Ifempiricalevidencealoneistobethetouchstoneforacceptability,thenthereisnoreasontopreferthisviewtothatconstructedbyEllis.Ellishasalsoargued,withRuthWilliams,thathomogeneityandisotropymaybeexplainedmorepowerfullyusingamodelsometimesreferredtoasa‘small’universe.Atitssimplest,suchauniversehasthetopologicalspatiallyclosedstructureofatorus;seeFigure24(p.89).32Aninterestingfeatureofthesmalluniverseisthatitsspatialclosuregivesusaccesstoalleventsinourpast.88\nACONVENTIONALWORLD?Thetorushasthetopologyofaring:thetopologyofthesmalluniverseisthatofthesurfaceofsucharing.Figure24TopologyoftorusWhenwelookintoourpastweseenotaninfiniteworldbutafiniteworldwithaninfinitenumberof‘images’.Whereverwelook,weseeahugecollectionofgalaxies;but,aswelookfurtherandfurtheraway,weseemultipleimagesofthissamecollection.Eachimageistheresultoflighttravellingafinitenumberoftimesaroundthetorus.Ifwelookinadifferentdirection,weseeadifferentsetofimages,butthesearestillimagesofthesameuniqueobject—thefinitecollectionofgalaxieswhichconstitutesthetotalmaterialcontentsofthetorusworld.Anydifferencesinappearancearesimplyduetotheperspectivefromwhichweviewthecontents.EllisandWilliamsmaintainthatasmalluniversecouldbeempiricallyindistinguishablefromaplaneworld.Forwhatmightseemtobeaninfiniteplanemightbeinsteadaninfinitesetofimagesofafinitedomain.Homogeneityandisotropymaynowbeexplainedinastraightforwardway.Weneednoextravaganthypothesesabouthowtheuniversehasevolvedfromitsinitialconditions.AsEllisandWilliamssay:‘theuniverselookshomogeneousandisotropicbecauseweareseeingthesameregionoverandoveragain’atdifferentstagesofitshistory;and‘thisisthesimplestreasonforapparenthomogeneityonecan89\nTIME,SPACEANDPHILOSOPHYimagine!’33So,insuchauniverse,theCosmologicalPrincipleistriviallytrue.34Inthisandthepreviouschapter,wehaveexaminedthreecentralproblemsconnectedwiththestructureofspaceandtime:whatstandardofsimultaneitymayweadopt;isthemetricofspaceandtimeEuclidean;andjusthowmuchcanwesayaboutthetopologyoftheactualuniverse?Wenowhaveasufficientlysecurebasistoclarifythephilosophicalproblemsassociatedwithconventionalismaboutspaceandtimewithrathermoreconfidence.THEUNDERDETERMINATIONOFTHEORYBYDATATheclaimthattheoriesmaybeunderdeterminedbytheevidenceandsohaveaconventionalcharacterhasitsoriginsintheworkofPierreDuhemaswellasofHenriPoincaré.35Writingintheearlyyearsofthiscentury,Duhemarguesthattherecanneverbeany‘crucialexperiment’whichallowsustomakeadefinitivechoicebetweentwotheories.Fornotheorystandsonitsown:itisalwayssupplementedbyauxiliaryhypotheseswhicharenotpartofthetheoryitself.Suchhypothesessetdowntheinitialandboundaryconditionswhichcanprovidetheessentialempiricalbackgroundforatheoryanditslaws.Ifatheoryischallengedbysomeadverseexperimentalresult,thenwemayalwaysblametheauxiliaryhypothesesratherthancondemnthetheoryitself.Soanappropriateadjustmenttoourauxiliaryhypothesescanalwayssavethetheory.ManyrecentwritershavefoundDuhem’smessagebothconvincingandattractive.Indeed,thelogicianQuinesupports(andgoesbeyond)Duhem’spositionbysayingthat:‘Anystatementcanbeheldtruecomewhatmay,ifwemakedrasticenoughadjustmentselsewhereinthesystem(ofbeliefs)…Conversely,bythesametoken,nostatementisimmunetorevision’(Quine1980:43).36Theseideashavebeenenshrinedwithinthe‘Duhem-Quinethesis’whichmaybestatedprovisionallyasfollows:becausescientifictheoriesareunderdeterminedbytheempiricalevidencewemayalwaysfindanincompatiblealternativetoanygiventheory.But,wemayask,atwhatcost?WhenobservationsoftheorbitoftheplanetMercurywereshowntobeatoddswiththepredictionsofNewtoniangravitation,scientistslookedforsomeproblemintheauxiliaryhypotheses.Afterall,suchatactichadproducedtremendousresultsfortheastronomersLeVerrierandAdamswhenasimilaranomalyintheorbitofUranusledtotheir90\nACONVENTIONALWORLD?jointdiscoveryofNeptune:thehypothesisthattheplanetsstoppedatUranuswasatfaultandNewtoniangravitywassaved—forthetheorywasabletoshowhowNeptune’spresenceaffectstheorbitofUranus.Soitwasnotentirelysurprisingwhen,inthe1850s,LeVerrierpostulatedtheexistenceofyetanotherplanet,calledVulcan,insideMercury’sorbit—yetanotheradjustmenttotheauxiliaryhypothesestokeepNewton’sheadabovewater.Butthistimenonewplanetcouldbefound.Evenso,otherreasonscouldbegivenforthefailuretodiscoverVulcan:forexample,anyplanetsoclosetotheSunwouldbeobscuredbytheglareoftheSun,makingadefiniteobservationextremelydifficulttoobtain.Itdoesindeedseemthatwemightalwaysfindsomereasontoholdontoourtheories,howeverdamningtheevidenceagainstthemmightappeartobe.Ofcourse,suchareasonmaylackconviction,theresultantcombinationoftheoryandsupportinghypothesesmightbecomplex,andthechangeinauxiliaryhypothesesmayquitejustifiablybecalledadhoc.AsKarlPopperandRichardFeynmanbothobserve,tointroducesomehypothesiswhichisdesignedtodonomorethansaveatheoryintheteethoftheevidencecanshowacallousdisregardforscientificintegrity:ifwedesignacrucialexperiment,thefiercesttestwhichwecandeviseforsometheoreticalposition,andiftheresultsofthatexperimentruncontrarytothetheory’spredictions,thenitisbettertoacceptthatthetheoryhasbeenfalsifiedthantoinventsomeadhochypothesistoaidourcontinuedbeliefinthetheory.37Doesconventionalismhaveanypositivemessage?Orisitmerelyanegativeresponsetothescientificrealist’sbeliefthatscientifictheoriescananddorepresenttheworldinastraightforwardandindependentway?BothQuineandReichenbachemphasisetheideaofempiricalequivalencewhentryingtoclarifythesupposedconventionalcharacterofourscientificbeliefs.38Butthisideaisnotaltogetherclearitself.Ifwearetogetaclearpictureofconventionalism,thenweneedtoknowjustwhenwemaysaythattwoalternativetheoriesareempiricallyequivalent.For,withoutsomesuchequivalencebetweenalternatives,wewillbeabletocitesomematteroffactwhichislikelytooffermoresupporttojustoneofthetwotheories.Andwealsoneedtobeconfidentthatthepresuppositionsinvolvedinanyclaimthattwotheoriesareempiricallyequivalentaregenerallysound.However,itisdifficulttostateanyconvincingthesisofempiricalequivalence;mostcandidateseitherlackforceandaretooglibtobeusefulor,ifforceful,maketoomanyproblematicorunpalatableassumptions.Consider,forexample,thisstatement:91\nTIME,SPACEANDPHILOSOPHYAT1isempiricallyequivalenttoT2ifboththeoriesareacceptableonthebasisofthesameavailableempiricalevidenceE.Thisstatementcapturestheideathattwoequivalenttheoriesshouldrelyonthesameobservationalevidence,butthetwotheoriesmightstillbeempiricallydistinct.Theymightmakequitedifferentpredictionswhichwehaveyettotestempirically.So,althoughthetheoriesmaybeacceptableonthebasisofthesameevidence,thereremainsthepossibilitythatsomeadditionalevidencewillpointunequivocallytojustoneofthetwotheories.Alsothestatementinvolvesnosenseofthetwotheoriesclashing—sothatbeliefinonetheorysomehowrulesoutbeliefintheother.TheideaofaclashiscertainlyinvolvedinPoincaré’sviews.Noristhereanyindicationoftheideathatwecanalwaysfindsomealternativetoanygiventheory,asseemstobesuggestedbytheprovisionalstatementoftheDuhem-Quinethesis.Sowemightbemoreinclinedtoacceptarathermoredetailedversionofthethesis:BForanygiventheoryTwhichisacceptableonthebasisofempirical1evidenceE,thereisatleastoneequallyacceptablebutincompatibletheoryTwhichisempiricallyequivalenttoTandwhichmakes21thesameempiricalpredictionsasT.391Althoughthisseemstobemoreinformativeandhelpful,threemajorquestionsneedtobesettledbeforeweshouldembraceBoranysimilarthesis:1Whatismeantbytheideaofempiricalacceptability?Generally,atheorymaybesaidtobeacceptable:aifitissupportedtoareasonabledegreebytheavailableevidence;andbifitisnotasyetfalsifiedbyanysuchevidence.However,whatwearetocountasareasonabledegreeofsupportdoesneedfurtherclarification—andthisshouldbedonewithinthecontextofasoundtheoryofconfirmation.Yetthedevelopmentofsuchatheoryisasubstantialproblem—fortheconventionalistjustasmuchasforthescientificrealist.2Whatpreciselyaretheconditionsunderwhichtwotheoriesareincompatible?Iftheideaofincompatibilityistohaveanyforce,then92\nACONVENTIONALWORLD?itmustthreatenthepositionadoptedbythescientificrealistwhobelievesthatwhatatheorysaysisatleastapproximatelytrueoftheworlditself.Hence,twotheoriesareincompatiblewhentheyimplytheexistenceoftworadicallydistinctrealworlds—sodistinct,infact,thattherealistcouldnotclaimthattheyarebothapproximationstothetruthatthetheoreticallevel.Thisimplicationcarrieswithittheassumptionthatatheoryisnotreducibletoitsobservationalconsequences.For,ifitweresoreducible,therewouldbenoradicaldifferencebetweenthetheories.403Withrespecttowhatkindofempiricalevidencecanwesaythattwotheoriesareequivalent?Therearethreemaincategoriesofevidencewhichmightbecitedhere:atheempiricalevidencewhichiscurrentlyavailable;bsomerestricteddomainofevidence,forexample:evidencewithinthecontextofagivendomainsuchaskinematics(ieexcludingdynamics);callpossibleempiricalevidence.Clearly,onlythelastcategorymightcauseasignificantproblemfortherealist.Forthefirsttwowillalwaysleaveopenthepossibilitythatwecoulddecideatleastinprinciplebetweentwoalternativeviewsonempiricalgrounds.Thesequestionsdemandsomechangesinthethesisiftheconventionalististomaintainaclearandforcefulpositionwhichcontrastssharplywiththerealistpointofview.Inordertoexpresssuchaposition,wemightmovetoastrongerversionofthethesis:CForanygivenacceptabletheoryT,therewillalwaysbeanalternative1theoryTsuchthat:2iboththeoriesareequallywellconfirmedbyallavailableevidence;iineithertheoryisfalsifiedbyavailableevidence;iiinopossibleempiricalevidencecouldsupportonlyoneofthetheories;ivthetwotheoriesimplytheexistenceofradicallydistinctrealworlds.Pointsiandiihelptheconventionalisttoclarifytheideaofempiricalequivalence.Pointiiiextendstheideaofequivalenceacrossallphysicalpossibilities.Andpointsiiiandivcarrytheconventionalistattackonrealism.Withoutiii,therealistcouldsaythatanyequivalencebasedonavailableevidenceismerelysuperficial,andthatfurtherevidencecould,93\nTIME,SPACEANDPHILOSOPHYatleastinprinciple,demonstratethatthealternativesarenotgenuinelyequivalent.Withoutiv,therealistcouldarguethattwoempiricallyequivalentalternativesarebothapproximationstothesamerealworldandthatanydifferencesbetweenthetheoriesaretrivialorposenothreattothebeliefthatovertimescientificbeliefsareconvergingonatruthfulaccountoftheworld.So,althoughtheconventionalistpositiondoesseemtobeanattempttoblocksomeofthenaturalinstinctsofscientificrealists,itiswrongtodismissitasamerelynegativestrategy.For,ifthesisCiscorrect,thenwehaveachievedapowerfulandpositivearticulationofthelimitsofrealismwithinthecontextofempiricalbelief.Therealistmay,ofcourse,challengeCbyquestioningtheimplicitassumptionthatwemayisolateanempiricaldomainwhichmightthenserveasareferencepointbetweendistincttheories.But,evenifweacceptC,therearetwomainoptionsbeforeus:1WemightadoptCwithathoroughlyconventionalistspirit:anychoicebetweenalternativeswouldthenbesimplyamatterofconvention—agiventheorywouldbeadoptedfreelyastruebyconvention.2Wemightarguethatnon-empiricalconsiderationsneverthelessallowustoselectjustoneofthealternatives,citing,forexample,greatersimplicityasthereasonforourchoice.Ifsuchaselectionismadeonpragmaticgrounds,thenwemightstillmaintainanessentiallyconventionalistposition.41Butifthereisanysuggestionthatthereasonforourchoicehassomeapriorifoundation—perhapsanaprioribeliefthatsimplicityisasignofthetruth—thenwewouldnotonlybepartingcompanywithconventionalism,wewouldbeintheinvidiouspositionofneedingacoherentargumenttosupportsuchastep.42Thereisanadditionalpossibilitywhichinvolvesafurthermodificationoftheideaofconventionalismbutwhichdropstheideaofunderdetermination:DForanygivenacceptabletheoryT1therewillalwaysbeanalternativetheoryT2suchthat:iboththeoriesareequallywellconfirmedbyallavailableevidence;iineithertheoryisfalsifiedbyavailableevidence;iiinopossibleempiricalevidencecouldsupportonlyoneofthetheories;ivthetheoriesarenotgenuinelydistinctsinceeachisreducibletoexactlythesameempiricalbasisgiventhatthetheoriesimplynomorethantheirobservationalconsequences.94\nACONVENTIONALWORLD?PointivhereissuggestedbyReichenbach’spositionongeometrynotedabove.Empiricalequivalenceiscentral.ButReichenbachassertsthatapparentlydifferentgeometricaltheoriesaremerelyexpressionsofthesametheory.Thisseemstoimplythatatheorymaybereducedtoitsobservationalconsequences,sinceReichenbacharguesthattwodifferenttheoriesamounttoessentiallythesame‘theoretical’view.Hemayonlysaythisifanydifferencesatthetheoreticallevelareeradicatedduringareductiontotheobservationallevel.However,ifweacceptReichenbach’sargument,wemaynolongersaythatatheoryisunderdeterminedbyitsobservationalcontent.Foratheoryisnomorethanitsobservationalcontent!DoesthesisDinvolveacommitmenttoafirm,absolutedistinctionbetweentheoryandobservation?Positivistsfoundtotheircosthowdifficultitistomaintainanyclear-cutdistinctionbetweentheoryandobservation.AsHanson,Kuhn,andothersargue,ourobservationalclaimsabouttheworldseemtodependtosomeextentuponourwidertheoreticalperspectives.43Sowecannotjustsimplyassumethatourobservationalstatementsarefreefromtheoreticalpresuppositions—weneedtoprovideconvincingargumentsforthis.44However,theabovethesisneednotinvolveanydisastrousassumptionsaboutthestatusofourempiricalbeliefs.Liketheearlierstatementsofunderdetermination,thesisDdoesmakeanimplicitdistinctionbetweenobservationandtheory.Eachthesisreferstotheideaofempiricalevidenceasifthiscanbeidentifiedindependentlyoftheoreticalpresuppositions.Butthisdistinctionneednotbeabsolute;Reichenbachneednotproposesomeabsoluteobservationaldomain.Instead,theempiricalevidencecitedinanassertionofequivalencebetweenanytwotheoriesmayberelativemerelytothetwotheoreticalalternativesinvolved.Anobservationaldomainisindeedrequiredifempiricalequivalenceistomakesense;butmanyobservationalreportsmaybecommontoboththeories,evenifthesereportsarenotfoundelsewhere.45IsReichenbachaconventionalist?Heisinoneimportantsense:itdoesnotmattertohimwhichoftwoacceptablealternativegeometricaltheoriesareemployedtodescribetheworld.Boththeoriesare‘correct’sinceeachtheoryamountstonomorethanitsobservational95\nTIME,SPACEANDPHILOSOPHYconsequences,andtheseconsequencesarethesame—givenempiricalequivalence.Sothechoiceofnon-Euclideangeometryisaconventionaldecisionevenifwehavegoodpragmaticreasonsforourdecision;Reichenbachsuggestsconsiderationsofdescriptivesimplicityasthemostpersuasivecriterionoftheorychoice.But,unliketheconventionalistwhoadoptsthesisC,Reichenbachmustprovideanexplanationofhowareductionfromtheorytosomeobservationalbasisistobeachieved.Eveniftheobservationalbasisisrelativetojusttwoalternativetheories,westillneedtospecifyhowthecomplexdomainoftheoreticaltermsmightbereducedinacoherentwaytoobservationallanguage.Thosewhohavetriedtojustifyanysuchreductionhavebeenplaguedbydifficulties.46Hence,thesisCappearstobetheconventionalistapproachwhichisthemostproblem-freeandwhichalsoposesthemostacuteproblemsfortherealist,givenitsstrongcommitmenttotheideaofunderdetermination.Howmanyofthespecific‘conventions’discussedintheearliersectionsofthisandinthepreviouschaptercarrytheforcedemandedbythesisC?Whichofthemposesaseriousthreattotherealist?Althoughwehavebeenabletoraisethequestionshere,itisbeyondthescopeofthisbooktoattemptanythingmorethanapreliminaryreviewofthesecomplexissues.Wehavemerelyfixedthecriteriaagainstwhichcandidatesforconventionsmaybejudged.However,tworelatedgeneralpointsaboutconventionalismmaybemade.Ifwereviseonepartofatheoryatatheoreticallevelintheattempttoconstructan‘equivalent’alternativetotheoriginal,thenrevisionsmustbemadeelsewhereinitstheoreticalcontext.Ifwechangeanimportantenoughpartofthetheory,thenthesubsequentrevisionsmaybesoradicalthattheentirecontextisfundamentallychanged.Thetwo‘equivalent’theorieswouldberadicallydistinctinalmosteveryway.Butwemustrememberthatourscientifictheoriesareembeddednotonlyinagenerallyacceptedscientificcontextbutalsoinourwidersystemofbeliefs.Wesimplycannotjudgebetweentwoalternativetheoriesinisolationfromtherestofourbeliefs.Consequently,itmightbefoolhardyforanyonetoguaranteethatsucharadicalshiftcouldnotinprincipleintroduceanempiricaldifferencebetweentwoalternativetheorieswithrespecttoouroverallsystemofbeliefs.However,ifthe96\nACONVENTIONALWORLD?twotheoriesdifferonlyinminorways,sothatthefundamentalideasoftwo‘equivalent’theoriesareessentiallythesame,thenanyconventionalfeatureresultingfromsuchamodestchangeinoutlookisunlikelytoembarrassthescientificrealist,whotypicallyfocusesonthestatusofthecentralfeaturesofatheory.Secondly,theconventionalistneedstosaywhyempiricalevidencealoneshouldbeusedtohelpusdecidebetweentwocompetingtheories.Considerjustoneproblemfortheconventionalist.Malament’scomparisonofstandardandnon-standardaccountsofsimultaneityrevealsthatsimultaneityinSTRdependsupontheconformalgeometryofitsspacetime.TheconformalstructureofMinkowskispacetimehasrathermorethana‘background’theoreticalrole.Itplaysacentralpartinthenetworkofmodelsandtheoriesusedtodescribetheworldkinematically(STR)anddynamically(GTRandelectromagnetism),andevenfeaturesindiscussionsofquantumgravity.AsMichaelFriedmanpointsout,suchtheoreticalconceptsseemtohavea‘unifying’andexplanatoryroleinoursystemofbeliefs—theyprovidekeylinkswhichunderwriteouranalysesofrelatedphenomenaindifferentphysicaldomains.47If,bymovingfromastandardviewofsimultaneity,weloseatleastsomeoftheselinks,thenwemayhaveapowerfulnon-empiricalreasontoresistthemove.Theconventionalistseemstobemotivatedbyadeepscepticismaboutthestatusoftheory.Thisscepticismalsomotivatestherelationist,whotriestoreducetheideasofspaceandtimetothemoreconcreteconceptsinvolvedinmaterialinteractionsandrelationships.Likeconventionalists,relationiststypicallytrytominimisetheroleoftheoreticaltermsinscience.Thisanti-theoreticalstrategyisfrequentlyadvancedbymeansofargumentsforsimplicity.Theconventionalistsaskustorestcontentwiththerelativesimplicityoftheempiricalcontentofatheory;theyseenoneedtogranthighstatustotheoryassuch,giventheirclaimsabouttheunderdeterminationoftheorybydata.AndtherelationistsfrequentlywieldOccam’srazor,tryingtocut‘superfluous’entitieslikespaceandtimefromourscientificvocabulary.48Bothtemptuswitha‘simpler’pictureoftheworldinwhichtheoryplaysatbestasecondaryrole.Althoughthereisclearevidencethatprinciplesofsimplicitydoconstrainandstructurethedevelopmentofscientifictheories,weshouldnotaccept97\nTIME,SPACEANDPHILOSOPHYtooreadilythepropositionthatideasofsimplicityaloneshouldgovernthewaywedoscience.Inthefollowingthreechapters,weshallexploretherelationiststrategyandassesstheargumentsforandagainstrelationism.98\n5NEWTONANDTHEREALITYOFSPACEANDTIMEINTRODUCTIONAfewpleasantriesinahandfulofletterswereallthatpassedbetweenthetwogiantsofscienceandmathematicsintheseventeenthcentury:IsaacNewtonandGottfriedLeibniz.Intheselettersthereislittleornosuggestionofthegulfbetweenthemontheirattitudestowardstherealityofspace,time,andmotion.OnlyinLeibniz’scorrespondencewithSamuelClarkeandChristianHuygensdoweseeanydetailedcontemporarydebateontheissueswhichdivideNewtonandLeibniz.1MuchofthisdebateisconcernedwiththeologicalissuesturningonthenatureandthepowersofGod,butthroughoutthereisastrongdesiretoclarifytheconceptsofspaceandtime.ThedebateisprovokedbyNewton’sPrinciplesofNaturalPhilosophy(Principia)andparticularlybyhisassertionthatspaceandtimeareabsoluteentities.Newton’sPrincipia,publishedin1687,presentsuswithapowerfulandpersuasiveanalysisofspace,time,andmotion.Spaceis,inNewton’sview,essentiallyanabsolute,independent,infinite,three-dimensional,eternallyfixed,uniform‘container’intowhichGod‘placed’thematerialuniverseatthemomentofcreation.2Timeisanabsolute,independent,infinite,one-dimensional,fixed,uniform‘framework’.Absolute(asopposedtomerelyrelative)motionismotionthroughspaceitself—anobjectwhichisreallymovingchangesitsabsolutepositioninspacecontinuously.Itisimpossibleforustosaywhetheranyobjectisatabsoluterestinspace;forthereisnoexperimentorobservationwhichwillallowustosingleoutanyoneframeofreferenceasabsolutelyatrest.Newton’slawsofmotionimplyanequivalencebetweenallnon-acceleratedframes99\nTIME,SPACEANDPHILOSOPHYofreference.Thisequivalenceistheessentialfoundationoftheclassical‘Galilean’principleofrelativity,whichstatesthatthelawsofmotionapplyinthesamewaytoallnon-acceleratedsystems.Sowecannotusethelawstohelpusdecidewhichifanyoftwoormorebodiesinrelativeuniformmotionisreallyinmotionorreallyatrest.The‘fixed’starsmightseemtobefixedandatrestinspace—but,forallweknow,theentirematerialuniversemightbemovingatauniformvelocitythroughspace.Butwemaysayofcertainobjectsthattheyreallyaremoving.Objectswhichareacceleratedexperience‘inertial’forces—thekindofforcesatworkwhenweareforcedbackintoourseatsasacarorplaneaccelerates.IfIamenclosedinaboxandexperiencenoinertialforces,thenIcandonothingtosaywhethertheboxisatrestormovingataconstantvelocityrelativetoanybackgroundframeworkIchoose—includingspaceitself.But,assoonasIexperienceinertialforces,thenImaybecertainthattheboxisacceleratinginsomeway;itmaybeacceleratinginastraightlineoritmayberotatingorboth.ThisempiricalfactprovidesthefoundationforNewton’s‘proofoftheexistenceofabsolutespace:aproofwhichmaybeeasilyextendedtoprovideanargumentforabsolutetimeaswell.LeibnizresistsNewton’sclaimsaboutspaceandtimestrongly.InhiscorrespondencewithSamuelClarke,Leibnizdefendstheadoptionofrelationalconceptsofspaceandtime.Althoughhisdefenceturnsontheacceptanceofarationalisticphilosophicalsystem,LeibnizneverthelesspresentssomeformidablechallengestoNewton’sabsolutistperspective.InthischapterwewillexploretheissueswhichdivideNewtonandLeibniz.Andthisdebatewillturnouttobeofmorethanmerelyhistoricalinterest,fortheproblemsdiscussedremaincentraltoourunderstandingofthenatureofspaceandtimeasdescribedinmodernphysics.ABSOLUTESPACEANDTIMENewton,inhiscelebratedrotatingbucketandtwoglobesthoughtexperiments,asksustoconsiderthegeneralpropertiesofrotatingsystems;seeFigure25(p.101).WemayreconstructhisargumentintheScholiumonspaceandtimeinthePrincipiaalongthefollowinglines:31Ourgeneralexperienceleadsustolinkoccurrencesofinertialforceswithaccelerationssuchasrotations.Sowemightargue100\nNEWTONANDTHEREALITYOFSPACEANDTIMEFigure25Newton’sthoughtexperiment:rotatingspheresthatanyrotatingsystemexperiencesinertialforcesasaconsequenceofitsrotation;forexample,asystemoftwoglobesconnectedbyacordrotatingaboutacommoncentrewouldexperienceatensionforcealongthecord.2Withasystemlikethetwoglobesandcordtherearetwopossiblesituations:eitherthereisatensioninthecordorthereisnotension.3Inbothcases,therelativepositionsoftheglobesandcordarealwaysthesame.4So,ifwerestrictourattentiontothesystemitself,thentheonlywaywemighttellthatitisrotating(oracceleratinginsomeotherway)isbycheckingfortensioninthecord.5Wemightsupposethatwecanalwayscheckforrotationbylookingforrelativemotionbetweenthesystemandsomebackgroundframeofreferencelikethefixedstars.6Butwecaneasilyimaginethesysteminanotherwiseemptyspace—inanimmensevoid,asNewtoncallsit.7Inthiscase,weareunabletorelyonamaterialbackgroundframeofreference.Butwecanstillbesurewhetherornotthesystemisacceleratingbycheckingforsignsoftensioninthecord.8Ifthereistension,thenwecanjustifiablysaythatthesystemisaccelerating‘absolutely’withrespecttospaceitself.9Andinsuchacasethesourceoftheinertialforcesmustlieinsome‘inertial’interactionbetweentheacceleratingsystemandspace.101\nTIME,SPACEANDPHILOSOPHY10Therefore,wecannotexplainthepresenceofinertialforceswithoutanessentialreferencetospaceitself.Inthissense,spacemaybesaidtobeabsolute—itisanirreducibleelementinourphysicaldescriptionofmatterandforces.Newton’sargumentforabsolutespacemaybereadilyextendedtogiveusa‘Newtonian’argumentforabsolutetime.4Inertialforcesintheglobesystemindicate:thattherereallyisarotation;and,therefore,thatthevelocityofeachglobeiscontinuouslychanging,becausevelocitydependsondirectionaswellasspeed.Thechangesindirectionandthereforeinvelocityarechangesintime.Butinanotherwiseemptyspacethereisnochangingmaterialframeworktowhichthischangemaybereferred.Sothechangeisrelativetoanon-materialtemporalstructure:namely,absolutetime.Steps6and7inthisargumentarecrucial.Theycarrywiththemtheassumptionthattheresultsofdynamicalexperimentswouldbenodifferenteveniftherewerenoothermatterintheuniverse.Ifweallowthisassumption,thenitishardtoseehowwemightruleoutspaceasanirreducibleentityinitsownright.ItmightseemthatNewtonisgoingtoofarbeyondtheevidenceinmakingthisassumption;butheissimplyrelyingonastrategycommonamongstphysicists.Becauseoftheapparentuniversalityoftheeffectsunderdiscussion,Newtonseesnoreasonwhyheshouldnotabstractfromactualconditionstomoregeneralcases.Indeed,hehaseveryreasontodoso.Lawsofnatureareusuallysaidtoholdincounterfactualaswellasfactualsituations.Afactualstatementdescribesasituationwhichhashappenedorishappening.Acounterfactualstatementdescribesasituationwhichmighthavehappenedormightbehappening,butwhichinfacthasnottakenplace.Indeed,wemightreasonablyarguethatitistheprovisionofsupportforcounterfactualswhichdistinguisheslawsfromaccidentallytruegeneralisations.5Thisprocessofabstractioninvolvingtheapplicationoflawstocounterfactualsituationshasmetwithtremendoussuccessthroughoutthescientificenterprise,giventhatitallowsustopredicthowthingswillturnoutinunfamiliarcircumstances.ErnstMach,writinginTheScienceofMechanicsin1883,providesperhapsthestrongestchallengetotheassumptionsbehindsteps6and7;weshallreviewhisargumentinthenextchapter.102\nNEWTONANDTHEREALITYOFSPACEANDTIMEWeshouldalsonotetheassumptionbehindstep8.Newtonregardsmotionasarelationbetweentwoobjects:whenoneofthese‘objects’isspaceitselfthemotionisabsolute.Butwhyshouldweacceptthatmotionisarelationalpropertyatall?LawrenceSklarsuggeststhatwemaytreatmotionasabrutefactaboutanobjectratherthanarelationbetweentwoormoreobjects.WeshallconsiderthisinterestingchallengetooneofNewton’sfundamentalassumptionsinthefinalsectionofthischapter:‘Absolutemotionwithoutabsolutespace?’Wemightbeinclinedtoaddtheclaimthatspaceisasubstanceconsistingin‘asubstratumofspacepointsorregionsthatunderliebodies’toNewton’sargument.6WeshallseelaterinthischapterthatSamuelClarkedoespreciselythisinhisdefenceofNewton.ThereissomejustificationforClarke’smove.NewtontalksfreelyintheScholiumonspaceandtimeinthePrincipiaof‘partsofspace’;andhedoesseemtotreatspaceasasubstanceofsomesortinitsownright.ButtheargumentintheScholiumbyitselffailstoprovideanybasisforthisfurtherclaimaboutsubstantivalpoints.Itisanargumentforabsolutemotionandforspaceasanirreducibleelementinourdescriptionofmotion.Itisnotanargumentforsubstantivalism,ifwemeanbythattheclaimthatspaceconsists(ofprobablyacontinuum)ofsubstantivalpoints.IfweacceptNewton’sargument,thenwemayhavetoconcedethatspaceisasubstanceofsomesort.Butweneedfurtherjustificationforthemovetotreatspaceasasubstratumofpoints.Ofcourse,thesuccessofNewton’sgeometricaldescriptionofmotionmaybecitedinfavouroftheclaimaboutpoints.Forthisapproachtotheproblemsofmotiondoesrelyontheideaofacontinuumofpoints.However,aswesawinChapter1,wehavesomereasontobesuspiciousofthestatusofthecontinuum.Sowemaybeinclinedtoreservejudgementontheclaimthatconfidenceintheexistenceofspaceasanessentialelementinourdescriptionofmotionshouldautomaticallyinclineustowardstheviewthatspaceisasubstratumofpoints.WeshalldiscussthisissuefurtherinChapters7and10.MATTERINTHENEWTONIANWORLDNewton’sconceptionofthephysicalworldisessentiallymechanicalandcorpuscular.Materialbodieswhichconsistoftinycorpusclesinteractwithoneanotherinavastspatialcontaineraccordingtothemechanical103\nTIME,SPACEANDPHILOSOPHYrulessetdowninthePrincipia.IncontrasttoDescartes’sviewof‘contact’materialinteractionswhichdominatedtheperiod,Newton’saccountofgravityallowsmattertoinfluencematteratadistance.7Hislawsofmotionandgravityaresaidtohold,notjustforourownlimitedregionofspace—thesolarsystem—buteverywhereandforalltimes.Andtheselawsgovernthebehaviourofcorpusclesorparticleswhichare‘solid,massy,hard,impenetrable,moveable’.8Newton’sviewsonmatter,wherearticulated,aregenerallyclosetothoseofRobertBoyleandJohnLocke.JohnLocke,whohadfollowedwithkeeninterestandwhohadevenassistedtheexperimentalworkofRobertBoyleinBoyle’sOxfordlaboratory,helpedtoconsolidatethelateseventeenth-centurynotionofmatterinhisEssayonHumanUnderstandingpublishedin1690.Boyle’sownpublicationsinthe1660s,includingTheOriginofFormsandQualities,hadapowerfulimpactuponLocke’sthinking.LockeacceptedBoyle’smaterialistandmechanicalexplanationsofaphysicalworldpopulatedwithcorpusclesorparticlesofmatter.Locke,againinaccordwithBoyle,realisedthatmanyofthethingsthatwesayaboutmatteraresuspectinthesensethattheymaynotbetrueofmatteritself.Theymaymerelybecommentsonthewaysweperceivematerialthings.SoLockeadoptsBoyle’sdistinctionbetweentheprimaryandthesecondaryqualitiesofmatter:betweentheprimarypropertiesgenuinelypossessedbymatterandthesecondarypropertieswhichariseonlyinourperceptionsofmaterialobjects.Locke,whocorrespondedregularlywithNewton,wasfullyinsympathywithNewton’sempiricalcharacterisationofthematerialworld.HeshowsusjustwhatourepistemologicalcommitmentsarewhenweaccepttheideaofmatterassumedinthePrincipia.Whenwethinkofanymaterialobject,wemustthinkofitintermsofitsprimaryqualities:asextended;asinsomestateofmotionorrest;ashavingasurfaceandaninterior;andasbeingeitherasingleentityorsomecollectionofentities.Butwhenwethinkofanobjectashavingtaste,forexample,thereisnothingintheobjectpersewhichimpelsustoregardsuchaqualityasinherentintheobject.Asweettastetoonepersonmaybeundetectablebysomeoneelse.Anobjectmaybeblueatonemoment,but,whenthelightisswitchedoff,nocolouristhereatall.Wemayreadilythinkofmatterwithoutconceivingitascolouredortasty.However,Lockebelievesthatitisimpossibleforustoconceiveofmatterwithoutthinkingofitintermsofitsprimaryqualities.Ifwetakeagrainofwheatandsubdivideitagainandagain,Locketellsusthatitwillstillpossessthesameprimaryqualities.Primaryqualitiesarethereforethemostgeneralandbasic104\nNEWTONANDTHEREALITYOFSPACEANDTIMEcharacteristicsofmatter.EachqualityisabasicbuildingblockoftheNewtonianconceptionofmatter;andNewton’smaterialuniversemaybesatisfactorilycharacterisedintermsoftheseessentialconceptualbuildingblocks.9AlthoughLockeandNewtonseemtobeinagreementonthebasicideaofmatter,Locke’searlierthinkingonspaceandtimeisatoddswithNewton’sviews.InhisJournalforJuly1676,Lockesuggeststhatspaceandtimemayberelationsamongstmaterialthingsandeventsandnotsubstantivalphysicalentities:Andaswecaninourimaginationapplythatmeasureoftime,whichisbutthemotionofsomebody,toduration,wherethereisneitherbodynormotion,sowecanapplythatmeasureofextension,whichisonlyinbodies,tospace,wherethereisnobodyatall,thoughthatdurationandthatspacewithouttheexistenceofanyotherthingbeinitselfreallynothing….Butwhenthingsreallyare,bothdurationandspace,thatisconsideredassomuchspacebetweenthem,arereallyarelationcommensurabletoourmeasuresoftimeandextension.(Locke1676)10However,by1690,Lockespeaksquitehappilyof‘purespace’andof‘durationinitself…goingoninoneconstant,equal,uniformcourse’evenbeforethemomentofcreation.11Theeventbeforethischangeinattitudewas,ofcourse,thepublicationofthePrincipia.LEIBNIZANDRELATIONISMLeibniz’smetaphysicalbeliefstendtoobscurehisattacksontheideaofabsolutespace.Leibnizdrawsadistinctionbetweenthe‘phenomenal’worldofappearancesandtheworldasitreallyis:oursensesgiveusaccesstoappearances,butonlyreasonallowsustothrowbacktheveilofperception.Althoughoursensesseemtotellusthattheworldconsistsofmaterialobjectsoccupyingspaceandpersistingthroughtime,hesaysthatwehavenoreasontotrustourperceptions.Insteadreasonshouldbeourguidetothestructureofreality.Leibniztellsusthatwemayunderstandhowthingsactuallyareonlybydeductionfromundeniableprinciples.Likemanyphilosophersbeforeandafterhim,heregardsthedeductivemethodsofmathematicsandgeometryasanattractivemodelforphilosophicalspeculation.Usingsuchclearly105\nTIME,SPACEANDPHILOSOPHY‘rational’methodsheconstructsamysteriousworldofmonads—simplemetaphysicalsubstances.Thisworldisneitherthespatio-temporalnorthematerialrealmwhichsenseexperiencesuggests.Itisaworldtowhichrationalitycangiveusaccess;butthisaccessislimitedbyourfinitenatures—onlyGodhastheinfinitecapacitiesneededforacompletegraspofreality.Indealingwiththeproblemsofspaceandtime,Leibnizmovessomewhatuneasilybetweentherealmofmonadsandthephysicalworld.Themonadicrealmissometimescharacterisedas‘spatial’and‘temporal’,butnotinthesensesweusuallyemployfortheseterms.Leibnizseemstoclaimthatwhatweseeasspatialandtemporalpropertiesaremerelyanalogoustotheunderlyingstructuresofreality;hencethatthephysicalworldseemstohavespatio-temporalpropertiesisaconsequenceofourimperfectperceptionsofthestructuresofthemonadicrealm.Leibniz’sdenialofanyindependent‘absolute’statusforspaceandtimeinthephysicalworldderivesfromhischaracterisationsofthemonadicrealm.Therelationsbetweenmonadsunderwritetherelationswhichweseeasholdingbetweentheobjectsofthephenomenalworld.Leibnizcomparestheserelationstothosewhichholdbetweenthemembersofafamilytree:nogivenrelationhasanyfundamentalstatusforanymemberofthetree,for‘hewhowasafather,oragrandfather,mightbecomeason,oragrandson’.12Butsincethemonadicrealmis,perhapsnecessarily,opaque,heconcentrateshisattacksonthenotionsof‘absolute’spaceandtimewithinthecontextofthephenomenalworld.So,whenhesaysthatspaceandtimeare‘merelyrelative’sincespaceis‘anorderofthings’whichcoexistandtimeis‘anorderofsuccessions’ofthings,Leibnizisnotthinkingof‘things’ashismetaphysicalmonadsbutastheregularmaterialobjectsofthephenomenalworld.13Leibniz’sargumentagainstabsolutespacehasitsclearestexpressioninhiscorrespondencewithSamuelClarkeduringtheperiod1715–16.ThecorrespondenceconsistsinfivelettersfromLeibnizandfiverepliesfromClarke,andaspiriteddebatewasonlybroughttoanendbyLeibniz’sdeath.MuchofthecorrespondenceconcernstherelationshipbetweenGodandtheworld,buttheirrespectivebeliefsaboutthenatureofspace,time,andmotionemergereasonablyclearlyandtheessentialdifferencesbetweenthemdonotrelyonanyparticularviewofGod.However,theirdebateisplaguedbynumerousmisunderstandings,mostlyonthepartof106\nNEWTONANDTHEREALITYOFSPACEANDTIMEClarke;anditalsosuffersfromLeibniz’sultimatefailuretoprovideanadequateanswertoNewton’sargumentforabsolutespaceasanessentialelementinourdescriptionofmotion.But,asweshallseeinthenexttwosections,anaturalextensionofLeibniz’sideasdoessuggestapossibleanswer:absolutemotionwithoutabsolutespaceortime.ClarkehadworkedcloselywithNewtonandthecorrespondencegavehimagoldenopportunitytodefendtheNewtonianworld-view.LeibnizarguesthattheNewtoniancasefor‘absolutespaceandtime’involvescontradictorybeliefsandmustthereforeberejected.InhisthirdlettertoClarke,Leibnizemploystwoimportantprinciples,thePrincipleofSufficientReason(PSR)andthePrincipleofIdentityofIndiscernibles(PII),asaxiomsofhisargumentsagainsttheNewtonianviewofspaceandtime:PSR‘nothingeverhappenswithouttherebeingacauseoratleastadeterminingreasonforit’;14and,asaconsequenceofPSR,PII‘therearenotinnaturetworeal,absolutebeings,indiscerniblefromeachother[byGod];becauseiftherewereGodandnaturewouldactwithoutreason,inorderingtheoneotherwisethantheother’andsotwoindiscerniblethingsareidentical—i.e.thetwoareinfactoneandthesamething.15Leibnizaddstwo‘plausible’assumptions(3and4below)totheseprinciples,andthentriestoshowthattheNewtonianpositionisincoherent:1ArationalGodmusthaveasufficientrationalreasontoactortomakeachoice(givenPSR).2IfAisindiscerniblefromB,thenAisidenticalwithB(givenPII).3Spaceisaninfinite,non-material‘absolute’containerformatter.4Ifspaceexistsasanindependentcontainer,thenmattermaybeplacedinthatspacehowsoeverGod(whoisomnipotent)chooses.5Godmightchoosetoplacematterinspaceinagivenpositionandorientation,thuscreatingauniverseXwithagivenconfigurationrelativetospace.107\nTIME,SPACEANDPHILOSOPHY6Hecouldhavechosentoplacematterinadifferentpositionandorientation,thuscreatingauniverseYsuchthattherelativepositionsofthematerialcontentstooneanotherarejustasinX,butinwhichthesecontentsareplaceddifferentlyinspace,e.g.thecontentsofYmayberotatedthrough180degreescomparedwiththoseofX.7Spaceisabsolutelyuniform,i.e.allitsconstituentpartsarealike;hencethematterdistributionsinbothXandYwillberelatedtothebackgroundspatialcontainerinthesameway.8Given7,thereisnodiscernibledifferencebetweenthetwopossibleuniversesXandY.9ThereforeXandYareindistinguishable.10Given2and9,XandYarethesamething.11Arationalchoicecannotbemadebetweentwoobjectswhichareinfact‘thesamething’sincethereisnorationalchoicetobemade.12Given1,10and11,arationalGoddoesnothaveasufficientreasontochoosebetweenXandY.13ThereforeGodhasnochoiceinhowmatterisplacedinspace.14Given4and13(thedenialoftheconsequentof4),theantecedentof4isalsofalse(bymodustollens).1615Butassumption3istheantecedentof4:henceassumption3(spaceisanindependent‘absolute’containerformaterialthings)mustalsobefalse.Leibnizappliesasimilarargumentagainst‘absolute’time.InsteadofaskingwhetherornotGodhasafreechoiceastohowmatterisplaced,LeibnizasksifGodmaychoosethemomentofcreation.Hethensaysthattherecouldbenorationalchoicebetweenauniverseinwhichmatteriscreatedatagiventimeandoneinwhichmatteriscreatedayearearlier.And,justasinthecaseofspace,thereseemsnodiscernibledistinctionbetweenthetwopossibleuniverses—theirhistorieswouldbeidentical.Sowemustalsorejecttheideaoftimeasanindependent‘absolute’backgroundframeworkforchangeinthematerialworld.17CLARKE’SDEFENCEOFNEWTONSamuelClarkerespondstoLeibniz’sargumentsbychallengingthestatusandpowerofPII.Hesays:108\nNEWTONANDTHEREALITYOFSPACEANDTIMEIntelligentbeingsareagents…theyhaveactivepowersanddomovethemselves,sometimesupontheviewofstrongmotives,sometimesuponweakones,andsometimeswherethingsareabsolutelyindifferent.Inthelattercase,theremaybeverygoodreasontoact,thoughtwoormorewaysofactingmaybeabsolutelyindifferent;[and]Twothings,bybeingexactlyalike,donotceasetobetwo.Thepartsoftime,areasexactlyliketoeachother,asthoseofspace:yettwopointsoftimearenotthesamepointoftime,noraretheytwonamesofthesamepointoftime.(Clarke1716)18So,althoughtheremaybenoapparentdifferencebetweentheuniversesXandYinLeibniz’sargument,Clarkeclaimsthattheyareneverthelessdistinct—allpointsofspaceandtimeareindeedalikebuttheyarecertainlynotthesamepoints.HencePIIdoesnothold:fortwoindistinguishablethingsarenotoneandthesamethingatall.ButhedoesnotdenyPSRsinceheacceptsthatGodwillalwayshavesomereasontoactorchoose.LiketheStoicsintheyearsafterAristotle,Clarkeseesnoprobleminimaginingtheentirematerialworldbeingmovedinaninfinitespace.19However,hequestionsthelinkbetweenPSRandPIIwhenhesaysthatachoicewillbepossibleevenincaseswheretwoobjectsareexactlyalike.Clarke’sdefenceoftheNewtonianpositionreliesonthestatementthat,despitetheuniformityofspaceandtime,theyneverthelesshavedistinctivepoints.Thisleadstotheclaimthatspaceandtimearequantitiesinthemselvesratherthanmererelationsbetweenobjects.ThemainargumentforthisclaimappearsinClarke’sfourthreplytoLeibnizanddependsuponNewton’sdistinctionbetweenrealandapparentmotion,usedtogoodeffectintherotatingbucketandtwoglobesthoughtexperiments.20Thisargument,likeNewton’s,doesnotdependuponthepowersofGodandhisinteractionswiththephysicalworld.ItturnsonanempiricalproblemwhichLeibnizmustconfrontifheistovindicatehisrelationalviewpoint.ThefollowingisareconstructionofClarke’sargument:1Ifthepointsorregionsofspaceandtimecanbeshowntohavedistinctcharacteristicswhichdonotthemselvesdependonmatterandtherelationsbetweenmaterialobjects,thenspaceandtimehaveanindependentrealityasquantitiesintheirownright.109\nTIME,SPACEANDPHILOSOPHY2Ifandonlyifanobjectoccupiesdistinctsetsofpointsordistinctregionsatdistincttimes,thenitisreallymovingthroughspaceandintime.3Ifandonlyifanobjectisreallymovingthroughspaceandintime,thenitexperiencesinertialforceswhenitsstateofmotionchanges.4Objectsdoexperienceinertialforceswhentheirstatesofmotionarechanged.5Given3and4,objectsreallydomovethroughspaceandintime.6Given2and5,objectsoccupydistinctplacesorregionsinspaceatdistinctmomentsoftime.7Hence,thepointsorregionsofspaceandtimehavedistinctcharacteristics.8Given1and7,spaceandtimehaveindependentexistences.Assumption2forgesalinkbetweentherealityofpointsorregionsinspaceandabsolutemotion.WeshallseeinthefinalsectionofthischapterthatpreciselythislinkischallengedbySklar’sargumentthatabsolutemotionmaybeabrutefactaboutanobjectratherthananassertionofarelation.Soweshouldnotleapfrom‘theobjectismoving’to‘theobjectismovingwithrespecttosomeotherobject’.Thisclearlyrequiressomefundamentalchangesinattitudetowardsmotion.Ifweweretomakesuchchanges,thenneitherClarke’snorLeibniz’sspecificapproachestotheprobleminhandwouldbeappropriate,for,asweshallsee,thetermsofthedebatewouldbechangeddramatically.21Assumption3abovereliesonNewton’sanalysisonrealandapparentmotionandaswehaveseentheproblemofinertialforcesismostacuteinthecontextofrotation.Thedifferencebetween(a)alargewheelturningfromanobserver’spointofviewand(b)theobserverappearingtomovefromthepointofviewofsomeoneattachedtotheouterrimofthewheelmaybetracedtothefactthatonlythewheelandits‘rider’experienceinertialforces.Thatiswhywemaysaythewheelisreallymovingandtheobserverisnot.22LeibnizhadalreadyconsideredtheproblemsofinertialforcesindetailduringacorrespondencewiththephysicistChristianHuygensduring1695–5.23Theirlettersindicatethattheyhadreachedameasureofagreementconcerningtherelationalnatureofmotion,althoughHuygensallieshimselfinspiritandwordwithmuchoftheNewtonianview.Eachclaimstohaveresolvedtheproblemofinertialforces,butonlywhatseemstobeHuygens’purportedsolutionhasbeendiscovered.Inthis,Huygensshowsthatheappreciatestheneedforanaccountoftheoriginofinertialforces,butsimplyrestatesthe110\nNEWTONANDTHEREALITYOFSPACEANDTIMEproblemintermsofvelocitydifferences.Hearguesthatanyforceswhichareexperiencedbyanobjectattachedtoarotatingobjectareduetorelativemotionalone:thefactthatanobjectattachedtothecircumferenceofaspinningwheelexperiencesinertialforcesisexplainedbytherelativemotionbetweenthisobjectandtherestofthewheel.Althoughtheobjectandsomepartsofthewheelmaybemovingtotheleftatagiveninstant,otherpartsofthewheelwillbemovingtotheright.Soaspinningwheelexperiencesinertialforcesnotbecauseofitsrotationrelativetoanyabsoluteframework,butbecauseoftherelativemotionsofthevariouspartsofthewheel.Huygens’solutiondoeshavethevirtueofrecognisingtheobjectivesignificanceofdifferencesinvelocities:suchdifferencesmayobtainevenwhentherelativepositionsofbodiesorofpartsofbodiesremainthesame,asinthespinningwheelexample.Twodistinctpointsonthecircumferenceofthewheelmayhavethesame‘speed’buttheywillhavedifferentvelocitiesbecausetheyaremovingatanygiveninstantindifferentdirections.ButHuygensfailstograsptheproblemofinertialforcescompletely.Forwemayeasilytransformawaytherelativemotionsbyviewingthespinningwheelfromaframeofreferencewhichisrotatingalongwiththewheel.Thereisnorelativemotionand,asfaraswecansee,therearenovelocitydifferences.Butthereareplentyofforces!This,ofcourse,isexactlytheproblemraisedinNewton’sthoughtexperiments.UnlessLeibniztacklesthisfundamentalproblem,wecanhardlyregardhimasvictoriousinhisdisputewithClarke.ClarkedirectsLeibniztoNewton’sanalysisinthePrincipia,whichtakesusrightbacktotheheartofNewton’sargumentforspaceasan‘absolute’independentelementofreality.And,aswehaveseen,asimilarargumentmaybeconstructedforabsolutetime.Clarke’sargumentrestsonthesamepremiseasNewton’s:inertialforcesprovideaclearindicationof‘absolute’motionthroughspaceandintime.Clarkeisrathervagueabouttheideaofspaceandtimeasabsolute‘quantities’;butitseemsreasonabletosupposethathetakesthenaturalinferenceofNewton’sargumenttobethatspaceandtimehaveessentiallythesameontologicalstatusasmatteritself.Theyarenotqualitiesofotherthings;buttheyareafundamental(ifnon-material)quantity.24Justasamaterialbodycanhaverealeffectsonotherbodies,sospaceandtimehaverealeffectsonobjects.Hence,space,time,andmatterareallinsomesenserealquantities.25111\nTIME,SPACEANDPHILOSOPHYInhisfifthandfinalletter,LeibnizsaysthathecanfindnothinginNewton’sdiscussioninthePrincipia‘thatproves,orcanprove,therealityofspaceinitself.26Thisisbecauseherefusestoacceptanylinkbetweeninertialforcesandmotioninspace.Hegrantsthatthepresenceofinertialforcesinabodyindicatesthatthebodyisinmotion,butthismotionismerelyrelativetootherobjectsratherthantospaceitself.Thecauseofinertialforcesneednotbetracedbacktomotionwithrespecttospaceandtime;forwhenabodyisreallyinmotion‘theimmediatecauseofthechangeisinthebody’itself.27Clarkedismissesthisresponseasinadequate,andrefersthereaderbacktoNewton’sargument.IntheabsenceofLeibniz’smissing‘solution’,whatisneededisastraightforwardanswertothequestion:howmayweexplainthepresenceofinertialforcesinsomebodiesandtheirabsenceinotherswithoutinvokingspaceandtimethemselves?ThecluetoapossibleanswerliesinLeibniz’sratherobscurecommentabovewhichsuggeststhatthesourceofinertialforceslieswithinbodiesthemselves—undoubtedlyareferencetohisnotionofvisviva,anactiveinternalforce,thepossessionofwhichisanindicationoftruemotion.28Thissuggeststhepossibilityoftrue‘absolute’motionwithouttheneedforabsolutespaceasaglobalreferencebackgroundformotion—and,aspromised,thispossibilitywillbeconsideredinthenextsection.Leibnizoffersusaphysicalworldinwhichonlymaterialthingshaveanymetaphysicalbasis.Hisprogrammeisreductionistinthesensethatallspatialandtemporalrelationsmaybereducedatthephysicalleveltorelationsbetweenmaterialbodies(andatthemetaphysicalleveltorelationsbetweenmonads).29Inasense,Leibnizproposesasimplemonistpictureoftheworld,evenifthismonismisonlyfullyeffectiveatthemonadiclevelwherethemonadsthemselvesarecharacterisedaspurespirit.ButNewtonpresentsuswithamorecomplexphysicalworld:notonlymustweacceptmaterialthingsasfundamental,butspaceandtimetooareirreduciblenon-material‘substances’.ItistemptingtoregardNewton’sphysicalworld-viewasessentiallydualist,andmanywritershavedonejustthis.30ButNewtongivesusnoreasonatalltosupposethathethinksthatspaceandtimearethesamekindsofthing.Hisphysicaluniversehasthreedistinctkindsof‘substance’:matter,space,andtime.Andatthemetaphysicallevel,spiritisaddedtothesethree.31AlthoughtheproblemsofNewtonianspaceandtimearemostfrequentlysetinthecontextoftheLeibniz-Clarkedebate,thereisagooddealofevidencetosuggestthatNewton’sconceptionofthephysicalworldarisesatleastinpartfromhisoppositionto112\nNEWTONANDTHEREALITYOFSPACEANDTIMEDescartes’sphysicaltheories.32Indeed,theideasofBoyleonmatteranditsproperties,theinfluenceofwhichmaybeseeninbothLockeandNewton’sthinking,areinpartaclear,criticalreactiontotheCartesianview.Descartesarguesforareductionistpointofview,but,unlikeLeibniz,hetakesspatialextensiontobenotjustthemostfundamentalaspect,butalsothedistinguishingfeatureofmatter;hencemattermaybereducedtospatialconcepts.Forabrieftime,untilNewton’sviewsgainedwideacceptance,Descartes’spositionremainedpopular.ABSOLUTEMOTIONWITHOUTABSOLUTESPACE?Tworocketsinspace,farfromanystar,arealongsideeachother.OnerocketcarriesPeter;andPaulaisintheother.Theyseemtobequiteweightless,floatinginaforce-freeenvironment.Theclassicalprincipleofrelativityimpliesthatthereisnowayforthemtodiscoverwhethertheyareatrestwithrespecttospaceitself,orwhethertheyarerushingalongatanenormousbutuniformvelocity.ThenPeterlooksoutatPaula’srocketandseesitacceleraterapidlyaway.AndPaulalooksoutandseesPeter’scraftacceleraterapidlyaway.ThereisnokinematicdifferencebetweenPeter’sperspectiveandPaula’s:bothastronautsmeasurethesamerelativeaccelerationbetweenthetworockets.Buttheremaybeadynamicaldifference.PetermayfeelaninertialforcewhilstPaularemainsforce-free,orviceversa.IfPaulaexperiencessuchaforce,thenshemaysaythatsheisacceleratingabsolutely.ButisthisaccelerationabrutefactaboutPaulaandherrocket?NewtonandClarkedonotevenquestiontheirbasicassumptionthatmotionisalwaysrelativetosome‘thing’—whetherthisisamaterialobjectorspaceitself.However,SklarsuggeststhatwemightsaythatPaulasimplyfeelsinertialforcesandthatanyobjectwhichexperiencessuchforcesis‘inmotion’.33ThissuggestionseemsanaturalextensionofLeibniz’sratherundevelopedviewofinertialforcesasarisingfromtheinternalcharacteristicsofmaterialobjectsandnotfromsomedynamicrelationtoanexternalframeofreference.Newtonexplainsthepresenceofinertialforcesbyinvokingabsolutemotion(i.e.motionrelativetospaceitself).Newton’saccountsuggeststhatthereissomekindofcausalinteractionbetweenspaceandmatter:spaceactsonanacceleratingobjectinsuchawayastoproduceinertialforces.Itisinterestingtonotethat,contrarytoanyexpectationthatNewton’sthirdlawofmotionmightgiveus,thereisnoreactionof113\nTIME,SPACEANDPHILOSOPHYmatteronspace,sinceNewtontellsusthatspaceisnotinfluencedinanywaybyitsmaterialcontents.34Sklardoesnotseekanexternalexplanationoftheinertialforces:‘real’motionisjustsomethingwesayishappeningtoanobjectwheninertialforcesareexperienced.Butthis‘real’motionisnomorerelationalthantheforceitself.Justasitisabrutenon-relationalfactaboutanobjectthatithassomanyatomsandthereforesomuchmass,sowemaythinkoftheexperienceofaforceasabrutefactaboutanobject.Materialobjectsjusthappentobedividedintotwosets:1thoseobjectswhichexperienceinertialforces;and2thosewhichdonotexperienceinertialforces.Sklarsuggeststhattheremaybenothingtobegainedbyaddingtheclaimthatobjectsinset1arereallyinmotionwithrespecttospace.Newtonmightrespondbysayingthatwhatwegainisthepowerfulconceptofaninertialframeofreference.Acceleratedmotionisalwaysmotionrelativetoallinertialframes,oneofwhichisspaceitself.But,sincetheconceptisintroducedinordertodistinguishthemembersofset1fromthoseof2,wemightarguethatthereisanelementofcircularityhereandthatweneedsomeindependentreasontosupposethatabsolutespaceisrequiredinourdynamicaldescriptions.However,Sklar’sviewseemstodemandaradicalshiftinthewaywethinkaboutmotion.Sowemightbeinclinedtodismisstheideaofmotionasabrutefactsimplybecauseitdoesnotcaptureourbasicintuitionthatmotioningeneralmakessenseonlywhenwethinkofitintermsofsomekindofrelativemovement.MichaelFriedmanarguesthatspaceinclassicalphysicsprovidesaunifyingprinciple,drawingtogethertheotherwiseseparatedomainsofclassicalmechanicsandelectromagnetism.35Allmotionsmaythenbereferredtoexactlythesamebackgroundframework:space.Similarly,motionsinmodernphysicsmaybereferredtoaspacetimeframeworkwhichunifiesrelativityandelectromagnetism.WhatmakestheideaofabsolutespaceespeciallypowerfulinFriedman’sviewisthefactthatitplaysthesameexplanatoryroleindistincttheories.EvenifweacceptFriedman’sclaimthatunificationbringsbetterscientificexplanations,therolesofspaceandspacetimeasunifyingframeworksmaystillbechallengedbyMach’sargumentthattheonlyrelevantbackgroundarenaformotionisthematerialuniverse.WeshallthereforeconsiderthisargumentinChapter6.AlthoughSklar’ssuggestionmayseemtolackexplanatorypowerandmayseemtoinvolveanextremely114\nNEWTONANDTHEREALITYOFSPACEANDTIMEcounterintuitiveideaofmotion,JohnEarmanclaimsthatthedifficultiesbeforestandardabsolutistandrelationistprogrammesaresogreatthatweshouldconsiderthesuggestionaspossiblythebestanswertotheproblemsoftherealityofspaceandtime.ThisclaimwillbeconsideredingreaterdetailwhenweexamineEarman’scritiqueofabsolutistprogrammesinthefinalsectionofChapter7,togetherwithsomefurtherdiscussionatthecloseofChapter10.36115\n6MACHANDTHEMATERIALWORLDINTRODUCTIONInhisstudyofmotion‘Demotu’writtenin1721,BishopGeorgeBerkeleydismissesthenotionsofabsolutespaceandtimeasbeingwithoutmeaning.Berkeleybelievesthatonlysenseexperiencemayunderwritemeaning;and,sincespaceandtimehavenofoundationinoursenseexperience,wehavenoreasontoacceptthemasmeaningfulwords.ThislineofargumentisonepursuednearlytwocenturieslaterbyErnstMachandbyothersofapositivistpersuasion.However,Machaddshisownuniquetouchtotheargument—combiningtheempiricismofBerkeleywithadeeprespectforsimplicityinscience.1Mach’sargumentisdrivenbytwodistinctmotivations:thedesire,likethatofBerkeley,nottogobeyondtheevidenceofdirectsenseexperience;andthewishtoachievethegreatestpossiblesimplicityincharacterisingthephysicalworld.Thereisnodoubtthathisdislikeoftheideaofabsolutespacestemsinpartfromthefactthatspaceitselfseemstobeunobservable.Machsaysthatonlytheobjectsofsenseexperiencehaveanyroleinscience:thetaskofphysicsis‘thediscoveryofthelawsoftheconnectionofsensations[perceptions]’;and‘theintuitionofspaceisboundupwiththeorganisationofthesenses…[sothat]wearenotjustifiedinascribingspatialpropertiestothingswhicharenotperceivedbythesenses’.2AndMachseemstobemorethanwillingtowieldOccam’srazortoexciseanysuperfluousentityfromourscientificdescriptions.Hisremarksonsimplicityinscienceshowthathetakeseconomytobeofcentralimportance;hesaysthat‘thefundamentalconceptionofthenatureofscience[isthe]economyofthought’andscience’s‘goalisthesimplestandmosteconomicalabstractexpressionofthefacts’.3116\nMACHANDTHEMATERIALWORLDSuchassertionsformthephilosophicalbackgroundtoMach’sattackonhypotheticalentities.AtomsaswellasabsolutespacereceiveastrongrebufffromMach.Hisattitudemaybesummedupthus:ifyoucannotseeorotherwisesenseapurportedentityandifitisnotanessentialelementofourphysicaldescriptions,thendonotgrantanythingmorethaninstrumentalstatustotheentity—atbestanyreferencetotheentityisshorthandforacollectionofobservationaldescriptions;atworstreferencestoanentitymaycarrywiththemtoomanymetaphysicalassociationstoadmitthemtoourscientificdescriptions.Mach’smostimportantroleinthedebateaboutspaceandtimemaywellbeasanagentprovocateurratherthanasanactiveparticipantinhisownright.Yes,hischallengetoNewton’sargumentissignificant.However,themainsignificanceprobablyliesinhowothershavebeenmotivatedbythechallenge.ForMachwasunabletofulfilhisowndreamofatheoryofmotionwithoutspaceandtimeasessential‘absolute’elements.Many,includingEinstein,admiredMach’sfiercescepticismaboutthestatusoftheoreticalentities,eveniffewagreedfullywithMach’sgeneralphilosophyofscience.AlthoughsomewritersclaimthatMachultimatelyfailedinhisattacksuponNewtonandthathepropoundedadisastrouslyflawedphilosophy,thereislittledoubtthatMach’sinfluencehasbeenbothfar-reachingandlong-lasting.4Threeaspectsofthisinfluencearenoteworthy:1EinsteinwasdeeplyimpressedbyMach’sScienceofMechanics,and,forsomeyears,hebelievedthatrelativitymightbetherealisationofoneofMach’saims:atheoryofdynamicswithnoessentialreferencetospaceitself.52DespiteEinstein’sadmissionthathisowntheoryseemedtofallshortofthisaim,Mach’sideasalsoinspiredeminentphysiciststhroughoutthetwentiethcenturytoconstructvariationsofrelativitywhichmightfulfilthatsamedream.63Machalsostimulatedmanyintheworldofphilosophy:inparticularlogicalpositivists,formingoneofthemostinfluentialphilosophicalmovementsofthemid-twentiethcentury,attimeslentheavilyuponMach’spositivisminformulatingtheirphilosophicalprogramme.7EvenifMach’sscientificdreamsareneverrealised,hisroleasaninspirationtoagenerationormoreofscientistsandphilosophersmarkshimoutasoneofthemostimportantcontributorstoourideasaboutspaceandtime.117\nTIME,SPACEANDPHILOSOPHYMACH’SRELATIONISMPassagesthroughoutTheScienceofMechanics,publishedin1883,showMachtobeunsympathetictoNewton’sconceptofspaceasanabsoluteframeofreferenceformotion.But,unlikeLeibniz,hedoestrytoprovideacogentresponsetoNewton’sempiricalargumentforabsolutespace.HegoesrighttotheheartofNewton’sargumentandchallengesthebasicpresumptionbehindsteps6and7,whichwemayrecallfromthelastchapterare:6Butwecaneasilyimaginethesysteminanotherwiseemptyspace—inanimmensevoid,asNewtoncallsit.7Inthiscase,weareunabletorelyonamaterialbackgroundframeofreference.Butwecanstillbesurewhetherornotthesystemisacceleratingbycheckingforsignsoftensioninthecord.NewtonhasnoqualmsaboutassumingthatthedynamicaleffectsobservedinanEarth-boundlaboratorywillapplyinallcases,possibleaswellasactual.Thefixedstarsmayactasaconvenientglobalframeofreferenceformotion,butNewtonclearlybelievesthattheirpresenceismerelyincidentalandthereforetheyshouldnothaveanyprimaryroleintheanalysisofmotion.ButMachobjectsthatwehavenowayofdeterminingwhatmighthappenincircumstancesradicallydifferentfromthoseobserved—wheretheentirebulkoftheuniverseisremoved.Sowhatisthesenseintalkingabout‘emptyspace’withnomaterialcontents?Wemustdealwiththeuniverseinwhichweliveandnotbuildspeculativecastlesintheair.Hesaysthat:ItisscarcelynecessarytoremarkthatinthereflectionsherepresentedNewtonhasagainactedcontrarytohisexpressedintentiononlytoinvestigateactualfacts.Nooneiscompetenttopredicatethingsaboutabsolutespaceandabsolutemotion;theyarepurethingsofthought,purementalconstructs,thatcannotbeproducedinexperience.Allourprinciplesofmechanicsare,aswehaveshownindetail,experimentalknowledgeconcerningtherelativepositionsandmotionsofbodies….Nooneiswarrantedinextendingtheseprinciplesbeyondtheboundariesofexperience.Infact,suchanextensionismeaningless,asnoonepossessestherequisiteknowledgetomakeuseofit.(Mach1883:280)8118\nMACHANDTHEMATERIALWORLDMachdoesnotseewhyweshouldacceptNewton’sbeliefthatthefixedstarsareincidentalintheanalysisofmotion.SoNewtonisaccusedofignoringtheroleofthefixedstarswhenwhatisatissueispreciselythatrole.Ourscientificreasoningshouldbelimitedtotheworldasitis,theworldofobservationandexperiment.Hence,wehavenoreasontotracetheoriginofinertialforcestoanythingotherthanmaterialbodies.Todootherwiseistoengageinfruitlessandprobablymeaninglessspeculation.Onaglobalscale,ourreferenceframemustbe‘theentireuniverse’andweshouldthereforelookforadynamicaltheorywhichaccountsforinertialforcesinmaterialtermsandwhichthereforedoesnotrelyontheprobablysuperfluousideaofabsolutespace.9InattackingNewton,Machpresentsaforcefulchallengetotheideathatourlawsmightoperateinanycounterfactualcircumstances.HearguesinTheScienceofMechanicsthatweshouldrefermotionstoamaterialframework—thatofthematerialuniverseasawhole.Heseesnocompellingreasonwhyweshouldrelyupon‘metaphysical’artefactslikeabsolutespacewhenallmotionsmightbereferredinsteadtoaphysicalframeofreference.Theinertialforcesexperiencedbyacceleratingbodiesmightthenbetheresultofsomekindofglobalinteractionwiththerestofthematterintheuniverse.Machspeculatesmuchaboutthepossiblematerialinteractions,localaswellasglobal,whichmightgiverisetoinertialforces.Heasks,forexample,whetheranenormouslythick-sidedbucketsetinrotationmighthaveimmediateandobservablelocalinertialeffectsonany‘stationary’waterwithinduetotheresultingrelativerotationbetweenbucketandwater.However,Machoffersusnodetailedpositiveexplanationofhowsuchamaterialinteractionmightoperate.Machfailstoraise,letaloneanswer,suchimportantquestionsas:doinertialforcesfalloffwithdistanceaccordingtoaninversefunctionoraninversesquarefunction?HesimplystopsshortathisnegativecritiqueofNewton’sviews.Einstein,whohadstudiedTheScienceofMechanicsasayoungman,admiredMachforhisscepticismbutwasscepticalhimselfaboutthepositivesideofMach’sideas.Einstein,perhapsmorethananyoneattheturnofthecentury,realisedthatmuchworkhadtobedonebeforesciencecouldprovideacoherentaccountofinertialforceswithoutreferencetospaceandtimethemselves.Machdidnotachievehisdesireforaneconomicalaccountofdynamicscontainingnoessentialreferencetospaceandtime.119\nTIME,SPACEANDPHILOSOPHYNewtoniantheorydemandssuchreferences;andthereforethebestthatmaybesaidforMach’sargumentisthatitcouldpointthewayeithertoanewtheoryoranewvariationofNewtoniantheoryinwhichmatteritselfisthesourceofinertia.Butweneedmorethanrhetorictopersuadeustogiveupanestablishedandsuccessfulscientifictheory.Twomainalternativesfaceus:1Weneedtobeshownthatirreduciblereferencestotheoreticalentitiessuchasspacearewronginprinciple,sothatanytheorycontainingsuchreferencesmayberejectedonthosegroundsalone.Ifwetakethispath,thenweshallrequireaverystrongstatementagainsttheoreticalentitiesandstatements.2Weneedtolookforanalternativetheorywhichmakesnoessentialreferencetotheoreticalentitiessuchasspaceandtime,achievingwhatMachmightregardasasimpleraswellasanobservationaldescriptionofthephysicalworld.Inthiscase,weshallstillhavetojustifytheimplicitpreferenceforobservationalovertheoreticalstatements.So,ifweareinclinedtobesympathetictoMach’sapproach,weneedtoexplainwhyweshouldadopthisempiricistbias.TherearetwocentralelementsinMach’sapproachtotheproblemofmotion:hispositivisticoutlookandhisideasaboutsimplicity.Thereissomereasontothinkthathistheoryofeconomyinsciencebothconstrainedandinformedhispositivisticoutlook.So,beforeexaminingthisoutlookfurther,weshallexplorethemainideasinhistheoryofeconomy.SIMPLICITYANDSCIENCEMach’sideasonsimplicitybegantoemergeasearlyas1861,sometwentyyearsbeforethepublicationofhisattackonNewtoninTheScienceofMechanics.10Inthattwenty-yearperiodMachdevelopedperhapsthefirstdetailedaccountoftheroleofeconomyinscience.HisadmirationforbothGalileoandNewtonwasbasedinpartontheelegantandsimpletreatmentwhichthesetwogiantshadgiventotheirrespectiveideasofmechanicsandmotion.And,intheirworkandintheworkofothergreatscientists,Machsawthreedistinctbutinter-relatedkindsofeconomicalattitudesatwork.120\nMACHANDTHEMATERIALWORLD1Machrecognisedthatthescientificenterpriseisfartoovastandcomplextobemanagedifwearedistractedbytoogreatanattentionforthefinedetailofthephysicalworld.Scientistslooktosimplifyingrelationshipsandformulaesothattheyhavesomechanceofsuccessintheirattempttocharacterisetheworld.Machreadilyadmitsthat‘Weneverreproducethefactsinfull,butonlythatsideofthemwhichisimportanttous,movedtothisdirectlyorindirectlybyapracticalinterest.Ourreproductionsareinevitablyabstractions’(Mach1883:578–9).11Andso,inanattempttocopewithboththedetailandthemysteriesofourexperienceofthephysicalworld,wearedriventogeneraliseandtosymbolise:anecessarycompromisebetweenthedesiretounderstandallandtherecognitionofourlimitationsashumanobservers.Itisanecessary‘sacrificeofexactnessandfidelity’.12Machseesthatscientistsshouldnotwastetheirtimeunnecessarilyontediousprocedures.Theyshouldeconomiseasmuchaspossibleontheirtimeandtheirefforts.Andsymbolicgeneralisationsandabstractionshelpthemtoachievethisgoal.2Mach’sregardformathematicsisenormous.Withoutmathematics,thetaskofdescribingthephysicalworldinapowerfulwaymightindeedseemfutile.Theveryformsofmathematicsandgeometryareamongstthegreatestalliesofscience.Andsoscientistsshouldbeconcernedtocapitaliseontheflexibilityandpoweroftheseforms.Economyintheformofadescriptionshouldbethegoalofeveryscientist.MachfrequentlycitedNewtonianmechanicsasanexemplarymodelforallscientifictheories,giventheelegantandeconomicalformofitsmathematicalandgeometricaldescriptions.3Machbelievesthatscientistsshouldbeeconomical,notjustwiththeirlabourandintheformofthedescriptionstheyproduce,butalsowithregardtothecontentofthosedescriptions.Scienceshouldaimtominimiseitsepistemologicalcommitments:forexample,ifbeliefinthreebasickindsof‘stuff’issufficienttodescribetheworld,thenweshouldnotaddtothosebeliefs—ifwesupposeanyadditionalentitytoexist,thenthatsuppositionislikelytobenomorethanmetaphysicalspeculationabouttheworld‘beyond’thefacts.Machseeslittletobegainedbyengaginginmetaphysicalgames.Hedoesrecognisetheimportanceofusingtermslike‘atom’forsummarisingkeyfeaturesofourexperience.Butuseofsuchtermsimpliesnoepistemologicalcommitmenttounobservableentitiesassuch:theyaremerelyshorthandtermsstandingforcomplexdescriptions.SoMachurgesscientiststowieldOccam’srazortocutawaysuperfluousentities121\nTIME,SPACEANDPHILOSOPHYfromtheiressentialcommitments:thebuilding-blocksofscienceshouldbekepttoaminimum.AlthoughMach’sideasaboutsimplicityandeconomyblendwellwithhiscommitmenttoobservationandourdirectexperienceoftheworld,theseideascarryawider,deepermethodologicalmessageforourapproachtoscience.Scientificmethodsthemselvesshouldbeconstrainedandenrichedbythesearchforsimplicity,which,Machtellsus,isthe‘fundamentalconceptionofthenatureofscience’.13Mach’sideasoneconomyarenomorethanasketchforamoredetailedaccount.Hefailstorecognisethatthevariousgoalsofeconomyinlabour,form,andcontentmightpointusinquitedifferentdirectionsintryingtocharacterisetheworld.14Amoreseriousproblemishisfailuretodistinguishbetweensubjectiveandobjectiveideasofsimplicity.15ButperhapsthemostseriousproblemisraisedbyMichaelFriedman,whoarguesthatadditionstoourbasicbuilding-blocks,whichMachmightregardasmetaphysicalextravagances,cansometimesplayaunifyingandsimplifyingroleinourgeneralaccountsofthephysicalworld.16Friedmanseestheconceptsofspaceandtimeasplayingsucharole,helpingtobringtogethergravityandelectromagnetismbyreferringallphenomenainthesedistinctdomainstothesamebackgroundframework.AlthoughMachcertainlycapturesanessentialfeatureofthescientificenterprise,hisviewsonsimplicityarejusttoobroadandhazytohelpusdeterminespecificcriteriaforscienceanditsmethods.Buthisviewsdohelpustoappreciatehisattackonabsolutespaceandtimeas,atleastinpart,motivatedbymethodologicalgoalsofsimplicity.POSITIVISMINACTIONMachisfrequentlyregardedasthechiefinstigatorofscientificpositivism,aphilosophyofsciencewhichregardsthepossibilityofobservationaland/orexperimentalverificationasthedefiningcharacteristicofallscientificstatements.HisempiricistpolemicechoedtheviewsofearlierphilosopherssuchasBerkeleyandHume.Andothersduringthenineteenthcenturyvoicedsimilarbeliefs.Mach’sinfluenceinthescientificworldwasfar-reaching.Manyscientistswereinitiallyexcitedbywhattheybelievedtobethepossibilityoffreeingtheirdomainonceandforallfrommetaphysicalspeculation.They122\nMACHANDTHEMATERIALWORLDwelcomedMach’sattackonNewton,aswellasthatonatomism.Andtheywereimpressedwithhiscondemnationofthosewhoaretemptedtogotoofarbeyondthephenomenallevelofsenseexperience.LikeMach,theymaintainedthatwhatcannotbeobservedbythesensescannothaveanymorethaninstrumentalvalueinscience.Manyinthephilosophicalcommunitywereperhapsevenmoreenthusiastic.TheylookedforwaysinwhichtogiverigorousexpressiontoMach’sphenomenalisticphilosophyofscience.Empiricism,avigorousanti-metaphysicalapproach,andacloseacquaintancewithlanguageandlogicwereallcombinedtoproducelogicalpositivism—aphilosophyaccordingtowhichstatementsaremeaningfulonlyinsofaraswecanverifythem(atleastinprinciple)bymeansofoursenseexperience.Fromtheseideas,wemaydistilthreeessentialaspectsofpositivismprogrammes:1Weshouldregardsenseexperienceastheonlyadmissibleguarantorofourphysicaldescriptions;hencestatementsinvolvinganessentialreferencetotheoreticalorunobservableentitiesmayhaveatbestaninstrumentalstatusinouraccountsoftheworld.2Ourknowledgeabouttheworldmayonlyberegardedassecureifitmaybecheckedagainstobservationandexperiment.3Weshouldnotseekanythingmorethancompletedescriptivepowersinouraccountsofthephysicalworld;‘fundamental’explanations,particularlythoseinvolvingsupposedcausalconnectionsormetaphysicalentities,shouldhavenoplaceinscience.Apartfromthelackofapositivetheoryforinertialinteractions,tworelatedobjectionstoMach’spositivisticviewsmaybemade.First,hisconvictionthatobservationandexperimentprovidetheonlybasisforscienceissuspect.For,aspositivistshavediscovered,itishardtomakeanycleardistinctionbetweenobservationandtheoreticalstatementsorbetweenobservableandunobservableentities.Secondly,hisviewthatcounterfactualsuppositionsshouldbeconstrainedbyobservationalevidenceisproblematic.WeneedaclearaccountofwhyourphysicaltheoriesshouldberestrictedinscopeandMachdoesnotprovidethis.Thefirstdifficultyismoreseriousandhowwerespondtothiswillalsodetermineourresponsetothesecond,sincethistooreliesontheviewstakenbothofthestatusoftheoreticalstatementsandofanydistinctiontheremaybebetweenobservationandtheory.123\nTIME,SPACEANDPHILOSOPHYPositivismhassufferedmanyblows;butperhapsthefiercestistheclaimthatobservationstatementsaretheory-ladenortheory-dependent.GiventhatMachreliesonthebasicpresumptionthatobservationisepistemologicallypure,thisblowstrikeshimjustashardasthelaterpositivists.Ifwhatwesayabouttheworldisalwaysdependentuponournon-observationalbeliefs,thenitmaybehardtojustifygivingobservationalstatementsanypreferredstatus.ThomasKuhnisoneofmanywhoarguethatourobservationalclaimsdependuponourtheoriesandbeliefs.InTheStructureofScientificRevolutions,hediscusseshowscientificeducationeventuallytransformsastudent’svisionoftheworldinaprofoundandradicalwayandhowrevolutionsinscienceproducesimilar‘Gestalt’shiftsinthinking.Toillustratehisgeneralclaimaboutthewayweseetheworld,KuhnconsidersWittgenstein’sexampleofadrawingwhichmightbeseeneitherasaduckorarabbit:Whatwereducksinthescientist’sworldbeforetherevolutionarerabbitsafterwards….Transformationslikethese,thoughusuallymoregradualandalmostalwaysirreversible,arecommonconcomitantsofscientifictraining.Lookingatacontourmap,thestudentseeslinesonpaper,thecartographerapictureofaterrain.Lookingatabubble-chamberphotograph,thestudentseesconfusedandbrokenlines,thephysicistarecordoffamiliarsubnuclearevents.Onlyafteranumberofsuchtransformationsofvisiondoesthestudentbecomeaninhabitantofthescientist’sworld,seeingwhatthescientistseesandrespondingasthescientistdoes.Theworldthatthestudentthenentersis…determinedjointlybytheenvironmentandtheparticularnormal-scientifictraditionthatthestudenthasbeentrainedtopursue.(Kuhn1970:111–12)17Kuhn’smessageis:whatweseeandcertainlyourreportsaboutwhatweseearedependentuponus—uponoureducation,oursocialcontext,ourculture,ourgeneralbeliefs,ourscientifictheories.Becausewecanfindmanyexamplesofdifferentpeopleindifferentplacesandtimeswhowoulddisagree(sometimesstrongly)withevenourmostbasicobservationalreports,thereisnojustificationfortheclaimthatwhatwesayweseeiscorrect.Wecannotevensaythattwopeoplefromdifferentcontextsarelookingatthesamethingandthatanydifference124\nMACHANDTHEMATERIALWORLDofopinionabouttheobjectissimplyabouttheprecisedescriptiontobegiven.Forwemayfindthatthedisagreementisovertheveryexistenceoftheobject.18Inthefinaltwosectionsofthischapter,weshallconsiderthestatusofobservationalandtheoreticalentitiesandstatementsingreaterdetail.ThiswillallowustojudgewhetherwehaveanyfirmempiricistgroundsforrepudiatingNewtonianphysicsanditsapparentcommitmenttospaceandtimeasabsoluteentities.Inthenextchapter,weshallconsideralternativestoNewtoniantheory.ThereweshalldiscusstheextenttowhichMach’svisionhasbeenvindicatedbytwentieth-centurytheories.CANWESEESPACE?TheideasofKuhnandothershaveprovokedthreebasicresponsestotheproblemofthestatusofobservation.1Thespectrumview.Althoughwecannotmakeafirmdistinctionbetweenobservationandtheory,wemayneverthelessrankourscientificstatementsinaleaguetableorviewthemasbelongingtoaspectrum:atoneendthehighlytheoreticalclaimsaboutsuch‘entities’asquarksandblackholesandspace,andattheotherverybasicobservationalclaimsaboutsuchimmediateexperiencesascolourandlengthandmotionrelativetous.19Wemaythenarguethatwehaveeveryrighttobemoreconfidentinourassertionsattheobservationalendofthespectrum.Foritisrationaltohavestrongdoubtsaboutstatementswhichareonlyindirectlyandlooselyconnectedwithourday-to-dayexperience.Butitisfarlessplausibletoquestionthismoredirectexperience—which,throughitsdeepentrenchmentinourbeliefsystem,becomes‘contingentlyincorrigible’,i.e.althoughwemaydoubtourmostbasicstatements,itisoftendifficulttoadvanceconvincingreasonsforsuchdoubtsandthereforesuchdoubtsmaynotbealtogetherrational.20Thisapproachgivesusareasonforpreferringatheorywithoutreferencestotheoreticalentitiessuchasspaceandtime.ButwehavenojustificationforrulingoutNewtoniantheorysimplybecauseitdoesmakeessentialreferencestospaceandtime.Sothebestwecandoistolookforanalternativetheorywithoutsuchreferences.2Allstatementsaretheoretical.PaulFeyerabendsaysthateventhemostbasic‘observational’claimsaretheoretical.21Ourviewoftheworldisnotanessentiallyobservationalperspectivewhichisthen125\nTIME,SPACEANDPHILOSOPHYinfluencedbyourtheoreticalbeliefs.Rather,ourviewistheoreticalineveryway.Consequently,twopeople,fromdifferenttheoreticalbackgrounds,lookingatascientificexperimentwouldseenotoneexperimentbuttwo.Eachobserverwouldseemeterreadingsandothersuchdatafromtheirownparticulartheoreticalperspective.Eveniftheyagreedbeforeanexperimentthatthepointerofametermovinginaspecificdirectionwouldconfirmoneviewandbeanomalousintheother,theexperimentwouldnotprovidean‘objective’testbetweenthetheories.Ifeventhemostbasicdataareboundupwiththetheoreticalworld-viewofanobserver,itwouldbeamistaketothinkintermsofaspectrumrangingfromtherelativelyobservationaltotherelativelytheoretical.Ifwefollowthismoreextremelineofargument,thentheoptionopentousabove(tosearchfora‘non-theoretical’alternativetoNewtoniantheory)isnowclosed.Forthereisnothingtobegainedbyadoptingatheorywhichisless‘theoretical’incontent.3Wecanseeifwecansee.Althoughwecannotclearlydistinguishbetweenobservationalandtheoreticalstatements,wemaymaintainafirmdistinctionbetweenobservableandunobservableentities.BerkeleyseemstobeonsafegroundwhenhesaysthatNewtonianspaceandtimeareunobservable.ButwhatmakesNewton’sargumentsopersuasiveistheclaimthatspacehasobservableeffects.Wemightnotseespacedirectly,butwecan‘see’itindirectlybythewayitactsonmaterialobjects.Scientistsoftenrelyonsuchindirectapproaches:frominferringthenatureofthecoreoftheEarthorofadistantplanetfromsurfaceactivitytousingX-rayphotographytodetect‘unseen’tumours.InTheScientificImage,BasvanFraassenarguesthatthereisanaturallimittotheseindirectinferences.Hesaysthat,whenIlookatthevapourtrailofajetinthesky,eventhoughitisoutofsightthejetisobservablebecauseintherightcircumstancesIcouldobservethejet,e.g.ifIwerecloseenough.But,whenIexaminethetrailofamicro-particleinabubble-chamber,IcannotcounttheparticleasobservablebecausethereisnocircumstanceinwhichIcouldobservetheparticle.Thepowersandcapacitiesforobservationpossessedbyhumanbeingsdeterminewhatisandwhatisnotobservable.Thereisnowayinwhichspaceandtimemaybeobservedbyhumans.‘Seeing’somethingviaitseffects,whetherthisisatrailinabubble-chamberoraninertialforce,doesnotamounttoacaseofobservationunlesswecouldinprinciplegenuinelyobservethethingviathesenseswhichwehappentohave.126\nMACHANDTHEMATERIALWORLDNewtoniantheorymayinvolveanessentialreferencetospace,butthereisnonecessityforustobelieveallthatthetheorytellsus:wemayinsteadacceptthetheoryonthebasisofwhatithastosayaboutobservables,ofits‘empiricaladequacy’.Hesays:‘Scienceaimstogiveustheorieswhichareempiricallyadequate;andacceptanceofatheoryinvolvesasbeliefonlythatitisempiricallyadequate’(vanFraassen1980:12).22Science,forvanFraassen,isprimarilyaboutobservables.SoacceptanceofNewton’stheorydoesnotnecessarilycommitustoanybeliefinspaceandtimepersesincetheyarenotobservablesinvanFraassen’ssense.ThisseemsapromisingplausibleapproachforaMachiantotake—itunderwriteshis/herobservationalbias;butitdoesnotgivetheMachianareasontolookforanalternativetoNewtoniantheory.TheMachianmaysimplyacceptNewtoniantheoryandbecontentthatthisacceptancedoesnotcommithim/hertoabeliefintheexistenceofspace.Therefore,wemightbeconcernedthatanynaïveacceptanceofvanFraassen’sbasicapproachcouldbearecipefortheoreticalstagnation.WemightalsobeconcernedaboutvanFraassen’sfaithinourabilitytoprovideanobjectiveaccountofthepowersandcapacitiesofobservinghumans.Headmitsthatthisisanempiricalproblem.Buthefailstorecognisethat,likeallempiricalproblems,ourresolutionmaydependuponourwidersystemofbeliefifnotdirectlyuponourtheoreticalprejudices.Inotherwords,ouraccountofobservabilitymayitselfbetheory-dependent.EXPERIMENTANDINTERVENTIONIanHackingarguesthattheabovetalkofobservationalandtheoreticalentitiesandstatementsdoesnotgiveusasatisfactoryaccountofthenatureofscience.Hesaysthatthreeissuesmustbefaced:1Whenweobserve,weshouldnotusejustoureyes;weshouldnotbemerespectators.Thesecretsofamicroscopeslidewillonlyberevealedtothosewholearntointervene,tointerferewiththespecimenunderscrutiny.Justtopeerdownthemicroscopepassivelywilltelluslittleornothing.Welearnbyexperimentandexperimentationdemandsactiveintervention—itisnotaspectatorsport.Therefore,amorecompletepictureofthescientificenterprisedemandsanaccountoftheactiverelationshipbetweenobservationandexperiment.23127\nTIME,SPACEANDPHILOSOPHY2Wemustrecognisethefactthatobservationisnotsocloselyrelatedtoordependentupontheoryassome,includingFeyerabend,havesupposed.WhenIseeanopendoor,myobservationdoesnotdependontheoriesabouthowlighttravelsfromthedoortome.Ineedknownothingatallaboutthenatureoflighttofeelconfidenceinmyclaimthatthedoorisopen.Ofcourse,theremaybeastrangeforce-fieldwhichaffectsthelightinsuchawayastomakeacloseddoorappeartobeopen.But,sincetherearenosuchtheoriesaboutsuchfields,myobservationscanhardlybesaidtodependonthem.Iftheory-dependenceistomakeanysenseatall,thenastatementmustdependonarticulatedtheories.Isimplydonothaveatheoryaboutdeep-seacreatureswhichinfluencethepicturesonmytelevisionsetbytheirextraordinarymentalpowers.Soitwouldbefoolishtosuggestthatmyobservationofapoliticianonthescreendependsuponmydenialofsuchatheory.3‘Experimentationhasmanylivesofitsown’;formanyexperimentaltraditionssurvivequitehappilywithoutanytheoreticalbasis.Hackingcitesthehistoryofsuperconductivityasprovidinganexcellentexampleofthis:46yearspassedbeforetheexperimentaldomainofsuperconductivitygainedatheoreticalbaseinquantummechanics.24Experimentalsciencecanhaveanindependentconcreteexistencewithitsownsubjectmatterandassumptions,instruments,anddataandthevariousoperationsperformedonthedata,togetherwithadistinctsocialcontextforexperimentalscientists.Clearlytheory,observation,andexperimentdointeract—butgiventheseparate‘lives’theylead,thesethreedomainsmaybebestdescribedaspartiallyautonomous.25Hackingregardstheactiveinterventionoftheexperimentalscientistastheprimarybusinessofscience.Herecognisesthattheoryis‘crucialtoknowledge,tothegrowthofknowledge,andtoitsapplications’.26But,ifwewantthefirmestofgripsonthephysicalworld,weshouldnotthinkintermsoftheoreticalrepresentations.Toomanyproblemsfacetherealistwhoregardstheoriesascandidatesforrepresentationsoftheworld.Amajorproblem,whichwehavealreadynotedforthiskindofrealist,isthattheoriesmaybegenerallyunderdeterminedbydata;seeChapter4.Butthehistoryofsciencealsoshowsthattheoriesprovidenocompleteguaranteeofstability.Forreferencestounobservableentitieslikephlogistonsometimesdisappearafterachangeoftheoreticaldirection:and,ifonereferencecandisappear,whyshouldwehaveconfidencethat128\nMACHANDTHEMATERIALWORLDanyofourtheoreticaltermsrefertothingsintheworld?Thebestthatthisbrandofrealistseemsabletodoistoaskforcharityandtoofferaprincipleoffaithandhope:givemethebenefitofthedoubt—Iamusuallyright!27Hackingwantsrathermoresecurityfortheknowledgebaseofscience.Andhetriestopersuadeustomoveawayfromrepresentationalrealismtoarealismaboutentities.Theaddedbonuswegetfromthismoveisahighdegreeofstabilityforthescientificenterprise.Hearguesthatitistheabilitytomanipulateentitieslikeelectronsinordertoproduceindependentresultsinexperimentalcontextswhichunderwritesourbeliefintheexistenceoftheelectron.Wemightbewronginourdescriptionsofelectrons,butwecannotbewrongabouttheirexistence.ThusHackingasksustoberealistsaboutentities,butscepticsaboutthetheoreticaldescriptionsoftheseentities.Whenwecanmanipulateanentity,wegetagriponsomethingintheworldwhichwillnotbedisruptedthroughtheorychange.So,likevanFraassen,Hackingviewstheoretical‘knowledge’assecondaryinthescientificenterprise.Withhis‘positivistic’biastowardsexperimentationandhisbeliefthatlow-levelobservationisfreefromtheoreticalprejudice,HackingofferssomecomforttotheMachian.Spaceandtimearenotmanipulableandthereforetheyarenot‘observable’inHacking’ssense.Andsowehavenoreasontobelievethatspaceandtimeactuallyexist.WhenweaddtothisHacking’srecognitionoftheimportanceoftheoryforthedevelopmentofknowledge,wehaveatlastastrategywhichtheMachianmightadoptinasearchforaconsistentrelationisttheoryofinertiaandmotion.Buttodosorequiresacompromise.ThecostfortheMachianmaybethattheyhavetoadmitthatentitiesdopossessintrinsiccausaldispositionsandpowers:forsuccessfulmanipulationinavarietyofcontexts,usingunobservableandobservableentitiesalike,seemstocommitustothebeliefthattheseentitieshavedefinitecausalcharacteristics,evenifanyparticulardescriptionofthesepropertiesmayitselfbesuspect.AndHacking,unlikethepositivistsofold,seesnovirtueinrestrictinghisattentiontothelow-levelendoftheobservation-theoryspectrum.AlthoughHackingadmitstodoubtsaboutthestatusofblackholes,unobservableortheoreticalentitiesarenotbanishedfromscience.Forthehistoryofscienceshowsusthatwhatisunobservabletodaymaybemanipulabletomorrow.Couldwewithsomeadvancedtechnologymanipulatespaceandtime?Ithinkwecannotrulethatpossibilityout.Andso,despiteHacking’spositivisticcommitmenttoobservationandexperiment,heprovidesastrategywhichmightdismaytheMachianwhoisdeterminedtorepudiatespaceand129\nTIME,SPACEANDPHILOSOPHYtimecomewhatmay.ButthesamestrategyoffershopetothelessdogmaticMachian,whoseesthematerialworldasthecentralfoundationforourphysicaltheories.Ifwegraspthisworldthroughintervention,andspaceandtimeturnouttobemanipulable,thenspaceandtimewouldbeaspectsofthismaterialworld.IthinkMachhimselfwouldhaveseenaglimmerofhopeinHacking’sapproach.130\n7EINSTEINANDABSOLUTESPACETIMEINTRODUCTIONTheGeneralTheoryofRelativity(GTR)hasfrequentlybeenregardedastheultimatedefeatforabsolutism.Forsometimeafterthedevelopmentofthetheory,writersfollowingtheleadgivenbyHansReichenbachcontinuedtoclaim:thattheideasofLeibnizandMacharevindicatedbyGTR;thatAlbertEinsteinovercametheneedforirreduciblereferencestoabsolutespaceandtimeinourphysicaltheories;andthatinertialeffectsmaybeunderstoodintermsofmaterialinteractionsalone.1ButasuccessionofcontemporarywritershaveshownthatsuchclaimsarenotfullysupportedbyGTRasnormallyunderstood.2Inordertoexploretheissuesinvolved,weneedtofocusclearlyontheideaofamodelofGTR.Weneedmodelstocharacteriseatremendousrangeofgravitationalcontexts,fromplanetsorbitingtheSuntodistantrotatingblackholes.RelativiststypicallyconstructandanalysemodelsofGTR—sometimescalledsolutionsofthefieldequations—whichconsistinthefollowingelements:1afour-dimensionalgeometricalbackgroundframeworkor‘manifold’:spacetime—acontinuumofspacetimepoints;2amatter‘field’whichrepresentsthedistributionofmatterandenergyinaspacetime—movingbeyondtheNewtonianconceptionofmatteraspersisting,distinctiveparticlesandinvokingEinstein’sideathatmatterandenergyshouldbeinter-relatedinrelativistictheoriesbythefundamentalequation:E=mc2;3ametricwhichhasa‘flat’(Lorentz)characteratleastlocally;131\nTIME,SPACEANDPHILOSOPHY4asetoffieldequations,withatleastastrongsimilaritytotheEinsteinfieldequationsof1915–16,whichrelatethemetricandthematterfield,andwhichincorporatestheideaofanaffineconnectionthatdefinestheideaofforce-free‘straight-line’motioninthespacetime.3ThekeytoGTRisoftensaidtobeEinstein’suseofthelocallyflatLorentzmetric,whichallowsustodecipherthecomplexitiesofgravitationandmotionbyreferencetolocalinertialframesratherthantoglobalframes.Thelocalinertialframeprovidesuswithastandardforinertial,force-free‘straight-line’motion.Inthelocallimit,i.e.aninfinitesimallysmallregion,wemayneglectmostgravitionaleffects.Gravitationtendstobeassociatedwiththemore‘global’curvatureoflargerscaleregionsinspacetime.4ThemathematicalapparatusofGTRlinkslocalandglobalgeometryandtherebyguaranteesthatthegeodesicofanobjectincurvedspacetimecorrespondstothelocallydefined‘straight-line’motionoftheobject.5Theideaofstraight-linemotionisdefinedintermsoftheaffinestructureofspacetime,andinthelocallimittheaffine‘connection’iszero,signifyingtheabsenceofforces.TheaffineconnectionisageometricalquantityemployedinGTR’sequationsofmotiontodescribethemotionofparticlesinspacetime.SeeChapter3,Figure11(p.54),forabriefreminderofthevariouslevelsofstructure,includingaffine,involvedinspacetimegeometries.Newtonassertedthatmotioninourlocalenvironmentmaybereferredtotheglobalframeworkofspace;Macharguedthattheonlyadmissibleglobalframeworkisthatofthefixedstars.AlthoughEinsteinrefersmotiontoalocalinertialframeworkinGTR,theintimateandessentialrelationshipbetweenlocalandglobalgeometryallowsustoasktwoquestions:specifically,whethertheaffinestructureofspacetimeandthereforeourideaofinertialmotionisdeterminedunambiguouslybythedistributionofmaterialinspacetime;and,moregenerally,whetherthedescriptionsofglobalgeometrymaybereducedwithoutremaindertomaterialterms.DespitethehopesofrelationistslikeReichenbach,GTRmaybeusedtoconstructmodelswhichdoseemtoinvolveanirreduciblecommitmenttospacetimeasanindependententity.Thereareanumberofimportantmodelsinvolvingsuchacommitment:emptyandalmostemptymodels,includingMinkowskispacetimeandtheSchwarzschildsolutionofGTR’sfieldequations;andmodelsinwhichthematerialcontentsexhibitthesamekindofoverallrotationwhichNewtondescribesinhis‘rotatingbucket’thoughtexperiment.132\nEINSTEINANDABSOLUTESPACETIMEHowever,othermodelsofGTR,includingthestandardFriedmanncosmologicalsolutionsofthefieldequations,donotseemtorequireanytotalabsolutistcommitment,atleastwhenwefocusontheproblemofinertialmotion.Andweshallseethatthechampionsofrelationisminthetwentiethcenturyhavetriedtofocusourattentiononthese,oftenclaimingsomepre-eminentempiricalstatusforthem.‘Non-absolutist’modelsaresaidtoinclude:infiniteworldswithaninfiniteamountofmaterial;spatiallyclosedworlds;andmodelswithadditionalfields.SometimessuchmodelsaregeneratedentirelywithinGTRascharacterisedinstandardtextbooks,sometimeswithinvariationsofGTR.But,despitetheingenuityofthosewhotrytorealisetherelationistdream,itisfarfromclearthatallsuchmodelsarefreefromanessentialcommitmenttospacetimeasafundamentalandirreducibleentityinourexplanationofthesourceofinertia.And,evenifwecanshowthatsomemodelswithasoundempiricalpedigreedosatisfyrelationistdemands,westillneedtobeshownwhyweshouldabandonallother‘absolutist’models,especiallygiventhatmanyofthesehavegoodempiricalcredentialsaswell.MACH’SPRINCIPLETheessentialideasbehindMach’sPrincipleasusedwithinthetheoreticalcontextofGTRderivefromErnstMach’sattacksonabsolutespaceandtime.WesawinthepreviouschapterthatMachbelievedstronglythatscienceshoulddealonlyinobservablephenomenaandthatouraccountsofthephysicalworldshouldbeaseconomicalaspossible.Machthereforesawnoreasontoembraceanentitywhichcouldnotbeseenandwhich,inhisopinion,isanunnecessaryandextravagantelementinourgeneralaccountofinertiaandmotion.Einsteinwasthefirsttoreferexplicitlyto‘Mach’sPrinciple’inapaperof1918;hesaidthathechosethename‘becausetheprincipleimpliesageneralisationofMach’srequirementaccordingtowhichinertiashouldbereducedtotheinteractionofbodies’.6Inapaperwrittenin1917,Einsteinfoundastatic,spatiallyclosedsolutiontohisfieldequations,whichhebelieved,forashorttime,mightbethoroughlyMachian.ButworkbytheDutchphysicistdeSitterdemonstratedthatEinstein’sideaswerenotfullyconsistentwithMach’sPrinciple.WeshallreviewEinstein’ssolutionandhisdebttoMachinChapter10.Manyotherphysicists,usuallypursuingMachianprogrammes,havegiventheirownversionsoftheprinciple.Wheelergivesatypicalstatementoftheprincipleasfollows:‘Thegeometryofspacetimeand133\nTIME,SPACEANDPHILOSOPHYthereforetheinertialpropertiesofeveryinfinitesimaltestparticlearedeterminedbythedistributionofenergyandenergy-flowthroughoutallspace’(Wheeler1964:305).7Sincewearedealingwiththeinertialpropertiesofobjectsinmotion,andthesearedeterminedbytheaffinestructureofspacetime,itseemsreasonablethatweshouldaccountforthisfactinanystatementoftheprinciple.SowemightrestateMach’sPrincipleasfollows:MPTheaffinestructureofspacetimeisuniquelydeterminedbythedistributionofmatter/energyintheuniverse.8Thishastwoimmediateimplications:first,inanemptyspacetime,thereisnomeaningtotheideaofinertia—since,withoutmaterialcontents,therecanbenoaffinestructure,andinertialpropertiesonlymakesenseintermsofaffinestructure;secondly,theonlymeaningfulideaofmotionismotionrelativetoothermaterialobjects—hencetheideaofanoverallrotationofthematerialcontentsoftheuniversejustdoesnotmakesense.Hence,ifGTRgivesanaturalandstraightforwardmeaningtoeitherofthesepossibilities,wewouldhavelittlereasontosaythatGTRincorporatesMP.MuchoftheconsiderabledebateaboutMPturnsonjustthesetwopossiblesituations.WeneedtoexaminethemodelsofGTRinvolvingtheabovefeaturesbeforewemayarriveatanyconsideredconclusionaboutthestatusoftheprincipleinGTR.ABSOLUTELY,PROFESSOREINSTEIN?ThesourceofmuchoftheconfusionaboutwhetherornotGTRvindicatesrelationismliesinthefactthatthereareseveralwaysinwhichtheword‘absolute’isemployedwithinthecontextofspacetimetheories.Insomesenses,GTR’sspacetimeisclearlynotabsolute:forexample,inbothGTRandtheSpecialTheoryofRelativity(STR)thetimemeasuredbetweentwogiveneventsdependsuponthespacetimepathtakenbetweenthoseevents—aswesawinthecaseofthe‘paradox’ofthetwinsinChapter2.And,breakingfreefromNewtonianbeliefs,inGTRasinSTR,wemaynolongeridentifyauniquetime-slicewhichrepresentstheentireuniverseataninstanttowhichalleventsnearandfarmaybereferred.ButinothersensesGTRdoesseemtobeabsolute:forexample,theinfinitesimalspacetimeintervalbetweentwoneighbouringpointsisinvariant.Differentobserversmightmeasuredifferentspatialdistancesanddifferenttimesbetweenthetwopoints,buttheywillalwaysagreeuponthespacetimedistance,i.e.upontheinfinitesimalspacetimeinterval.134\nEINSTEINANDABSOLUTESPACETIMESo,beforewebegintoexploretheconflictbetweenabsolutismandrelationismwithinthecontextofrelativity,weneedamuchsharperaccountofthevariouswaysinwhichaspacetimetheorymaybeabsolute.WehaveseenthatNewton’saccountofspaceandtimeinvolvesthefollowingclaims:1spaceisathree-dimensionalarenainwhichobjectsarelocatedandeventstakeplace;noobjectandnoeventhasanyeffectonspaceitself—hence,itisdynamicallyindependentofallthedynamiceventstakingplacewithinit;2timetooisindependentofalltheeventstakingplaceintime—anditprovidesanindependentandglobaltemporalframeworktowhichalleventsmaybereferredinthesameway;3becauseoftheindependentnaturesofspaceandtime,wemayalwaysspecifythedistancesandtimesbetweeneventsinanunambiguousway;4hence,wemaygiveanunambiguoussensetotheideaofsimultaneitysothattwodistanteventswillberegardedassimultaneousregardlessofthestateofmotionofthepersonwhomakesthejudgementofsimultaneity;5whenanobjectexperiencesinertialforces,wemaysaythattheobjectisindeedinmotionrelativetospaceitself,sothataccelerationisaninvariantquantitywhichcannotbetransformedawaybyachangeofreferenceframe.Newton’spositionallowsustodistinguishthreedistinctwaysinwhichspaceandtimemaybesaidtobe‘absolute’:1spaceandtimemayexhibitan‘absolute’independenceofobjectsandevents;2spaceandtimemaybe‘substances’ofsomesortwithdistinct‘absolute’invariantproperties;3spaceandtimemayberequiredasirreducibleandessential‘absolute’elementsinourgeneralaccountofmotion.Newton’sideasofspaceandtimeclearlyinvolveallthreesensesof‘absolute’here:spaceandtimeareindependent—notonlyofmaterialobjects,butalsoofeachother;theyareentitieswithwell-definedinvariantproperties;andaccelerationinNewtonianspacemayonlybeproperlycharacterisedwithrespecttospaceitself.SoNewton’sbeliefsaboutspace135\nTIME,SPACEANDPHILOSOPHYandtimeallowustoidentifythesethreeimportantsensesoftheterm‘absolute’:9AE‘absolute’asindependententitiesorsubstances;AP‘absolute’assubstancespossessinginvariantproperties;AM‘absolute’asirreducibleelementsinourgeneralaccountofmotionandobjectsinspace.WeshouldnotthinkthateveryconceptofNewtoniantheoryis‘absolute’.Forexample,itiscertainlynottruethatallconceptsinvolvedintheNewtonianaccountofspace,time,andmotionareinvariant.Simultaneityisaframe-independent,i.e.invariant,notionbecause,inNewtonianphysics,twoobjectswillbejudgedassimultaneouswhatevertheframeofreferenceinwhichwemakeajudgementofsimultaneity.Buttheconceptofvelocityisframe-dependent.Ifanobjectisfreefromtheinertialeffectsassociatedwithacceleration,wearefreetochooseanyinertialframeasthereferencepointforourcalculationofvelocity;andsotheuniformvelocityweascribetoanobjectdependsupontheframetowhichwereferthemotion—allwemaysayisthattheobjectismovinguniformlyrelativelytothisorthatframeofreference.Hence,velocityinNewtonianphysicsisa‘relative’ratherthan‘absolute’(AP)concept.ThisfollowsfromthekeyideaofGalileanrelativityofmotionwhichliesattheheartofNewtonianphysics:thelawsofmechanicsapplyinallinertiallymovingsystemssothateverymechanicalexperimentineverysuch‘force-free’systemhaspreciselythesameresults.Sowecannotfindexperimentalevidencewhichallowsustopointtosomeuniqueprivilegedinertialsystemandsay:thatsystemisreallymovingatsomewell-definedvelocityorisreallyatrestwithrespecttospaceitself;andsowecannotsingleoutaprivilegedinertialframeofreferencewhichisatrestwithrespecttospaceitself.Ifwearetocharacteriseabsolutisminarelativisticcontextsuccessfully,weneedtotakeaccountofthegeometricallanguageofSTRandGTR.Inthesetheories,wemovefromtalkofspaceandtimetotheideaofspacetime—andsoouraccountofabsolutismmusttakethisintoaccount.ThismovefromspaceandtimetospacetimeneednotdisruptourviewofNewtoniantheory,foritisastraightforward(iftechnical)mattertoexpressNewtonianphysicsasaspacetimetheoryusingthegeometricallanguageofmodernspacetimetheories.Ifanything,suchamovehelpsustoappreciatethesimilaritiesaswellasdifferencesbetweenrelativisticandclassicalperspectivessincebotharecouchedintermsofthesamegeometricallanguage.10OneofthedistinguishingmarksofNewtonian136\nEINSTEINANDABSOLUTESPACETIMEtheoryis,ofcourse,thefactthatwemayconstructuniqueglobaltime-slicesthrougheachpointtowhichwemayreferalleventsinthespacetime.SowemaycharacterisetheNewtonian‘separateness’oftimefromspacebypointingtotheappropriatefeaturesofspacetimegeometrywhichallowustoconstructsuchtime-slices.Consequently,thereismuchtobegainedbysettingdiscussionsofabsolutisminthecontextofspacetimetheoriesingeneral,sincethediscussionwillthenapplyequallywelltorelativisticandnon-relativistictheories.However,weshouldbecautiousaboutusingspacetimeratherthanspaceandtimewhenweconsiderhistoricalcasestudies.Forexample,ifwewishtoprovideanaccurateinterpretationofNewton,weshouldnottranslatehisideaofone-dimensionaltime‘flow’intoafour-dimensional‘static’spacetimecontext.Newtoniantimeassuchisuniform,fixed,andcertainlyflowinginsomesense;butitisnotpartofaspacetimeframework.IfwecouldbringNewtonbacktolife,hemightwellbepersuadedthattheScholiumandthePrincipiaareinadequatewhenfacedwiththeelegantandpowerfulmathematicalformalismusedbytwentieth-centuryrelativists,butthehistoricalNewtonknewnothingofspacetimeassuch.WecanrewriteNewtoniantheoryinamodern‘spacetime’way;butwecannotrewritehistory.11Clearly,themostimportantsenseof‘absolute’forsomeonewhowishestoassesstheroleofspacetimeinouraccountofmotionisthethird(AM).Andthissensemustalsoprovidethefocusforanyrelationistattackonabsolutismaboutspacetime.For,ifalltalkofspacetimecouldbereducedtomaterialterms,thenwemightsaythatallapparentpropertiesofspacetimeareinfactreducibletopropertiesofmatterandthatspacetimecannotbegenuinelyindependentinanywaysinceitisnomorethanitscontents.Therefore,theapparentindependenceofaspacetimeandwhetherornotapropertyisinvariantmayultimatelybetracedbacktomaterialfactors.Consequently,therelationistisnotimmediatelythreatenedbythefactsthatsomepropertiesofspacetimeareinvariantandthatspacetimeisatfirstsightanindependententity.Wemightthereforecharacterisethestrongestrelationistpositionasonewhichopposesthekey,thirdsenseof‘absolute’.Arelationist,perhapsacceptingMachiandemandsforanaccountoftheworldinpurelyobservationalandmaterialterms,mightthereforesay:RMallspatio-temporalelementsinvolvedinouraccountofmotionarewhollyreducibletomaterialterms.137\nTIME,SPACEANDPHILOSOPHYAnaturalconsequenceofRMisthemorerestrictedideathatthosespatio-temporalelementsinvolvedindescriptionsoftheinertialpropertiesofmatterareuniquelydeterminedbythematerialdistribution,i.e.RMimpliesthatMach’sPrinciple(MP)holds.Hence,ifwecanshowthatMPdoesnotholdinGTR,thenitisclearthatRMcannotbetrueofGTR.However,weshouldnotethatevenifwecandemonstratethatGTRinsomesenseincorporatesMP,wemaystillfallshortofshowingRMtobetrue,fortheremaybemoretospacetimethanthoseelementsinvolvedindescriptionsofinertia,e.g.topologicalelements.12Hence,ifaMachianwishestogiveawhollymaterialaccountofallthepropertiesofspacetimethens/hewouldneedtoshowthatRMholds.WeshouldnotethatSTR’sMinkowskispacetimeistypicallyemptyofallmaterialotherthan‘massless’testparticles.Butaparticlemovinginertiallythroughthisspacetimehasits‘straight-line’motiondeterminedbythespacetime:itwilltravelalongageodesicwherethespacetimeintervalsalongthegeodesicaredeterminedbytheaffinegeometryofthespacetime.ThespacetimeintervalisaninvariantpropertyofSTR’sspacetime,sincewheretheparticlemovesnextinspacetimeisafactwhichmaybeagreedfromallpointsofview,fromallframesofreferenceinthespacetime.Thereisnoothermatterinthespacetime;soanirreduciblereferencetothegeometricalpropertiesofthespacetimeisrequiredinacompletedescriptionoftheparticle’smotion.Therefore,inSTR,wecantalkaboutthemotionofatestparticlewithoutreferencetoanyothermaterialinspacetimewithperfectsense.Hence,RMcannotbetrueofSTR.WetreatSTR’sspacetimeasasubstancewithspecificgeometricalproperties.AlthoughitspropertiesareclearlydifferentfromthoseofNewtonianspaceandtime,STRstillincorporatesgeometricalstructureswhichunderwritesuchabsolute(AP)conceptsasaccelerationandthespacetimeinterval.AndsinceSTR’sspace-timestructuresareunaffectedbyanymaterialinthespacetime,wemayalsomaintainthatAEistrueofSTR.GTRcertainlyoffersmorehopetotherelationist.ForEinstein’sfieldequationsof1915–16rejecttheideaofaspacetimewhichisindependentofitsmaterialcontents.Matter,understoodintermsofa138\nEINSTEINANDABSOLUTESPACETIMEmass-energyfield,andthegeometryofspacetimearerelatedinadirectway:thedistributionofmatterinfluencestheoverallaffineandmetricalstructuresofspacetime;and,conversely,thegeometryofspacetimedeterminesthepathstakenbyinertiallymovingparticles.WhenwespeakofmatterproducingcurvatureinGTRspacetimes,weusuallymeancurvatureofthemetric;and,ofcourse,measurementsincurvedspacetimetypicallydifferfromthoseinflatspacetime.Butacurvedmetricalsoindicatesthattheaffinestructurehasbeenaffectedbythepresenceofmatter.Theeffectontheaffinestructureisparticularlyimportantwhenweconsidertheproblemofmotion,giventhattheaffinestructureofspacetimeallowsustocharacterisethe‘straightestpossiblepaths’whichforce-freeparticlesfollowinspacetime.Hence,inGTR,spacetimeisnotacompletelyindependententity,i.e.itisnotAE,atleastwhenwethinkonlyofaffineandmetricalstructures.However,wemustrememberthatother‘deeper’structuresareinvolvedinafulldescriptionofanygivenspacetime:forexample,weneedtotakeaccountoftopologicalproperties.Therefore,thefactthatGTRinvolvesanexplicitrelationshipbetweenaffineandmetricalstructuresandmatterhasnoimmediateimplicationsconcerningthetopologicalindependenceofspacetimeinGTR.Therelationshipbetweenmatterandgeometrygivessomehopetotherelationistwhowishestoreducespacetimetomaterialterms.Butthisdynamicrelationshipprovidesmerelyapossiblemechanismforthereduction.Wemayneverthelessrequirespacetimeconceptsinourgeneralaccountofmotion.UnlesswecandemonstratethatasoundtheoryofspacetimeandmotionsuchasGTRmaybespeltoutfullyinmaterialterms,relationismseemsdestinedtobenomorethanapiousempiricalhope.EMPTY,ALMOSTEMPTY,ANDROTATINGWORLDSItmightseemempiricallyunsoundtoconsideremptymodelsofspacetimetheoriesashavinganysensibleimplicationsatallforourviewofthe‘real’world.Wemightjustdismissthemas‘unrealistic’:ifourworldisanything,itdoesnotseemtobeempty!139\nTIME,SPACEANDPHILOSOPHYHowever,therelationistcauseisnotparticularlyhelpedbytakingupsomestrongMachianlineandsayingthatweshouldtakeseriouslyonlythosemodelswhichcorrespondcloselytotheactuallyobserveduniverse.Fortheactualobservationsseemtobeuncertainenoughtogivecosmologistsatremendousamountofleeway.Wecaneasilyimagineauniverse,resemblingourowninmanyways,whicheffectivelyamountstoanemptyworld.Ifthereisafiniteamountofmatterintheuniversecongregatedatthe‘centre’ofsomevastinfinitearena,thenthisworld,consideredfromatrulyglobalperspective,approximatestoasysteminwhichasinglemassislocatedinanotherwiseemptyinfinitespacetime.Althoughpresentcosmologicalevidencesuggeststhattheuniversedoesnothavethisparticularstructure,theuncertaintyofthis‘locally’gatheredevidenceshouldbesufficienttopreventanyhastydismissaloftheideathatthiseffectivelyemptyworldisapossiblescenariofortheactualuniverse.And,ifweconsiderthesinglemassinanotherwiseemptyuniverseasaparticlewithoutgravitatingpropertiesandwefocusonnon-gravitationalaspectsofmotion,theneventheemptyMinkowskispacetimeofSTRmaybeconsideredasaplausibleempiricalmodelfortheuniverse.Infact,somerelativistsviewthe‘flat’spacetimeofSTRasa‘natural’geometricalbackgroundformaterial.13Theintroductionofafiniteamountofgravitatingmaterialintothisspacetimeproducessignificantcurvaturelocally(andthereforetakesusintoGTR’stheoreticalcontext),butfarawayfromthematerialthecurvatureeffectsarenegligibleand,atinfinity,arezero.Globallysuchaspacetimeiseffectively‘flat’.WehavealreadynotedthatanyaccountofmotioninMinkowskispacetimeinvolvesanirreduciblereferencetothegeometricalpropertiesofspacetime.InSTR,theaffinestructureofspacetimedetermineshowanyfreeparticlemaymove;butwemaynotexplaintheaffinestructureinmaterialterms.ButMinkowskispacetimeisjustoneofthemanymodelsofGTR,a‘special’caseina‘general’theoreticalcontext.Hence,atleastonemodelofGTRseemstobeabsoluteineverysensethatSTRisabsolute.14But,strictlyspeaking,Minkowskispacetime,asamodelofGTR,shouldnotbeconsideredasanentitywhichisindependentofitsmaterialcontents.GTR’sfieldequationsareusedtoconstructthemodel,andsothegeometricaltermsintheequationdescribingtheaffineandmetricalpropertiesofitsspacetimearedeterminedinpartbythetermsdescribingthedistributionofmatter.Thefundamentalrelationshipbetweenmatterandgeometryexpressed140\nEINSTEINANDABSOLUTESPACETIMEbythefieldequationscannotbeignoredjustbecausethetermsdescribingthematterdistributionindicateanemptyspacetime.Hence,allmodelsofGTRarerelationalinthesensethatspacetimeisnotanabsoluteindependententity,atleastintermsofitsaffineandmetricalproperties,i.e.theydonotimplyAEattheselevelsofstructure.Theymay,ofcourse,stillbeabsoluteentitiesorsubstancesatatopologicallevel.TheSchwarzschildsolutionofthefieldequationssuppliesuswithastandardmodelofGTRwhichmaybeusedtoapproximateavarietyofphysicalsituations,includingthesolarsystem.Asinglesphericalgravitatingmassislocatedinaspacetimewithsphericallysymmetricspatialproperties.Testparticlesinthisspacetimemaybeusedtorepresentthebehaviourofobjectsmovingclosebythecentralmass,whichmayitselfrepresenttheSun.Thecentralmassproducesmarkedlocalcurvatureinthemetricofspacetime,andsoanyfreelymovingtestparticlewillfollowgeodesiesinthiscurvedspacetime.AndsowemayusethemodeltopredictthebehaviourofobjectsinthevicinityoftheSun.However,atconsiderabledistancesfromthecentralmass,thegravitationalinfluenceontestparticlesisnegligible.Howtheymovenolongerdependsuponthegravitatingpropertiesofthecentralmass,for,atgreatdistancesfromthismass,spacetimebehavesasifitwereempty.Thespacetimepathsofsuchtestparticlesaredeterminedbytheoverallspacetimegeometry.Thisisthecaseinallsuch‘almostempty’spacetimes—whereafiniteamountofmatterislocatedinanotherwiseemptyinfinitearena.However,theoverallgeometryofaspacetimemodelisnot,ingeneral,fixedeventhoughwespecifythematerialdistribution.Inordertodeterminetheaffinestructure,wemustalsospecifytheboundaryconditionswhicharetoobtaininthemodel.Forexample,wemightsaythatthespacetimeinwhichmaterialislocatedisspatiallyunbounded—infiniteinthesensethatobjectsmaybelocatedatinfinitedistancesfromeachother;orwemightsaythatspacetimeisspatiallyboundedsothatspacemaybethoughtofasathree-dimensionalanalogueofthesurfaceofasphere.Ifweallowinfinitelyextendedspacetimeswithafiniteamountofmaterialinthem,asintheSchwarzschildsolution,theningeneralthepathsofobjectsareeffectivelydetermined,notbythepresenceofmatterinsomelocalenvironment,butbytheinfinitelyextendedstructureofthealmostemptyspacetime.Weimposeanoverallgeometricalstructureonthespacetimewhen141\nTIME,SPACEANDPHILOSOPHYwespecifytheboundaryconditions,andthisstructureisthenessentiallyandirreduciblyemployedwithdefinitemetricalandaffinepropertiesinouraccountofmotioninthespacetime.Therefore,almostemptyandspatiallyunboundedspacetimesare,likeemptyspacetimes,absoluteinthesensemostthreateningtotherelationistprogramme:theyareirreducibleelementsinouraccountofmotion(AM).Thesecondmajorproblemforrelationistsconcernsthepossibilityofglobalrotationsinspacetime.Iftheentiremattercontentoftheuniverseisrotating,thentherotationcouldnotbewithrespecttosomenon-rotatingmaterialbody,foreverythingisrotating.Therotationcouldonlybewithrespecttosomeelementofspacetimeitself,andsowecouldonlymakesenseofthemotionofmaterialwithanirreduciblereferencetosomeelementofspacetimeitself.DosuchglobalrotationsmakesenseinGTR?ThereareseveralmodelsofGTRwhichseemtoallowthepossibilityofanoverallrotation,including:1Gödel’ssolution,inwhichmatterisrepresentedasaperfectfluidthroughoutthespacetimeandinwhichspaceisrotationallysymmetricaboutanypointinthespacetime;2Kerr’ssolution,describingthepropertiesofanisolated,rotatingmass;and3theOszvàthandSchückingsolution,whichcharacterisesagloballyrotating,homogeneous,andspatiallycloseddust-filledspacetime.15Therelationistmightfinditeasytobesuspiciousofthefirstsolutionabove;butitismoredifficulttochallengethestatusoftheothertwo.Gödel’ssolutioninvolvesthepossibilityofclosedloopsintime,which,asweshallseeinChapter8,maywellcommitustophysicallyimplausiblesituations,ifnotoutrightlogicalcontradictions.SowemightbejustifiedinadoptingaseverelyscepticalattitudetowardsGödel’ssolution.But,ifwearetodescribethepropertiesofanisolatedrotatingobject,thenweclearlyneedtousesomemodelsuchasKerr’ssolution.And,apartfromtherotationwhichitallows,theOsvàthandSchückingsolutionshouldnotbeimmediatelyobjectionabletotherelationistsinceitsspatiallyclosedboundaryconditionsavoidthekindofprobleminvolvedinalmostempty,spatiallyunboundedsolutionswherespacetimeconditionsat‘infinity’determinethewayobjectsmoveinspacetime.142\nEINSTEINANDABSOLUTESPACETIMERELATIONISMANDRELATIVITY:ANEMPIRICALVIEW?DespitethenumberandvarietyofmodelsofGTRwhichseemtoinvolveanirreduciblereferencetospacetimeitself,thereareseveralmodelswhichseemtobegenerallyinlinewiththerelationistprogramme.ThemostcelebratedofthesearetheFriedmanncosmologicalmodels,initiallydevelopedbytheRussianmathematicianAlexanderFriedmannintheearly1920s.16FriedmannusestheEinsteinfieldequationstoconstructasetofequationswhichcharacteriseavarietyofhomogeneousandisotropicdust-filleduniverses.Thesemodelshaveahighempiricalstatus,giventhattheyallowthepossibilitiesofglobalexpansionandcontractionofthematerialcontentsoftheuniverseinbothspatiallyopenandclosedcontexts.Theydonotexhibitanyoverallrotationandtheiraffineandmetricstructuresareuniquelydeterminedbythedistributionofmaterialinthem.Hence,theyseemtosatisfyMP.17ButtheydonotsatisfyRMbecausethetopologicalpropertiesofthespacetimesarenotuniquelydeterminedbythedistributionofmaterial.AswenotedinChapter3,theconditionsofhomogeneityandisotropyfor,say,anexpandingmatter-filleduniversedonotforceustoadoptaspatiallyunboundedtopology:wemaydecidetoadoptthetopologyofatorusinstead,asisdoneinthecaseofthe‘smalluniverse’.TheempiricalpedigreeandtheMachiancharacteroftheFriedmannmodelshaveencouragedsomerelativiststousethemasafocusfortheirrelationism.FollowingworkdonebyDennisSciamaandothers,DerekRaineshowsthatwemayconstructaMachianvariantofGTRwhichidentifiesalimitednumberofsolutionsofGTRinwhichtwoconditionsaremet:first,ifwespecifycurvature,thenauniquemetricshouldbedetermined;and,secondly,theWeylcomponentofspacetimecurvatureshouldbedetermineduniquelybythematterdistribution.18Togethertheconditionsensurethatthematterdistributiondeterminescurvature,whichinturndeterminesthemetricalandaffinestructure.TheequationswhichheemploysdifferfromGTRinasmuchastheyallowustosumtheeffectsofdistantmatteronanygivenmaterialobject:onlylineartheoriesallowthisintegralapproachandGTRisnon-linear.ButRaine’smodifiedequationsdohavethe‘usual’fieldequationsasalimitingvalue.And,givenourremarksabove,itisperhapsnosurprisethatthemaingroupofsolutions,consistentwithRaine’sMachianconstraints,isformedbytheFriedmannmodels.Empty,almostempty,androtatingsolutionsareallruledoutbythe143\nTIME,SPACEANDPHILOSOPHYconditions.AlthoughRaine’sapproachiscertainlyinventive,hisstrategyhasmuchincommonwiththeapproachofotherrelationistswhostickwiththeusualfieldequations.Mach’sPrincipleisusedasacriterionfortheselectionof‘permissible’modelsofGTR.TheadvantageofRaine’sintegralformulationofGTRisthatitgivesusclear,ifcomplex,mathematicalguidelinesonhowthecriterionistobeapplied.Butwestillmightdecidethattheoverallstrategyhereissomewhatadhoc:wewillonlyadmitthosesolutionsinwhichMPholds;and,ifGTRitselfdoesnotprovidesufficientreasonforustothrowoutundesirablemodels,thenweshalladoptamoreobligingvariantofthetheory.19TheMachianrelationisttriestoforceourattentiontowardsalimitednumberofglobalcosmologicalsolutions,perhapsallowingothermodelsamerelyinstrumentalstatusinourgeneralaccountofinertiaandmotion.WemayusemodelsinwhichMPdoesnothold,butweshouldnotexpectsuchmodelstoapplytotherealworld.TheMachianseesnoprobleminthis,fortheapproachisjustifiedbyastrongbeliefintheprimacyofobservation.Wedonotseeanemptyuniverse,soMinkowskispacetimecarriesnoimplicationsaboutthestructureoftherealworld.Wedonotseeanisolatedmassinanotherwiseemptyuniverse,sotheSchwarszchildsolutionhaslimitedvaluealso.Wedonotobserveanyglobalrotation,sotheOszvàthandSchückingmodeldoesnotapplytotheactualuniverse.20Wedoseeahomogeneousandisotropicmatter-filleduniverse,soanymodelwhichfitsthesefactsispotentiallyacceptable.Hence,therelationistnarrowstheavailableoptionsconsistentwithGTR’sfieldequationsdowntothosesolutionswhichcorrespondwiththeobservedconditionsinthephysicalworld.Buttherangeofempty,almostempty,androtatingsolutionsshouldcertainlynotberegardedas‘second-class’models:withoutthemwecannotmakesenseofsuchlocalirregularitiesasthesolarsystem!TheincorporationofMPintothegravitationalcontextofGTRseemstodemandarathercavalierattitudetowardsthegeneralityofthefieldequations.Weareaskedtorejectarichvarietyofsolutionsonthegroundsthattheydonotcorrespondwithactualglobalobservations.Thefactthatcertainaspectsofamodelmightapplytoalocalsituationdoesnotgiveussufficientreasonforacceptingthemodelasempiricallyvalid.Ifmattereverywheredetermineswhatishappeninginanygivenlocality,thentheonlysolutionswhichcarryanysignificantweightwillbethosewhichcharacterise,atleastapproximately,theactualglobalmatterdistribution.Consequently,theMachianrelationistseemsto144\nEINSTEINANDABSOLUTESPACETIMErequirethatthefieldequationsshouldhavealimitedrangeofapplication,definedbytheavailableempiricalevidence.21Butmanyscientistsregardthemajorstrengthofphysicallawsastheirgenerality,theirapplicabilitytoallpossiblephysicalsituations.ThiskindofgeneralityunderwritesNewton’sargumentforabsolutespaceinwhichhefeelsfreetoapplyhislawstoanalmostemptyuniverse.AnditisthisunrestrictedgeneralitywhichMachresistswhenheasksustofocusontheactualuniverseandnotsomeimaginaryartefact.AlthoughMachcertainlyallowsthatlawsshouldapplyinareasonablybroadrangeofcircumstances,hewouldnodoubtagreewithHermannBondi’sassessmentofgeneralrelativityandtherangeofapplicabilityofitslaws:Itakeanextremelyempiricalviewofgeneralrelativity.Itisatheorylikeallotherphysicaltheories,foundedonexperimentandobservation.Weexpectadecenttheorytogoalittlefurtherthantheexperimentsandobservationsonwhichitisbased,toaccountforsomethingalittlemoregeneralthantheparticularcircumstancesinwhichthetheorybecameestablished.Butweneverexpectourtheoriestoholdincircumstancesutterlyandcompletelydifferent.(Bondi1967:75–6)22ThemajorproblemforempiricistslikeBondiishowtodrawthelinebetweentheempiricallyacceptableandtheunacceptable.Relativistsareindeedonlytoowillingtoruleoutsomelogicallypossiblesituationsashighlyimplausible.Forexample,HawkingandEllisspecificallyrestricttheadmissiblesolutionsofEinstein’sfieldequationstothose‘exactsolutions’whichsatisfytwoempiricallybasedconditions:first,allcausalsignalsinadmissiblesolutionsmusttravelalongnon-spacelikepaths;and,secondly,localkineticenergyshouldneverbenegative.23Wehaveeveryreasontosupposethattheseconditionsholdforallknownformsofmatter.24ButtheseconditionsstillgiverelativiststremendousflexibilityintheirconstructionofmodelsofGTR.Therelationistmightnowsaythatwehaveeveryrighttoimposefurther,tighterrestrictionsinlinewithobservationalevidence.Buttheevidenceitselfsuggeststhatthisrelationistploycannotdothejobtheyaskofit.WhenMachaskedustorecognisethatwecannotneglectthebackgroundmaterialreferenceframeofthefixedstarsinouraccountofmotion,everyonethoughtthatthestarsreallywerecosmicfixtures,thattheuniverseisgloballystatic.Buttheevidenceofastronomers,145\nTIME,SPACEANDPHILOSOPHYfromEdwinHubble’sdiscoveryofredshiftin1928onwards,stronglysuggeststhatnothingisfixedinadynamicuniversewhichisevolvingfromsomethinglikea‘bigbang’.Thereisstillconsiderabledebateabouttheconditionsintheveryearlyuniverse,andthereissomedebateaboutthelikely‘final’stateoftheuniverse.Theempiricalevidenceissimplyinsufficienttoruleoutallofthe‘absolutist’modelsofGTRasphysicallyimplausible.Forexample,wehavenoreasontoruleoutaglobalrotationintheinitialturbulentconditionsofthe‘bigbang’;astheuniverseevolves,therotationslows.Thisisatleastasphysicallyplausibleasmanyoftheweirdandwonderfultaleswhicharetoldabouttheearliestoftimes.25Hence,theobservationalevidenceforanevolvinguniverserequiresthatwetakeaflexibleratherthanrestrictiveattitudetowardsthephysicallyadmissiblemodelsofGTR.Weshouldnotdismissthepossibilitythat,atsometimeinthepast,conditionswere,asBondisays,‘utterlyandcompletelydifferent’—oratleastdifferentenoughtomaketherelationistFriedmannmodelsredundant.Ofcourse,itmightturnoutthattheearlyuniverseturnsouttobeaMachiankingdom.But,withouthardevidence,wehavelittlereasontouseMPasacriterionforselectingtheonlyadmissiblesolutionsofGTR.THEHOLEARGUMENTANDSPACETIMEPOINTSSofartheinitiativeseemstobewithabsolutists.However,oneofthedevicestheyrelyonmaybeturnedagainstthem.AbsolutistsremindusthatGTRallowstheconstructionofsolutionsinwhichthedistributionofmatterandenergydoesnotdeterminethemetricgeometryunambiguously.Theythenarguethat,becauseofthis,MPandthereforeRMfail.Fortheysaythatthereisclearlymoretospacetimethanmaybeexpressedinmaterialtermsalone.Buttheabsolutistsmustfacetheholeargument,originallyusedbyEinsteininthegenesisofGTR,andnowrefinedbyanumberofwritersworriedaboutthestatusofabsolutism.HereIshallfocusontheargumentasdeveloped(technically)byJohnEarmanandJohnNortonand(rathermoreinformally)byPaulTeller.26GTRistypicallypresentedbymanyrelativistsasaclassical‘deterministic’theory,inwhich(inprinciple)theequationsallowustopredictandretrodictwithcertainty,solongaswehavesufficientinformationtofeedintotheequations.27Imaginethatweknoweverythingweneedtoknowtoconstructa‘global’time-sliceofspacetime—allspaceataninstant,rememberingthatsuchsurfacesof146\nEINSTEINANDABSOLUTESPACETIMEsimultaneityarenotgenerallyuniqueinrelativity:i.e.weknowallthefactsaboutthephysicalconditionsontheslice.Isthisinformation,togetherwiththefieldequationsofGTR,sufficienttoallowustoconstructacompletehistoryofthespacetime,i.e.todeterminethepastandthefutureoftheslice?28However,asweshallsee,itisnotquitesoeasytofixthehistoryofthesliceinGTR,andthisproblemtakesustotheholeargument.OneofthemainfeaturesofGTRisthatitsequationsaregenerallycovariant.GTRallowsmeadoptanycoordinatesystem.Suchasystemprovidesmewithawayofdecidingdistancesanddirectionsbetweenobjects.But,ifIchooseanotherframe,thentheequationsofGTRshouldremainessentiallythesamewhenIapplythemtothisnewframeandsystem.Iftheequationsremainessentiallythesameasweapplythemtodifferentframesofreference,thentheyaregenerallycovariant.TheequationsofGTRarewrittenintensorform,becausetensorshavetherequiredpropertyofgeneralcovariance.29Althoughgeneralcovarianceisapurelyformalrequirement,weshouldnotethattensorsdescribephysicalstatesofaffairsgeometrically.30Sotheuseoftensorscommitsustoafundamentallygeometricalperspectiveonthephysicalworld.TheadvantageofgeneralcovarianceinGTRisthatwecanadoptanycoordinatesystematalltodescribethephysicalsituationinthespacetimeframeworkormanifold.However,becausedistancesanddirectionswilldependupontheparticularcoordinatesystemchosen,theuseofgenerallycovariantequationsinGTRtocaptureanycoordinatesystemwhatsoeverimpliesthatthespecificationofdistanceanddirectionbetweenpointsisnotfixedbutvariable.Letusimaginethatwehavejustmadesuchachoiceofcoordinatesystem.Maywechangecoordinatesystemwithoutaffectingthephysicsofthespacetime?Generalcovarianceguaranteesthatwemaydojustthis.Butmaywemakechangestothedistributionofmaterialinaspecificregion(or‘hole’)inspacetimesothattheresultisanewspacetimebutonewiththesamephysicaldescriptionastheoriginal?Generalcovariancealsoallowsthis!Wemayconstructalternativespacetimeswithnoimmediatephysicaldifferencessimplyonthebasisoftheformalpropertiesofgeneralcovariance.Clearly,thepossibilityofproducingtwophysicallyequivalentworldsassumesthatwemaypickoutindividualspacetimepoints.Tellercallsthenewworlda‘Leibnizalternative’:fortheidentityofindiscerniblesseemstoapplytotheoriginalanditsalternative.31Eachoftheseworldshasthesamematerialcontentandthesamerelations147\nTIME,SPACEANDPHILOSOPHYamongstobjects.Theyseemtobeindistinguishableataphysicallevel.Buttheydifferintermsoftheirattachmenttothepointsofthespacetimemanifold.Atfirstsight,thismightappeartobeyetanotherargumentforsomeformofabsolutism.Onceagain,thereseemstobemoretospacetimethanmaybespecifiedinmaterialterms.Herethespacetimestructuresweareencouragedtoembraceareatthetopologicallevel.Butabsolutistsmusthesitatebeforeacceptingthisargument.FortheEinstein‘hole’storyastoldbyEarmanandothersseemstoinvolvethedownfallofdeterminism:wemaynolongerbesureaboutthefutureevenwhenweknowallthereistoknowaboutthehistoryofspacetimeuptonow.Fordifferentfuturesmaybegeneratedfromasinglesetofinformationabouttheworlddespitethefactthattheequationsusedareessentiallydeterministic.Imaginethataspacetimeslicecontainsasmallregionor‘hole’.GTRisrichenoughinstructuretoenableustoconstructdifferentmodels,withexactlythesamedistributionofmaterialontheslice,butwithdifferenttopologicalfeatures.Wemaydothisbyalteringthewaythephysicalfieldsattachtothepointsinspacetimetoallowtheconstructionofdistinctmodels.Thisprocedureallowsustobuild,onthebasisofinformationgiven,(forexample)twodistinctmodelswithexactlythesamepastsandpresents(asdefinedbythetime-slice)butwithdifferentfutures.32Manipulationsofpointsinthespacetimemanifoldcertainlymakesenseinabstractmathematicalterms.Butmustwecommitourselvestothebeliefthatthespacetimemanifoldisa‘substratum’withanindependent‘absolute’reality,i.e.thatspacetimepointshavea‘substantival’realityoverandabovethemetricofspacetimeanditsmaterialcontents?Iftalkofspacetimepointsisnottobeunderstoodliterally,thenweneednotdrawanyconclusionsaboutphysicalrealityonthebasisofanymanipulationofthosepoints.Thisisthepaththatrelationistsmightbeinclinedtotake.Theymightsimplysaythattheonlysignificantfeaturesofspacetimeareatthephysicallevel:soweshouldonlytalkintermsofmaterialobjectsandtheirrelationshipstoeachother.Theymightthenarguethatthe‘indistinguishable’worldsarenotreallytwoworldsatall:theidentityofindiscerniblesoperatingatthephysicallevelguaranteesthatwearedealingwithasingledomain.Whatdoestheholeargumentleaveforabsolutists?Asyettheissuesarecomplexandunresolved.Somewritersbelievethatwemaylegislateagainstchangesatthetopologicallevelbyappealsto148\nEINSTEINANDABSOLUTESPACETIMEtheessentialpropertiesofspacetimepoints;othersarguethat,whenspacetimepointsandrelationsareproperlyunderstood,wewouldfindnorealdistinctionsbetweenapparentlydifferentworldsconstructedbyappealstothewayfieldsmayattachtospecificregionsinspacetime.33Findingthesesubstantivalistmanoeuvresunpromising,Earmansuggeststhatthewayoutofthedilemmamaybetoacceptthatabsoluteaccelerationstakeplacebuttodenythattheinertialstructureuponwhichsuchaccelerationsdependisnotembeddedinasubstratumofspacetimepointsassuch.Inotherwords,hebelievesthatSklar’sideaofabsoluteaccelerationwithoutabsolutespacetime,accordingtowhichaccelerationisregardedasaprimitivepropertyorbrutefactaboutphysicalbodies,mayturnouttobethebestoptionfortheabsolutist.Sklar’sapproachhastheadvantageofbeingatleastapartialcompromisebetweenabsolutismandrelationism.Absolutistsmayholdontotheirclaimthataccelerationeffectsmayonlybecapturedthroughanabsoluteinertialstructure;andrelationistsmayretaintheirbeliefthattalkofspacetimepointshasnosignificancefortheactualworld.Earmantracesthedebatebetweenrelationistsandabsolutistsfromaproblemaboutinertialforcestoworriesaboutdeterminism.Hesaysthathewantsdeterminismtobegivenafightingchanceandsohechallengessubstantivalism.However,theforceofhisargumentdependsuponhisdecisiontosetthedisputebetweenabsolutistsandrelationistsattheontologicallevel.Clearly,ifwhatisatstakeistherealityofspacetimepoints,thentheholeargumentshouldmaketheabsolutisthesitatebeforeembracingsubstantivalism.However,thoseabsolutistswhotakeamoreinstrumentalistattitudetowardsspacetimetheoriesneednotbeembarrassedbyEarman’sattack.TheymightagreethatatheorysuchasGTRinvolvesspacetimestructures—fromtheaffineconnectiontothedifferentiablemanifoldofpoints—asirreduciblefeatures.Buttheymightseenoreasontoaddanyclaimabouttherealityofthesestructures.Earmanencouragesustoacceptthatabsolutismwithandabsolutismwithoutsubstantivalismaretheonlyviablepositionsforthosewithanti-relationistsentiments.Solongasweconcentrateontheproblemofabsolutespacetimeasanontologicalproblem,thereissomejustificationforEarman’sposition.However,wemayalsoviewtheabsolutist-relationistcontroversyasadisputeabouttheextentofourepistemologicalcommitments.Wemaythereforeidentifyafurther‘instrumentalist’positioninwhichtheabsolutist(a)avoidsdiscussionattheontologicalleveland(b)arguesthattalkofspacetimewithinagiventheorymaynotbe149\nTIME,SPACEANDPHILOSOPHYreducedtotalkofobjectsandevents.i.e.thatspacetimeisanepistemologicallyessentialstructureinthattheory.Indeed,manyphysicistsinvolvedinGTR,e.g.RogerPenrose,workconfidentlywithspacetimestructuresasessentialelementsintheiraccountsofmotionbutneverthelessexpressdoubtsabouttheontologicalstatusofthosestructures.TheabsolutistwhoendeavourstokeepdiscussionattheepistemologicallevelneednotshareEarman’sconcernsaboutembracingsubstantivalismandtherebyrepudiatingdeterminism.Foranysuchrepudiationwouldberelativeonlytoagiventheoreticalcontext,whetherthisbeGTRorsomeotherspacetimetheory.So,ifweweretotreatthespacetimeofGTR‘substantially’—asanessentialartefactbutnotarealsubstratum,wewouldnotbecommittedonceandforalltoindeterminism.WhetherornottoembracesubstantivalismandthereforeindeterminisminGTRwouldbeweighedagainstotherconsiderations.And,asweshallfindinChapter10,theredoesseemtobeagooddealofsupportforanindeterministicapproachwithinthegeneralcontextofGTR,giventhefactthatmodernspacetimetheoriesmustdealwitharangeofessentiallyindeterministicphenomenasuchasblackholes,nakedsingularities,andtheearlycosmos.150\n8TIMETRAVELINTRODUCTIONWhenH.G.WellswroteTheTimeMachine,herelatedtheastonishmentofhistimetravellerasthefatefulleverisdepressed,themachineissetinmotion,andthefutureunfoldsitssecrets:Iseemedtoreel;Ifeltanightmaresensationoffalling;and,lookinground,Isawthelaboratoryexactlyasbefore.Hadanythinghappened?ForamomentIsuspectedthatmyintellecthadtrickedme.ThenInotedtheclock.Amomentbefore,asitseemed,ithadstoodataminuteorsopastten;nowitwasnearlyhalf-pastthree!…Ipressedtheleverovertoitsextremeposition.Thenightcameliketheturningoutofalamp,andinanothermomentcametomorrow.(Wells1958:20)1ManyofushavebeenintroducedtothefantasyworldoftimetravelbyWells’masterfulnarrative.Butdowehavegoodphysicalreasonsforacceptingsuchexoticstories,orshouldwetreattimetravelasamerefiction?Canweexplainanyallegedexampleoftimetravelintermsofour‘normal’viewsofeventsinspaceandtime?Wouldwebeforcedtodropthecommonbeliefthatcausesmustprecedetheireffects?Doestheideaoftimetravelinvolvelogicalorphysicalcontradictions?Indeed,manywritersregardthepossibilityoflogicalcontradictionsasthemostconvincingreasonwhyweshouldregardtimetravelasanabsurdfiction:ifwecangainaccesstoourpastandchangeit,thenwemightevenbeabletokillallourgreatgrandparentsintheiryouth!But,ifwedidcommitsuchacrime,wewouldnothavebeenbornsohowcouldweexisttogobackintimeandcommitit?However,themicroscopiclaws151\nTIME,SPACEANDPHILOSOPHYofphysicsarecertainlyindifferenttotimedirection:givenfullinformationaboutaregionofspaceatagiventime,wemayusethemicroscopiclawsofphysicstoshowusboththefutureandthepastofthatregioninaneven-handedway.Sowemightthink,fromamicrophysicalpointofview,thatourreluctancetoconsiderthepossibilityoftravelbackwardsintimeisapeculiarlyhumanaffectationarisingfromourlimitedmacroscopicperspective.Despitethefactthattalkofspaceandtimedoesnotappeartobereducibletotalkofmaterialobjectsandtherelationsbetweenthem,ourideasoftime,change,andcausalityareintimatelyconnected.So,althoughwemightacknowledgetheindependenceofspaceandtime,itstillseemshardtodiscussideasoftimeandthedirectionoftimewithoutreferencetobothchangeandcausationinthematerialworld.Nevertheless,bythinkingintermsofthebasicconceptsof‘earlier’and‘later’,wedoseemtobeabletodiscuss‘thedirectionoftime’inawaywhich,ifnotindependentofobjectsaltogether,givesusafundamentalgraspofthepropertiesoftime.D.H.MellorpresentssuchapositioninhisbookRealTimewhenhesays:Thedirectionoftimeisthedifferencebetweenearlierandlater,adifferencenoteasytopindown.Earlierandlaterarenotdifferentlyrelatedtoeachother,foreachisjusttheother’sconverse….Ihaveremarkedhowwecanoftenseethatoneeventisearlier,notlaterthananother.Torevivetheoldexample,seeingaclockhandmoveclockwiseisinteraliatoseeitpass‘1’justbefore,notjustafter,itpasses‘2’.Beingearlierinthiscaseisaperceptiblerelationbetweenevents,asspatialdistanceoftenisbetweenthings.Itnomoreneedsdefiningthancolourdoes:wejustseeit.(Mellor1981:140–1)2Asweobserveeventsandthingsintimeweobservevariations,andwetalkofsuchvariationsusingthelanguageofchangeandcausality.Changesarechangesinthings,andchangeinvolvescausation.Althougheventsdonotchangeperse,westillmayspeakofeventsasbeingcausesandeffectsofotherevents.3Thenotionthatacausecannotoccurlaterthanitseffectsseemstobedeeplyembeddedinthewaywethinkabouttheworld.Doesthedirectionoftimedeterminethatofcausation?Ifitdoes,thentheideaofbackwardscausationassuchwouldbeuntenable.Forwecouldthenstipulatebydefinitionthatcausescouldneveroccurlaterthantheireffects:causes152\nTIMETRAVELwouldalwaysoccurearlierthantheireffects.Thedirectionofcausationwouldbeunambiguouslyfuture-directed.Ofcourse,wemightneverthelessbeabletoreachpointswhichlietoourpast,butonlybyfollowingafuture-directedpath:thismightoccurinthoseuniverseswhichpermitclosedtimelikeornullcurves,e.g.inacylinderworld.4So,althoughwemightbeabletoruleoutbackwardscausationperse,wemightstillallowthepossibilityoftimetraveltothepastviathefuturealongclosed‘loops’.However,ifcausaldirectionisindependentoftimedirection,thenwemighthavetograntthepossibilityofbackwardscausationaswell.Inthischapter,weshallassessthecasesforandagainstbothbackwardscausationandtravelalongclosedtimelikepaths.Butweshallalsomeetother,perhapsmoreexotic,variationsuponthethemeoftimetravelarisingprincipallyfromideasinrelativity.TheconceptualapparatusoftheGeneralTheoryofRelativity(GTR)allowsustoarticulatetwomainsensesinwhich‘timetravel’mightbesaidtotakeplace,inadditiontotheideasoftachyonsandspaceliketraveloutsidethelightconediscussedinChapter3:1backwardstimetravelandcausation:bymovingbackwardsintimebetweentwopoints,allowingthepointstobeconnectedby‘causal’signalstravellingatspeedslessthanorequaltothatoflight;and2closedtimelikeornullcurves:bymovingforwardsintimebuteventuallyreachingapointtothepastofthestarting-point,sothataclosed‘loop’intimemaybeformed;sincetachyonsmightalsoformsuchloops,atleastinprinciple,wemightusefullyassociatetheireffectswiththiskindoftimetravel.However,weshallfindthatmanyscientistsandphilosophersstronglyobjecttothegeneralideaoftimetravel.Therearethreemainkindsofobjection:1theideainvolvesphysicalassumptionswhicharehighlyimplausibleandmaythereforeberuledoutonempiricalgrounds;2timetravelmayinvolvethepostulationoflogicalcontradictionsandmaythereforeberuledoutonrationalgrounds;and3anystoryoftravelintothepastmayberetoldintermsof‘forwards’travel.153\nTIME,SPACEANDPHILOSOPHYSo,inthischapter,weshallreviewthetwomaincandidatesforadoptionasgenuinecasesoftimetravel;andweshallseehowtheyfareagainstthevariousobjectionswhichhavebeenraisedbeforepassingafinaljudgementonthecasefortimetravel.Beforelookingatthevariouspossibilitiesindetail,weshallfirstdiscusssomeofthefeaturesofspacetimestructurewhichhaveabearingontheproblemoftimetravel.SPACETIMESTRUCTURESpacetimeisgenerallywellbehaved.Wedonotseestrangesequencesofeventswhichmightonlybeexplainedintermsofbackwardscausation.Butwealsodonotseemtoencounterglaringexamplesoftimedilationinourlow-speed‘Newtonian’lives.Whyshouldwebeinclinedtoruleoutcausalanomaliesbutacceptsuchastrangephenomenonastimedilation?Perhapsbecauseweseemtohaveadeeplyentrenchedbeliefthattimeandcausationmarchtogetherinonedirection.Butisthisbeliefjustified?Inordertodiscusscausalbehaviourinanyspacetime,weneedtofilloutthestructureofthespacetime—andeachdetailweaddtothestructureactsasaconstraintontheeventsandphenomenawhichmightoccurinspacetime.Wecanassessthecaseforeachelementofstructureandinsodoingprovideananalysisoftherelationshipbetweenspacetimeandcausation.Thenweshallbeinapositiontoclarifytheideasofbackwardscausationandclosedcausalloopsandtomakeatleastapreliminaryjudgementontheirlikelihood.ThefollowingelementsaretypicallyregardedasfundamentalpartsofthecausalstructureofspacetimesinSTRandGTR:1Time-orientability:theleastweshouldbeabletodoinaspacetimeisdefinetheideaofadirectionoftimeinanunambiguousandcontinuouswayatanygivenpoint;aspacetimeissaidtobetime-orientablewhenwecandivide‘arrows’oftimeassignedtothepointintotwogroups—thosepointing(i.e.oriented)forwardsandthosepointingbackwards;hence,thetimedirectionateverypointofspacetimemaybefixedwithreferencetothesetwogroups.2Timedirection:wemaythenchooseaparticulardirectionatanygivenpoint,perhapsrelyingonthermodynamicorotherphysicalconsiderations;forexample,wemightbelievethatthegeneraltendencyofsystemsandstructurestobreakdownwithaconsequent154\nTIMETRAVELincreaseinentropy(oftencalledanincreasein‘disorder’)setsthedirectionoftime’sarrow,andsoalldecisionsontimedirectionmayappealtosuchconsiderations;or,followingRogerPenrose,wemightappealtotheinitialgravitationalconditionsatthebigbangastheexplanationforthedirectionoftime.53Precedenceandcausalprecedence:wenowallowthepossibilityofcurvesorpathsbetweenpointsinspacetimeandwedefineatemporalorderonthecurve;ifwewishtoruleoutsuchexoticpossibilitiesastachyonsfromthespacetime,thenwemuststipulatethatonlynon-spacelikecurvesmayhavepointsonthemwhicharetemporallyordered;anon-spacelikecurvebetweentwopointsmaythenbeginatonepoint‘p’andendatanother‘q’suchthatpprecedesq.Whenweallowcausalsignalstofollowsuchnon-spacelikepathsinspacetime,wemaythenspeakofaneventatpcausallyprecedinganeventatq.Thecausalfutureofanygivenpointpisthenthesetofalleventscausallyfollowingp,i.e.whichmaybeconnectedwithpbynon-spacelikecurvesbeginningatpanddirectedtothefutureofp.Similarly,thecausalpastmaybedefinedintermsofthesetofallpointscausallyprecedingp.4Lightcones:giventhe(empiricallyplausible)assumptionthatthespeedoflightinavacuumisthesamewhenobservedfromanyinertialframeofreference,wemayintroducetheideaoftwolightconesspreadingouttothefutureandthepastfromanobservermovingthroughanygivenpointinspacetime;theseconesmaybedefinedbythepathsthatlightraystakethroughthatpoint:oneconecontainsitscausalfuture,andtheothercontainsitscausalpast.Weshouldnotethatelement3rulesoutthepossibilityofbackwardscausation,i.e.ofacausalsignaltravellingbackwardsalonganon-spacelikecurvefromfuturetopast.Thisprohibitionwillbethesubjectofthenextsection:‘Backtothepast’.Butwedonottherebyruleoutthepossibilityofclosedcausalloops.Thisisbecausesofarwehaveonlydefinedthedirectionoftimeinlocalterms.Locallythepastisalwayspast;butgloballysomecurvemayeventuallyfinditswaytothepastviathecausalfuturealonganon-spacelikecausalcurve.However,thereisnoquestionofthenormaltemporal(andthereforecausal)orderofprecedencebeingreversed:eachneweventtobeencounteredliestothefuture.155\nTIME,SPACEANDPHILOSOPHYNevertheless,suchcurvesareclearlyanomalous,inthesensethatbyfollowingthecurvearoundtheclosedloopwemayhelptodeterminethepast.Thisproblemwillbeatissueinthenextbutonesection:‘Forwardtothepast’.Butwemaynoteatthisstagethatclosedcausalcurvesmayberuledoutbyanadditionalstructuralconstraintonspacetime,namelythedemandthatspacetimebe‘causallystable’.5Stablecausality:whereastime-orientabilityprovidesalocalsensetothedirectionoftime,stablecausalityinvolvestheassignmentofaglobaltimesenseinaspacetime.Thisallowsustoconstructsurfacesofsimultaneityor‘time-slices’througheverypointofacausallystablespacetime.Agiventime-slicedividesallspacetimeintotwocategories:pastandfuture—eventhoughweshouldnotethat,ingeneral,thereisnouniquewayofmakingsuchdivisions.6Hence,acurvepassingthroughanygivenpointtoitsfuturewillremaininthefuture;and,sincethisfutureisgloballydefined,thereisnowayinwhichthecurvecanfinditswayintothepast.Thisideaisoneofthefundamentalcharacteristicsofspacetimetheories.Forwemaydetermineboththedifferentiableandtheconformalstructuresofaspacetimegiventheconditionofstablecausality.Furtherconstraintsoncausalstructuremaybeaddedtothesefivefundamentalelements.WeshallseeinthediscussionofsingularitiesinChapter10thatsuchadditionsmayenhancethepredictivepowerofspacetimetheories.BACKTOTHEPASTBackwardscausationmaybedefinedasacausalsignaltravellingbetweentwoeventsinspacetimesuchthatthetimedirectionoftravelisfrom‘later’to‘earlier’,i.e.therelationofcausalprecedenceisoppositetothatoftemporalprecedence.Sobackwardstimetravelmaybedescribedastravelfromlatereventstoearlierevents.Inaseriesofimportantarticlesonthesubject,MichaelDummettexplorestheideasinvolvedinandtheimplicationsofbackwardstimetravelandcausation.7Inthemostrecentofthese,‘Causalloops’,Dummettpresentsuswithanumberofstoriesconcerningtheappearancesanddisappearancesofa‘timemachine’andhenotesthatwemightfindithardtoexplainexactlywhatishappeningwithoutinvokingtheideaoftravelbackwardsthroughtime.Buthealsopointsoutthatthereisacost:wemaynotbeabletofindanexplanation156\nTIMETRAVELforeveryphenomenonassociatedwithasituationinwhichbackwardscausationisatwork.Here,weshallexaminethreestoriesbasedonDummett’saccountin‘Causalloops’.Dummettasksustoimaginesomethinglikethefollowingsituation:Aat6p.m.onFridayasmall‘timemachine’isplacedonthelefthandsideofanemptytabletopinastudy;ithasaclockattachedwhichreadszero;themachineisactivatedandimmediatelythemachinedisappears;Bat6p.m.onThursday(i.e.24hoursearlier)themachine‘materialises’ontheleft-handsideofthetableinthestudywithitsclockreadingzero;Cat10a.m.onFridaysomeoneenterstheroomandmovesthemachinefromtheleft-totheright-handsideofthetable;Dthepersonactivatingthemachineontheleft-handsideofthetablenoticesthattherethemachineisontheright-handsideofthetableintheroom;thisclockstaysinpositionandshortlyafter6p.m.itsclockreadsjustover24hours.ThecausalsequencehereseemstobeABCD:theinitialcauseofthechainofeventsseemstobetheactivationanddisappearanceofthemachine(A),followedby—itsappearanceatanearliertime(B)—itsremovaltoanotherpartofthetable(C)—andsomeoneseeingthemachineinthisnewlocation(D).But,quiteobviously,thisisnotidenticalwiththetemporalsequenceBCAD:appearance(B)—relocation(C)—disappearance(A)—seeingmachineinnewlocation(D);seeFigure26(p.158).Howmightweexplainthesuddenappearanceofthemachineat6p.m.onThursdaywithoutinvokingbackwardstimetravel?Clearly,wewouldhavetochangeourmindstosomeextentaboutthephysicsgoverningourworld—especiallyifsuchthingshappenfrequently.However,itseemsthatbackwardscausationcommitsustosomestrangebeliefsabouttheeventsinsuchasequence.Thefactthatthemachineisplacedonanemptytabletopseemstorequirethat,atsometimebetweenitsappearanceonThursdayat6p.m.andthemachinebeingbroughtintothestudy24hourslater,themachinemustshiftitsposition!DoesthismeanthatwhoevercomesintotheroomonFridaymorningiscompelledtomovethemachine?IfA,B,andCinthestoryabovearetohold,thensomethingmusthappentorelocatethemachine.Butwhatifwe157\nTIME,SPACEANDPHILOSOPHYFigure26Timemachine1weretomakeadeterminedefforttopreventanyonemovingthemachine—forexample,wecouldlockthestudyforthe24hoursbeforethemachineisactivated.Nownoonecangetinandmovethemachine.Therearetwopossibleoutcomestosuchamanoeuvre:1despiteourdetermination,eithersomeoneisabletogetintotheroomandmovethemachineorthemachineitselfchangesitspositioninsomeway;or2oureffortssucceed,andthemachinestaysinitspositionontheleftsideofthetable.However,thissecondpossibilityclearlyrequiresthatthemachineshouldbeontheleft-handsideofthetabletopat6p.m.onFriday.Ifthe158\nTIMETRAVELmachineisnotthere,then,whateverwedo,wecannotsucceedinanyattempttokeepthemachineontheleft-handsideofthetable.8Ifthemachineisontheleft,thenadifferentstorymustbetoldwhichisconsistentwiththisconstraint,forexample:Aatimemachine,withaclockattachedwhichissetatzero,isbroughtintotheroomjustbefore6p.m.onFriday;ontheleft-handsideofthetablethereisanothermachinewhichisthesameinallrespectsexceptthatitsclockreadsalmost24hours;Bat6p.m.preciselythemachinewiththeclockreadingzeroisactivated:onactivationitsclockbeginsanditappearstomergewiththemachineonthetable;Cat6p.m.onThursdayamachinewithaclockreadingzerosuddenlyappearsontheleft-handsideoftheemptytabletop—itremainstherefor24hourswithitsclockrunningnormally;Dafter6p.m.onFridaythismachinestaysinplaceontheleft-handsideofthetabletopwithitsclockrunningonnormallybeyond24hours.ThetemporalsequencehereisCABD;thecausalsequenceisABCD—bringingthemachineintotheroomandactivatingitseemstobethecauseofanearlierevent;seeFigure27(p.160).Althoughtheremightseemtobetwomachinesinthisstoryofbackwardstravel,thereisinfactonlyone.Themachinebroughtintotheroomtravelsbackwardsintime,mergeswithitself,appearsonthetable,andremainsthereforatleast24hours.Themachinemustmergewithitself,orwecannotexplainwhythemachineisfoundonthetablewhenenteringtheroom.Thissecondstorydemandsacceptanceoftwostrangefacts:first,assumingthatthemachinehadbeenconstructedsometimebefore,anobjectis‘bilocated’for24hours—i.e.oneandthesameobjectisintwoplacesatthesametime;and,secondly,anobjectmaymergewithitself.Whicheverstorywechoose,weseemtobecommittedtoacceptingsomeratheroddbeliefsabouttheworld—andifweembracetheideaofbackwardscausationwehavenoimmediatereasontoobjecttoeitherstory.Sowemustacceptthatcertainpastactionsoreventsarenecessitatedbyfutureactionsorevents,evenwhentheseseemincidentaloraccidental—butthis,ofcourse,mightseemstrangeonlyinasmuchaswearenotusedtothinkingintermsofbackwardscausation:someonemightthinktheyhavetheoptionofmovingornotmovingthemachinefromthetable,butinrealitytheyhavenochoiceatall.Hence,ourfreedomtoact159\nTIME,SPACEANDPHILOSOPHYFigure27Timemachine2maybeconstrainedbyfutureevents.Andwemustacceptthephysicalpossibilitiesofbilocationandmerging.Wenotedabovethatwemighthavetochangeourphysicstoaccountforsuddenappearancesofobjectswithoutinvokingbackwardscausation.Butitnowseemsthatweshallhavetochangeourphysicsinanycase.Ineachofthetwostoriessofar,wehaveconsideredwhatseemstobean‘instantaneous’leapbackwardsbythetimemachine.Sucha‘journey’doesnotseemtofollowanykindofspacetimepathatall.Hencethese‘journeys’seemtobeinconsistentwithourideasoftravelinspacetime,whichrequiressomesortofsmoothpassagealongaspacetimepath.Itmightbethatthe‘instantaneous’journeytakesplaceviaatopologicalanomaly—perhapsawormholeinspacetime:althoughthetwolocationsinspacetimedonotseemtobeadjacent,thetopologicalfeaturesofspacetimemay160\nTIMETRAVELOnatopologicallyflatplane,thereisnoshort-cutfromPtoQbutiftheplaneisfoldedoversothatPlies‘above’Qandthe‘higher’and‘lower’partsoftheplanemakecontactatPandQ,thenamovemaybemadefromPtoQwithoutthenecessityforalongjourneyalongnormalspacetimepaths:Figure28Ideaofwormholeinspacetimebesuchastobringthetwolocationstogetherviaawormhole;seeFigure28aboveforanillustrationofthisidea.9If,however,weimagineatimemachinewhichtravelsmoresmoothlybackintime(alongapossiblespacetimepath,butinthepast-directeddirection),furthercomplicationsarise.Imagineamachinewhichtravelssteadilybackwardsintimefor24hours,andthentravelsforwardsintime.Wemightcharacterisesuchassituationasfollows:AamachineM2isinpositionontheright-handsideofthetabletopjustbefore6p.m.onFriday—ithasaclockattachedwhichreadsalmost48hours;ontheleft-handsideofthetablethereisanothermachinewithaclockreadingjustafewsecondsfromzero—countingdowntowardszerosothatat6p.m.itwillreadzero;BafurthermachineM1withaclocksetatzeroisbroughtintotheroom,heldabovethemachineontheleftcountingdowntozero,and,at6p.m.,M1’sclockisactivatedanditisdroppedontoandmergeswiththemachinebelow—M1istotravelbackwardsintime;asthemachinesmergeat6p.m.,theydisappearleavingjustM2ontheright-handsideofthetable;Catanytimeinthe24hoursbefore6p.m.onFriday,twomachinesmaybeseenonoppositesidesofthetable:M2’sclockrunsforwardsfrom24to48hours,butM1’sclockrunsbackwardsintime(i.e.this161\nTIME,SPACEANDPHILOSOPHYclockwillappeartoanynormalobservertoberunningbackwardsfrom24tozero);Dbefore6p.m.onThursdaythetabletopisempty;at6p.m.preciselyasinglemachineinthecentreofthetablewithaclockreading24hoursappearstosplitintotwoidenticalmachinesonoppositesidesofthetable—theonlydifferencebetweenthemachinesseemstobethatoneisrunningforwardsandtheotherbackwards;Eafter6p.m.onFridayM2remainsinplaceonthetabletopwiththeclockrunningonnormallybeyond24hours.Onceweacceptbackwardstimetravelandthepossibilitiesofmergingandbilocation,theonlyadditional‘strange’occurrenceinthisstoryisthesuddenappearanceofatimemachinewhichspontaneouslysplitsintotwonearlyidenticalmachines;seeFigure29(p.163).Whydoesbackwardstravelinvolvesuchasplit?Thestoryconcernsjustonemachine:M1isthemachinewhenitisrunningbackwardsintime;M2isthemachinerunningforwardsintime.Theothermachineonthetablejustbefore6p.m.onFridayisinfactM1travellingbackwardsintime.At6p.m.onFriday,themachineisplacedasM1ononesideofthetable—bymergingitwiththemachineseeninthatpositiononthetable;but,sinceM2isontheothersideofthetableat6p.m.,itisobviousthatatsometimebeforethenthemachinemustshiftposition.Thesplittingprocessisinfactthemachinewhichhasbeenrunningbackwardsmovingacrossthetabletostartits‘journey’forwardsintime.Ifweentertheroomduringthe24hoursbefore6p.m.onFriday,thenwewillfindthatanumberofactions,whichwemightattempt,turnouttobeimpossible.Forexample,ifonenteringtheroomweseetwomachinesonthetable,wecouldnotdestroythemachinetravellingbackwardsnomatterhowhardwetry.Evenifweprogrammedarobottoentertheroomanddestroythemachinewiththebackwards-runningclockifandandonlyifitdetectstwomachinesonthetable,therobottoocouldnotsucceedinthistask!Fortodestroythemachinewiththeclockrunningbackwardsimpliesthatthemachinetravellingforwardswouldnotbepresent—fromthemachine’s‘pointofview’thebackwardsphaseisjustanearlierpartofitshistory.Somecoursesofactionwouldbepossible.162\nTIMETRAVELFigure29Timemachine3Wecouldmovethemachinetravellingbackwards—thespontaneous‘splitting’wouldthentakeintoaccountthenewlocation.Ifwedestroyormovethemachinefromthetableinitsforwardsphase,thentherewouldsimplybenomachinealreadyonthetablewhenthetimemachineisactivatedat6p.m.onFriday.Therearemanysuchstorieswhichwemighttellabouttravelbacktothepast.Toacceptthemasphysicallypossiblemeansthatwemustbepreparedtochangeourbeliefsaboutboththephysicsinourworldandourfreedomtoact.However,wemightbeinclinedtoregardallsuchstoriesasmerelyfictionalentertainment.163\nTIME,SPACEANDPHILOSOPHYFigure30PositronsandelectronsIndeed,wemightbeentirelyjustifiedindoingjustthis—iftherewerenoempiricalbasiswhatsoeverforbackwardstimetravel.However,somephysicistshavesuggestedthatcertainphysicaleventsarefarmoreelegantlyexplainedifweusetheideaofbackwardstimetravel.RichardFeynman’s‘positrontheory’isonesuchaccount.10WemaytellthestoryillustratedinFigure30aboveintwoways.1Agammaraycreatesanelectronanditsantiparticle,apositron;thepositroneventuallymeetsanotherelectronandbothoftheseareannihilatedresultingintheemissionofasecondgammaray.2Anelectroninnormalmotionemitsagammarayandimmediatelystartstomovebackwardsintimeuntilitmeetswithandabsorbsanothergammaraywhentheelectrononceagainmovesforwardintime.Thepositroninthefirstaccountmovesforwardsintime;butFeynmancharacterisesthepositronasanelectronmovingbackwardsintime.Whatadvantagesdoesthisalternativedescriptionhave?Certainly,164\nTIMETRAVELthereisasenseinwhichFeynman’saccountissimplerthantheorthodoxversion,sinceonlyonekindofparticleisrequired:anantiparticleissimplyaconventionalparticlemovingbackwardsintime.ButthereisalsoalingeringsuspicionthatFeynman’salternativedescriptionisnomorethananeedlessredescription.Fortheeconomygainedwhenwereducethenumberofparticlesiscounter-balancedbythelossineconomywhichresultsfromourneedtothinkintermsofbackwardsaswellasforwardstimetravel.So,unlesswearepresentedwithindependentevidenceforbackwardstimetravel,wemightbeinclinedtostickwiththeorthodoxversion.Thereareotherpossibleinstancesofbackwardstimetravel,but,ratherthanconsidereachpossibilityinturn,weneedtolookatHughMellor’sargumentagainstbothbackwardstravelandtimetravelingeneral.11Mellormaintainsthatitisalwaysbettertoholdontoournormalaccountoftimeandcausationandtoaccept,ifnecessary,theneedforrevisionstoourphysicaltheories.Thisargumentturnsinpartontheclaimthatallstoriesaboutbackwardstimetravelinvolveclosedtimelikeornullloopsaroundwhichcausalsignalsmaytravel,atleastinprinciple.Clearly,thethirdtimemachinestoryabovedoesinvolvesuchaloop,forthemachinecompletesa‘round-tripintime’.But,evenifamachinetravelsbackwardsanddoesnotreturntothepresent,itwillmeetotherobjectswhichinprinciplecouldfindtheirwaytothemachine’soriginallocationinspaceandtime,thusprovidingacontinuousclosedloop.Thefirstandsecondstoriesmayinvolvea‘discontinuity’intime—wecannottraceanycontinuous‘path’forthemachinesbackwardsintime—theymayjustdisappearfromthelatertimeandreappearattheearliertime,unlessthemachinesinthesestoriestravelviasomethinglikeawormholeinspacetime.But,evenwithoutwormholes,wemightstilltraceaspacetimepathdirectedbackwardsintimewhichconnectsthelaterandearlierspacetimelocationsofthemachines.So,inasense,wemaystillthinkintermsofclosedloopsintimeeveninsuchcases—providingwehavereasontobelievethatpast-directedtravelisinvolved.12However,thereareother,perhapsmoreplausible,waysofformingclosedloopsintime—bytravelforwardstothepast.SoweneedtodiscussthisideabeforeturningtoMellor’sargument.165\nTIME,SPACEANDPHILOSOPHYFORWARDTOTHEPASTAnumberofspacetimesinGTRseemtoallowthepossibilityofaparticlewhichfollowsacontinuousfuture-directedpathfromagivenstarting-pointeventuallyfindingitswaytootherpointswhicharelocatedinthecausalpastofthestarting-point.And,fromthecausalpastofthestarting-point,theparticlemaythenreachthestarting-pointitself.Hence,wemayconstructclosednon-spacelike(i.e.timelikeornull)pathsalongwhichcausalsignalsmaytravelinsuchspacetimes.Aswehaveobserved,eventhoughtheideasoftemporalandcausalprecedencemaybeinviolateinthesespacetimes,thepossibilityofclosednon-spacelikecurvesviolatestheconditionofstablecausality.Thesimplestwaytoenvisagethepossibilityoftravelforwardstothepastviaclosedloopsintimeisinthecylinderworld,asdescribedinChapter4.13Althoughaparticlereleasedfroma‘starting-point’insuchuniversesmightalwaystravelalongfuture-directedcurves,therealwaysremainsthepossibilitythatitmightreturntoitsimmediatecausalpastandperhapscollidewithitselfatthestarting-point;seeFigure31below.Weshouldrememberthataparticlemakingsucha‘round-trip’isnot‘movingthroughtime’inthewaythatwetypicallythinkofobjectsmovinginspace.Theround-tripinspacetimeischaracterisedbytheworldlineoftheparticle.Aparticledoesnotmovealongaworldline:rathertheworldlinerepresentstheparticle’smotioninspacetime.Theworldlinedescribestheentirehistoryoftheparticlearoundtheclosedpath:itrepresentstheparticle’spathinspacetimesuchthateachpointontheworldlinerepresentsthespatiallocationoftheFigure31Closedtimelikeloop166\nTIMETRAVELparticleatadifferentpropertime(i.e.timeasmeasuredfromtheparticle’spointofview).Tofocusonagivenpointofthepathistofindtheparticlethereataspecificpropertime.Soitisinthissensethattheparticleis‘always’presentatthestarting-pointasitispresentateverypointofitsworldline.Hence,aparticlemakingsucha‘round-trip’reallycouldmeetitself!Butthismeetingisauniqueevent.Thereisnoquestionoftheeventhappeningoverandoveragain.Foreacheventuponaworldlineisauniqueeventinthehistoryofaparticlefollowingtheworldline.And,ifaparticlecouldfollowaclosedpathintime,thenwhynotaspeciallydesignedmachinewithorwithoutapersoninsideit?Thispossibilityallowsustoconstructallsortsofpuzzlingsituationswhichsomearguegiveussufficientreasontodismissclosednon-spacelikeloopsasabsurd.Forexample,wemayimaginethefollowingsituation.14Firstwemaketwobackgroundassumptions:1TwopointsPandQlieonaclosedcausalloop:…P…Q…R…S……Petc.2ThetemporalorderofthesepointsisPQ,i.e.afuture-directedcurvefromPmayreachQ,andafuture-directedcurvefromQmayalsoreachPbytrackingaroundtheclosedloopinasingletimedirection.Thenwecharacterisea‘possible’situationonthebasisofthesetwoassumptions:3AdevicelocatedatQhastwopossiblestates:‘on’or‘off’.4ThisdeviceatQis‘on’ifandonlyifaparticlebeamfromPactivatesitandswitchesiton;ifnobeamactivatesitthenitremainsoff.5ThedeviceatQis‘on’.6WhenthedeviceatQis‘on’,aparticlebeamisemittedwhichtravels(viaR,S,…andsoon)towardsaseconddeviceatP;whenthedeviceatQis‘off’nobeamisemitted.7Therefore,given3,4,5,and6,abeamfromQisemittedintheforwardtimedirectiontowardsP—clearlythisispossibleonlyifthebackgroundassumptions1and2alsohold.8ThedeviceatPhastwopossiblestates:‘on’or‘off’.9ThisdeviceatPis‘off’ifandonlyifaparticlebeamfromQswitchesitoff;but,ifnobeamarrivesfromQ,thenthedeviceremains‘on’.167\nTIME,SPACEANDPHILOSOPHY10WhenthedeviceatPis‘on’,aparticlebeamisemittedwhichtravelstothedeviceatQ;whenthedeviceatPis‘off,nobeamisemitted.11Given7,8,and9,thedeviceatPis‘off’.12Therefore,given10and11,thedeviceatPdoesnotemitabeam.13Consequently,given3,4,and12,thedeviceatQis‘off’.Clearly,13contradicts5.AndmakingtheassumptionthatQis‘off’atstep5doesnothelp.ForweshallthenbeforcedtosaythatthedeviceatPis‘on’,emitsabeam,andactivatesQ,switchingit‘on’.WhateverwesayaboutQresultsinanoutrightcontradiction.Withoutthetwobackgroundassumptionswhichpermitclosedloops,thenweobviouslycouldnotcharacteriseanysuchsituation—sinceitdependsforitscontradictoryforceuponstep7,whichitselfdependsupon1and2.Hence,therootofthecontradictionseemstobejusttheseassumptions.Formanythiskindofargumentisareductioadabsurdumoftheideaofclosedcausalloopsandthereforeofanykindoftraveltothepast,giventhefactthattravelbackwardsintimealsoallowssuchloopstobeconstructed.15Sometimespuzzlesratherthanparadoxesmaybegenerated:wemightreturntothepastwithaportfolioofcolourphotographsofallVanGogh’sworkandleaveitwithhimwhenheisstillachild;hethenspendstherestofhislifecarefullycopyingthephotographs!16Sometimestheproblemsinvolvingpeoplemayleadtoparadox:ifwecantraveltothepast,thenwecouldkillourparentsbeforetheyconceiveus!WhatfollowsisbasedononesuchstorytoldbyJonathanHarrison.17Dumwakesuponedayinsidealargemetalcabinet,rememberinglittleaboutwhathashappenedtohim,buthedoesrememberthathisnameisDum.HenoticesabookclosebywiththetitleHowtoBuildaTimeMachine.Heleavesthecabinetandpushesitintoadeepriver.Hetakesthebookwithhim.HemeetsandmarriesawomancalledJocasta.TheygivebirthtoasonwhomtheynameDee.WhenDeeis21,hefindsDum’sbookand,followingtheinstructions,buildsatimemachine—itlookslikealargemetalcabinetfromtheoutside.DeepersuadesDumtoenterthemachine.Theysetitin‘motion’.WhenDeerealisesthatthejourneyisgoingtobealongone,hekillsandeatsDumtoconservesuppliesoffood!Havingeatenhisfather,DeedecidestochangehisownnametoDum.Hethensettlesdowntosleep.Dumwakesuponedayinsidealargemetalcabinetrememberinglittleaboutwhathashappenedtohim,buthedoesrememberthathisnameisDum…!168\nTIMETRAVELDoesthisstoryinvolveanyoutrightcontradictions?Certainly,wemightbegintowonderhowmanypeoplethereareinthestory,andwheretheycamefrom.ItlooksasthoughDumandDeearethenamesusedtorefertodifferentperiodsofasingleperson’slife:thepersonisbornasDee,murderssomeonecalledDumwhilsttravellingtothepast,changeshisnametoDum,hasasoncalledDee,andiseventuallymurderedbyhisson.Clearly,DeeasDummarrieshisownmother.Butthereseemstobenologicalcontradictioninvolvedinthestory.Somethingsinthestorydonotseemtohaveanydefinitephysicalorigin.Forexample,thebookisneverwrittenbyanyone;itjustexistsanditshistoryisdescribedbytheeventsonaclosedloopintime.AndalthoughDeeisborninthenormalway,wemightalsobepuzzledabouttheoriginofthegeneticmaterialpassedontoDeebyhisfatherDum.Therearenopaternalgrandfathersorgrandmothersinthisstory.ThesinglepersonisalsobilocatedasDeeandDumforpartofthatperson’shistory:strictlyspeakingthebilocationoccursatthebeginningofDee’slifeandtheendofDum’slife;seeFigure32(p.170)whichillustratesDee/Dum’sworldline.However,thesearephysicalnotlogicalpuzzles.Wecouldtrytoconstructacontradictionbychangingthestoryalittle.Forexample,wemightstipulatethatDee,remorsefulaftermurderinghisownfather,commitssuicide.Ifthesuicideattemptissuccessful,thennoonewillemergefromthecabinetandnoonewillmarryJocastaandfatherDee.So,ifthereisasuicide,therecannotbeasuicide,forDeewillneverbearoundtomaketheattempt.Hence,ifDeedoescommitsuicide,thenwedoseemtohaveaparadoxicalsituation.Theexistenceofparadoxesinsomecasesisenoughforsometosaythattimetravelisimpossiblebecauseitislogicallyimpossible—wejustcannotacceptanystoryaspossiblebecausewecanfindsomestoriesinvolvingsuchcontradictions.Othersaddtothistheassertionthattimetravelwouldbeeffectivelyimpossibleinmanycausallyunstableworlds—atleasttravelforanyperson,sincefarmoreenergywouldberequiredforsuchajourneythananytechnologywemightimaginecouldactuallydeliver.18Ifwewishtodefendtimetravelagainsttheaccusationthatitisimpossible,thenwemighttrytolimitthepossiblesituationswhichmightarisefromtimetraveltonon-contradictoryones.In169\nTIME,SPACEANDPHILOSOPHYIfwelookatthissequencefromthepointofviewofasinglepersonbeingborn,livingalife,thendying,wewouldtracetheworldlineofDee/DumasCEFABD.Figure32Worldlineofthetimetravellerwhokilledhimselfparticular,wemighttrytoblocktravelbypeopleintotheirlocalpastinordertoattempt‘impossible’taskssuchasmurderingthemselvesaschildren,i.e.auto-infanticide!PaulHorwich,forexample,arguesthatwecanacceptthepossibilityoftimetravel,butthenwemustacknowledgethatcertainsituations—particularlythoseinvolvingcontradictions—willnotbepossible,giventheobservedbehaviourofthephysicalworld.19Butthis‘defacto’argumentmightseemratherunsatisfactory:forweare170\nTIMETRAVELnotshownwhycontradictorysituationscannotarise,wearesimplytoldthattheycannotbecausetheydonot.Nevertheless,thatmightbethewaytheworldhappenstobe.Inthatcase,itmightseemchurlishtocomplainthattherearebrutefactsabouttheuniverseforwhichwehavenoconvincingexplanation.20CORRELATIONSANDBACKWARDSCAUSATIONIfanactioncarriedouttodaycanbeacauseforaneventwhichtookplaceyesterday,thenitseemsnaturaltoaskwhatmighthappenifwedonotcarryouttheaction.Couldtheeventstillhavetakenplace?Yes,itcould;butthensomeotheractionmusthavebroughtitabout.Butwhatifwecanshowastraightforwardcorrelationbetweenthiskindofactionandthatkindofeventsuchthattheactionisalwayssome24hourslaterthantheevent.Whentheactionisnotcarriedout,wefindnoevidenceoftheeventhavingtakenplace;and,whenitiscarriedout,wealwaysfindthattheeventdidindeedoccur.Forexample,myactionmightbetomakeawish(outloud)foraletterfrommybrotherjustbeforethepostmanarrives.But,formywishtobefulfilled,mybrothermusthavepostedhisletterthedaybefore(atleast!).Ifindthat,wheneverImakethiswishoutloud,thereisalwaysaletterfrommybrotherinthepostman’sbag;however,ifIfailtomakethewishoutloud,thenthereisneveranyletterfrommybrother.MustIconcludethatitismymakingawishoutloudtodaywhichcausesmybrothertoposttheletteryesterday?Imightdecidethatitismerelyacoincidence.But,inthiscase,IwouldhavetosaymostpairsofeventswhichIregardascausallyconnectedaremerelycoincidences—forallwehaveinsuchcasesaretheinvariablecorrelationsbetweenthepairedevents.Whateverwethinkaboutcorrelationandcausalconnections,weshouldnotapplyoneruletoforwardscausationandoneruletobackwardscausation.Thefactthatcorrelationsdoholdisusuallyseenasanindicationthatcorrelatedeventsarecausallyconnectedinsomeway.Thestrongerthecorrelation,themorelikelywearetotakethepairofeventsascausallyconnected—ifnotascause-effect,thenprobablyasthejointeffectsofsomecommoncause.Sowehavenoimmediatereasontodismissthecorrelationbetweenmywishandthearrivalofaletterasapairofeventswhicharenotcausallyconnected.And,ifothersfindexactlythesamekindofcorrelationbetweentheirmakingwishesoutloudforaletterfromtheirbrotherandthearrivaloftheletterwishedfor,thenwemightbeinclined171\nTIME,SPACEANDPHILOSOPHYtoacceptthatthecorrelationindicatesthatthereisindeedagenuinecausalconnection.OneofthereasonswhywemightacceptacorrelationasagenuinecausallinkisonthebasisofMellor’sassertionthat‘causesmaketheireffectsmorelikelythan,inthecircumstances,theywouldotherwisehavebeen’.Mellorgoesontoexplainhowthisisrelatedtotheideasofcorrelationandcausality:Inordertodiscovercausesandeffects…sciencemustatleastestablishstatisticalcorrelationsbetweenkindsofeventswhichitclaimstobecausallyrelated.Thecorrelationsmustmoreoverbemorethanmerecoincidence,i.e.theymusthavetheforceofstatisticallaws,howeverlimitedinscope.Otherwisenothingfollowsabouthowlikelyaneventoftheeffectkindistofollowaneventofthekindsupposedtocauseit.Ifhittingawindowcausesittobreak,itmustbemorethanacoincidencethatwindowsofthatsortmoreoftenbreakwhenhitlikethatthanwhennotsohit.(Mellor1981:123)So,for‘sensibledecision-making’aboutcausesandeffects,Mellorarguesthatwhateverelsewemightsayaboutstatisticalrelationshipsbetweenevents,wemustinvokeatleastthisnotionthatcausesmaketheireffectsmorelikely.ButnowMellorusesthislow-keyanalysisofcorrelationandcausalitytoshowthatthecorrelationsbetweensucheventsasmywishingforaletterandthepriorpostingofthelettershouldalwaysbeexplainedintermsoffuture-directedcausation.Imaginethatwehavealargegroupofpeoplewhohaveprovedsuccessfulinthepastingettingaletterfromtheirbrothersafterwishingoutloudforit.Wedecidetoinvestigatetheallegedcausalconnectionbetweenwishingoutloudandreceivingthedesiredletter.Onagivenday,sayMonday,wefindoutwhichpeopleinthisgrouphavebrotherswhohavepostedalettertothemthatsameday.Eachletterpostedhasanelectronicidentificationtagandminiaturetransmitter.Thisallowsanautomatedcomputersystemtokeeptrackofeachletterinthesystem;thissystemprovidesacontinuousinformationprint-outofthelocationofeachpostedletter.WethendividethegroupintotwosectionsAandBbasedonMonday’sinformation.Ifsomeonehasbeensentaletterbytheirbrother,thentheyareplacedinsectionA.AndthosetowhomnosuchletterhasbeensentareplacedinsectionB.Wedonotrevealtoanyonein172\nTIMETRAVELwhichsectiontheyhavebeenplaced.Wethensub-dividethetwosections.Weaskhalfofthemembersineachsectiontomakeawishoutloud,andweasktheotherhalfofthemembersineachsectiontodoanythingbutmakeawishoutloud.Mellorsuggeststhatinsuchsituationsthefollowingoutcomesarepossible:1WefindthatthemembersofsectionAdoexactlyasinstructed—halfwishingoutloudandhalfnotwishingoutloud;butnowthestrongpreviouscorrelationbetweenwishingoutloudandthearrivalofthedesiredletterislost—forhalfofthemembersofthissectiondonotwishbutstillreceivealetter;andwefindthateveryoneinsectionBalsocarriesouttheirinstructions,sothatanumberofpeoplewishingoutlouddonotreceivealetter;hence,wewouldbefarlessinclinedtosaythatthetwoeventsarecausallyrelated.2Everyonecarriesouttheirinstructionsandthestrongcorrelationismaintained.Contrarytoourexpectations,formedonthebasisofinformationcollectedaboutwhatwasandwasnotposted,everyonewishingoutloudreceivesaletterandeveryonenotdoingsodoesnot;somehowwewerepreventedfromgivingacorrectaccountofwhatwasandwhatwasnotposted—butthisseemsextremelystrange,sincemuchintheautomatedprint-outwouldnowbeincorrectdespitethefactthatthesystemprovidedtheprint-outonthebasisofthetransmittedlocationsofeachandeveryletterinthepost.21And,asMellorpointsout,ourjudgementthatouraccountofagiveneventisreliableseemstodependupontheeffectsofthateventratherthanuponitscauses:ifwewanttobesurethatanearthquakehasoccurredintheSanFranciscoarea,wedonotrelyoninformationaboutthecausesofearthquakesbutontheseismographreadingsinthatarea,i.e.welooktotheeffectsofearthquakes.Similarly,theautomatedprint-outprovidesuswithareliableguidetowhetherornotaletterhasbeenpostedandtowhatthenhappenedtoapostedletterinthepostalsystem.Whyshouldallthisevidencenowbedoubted?Toacceptthisasapossibleoutcomewouldrequireustoquestionthereliabilityofallempiricalevidenceevenwhenthisevidenceisasobjectiveasmightseempossible.Wemightalsoquestionthereliabilityofourevidencethatthewishingevertookplace:whyshouldwenowacceptanyofthe‘facts’aboutwishingasevidenceforacorrelation?Hence,weshouldbereluctanttoacceptthisoutcomeasplausibleinanyway.173\nTIME,SPACEANDPHILOSOPHY3WefindthateveryoneinsectionA,contrarytoinstructions,wishesoutloud;andthateveryoneinsectionB,againcontrarytoinstructions,doesnotwishoutloud.Despiteourefforts,thestrongcorrelationismaintained.But,sincenoneofthesectionmembersknowstowhichsectiontheyhavebeenassigned,wecannotsaythattheyactedonthebasisoftheirknowledgeaboutwhetherornotaletterhadinfactbeenpostedtothem.Thislookslikeamuchmorereasonablepossibility,forhereweneednotreviseourbeliefsaboutwhatistocountasevidence.Butwedoneedtoaccountforthefactthatthecorrelationismaintainedonlybecausetheinstructionsarenotuniversallyobeyed—soweneedtoexplainthisdisobedience.Infact,wemighttrytosetupthesituationsothatdisobedienceseemsoutofthequestion.Forexample,allmembersofsectionAmightbegivenastronglong-termsedativeonMonday,‘guaranteed’toknockthemuntilwellafterthepostmanhasarrivedonTuesdaywiththelettersfromtheirbrothers.Forthestrongcorrelationtobemaintainednow,weneedmorethandisobedience:eitherourguaranteedsedative,whichhasworkedoneveryoccasioninthepast,mustfailtoworkinthiscase,orthemembersofthesectionmustwishoutloudintheirsleep—evenifwetapetheirmouths,theywillripthetapeoffintheirsleep,breakinganyotherbondstodothis—orwemustbepreventedfromadministeringthesedative.Allofthesepossibilitiesmightseemhardtoexplain.Butstillwecanimaginethatthecorrelationisneverthelessmaintainedbyoneorallofthesestrangeoccurrencestakingplace.Wouldwethenbeobligedtosaythatbackwardscausationisatwork:thatwishingoutloudtodaycanreallycausesomeonetopostaletteryesterday?Mellorsaysthatweneednotdepartfromournormalcausalthinking,foreverythinginthispossibleoutcomemaybeexplainedintermsofMonday’seventsbeingthecauseofwhathappensonTuesday.Forexample,wemaysaythatthewishismadebecausetheletterisposted,orthatthesedativefailsbecausetheletterisposted,andsoon.Ofcourse,weshouldstillbepuzzledabouttheratheroddcorrelationsinsuchacase.Forexample,whyshouldmybrother’sactionofpostingaletteryesterdayinfluencemetoday—istheresomekindof‘telepathic’signalinvolved?Butevenwiththebackwardscausalaccountwearepuzzled.Forexample,howisthewishtransmittedbackwardsintime,whydoesthisworkwithwishesaboutlettersandnotaboutotherthings,howdoeswishingtodayconstrainthe174\nTIMETRAVELapparentlyindependentactivitiesofthepostalserviceyesterday,andsoon?Inbothcaseswemayneedtorevisebothourphysicaltheoriesandourgeneralbeliefsabouttheworld.22Ifouraccountintermsofforwardscausationseemsproblematicinanyway,thenatleastwehavethecomfortofknowingthattheaccountintermsofbackwardscausationisatleastasunbelievable,ifnotmoreso.Whetherwearedealingwithwishesandlettersortheappearancesanddisappearancesoftimemachines,wemayalwaysgiveanaccountintermsofforwardscausationandthereseemtobenoseriousdisadvantagesinvolvedwhicharenotsharedbyaccountsintermsofbackwardscausation.Giventhatwemaydothisinprinciple,thenwemightbeinclinedtoregardalltheweirdandwonderfulinventionsinthischapterasjustthat—inventionswithnoempiricalbasis.Iftheworlddidbehaveasoddlyastheexamplesoftimetravelwhichwehaveencounteredseemtodemand,thenwemightthinkagainabouttheempiricalbasisforbackwardscausation.Butwestillmightquestionitsrationalbasisandconcludethataccountsintermsofforwardscausationcertainlyleaveusnoworseandpossiblyratherbetteroff.175\n9EINSTEIN’SGREATESTMISTAKE?INTRODUCTIONWhenconfrontedbytheeleganceandsimplicityofaphysicallaw,wemightbeforgivenforoverlookingthepresenceofaconstantintheequationbeforeus.Sometimesaphysicalconstantmightbeimposeduponalawafterexperimentalinvestigations—asinthecaseoftheNewtonianconstantofgravitationG.However,sometimesthereasonfortheadditionofaconstanttofundamentalequationsmightseemtobelessconvincing.Einstein’sintroductionofthecosmologicalconstantintothefieldequationsoftheGeneralTheoryofRelativity(GTR)hasoftenbeencriticised.ThephysicistdeSitterclaimedin1917thattheintroductionoftheconstantintoGTR‘issomewhatartificial,anddetractsfromthesimplicityandelegance’oftheoriginalfieldequations,andrecentwriters,suchasAbrahamPaisandStephenHawking,emphasisetheadhoccharacteroftheconstant.1AndEinsteinhimselfisreportedtohavesaidthattheconstantwashisgreatestmistake—hehadrecommendedgettingridoftheconstantin1931.2Inthischapter,weshallexaminehowthecosmologicalconstantaffectedthedevelopmentofGTRandassessthetwocriticismsabove.Ishallconcludethattheintroductionoftheconstantwasneitheradhocnorablunderandfurthermorethatwecanlearnmuchaboutthestatusoffundamentallawsfromthehistoryoftheconstant.Weshallalsofindthattheproblemofthecosmologicalconstantintroducesustoanotherimportantareaofdebatewithinmoderncosmology:doestheuniverseobeyaprincipleofanthropocentricity?Thecosmologicalconstanthasplayedastrangepartinthehistoryof176\nEINSTEIN’SGREATESTMISTAKE?moderngravitationalphysics.WelcomedintothefoldofGTR,thenrejected,thenrehabilitated,theroleofthisconstanthasimportantimplications—philosophicalaswellasphysical.TheconstantfirstappearedinGTRinaseriesofpapersbyEinsteinin1917and1918.3Einsteinwasthenconcernedwiththeglobalimplicationsofhisgravitationalfieldequations—thelawsgoverningmotioningravitationalfields.Inparticular,hewastroubledbyfourinter-relatedproblemswhichhadraiseddifficultiesformany—fromNewtonhimselftonineteenth-centuryphysicistssuchasErnstMach,CarlNeumann,andHansSeeliger:1Isspaceaninfinitelyextendedthree-dimensionalcontainer?2Doestheuniversehaveafinitemattercontent?3Isspaceitselfthesourceoftheinertialforcesexperiencedbyacceleratingobjects?4Arethe‘fixed’starsreallyfixed—istheuniverseanessentiallystaticplacewiththestarsremainingroughlythesamedistancesapart?WhenEinsteinintroducedthecosmologicalconstant,hebelievedthathehadfoundasatisfactorysolutiontotheseproblemswithinthecontextofGTR.BeforeexaminingEinstein’ssolution,weshalldiscussnotonlyNewton’sownreactionstotheissuesraisedbytheseproblemsbutalsothevariousresponsesofMach,Neumann,andSeeliger.SPACEANDINFINITYInalettertoRichardBentleyin1692,Newtonexplainedhisreasonsforregardingtheuniverseasaninfinitearenacontainingmattereverywhere:Ifthematterofoursunandplanetsandallthematterintheuniversewasevenlyscatteredthroughoutalltheheavens,andeveryparticlehadaninnategravitytowardsalltherest,andthewholespacethroughoutwhichthismatterwasscatteredwasbutfinite:thematterontheoutsideofthisspacewould,byitsgravity:tendtowardallthematterontheinsideand,byconsequence,falldowntothemiddleofthewholespaceandtherecomposeonegreatsphericalmass.Butifthematterwasevenlydiffusedthroughoutaninfinitespace,itwouldneverconveneintoonemass;butsomeofit[would]conveneintoonemassandsomeintoanother,soastomakeaninfinite177\nTIME,SPACEANDPHILOSOPHYnumberofgreatmassesscatteredatgreatdistancesfromoneanotherthroughoutallthatinfinitespace.(Newton1692)4Afiniteamountofmatterinaninfinitespacewouldeffectivelyamounttoatinyclusterofstarsinanimmensevoid;and,asNewtonrealised,thisclusterwouldtendtocollapseinuponitselfasaresultofthegravitationalattractionsbetweenthestars.Thepostulationofafinitespatialcontainerformatterwouldnothelp.Forthematerialinthisuniversewouldalsogravitateinwards.Thereseemedtobeonlyonestraightforwardresolution:aninfinitenumberofstarsmustbespreadreasonablyevenlythroughoutspacesothattherewouldbenocentreofattraction.Everystarwouldbepulledequallyinalldirections;andsoanequilibriumwouldbemaintained.Theonlydrawbacktothissolution—apartfromthepostulationofaninfiniteamountofmatter—istheconsequentlackofsensefortheconceptsofgravitationalpotentialandforce.Physicscanonlydealsensiblywithfinitequantities.Sincetherewouldbeaninfinitenumberofstarsspreadevenlythroughouttheuniverse,thepullonanygivenstarwouldbeinfiniteinalldirections.However,thisimpliesthatthepotentialatanypointandtheforceactingonabodyatthatpointcannotbedefined.Forwecangivewhatevervaluewechoosetotheresultof‘infinityminusinfinity’.ThisanomalyintheNewtonianperspectivewastackledinthemid-1890sbyNeumannandSeeliger.5InhisbookonNewtonianmechanics,Neumannremainedfaithfultotheideaofaninfinitespace,buthesawawaytopreventthegravitationalcollapseofaclusterofafinitenumberofstars.Heintroducedacosmologicaltermintotheequationforgravitationalpotential.Poisson’sequationforthepotentialwaschangedtowherelisthecosmologicalconstant.Theeffectofthecosmologicalterm—lfistobalancetheattractiveforceofgravitywitharepulsivecosmologicalforce.Sothepossibilityofgravitationalcollapseispreventedbytheexistenceofaforceofrepulsionactingoverlongdistances.178\nEINSTEIN’SGREATESTMISTAKE?Theideaofafiniteamountofmatterinaninfinitespacepresentedafurtherproblem.Suchauniverseapproximatestoasinglesysteminavastemptyspace.ThisisexactlythescenarioassumedbyNewtoninhisargumentinthePrinciplesofNaturalPhilosophyleadingtowardsthepostulationofspaceasanabsoluteentity.6AswehaveseeninChapter6,ErnstMach,inTheScienceofMechanics,respondsdismissivelytoNewton’sconceptofspaceasanabsoluteframeofreferenceformotion:hegoesrighttotheheartofNewton’sargumentandchallengesthebasicpresumptionthatourlawsmightoperateinanycounterfactualcircumstances.Hearguesinthatbookthatweshouldrefermotionstoamaterialframework—thatofthefixedstars.Heseesnocompellingreasonforustorelyuponmetaphysicalartefactslikeabsolutespacewhenallmotionsmightberelativetoaphysicalframeofreference.Theinertialforcesexperiencedbyacceleratingbodiesmightthenbetheresultofsomekindofinteractionwiththerestofthematterintheuniverse.However,Machoffersusnodetailedpositiveexplanationofhowsuchamaterialinteractionmightoperate;hestopsshortathisnegativecritiqueofNewton’sviews.Butthiscritiqueinvolvesanimplicitattackontheideaofunrestrictedabstraction.AlthoughMachemploystheprocessofabstractionhimself,hecannotbringhimselftoextendittowhathebelievesareabsurdconditions.Itisonethingtoneglectfrictionalorelectricalforceswhenmakingpredictions,buthethinksitisquiteanotherthingtoneglecttheentirecontentsoftheuniverse.However,asweobservedinChapter7,weneedaclearindicationofwherewemightdrawthelinebetweenpermissibleandoutlawedabstractions.Machfailsus,preferringpolemicalassertionstocarefulargument.So,atthebeginningofthetwentiethcentury,therewasnoclearreasonforanyphysicisttodismissNewton’sargumentforabsolutespaceoutofhand.AfinalproblemfacingEinsteinwasthatthereseemedtobearatherfar-fetchedpossibilitypresentedbyhisfieldequationsofgravity.Theyallowedforadynamicuniverse—auniverseinwhichspacetimeitselfmightexpandorcontract;butalltheastronomicalevidenceavailableintheearlytwentiethcenturysuggestedthatweinhabitastaticuniverse.Starsmightliveanddie,butthestructureofspacetimeseemedtobeunchanging—atleastglobally.AlthoughEinsteinhadlinkedthestructureofspaceandtimetothedistributionofmatterandenergyintheequationsofGTRviathemetrical179\nTIME,SPACEANDPHILOSOPHYnotionofcurvature,hesawnoreasonatalltodepartfromNewton’sbeliefthatthenatureofspaceasawholeiseternalandunchanging.EINSTEIN’SUNIVERSEEinstein,asheturnedtocosmologicalissues,wasthereforeconfrontedwithanumberofbeliefswhichseemedtobeconsistentwithNewtoniangravitationandwiththeavailableempiricalevidence:1Theuniverseisspatiallyinfinite.2Thereisafinitemattercontent.3Themattercontentisgloballystatic.4AsNeumanndemonstrated,acosmologicaltermmaybeintroducedintothegravitationalcontext.5Thematerialuniverseapproximatestoasinglesysteminanotherwiseemptyspace.6GivenNewton’sargumentinthePrincipia,thepresenceofinertialforcesinacceleratingbodiesandsystemsmustbeexplainedintermsofmotionrelativetospaceitself.Einstein’sapproachtothesesixstatementswasconstrainednotjustbytheempiricalevidencebutalsobyhislong-termregardforMach’santi-metaphysicalphilosophyofscience.Theavailableempiricalevidencesuggestedthatthematerialcontentoftheuniverseisbothfiniteandstatic.And,onceweacceptstatements2and3,weareleddirectlytostatement4:afinitebutgloballystaticmattercontentwouldcollapseinuponitself,soweneedtofindsomereasonwhysuchacollapsedoesnotseemtobehappening.Hence,wemayviewtheintroductionofthecosmologicaltermasapurelyempiricalmatter;andthisisso,regardlessofwhetherthecontextisNewtonianorrelativistic.HenceEinsteinwasinclined,likeNeumannandSeeligerbeforehim,toaddaterm—thecosmologicalconstant—tohisgravitationalequations.AsintheNewtoniancase,thevaluegiventotheconstantwassuchthatitseemstoactratherlikealong-distanceforceofrepulsion.Withinthecontextofrelativity,theadditionoftheconstantproducesasourcelessfieldwhichmaycounteractthe‘attractive’gravitationalfieldgeneratedbythemattercontent.Setattherightvalue,theconstantcouldproducethestaticglobalbalancedesiredbyEinstein.ButnowEinsteinwasfacedwithaproblem:toacceptstatement1aswellas2leadsusnaturallytostatement5andthento6.Einstein180\nEINSTEIN’SGREATESTMISTAKE?wasextremelyreluctanttoacceptstatement6.Todosowouldbetoembracespacetimeasanirreducibleelementinhisaccountofgravitation.TheequationsofGTRallowthepossibilityofafiniteamountofmatterinaninfinitespace,and,aswehavealreadynoted,insuchauniversespaceitselfisanirreducibleelementinourdescriptionsofmotion.Itisonethingtoexplaingravityingeometricaltermsandtodemonstratethatspacetimeisadynamicstructurewhichisinfluencedbythepresenceofmatterandenergy.Itisquiteanothertoadmitthatthestructureofspacetimehasafundamentalandirreduciblestatus.Forspacetimeasawholewouldbeabsoluteinthissense:allinertialforceshaveastheirultimatesourcenotmatterbutspacetime.ThisrancountertowhatEinsteincalls‘Mach’sPrinciple’inhis1918paper‘Principlesofgeneralrelativity’.7Thisprincipleimpliesthatanyinertialforcesexperiencedbyacceleratingbodiesandsystemsmusthavetheirsourceinaninteractionwiththematerialcontentsoftheuniverse.ThisisclearlyinaccordwithMach’sconvictionthatmatterandnotspaceisthesourceofinertialforces.Einstein,whohadreadTheScienceofMechanicsasayoungman,wasmuchimpressedbyMach’sreluctancetoembracetheconceptofabsolutespace.Hence,itwasnaturalforEinsteintochallengestatement6byattackingstatement1;andhedidthisbyconstructingaspacetimeconsistentwithhisequations,inwhichspacetimeisspatiallyclosed,i.e.inwhichspaceisfinite.The‘Einsteinuniverse’of1917isasphericallysymmetricuniversefilleduniformlywithmatter—ahomogeneous,vast,yetfinitesphere.ButthesameproblemwhichfacedNewtonstillconfrontedEinstein:ifthemattercontentisgloballystatic,thenthisspherewillcollapse.Thispropensitytocollapsemaybepreventedifthereissomerepulsiveforceatworktocountertheattractivegravitationaleffects.Andthisrepulsionmaybeprovidedbythecosmologicalterm.However,aswehaveseen,thequestionofcollapsealsoarisesinanystaticuniversewithoutaninfiniteamountofmatter.ItisnotaproblemwhichispeculiartoEinstein’sstaticuniverse.Ifwedonotassumethatthemattercontentisstatic,thenthereisnoimmediateneedforthecosmologicalterm.Althoughwemightacceptthegeneralpoint,madebyPaisandothers,thatEinstein’sstaticuniversewasmotivatedbyaMachiandesiretorepudiateabsolutespace,weshouldnotacceptthattheintroductionofthecosmologicaltermassuchintothegravitational181\nTIME,SPACEANDPHILOSOPHYphysicsofthisuniversewassimilarlyinspired.8Fourimportantfactorsneedtobeconsideredcarefullyhere:1Thepaperof1917inwhichEinsteinintroduceshisstaticuniversebeginsbyfocusingonpreciselythesameproblemasNeumannandSeeligerintheirworkof1895–7.92And,justasthesetwophysicistswerecompelledtoconfronttheproblemoftheextentofspaceintheNewtoniancontext,EinsteintoocouldnotescapethiscosmologicalproblemwithinGTR—andthiswouldhavebeenthecaseevenifhisbasicintuitionshadbeenanti-Machian.Hence,the1917paperdiscussesthisproblemindetailbeforemovingontohisworriesaboutthesourceofinertialforces.3WemustdistinguishcarefullybetweenthedesiretoestablishMach’sPrincipleandthedesiretoincorporatethecosmologicalconstantintoGTR.Einsteinneededtomakeaboldmovetofulfilthefirstdesire,sincetherewasnodirectempiricalevidenceatallforafinitespace.Butacceptedobservationsbackeduptheseconddesire:thestarsdidindeedseemtobegloballystaticandthereseemedtobeafinitenumberofthem.4Becausethecosmologicalconstantrepresentsasourcelessenergyfield,theconstantitselfishardlyaMachianartefact:Machwasstronglyopposedtoanyputativenon-materialphysicalentityorphenomenon.Sotheprimemotivationsfortheintroductionoftheconstantwere:1theavailableempiricalevidence;and2theneedtotackleNewton’sproblemabouttheextentofspaceanditscontents.Atworst,wemightsaythattheconstantisguiltyonlyofassociationwithMach’sPrinciple:forwiththestaticuniversenotonlydidEinsteinofferaresolutiontoNewton’sproblembuthealsobelievedthathehadfoundaneconomicalwayofrealisingMach’sdreamofrepudiatingspaceasanirreducibleelementinourdynamicaldescriptions.However,itsoonturnedoutthatananti-Machianuniverseincorporatingtheconstantcouldbeconstructed:deSitterfoundsuchasolutioninwhichthesourcemasscontentisnilandthereforeinwhichallmotionsmustbereferredto‘absolute’spacetimeitself.10ItisimportanttonotethatEinsteindidnotrejecttheideaoftheconstantimmediatelyafterde182\nEINSTEIN’SGREATESTMISTAKE?Sitter’santi-Machiandiscovery.Hadhedoneso,wemightthenhavesomereasontosupposethatthemotivationbehindtheconstantwasMachian.ThisgivesfurthersupporttotheargumentaboveinfavourofmakingadistinctionbetweenEinstein’sMachianmotivationsandhisbroaderempiricalintuitions.Nevertheless,theintroductionofthestaticuniversedoeshighlightacertaintensionamongstEinstein’scommitmentsandbeliefs.Ontheonehand,therearedefiniteMachianinfluences—ahighregardforempiricalevidenceandastrongdesireforsimplicityinourdescriptionsofthephysicalworld.11Butthereisalsoacommitmenttoa‘constructiveandspeculative’andmoreintuitivelytheoreticalapproachnottotallyconstrainedbyobservationalevidence.12Aswehaveseen,theconstantisclearlymotivatedbyempiricalconsiderations.ButitsintroductionhasimmediateeffectsonthestructureofGTR:thereisadefinitegainingeneralitywiththefieldequationsnowbeingabletogenerateamuchwiderrangeofsolutions—includingEinstein’sstaticuniverse;however,thereisacorrespondinglossinthemathematicalbeautyofthefieldequations—asimple,elegantpictureofgravitationisreplacedbyamorecomplexviewwhichaddsanon-materialenergyfieldtothetoolsweneedtodescribetheworld.13,14Furthermore,thisfieldisclearlyatheoreticalartefact,lackingasitdoesanyindependentobservationalbasis.Thisdoesnotmeanthattheconstantisanadhocaddition,butitdoesillustratethatEinsteinwaswillingtoembracetheoreticalentitiesdespiteMach’sinfluence.Anadhocchangeismostfrequentlytakentobeaspecificadjustmenttosaveatheoryinthefaceofsomethreatening(usually)observationalevidence.AsPopperpointsout,suchchangesinvariablyallowatheorytoescaperefutation.15However,GTRwasinnorealdangerofrefutationasfarasthequestionoftheglobalmotionsofthestarswereconcerned.Yes,Einsteindidhaveanempiricalmotivationforaddingtheconstant,butthiswasnotcontradictoryevidenceassuch.TherealproblemwasthatGTR’sequationsseemedtoleadtotheunwelcomepredictionofgravitationalcollapse.GTRfarednobetterthanNewtoniantheoryinthisrespect,anditwasnaturalforEinsteintoofferthesameremedyasNeumannandSeeliger.Hisreactionwastointroduceaconstant,thevalueofwhichwoulddeterminethelarge-scaledynamicsoftheuniverse.Theresultwasatheorywithincreasedscopewhichaddressedaveryrealproblem.IndeedboththeneedtoresolveNewton’sproblemandthegeneralitydeliveredbytheadditionoftheconstantprovidepersuasiveevidenceagainsttheclaimthatEinstein’smovewasadhoc.183\nTIME,SPACEANDPHILOSOPHYTHECOSMOLOGICALCONSTANT:DIDEINSTEINBLUNDER?Einsteinmayhaveconsideredtheconstantamistake.Butwasitsintroductionreallyablunder?ItisfrequentlysaidthathispreoccupationwiththeconstantstoppedhimfromexploringamuchmorerewardinganswertoNewton’sproblem.Althoughthestaticuniversepossessedahighempiricalpedigree,Einsteinhadshownthathewasimaginativeenoughnottobetiedtoconventionalwisdom.Hadhetakenabolderapproach,aswriterslikePaismaintain,thenEinsteinmighthaveanticipatedtheoreticallyHubble’scelebratedobservationaldiscoveriesin1929oftheglobalrecessionofgalaxies,thuspavingthewayforthehypothesisofanexpandingdynamicuniverse.Yet,howevermuchEinsteinmayhavewishedthathehadnotbeensopreoccupiedwiththeideaofthestaticuniverseandthathemighthavehadtheforesighttopredictanexpandinguniverse,tocitetheauthorityofEinstein’sownretrospectiveantagonismtotheconstantissurelyagoodexampleofasuperficiallypersuasivebutillicitappeal!AndhewassurelynotaloneinmissingthechanceofanticipatingHubble.Newtonmightsimilarlybeaccusedofaterrificblunderinnotofferingthepossibilityofanexpandingmaterialuniverseasawayoutofhisdilemma.Ifanything,theuncertainastronomicalevidenceaswellasthegeneralscientificclimateofNewton’sdaymightbesaidtoberathermoresupportiveoftheidea.Itmightbesaid,againstthis,thattheideaofadynamicexpandinguniverseismorenaturalwithinthecontextofEinstein’stheorythanwithinNewtoniangravity.ForEinstein’sconceptofthemetricofspacetimeisessentiallydynamic.However,theequationsofGTRfocusonthelocaldynamicsofthemetricofspacetime.Yes,wemaybuildupaglobaldynamicpicture—butwemayalsobuildagloballystaticuniverse.Withoutempiricalcosmologicalevidence,thereisnoreasonwhyanyparticularviewshoulddominate.AndsowemustrecallEinstein’scommitmenttoempiricalevidence.Itwouldhardlyhavebeenincharacterforhimtogoagainstthethenconvincingobservationalevidenceofafinitemattercontentandastaticuniverse.16Ofcourse,GTRmakessometremendouspredictions.ButweshouldnotethatthetwomajorpredictionsmadeintheearlydaysofGTRwererelated184\nEINSTEIN’SGREATESTMISTAKE?toempiricalproblemssetfirmlyintheNewtoniancontext.17ThecalculationsdeliveredbythefieldequationsforMercury’sorbitandforthecurvedpathoflightclosetotheSunwereinbothcasesvariationsonaNewtoniantheme:1ThefailureofNewton’sequationstogiveanaccurateaccountofMercury’sorbithadledtonumerousattemptstosavethetheorybyproducingoutofthetheoretician’shatahostofplausibleexplanations,includingLeVerrier’spostulationofanextraplanetclosetotheSun.2Newton’stheorytogetherwithacorpusculartheoryoflightcertainlypermitsthepredictionoflightbending.Einsteinhimselfhadshownin1911—wellbeforetheformulationofGTR—thatlightraysmustdeflectinagravitationalfieldgiventheconstancyofthevelocityoflightandtheprincipleofequivalence.ItisinterestingtonotethatEinstein’s1911predictionforthedeflectionofalightrayclosetotheSunisnumericallythesameasthatpredictedbyNewtoniantheory.OnlylaterdidEinsteincometorealisethatthecurvatureofspacetimewouldaffecttheamountofdeflection,andbytheendof1916Einsteinhadcalculatedadeflectiontwicethatofhisearlierprediction.18Hence,thereiseveryreasontosupposethatEinstein’simaginationwasconstrainedbyhisperceptionofthecontemporaryphysicalevidence.Andtoactentirelywithincharacterishardlyamistakewhenthatcharacterhadproducedsuchextraordinaryadvances.Wemightalsoarguethattheadditionoftheconstantwasinspiredinthesensethatitsettheclimateforfurtherresearchwhichwasimportantnotonlyforthedevelopmentof‘classical’GTRbutalsoforthedifficultproblemofunderstandinggravityintheextremeconditionsclosetosingularities.Ofcourse,weshouldnotdismissthefactthatcurrentobservationssetanupperlimitofabout10-32eVonthevalueoftheconstant.Evenmeasurementsdesignedtoconfirmthatthephotonhaszeromassonlyputanupperlimitonthatmassof10-16eV.Infact,thecosmologicalconstantisknowntobeclosertozerothananyotherphysicalquantity.Yet,althoughwemayappealtogaugeinvariancetosetthephotonmassatzero,thereisnosimilar185\nTIME,SPACEANDPHILOSOPHYtheoreticalreasonforustosettheconstantatzero.19Furthermore,considerthefollowingfivepointswhichindicatetheimportantroleplayedbytheconstant:1TheconstantundoubtedlyincreasesthegeneralityofthefieldequationsofGTR—byadjustingthevalueoftheconstant,weareabletoproducestatic,expanding,andcontractingsolutions.Giventheuncertaintyofastronomicalevidence,especiallyatthetimeEinsteinincorporatedtheconstant,thismustbeseenasadefiniteplus.WemightnotethatthepaperinwhichEinsteinintroducestheconstantdoesnotspecifyanyparticularvaluefortheconstant;hemerelydemandsthatitbe‘sufficientlysmall’tobe‘compatiblewiththefactsofexperiencederivedfromthesolarsystem’.202Themathematician,AlexanderFriedmann,capitalisedonthisflexibilitywhenhedevelopedthe‘Friedmann’modelsof1922–4whichgivetheoreticalrigourtothesevariouscosmologicalpossibilities.21Thesemodels,welcomedbyEinsteininabriefnotein1923,havebeenofvitalimportanceindevelopingourviewoftheuniverse’soriginsandglobalstructure.223DespiteEinstein’sdismissaloftheconstantin1931andthelackofanydirectobservationalevidenceforanon-zerovaluefortheconstant,itsimplyrefusedtodisappearandlikea‘mischievousgenie’continuallyreturnedtohauntcosmologists.23TheconstantappearsinnumeroustheoreticallyimportantsolutionsofGTR,suchasGödel’ssolution,whichraisesthepossibilityoftraveltothepastviaclosedcausalloops,andtheOszvàthandSchückingmodel,aswellasintheearlierFriedmannmodels.244TheconstanthasalsobeenusedtohelpphysicistsinvestigatepossiblelinksbetweenGTRandelectromagnetism;see,forexample,Weyl’spaperof1918aswellasmuchmorerecentworkinquantumgravity.255Indeed,manycosmologistshavecometoregardtheconstantasanindispensableelementintheirdescriptionoftheuniverseatearlytimes.Inquantumfieldtheories,avacuumisdefinedasthelowestpossibleenergydensity;sophysicistsareabletocontemplateavacuumwhichisemptyofregular‘source’particlesbutnotoffields!Andsuchfieldswillhavepreciselythesamesourcelesscharacterasthatproducedbythecosmologicalconstant.26Itisevensuggested,inthepowerful‘inflationary’modelsoftheuniverse,thattherapid,earlyexpansionoftheuniversemayhavebeenproducedfromavacuumenergyinvolvingaveryhighvaluefortheconstant.27186\nEINSTEIN’SGREATESTMISTAKE?AllofthesefactorsindicatethattheconstanthasplayedafarfromnegligibleroleinthedevelopmentofbothclassicalandquantumGTR,stimulatingratherthanhinderingprogress.LAWSANDTHEORETICALCHANGEIbelievethatthetendencytoregardthecosmologicalconstantasanadhocmistakederivesfromamisplacedreverenceforfundamentallaws.Ifwetreatafundamentallawastheoneelementofatheorywhichisimmunetorevision,thenanysuggestedchange,nomatterhowwell-motivated,willberegardedasadhoc:theonlychangestobetoleratedarethosewhichbringrevolutionarynewtheorieswithradicallynewlaws.However,mostscientifictheoriesundergoalmostcontinualscrutiny.Scientistsareinclinedtoreviewnotonlythewayinwhichthetheorymaybeappliedbutalsothebasicideasofthetheory.Theirreviewsarenotnecessarilythepreludetoacrisisofconfidenceinthetheory;morefrequentlytheyareattemptstoextendthescopeofthetheory.Suchextensionsmaybeachievedeitherbymodifyingthelawsthemselvesorbyaddingtothetheoreticalcontextnewbutrelatedlawsassupplementstotheoriginalequations.Thecaseofthecosmologicalconstantshowsjusthowwillingscientistsaretodothis;buttherearemany,manyotherpossibleillustrationsofthis;forexample:1Newcomb’ssuggestionthatwemayonlyaccountfortheevidenceofplanetaryorbitsbyrecognisingthattheinversesquarerelationshipofgravitationisonlyapproximateandthatveryslightcorrectionsmustbemade;2themultiplicityofformulationsoftheequationsofquantumtheory;3thechangeswhichelectromagnetismrequiredforsoundmicroscopicdescriptionsofsuchphenomenaasthatofdispersion;and4theadjustmentsmadetothelawsofclassicalmechanicstoallowthemtodealwithfarmorethanmasspoints,rigidbodies,perfectfluids,andsmoothmotions.Hence,thereisnothingunusualorhereticalinEinstein’sdecisiontoaddtheconstanttohisfieldequations,therebychangingtheformandcontentoftheoriginallawsof1915–16.Lawsarenot,afterall,untouchables.Theyaresetwithinachangingtheoreticalcontextandtheythemselvesmaybemodifiedwithoutdoingirreparabledamagetothetheoryitself.Fortheoriesarecomplexconstructionsofwhichlawsareonlyapart—but,granted,animportantpart.GTR,forexample,is187\nTIME,SPACEANDPHILOSOPHYanintricate,interlockingcontextincluding:fundamentallaws,empiricalandphilosophicalprinciples,derivedtheoreticalandexperimentalequations,localandcosmologicalsolutions,fundamentalcommitmentstogeometricalandotherentities,models,analogies,andphysicalfactsandconstraints,togetherwithmanylinkstootherphysicaltheories;weshallexplorethiscontextingreaterdetailinthefinalchapter.28Iftheoreticallawscould,eveninprinciple,deliverthetruthaboutthephysicalworld,wemighttakealesstolerantlineonusingarangeoflawswithinthesametheoreticalcontext.Forwewouldthenhaveavestedinterestinmatchingapreferredlawtotherealworld.However,NancyCartwrightarguesthattheoreticallawscannotbestrictlytrueinanyrealcircumstancesincesuchlawsinvariablydealwithabstractsituationswhichcannotbefoundinanyempiricalcontext.29Suchlawsmaybeaccuratewithintheiridealiseddomains,buttheymayonlyserveasguidestowhathappensinrealisticsituations.HawkingandEllisprovideausefulsummaryofboththepowerandthelimitationsofthefieldequationsofGTR:Becauseofthecomplexityofthefieldequations,onecannotfindexactsolutionsexceptinspacesofratherhighsymmetry.Exactsolutionsarealsoidealizedinthatanyregionofspacetimeislikelytocontainmanyformsofmatter,whileonecanobtainexactsolutionsonlyforrathersimplemattercontent.Nevertheless,exactsolutionsgiveanideaofthequalitativefeaturesthatcanariseingeneralrelativity,andsoofthepossiblepropertiesofrealisticsolutionsofthefieldequations.(HawkingandEllis1973:117)30Hence,thefieldequationsapplydirectlyonlytohighlysymmetricalandsimplifiedabstractcontexts.Anadequatecharacterisationofanyrealisticsituationwillrequirealessabstract,less‘exact’approach,butonewhichwillhaveagreaterchanceofdirectapplicabilitytosuchasituation.Realsituationsdemandamoreflexibleapproach,drawingonperhapsseveraltheoreticalperspectivestoproduceanexperimentalortheoreticalconcoctionwithwhichwemaycharacterisethecontextunderexaminationmoreprecisely.Suchmanoeuvresareuseddailyincontextsfromtheproblemsofexperimentalparticlephysicstothetheoreticalbehaviourofsingularities.Hence,thereisevenlessreasonforustostickrigidlytoasinglelaworsetoflawswhichsomemaynarrowlyviewasthelaworlawsofagiventheory.Andthereisallthemorereasonforus188\nEINSTEIN’SGREATESTMISTAKE?toexploitwhateveradvantagesabroadanddynamictheoreticalcontextmightoffer.ThecosmologicalconstantenrichesthetheoreticalcontextofGTR.Themaindrawbackisthelackofdirectobservationalevidenceforanon-zerovaluefortheconstantinthepost-‘bigbang’era.Wemayhavestrongreservationsaboutthelikelihoodofthesourcelessfieldimpliedbytheconstant,butassertionsabouttheadhoccharacteroftheconstantareneitheraccuratenorhelpful.Fortheydeflectusfromgivingfullcredittoatheoreticalrolewhichcertainlyjustifiesusinregardingtheconstantasausefuland,oncurrentevidence,stillapotentiallyfruitfuladditiontoGTR.THEANTHROPICPRINCIPLEThefactthattheobservedpresentvalueforthecosmologicalconstantissoclosetozeroandthefactthatsignificantpositiveornegativevaluesfortheconstantwouldresultinuniversesnotfitforhumanhabitationhaveledsomewriterstosuggestthatthevalueoftheconstantislowsimplybecauseithastobelowforlifetodevelop,andtheygoontoproposeaprincipleofanthropocentricity.31Anthropicprinciplesfocusourattentionongoodcosmicbehaviour—where‘good’isdefinedintermsofwhatisconsistentwithlifeintheuniverse.Therearetwomainversionsoftheanthropicprinciple:strongandweak.Eachprincipleassertsalinkbetweenthephysicsoftheuniverseandtheconditionswhichmustobtainforlifetobepossible.BrandonCarter,whoseideasaboutanthropocentricitystimulatedthislatestroundofdebateabouttheroleofpeopleintheuniverse,characterisestheStrongAnthropicPrinciple(SAP)asfollows:SAP‘Theuniversemustbesuchastoadmitthecreationofobserverswithinitatsomestage’ofitsevolution.32TherationaleforSAPdependsinpartupontheimplicationsofquantummechanicsforthestatusoftheobserver,inpartonthedimensionalityofspacetime,inpartonthepossibilityofmultipleworldswithinanall-encompassinguniverse,andinpartuponanumberof‘coincidences’.33Thesecoincidencesturnonthevaluesoffive‘fundamental’parameters189\nTIME,SPACEANDPHILOSOPHYinvolvedintheevolutionofthephysicaluniverse,includingthecosmologicalconstant.34Ifthecosmologicalconstantandtheotherparameterswerenotveryclosetotheirobservedvalues,thentheuniversewouldhaveeitherrecollapsedveryearlyinitshistoryorexpandedsorapidlythatgalaxiescouldnothaveformed—inbothcaseslifecouldnothaveevolved,fortheevolutionoflifeneedstime,theavailabilityofcomplexatoms,andreadilyaccessibleenergysources.Soafine-tunedcosmicbalanceisneededforlifetobepossible.Isitmerelyacoincidencethatthecosmologicalconstantandotherimportantphysicalparametershavethevaluestheydo?Ormayweseekanexplanationintermsoflifeitself?SupportersofSAPattempttoseduceuswithtwoappeals:first,weareaskedtofocusonthisuniverseinanarrowway—asitappearstobeaccordingtoourcurrentobservations;andthenweareaskedtodismisstheideathatobservedconditionsandvaluescouldbemerecoincidencesandthatlifeisthereforeanaccidentalfeatureoftheuniverse.SAPistypicallyjustifiedbydefactoappealstotheactualuniverseanditsstructures:weonlyhaveoneuniverse;iftheactualconditionsandvaluesforphysicalconstantswerenotclosetotheobservedconditionsandvalues,thenlifewouldnotbepossible;observersexistinthisuniverse;itishardtoexplainwhytheconditionsandvaluesareastheyarewithoutreferencetotheexistenceofobservers;hence,theobserverisanessentialpartoftheuniverseasgiven.PaulDaviesclaimsthatSAPismotivatedbypositivismbecause,hesays,itboilsdowntothedemandthat‘onlythatwhichisperceivedenjoystruereality’withtheconsequencethat‘auniversewhichdidnotadmitobserversismeaningless’.35Althoughthereisanelementofpositivisminthedesiretocharacterisetheworldasitappearstobeinobservation,SAPtakesastepwhichnogenuinepositivistcouldcondone:physicalnecessityisinvokedtoexplaintheconnectionbetweenlifeandtheobservedconditions;asJohnEarmanhaspointedout,SAPmakesamovefrom‘is’to‘must’,fromhowthingsaretohowthingsmustbe.36Positiviststypicallyremaincontentwiththelevelofmeredescription.Therecanbenonecessitywhichattachestotheuniverseorprocessesandeventswithintheuniverse.Tomakeclaimsabouthowthingsmustbe,whichistheessenceofSAP,is,forthepositivist,toengageinmeaninglessmetaphysicalspeculation.Hence,SAPseemstobemotivatedmorebythedesiretoprovideafirmmetaphysicalfoundationforcosmic190\nEINSTEIN’SGREATESTMISTAKE?coincidence.Indeed,somesupportersofSAPengageinwhatmightseemtoberatherexoticmetaphysicalspeculationwhentheyarguethat(a)theuniverseconsistsinacollectionof‘worlds’whichtogetherrepresentallphysicallypossibleuniverses;(b)theuniverseweinhabitisreallyoneofmanyworldswithinthislargeruniverse;and(c)ourownworldmustexistandthereforelifemustexist,becausethisonemustbepresentamongstthecollectionofpossibleworlds.Althoughsomecosmologistsfavoursuchaworlds-within-worldscenarioconsistentwithinflationarycosmology,thereislittleevidencetosuggestthatthisviewshouldbetakentooseriously.37Forthesereasons,SAPisahighlyspeculativeideatowhichfewcosmologistswouldwishtosubscribe.Itcarrieswithitperhapstoomanyegocentricovertonesformostcosmologists:‘theuniverseexistssothatwemightexist’isalittletoomuchtoswallow,evenifasaresultweareleftwithonegiganticsetofcosmiccoincidences.Whyshouldwehavetoexplaineverycoincidence?And,moreproblematically,shouldwebesatisfiedwithan‘explanation’whichspellsouttheconditionsinanddevelopmentoftheuniverseintermsofourexistence—notbyprovidingsomestraightforwardmechanismlinkingourexistencetotheactualconditionsanddevelopment,butbyassertingthelinkasan‘irreducible’fact?AsPaulDaviespointsout:Fromthestrictlyphysicalpointofviewitseemsmysterious,tosaytheleast,thattheexistenceofconsciousbeingscanactuallybringaboutthecelebratedcoincidences.Clearlyanydirectcausalconnectionisimpossible.Specialphysicalconditionsmayproduceman,butmancanhardlybeattributedthecreditforestablishinghisownenvironmentalrequirements.(Davies1982:122)Theweakerprincipleislesscontroversialandmorewidelyaccepted,foritacknowledgesthefinebalancerequiredforlifetoexist,butdoesnotseektoimposethisbalanceontheuniverseasamatterofnecessity.StephenHawkingstatestheWeakAnthropicPrincipleasfollows:WAP‘Intelligentlifecanexistonlyincertainregionsofagivenuniversewithgivenphysicallaws.’38191\nTIME,SPACEANDPHILOSOPHYThisprincipleisbasedonCarter’sassertionthat‘whatwecanexpecttoobservemustberestrictedbytheconditionsnecessaryforourpresenceasobservers’.39WAPdoesnotforceustoexplainthelawsandconditionsintheuniverseintermsoflifeandconsciousness;theexplanationshouldbetheotherwayround:lifeandconsciousnessdependuponspecificconditionsandlawsholdingatleastincertainregionsofspacetime.Hence,thefactthattheobservedvalueofthecosmologicalconstantiswhatitismayhaveacentralpartintheexplanationofwhylifeispossibleinthisuniverse.CartermaintainsthattheessentialfunctionofWAPistoalertusto‘theriskoferrorintheinterpretationofastronomicalandcosmologicalinformationunlessdueaccountistakenofthebiologicalrestraintsunderwhichtheinformationwasacquired’.40Thecosmologicalconstantisjustoneiteminalistofpotentialproblemsforthewell-behaveduniverse,someofwhichmightadverselyinfluencethepossibilitiesforintelligentlifeandotherswhichmightsimplyinterferewithourabilitytomakepredictionsaboutphysicalphenomena.Blackholes,thebigbang,andtopologicalanomaliesinspacetimeallpresentproblemsforGTR.Althoughthefocusofthenextchapterwillbeonthebeginningoftheuniverse,weshallalsoreviewthecausalproblemsassociatedwithblackholesandother‘singularities’inspacetime.Indoingso,weshallobtainafarmoredetailedvisionofthecomplexityandrangeofGTR’stheoreticalcontext.192\n10COSMOLOGICALCONUNDRUMSINTRODUCTIONTheNewtonianuniversehasabeginningintimeformatter,butnotfortimeitself.Godissupposedtohavecreatedthematerialworld,probablyinstages,startingatsomedefinitepointinthepast.Beforematterappeared,therewasamaterialvoid:anemptyandeternalspatialarena.WemightasktheNewtonian:whathappenedbeforecreation?Forthecreationisamomentinatemporalframeworkwhichstretchesinfinitelytothepastandtothefuture.God’seternalcharacteristypicallyrelatedtothisinfinitespanfortime.Space,time,andGodallexistedbeforethecreation.Godchoseaparticularmomenttocreatethematerialuniverse;andGod,liketimeandspace,willpresumablycontinuetoexistaftertheendofthematerialuniverse.WithoutGod,theappearanceofmatteratagiventimemightseemsomewhatmysterious.But,ofcourse,anyreferencetoGodascreatorsimplyreplacesonemysterybyanother.LeibnizfoundmuchaboutthisNewtonianviewobjectionable.ThemaindifficultyhighlightedbyLeibnizisthis:whydidGodcreatethematerialuniverseatonetimeratherthananother?WhatpossiblereasoncouldGodhavehadforchoosingoneinstantoutofeternityoveranyothertime?Asinthesimilarproblem,discussedinChapter5,abouttheindiscernibilityoftwouniverseswhicharedistinctonlyintheirrespectivespatialpositions,LeibnizappealstothePrincipleofSufficientReason:Inthingsabsolutelyindifferent,thereisnochoice;andconsequentlynoelection,norwill;sincechoicemustbefoundedonsomereason,orprinciple.Amerewillwithoutanymotive,193\nTIME,SPACEANDPHILOSOPHYisafiction,notonlycontrarytoGod’sperfection,butalsochimericalandcontradictory….SinceGoddoesnothingwithoutreason,andnoreasoncanbegivenwhyhedidnotcreatetheworldsooner;itwillfollow,eitherthathecreatednothingatall,orthathecreatedtheworldbeforeanyassignabletime,thatistheworldiseternal.Leibniz(1716)1SoLeibnizarguesthat,ifGodcreatesmatteratall,thentheNewtonianmustacceptthattherehasalwaysbeenmatterandgiveupthebeliefthatGodcreatedthematerialworldinapre-existingspatialarena.ButthismovepresentsobvioustheologicaldifficultiesforsomeonewhowishestoacceptsomethinglikethefreecreatorofGenesis:ifmatteriseternal,thennotonlyisthecreationstoryofGenesisexplicitlyattacked,buttheexistenceofmatterbecomesanecessaryfeatureofGod’splan,ratherthansomethingwhichiscontingentuponGod’swill.LeibnizbelievesthattheNewtonianmustthereforeabandonhisposition.Clarke,indefenceoftheNewtonianview,appealstoGod’seternalpointofviewandtohisinscrutabledesign:ItwasnoimpossibilityforGodtomaketheworldsoonerorlaterthanhedid:norisitatallimpossibleforhimtodestroyitsoonerorlaterthanitshallactuallybedestroyed.Astothenotionoftheworld’seternity;theywhosupposematterandspacetobethesame,mustindeedsupposetheworldtobenotonlyinfiniteandeternal,butnecessarilyso.2…ButtheywhobelievethatGodcreatedmatterinwhatquantity,andatwhatparticulartime,andinwhatparticularspaceshepleased,arehereundernodifficulty.ForthewisdomofGodmayhaveverygoodreasonsforcreatingthisworld,atthatparticulartimehedid;andhemayhavemadeotherkindsofthingsbeforethismaterialworldbegan,andmaymakeotherkindsofthingsafterthisworldisdestroyed.Clarke(1716)3So,Clarkeasks,whoarewetoquestionhowGodactsandwhenhechoosestoact?LeibnizbelievesthatthisresponseisinadequateanddoesnotdojusticetoGod’sinherentlyperfect,rationalwill.RatherthanadmitthatGodmayhavehisownreasonsforacting—reasonswhichareinaccessibletothefinitehumanmind,Leibnizarguesthatspaceandtimearemereappearances.Wedonotseerealityasitis.Weviewreality194\nCOSMOLOGICALCONUNDRUMSasaspatial,temporal,andmaterialdomain.Butthisisnotitsrealnature.Realityistherealmofmonads:thebasicmetaphysicalstuffwhichformsthebasisofallcreation.Spatial,temporal,andmaterialpropertiesarenotessentialfeaturesofmonadsthemselves,theyarefeaturesofthewayweexperiencemonads.HencecreationforLeibnizisnotcreationintime,itiscreationofallthatgivesrisetoexperience,includingtime.Fromahumanpointofview,Godactstocreateallmatter,space,andtime‘allatonce’.ItisasifGodsculptssomevastfour-dimensionalimage,atotalhistoryoftheuniverse,inasingledivine‘act’.Clearly,thisactcannotbethekindofactionwhichyouorIperform,sincesuchactionsalwaystakeplaceintime.Theactofcreationhereinvolvesthecreationoftimeitself.DoesLeibniz’ssuggestionmakesense?Ifwesetasidetheobscuremetaphysics,wemightidentifythecentralideamoreclearly:space,time,andmattercomeasapackagedeal.ThefundamentallinkbetweenmatterandspacetimeinGTRmightprovidesomesupportfortheessentialideahere.Whateverwetaketobecreationisthecreationofeverything,includingtimeitself.Ifweinsistonthinkingintermsofanactualcreator,thenthatcreatoristranscendentor‘outside’timeandspace,i.e.hasacharacterwhichisneitherspatialnortemporal.So,fromsuchacreator’spointofview,itismoreappropriatetothinkofthecreationofallhistoryinasingle‘act’ratherthanofthecreationofthematerialworldinaspatialarenaatagiventime.Ifwearetoappreciatejusthowsensibletheseideasmightbewithinthecontextofmoderncosmology,thenwemustreviewtheclaimsmadebycosmologistsaboutthebigbang—theinitialsingularity—sometimessaidtobethebeginningoftimeitself.Andtheproblemofthebigbangwillthenleadustotworelatedcosmologicalconundrums:1Wheredidthemattercontentoftheuniversecomefrom?Shouldweacceptthecentralclaimofrecentworkininflationarycosmologythatitscreationwasliterallyfromnothing?2Cansingularitieslikeblackholesbringcausaldisruptionandindeterminismtotherestofspacetime?Orareweallowedtoadvocatethecosmiccensorshiphypothesisanddrawanimpenetrablecloakaroundeverysingularity?Althoughfewcosmologistsdaretoofferfinalanswerstosuchproblems,byexploringtheirideas,wemayneverthelessobtainaclearerviewofthequestionswhichweshouldbeaskingaboutthelarge-scalestructure195\nTIME,SPACEANDPHILOSOPHYandhistoryofspacetime.Andwemightalsolearnwhysomequestionsjustdonotmakesense.THEBIGBANG:ASINGULARIDEAThestandard‘bigbang’viewofmoderncosmologystartsfromthreebasicassumptions:1TheFriedmannmodelsofGTRareessentiallycorrectatleastintheirlarge-scalecharacterisationoftheuniverse.Theuniverseishomogeneousandisotropic:thesameeverywhereandineverydirection.AswenotedinChapter3,thisassumptionhasahighempiricalpedigree.2Eachclusterofgalaxiesseemstobemovingawayfromeveryotherclusterofgalaxies:sothattheuniverseis‘expanding’onaAsspacetimeexpands,thematerialintheuniversebecomeslesscompressed.Thetwosequencesaboveillustratetheexpansionofaspatiallyclosedsphericaluniverse(ratherlikeaballoonexpanding)andtheexpansionofaspatiallyopenplane(ofwhichonlypartisshownhere)Figure33Expandingsphereandplane196\nCOSMOLOGICALCONUNDRUMSgrandscale.ThefieldequationsofGTRtiethegeometryofspacetimetothedistributionofmatter.So,intheFriedmannmodelsoftheuniverse,thematterdistributionisexpandingbecausespacetimeisexpanding.Cosmologistsfrequentlycitetheanalogyofanexpandingballoontoillustratethegeneralideainvolved:imagineaballoonwithspotsspreadevenly(homogeneously)overitssurface;astheballoonexpands,thespots‘move’furtherapart;seeFigure33(p.196).Thespotsactlikeclustersofgalaxies,andtheballoonlikethestructureofspacetime.Theexpansionisofspacetimeitself.3Clearly,ifspacetimeisexpanding,thenatsometimeinthepastspacetimeandmattermusthavebeentremendouslycompressed.Matterinsuchacondensedformwouldhavearatherdifferentcharactercomparedwiththematteraroundusnow:eventhesimplestatomswouldbebrokendownintotheirfundamentalpartsinaveryhotandextremelydense‘radiationsoup’,farmoreactiveandviolentthanthehotteststar.Evidenceforthisassumptionwasprovidedin1965byArnoPenziasandRobertWilson,whofoundthe‘microwavebackground’:thecosmicremnantsofthisearly,radiation-dominatedphaseoftheuniverse;seeFigure34(p.198).4SoGTR,togetherwithsomeveryplausibleempiricalassumptions,leadsusdirectlytothehypothesisthatthematter(orradiation)contentoftheuniversebecomesmoreandmoretightlycompressedaswelookfurtherandfurtherbackintothepast.Theearlieststageofallmaybecharacterisedastheinitialsingularity:allmaterialcompressedintonospaceatall—a‘point’ofinfinitedensityandthereforewithinfinitecurvature.5Theexpansionofmaterial(andthereforespacetime)awayfromtheinitialsingularityisrapidandexplosive—hencetheideaofabigbang.Amoreprecisedefinitionmaybegiventotheideaofasingularityusingtheconceptofageodesic.Theworldlineofafreelymovingobjectiscalledageodesic,andthemotionoftheobjectmaybedescribedbyits‘affineparameter’.Iftherearenosingularitiesinspacewhichmightmarkthe‘end-points’ofgeodesies,thenwewouldexpecttheaffineparameterwhichdescribesthegeodesictoassumeallvaluesfromplustominusinfinity,thussignifyingthatthegeodesicgoeson‘forever’.Ageodesicallyincompletespacetimeisoneinwhichatleastonegeodesiccomestoahalt.Thegeodesicwouldreachan‘end-point’197\nTIME,SPACEANDPHILOSOPHYAnextremelybriefhistoryoftheuniverse:Note:nottoscale!Figure34Backtothebigbangifspacetimecomestoanend.Andsingularitiesare‘places’wherethiscanhappen.Wecandrawananalogybetweenphysicalsingularitiesandtopologicalholestomakethisideaclearer.Ifaspacetimeweretohavesetsofpointsliterallycutoutfromitproducingatopologicalholeinspacetime,thenanygeodesicarrivingattheexcisedregionwouldhavenowheretogo.Thetopologicalholewouldbeaterminusforsomegeodesiesandastarting-pointforothers.Aphysicalsingularitybehavesinessentiallythesamewayasfarasgeodesiesareconcerned—thesingularityprovidesadefiniteterminalboundaryforgeodesies;seeFigure35(p.199).Hence,wemaydefineasingularityasaphysicalphenomenonwhichproducesageodesicallyincompletespacetime.6Somesingularities,suchasblackholes,maybe198\nCOSMOLOGICALCONUNDRUMSFigure35Geodesiesmeetingasingularitythroughoutspacetime.Buttheinitialsingularityisafixedterminusforallspacetime.THEBEGINNINGOFTIME?Itmightseemverytemptingtoaskatthispoint:whathappenedbeforethebigbang?Threemainresponseshavebeengiventothisquestion:1Thefirst(andnowstandard)responseistosaythatthequestiondoesnotmakesense!Sincespacetimebeginsattheinitialsingularity,thereisnosenseintheideaoftimebeforethesingularity.Foralltimeisbydefinitioncontainedwithinaspacetimewhichis‘thisside’ofthesingularity.Itjustdoesnotmakesensetotalkabout‘theotherside’oftheinitialsingularity,forourcharacterisationofspacetimeissuchthattheresimplyisnootherside,noearliertime.Similarly,ifthereisafinalsingularity,whenallmatterandspacetimecollapsesina‘bigcrunch’,therewouldbenosenseintheideaofatime‘later’thanthefinalsingularity.Ifweaccepttheideaofaninitialsingularity,whatwecannotsayisthattherewasanyearliertime.Similarly,thegeneralideaofspacetimepreventsusfromgivinganysensibleanswertothequestion:whatisspacetimecontainedin?Fortheideaofcontainmentisitselfspatial.Wecannotinventhigherspatialdimensionsinwhichtolocatespacetime.Theonlyspacewehaveisalreadygivenfullyinspacetime.Soallwehaveisspacetime,andthereisnosenseintheideasofatimebeforeoraspacebeyond.2However,wemightarguethereisnogenuineinitialsingularity,thattherapidexpansionoftheuniversefollowsimmediatelyafteramassivecollapse,andthattheuniverse‘bounces’fromcollapsetoexpansionwhensomeminimumvolumeisreached.Whathappenedbeforethe199\nTIME,SPACEANDPHILOSOPHYbigbangwouldthenbeanintelligiblequestion.Wemightsay,forexample,thatthepresentexpansionisjustonephaseinauniversewhichhasexistedalreadyforaninfinitetime,thattheuniversecontinuallyexpandsandcollapsesoverandoveradinfinitum.Orwemightbemorecautiousandsaythatthereweretimesearlierthanthebigbang,butthatallwemaysayabouttheuniversebeforethebigbangisthatitwasinastateofcollapsejustbeforethebigbang.3Wemightalsoaccepttheideaofoverallexpansionbutdenythattherewasanyinitialhighlycompressedstate.Thesteady-statetheory,popularuntilthediscoveryofthemicrowavebackground,askedustoacceptauniverseinwhichmatterisbeingcreatedcontinuouslythroughouttheuniverse,inastableanduniformmanner.Thesteady-statemodeldevelopedbyBondi,Gold,andHoylewasbasedinpartonthestaticuniversedevisedbydeSitterin1917.7Theexpansioninthesteady-statemodeliscausedbyarepulsiveforcelinkedwiththecosmologicalconstant.Inthisviewtoothereisnofirstmoment:thecreationofmatterhasalwayshappenedandwillalwayshappen.Butonthisviewthereisnoreasontosupposethattherewillbeanywidespreadremnantsofanearlyphaseoftheuniverse,dominatedbyradiation.Hence,itwasdroppedlikeahotpotatobymostcosmologistsassoonasPenziasandWilsonfoundthemicrowavebackground.Thefirstresponse,forallitsaudacity,isthemostconservativeofthethree.Forwehavenofirmevidencetosuggestthattheuniversehasever‘bounced’fromcollapsetoexpansion.Indeed,itishardtoimaginewhatkindofobservationalevidencemightindicateabounceratherthanasingularity.Noristhereanyconvincingempiricalevidencefortheideaofcontinuouscreationdespitethefeverish,adhocadjustmentsmadetothesteady-statetheorybyitssupportersintheirattemptstomakeitmoreacceptableafterthediscoveryofthemicrowavebackground.Whyshouldanyonewishtoresisttheideaofauniverseevolvingfromaninitialsingularity?Thismaybebecausetheideaofauniversewhichisinfiniteintime,pastaswellasfuture,hasapleasingsymmetry.Thestandardcosmologicalviewinvolvesauniverseexpandingfromaninitial‘boundary’forever.Thefutureisindeedinfinite.Butthepastisfinite.Althoughtherearesuggestionsthattheuniversecouldcollapsein200\nCOSMOLOGICALCONUNDRUMSonitself,thereisasyetnostrongobservationalevidencetosupportthisview,aswenotedinChapter3.Hencetheuniverseseemstobecuriouslyasymmetrical.However,evenifthereissufficientmatterintheuniversetoinduceaglobalcollapse,themostlikelyresultwouldbeauniversewhichisfiniteintime,futureaswellaspast.So,evenwithsingularities,theuniversecouldstillexhibitaglobalsymmetryintime.Somemightdecidetofavourthesecondandthirdresponsesbecauseoftheimplicitrefusaltocontemplatea‘first’momentintime.Theideaofafirstmomentintimecarrieswithit,insomeeyes,thesuggestionofacreation.Soanalternativeviewoftheuniverse,involvingnosuchfirstmoment,mightbringwithittheadvantageofnocreationandsonocreator.Butthismanoeuvrefails:forwecanstillimagineafour-dimensionaluniversebeingcreatedasapackagedealalongthelinessuggestedbyLeibniz.ItmaybeanunnecessaryextravagancetothinkoftheuniverseascreatedbyGod.Theworldmightsimplyexistandrequirenosuchjustification.8ButnothingpreventsbelieversinGodfromaddingtheclaim‘andallspacetimewascreatedinasingledivineact’totheirotherbeliefs.Sothereseemslittlereasontoresisttheideaofafirstmomentjustbecauseofsomegeneralantagonismtothepossibilityofadivinecreation.AmoreplausiblereasontopreferthesecondorthirdresponsemightinvolvetherecognitionthatGTR,asaclassicaltheoryofspacetime,justcannotdealwitheitherthephysicalinfinitieswhichariseatsingularitiesandthequantumeffectswhicharelikelytooccurintheirneighbourhood.StephenHawkingdescribestheproblemasfollows:workonsingularitiesinthe1960sandtheearly1970sproducedanumberofsingularitytheoremswhichshowedthatifclassicalgeneralrelativitywerecorrect,therewouldinevitablybeasingularityatwhichallphysicallawswouldbreakdown.Thusclassicalcosmologypredictsitsowndownfall.Inordertodeterminehowtheclassicalevolutionoftheuniversebeganonehastoappealtoquantumcosmologyandstudytheearlyquantumera.(Hawking1987:632)9Sosomemightbeinclinedtolookforacosmologicalaccountwhich201\nTIME,SPACEANDPHILOSOPHYdoesnotinvolvesingularitiesandthereforedoesnotleadtothebreakdownofclassicalrelativity.Butitseemsabsurdtoexcludetheinitialsingularitywhenothersingularities,includingblackholes,areanessentialelementinourgeneralaccountoftheuniverse.Blackholes,likeallsingularities,arelikelytoinvolvequantumeffectsinthehot,extremelydenseenvironmentwhichtheyproduce.And,liketheinitialsingularity,blackholesarepredictedbyGTR.So,howeverproblematicsingularitiesmightseemtobe,wehavenoreasontotrytoavoiddealingwiththequantumeffectswhichareassociatedwiththem.However,wemightstillbeperplexedbytheideaofaphysicalsingularitywithinspacetime:foritimpliesthatthereexistsanactualinfinitequantityinagivenlocation.Canweacceptthatcosmologymightinvolvequantumeffects,thattheuniversereallyisevolvingfromextremelyhot,denseinitialconditions,andthatthereisnoprevious‘pre-bounce’universe,butalsosaythatthereisnoinitialsingularityassuchinoratleastontheedgeofspacetimeitself?StephenHawkingandJimHartlesuggestawayforustodojustthat.Theyaskustoaccepttheideaofaninitiallyhighlycompressedstateanddenythepossibilityofanactualfirstmoment.10Theirexplanationisbasedonthefactthatspacetimeneednotbeclosedattheinitialsingularity:itmaybeopeninstead.Theideathatasingularityprovidesastarting-(orfinishing-)pointforageodesicrequiresustoconsiderthesingularityasaphysicalentityofsomesort.Theinitialsingularitymaythenbethoughtofasprovidingadefiniteboundarytospacetime.ButHawkingandHartlesuggestthattheremaybenoboundaryandnosingularity:‘theboundaryconditionoftheuniverseisthatithasnoboundary’.11Onlyifspacetimeisclosedattheinitialsingularitywilltherebeadefinitesensetotheideaofafirsttime.Ifwepickouttwodefinite‘end-points’foranintervalonametrerule,say10cmapart,thentheintervalbetweenthesepointsmaybeopenorclosedatoneorbothends.Ifweincludetheend-pointsintheinterval,thentheintervalwouldbeclosedatbothends.But,ifwedonotincludetheend-points,theintervalwouldbeopenatboth‘ends’:thereisnoend-pointatallbutaninfinitenumberofpointseachclosertothe‘end’butneverreachingthe‘end’—andthisisbecausethereisliterallynoend;seeFigure35(p.199).Ifwetrackageodesicbackthroughtime,thenthegeodesicmaysimplyapproachtheso-calledinitialsingularityandneverreachit.Itdoesnotreachitbecausethereisliterally202\nCOSMOLOGICALCONUNDRUMSnothingtoreach.Thegeodesicneednothaveadefiniteend-pointatthesingularityforinsuchanopen-endedspacetimethereisnodefiniteboundarywithasingularity.Thisideaallowsustoconsiderthepossibilityofanopen-endedspacetimewhichhasnodefinitebeginning.12However,itmightbeappropriatetorecallsomeoftheworriesaboutthestatusofthecontinuumraisedinChapter1.Whyshouldwerelyonthereceivedwisdomaboutthemathematicalpropertiesofspacetimestructureandallowthecontinuummodel,whichunderwritestheaccountofopenandclosedintervals,todominateourviewofearlycosmology?Evenifweacceptthatthecontinuummodelprovidesafairlygoodapproximationtothespacetimearoundusnow,wemayreserveourjudgementaboutsuchearlytimes.ModerncosmologyallowsustoapproachthedebatebetweenLeibnizandClarkewithawidervarietyofoptions.Ifweinsistonsomesortofdivine‘creator’,thefour-dimensionalperspectiveofGTRdoesseemtosupportLeibniz’s(non-metaphysical)ideasrathermorethantheNewtonianaccount.Clarke,likeNewton,didnotimaginethatmatter,space,andtimemightcomeasapackagedeal.Thecreationofmatterwas,forthem,amysteriousevent.Andtheexistenceofan(uncreated)eternalspace,whichhasrealphysicalpropertiesintheNewtonianview,wasalsoamystery.GTRprovidesuswithawayofcharacterisingspace,time,andmattersuchthatthequestionofcreationmaybeavoidedifwechoosetodoso.Talkofwhatliesoutsidespacetimehasnophysicalmeaning;sotalkofsomethingnotinspacetimecreatingspacetimehasnophysicalmeaning.Ofcourse,itisuptoyoutodecidewhetherthereisatranscendentdeity‘beyond’spacetimewithsomekindofnon-physicalcharacterresponsibleinsomesenseforthephysicaldomain.Butthatdecisionjustcannotbeanythingotherthanmetaphysicalspeculation,perhapsbackedbysomesupposednon-physical‘experience’ofGod.However,moderncosmology,asaphysicaltheory,mustconfineitselftothephysical,towhatlieswithinspacetime.Yetitishardtosuppressthequestion:wheredidallthematerialintheuniversecomefrom?Butthereisnoneedtoansweronceweacceptthatspacetimeanditscontentsmightsimplyexistasafour-dimensionalentity,withorwithoutboundaries.Thepresenceofmatterwouldthenbejustacharacteristicofthisfour-dimensionaluniverse.However,recentworkininflationarycosmologyindicatesthatwemightprovideananswer203\nTIME,SPACEANDPHILOSOPHYwhichfurtherunderminestheneedforametaphysicalcreatortoexplainthepresenceofthephysical.Wheredidallthemattercomefrom?Theanswermightbesimply:fromnothing!INFLATIONARYCOSMOLOGY:SOMETHINGFORNOTHING?Amodificationtothestandard‘bigbang’viewwasproposedbyAlanGuthin1981inhisinflationarycosmologicaltheory;duringthe1980sthistheorywasrefinedbyseveralcosmologists,includingAndreiLindeandPaulSteinhardt.13Guthandtheothercosmologistswereawareofanumberofphysicalproblemswhichthestandardviewdoesnotreallyresolve.Mostoftheimportantquestionsconcerntheinitialconditionsintheearlyuniverse.Whyaretheearlyconditionssosmooth?Whydoestheexpansionfromtheinitialsingularityproceedattheobservedrate?Whatcontrolledtheexplosiveforceofthebigbang?Whydidtheexplosionandexpansionhappenanyway?14Theinflationarytheorytriestoanswerthesequestionsinanaturalandconvincingway,andindoingsothetheorymakestheapparentlyoutrageoussuggestionthatmatteriscreatedfrom‘nexttonothing’.Thecentralideaofthetheoryinvolvesaveryrapid‘inflationary’expansionfuelledbymassiverepulsiveforcesintheearlyuniverse.Guthsaysthattheinflationaryuniversedevelopsfromtheinitialsingularity,butremainsvagueastotheearliestconditions.Headmitsthat‘aswithothercosmologicalscenarios,thestartingpointisamatteroftasteandphilosophicalprejudice’—giventhefactthattheconditionsoftherapidexpansionintheinflationaryuniversepreventusfromobtaininganysubstantialinformationaboutthepre-inflationuniverse.15However,hesaysthattheinflationarytheorystartsfromtwosimpleandreasonableassumptions:theearlyuniverseshouldbeextremelyhot(over1027degreesKelvin)andatleastsomeregionsofspacetimeshouldbeexpandingrapidlyandthereforecoolingastheyexpand;healsoestimatesthatmaterial(intheformofenergy)amountingtojust10kilogramswouldbesufficienttoprovidethebasicingredientsfortheentireuniverse.16Thefigureof1027degreesKelvinisimportantbecauseitissupposedthatatthistemperaturetheuniverseinanormalvacuumstatewouldundergoaphasetransition.Above1027degreesKelvin,fundamentalnuclearandelectromagneticforcesandparticlesarethought204\nCOSMOLOGICALCONUNDRUMStobeindistinguishable:particlephysiciststalkofahighdegreeof‘symmetry’betweenforcesandparticlesandthinkintermsof‘grandunifiedtheories’whichareneededtocharacterisethephysicsinvolvedinsuch‘uniform’situations.17However,withoutsomemechanismtodelaythephasetransition,atthiskeytemperaturethesymmetriesarebrokenandthefundamentalforcesandparticlesbecomedistinct.Astheuniversecools,thesedistinctforcesandparticlesproducethemorefamiliarmaterialsituationswhichwenowobservearoundus.Notingthatmaterialoramatterfieldinanygivenregionmaybecharacterisedinrelativitytheoryintermsofitsenergydensity,Guthfollowsparticlephysicistsinhighlightingaverystrangestateformaterialpossibleintheearlyuniverse:the‘falsevacuum’.Weshouldnotbemisledbytheuseoftheterm‘vacuum’:suchregionsarenotinvariablyempty.Vacuumstatesdescribethelowestpossibleenergydensitylevelsforafieldinagivenregionofspacetimeatagiventemperature.Eveninthelowestpossibleenergystate,thatofthetruevacuum,theremaybenon-zeromatterfields.Althoughtheaveragevalueofafieldinaparticularregionmayindeedbezero,quantumtheorypredictsthattherewillbefluctuationsaroundthiszerovalue.Eachfluctuationsignifiesthebriefappearanceofa‘virtual’particle.Hence,eveninatruevacuum,matterfieldsmayappearbriefly.Evenifthematterfieldsinvolvedinthevacuumstateareratherpeculiarandcertainlynotobservableinthesensethat‘real’particlesare,itisamistaketothinkofanyphysicalvacuumassomeabsolutelyempty‘void’.Ifthetruevacuumstateintheveryearlyuniverseismaintainedeverywhere,thenduringthesymmetry-breakingphasetransitionlargenumbersofexoticparticles,knownasmagneticmonopoles,wouldhavebeenproduced.Butthereisnoevidencetoindicatethatthisdidhappen.Sosomethingmusthavepreventedthephasetransitionfromtakingplaceinthetruevacuumstate.Inflationtheorysuggeststhatatleastsomeoftheenergyinspacetimebecamefixedatahigherbutunstableenergylevel:thatofthefalsevacuum.Inthefalsevacuumstate,energyisabletopostponeitstransitionfromthehighlysymmetricalsituationdescribedingrandunifiedtheoriestoa‘normal’situationwhichmaybedescribedintermsofmorefamiliarelectromagneticandnuclearparticlesandforces.Duringthedelay,thetemperatureofmaterialinthefalsevacuumstate‘supercools’toatemperaturebelowthatofthephase205\nTIME,SPACEANDPHILOSOPHYtransition.Thisdelayiscriticalintheoveralldevelopmentoftheuniverse:duringashortbuteventfulperiod,massiverepulsiveforcesareproducedwhichactonnewlyproducedparticlesandantiparticles:Whiletheinflationarymodeldoesnotattempttoexplaintheformationoftheinitiallyhotexpandingregionswhichsubsequentlysupercoolintothefalsevacuumstate,itdoesexplaintheoriginofmostofthemomentumofthecosmicexpansion:thebigbanggetsitsbigpushfromthefalsevacuum.(BlauandGuth1987:543–4)18Thephysicalpropertiesofvacuumfieldsingeneralandthefalsevacuuminparticularprovidethekeytounderstandinginflationarytheory:1Inthefalsevacuum,thefundamentalsymmetriesremainunbrokenforatleastsometime,evenbelowthephasetransitiontemperatureof1027degreesKelvin.Althoughthisunstablesituationcannotbemaintainedindefinitely,advocatesforinflationtheorysaythatitholdsjustlongenoughtotriggerbothamassiveexpansionandtheproductionofanenormousamountofmaterial.2Theenergydensityofthefalsevacuumistremendous.‘Itistheenergydensitythatalargestarwouldhaveifitwerecompressedtothesizeoftheproton.’19Thishighdensityderivesfromtheexcessenergyinrapidlycoolingregionsinwhichthesymmetriesbetweenthefundamentalnuclearandelectromagneticforceshaveyettobebroken.Thesituationheremaybecomparedwiththeexcessenergyinsupercooledliquidsheldattemperaturesbelowtheirfreezing-pointsbutinwhichthegeneralsymmetriesassociatedwithliquidshavenotbrokendowntothelesssymmetricalstateofasolid.203Theenergydensityofthefalsevacuumisfixed.Foritistheminimumvalueforenergyinthatstate.Hence,iftheregionoccupiedbyafalsevacuumexpands,thentheenergyassociatedwiththatregionmustincrease:morefalsevacuum,sameenergydensity,thereforemoreenergyoverallcontainedwithinthefalsevacuum.4Thepressureinthefalsevacuumisverylargeindeedandnegative.Aregionwithnegativepressurehasinterestinggravitationalproperties.Theneteffectofsuchapressureistoproduceatremendousrepulsive206\nCOSMOLOGICALCONUNDRUMSgravitationalforce.Henceanalreadyexpandinguniverseisgivenan‘inflationary’boost.Duringthefalsevacuumstate,theuniversedoublesinsizeevery10–34seconds.5Thefalsevacuumstateisunstable:asthetemperaturefalls,thesymmetriesbetweenforcesandparticlesaremorelikelytobebroken.Thenthemassiveamountofenergystoredinthefalsevacuumstateisreleased.Thisenergyreleaseistypicallyintheformofparticlesandantiparticles.Thismaterialmovesrapidlyapart,giventheenormousrepulsivegravitationalforcesinherentinthefalsevacuumstate.Theobservedglobalexpansionoftheuniverseistheresultofthisinitial‘inflationary’push.Perhapsthemostcelebratedaspectoftheinflationarymodelisitsassertionthattheuniverseisa‘freelunch’,thatallmatterwascreatedliterallyfromnothing:acreationexnihilo.21Butthereissometimesequivocationaboutwhetherthelunchisfreeornot,andGuthwiselyaddstherider‘fromnexttonothing’whendiscussingthesourceofmaterialintheuniverse.Buthowcouldtheobservedmatterandenergycontentappearevenfromnexttonothing?Theabovepropertiesofthefalsevacuumgiveustheclueweneedtosolvethispuzzle.Theenergyavailablefromthefalsevacuumstateseemstobearrivingfrom‘nowhere’andatnoenergycost!Whatishappeningmaybedescribedinastraightforwardphysicalway,giventhepropertiesofthefalsevacuum.Thetotalenergyinvolvedinboththeproductionofmatterandthefuellingofinflationaryexpansionisliterallyzero.Yes,theamountofenergyneededtocreatethetremendousquantityofmaterialintheuniverseisanenormouspositivefigure.Buttheamountofenergyassociatedwiththegravitationalrepulsionisanequallyenormousnegativefigure.Thesetwofiguresbalanceandcancelexactly.Sothefalsevacuumprovides‘equalandopposite’resources:positiveenergyforthecreationofmaterialintheformofparticlesandantiparticles;andanequalamountofnegative(gravitational)energywhichproducestherapidinflationaryexpansion.Sowemaysaythattheobserveduniversehasitssourceinthefalsevacuum,whichseemsprettyclosetonexttonothing.Theappearanceofmatterisinasenseafreelunch,forthefalsevacuumpicksupthebillformatterandpaysincreditfromthefriendlygravitationalbank!ButweshouldrememberthatGuthadmitsthatwe207\nTIME,SPACEANDPHILOSOPHYneedabout10kilogramsworthofenergyintheuniversebeforetheexpansionandmatterproductionassociatedwiththefalsevacuumreallygetsunderway—wemightthinkofthisasanenergycovercharge.ButGuthevensuggeststhattheinitial10kilogramsoffieldenergymayitselfbebalancedoutbyanequalandoppositeamountofgravitationalenergy.Therewouldthenbenocovercharge,andthetotalenergycostoftheuniversewouldbepreciselynothing.Maywesaythattheinflationaryuniverseiscreatedfromabsolutelynothing,thatinflationarytheorygivesusamodelforcreationexnihilo?Thiswouldbeunjustified.For,evenifthetotalenergycostfortheuniverseiszero,thisdoesnotimplythattheuniverse‘starts’fromanabsolutevoid.Aftertheinitialsingularityandbeforetheinflationaryexpansion,therewasafluctuatingandexpandingvacuumfield.Eveniftheaverageenergycontentofthefieldiszero,thereisstillafieldwithallitsquantumfluctuations.Avacuumfieldmaynotbemuch,butitissomething!Thecauseoftheexpansionandtheproductionofmaterialintheinflationaryuniverseisclearlyphysical:thereisnoneedtoinvokesomeexternalnon-physical‘divine’cause.22Toaskforthephysicalcauseoftheinitialvacuumfieldisinappropriate.Allinflationarytheoryallowsustosayisthis:fromtheinitialsingularityuntilthefalsevacuumstate,therewasaslowlyexpandingandfluctuating(andthereforehot)vacuumfield.The‘source’ofthisexpansionliesatthesingularity.But,asinthestandardbigbangmodel,ifthesingularityisadefiniteboundarytospacetime,thereisnoquestionofanearliermomentintime;and,ifspacetimeisopenattheinitialsingularitywiththissingularityasalimit,thentheremaybenobeginningtotimeatall.Aswenotedintheprevioussection,thereisnoneedtoseeksomemetaphysicalexplanationfortheuniverse:wemayregarditisassimplyanexistingself-containedentity.But,asbefore,thereisnothingtopreventusfrominvokingsomedivineagency,exceptperhapsconsiderationsofsimplicity.Manycosmologistsregardinflationarytheoryassoundbecauseitprovidessomeveryplausibleanswerstothemanyquestionsnottackledeffectivelybythestandardbigbangmodel.Someofthesequestionshavealreadybeentackledabove.Othersinclude:1Whyistheuniversesosmoothglobally?Duringtheinflationaryphase,asStephenHawkingtellsus:‘anyirregularitiesintheuniverse208\nCOSMOLOGICALCONUNDRUMSaresmoothedoutbytheexpansion,asthewrinklesinaballoonaresmoothedawaywhenyoublowitup’.23Thebeautyofthisideaisthatitallowscosmologiststoplacefarlessemphasisupontheinitialconditionsholdingbeforethephasetransition:solongasweacceptGuth’sbasicassumptions(thattheuniverseishotandexpandingwithunbrokensymmetriesbetweenparticlesandforces),howeverirregulartheearlyuniversemaybe,theinflationaryphaseleadstothesamegenerallysmoothendresult.Hence,thereisnoneedtoprovideadetailedaccountoftheveryearlyuniverse.Theformationofgalaxiesmaythenbeexplainedintermsoflocalfluctuationsinthegloballysmoothenvironment.2Whyistheobservedexpansionratesoclosetothatfora‘flat’universe?Thisarisesfromtheperfectbalancebetweentheenergiesinvolvedin(i)theproductionofmaterialand(ii)thegravitationalrepulsion.Thereisjustenoughenergyintheformofgravitationtobringtheexpansiontoahaltat‘infinity’:i.e.thedensityofmatterintheuniverseisnotsufficienttobringabouteventualrecollapse,norisitsmallenoughtoallowanacceleratedexpansion.Variationsonthethemeofinflationalsotacklequestionssuchastheuniquenessoftheuniverseandthenecessityofathree-dimensionalworld.AndreiLinde’schaoticinflationtheorysuggeststhattheobserveduniverseisjustonedomainor‘bubble’.Eachdomainisreallya‘mini-universe’withinavastseaofbubbleswhichconstitutestheuniverseproper.Inflationisolatesthedomainsfromeachother,andourobservationsarerestrictedtoourowndomain.Andonlyincertaindomainswilltheorbitsofelectronsaroundatomsorplanetsaroundstarsbestable:i.e.indomainswiththreespatialdimensions.24BLACKHOLESThethinkingbehindblackholesisnotarecentinvention.ThemathematicianLaplacediscussed,in1799,thepossibilityofastar’sgravitationalattractionpreventinglightfromleavingitsneighbourhood,sayingthat‘theattractiveforceofaheavenlybodycouldbesolarge,thatlightcouldnotflowoutofit’.25LaplacearguesthatthiscouldhappenifastarwithadensityclosetothatoftheEarthwereroughly250timesthediameteroftheSun.And,in1932,theIndianphysicistChandrasekhar209\nTIME,SPACEANDPHILOSOPHYtried(andfailed)topersuadeSirArthurEddingtonthatgravitationalcollapsetoa‘singularity’couldnotberuledoutinarelativisticuniverse.26Furthermore,theideaofauniverseexpandingfromaninitial‘bigbang’demonstratedthatGTRmayinvolvesingularities:theinitialconditionsinthebigbangscenarioaresingularbecause,asthevolumeoftheuniversedecreases,thedensityofthe(fixed)mattercontentapproachesinfinity.ButonlysincetheworkofStephenHawking,RogerPenrose,RobertGeroch,andothersinthe1960shavewehadadetailedappreciationofthepropertiesofblackholes.Blackholesaresingularitiesofgravitationalcollapse.Wheelercoinedthephrase‘blackhole’tocapturetheideathatnothingcouldescapethegravitationalattractionofthesingularity.Theyaresingularitiesbecause,asthesurfaceareaandthereforethevolumeofacollapsingstarapproacheszero,thedensityoftheholeapproachesinfinity.Anobjectwithinfinitedensityisasingularityand,aswehavealreadynotedinthischapter,asingularityisgenerallyunderstoodastheterminusforgeodesies.Animportantfeatureoftheblackholeisitseventhorizon.Whenastariscollapsing,therewillbeacriticalstageinitscollapsewhichrepresentsthepointofnoreturn.Thisstageisreachedwhenthestarhasadiameterandmasswhicharesufficienttoinducetotalcollapsebythemselves.Beforethiscriticalstageisreached,theremaybejustachancethatthestarmayexperiencesomeinternalorexternalstimuluswhichpreventstotalcollapse.But,oncethecriticalvaluesforsizeandmassarereached,completegravitationalcollapseisguaranteed.Thediameterassociatedwiththiscriticalstagerepresentsthediameteroftheblackhole’seventhorizon.Whenanyparticleenterstheregiondefinedbytheeventhorizon,wemaynolongerseewhathappenstotheparticle.Wehavenowayofdetectingeventswithinthehorizon.Foranysuchparticleisthendrawninexorablytowardsthehole,andanysignal(e.g.light)whichmightbeusedtotransmitinformationabouttheprogressoftheparticleislikewisedrawninwards.Hence,allinformationabouttheparticleiscloakedbythehorizon.StephenHawkinghasshownthatblackholesdonotsimplygrowfatonthematerialdrawnintothembygravitationalattraction.Theyejectparticlesinordertomaintainequilibrium.Theblackholeactslikeastrangekindofpump,suckinginnearbymaterialby‘classical’gravitationalattractionandejectingitto‘infinity’viaquantumprocesses:210\nCOSMOLOGICALCONUNDRUMSOnecanthinkoftheemittedradiationashavingcome,frominsidetheblackholeandhavingquantummechanicallytunnelledthroughthepotentialbarrieraroundtheholecreatedbythegravitationalfield,abarrierthatcouldnotbesurmountedclassically;…itispossibleforablackholetoemitatelevisionsetorCharlesDarwin…buttheoverwhelmingprobabilityisfortheemittedparticlestohaveanalmostthermalspectrum.(HawkingandIsrael1979:19)27Thephysicsofthisprocessisdescribedintermsofthreetheories:GTR,quantummechanics,andthermodynamics.Hawkingsuggeststhattheemissionprocesssendsparticlesandradiationto‘infinity’,butRogerPenrosearguesthataparticleorgroupofparticlesmightbeinterceptedjustaboutanywheresincethereisnowaytoruleoutsuchparticleshavingeffectslocallyaswellasatadistance.28Sotheeventhorizonaroundablackholeactsasaneffectivebarriertoanyparticleorsignaltryingtoescapethegravitationalpullofthesingularitybyclassicalmeans,butitdoesnotblocktheprogressofparticlesdeterminedtofindtheirwayoutthroughquantumtunnels.However,theeventhorizonhasanotherrole.Becausephysicistsgenerallyexpecteventsclosetothecentreoftheholetobeextremelyhot,dense,andviolent,theyalsoexpectthataquantumapproachwillbeneededtoexplainwhatishappeninginsidetheeventhorizon,justastheybelieveitwillberequiredtogiveafullaccountofeventsclosetotheinitialsingularity.Thehorizonthereforeissaidtoactasaclassicalcloakaroundquantumeffects.Hence,classical(i.e.non-quantum)GTRcannotbeusedtocharacterisethesingularbehaviourinsidethehorizon.Consequently,GTRpredictsphenomenawhichitcannothandle:inasense,thetheorypredictsitsownbreakdown.ButtheexistenceofthehorizonallowsustouseclassicaltheorieslikeGTRuptothehorizonandleavetheproblemofwhatishappeninginsideinabeyance.COSMICCENSORSHIPRogerPenrosehassuggestedthattheexistenceofhorizonsisnotahappyaccident,butageneralconsequenceofthecosmiccensorshiphypothesis,whichrequiresthatexternalobserversshouldalwaysbeprotectedfromthequantumgravitationaleffectsassociatedwithblackholesbyevent211\nTIME,SPACEANDPHILOSOPHYhorizons.ThishypothesisdependsonthedefinitionofaCauchyhypersurface.Soweshallexploretheideasbehindthishypersurfacebeforeturningtoananalysisofcosmiccensorship.InChapter8,weinvestigatedvariouslevelsofcausalstructurewhichmightbeimposedonaspacetime.Thestrongestconditiondiscussedtherewasthatofstablecausality.Acausallystablespacetimeisoneinwhichclosedtimelikeloopsareimpossible.Butwemayplacefurtherconstraintsonspacetime.Inparticular,wemaydemandthatnothingshouldpreventusfrompredictingthefutureofaregionofspaceinwhichnopointinthespacecausallyprecedesanyotherpoint.Sucharegioniseffectivelyalocal‘time-slice’,calledanachronalslice.Ifweknowallthereistoknowaboutthephysicalconditionsonthissliceandtheseconditionsarenotsingular,thenwemayusethefieldequationstodeterminethecausalfutureofthisslice.Justaswemaydeterminethefutureonthebasisoftheconditionsonthesliceandthelawswhichapply,symmetryconsiderationsshowthatwemayalsodeterminethepast.Clearly,aswemovefurtherintothefuture,thescopeofourpredictionsgrowseversmaller.Thisisbecauseaswemovealongaworldlinepassingthroughtheslice,thechancesofmeetingaworldlinewhichdoesnotpassthroughthesliceincrease.However,thepossibilityofmeetingaworldlinefromoutsidethesliceisrestrictedbythefactthatnocausalsignalmaytravelfasterthanlight.Sowemaystillconstructaregioninwhichalleventsmaybedetermined:sucharegioniscalledthedomainofdependenceoftheachronalslice;seeFigure36(p.213).Ifwecanconstructanachronalslicethroughtheentirespacetime,thenpredictionisnotrestrictedtojustlocalevents,butmayapplytoallspacetime.Suchaglobaltime-sliceiscalledaCauchyhypersurfaceanditsdomainofdependenceistheentirespacetime.Thereforethisglobalhypersurfaceseemstoallowustodetermineeveryeventinaspacetime.Thiskindofdeterminismislinkedwiththeideaof‘Laplaciandeterminism’,aperspectivesuggestedbyLaplaceinwhichacompletesetofinformationaboutthepresenttogetherwiththetruelawsofnaturecouldprovideusacompletehistory(past,present,andfuture)fortheuniverse.29SpacetimeswhichpossessCauchyhypersurfacesincludeMinkowskispacetimeandtheFriedmannmodels.30212\nCOSMOLOGICALCONUNDRUMSAlthoughaworldlinecanreachRfrommanypointsontheachronalslice,includingpointP,nosignalfromanypointoutsidetheslice,suchasQ,canreachR—unlessofcourseittravelsfasterthanlight!Ifweexcludethispossibilitythenalleventswithinthedomainofdependenceoftheslicemaybeinfluencedonlybyeventsontheslice.Figure36AchronalsliceanddomainofcausaldependenceHowever,theCauchyhypersurfaceandthedeterministicequationsofGTRmaynotbesufficienttoguaranteecompletepredictivityinaspacetimebythemselves.WenotedinthefinalsectionofChapter7thatthosewhotakeasubstantivalistviewofspacetimepointsmaybeunabletoholdontodeterminism,giventhe‘holeargument’.Butwemustalsofacethepossibilitythatanysingularitywithoutaneventhorizonmayintroduceindeterminismviaquantumeffectsintothedomainofdependence.Althoughwemightstipulatethattheconditionsonanachronalslicebenon-singular,asingularityofcollapsemaywellevolvetothefutureoftheslice,giventheconditionsontheslice.Thisiswheretheeventhorizoncomesin.Foraneventhorizonissupposedtoshieldspacetimefromthequantumeffects,sothatthecompletecausaldeterminationofeventswillnotbedisrupted.Therefore,PenrosesuggeststhatthecosmiccensorshiphypothesismustholdinGTRifwearetohaveachanceofachievingcompletepredictivity.Couldwehaveasingularitywithoutaneventhorizon?Andwouldtherebeanysignofsuchasingularityinthephysicalconditionswhichobtainonanachronalslice?Itturnsoutthattherearetwomainwaysinwhichthismaybepossible:213\nTIME,SPACEANDPHILOSOPHY1ChrisClarkeandothershaveshownthatnakedsingularitiesmayevolveinanapparentlywell-behavedspacetime;thesearephysicalsingularitieswithouteventhorizons.Particlesmayemergeinanon-predictablewayfromnakedsingularitiestoourfutureandinterferewithevents,thusdisruptingattemptstoprovideacompletecausaldeterminationofthefuture;symmetryconsiderationsshowthatwewouldlikewisebehinderedinanysuchattempttodeterminethepastfully.Clarkehasshownthatnakedsingularitiesmaydeveloptothefutureofanon-singularachronalslice.312Theremaybetopologicalholespresentinaspacetime—aspacetimewithregionsliterallycutaway(donotconfusethemwiththe‘holes’intheholeargumentofChapter7);althoughthesearenotphysicalentitiesinthesenseofnakedsingularities,theymaybethestarting-pointsfornon-predictablegeodesiesandsotheycouldbejustascausallydisruptiveasnakedsingularities.32Wemightdecidetoruleoutsuchentitiesonempiricalgrounds:thereseemstobelittleornoempiricalevidencefortheexistenceofnakedsingularities—alltheyhaveistheoreticalbacking;andtopologicalholesmightbedismissedasbeingnomorethanmathematicalcuriositieswiththeweakestofempiricalfoundations.Suchdecisionswouldbeinlinewiththecosmiccensorshiphypothesis.Therefore,theclaimthatcosmiccensorshipholdsisessentiallyempirical.ItismotivatedbythedesiretomaximisecausaldeterminationwithinthecontextofGTR.However,wecannotruleouttheinitialsingularityandblackholesonempiricalgroundssoeasily.Andthebigbangandblackholesmayproduceeffectswhichthreatencausaldeterminationalmostasmuchasnakedsingularitiesandtopologicalholes.Ifquantumeffectsdominatethetimesjustafterthebigbang,thenGTR’sfieldequationscannottelluswhatishappeningthen.Ifblackholesmayejectmaterialintospacetimequantum-mechanically,thenGTR’sfieldequationscannothelpuspredictwithcertaintywhatliestothecausalfutureofablackhole.Toensurecompletecausaldeterminationinspacetime,wewouldhavetoruleoutallsingularitiesandadvocatea‘strong’versionofthehypothesisrulingoutanythinglikelytointerferewithcausaldetermination;and,ifwetaketheholeargumentinChapter7seriously,absolutistswouldalsoneedtorevisetheirattitudetowardsspacetimepoints.Giventhatwehavegoodempiricalgroundsfortheinitialsingularityandforblackholes,thereseemstobelittlereasontopushcosmiccensorshiptosuchlimits.But,ifweapplycensorshiptojustnakedsingularitiesandtopologicalholes,thenthehypothesisinthis214\nCOSMOLOGICALCONUNDRUMS‘weak’formappliestojustthosethingswefindempiricallyobjectionable;and,indoingso,theweakerhypothesisfailstoachievethegoalofthecosmiccensorship:causaldetermination.ThehypothesismightbeausefulwayofgroupingthosesolutionsofGTRwhichare‘classical’ineveryway;anditmayhelpustoexplorethegeneralpropertiesofsuchsolutions.ButthereseemslittlereasonforustoinsistthatGTRshouldbelimitedbythehypothesisineitheritsweakerorstrongerform.ThefieldequationsofGTRmaybedeterministicinaclassicalsense,butthephenomenawhichtheypredictandwithwhichtheydealaresometimesindeterministic.Weshouldnotdriveawedgebetweenthetheoryanditstheoreticalandempiricalcontext,nomatterhownicelythecosmiccensorasks.DETERMINISMVERSUSINDETERMINISMSingularitiesandthecausaldisruptionassociatedwithsingularitiesmayturnouttobeexplainedfullyintermsofthematerialpropertiesofobjectsinspacetime.Theydonotposeanyimmediatethreattoarelationistposition.Butatopologicalholerepresentsanirreducibleelementofspacetimestructure;andsuchholesinspacetimemaybejustascausallydisruptiveasphysicalsingularities.Toacceptthepossibilityoftopologicalholesistoconcedethatspacetimemayhavestructureoverandaboveitsmaterialcontents.Suchholesarelikelytohavedirecteffectsontheaffinestructureofspacetime,sincetheyarethestarting-andfinishing-‘points’forobjectsmovingalongthegeodesiesofspacetime;and,aswesawinChapter7,howobjectsmoveinspacetimeisdefinedintermsoftheaffinestructure.Thereforetherecognitionthatasuccessfultheoryofgravitationmustdealdirectlywithindeterministicphenomenamayleadtotheconcessionthatthespacetimeofsuchatheoryislikelytobeessentiallyabsoluteinthesensethatitsstructuresmaynotbecompletelyreducibletoitsmaterialcontents.Soarelationistwouldbeeagertoruleouttopologicalholesascandidatesforexistence.However,relationistsdonotseemtohaveaparticularlystrongargumentforthismove:iftheysaythatwemustdenytheirexistencebecausesuchholesareyettobeobserved,thentheymustalsoaskustoruleoutatremendousvarietyoftheoreticallyrespectableentitiesascandidatesforexistence;iftheyadoptadeterministpositionandurgesomeversionofthecosmiccensorshiphypothesis,thenwewouldbeaskedtoneglectanumberofplausiblegravitationalphenomenasimplybecausetheyaretarredwiththesameindeterministicbrushastopological215\nTIME,SPACEANDPHILOSOPHYholes.Theproblemhereaselsewherefortherelationististhatthestructuresofspacetimearerichenoughandcomplexenoughtoallowustogeneratearangeofinterestingfeatureswhicharecharacteristicsofspacetimeitself.Absolutistshavetheseeminglylimitlessingenuityofthemathematicianontheirside.Relationistsmustresorttooftenineffectualcomplaintsaboutthelibertiestakenbymathematicianswhendealingwiththephysicalworld.33Thisleadsustoaninterestingproblemconcerningtheholestory.JohnEarmansaysthattakingspacetimepointstooseriouslyleadstheabsolutisttoadilemma.WehavenotedthatthosewhoaresubstantivalistsaboutspacetimepointsmustconcedethepossibilityofasingleCauchyhypersurfaceallowingtheconstructionofalternativemodelspacetimeswithdifferentfutures.Butthismeansthatdeterminismis,atleastforsubstantivalists,alostcause—fortheinformationspecifiedonthehypersurfacedoesnotallowustopredictthefutureunambiguously!But,ifwetakespacetimestructureseriously,thenwearealsolikelytohavenofundamentalobjectionstotopologicalholesandotherindeterministicphenomena.Soanabsolutistperspectiveonspacetimestructureseemstoleaddirectlytowardsindeterminism.EarmantriestopersuadethosewithabsolutistinclinationstodroptheirsubstantivalismaboutpointsandtofollowthepathsuggestedbySklar’snotionof‘absolutemotionwithoutabsolutespacetime’.ThismightseemattractivetothosewhodreamLaplaciandreams.However,thoseofuswhoareconvincedthattheworldisessentiallyindeterministicaremorelikelytocontinuetousethosemodelsandstructureswhichenableustodescribethisapparentlyindeterministicworldascoherentlyaswecan.Wedoso,notbecauseofanyabsolutistprejudices,butbecausetheempiricalevidencesuggeststhattheworldisessentiallyindeterministic.216\nCONCLUSION:RELATIVITY—JUSTANOTHERBRICKINTHEWALL?INTRODUCTIONOnemajorproblemforthephilosopherofspaceandtimeisthefactthatscientistskeepchangingtheirminds.Theydosowhentheymake‘revolutionary’leapsfromtheorytotheory;andtheycontinuetodosoastheydeveloptheideasofagiventheory.Sometimestheendproductofsuchadevelopmentmightbejustasrevolutionarywhencomparedwiththeoriginalformulationofthetheoryasanentirelynewtheory.Sometimesthechangesmadearenotquitesodramatic.Allthisleadsusintodifficulties.Whichsetofideasshouldweinvestigate?Whichtheorytellsusaboutthespaceandtimeoftheactualworld?Shouldweacceptthatwearepartofaprocessleadingtowardsthetruth,butthatwehavealongwaytogo?ShouldweacceptStephenHawking’ssuggestionthattheendoftheoreticalphysicsmaybeinsight?1Orshouldwetakeeachtheorywithapinchofsalt:justonemoreinventionofsomeratherbrilliantmathematicianswithnothingtosayabouttherealworld?Anyattempttoanswersuchquestionstakesusfirmlyintotheterritoryofthephilosophyofscience.Butourdeliberationsaboutthenatureofspaceandtimemayhelpustounderstandhowwemightapproachsuchquestions.Nobodyislikelytoclaimthatrelativitytheorytellsusthewholetruthaboutspace,time,andmotion.Therearetoomanyunsettledscientificproblems,toomanyanomalies,foranyonetobesobold(orsofoolish).However,whenwebegintoinvestigatethestructureandcontentsofrelativitytheory,westarttofindthatrelativityisnotsomemonolithic,moribunddogma.Itisavibrant,changing,wide-rangingcontextinwhichempirical,theoretical,andphilosophicalissuesaresubjecttocontinualdebate.Butthisshouldmakeusallthemorecautiousintrying217\nTIME,SPACEANDPHILOSOPHYtogivefinalanswerstomanyoftheproblemsofspaceandtime.Wemaycertainlyclaim‘relativityinterpretedinsuch-and-suchawayhassuch-and-suchimplicationsforthenatureofspaceandtime’.Wemaynotconcludethatanysuchassertionisthefinalwordaboutspaceandtime.Ofcourse,wemayregardtheempiricalsuccessofrelativityasevidenceforatleastitsapproximatetruth;wemaythinkthatthetheorycapturessomethingofthephysicalworld.Butrelativitytheoryisnotaloneinthis:othertheoriesinvolvingspace,time,andmotionseemtoapplytothephysicalworldinmuchthesameway,ifnotalwayswiththesamedegreeofsuccess.Isrelativityjustonemoretheory,justonemorebrickinthewallofscience?Willitsfatebebanishmenttothearchives?Orshouldweseeitaspartofalong-termevolution,withitsessentialideasinvolvednotjustintheSpecialandGeneralTheoriesofRelativity(STRandGTR)asoriginallyconceived,butalsointhelonghistoryofthoughtonspace,time,andmotion:past,present,andfuture?Howweanswersuchquestionsdependsuponourviewofspacetimetheoriesinparticularandscientifictheoriesingeneral.And,withaclearerviewofthestructureoftheories,wemaybecomeevenmorecautiousbeforeprovidingfinalanswerstothemanyquestionsconcerningspaceandtimebeforeus:thestatusofthecontinuum;theproblemsofconventionalism;therationaleforabsolutismandrelationism;theissueoftimetravel;thebeginningoftime;andthenatureofsingularities.WHATISATHEORY?Physicaltheoriesarefrequentlycharacterisedasstructuresbuiltaroundstablefundamentallaws.WhetherweadoptKuhnian,realistorfalsificationistviewsofscience,thelawsofatheoryareseenasthefocalpointsofthetheoryandconsequentlyphilosophicalinvestigationsandreconstructionshavetypicallyconcentratedonthecentralroleoffundamentallawsinphysicaltheoriesandtheimplicationsofthoselaws.However,manywritersnowwarnagainstplacingundueemphasisuponthetheoreticalcontextandthetheoreticallawsofscience.Cartwright,Galison,andHackingattacktheprimacyoftheoryanddirectustotheexperimentalcontextinwhichtheoftenclumsyconcoctionsoftheexperimentalistsreplacetheelegantgeneralisationsofthetheorists.2Philosophershaveindeedneglectedtheexperimentaldomainanditisrightthatthisshouldberectified;butweshouldbecarefulnottounder-emphasisetheinteractionsbetweentheoryandexperiment.Ifweperceiveatheoryasconsistinginjustalaworsetoflaws,thenitiseasytodrive218\nCONCLUSIONawedgebetweentheoryandexperiment.However,wemayadoptaviewofphysicaltheorieswhichwillhelpustoprovideabridgebetweenthosewhoseetheoryastheprimarydrivingforceofscienceandthosewhowishtoemphasisetheautonomousoratleastpartiallyautonomousandpivotalroleofexperimentation.TheviewwhichIshalldiscussheredrawsontheideasinvolvedbothinThomasKuhn’sviewofadisciplinarymatrixandinImreLakatos’notionofaresearchprogramme.3Butweshallfindthatthereareanumberofsoundreasonswhyweshouldadoptanevenbroaderperspectiveofphysicaltheories.Weshouldavoidthetemptationtocharacteriseaphysicaltheoryashavinga‘limitednatureandscope’—insteadweshouldregardatypicaltheoryasacomplexandwide-rangingtheoreticalstructure.4Ifwefailtodothis,thenwemaydeveloparatherlimitedphilosophicalappreciationnotonlyoftherelationshipbetweentheoryandexperimentbutalsoofsuchproblemsasthetheory-dependenceofobservationandthecontinuityoftheoreticalknowledge.Maturephysicaltheoriesarecertainlynotsimplestructures:theyinvolvemorethanfundamentallaws—thismuchhasbeenlearntfromsuchwritersasKuhn,Hesse,Lakatos,andLaudan,whovariouslydirectustotheimportantrolesofstandardderivationsandproblems,models,methodologicalconventions,andvalues.5Inparticular,Kuhn’snotionofanestablishedtheory(thedisciplinarymatrix)involves:1stablelaw-likesymbolicgeneralisationswhichmayalsofunctionasdefinitionsofthesymbolsusedinthem;2epistemological(andpossiblymetaphysical)commitmentstocertainphenomenaandentitiesandalsotoanalogies,metaphors,andmodels;3methodologicalandpragmaticvalues;and4exemplars—theconcretesolutionsofcentralproblems.6AndLakatosmakesaninterestingdistinctionbetweenthenegativeheuristicandpositiveheuristicofaresearchprogramme:1thenegativeheuristicistheunfalsifiablecentral‘hardcore’ofgeneralisationsinatheoreticalresearchprogrammesothatthedevelopmentofafruitfulprogrammeisstronglyfocusedbyafirmcommitmenttothelawsofthetheory;torejectthishardcoreistorejecttheentiretheory;henceitisthenegativeheuristicwhichunderwritesthestabilityandcontinuityofagivenprogramme;219\nTIME,SPACEANDPHILOSOPHY2thepositiveheuristicactsasa‘protectivebelt’ofsupportingassumptionsandideasaroundthehardcore;whenatheoryischallenged,theauxiliaryassumptionsinthepositiveheuristicaretypicallymodifiedtohelpmaintaintheoreticalstability.ThetheoreticalcontextsofGTR,QuantumMechanics(QM),andotherestablishedtheoriesexhibit,toacertainextent,theseKuhnianandLakatosianfeatures.Infact,athoroughreviewofanyestablishedtheorywillrevealatremendousarrayofpossibleconstituents.PeterGalisonsaysthatthestabilityofsciencederivesfromtheamorphousnatureoftheenterprise.7Acrystaliseasilysplitapart;butamorphousmaterialslikefibreglass,withitsdistinctbutinterweavingthreads,aremuchmorerobust.Hesaysthatscienceislikeabrickwall:withthreekindsofinterlockinglayers—theoretical,observational,andexperimental.Eachofthesethreedomainsofthescientificenterprisehavepartialautonomyfromtheothertwo.Thisappearstobecorrect.Wemaycertainlyidentifypeculiarlyexperimentalfeaturesortheoreticalaspectsofanymajorphysicaldisciplinelikegravitation.Butwemustalsorecognisethatthereisahighdegreeofinteractionbetweenthesethreedomains.Hence,itmaybeamistaketofocusasstronglyasKuhnandLakatossometimesseemtodoonthecentraltheoreticalgeneralisationsofatheoryastheload-bearingstructuresinthescientificenterprise.Observationandexperimentationalsoplayimportantpartsinprovidingstabilityandcontinuity.Galison’sgeneralviewofsciencemayalsoapplytoeachindividualdomain.Eachdomainmayberegardedasanamorphousstructure,withavastarrayofinterlockingelementsandideas.WehaveseenthatatheorylikeGTRisacompositestructure.Anditsinternalstabilityisprovidedbytheinteractionofallitsvariouselementsthroughtime.Noelementisimmunetochange,noteventhecentrallaws.Thoseelementswiththegreateststabilitytendtobe:generalprinciplesofsymmetryandinvariance,whichmayapplyequallywelltoalternativetheories;andthoseideaswhicharecommontoseveraldominanttheoriesoftheday.SoGalison’smetaphorofawallisilluminatingbutperhapstoosimplistic.InsteadwemightregardGTRasapartofastrongropeofinter-weavingthreads:somethreadsareshort,somearelong.Therope220\nCONCLUSIONisscience.Thethreadsaretheconstituentelementsoftheinteractingdomainsofobservation,experiment,andtheory.Thestrengthoftheropeliesintheweaveofallthesediverseelements..Weneedtoinvestigatetheideasinvolvedinatheoreticalcontext.Thiswillenableustocapturesomethingofthenatureandscopeofphysicaltheories,especiallythoseconcernedwithspaceandtime,whichprovideourmainfocushere.Thisshouldhelpustoseealittlemoreclearlywhyscienceisanessentiallystableenterprise.Butitwillalsohelpustoseewhyweshouldbecautiousinourattitudetowardstheproblemsofspaceandtime.THESTRUCTUREANDSCOPEOFSPACETIMETHEORIESTherearesixmainingredientsofphysicaltheories:laws;principles;epistemologicalcommitments;analogies;theresourcebase;andvalues.WeshallconsidereachinturnandseethateachelementplaysanimportantpartinspacetimetheoriessuchasNewtoniangravitationandGTR.1Laws.Therearethreemaintypesofphysicallaw:aFundamentallawssuchastheEinsteinfieldequationsof191516.bLawsderivedfromthefundamentallawsusingspecificinitialandboundaryconditionstofocusonaparticularproblem:suchastheequationsusedintheSchwarzschildsolutionofGTRtocharacterisethegravitationalfieldaroundasingle,isolatedmass.Suchderivationsarefrequentlyusedtogenerateobservationalandexperimentalpredictions,cInter-theoreticallawswhichdrawonmorethanonetheorytohelpuscharacteriseeitheratheoreticaloranexperimentalcontextinasrealisticamanneraspossible.TheHawking—Penrosetheoremsof1970andtheHawkingequationsof1974helpustotheoriseaboutthebehaviourofblackholesbydrawingonthermodynamicsandQMaswellasonGTR.Andtheequationsproducedtohelpusestimatetheeffectsofgravitationalwavesonlargealuminiumcylinders—typicalfeaturesintheearlyexperimentalapparatususedinthesearch221\nTIME,SPACEANDPHILOSOPHYforsuchwaves—takeintoaccountthebulkelasticpropertiesofmaterials.2Principles.Therearethreemainkindsofprinciple—butthedivisionsbetweeneachkindarenotsoclear-cutasinthecaseoflaws:aEmpiricalprinciples.Forexample,theprincipleofconservationoflocalenergy-momentumisanessentialpartofGTRjustastheconservationprinciplesforenergyandofmomentumaredeeplyembeddedinNewtoniantheory,bMethodologicalprinciples.Principlesofequivalence,invariance,andgeneralcovarianceactedasaguidetoEinsteininhissearchforthefieldequationsofGTR.Andsymmetryprinciplesplayanimportantmethodologicalrole,forexample,inhelpingustocapitaliseonsymmetricalfeaturesofmodelsofGTRsuchasSchwarzschild’s(sphericallysymmetric)solution.cPhilosophicallymotivatedprinciples.TheStrongAnthropicPrincipleandMach’sPrincipleareusuallyregardedasphilosophicalprinciples,basedperhapsonsomevisionoftheroleofconsciousnessoraparticularphilosophicalperspective,likepositivism.Itishardtomakeacleardistinctionbetweenthesethreekindsofprinciple,formethodologicalprinciplesfrequentlyhavestrongempiricalpedigrees,asisthecasewiththeprincipleofequivalence,andtheyoccasionallyhavephilosophicalovertones,asisthecasewiththeprincipleofgeneralcovariance,whichdemands‘simplicity’intheformulationofGTR’sequations.Again,therearehardchoicesincasessuchasthecosmologicalprinciple,whenitisdifficulttodecidewhetheritshighempiricalstandingcompletelyoutweighsitsCopernicanphilosophical‘prejudice’againstanyformofanthropocentricity.Indeed,thesituationiscomplicatedfurtherbythefactthatthestatusofprinciplesmaychangeduringthedevelopmentofatheory—asindeedhappenedinthecasesofbothMach’sPrincipleandtheCosmologicalPrinciple,theformergraduallyfallingoutoffavourandthelatterrisinginstatus.3Epistemologicalcommitments.Wejustcannotmakesenseofeitherourlawsorourprinciplesifwefailtogivesomemeaningtotheelementswhichtheycontain.GTRcontainsarangeofbasic‘building-blocks’whichenableustointerpretthevariouslawsofthetheory;theseare:222\nCONCLUSIONaFundamentalcommitmentstoentities.Commitmentstobasicgeometricalconceptual‘building-blocks’areneededtoconstructthemetricsandenergyfieldsofGTRspacetimes.WhenLockecompiledhislistofprimaryqualities,heprovidedareviewofthebasiccommitmentswhichhesawasessentialtothemechanicalperspectiveinherentinNewtoniantheory.Similarly,weneedtoassimilatesuchbasicgeometricalelementsastheaffineconnectionifwearetooperatewithinGTR’scontext.bCommitmentstoconstantsusedinthelawsofthetheory.Sometimesthesecommitmentsaredeep-seated,asinthecaseoftheNewtoniangravitationalconstantorPlanck’sconstant.Occasionallytherearemixedreactionstoaconstant:asinthecaseofthecosmologicalconstant,whichEinsteinfirstaddedtohisfieldequationsin1917andlaterdroppedwhenempiricalevidencecalleditsuseintoquestion.cCommitmentstophenomenaorentitieswhichappearinawiderscientificcontext.GTR’scommitmenttoafinitespeedinvacuaforlightwhichisconstantforallinertialobserversisalsoabasicdemandinbothSTRandelectromagnetism.Suchcommitmentsneednotalwaysbeatahightheoreticallevel:wemayalsoholdlower-level‘observational’commitments,forexample,tothe‘fact’thatlocallylighttravelsinstraightlines.Ofcourse,thesearenottheonlyepistemologicalcommitmentsmadeinatheory.Wewillalsobecommittedtolaws,principles,models,andsoon.Butwhatdistinguishesthefundamentalcommitmentslistedaboveisthefactthattheyareepistemologicallypriortothelaws,principles,andmodelsofthetheory.Withoutsuchbasiccommitments,wecouldneitherusenorgrasp(letalonedevelop)atheory.4Analogies.Therearethreebasictypicalkindsofanalogytobefoundwithintheoreticalcontexts:aBasicmetaphorsareparticularlyusefulintheoriesforseveralreasons.First,theyprovidelinksbetweenourmacroscopicappreciationoftheworldandourmicroscopicperspective,allowingustograspandelucidatethelatterbyreferencetotheformer;forexample,whenweusetheideaof‘spin’tocharacterisethebehaviourofparticlesinQM,wemaybegintoexplaintheideabypointingoutthattwoidenticalrotationsthrough360degreesforapersonwouldbetwodistinctrotationsforaparticlewithspin1/2.223\nTIME,SPACEANDPHILOSOPHYSecondly,theycanhelptostimulateimaginativeresearchwithinatheoreticalcontext;forexample,whensimilaritiesarespottedamongstsituationsindistincttheoreticalcontexts,asisthecaseinHawking’sanalysisofblackholes,whichhetreatsasakindofthermodynamic‘pump’absorbingparticlesclassicallyandthenemittingthemquantum-mechanically.Thirdly,acontextmaybefruitfullydominatedbyasinglemetaphor,asintheten-dimensionalsuperstringtheoryofGreenandSchwartz,inwhichdifferentparticlesaredepictedasdifferentmodesofvibrationonafundamental‘string’.8bPhysicalmodelscanhelpustounderstandunfamiliarconcepts.The‘expandingballoon’modelofmoderncosmologyoffersawayofvisualisinganexpandingspatiallycloseduniverse,asinthecaseofbasicmetaphors.Suchmodelsfrequentlyprovidealinkbetweenproblemswithinthetheoryandmorefamiliarphysicalcontexts.cConceptualmodelsfrequentlyhaveacentralroleinhelpingtodevelopatheoreticalidea.Suchmodelsallowustoexploretheimplicationsofthosehigh-leveltheoreticalideaswhichareremotefromthedomainsofobservationandexperiment:forexample,theinflationarycosmologicalmodeloftheearlyuniverse,inwhichaninitialrapidexpansionisdrivenbytheenergyinherentinthevacuumstate—anexpansionfromasneartonothingaswecanget.9Modelsandmetaphorsmayhavetheirlimitationsinsofaras:first,theyfocusonthekeypositiveanalogiesandoftenpushtoonesidenegativeaspectswhichmightneverthelessbeofcriticalimportance;and,secondly,theyintroduceamarkedlyimaginative,intuitive,andtherefore,insomeeyesatleast,undesirablesubjectiveelementintotheoreticalcontexts.However,withoutmodelsandmetaphors,theoriesandtheorisingwouldbeseverelyimpoverished.5Resourcebase.Inordertooperatewithinthetheoreticalcontexteffectively,moreisneededthanmodels,metaphors,andbasicepistemologicalcommitments.Weneedaresourcebasewhichcontainstheessentialtoolsrequiredtodevelopandtoenrichthetheory.Suchabasisallowsustoexploretherangeofapplicationofthetheoryandtoprojectthetheoryintobothfamiliarandunfamiliarphysicalsituations.Theresourcebaseofphysicaltheoriestypicallyincludes:abstractions;solvedproblemsandstandardapplications;andthoughtexperiments:224\nCONCLUSIONaAbstractionsallowscientiststofocusonproblems,helpingtoavoidtoomanydistractions.WhenweusethekeyideaofalocalinertialframeinGTR,weconsiderthe‘limiting’situationofaninfinitesimalregion.Withoutsuchsimplifyingmanoeuvres,itishardtoseehowwecouldploughthroughwhatwouldbeamathematicalquagmire:GTRwouldbepracticallyuseless.Furthermore,someabstractmathematicalmodelstakeonacentralroleintheresourcebase,forexample,Friedmann’scosmologicalsolutionsofEinstein’sfieldequations,whichdependupontheassumptionthattheuniverseiseverywherefilledwithdust,areregardedastheyardstickforanyglobalviewoftheuniverse,bSolvedproblems—Kuhn’sexemplars:GTRincludesanumberofsuccessfulconcretesolutionstopre-existingproblemssuchastheaccountofMercury’sorbitandalsotonewproblemssuchasthephenomenonofredshift,cMathematicaltoolsareofcourseessentialforanyphysicaltheory.ItisinterestingtonotethatatheorysuchasGTRmaybeformulatedmathematicallyinseveralways.Forexample,wemayproduceLagrangianorHamiltonianformulationsofGTR,whichareparticularlyusefulinaddressingquantumgravitationalproblems.10Hence,atheoreticalcontextmaybeenrichedbyhavingaccesstoarangeofmathematicaltools.Indeedwithoutsuchclassicalmathematicaltoolsascanonicalformalism,thedevelopmentofQMwouldhavebeenseriouslyimpeded.11ElieZaharalsoremindsusoftheimportanceofformalismwhenhearguesthatmanyinterestingcontributionstorelativity(STRandGTR)werestimulatedbytheconceptualpotencyandpotentialoftheavailablemathematicaltechniques,whichactedasapowerfulheuristic.12dThoughtexperimentsoftenplayapivotalroleinspellingouttheoreticalideas.Sometimesthoughtexperimentshelpustovisualisewaysinwhichatheorymaybetested.Sometimestheyhelpustomakeaconceptualpointaboutatheory,asinthecaseofNewton’srotatingbucketexperiment.Andsometimestheymaybeusedtoindicatecontradictionsorincoherencewithinatheory.136Values.Theseingredientsofatheoreticalcontextaresupplemented,informed,andconstrainedbycommitmentstoarangeofinternalandexternalvalues:methodologicalvaluesofobjectivityorof225\nTIME,SPACEANDPHILOSOPHYverificationandfalsifiabilityorofsimplicity;pragmaticvaluesofeconomyorofusefulness;motivationalvalueswhichmaydriveindividualscientistsintheirsearchfortruthorcoherenceorconsistencyortheoreticalunity;social,ethical,andpersonalvalueswhichmaybindanddirectthescientificcommunity;andsometimes‘metaphysical’valueswhichmayinfluenceascientificdisciplineforatime,suchastheconvictionbehindtheAnthropicPrinciplethathumanityhasafundamentalplaceinthecosmological‘scheme’.14THELASTWORD?BothLakatosandKuhnregardfundamentallawswithagooddealofreverence.Theyareseenasfocalpointsofatheoreticalcontext.Butwehavestrongreasonsfortakinglawsalittlelessseriously:1ThecomplexityanddiversityofatheoreticalcontextsuchasGTRindicatethatfundamentallawsshouldberegardedasanintegralpartofsuchacontext.2Wemustrecognisethatevenfundamentallawshaveexplicitrestrictionsplaceduponthem:thefieldequationsofGTRaresubjecttoenergyandcausalconditionswhichfocusthemuponphysicalpossibilitiesratherthanmathematicalcuriosities.3Andfundamentallawsalsoseemtohaveimplicitrangesofapplication:thereisnoexpectationthatthelawsshouldapplyinallphysicallypossiblesituations—nooneexpectsGTRtoapplyaccuratelytotheveryearlyuniverseortoconditionswithinablackhole.4Principlesofinvarianceandsymmetryseemtodominatethetheoreticalcontextsofphysicsrathermorethanfundamentallawsassuch:ideasofsymmetry,covariance,andinvarianceallowustofocusupontheconservationofquantitiesinanextremelygeneralway.155Fundamentallawsarenotstaticmuseumpieces:theymaychangeinform,incontent,andintheirrangeofapplicationovertime.ThelawsofNewtonianmechanics,aswenotedinChapter9,havegonethroughanumberofmetamorphoses;andthefieldequationsofGTRhavenotbeenasstableassomewouldliketothink.6AsNancyCartwrightpointsout,fundamentallawsarenotstrictlyspeakingtrueofanythingintheworld:theymaybeaccuratewithintheirabstractdomains,butphenomenologicalandexperimental‘laws’aretypicallyusedtocapturefullythecharacterofthephysical226\nCONCLUSIONworld.16Hence,fundamentallawsshouldbeseenaspartialingredientsforour,frequentlyinter-theoretical,phenomenologicaldescriptionsoftheworld.7Toomuchemphasisuponfundamentallawsisolatesphysicsfromotherscientificfields.Manytheoriesinotherfields,suchasgeologyorbiology,sharemanyofthefeaturesofthetheoreticalmixdescribedabove.Buttheyrarelyhaveanythinglikefundamentallaws.IfCartwrightisrightinthinkingthatfundamentallawscannotbetrue,thenitmaybefoolishtodreamofthedaywhenwecanprovidefundamentallawsforeveryfieldofscience.Theseconsiderationssuggestthatweshouldadoptascepticalattitudetowardsanyclaimforthepre-eminenceofthelawsofatheory.Lawsdohavearoleinscience;butthisroleiswithinthecontextofscienceasawhole.Thiscontextisanevolving,historicalprocess.Hence,anyconclusionswedrawabouttheoriesmusttakethisintoaccount.GTR,likeotherphysicaltheories,hasdevelopedandchangeditscharactermanytimesandinmanywayssinceitsoriginaldevelopment.Theformandcontentofthefieldequationshavechangedandchangedagain.Constantshaveappearedanddisappearedandreappeared.Classicalapproacheshavegivenwaytocompositeclassical—quantummethods.Mach’sPrinciple,theAnthropicPrinciple,andtheCosmologicalPrinciplegoinandoutoffashion.Suchdynamismmakesitdifficulttoresolvemanyofthequestionsraisedinthisbookaboutthecharacterofspace,time,andmotion.Wemightpointtoaparticularisolatedpartofatime-sliceofthetheoryandsay:yes,Einstein’stheory(asinterpretedbyphysicistXinthe1970s)commitsustospacetimeasanirreducibleelementinourdescriptionofgravitationalphenomena;oryes,thecosmiccensorshiphypothesisappliestoGTRconceivedclassically.But,assoonasweseeatheoryofspaceandtimeasanevolvingtheorysetinamoregeneralevolvingscientificcontext,ifsuchassertionsaremeanttoapplytotheentiredynamiccontext,thentheyareratherlessthanaccurate;seeFigure37(p.228).ToomuchattentiontoanyonenarrowcharacterisationofatheorysuchasGTRcarrieswithitnotjustthedangerofinaccuracy.Wemayalsobeledintoadogmaticframeofmind.Withoutdiversitywithinatheoreticalcontext,without227\nTIME,SPACEANDPHILOSOPHYGeneralculturalbeliefs,politicalandeconomicconstraints,scientificinstitutions,technologicalbaseforscience,educationalpractice,generalvalues—allsuchinfluencesconstrainandinformthescientificenterpriseFigure37Theoriesincontextcommitmentstoavarietyofprinciplesandproceduresandformalisms,weruntheriskofintellectualstagnation.RelativistsaretobecongratulatedforthecomplexnatureofGTR.Yes,individualscientistsmayhavedogmaticcommitmentstoagivenpointofview.Buttheentirecommunityshouldbeencouragedtoremainasdiverseaspossible.GTRisatestamenttointellectualdiversity.Althoughwemaybefrustratedbytheinabilityofthetheorytogiveusclear-cutanswerstomany‘basic’questions,thereissufficientcohesioninthetheoreticalcontextasawholetoallowareasonablystraightforwardaccountofmanyproblems.Thatcohesionistypicallynotprovidedbyanyoneelementinthecontext.OurabilitytoaddressphysicalandphilosophicalissuesingeneraldependsonthecontinuityandstabilitywhichGTRprovidesasadynamictheoryinawiderscientificcontext.228\nNOTESINTRODUCTION1ThefulltextofthisreportappearsinPais,A.(1982)SubtleistheLord:theScienceandLifeofAlbertEinsteinOxford:OxfordUniversityPressp.307.1ZENOANDTHELIMITSOFSPACEANDTIME1AristotlePhysics239b5–240a18and233a21–31.2SeeSalmon,W.C.(ed.)(1970)Zeno’sParadoxesIndianapolis:Bobbs-Merrillp.34;andPenrose,R.(1971)‘Angularmomentum:anapproachtocombinatorialspace-time’inBastin,E.(ed.)QuantumTheoryandBeyondCambridge:CambridgeUniversityPress;andalsoPenrose,R.(1987)‘Newton,quantumtheory,andreality’inHawking,S.W.andIsrael,W.(eds)300YearsofGravitationCambridge:CambridgeUniversityPress,inwhichPenroserenewshisinterestinthenon-continuous‘twistor’approachtospacetime.3See,forexample,Peirce,C.S.(1935)CollectedPapersCambridge,MA:HarvardUniversityPress;PeircesaysthattheAchillesparadox‘presentsnodifficultytoamindadequatelytrainedinmathematicsandlogic’,vol.VI,p.177.4Stewart,I.(1987)TheProblemsofMathematicsOxford:OxfordUniversityPress.5Owen,G.E.L.(1957)‘Zenoandthemathematicians’ProceedingsoftheAristotelianSociety58p.199.6Salmon,W.C.(ed.)(1970)op.cit.,p.141.7Simpliciuswasasixth-centuryADcommentatoronAristotle.8Barnes,J.(1979)ThePresocraticPhilosophersLondon:Routledgepresentsadetaileddiscussionofthisparadox;seevol.I,ch.12.9Cauchy,A.(1821)Coursd’AnalyseParis.10fcourse,ifAchillesisrunningatasteadyspeed,hisspeedisalwaysgreaterthanthatofthetortoise;andwecanthereforepredictthat,whenAchillesapproachesthetortoise,hewillindeedovertakeit.11Thomson,J.F.(1954)‘Tasksandsuper-tasks’Analysis15pp.1–13.12Sainsbury,R.M.(1988)ParadoxesCambridge:CambridgeUniversityPress.13Benacerraf,P.(1962)‘Tasks,super-tasks,andthemoderneleactics’JournalofPhilosophy59pp.765–84;seealsoBerresford,G.C.(1981)‘AnoteonThomson’slamp“paradox”’Analysis41pp.1–7.14Sainsbury,R.M.(1988)op.cit.,p.1.229\nNOTES15Black,M.(1950)‘Achillesandthetortoise’Analysis11pp.91–101.16Salmon,W.C.(ed.)(1970)op.cit.,p.34.17Hesse,M.B.(1966)ModelsandAnalogiesinScienceNotreDame,IN:NotreDameUniversityPressprovidesaclassicstatementoftheroleofmodelsinphysicalscience.18Newton-Smith,W.H.(1980)TheStructureofTimeLondon:Routledge,pp.68–73,121–6.19ThethesismaybetracedtoPierreDuhem’sideasinhis(1906)Latheoriephysique:sonobjetetsastructureParis;thequotationhereisfromQuine,W.V.O.(1970)‘Onthereasonsforindeterminacyoftranslation’JournalofPhilosophyLXVII6pp.178–83.SeealsoHesse,M.B.(1974)TheStructureofScientificInferenceLondon:Macmillan,andHookway,C.(1988)Quine:Language,ExperienceandRealityCambridge:PolityPress.2CLOCKS,GEOMETRYANDRELATIVITY1TheideaofaninertialframeofreferencewillbeexplainedinFigure4(pp.27–32)andinthefollowingsection.2Einstein’s(1905)paperintroducingSTRappearsintranslationinStachel,J.J.(ed.)(1989)TheCollectedPapersofAlbertEinsteinvol.IITheSwissYears:Writings,1900–1909Princeton:PrincetonUniversityPress,EnglishtranslationtextbyAnnaBeck;seealsoEinstein,A.,Lorentz,H.A.,Weyl,H.,andMinkowski,H.(1923)ThePrincipleofRelativityLondon:Methuen;theoriginalpaperappearsunderthetitle‘ZurElektrodynamikbewegterKorper’AnnalenderPkysik17(1905)pp.891–921,alsoreprintedinStachel,J.J.(ed.)(1989)op.cit.(maintext),pp.276–310.3SeeBergson,H.(1976)‘Discussionoftheparadoxofthetwins’inCapek,M.(ed.)(1976)TheConceptsofSpaceandTimeDordrecht:Reidel.4Wemeasurethelifetimeofsuchparticlesintermsoftheirhalf-life—althoughwecannotsaywhenanygivenparticlewilldecayfromitsunstableform,wecansaywithhighaccuracythatalargegroupofsuchparticleswillbereducedtohalftheoriginalnumberinaspecifictimeandwillbereducedtoaverysmallquantityindeedwithinadefiniteperiod.FordetailsofthisexperimentseeFrisch,D.H.andSmith,J.H.(1963)‘Measurementoftherelativistictime-dilationusingmu-mesons’AmericanJournalofPhysics31p.342;andalsoFrench,A.P.(1968)SpecialRelativityLondon:VanNostrandReinhold,whichreportstheexperiment:pp.98–9.5SeeSalmon,W.C.(1980)Space,Time,andMotionsecondeditionMinneapolis:UniversityofMinnesotaPressforamoredetailedreportofthisexperiment.6Ofcourse,wewillonlybelievethepredictionsoftheLTsolongasthelawsofphysicsarenotviolatedinanyway.STRneedstobeconsistentwith,forexample,theacceptedlawsofelectrodynamics.ThisconstraintuponSTR—theso-calledPrincipleofRelativity,demandingthatthelawsofphysicsshouldholdgoodinallinertialframes—wasoneofthebuildingblocks—ofthetheory.7Oneofthereasonsforadoptingarelativisticperspectiveisthefactthatdistanteventscannotbesaidtobesimultaneousgiventhelackofanysignalcapableoftravellingbetweenclocksinstantaneously.Theideaofsimultaneity230\nNOTESmayonlybeattachedtonearbyevents.So,becausethetimethelighttakestotravelfromBigBentomeisfinite,whenIsynchronisemyclockwithBigBen,‘mytime’isalways‘behind’byafactorrelatedtothedistancebetweenmeandBigBenwhenthesynchronisationtookplace.Buttime-lagisfixed;solongasIstayputmyclockwillbejustasmuchbehindBigBenattheendofthehourasitwasatthebeginning.8Wherethefactorb=(1-v2/c2)-1/2.Becausevcannotbegreaterthanc,itisclearthatbmustbegreaterthanorequalto1.9SofarIamallowingSTRthisluxury—butinChapters3and4,whichfocusonconventionalism,weshallexaminethethinkingbehindtheassumptionthatclocksinthesameframemaybesaidtosynchroniseperfectly.10Thespacetimeinterval(Ds)betweentwoeventsisgivenby:Dsthereforedependsonthespatialdistances(Dx,Dy,Dz—thedifferencesinthex,y,zcoordinates)andthetemporalduration(Dt)asobservedfromanyinertialframeofreference(withunitschosensothatc=1).Afinitespacetimeintervalisinvariantintheflat,pseudo-EuclideanspacetimeofSTR;but,ingeneral,incurvedspacetimeorinflatspacetimewithanon-Euclideantopology(e.g.aplanewrappedaroundacylinder),onlytheinfinitesimalintervalbetweentheneighbouringpointsisinvariant.11Inpracticetheforcesinvolvedwouldbetremendousiftheaccelerationsinvolvedaretoosudden.Notonlywouldmostlivingcreaturesfailtosurviveanytoorapidchangeofdirection,butphysicalobjectslikeclockswouldalsoberippedapart.Generally,physicistsovercomethis‘technical’problembyspeakingofidealstandardclocks,whoseconstructionwouldnotbesoaffected.STRmaythenbeusedtopredictwhatwouldhappen.Wecouldofcourseslowthechangeofdirectiontoapointwheredestructiveforceswouldnotruintheexperiment.ThisisexactlywhatwasdoneintheHalfele-Keatingexperimentmentionedintheintroductiontothischapter.But,iftherelativespeedsaretoolow,asintheHalfele—Keatingexperiment,then,althoughwewillbeabletomeasuredifferencesbetweensomekindsofphysicalclocks,thechancesofbeingabletodetectadifferenceinageinginlivingcreaturesareindeedremote.Soanymeasurableinfluenceontheageingofatravellerwouldrequireahighrelativevelocity.Althougharound-triptonearbystarsisindeedpossibleinprinciple,suchajourneywouldrequireaverylargeamountoffuel.Moreover,therocketcarryingatravelleronsuchajourneywouldrequireatremendousamountofshieldingtoprotectthetravellerfromcollisionswithinter-stellarmaterial.Hence,thechancesofmakingsuchatriparepracticallynil.SeeTaylor,E.F.andWheeler,J.A.(1963)SpacetimePhysicsSanFrancisco:W.H.Freeman,especiallytheexerciseonp.174andtheexerciseanswerinthesupplement,pp.60–1.12Thisstoryofthetwinshasbeensimplifiedagooddeal.Forwemustrememberthattheearthrotatesonitsaxisaswellasaroundthesun.And,sincerotationsareaccelerations,givingrisetoinertialforces,thetwinontheEarthtoomustchangeframe,notjustonce,butallthetime.Butwecouldretellthestorywithoutproblemsandleadingtothesameessentialresultifwetakethetwinwhostays‘behind’tobelocatedinaframeofreferencefixedbythebackgroundmicrowaveradiation—aremnantfromtheveryearly231\nNOTESuniverse—whichisnotrotating,accordingtotheavailableobservationalevidence.Thisstoryhasbeentoldmany,manytimesinalmostasmanyways.Furtherdetailsmaybefoundinalmostalloftheseaccounts;Ihavemyownfavouriteversions—allofthemgenerallyclearandfreefromerror:Bondi,H.(1967)AssumptionandMythinPhysicalTheoryCambridge:CambridgeUniversityPressforasimpleversion;French,A.P.(1968)op.cit.andTaylor,E.F.andWheeler,J.A.(1963)op.cit.formoredetailedandmathematicalaccounts;andalso,foradetailedphilosophicalanalysis,Newton-Smith,W.H.(1980)TheStructureofTimeLondon:Routledge;finally,foranextremelythoroughhistoryofthevariouspositionstakenupontheparadox,Marder,L.(1971)TimeandtheSpace-TravellerLondon:GeorgeAllenandUnwin.13Thisthemewillbeexploredseveraltimesinthisbook—foritrecursinmanyofthedifferentissueswithinthephilosophyofspaceandtime.14Strictlyspeaking,objectsdonot‘movealong’worldlines,fortheworldlineisitselfarepresentationoftheobject’smotion;seeFigure4(pp.27–32).15Thetripletsinvolvedinthisthoughtexperimentcouldnotbegenuinetriplets,forsomeprioraccelerationswouldbeneededtosetthetwotravellingtripletsoffontheirinertialtrajectories—ifwemaketheassumptionthatatsometimeinthepastallthreeandtheirclocksweretogetheronEarth.Thenwecouldnotruleoutthoseaccelerationsasthecauseofthedifferencesinageing.16Althoughsuchsynchronisationsandcheckswouldnotbeeasy,theywouldnotinvolveanysynchronisationorcheckatanygreatdistance.Iftheworldlinescomeclosetoagenuineintersection,then,asthetripletsandtheirclockspasseachother,areasonablytellingglimpseofeachotherandafairlyaccuratereadingofeachother’sclockscouldbemade.Solongasthespeedsinvolvedanddistancestravelledintheexperimentaresufficientlylarge—keepingthemarginoferrorsmallinproportiontotheresultsoftheexperiment—thenthereislikelytobeagoodmatchbetweentheoreticalpredictionsandactualobservations.17MorecomprehensivedetailsofthisversionoftheparadoxmaybefoundineitherBondi,H.(1967)op.cit.orNewton-Smith,W.H.(1980)op.cit.Wemayalsoachievethesamegeneralresultwithtwinsiftheyinhabitauniversewhichiscylindrical—onetwinstaysinthesameposition(representedbyaverticallinerunningupthesideofthecylinder)andtheothermovesawayinertially(representedbyastraightlinerunningaroundandupthecylindertointerceptthefirstworldlineafteronecompletecircuitofthecylinder).Noaccelerationneedbeinvolved.Theirspacetimejourneysdiffer,sowehavenoreasontosupposethattheirclocksandageswillagreewhentheyreunite.18Bohm,D.(1965)TheSpecialTheoryofRelativityNewYork:W.A.Benjamin;seeespeciallypp.165–67.19French,A.P.(1968)op.cit.20Ingeneral,GTRdoesprovidearathermoreelegantwayofdealingwithacceleratedmotionthanSTR.21Newton-SmithinhisversionadmitsthattheclocksCandCwillnotagree13atpointR,butarguesthatthereisneverthelessareciprocalretardationbetweenthetwospacetimepaths.TheswitchfromCtoCinvolvesa23discontinuityinwhichacontinuousseriesofeventsinC’shistoryistaken1232\nNOTEStooccursimultaneouslyfromthepointofviewofwhathecallsthecompositeclockconsistingofCandCtakentogether.Sothecompositeclockfailsto23recordthistime,whichisshowntobeequaltothedifferenceintimesontheclocksmeetingatR.Thisstrikesmeasasoundexplanationofwhydifferentspacetimepathsdonotingeneralagree,butthereciprocityinvolvedhereseemstodonomorethanclarifythedifferencesbetweenthetwopaths.For,ifweadmitthefactthattheclocksmeetingatRdoinfactdisagree,thereisarealphysicaldifference,arealasymmetry,whichcannotbeexplainedbyforcesperse,butonlybyspacetimegeometry.SeeNewton-Smith,W.H.(1980)op.cit.,pp.192–5.3TRAVELLINGLIGHT1Thisexperiment,carriedoutin1849,isdescribedclearlyinHoffman,B.(1983)RelativityandItsRootsNewYork:ScientificAmericanBookspp.49–50.2Salmon,W.C.(1977)‘Thephilosophicalsignificanceoftheone-wayspeedoflight’Nous11,3pp.253–92.3Ellis,G.F.R.andWilliams,R.M.(1988)FlatandCurvedSpace-TimesOxford:OxfordUniversityPresspresentsanexcellentdiscussionofthephysicsinvolved.4Notethattheconventionalityofsimultaneityisquitedistinctfromtherelativityofsimultaneity.Therelativityofsimultaneityarisesfromthefactthatdifferentobserversviewingtwoeventsinthesameframefromdifferentperspectiveswillingeneraldisagreeaboutthetemporalrelationsbetweenthoseevents.Butifsimultaneityisalsoconventionalanygivenobservercannotbesureaboutwhicheventsinthesameframearesimultaneouswithsomeeventlocaltotheobserver.5Theseareonlythreeofthetypicalbeliefsofonekindofscientificrealist;thereareotherbeliefsandthereareotherkindsofscientificrealist;weshallbeconsideringsomeoftheseinlaterchapters,especiallyChapters7,9,and10.Forfurtherdetailsontheissuesinvolvedinscientificrealism,seeHarré,R.(1986)VarietiesofRealismOxford:BasilBlackwell;Leplin,J.(ed.)(1984)ScientificRealismBerkeley:UniversityofCaliforniaPress;andDevitt,M.(1984)RealismandTruthPrinceton:PrincetonUniversityPress.6SeeQuine,W.V.O.(1975)‘Onempiricallyequivalentsystemsoftheworld’Erkenntnis9pp.313–28.7Laterinthissectionweshallexamineanotherattempttoestablishtheone-wayspeedoflightasanobjectiveempiricalfact:slowtransportclocksynchrony.8Foranexcellentaccountofvariousattemptstodiscovertheone-wayspeedoflightdiscussedhere,seeHoffman,B.(1983)op.cit.,ch.4,aswellasSalmon,W.C.(1977)op.cit.andSalmon,W.C.(1980)Space,Time,andMotionsecondeditionMinneapolis,MN:UniversityofMinnesotaPress;ReichenbachH.(1957)ThePhilosophyofSpaceandTimeNewYork:Doveralsodiscussesanumberofwaystodeterminetheone-wayspeedoflightandtotherebyprovideanobjectivedefinitionofsimultaneity.9Reichenbach(1957)op.cit.,section22.233\nNOTES10SeeGrunbaum,A.(1969)‘SimultaneitybyslowclocktransportintheSpecialTheoryofRelativity’PhilosophyofScience36pp.5ff.;Grunbaum’sarticleisreprintedwithanintroductionco-authoredbySalmoninGrunbaum,A.(1973)PhilosophicalProblemsofSpaceandTimesecondeditionDordrecht,Holland:D.Reidel.11Winnie,J.A.(1970)‘Specialrelativitywithoutone-wayvelocityassumptions’,PartsIandII,PhilosophyofScience37pp.81–99,223–38;seealsoGiannoni,C.(1978)‘Relativisticmechanicsandelectrodynamicswithoutone-wayvelocityassumptions’PhilosophyofScience45pp.17–46.12Malament,D.B.(1977)‘Causaltheoriesoftimeandtheconventionalityofsimultaneity’Nous11pp.293–300;seealsoBrown,H.R.(1990)‘DoestheprincipleofrelativityimplyWinnie’s(1970)equalpassagetimesprinciple?’PhilosophyofScience57pp.313–24;andFriedman,M.(1983)FoundationsofSpace-TimeTheoriesPrinceton:PrincetonUniversityPress.13Salmon,W.C.(1980)op.cit.,p.119.14Salmon,W.C.(1980)op.cit.,p.120.15Weshouldnotethatthesenseoftimetravelhereoughttobeunobjectionabletothosewhoopposetheideaoftimetravelonthegroundsthatitinvolvesclosedcausalloops.Nosuchloopsareinvolvedhere,sinceallthatisdoneisto‘skew’theplaneoftransmissioninNewtonianspacetime.16Ofcourse,thesameproblemwouldariseforSTRifthespacetimestructureweretobechangedtoallowthepossibilityofasimilarkindoftimetravel.17Ellis,B.andBowman,P.A.(1967)‘Conventionalityindistantsimultaneity’PhilosophyofScience34pp.116–36;andalsoAngel,R.B.(1980)Relativity:theTheoryandItsPhilosophyOxford:Pergamon.18Descartes,R.(1988)SelectedPhilosophicalWritingCambridge:CambridgeUniversityPress,translatedbyCottingham,J.,Stoothoff,R.,andMurdoch,D.;seep.58andalsop.42.19AsHoffman,B.(1983)op.cit.pointsout,Rømer’scalculationofthetimetakenissome5minutesshortoftheactualfigure.20See,aswellasSalmon(1980)op.cit.,Grunbaum,A.(1969)op.cit.,pp.5ff.;hisremarksinthispaperaredirectedtowardsEllis,B.andBowman,P.A.(1967)op.cit.21See,forsuchastandardpositiononrelativity,Hawking,S.W.andEllis,G.F.R.(1973)TheLargeScaleStructureofSpace-TimeCambridge:CambridgeUniversityPress;ofcourse,thisstandardviewneednotmaketheclaimthatnothingtravelsfasterthanlightisactuallyapostulateofthetheory:wemaysaythatrelativityisdistinguishednotbyabeliefinthefactthatnothingmaytravelfasterthanlightbutbytheassertionthatthespeedoflightisaninvariant;seeChapter1.22SeetheexcellentreviewandanalysisinRecami,E.(1986)‘Classicaltachyonsandpossibleapplications’NuovoCimento9,6pp.1–178.23See,fordetailsofthebehaviouroftachyonsinrelativisticspacetimes,Friedman,M.(1983)FoundationsofSpace-TimeTheoriesPrinceton:PrincetonUniversityPress;andRecami,E.(1986)op.cit.NotealsoEarman,J.(1972)‘Causalpropagationoutsidethenullcone’AustralianJournalofPhilosophy50pp.222–37;Earmanraisesanumberofobjectionstotheideaofspaceliketravel,asweshallseeinlatersectionsofthischapter.234\nNOTES24Tolman,R.C.(1917)TheTheoryoftheRelativityofMotionBerkeley:UniversityofCaliforniaPress;andBohm,D.(1965)TheSpecialTheoryofRelativityNewYork:Benjamin;seeRecami,E.(1986)op.cit.forananalysisofthisandothertalesabouttachyons,pp.64–77.25Feynman,P.R.(1949)‘Thetheoryofpositrons’PhysicalReview76pp.749–59.26Recami,E.(1986)op.cit.,p.66.27Elliot,R.(1981)‘Howtotravelfasterthanlight’Analysis41,1pp.4–6;seealsoRay,C.(1982)‘Canwetravelfasterthanlight?’Analysis42,1pp.50–2.28Elliot,R.(1981)op.cit.,p.5.29Theproblemofpersonalidentityisamare’snestwhichweshallleaveforotherbraversouls;hereweshallbeascavalierasElliotandmakesome‘courageous’assumptionsaboutwhatconstitutesaperson’sidentity;butseeParfit,D.(1984)ReasonsandPersonsOxford:OxfordUniversityPressforamoreconsideredaccountofidentity—heincludesadetaileddiscussionoftherelationshipbetweenidentityandpsychologicalcontinuity,pp.219ff.4ACONVENTIONALWORLD?1SeeNerlich,G.(1976)TheShapeofSpaceCambridge:CambridgeUniversityPressforacomprehensivestudyofEuclideanandnon-Euclideangeometries;NerlichremindsusthatEuclideangeometryisconstructedfromthesebuilding-blocks:‘definitionsofthevariousgeometricalfigures…axiomssupposedtobetooobvioustoadmitofproof…andpostulateswhichadmitofnoprooffromsimplerstatements,butwhicharenottakenasobvious’;p.51.2Poincaré,H.(1952)ScienceandHypothesisNewYork:Dover,atranslationbyGreenstreet,W.J.,oftheoriginalFrencheditionof1902.3Reichenbach,H.(1957)ThePhilosophyofSpaceandTimeNewYork:Dover;thisisatranslationofReichenbach’s(1927)PhilosophiederRaum-Zeit-LehrebyMariaReichenbach,hiswife.4TheideathattwoempiricallyequivalenttheoriesareessentiallythesameisalsoreflectedinvanFraassen,B.C.(1980)TheScientificImageOxford:OxfordUniversityPress.5SeethearticlesbySchwartzandLindeinHawking,S.W.andIsrael,W.(eds)(1987)300YearsofGravitationCambridge:CambridgeUniversityPress.6Weshouldnotethattherearefourdistinctbutinter-relatedlevelsofstructureforspaceandtime:metricalgeometry,affineandconformalstructures,andtopologicalstructure;seeFigure11(p.54).7SeetheaccountsinReichenbach,H.(1957)op.cit.;andStewart,I.(1987)TheProblemsofMathematicsOxford:OxfordUniversityPress;Gaussissaidtohavediscoveredtheessentialideasbeforethejointbutindependent‘discovery’bytheothertwo,butinexplicablyGaussdidnotpublishhisresults.235\nNOTES8Riemann,G.(1873)‘Onthehypothesesthatlieatthefoundationsofgeometry’Nature8p.14;Riemann’s(doctoral)paperof1854wastranslatedbyClifford,W.K.,theCambridgemathematicianresponsibleforthepropheticsuggestionthatmattermightbenomorethanhill-likeripplesinanotherwiseflatspace.9Minkowski,H.(1908)‘Spaceandtime’reproducedintranslationinEinstein,A.,Lorentz,H.A.,Weyl,H.,andMinkowski,H.(1923)ThePrincipleofRelativityLondon:Methuen.AlthoughthespacetimeofSTRisnotEuclideanperse,itsglobalflatness,guaranteedbytheabsenceofgravitatingmatter,givesitaEuclideancharacter.SoSTRspacetimesareoftencalled‘pseudo-Euclidean’.10ReportedinCalder,N.(1979)Einstein’sUniverseLondon:BBCBooks.11Eddington,A.S.(1920)Space,Time,andGravitationCambridge:CambridgeUniversityPress;ch.7givesafullreportoftheconfirmation.12Reichenbach,H.(1957)op.cit.;Reichenbach’sideaissimilarinmanyrespectstoanearliersuggestionbyPoincaré.SeeSklar,L.(1974)Space,Time,andSpacetimeBerkeley,CA:UniversityofCaliforniaPressforadetailedaccountofPoincaré’soriginalsuggestion.ThebasicideaofReichenbach’sapproachisgivenhere.Inordertohelpusminimiseproblemsofvisualisation,Reichenbachasksustoconcentrateonatwo-dimensionalworld.Heconstructsaworldinwhichtwoisolatedcommunitiesexistonthetwooppositesurfacesofaglassblock.Eachmemberofthetwogroupsisimaginedtobeatwo-dimensionalcreaturestucktohisorherparticularsurface.Sonoonecan‘standup’fromthesurfaceandtakeaprivilegedthree-dimensionallookaroundinordertodeterminethegeometryoftheworldtheyinhabit.However,eachgroupmaycarryoutaseriesofmeasurementsofdistancesonthesurfaceitselfusingrigidmetrerulesandothermeasuringinstruments.Bydoingsotheymaydeterminethemetricalstructureofthetwo-dimensionalsurfaceuponwhichtheymove.Eachgroupisabletoviewthemeasurementsmadeontheothersurfacesincetheglassblockistransparent.Ifthereisanycurvatureoneithersurface,thenwewouldexpectthemeasurementstakentoshowthistobethecase—justasmeasurementsmadeonthesurfaceofasphericaloranalmostsphericalobjectliketheEarthcandemonstrateitsoverallshape:onsuchasurface,forexample,theinternalanglesofatrianglewouldbegreaterthan180degrees.The‘top’surfaceisflatexceptforacentralhemispherical‘hump’.The‘bottom’surfaceisflateverywhere.WewouldthereforeexpectthefirstgrouptosaythattheirworldisgenerallyflatandEuclideanexceptforacentralregionwhichiscurved.AndthesecondgroupshouldsaythattheirworldisEuclideaneverywhere.However,theobservationsofthesecondgroupagreeperfectlywiththoseofthefirst!Indeed,afloodoflightshininguniformlydownwardsupontheworld,castsshadowsofthemetre236\nNOTESrulersonthetopsurfaceontothebottomsurfaceandtheseshadowscoincideexactlywithalllengthsonthebottom.Indeed,everything,includingthepeopleonthelowersurface,isaffectedinthesameway.13Reichenbachcallssuchforces‘universal’sincetheyaffectdifferentkindsofobjectsinexactlythesameway.Henotesthattheeffectsofuniversalforcescannotbeshielded.Hedistinguishesthemfrom‘differential’forceswhichdonotaffectallobjectsinthesameway:forexample,onedifferentialforceisamagneticforcewhichhasagreatereffectonsomematerialsthanonothers.SeeReichenbach,H.(1957)op.cit.,pp.24–8.14Reichenbach,H.(1957)op.cit.,pp.43–4.15Theproblemofvisualisationofgeometrieshereiscomplicatedbythefactthatthegeometriesincorporatedinmoderngravitationaltheoriesarefour-dimensional.AnimaginationconditionedtothinkofgeometryspatiallywillfindevenaEuclideanfour-dimensionalviewperplexing.ForfurtherdetailsofsuchproblemsseeNerlichonKant.TheGermanphilosopherKant,writingintheeighteenthcentury,suggeststhatwemustperceivetheworldspatio-temporally—inanimportantsenseweconstructtheworldinspaceandtime;andhealsoclaimsthatthisconstructionmustbeEuclidean.Headmitsthatourperceptionoftheworldassuchimpliesnothingabouttheworldasitreallyis—justthatwemustseeitassuch.SeeNerlich,G.(1976)op.cit.forafulldiscussionofKant’sthinking.Alsonotetheexcellentreviewofvariouswaysofvisualisingspatio-temporalworldsinChandler,M.(1990)‘Philosophyofgravity:intuitionsoffour-dimensionalcurvedspacetime’inHerget,D.(ed.)MoreHistoryandPhilosophyofScienceinScienceTeachingTallahassee:FloridaStateUniversity.16Noting,ofcourse,thattheideaofastraightlineisdefinedbytheaffinestructureandtheideaofanangleisdefinedbytheconformalstructure;bothofthesestructuresaremoregeneralthanthemetricalstructuresothatdifferentmetricalgeometriesmaybeconsistentwithmorethanoneoftheseotherstructures.SeethediscussioninFriedman,M.(1983)FoundationsofSpace-TimeTheoriesPrinceton,NJ:PrincetonUniversityPress.17Aspacetimeinwhichclosedpathsmaybeshrunkdowncontinuouslytoapointiscalled‘simplyconnected’.18ThetopologicaldiscussioninthissectionisbasedonReichenbach’sworriesabouttopologyinReichenbach,H.(1957)op.cit.;Reichenbachconcentratesontheempiricalequivalencebetweentheplaneandthesurfaceofatorustofuelhisdiscussion;theexampleofthecylinderworldanditscoveringspace,thestripworld,usedinthissection,maybefoundinSklar,L.(1974)op.cit.,ch.2.19Thisdispute,raisedbyNewton-Smith,W.H.(1980)TheStructureofTimeLondon:Routledge,isdiscussedinChapter1.237\nNOTES20Thesequenceofstripsontheplanerepresentsa‘coveringspace’forthecylinder-wemayeasilyimaginethestripsbeingwrappedaroundthecylindersothateachstripmatchesupwiththeeventsonthecylinderexactly.21Reichenbach(1957)op.cit.,pp.58–81.22Ehrenfest,P.(1917)‘Inwhatwaydoesitbecomemanifestinthefundamentallawsofphysicsthatspacehasthreedimensions?’ProceedingsoftheAcademyofNetherlands20p.200;andWeyl,H.(1922)Space,Time,andMatterNewYork:Dover:seep.284.23SeeDavies,P.C.W.andBrown,J.(eds)(1988)Superstrings:aTheoryofEverything?Cambridge:CambridgeUniversityPress;thedescriptionhereisbasedonSchwarz’sowncharacterisationofsuperstringtheory,whichhegivesinaninterviewreproducedinthebook.24SeeDavies,P.C.W.andBrown,J.(eds)(1988)op.cit.,pp.192–210forthecompletetextoftheinterviewwithFeynman.25SeeLinde,A.(1987)‘Inflationandquantumcosmology’inHawking,S.W.andIsrael,W.(eds)(1987)op.cit.26Barrow,J.D.(1983)‘Dimensionality’inMcCrea,W.H.andRees,M.J.(eds)(1983)TheConstraintsofPhysicsLondon:RoyalSocietypp.337–46.27Notealsothatopenspacetimeswithaplanestructurearenon-compactandclosedspacetimeswithasphericalstructurearecompact;seeWald,R.M.(1984)GeneralRelativityChicago:ChicagoUniversityPressfortechnicaldetails;alessdauntingdiscussionmaybefoundinSklar(1974)op.cit.28See,foraparticularlyclear‘verdict’onthisissue,Gribbin,J.andRees,M.(1989)CosmicCoincidencesLondon:Bantam.29SeePenrose,R.(1979)‘Singularitiesandtime-asymmetry’inHawking,S.W.andIsrael,W.(eds)GeneralRelativity:anEinsteinCentenarySurveyCambridge:CambridgeUniversityPress.30Acloseduniverseislikelytohavea‘sphericallyclosed’geometryandanopenuniverseislikelytohaveaflatornearlyflatgeometry—giventheavailableevidence;Ellis,G.F.R.andWilliams,R.M.(1988)FlatandCurvedSpace-TimesOxford:OxfordUniversityPressnotethatthepropertiesofhomogeneityandisotropymaybeneatlyexplainedintermsofthe‘multipleimage’torusuniverse—thesmalluniverse—inwhichthesamefinitecollectionofgalaxiesisviewedwhereverwelookoverandoveragain:hencethe‘appearance’ofaninfiniteworldinwhicheverywhereandeverydirectionseemstobethesame—itlooksthesamebecauseitisthesame!31Ellis,G.F.R.(1978)‘Istheuniverseexpanding?’GeneralRelativityandGravitation8pp.87–94.32SeeThurston,W.P.andWeekes,J.R.‘Themathematicsofthree-dimensionalmanifolds’ScientificAmericanJuly1984,andalsoEllis,G.F.R.andWilliams,R.M.(1988)op.cit.33Ellis,G.F.R.andWilliams,R.M.(1988)op.cit.,pp.287–8.238\nNOTES34Inthesmalluniverse,therecanbenosurprises,forwecanseeallitshistoryatleastinprinciple;buttheinfinite,planeworldcouldalwaysthrowupasurprise.For,inagenuinelyinfiniteworld,wecannotruleoutthearrivalofsomethingwhichmightdisturbourenvironmentfromsomepartoftheuniversetowhichwehavehadnoaccesssofar.Ourinformationaboutthepastisincomplete,andsopredictionwouldalwayscarryadegreeofuncertaintyevenifweweresureaboutthelawsgoverningtheuniverse.Inatorusworld,everypredictioncouldbespot-on(atleastinprinciple).Butthepossibilityofdifferencesinpredictivepowersuggeststhattheremaybesomesituationwhichallowsustodistinguishtheplaneandthesmalluniversesonempiricalgrounds.35Duhem,P.(1954)TheAimandStructureofPhysicalTheoryPrinceton:PrincetonUniversityPress.36Quine,W.V.O.(1980)FromaLogicalPointofViewsecond,revisededitionCambridge,MA:HarvardUniversityPress.NotetheastonishingextensionofDuhem’sbasicideahere:fromtalkofthestatementsofscienceandourgeneralexperiencetoallstatements,includinganalyticstatementssuchasthelawsoflogic.WemayindeedbethoroughlysurprisedbyQuine’scontentionthatthelawsoflogicandotheranalyticstatementscouldbefalse.37Popper,K.(1963)ConjecturesandRefutationsLondon:Routledgep.37;andFeynman,R.(1985)SurelyYou’reJoking,Mr.FeynmanLondon:UnwinHymanpp.338ff.38Quine,W.V.O.(1975)‘Onempiricallyequivalentsystemsoftheworld’,Erkenntnis9pp.313–28.39ThisthesisisbasedontheweakstatementoftheDuhem—QuinethesisinHesse,M.B.(1980)‘Thehuntforscientificreason’inProceedingsoftheConferenceofthePhilosophyofScienceAssociation1980,vol.II,Asquith,P.andGiere,R.(eds)EastLansing,MI:MichiganStateUniversityPress;themajorproblemsforthisthesiswhichfollowandtherestatementofastrongerversionareallsuggestedbyHesseinthisarticle.40Ifwedropthisassumptionthenwemightstilltakeananti-realistpositionbyarguingthatallequivalenttheoriesaremerelyexpressionsofthesametheorybutthatnoalternativeisthe‘true’theorysinceeachtheoryisreducibletoexactlythesameobservationalbasis;thatis,wewouldbedenyingthe‘reality’ofall‘theoretical’terms.ThisisessentiallythepositiontakenbyReichenbach:seethecommentsonversionDofthethesis.41Wewouldstillneedtobeawareofthecomplexitiesinvolvedincitingsimplicityorsomesuchotherpragmaticreasonforourchoice;note,forexample,thefactthatonetheorymaybesimplerthananotherontologically,butmightneverthelessbemorecomplexepistemologically.seeFine,A.(1986)TheShakyGameChicago:ChicagoUniversityPress.Seealsothe239\nNOTESbriefdiscussionofdifferentkindsofsimplicityinRay,C.(1987)TheEvolutionofRelativityBristol:AdamHilger.42Thisisavexedquestion.See,forexample,thediscussioninSklar,L.(1974)op.cit.,whoalsoprovidesanextremelyclearaccountoftheissuesunderconsiderationinthischapter.ThealternativesmentionedarebothsuggestedbySklaraspossibleroutesthroughthisepistemologicalminefield.ButseeHesse,M.B.op.cit.andalsoHesse,M.B.(1974)TheStructureofScientificInferenceLondon:MacMillanforanargumentinfavouroftakingsimplicityasthetouchstoneoftheorychoice.43Hanson,N.R.(1958)PatternsofDiscoveryCambridge:CambridgeUniversityPressandKuhn,T.S.(1970)TheStructureofScientificRevolutionssecondeditionChicago:ChicagoUniversityPress.44IshallexplorethevariousargumentsforandagainstthedistinctioninChapter6.45SeeHesse,M.B.(1974)forthedetailsofthisargument.46Papineau,D.(1979)TheoryandMeaningOxford:OxfordUniversityPressandHookway,C.(1988)QuineCambridge:PolityPressbothgiveclearaccountsofthedifficultiesinvolved.47SeeFriedman,M.(1983)op.cit.,pp.320ff.48Occam’srazor(sometimesOckham’srazor)isnamedafterWilliamofOckham’sprincipleofparsimonyorsimplicity.Ockham,writinginEnglandinthefourteenthcentury,frequentlyusedthisprincipleinhiswritingsonlogicandtheology;itsbestknownformisperhaps:youshouldnotmultiplyentitieswithoutnecessity,i.e.keepyourmetaphysicsassimpleaspossible.5NEWTONANDTHEREALITYOFSPACEANDTIME1Foracomprehensiveanddefinitivesurveyofthisdebate,seetheexcellentaccountbyBarbour,J.B.(1989)AbsoluteorRelativeMotion?vol.ICambridge:CambridgeUniversityPress.2AlthoughIamsympatheticuptoapointwithanyonewhomightsaythatthePrincipiadoesnotjustifyattributingthe‘container’ideadirectlytoNewton,IwouldjustifytheuseofthisideaintheNewtonianworld-view,givenhisbeliefthatGodcreatedthematerialworldandplaceditinspaceatagiventime.Yes,Newtondidresisttheideaofspaceasabackground‘substratum’;butthisresistancederivesfromhisdispleasurewithDescartes’snotionthatspaceistheprimarysubstanceandthatmatterismerelyanaspectofthissubstance.Asmentionedlaterinthischapter,themaintargetofNewton’sargumentsaboutspaceseemstohavebeenDescartes.3Steps1to8followtheargumentinthePrincipia(Scholiumtodefinition8)closely,butsteps9and10arenaturalinferencesfromthesubstanceandcontextoftheargument;seeRay,C.(1987)TheEvolutionofRelativityBristol:AdamHilgerch.1.240\nNOTES4ThereasonforNewton’sconcentrationonspaceratherthantimeandforhisfailuretoprovideaclearargumentforabsolutespacemaybeduetothefactthathistargetintheScholiumtothePrincipiawasnotLeibniz,whoregardedbothspaceandtimeasmereappearanceswithnoabsolutecharacter,butDescartes,whohadlittletosayabouttime;seeStein,H.(1967)‘OnNewtonianspacetime’TexasQuarterly10pp.174–200andRay,C.(1987)op.cit.,pp.10–11.5Thecounterfactualstatementthat‘ifVenushadamoon,thenitwouldhavearoughlycircularorbit’isgivensupportbyNewton’slawsofmotion;butthefactualstatementthat‘allPresidentsoftheUSAaremale’isnotgivensupportbyanylaw:itissupportedonlybytheaccidentallytruegeneralisationthatallPresidentstodatehavebeenmale.ForafulldiscussionofcounterfactualsandlawsseeArmstrong,D.M.(1983)WhatisaLawofNature?Cambridge:CambridgeUniversityPress.6ThisdefinitionisgiveninEarman,J.S.(1989)WorldEnoughandSpace-TimeCambridge,MA:MITPressp.12.7Thisleadstoanotoriousproblem,whichIshallnotconsiderhere:thatofNewton’sreluctancetoaccepttheideaofactionatadistance;forafulldiscussionseeKoyre,A.(1957)FromtheClosedWorldtotheInfiniteUniverseBaltimore:JohnsHopkinsPress.8FromQuaery31ofNewton’sOptics;seeHall,A.R.andHall,M.B.(eds)(1962)UnpublishedScientificPapersofIsaacNewtonCambridge:CambridgeUniversityPressp.192.ThiscollectionalsoincludesNewton’s‘Degravitatione’—whichmightberegardedasadraftfortheideasintheScholium.SeetheilluminatingdiscussionofNewton’sprogressfrom‘Degravitatione’totheScholiuminBarbour,J.B.(1989)op.cit.,pp.609–39.9SeeLocke’sessayinAyer,A.J.andWinch,R.(eds)(1952)BritishEmpiricalPhilosophersLondon:Routledgepp.57ff.:oranycomprehensiveeditionofLocke’sworks.10Locke,J.(1988)EssaysontheLawsofNature(editedbyvonLeyden,W.)Oxford:OxfordUniversityPresspp.258–9.11SeeAyer,A.J.andWinch,R.(eds)(1952)op.cit.,pp.69–78.12Alexander,H.G.(ed.)(1956)TheLeibniz-ClarkeCorrespondenceManchester:ManchesterUniversityPressp.71.13Leibniz’sthirdletter,paragraph4:inAlexander,H.G.(ed.)(1956)op.cit.,pp.25–6.14OneofmanyversionsoftheprinciplequotedinMates,B.(1986)ThePhilosophyofLeibnizOxford:OxfordUniversityPressp.152.15Mates,B.(1986)op.cit.,p.132.16Theruleofmodustollens,‘ifPthenQ;notQ;thereforenotP’,isavaliddeduction.17WemaydistinguishbetweentwoversionsofPSR:causalandtheological;asEarmanpointsout,weneedmorethanacausalreadingofPSRtodrivehomeLeibniz’sargument.SeeEarman,J.S.(1989)op.cit.,p.118.TellerarguesthatPIImightbesufficienttodrivehomeLeibniz’spoint:iftheinhabitantsofXhavenowayofdistinguishingXfromYeveninprinciple,thenXandYmustbeidentical—hencethenotionsofdifferentspatialpositionsandofdifferentpointsintimemeannothingbythemselves;seeTeller,P.(1987)‘Space-timeasaphysicalquantity’inAchinstein,P.and241\nNOTESKagon,R.(eds)Kelvin’sBaltimoreLecturesandModernTheoreticalPhysicsCambridge,MA:MITPress;andalsoTeller,P.(1991)‘Substance,relations,andargumentsaboutthenatureofspace-time’PhilosophicalReviewforthcoming,whichgivesasummaryofTeller’spositiononPII.18Alexander,H.G.(ed.)(1956)op.cit.,pp.45–6.19SeeCleomedes’reportsoftheStoicargumentsinSorabji,R.(1988)Matter,Space,andMotionLondon:Duckworth.20Alexander,H.G.(ed.)(1956)op.cit.,p.48.21However,wemightbeinclinedtoregardLeibniz’sideaofvisviva,discussedlaterinthissection,asasuggestionthatLeibnizwouldnothavebeenaltogetherantagonistictoSklar’snotionofmotionasabrutefact.22JohnEarmanidentifiestwomainaspectstothepleaforrelationism:(a)spaceandtimedonothaveelementsofanykindwhichallowustotalkintermsofabsolutemotion;and(b)spatio-temporalrelationsamongstobjectsandeventsarenotdependentuponanyunderlying‘substratum’ofspatio-temporalpoints.IagreewithEarmanwhenhesaysthat(a)entails(b),for(a)isclearlymoregeneralthan(b).HegoesontosaythatmanycommentatorsmaintainthatNewtonandhissupporters,presumablyincludingClarke,arguethat:because(b)ismistakenandtherearespatialpointsorregionsandtemporalmoments,(a)mustalsobewrong.WecanseefromthereconstructionofClarke’sargument,thatEarmanisrighttochallengethisviewoftheNewtonianargument.Thereisnoone-wayentailmentinClarke’sargument.Assumption2providesatwo-waylinkbetweenpoints/regionsandabsolutemotion;andassumption3providesasimilarlinkbetweenabsolutemotionandinertialforces.Thethreebasicideasherearebeinginter-defined.But,whenwelookatwhatgetstheargumentofftheground,inertialforcestakethelead.IntheoriginalargumentintheScholium,Newtonsticksatprovingthecaseforabsolutemotionintermsofinertialforces.Aswehavealreadynotednofurtherlinkismadebetweenmotionandthepartsofspace.ButClarkeisfacedwithachallengetothelinkbetweenmotionandthepartsofspace,givenLeibniz’sassertionthatspacecannothavedistinctivepartsinanyway.23AusefulsummaryofthiscorrespondenceappearsinHowardStein’s‘SomephilosophicalprehistoryofGeneralRelativity’inEarman,J.S.,Glymour,C.N.,andStachel,J.J.(eds)(1977)FoundationsofSpace-TimeTheoriesMinneapolis:UniversityofMinnesotaPress.24PerhapsthereasonforthelackofclarityisClarke’soccasionaltendencytodepartfromtheNewtonianstandpointandregardspaceandtimeasattributes:seeAlexander,H.G.(ed.)(1956)op.cit.,p.xxviii.25NoticethatNewton’sargumentdoesnotrelyonanyparticularconceptionsofGodandofhowGodinteractswiththeworld.26Alexander,H.G.(ed.)(1956)op.cit.,p.74.27Alexander,H.G.(ed.)(1956)op.cit.,p.74.28VisvivaisidentifiedbyLeibnizasthequantitymv2—aquantitywhichhebelievestobeconservedduringcollisionsbetweenobjects;foradiscussionofvisvivaanditsrelationtotheideasofmomentumandkineticenergyseeIltis,C.(1970)‘Leibnizandthevisvivacontroversy’Isis63pp.26ff.:andWilson,M.(1976)‘Leibniz’sdynamicsandcontingencyinnature’in242\nNOTESMachamer,P.K.andTurnbull,R.G.(eds)MotionandTime,SpaceandMatterColumbus,OH:OhioStateUniversityPress.29Oneinterestingquestion,arisingfromLeibniz’srelationism,askswhetherweshouldconsideronlyrelationsbetweenactualobjectsorevents,orweshouldtakerelationismtoinvolvepossiblerelationsaswell.FordetailsofthisissueseeSklar,L.(1974)Space,Time,andSpacetimeBerkeley:UniversityofCaliforniaPressforageneralreviewoftheproblemsconnectedwithpossiblerelations;Manders,K.(1982)‘Onthespacetimeontologyofphysicaltheories’PhilosophyofScience49pp.575–90;andMundy,B.(1989)‘Onquantitativerelationisttheories’PhilosophyofScience56pp.582–600.30SeeGraves,J.C.(1971)TheConceptualFoundationsofContemporaryRelativityTheoryCambridge,MA:MITPress.31GivenNewton’sunhappinesswiththeideaofactionatadistance,thereissomeevidencetosuggestthathethoughtofforcesasmediatedbyspiritintheabsenceofanysubstantialmaterialether;seeHall,A.R.andHall,M.B.(eds)(1962)op.cit.32SeeStein,H.(1967)op.cit.33Sklar,L.(1974)op.cit.,pp.225ff.34EinsteinunderstoodthisproblemandsaidthathisowndynamicalconceptionofspacetimedissolvedtheinconsistencyinherentinNewton’saccount.35Friedman,M.(1983)FoundationsofSpace-TimeTheoriesPrinceton:PrincetonUniversityPress;seealsoFriedman,M.(1982)‘Theoreticalexplanation’inHealey,R.A.(ed.)Reduction,Time,andRealityCambridge:CambridgeUniversityPress.36Earman,J.S.(1989)op.cit.,seepp.4,154,and170–4.6MACHANDTHEMATERIALWORLD1Here,IdisagreewiththoselikeJohnGribbinwhosaythatMachdidnotimproveonBerkeley’sargument;seeGribbin,J.(1984)‘Thebishop,thebucket,Newtonandtheuniverse’NewScientist1041435/36pp.12–16.2Mach,E.(1911)TheConservationofEnergyLaSalle,IL:OpenCourtpp.91,87.3Mach,E.(1960)TheScienceofMechanicssixtheditionLaSalle,IL:OpenCourtp.xxiiiandMach,E.(1943)PopularScientificLecturesfiftheditionLaSalle,IL:OpenCourtp.207.(OriginalpublicationdatesofGermaneditions1883and1895respectively.)4TheleastsympatheticofrecentwritersisprobablyJohnEarman;seeEarman,J.S.(1970)‘Who’safraidofabsolutespace?’AustralasianJournalofPhilosophy48pp.287–319;andEarman,J.S.(1989)WorldEnoughandSpace-TimeCambridge,MA:MITPress.5SeetherepeatedreferencestoMach’slastinginfluenceinEinstein,A.(1969)‘Autobiographicalnotes’,especiallypp.21,29,53,63,and67,inSchilpp,P.(ed.)AlbertEinstein:Philosopher-ScientistLaSalle,IL:OpenCourtpp.2–95.243\nNOTES6See,forexample,Brans,C.andDicke,R.H.(1961)‘Mach’sPrincipleandarelativistictheoryofgravitation’PhysicalReview124pp.925–35;Sciama,D.W.,Waylen,P.C.,andGilman,R.C.(1969)‘GenerallycovariantintegralformulationsofEinstein’sfieldequations’PhysicalReview187pp.1762–84;Raine,D.J.(1981)TheIsotropicUniverseBristol:AdamHilger;andBarbour,J.B.andBertotti,B.(1982)‘Mach’sPrincipleandthestructureofdynamicaltheories’ProceedingsoftheRoyalSociety382pp.295–306.7See,forahistoryandappraisalofthismovement,Hanfling,O.(1981)LogicalPositivismOxford:BasilBlackwell.ForadisplayofMach’sinfluenceonaphilosopherofscience,seeReichenbach,H.(1957)ThePhilosophyofSpaceandTimeNewYork:Dover.Aslogicalpositivismdeveloped,itsproponentsrealisedthatMach’sownbrandofpositivismwasinadequatefortheirpurposes;seeCarnap,R.(1934)‘Protocolstatementsandtheformalmodeofspeech’inHanfling,O.(ed.)(1981)EssentialReadingsinLogicalPositivismOxford:BasilBlackwell.But,intheearlydaysoflogicalpositivism,Machwasverymuchregardedasaguidinglight.Indeed,theViennaCircleoflogicalpositivistswasoriginallyknownastheErnstMachSociety.8Mach,E.(1960)op.cit.,p.280.9Mach,E.(1960)op.cit.,p.286.10Mach,E.(1960)op.cit.,p.591.11Mach,E.(1960)op.cit.,pp.578–9.12Mach,E.(1943)op.cit.,p.192.13Mach,E.(1960)op.cit.,p.xxiii.14Ray,C.(1987)TheEvolutionofRelativityBristol:AdamHilgergivesanexampleofcalculatingtheareaunderacurve:ifweuseSimpson’sruleorthetrapezoidalmethodtocalculatetheareawewouldbefulfillingthedemandforeconomyoflabour,butifweuseEuler’smethodwewouldsatisfythedemandforeconomyofform.Machgivesusnoadviceabouthowweshouldresolvesuchdifficulties.15SeeHesse,M.B.(1974)TheStructureofScientificInferenceLondon:Macmillanp.223;itslackofsomesuchdistinctionpreventsMach’stheoryfromgivingussatisfactoryandrigorouscriteriaofsimplicity.16Friedman,M.(1983)TheFoundationsofSpace-TimeTheoriesPrinceton:PrincetonUniversityPress,especiallythefinalchapter;alesstechnicalexpositionofhisideasappearsinFriedman,M.(1981)‘Theoreticalexplanation’inHealey,R.A.(ed.)Reduction,Time,andRealityCambridge:CambridgeUniversityPress;seealsothediscussiononNewtonandLeibnizinthelastsectionofChapter5above.17Theclaimsthatobservationistheory-ladenandthatthereisnoclearcutdistinctionbetweenobservationalandtheoreticalstatementsarealsodiscussedinastraightforwardwayinHanson,N.R.(1958)PatternsofDiscoveryCambridge:CambridgeUniversityPress;andNewton-Smith,W.H.(1981)TheRationalityofScienceLondon:Routledge.ThequotationgivenisfromKuhn,T.S.(1970)TheStructureofScientificRevolutionssecondeditionChicago:ChicagoUniversityPresspp.111–12.244\nNOTES18ImightbeaccusedoftakingKuhn’spositiontorelativistextremeshere—ofconstructingastrawman.InthePostscripttoKuhn,T.S.(1970)op.cit.,Kuhndeniesthatheisarelativist.Itakearelativist(aboutscientifictruth)tobesomeonewhobelievesthatthereisnowayofdecidingbetweenalternativetheoriesonthebasisoftheircorrespondencewiththeworld.Somepeoplemightwellberelativistsaboutallaspectsofscience:notonlymightanythinggoasregardstruth,anyoldmethodologymightbefine.Kuhnisnotoneofthese:hedoesrecognisetheneedforacoherentandstructuredmethodologytocontrolthedevelopmentofscience.However,sciencedevelops,forKuhn,notbygettingclosertothetruthabouttheworld,butbyproducingtheconditionsforsuccessfulpuzzle-solving.Undoubtedlysometheorieswillbebettersuitedtosolvingpuzzlesthanothers.SoKuhniscertainlynotarelativistasfarasmethodologyandpuzzle-solvingcapacitiesareconcerned.ButhismessageinKuhn,T.S.(1970)op.cit.isrelativistasfarastruthisconcerned.Hesays(p.206):‘thereis,Ithink,notheory-independentwaytoreconstructphraseslike“reallythere”;thenotionofamatchbetweentheontologyofatheoryandits“real”counterpartinnaturenowseemstometobeillusiveinprinciple’.MycharacterisationofKuhnisfleshandblood,notstraw.19SeeMaxwell,G.(1962)‘Theontologicalstatusoftheoreticalentities’inFeiglH.andMaxwellG.(eds)MinnesotaStudiesinthePhilosophyofSciencevol.IIIMinneapolis:UniversityofMinnesotaPress;thespectrumviewisdiscussedinNewton-Smith,W.H.(1981)op.cit.,ch.2.20Thephrase‘contingentinfallibilityorincorrigibility’issuggestedbyZahar,E.(1980)‘SecondthoughtsaboutMachianpositivism’BritishJournalforthePhilosophyofScience32p.268.21SeeFeyerabend,P.K.(1988)AgainstMethodsecondeditionLondon:Verso;andFeyerabend,P.K.(1981)PhilosophicalPapersvolsIandIICambridge:CambridgeUniversityPress.22vanFraassen,B.(1980)TheScientificImageOxford:OxfordUniversityPress.23ThisfirstissueisfullydiscussedbyHacking,I.(1983)RepresentingandInterveningCambridge:CambridgeUniversityPress:seeespeciallypartB.24Hacking,I.(1983)op.cit.,pp.164–5.25ThispointismadebyPeterGalisoninGalison,P.(1988)‘Philosophyinthelaboratory’JournalofPhilosophy85,10pp.525–7;seealsoGalison.P.(1987)HowExperimentsEndChicago:ChicagoUniversityPress;andHacking,I.(1988)‘Onthestabilityofthelaboratorysciences’JournalofPhilosophy85,10pp.507–14.26Hacking,I.(1983)op.cit.,p.165.27SeePutnam,H.(1984)‘Whatisrealism?’inLeplin,J.(ed.)ScientificRealismBerkeley:UniversityofCaliforniaPressfordetailsofthisstandardrealistposition;notealsoLeplin’sclearandconciseintroductiontotheissueofrealism.245\nNOTES7EINSTEINANDABSOLUTESPACETIME1ThecleareststatementofReichenbach’spositionappearsinReichenbach,H.(1924)ThetheoryofmotionaccordingtoNewton,Leibniz,andHuyghens’reprintedinReichenbach,H.(1959)ModernPhilosophyofScienceLondon:Routledge;see,foranexampleofsomeonewhofollowsReichenbach’sline,Alexander,H.G.(ed.)(1956)TheLeibniz-ClarkeCorrespondenceManchester:ManchesterUniversityPress.2See,forexample,Earman,J.S.(1970)‘Who’safraidofabsolutespace?’AustralasianJournalofPhilosophy48pp.287–319andEarman,J.S.(1989)WorldEnoughandSpace-TimeCambridge,MA:MITPress;Sklar,L.(1974)Space,Time,andSpacetimeBerkeley:UniversityofCaliforniaPress;Gardner,M.R.(1977)‘Relationismandrelativity’BritishJournalforthePhilosophyofScience28pp.215–33;Friedman,M.(1983)FoundationsofSpace-TimeTheoriesPrinceton:PrincetonUniversityPress.ThebestgeneralreviewofthephilosophicalbackgroundtoGTRisprobablyStein,H.(1977)‘SomephilosophicalprehistoryofGeneralRelativity’inEarman,J.S.,Glymour,C.N.,andStachel,J.J.(eds)FoundationsofSpace-TimeTheoriesMinneapolis:UniversityofMinnesotaPress.3Theoriginalfieldequationsmaybeexpressedincoordinateforminthefollowingway:wherethemetricalfunctionontheleft-handsideoftheequationrepresentsthegeometryofspacetime,andthestress-energyfunctionontheright-handsiderepresentsthedistributionofmassandenergy.Initsabstractgeometricalformulation,theequationmaybeexpressedasfollows:G=8pTSeeMisner,C.W.,Thorne,K.S.,andWheeler,J.A.(1973)GravitationNewYork:W.H.Freeman.4ThisfeatureofGTR’sspacetimeislinkedwiththeideasinvolvedinEinstein’sprincipleofequivalence.Thisprinciplewasmotivatedbytheestablishedobservationalequivalencebetweeninertialandgravitationalmassesandforces.Inastrongform,theprincipledemandsthatthoselawsgoverningthemotionofparticlesinSTRshouldalsoholdgoodinthecontextofGTR.SeeMisner,C.W.,Thorne,K.S.,andWheeler,J.A.(1973)op.cit.,pp.312–15.Thisisachievedbyconsideringasufficientlysmall‘local’portionofspacetimearoundtheparticleinwhichweareinterested.Ineffect,themotionisthenreferredtoalocalinertialframeinwhichtheparticlemovesinastraightlineatauniformspeed.ThisideaisthekeytoGTR:Einsteinmovesawayfromtheideaofaglobaltoalocalframeofreferenceformotion.Spacetimemaybegloballycurvedwithmotionsbeingconsequentlycomplex.ButlocallyweshouldtreatthespacetimeofGTRasifitwerethe‘flat’spacetimeofSTRwithanelegantandsimpledescriptionofallmotions.Thedynamicsofmotionmaythenbereferredtoalocallimitinginertialframeofreference.Thismathematicalandtheoretical246\nNOTESsimplicitymayonlybeachievedifweexcludeanyadditionalcurvaturetermsfromthefieldequations.Forsuchatermblocksthepossibilityof‘flat’spacetimeintheabsenceofmatter—evenlocallycurvaturewillbeapparent.AconsequenceoftheprincipleisthatphysicallawsshouldbeeffectivelythesameinthecontextsofSTRandofthelocalinertialframesofGTR.However,thereisnogeneralwayofpreventingcurvaturetermsfromappearingatalocallevelinphysicallawswhenwetrytousetheminGTR.5Misner,C.W.,Thorne,K.S.,andWheeler,J.A.(1973)op.cit.,pp.312–15.6Einstein,A.(1918)‘Principlesofgeneralrelativity’AnnalenPhysikLeipzig55pp.241–5;seenoteatfootofp.241.7Wheeler,J.A.(1964)‘Mach’sPrincipleasaboundaryconditionforEinstein’sequations’inChiu,H.Y.andHoffman,W.F.(eds)GravitationandRelativityNewYork:Benjamin.8Thedeterminationmustbe‘unique’,otherwisetwoormorespacetimestructuresmaybecompatiblewiththesamemass-energydistribution;hence,therewouldbemoretospacetimethanitsmaterialcontents.9Thesethreesensesof‘absolute’arebasedonMichaelFriedman’silluminatinganalysisoftheissueinFriedman,M.(1983)op.cit.,pp.62–70.10FordetailsofspacetimeaccountsofNewtonianphysicsseeFriedman,M.(1983)op.cit.;Stein,H.(1967)‘Newtonianspacetime’TexasQuarterly10pp.174–200;Malament,D.B.(1986)‘Newtoniangravity,limits,andthegeometryofspace’inColodny,R.G.(ed.)FromQuarkstoQuasars:PhilosophicalProblemsofModernPhysicsPittsburgh:UniversityofPittsburghPress;Cartan’sclassicattempttorewriteNewtoniangravityinspacetimeformisdiscussedinMisner,C.W.,Thorne,K.S.,andWheeler,J.A.(1973)op.cit.,pp.291ff.11WeshouldnotregardthepossibilityofrewritingNewtoniantheoryinamodernspacetimeformasaninvitationtorewritehistoryandstartimputingspacetimelanguagetoNewtonhimself,asEarman,J.S.(1989)op.cit.doesoccasionally;theexamplegivenhereconcernsNewton’sclaimintheScholiumthattimeflowsandEarman’sstatementthatNewtondidnotreallymeanthis;seep.8.EarmanalsoaccusesNewtonofmakingamistakewhenhecondemnsNewtonfornotbeingsharpenoughtorealisethathedidnotneedNewtonianspaceandtimeasabasisforabsoluteacceleration;hecouldhaveemployed‘neo-Newtonian’spacetimeinstead,withaninertialstructurebutnooverallabsoluteNewtonianframework;seeEarman,J.S.(1989)op.cit.,p.3.(Neo-NewtonianspacetimeincorporatestheNewtonianideaofsimultaneity,butdropsthenotionofbeingatthesamepointatdifferenttimesandthereforedropsthenotionofanoverall‘rigid’frameworkintheNewtoniansense—informallyonemaythinkofneo-Newtonianspacetimeasahalf-wayhousebetweenNewtonianideasandtheMinkowskispacetimeofSTR;seeSklar(1974)op.cit.forafulldiscussionofthepropertiesofthisspacetime.)But,historicallyNewtonhadnosuchoption.And,conceptually,itisfarfromclearhowNewtonmighthaveutilisedsuchastructure.SeethereviewofEarman’sbookbyHoefer,C.andRay,C.(1991)‘Earman:WorldEnoughandSpace-Time’BritishJournalforthePhilosophy247\nNOTESofScience423foradetaileddiscussionofthisandotherrelatedpointsarisingfromEarman’streatmentofspacetimephysics.12Notethewarningonp.725ofMisner,C.W.,Thorne,K.S.,andWheeler,J.A.(1973)op.cit.:evenifwecompletelydetermineallthelocalgeometricalpropertiesby,say,imposingademandforhomogeneityandisotropy,wedonottherebydeterminetheglobaltopologyofthespacetime.13SeeWeinberg,S.(1972)GravitationandCosmologyNewYork:Wiley;foraphilosopherofphysicswhofollowstheleadgivenbyWeinberg,seeSklar,L.(1974)op.cit.14Thisandothereffectively‘empty’modelsofGTRarediscussedinHawking,S.W.andEllis,G.F.R.(1973)TheLargeScaleStructureofSpace-TimeCambridge:CambridgeUniversityPresspp.117ff.15FordetailsoftheKerrsolutionandGödel’ssolutionofthefieldequationsseeHawking,S.W.andEllis,G.F.R.(1973)op.cit.,pp.162–70;theOszvàthandSchückingsolutionisdiscussedtogetherwithGödel’ssolutioninFriedman,M.(1983)op.cit.,pp.206–15.16Twoofhismostimportantpapers,Friedmann,A.(1922)‘Onthecurvatureofspace’andFriedmann,A.(1924)‘Onthepossibilityofaworldwithconstantnegativecurvature’,appearintranslationinBernstein,J.andFeinberg,G.(eds)(1986)CosmologicalConstants:PapersinModernCosmologyNewYork:ColumbiaUniversityPress;seealsoMisner,C.W.,Thorne,K.S.,andWheeler,J.A.(1973)op.cit.forfulldetailsofhismodelsplusp.751forabiographicalnote.17SeeRindler,W.(1977)EssentialRelativitysecondeditionBerlin:Springerp.243.18Raine,D.J.(1975)‘Mach’sPrincipleinGeneralRelativity’MonthlyNoticesoftheRoyalAstronomicalSociety171pp.507–28,andRaine,D.J.(1981)TheIsotropicUniverseBristol:AdamHilger;seealsoSciama,D.W.,Waylen,P.C.,andGilman,R.C.(1969)‘GenerallycovariantintegralformulationofEinstein’sfieldequations’PhysicsReview187pp.1762–84.19OtherattemptstoconstructMachianvariantsofgravitationaltheoryincludeBrans,C.andDicke,R.H.(1961)‘Mach’sPrincipleandarelativistictheoryofgravitation’PhysicalReview124pp.925–35;Barbour,J.B.andBertotti,B.(1977)‘GravityandinertiainaMachianframework’NuovoCimento38Bpp.1–27;Barbour,J.B.andBertotti,B.(1982)‘Mach’sPrincipleandthestructureofdynamicaltheories’ProceedingsoftheRoyalSocietyLondon382pp.295–306.20SeeCollins,C.B.andHawking,S.W.(1973)‘Whyistheuniverseisotropic?’AstrophysicsJournal180pp.317–34;andWill,C.W.(1981)TheoryandExperimentinGravitationalPhysicsCambridge:CambridgeUniversityPress.Currentestimatesforthemarginoferrorontheassertionthatthereisnooverallrotationarelessthan0.1%.21SeeRay,C.(1987)TheEvolutionofRelativityBristol:AdamHilgerpp.111–15foradiscussionoflawsandimplicitrangesofapplication.22Bondi,H.(1967)AssumptionandMythinPhysicalTheoryCambridge:CambridgeUniversityPress.23Hawking,S.W.andEllis,G.F.R.(1973)op.cit.,p.117.248\nNOTES24Ofcourse,wemightdiscovertachyonsorsomeothermaterialwhichbreakstheseconditions:butuntilwehavesomeevidenceforsuchstrangekindsofmatterthereislittlereasontoallowsolutionsinvolvingthemexceptasmathematicalcuriosities.25SeeChapter9foramoredetaileddiscussionoftheearlyuniverseanditsevolution.26See,forthegeneralbackgroundtoEinstein’sdevelopmentofGTR,Pais,A.(1982)SubtleistheLordOxford:OxfordUniversityPress;andEarman,J.S.andGlymour,C.(1978)‘Lostintensors’StudiesintheHistoryandPhilosophyofScience9pp.251–78.ForaccountsoftheholeargumentseeButterfield,J.(1989)‘Theholestory’BritishJournalforthePhilosophyofScience40pp.1–28;Earman,J.S.andNorton,J.(1987)‘Whatpricespace-timesubstantivalism?Theholestory’BritishJournalforthePhilosophyofScience38pp.515–25;Earman,J.S.(1989)op.cit.;Teller,P.(1991)‘Substance,relations,andargumentsaboutthenatureofspace-time’PhilosophicalReview.27Inpractice,thetechnicalitiesaresuchthatwemayonlydothisinhighlysymmetricalandsimplespacetimes.28Thereareanumberofcomplexissuesinvolvedinthepossibilityofsuchadeterministicpicture:wemayonlyconstructsuchaglobaltime-slice(aCauchyhypersurface)incertainwell-behavedspacetimes.Weshallconsiderthecomplications,connectedwiththeideasofclosedtimelikeloops,nakedsingularities,andtopologicalholes,inChapters8and10.29SeeMisner,C.W.,Thorne,K.S,andWheeler,J.A.(1973)op.cit.fordetailsofgeneralcovarianceanditsroleinGTR;andRay,C.(1987)op.cit.,especiallych.2,foraninformaldiscussionoftheideaandstatusofgeneralcovariance.30SeeKretschmann,E.(1917)‘Onthephysicalmeaningoftherelativitypostulate’AnnalenPhysikLeipzig53pp.574–614;andalsoFriedman,M.(1983)op.cit.31SeeTeller,P.(1991)op.cit.32Thisisasimplifiedaccountoftheargument,omittingalltechnicaldetails,whichmaybefoundinEarman,J.S.(1989)op.cit.,pp.175–80.33SeeTeller,P.(1991)op.cit.foradiscussionoftheseapproachestakenby,respectively,Maudlin,T.(1988)‘Substancesandspace-time:whatAristotlewouldhavesaidtoEinstein’(forthcoming)andButterfield,J.(1989)op.cit.8TIMETRAVEL1Wells,H.G.(1958)SelectedShortStoriesLondon:Penguin.2Mellor,D.H.(1981)RealTimeCambridge:CambridgeUniversityPress.3Foramplificationofthisideaofchange,applyingtothingsbutnotevents,seeMellor,D.H.(1981)op.cit.,p.9andpp.119ff.4Seethesection‘Conventionandtopology’inChapter4.5See,forexample,thediscussionoftheroleofthermodynamicsandentropyinquestionsoftimeasymmetryinSwinburne,R.(1981)SpaceandTimeLondon:Macmillan;andPenrose’sideasongravitationalcurvatureand249\nNOTEStimeasymmetrymaybefoundinPenrose,R.(1979)‘Singularitiesandtime-asymmetry’inHawking,S.W.andIsrael,W.(eds)(1979)GeneralRelativityCambridge:CambridgeUniversityPresspp.581–638;seealsotheextremelyhelpfulgeneralanalysesin(fromaphilosophicalperspective)Sklar,L.(1974)Space,Time,andSpacetimeBerkeley,CA:UniversityofCaliforniaPress;and(fromaphysicalperspective)Davies,P.C.W.(1974)ThePhysicsofTimeAsymmetryGuildford:SurreyUniversityPress.6Agivenspacetimemay,however,havemorethanonesurfaceofsimultaneitythroughagivenpoint,e.g.Minkowskispacetimehastwosurfacesofsimultaneity:theplane(forobserversatrestrelativetoeachother)andthehyperboloid(forobserversinrelativemotion).7Dummett,M.A.E.(1954)‘Cananeffectprecedeitscause?’ProceedingsoftheAristotelianSociety,SupplementaryVolume28pp.27–14;Dummett,M.A.E.(1964)‘Bringingaboutthepast’PhilosophicalReview69pp.497–504;Dummett,M.A.E.(1986)‘Causalloops’inFlood,R.andLockwood,M.(eds)TheNatureofTimeOxford:BasilBlackwellpp.135–69.8Suchmanoeuvresarecalled‘bilking’strategiesbyHorwich,P.(1987)AsymmetriesinTimeCambridge,MA:MITPress;seepp.91ff.9AninterestingreviewofwormholesappearsinRedmount,I.(1990)‘Wormholes,timetravelandquantumgravity’NewScientist126,1714pp.57–61;theinformationinFigure28isbasedontheexcellentgraphicsinthisarticle.10Feynman,P.R.(1949)Thetheoryofpositrons’PhysicalReview76pp.749–59;seealsoHorwich,P.(1987)op.cit.foraclearanalysisoftheimplicationsofFeynman’saccountofbackwardscausation.11Mellor,D.H.(1981)op.cit.12Itmaybepossibletoconstructsituationsinwhichtravelbackwardsintimedoesnotallowtheconstructionofclosedcausalloops.Forexample,theskewedNewtonianspacetimediscussedinChapter4inwhichlighttravelsforwardsintimewhenmovinginonespatialdirection,butbackwardsintimewhenmovingintheoppositespatialdirection:insuchacasethespacetimeisconstructedtogivealimitedsensetothenotionofbackwardstravel—itisbackwardsrelativetoanotionalplaneof‘instantaneous’simultaneity;butthespecificationsofthisspacetimeguaranteethatnothingcantravelalongaworldlinewithatiltlessthanthatdefinedbytheskewedplaneofsimultaneityasdefinedbylightsignals;hence,therecanbenosenseintalkingofclosedloopshere.13Thereareother,morecomplex,spacetimesinwhichclosednon-spacelikecausalcurvesmaybeconstructed,forexample:Gödel’suniverseandTaub-NUTspacetimes;fordetailsofthesecausallyunstablespacetimesseeHawking,S.W.andEllis,G.F.R.(1973)TheLarge-ScaleStructureofSpace-TimeCambridge:CambridgeUniversityPress,ch.5;foralesstechnicaldiscussionofGödel’suniverse,seeeitherHorwich,P.(1987)op.cit.orSklar,L.(1974)op.cit.;furtherphilosophicalanalysisofthetechnicaldetailsisgiveninMalament,D.(1984)‘TimetravelintheGödeluniverse’inAsquith,P.D.250\nNOTESandKitcher,P.(1985)ProceedingsofthePhilosophyofScienceAssociationvol.IIMinneapolis:UniversityofMinnesotaPress.14ThisisbasedonasimilarsituationcharacterisedinEarman,J.(1972)‘Causalpropagationoutsidethenullcone’AustralasianJournalofPhilosophy50pp.222–37.15SeeLewis,D.(1976)‘Theparadoxesoftimetravel’AmericanPhilosophicalQuarterly13pp.145–52,foraclassicdiscussionofthiskindofattackontimetravel.16SeeDummett,M.A.E.(1986)op.cit.17Harrison,J.(1979)‘Analysisproblemnumber18’Analysis39pp.65–6;seealsoHarrison,J.(1980)‘ReportonAnalysisproblemnumber18’Analysis40pp.65–9.18Malament,D.(1985)‘MinimalaccelerationrequirementsfortimetravelinGödelspace-time’JournalofMathematicalPhysics26pp.774–7.19Horwich,P.(1975)‘Onsomeallegedparadoxesoftimetravel’JournalofPhilosophy72pp.432–4,andHorwich,P.(1987)op.cit.,pp.111–28.20IremainuncertainaboutthestrengthofHorwich’spoint:IhavearguedinitsfavourinRay,C.(1987)TheEvolutionofRelativityBristol:AdamHilgerpp.127–9;butIsupposeIamnowlessthansatisfiedwithbrutefactsthanothersmightbe.21ThisstoryaboutwishingandlettersisbasedontheexamplesgivenbyDummett(1964)op.cit.andMellor(1981)op.cit.IhavetriedtocaptureDummett’sconnectionbetweenwishingthatsomethinghadbeenthecase(andactingonthiswish)andithavingbeenthecase,somethingwhichisabsentinMellor’sexample;butalsotoavoidtheassociationsofDummett’s‘dancingchiefexample,whichMellorquiterightlytriestoavoid.22Riggs,P.J.(1991)‘AcritiqueofMellor’sargumentagainstbackwardscausation’BritishJournalforthePhilosophyofScience42(forthcoming)arguesthatMellor’sviewsimplyanacceptanceofanyaccountintermsofforwardscausation‘regardlessofhowtenuousandimprobablesuchanexplanationmightbe’.Butthispointofviewseemstoneglectthetremendousdifficultiesinvolvedinmaintainingbackwards-typecausalexplanations.Idonotthinkwecouldwinwhateverweweretodecide.Butfortunatelythereseemslittleneedtomakeadecision,giventhetotalabsenceofempiricalevidenceforbackwardscausation.9EINSTEIN’SGREATESTMISTAKE?1deSitter,W.(1917)‘OnEinstein’stheoryofgravitationanditsastronomicalconsequences’MonthlyNoticesoftheRoyalAstronomicalSociety78pp.3–28;formorerecentattacksonthecosmologicalconstantsee:Gamov,G.(1957)‘Moderncosmology’inMunitz,M.(ed.)(1957)TheoriesoftheUniverseNewYork:FreePress;Pais,A.(1982)SubtleistheLordOxford:OxfordUniversityPress;Hawking,S.W.(1982)‘Thecosmologicalconstantandtheweakanthropicprinciple’inDuff,M.J.andIsham,C.J.(eds)QuantumStructureof251\nNOTESSpaceandTimeCambridge:CambridgeUniversityPress.Seealsotheexcellenthistoryofearlytwentieth-centurycosmology,whichfocusesondeSitterandEinstein’sdebatesaboutcosmology:Kerszberg,P.(1989)TheInventedUniverseOxford:OxfordUniversityPress.Anotherhistoryofcosmologywhichdiscussestheissuesraisedinthischapter,morewide-rangingthanKerszberg’s,isNorth,J.D.(1965)TheMeasureoftheUniverseOxford:OxfordUniversityPress;seeChapter2ofNorthforproblemsassociatedwiththeNewtonianaccount.2Einstein,A.(1931)‘ZumkosmologischenProblemderallgemeinenRelativitatstheorie’SitzungsberichtederPreussischenAkademiederWissenschaften142pp.235–7.3Einstein,A.(1917)‘Cosmologicalconsiderationsonthegeneraltheoryofrelativity’inEinstein,A.,Lorentz,H.A.,Weyl,H.,andMinkowski,H.(1923)ThePrincipleofRelativityLondon:Methuen.4Newton,I.(1961)Correspondencevol.IIICambridge:CambridgeUniversityPress.5Neumann,C.(1896)UberdasNewton’schePrinzipderFernwirkungLeipzig:B.G.Teubner,seepp.165ff.:seealsoSeeliger,H.(1895)‘UberdasNewton’scheGravitationsgesetz’SitzungsberichtederBayerischenAkademiederWissenschaften26pp.373–90andSeeliger,H.(1898)‘OnNewton’slawofgravitation’PopularAstronomy5pp.544–51.EinsteinacknowledgeshisdebttoSeeligerinEinstein,A.(1954)Relativity:theSpecialandtheGeneralTheoryLondon:Methuen:thisisatranslationofapopularaccountwrittenbyEinsteinin1917.6Newton,I.(1729)PrinciplesofNaturalPhilosophyLondon:Dawsons.7Einstein,A.(1918)‘Principlesofgeneralrelativity’AnnalenPhysikLeipzig55pp.241–4;notealsoMach,E.(1883)TheScienceofMechanicssixthedition(1960)LaSalle,IL:OpenCourt,p.207.8Pais,A.(1982)op.cit.9SeeEinstein,A.(1917)op.cit.10deSitter,W.(1917)‘Ontherelativityofinertia’ProceedingsoftheAcademyoftheNetherlands19pp.1217–25.11ErnstMach,aswehaveseen,continuallyremindedhisreaders,includingEinstein,oftheneedforeconomyinourdescriptionsofthephysicalworld.Hesaysthatthe‘fundamentalconceptionofthenatureofscience[isthe]economyofthought’and‘thegoal[ofphysicalscience]isthesimplestandmosteconomicalabstractexpressionofthefacts’;seeMach,E.(1943)PopularScientificLecturesfiftheditionLaSalle,IL:OpenCourt,p.207.12GeraldHoltoncharacterisesEinstein’sphilosophicaldevelopmentasapilgrimagefromMach’sphenomenalisticempiricismtoamorematurerationalisticrealism;seeHolton,G.(1973)ThematicOriginsofScientificThoughtCambridge,MA.:HarvardUniversityPress.Fine,A.(1986)TheShakyGameChicago:ChicagoUniversityPressarguesthat‘asearlyas1918,Einstein’sexpressionsofrealismarepresentedintermsofmotivationsforthepursuit252\nNOTESofscience’notsomuchtoseek‘realisttheories’buttobreakfreefromthe‘chainsofthe“merelypersonal’”;seepp.109–11.Butthereislittledoubtthatduringtheperiodupto1920Einstein’sworkwasstronglyinfluencedbyMachianstricturesonsimplicity;seeRay,C.(1987)TheEvolutionofRelativityBristol:AdamHilger.13AcompletediscussionoftheprinciplesofequivalencemaybefoundinWill,C.M.(1981)TheoryandExperimentsinGravitationalPhysicsCambridge:CambridgeUniversityPress;seealsoRay,C.(1987)op.cit.,ch.2.14Suchcurvaturetermsrepresentthecouplingofthelawwiththegravitationalfield;becauseoftheemergenceofthesetermsandotherdifficulties,ZahararguesthattheprincipleofequivalencecannotstrictlyspeakingbetrueofGTR,andthatwemustseeitasaheuristicprincipledirectingustowardsthefinalformoftheequationsofGTR.SeeZahar,E.(1980)‘Einstein,Meyerson,andtheroleofmathematicsinphysicaldiscovery’BritishJournalforthePhilosophyofScience31pp.1–43andZahar,E.(1989)Einstein’sRevolution:aStudyinHeuristicLaSalle,IL:OpenCourt.15Popper,K.(1963)ConjecturesandRefutationsLondon:Routledge.16EdmundHalleyhadconsideredNewton’sproblemabouttheextentofspaceanditscontents,butsuggestedthatNewton’sresolutionmighthaveinherentdifficulties:aninfinitenumberofstarsinastaticuniversewouldhaveresultedinatotallybrightnightsky—thisreasoningwaslaterdevelopedbyOlbers’intheearlynineteenthcenturyandtheproblemwasnamedOlbers’Paradox.Hence,Neumannandothernineteenth-centuryphysiciststooktheviewthatthenumberofstarsshouldbefinite.SeeHalley,E.(1720)‘Ofthefixedstars’RoyalSocietyTransactions31pp.22–6and,forafulldiscussionofOlber’sParadox,Sciama,D.W.(1961)TheUnityoftheUniverseNewYork:Doubleday.17Pais,A.(1982)op.cit.givesanexcellentaccountofthehistoryofGTR’spredictions;Will,C.M.(1988)WasEinsteinRight?Oxford:OxfordUniversityPressprovidesacontemporaryaccountoftheexperimentaltestsofGTR.18Einstein’s1911predictionappearsin‘Ontheinfluenceofgravitationonthepropagationoflight’inEinstein,A.etal.(1923)op.cit.pp.97–1080.19Hawking,S.W.(1982)op.cit.20Einstein,A.(1917)op.cit.21Friedmann,A.(1922)‘Onthecurvatureofspace’ZeitschriftfurPhysik10pp.377–86andFriedmann,A.(1924)‘Onthepossibilityofaworldwithconstantnegativecurvature’ZeitschriftfurPhysik21pp.326–32;thesepapersarereprintedintranslationinBernstein,J.andFeinberg,G.(eds)(1986)CosmologicalConstants:PapersinModernCosmologyNewYork:ColumbiaUniversityPress.22Einstein,A.(1923)‘AnoteontheworkofA.Friedmann’ZeitschriftfurPhysik16p.228.23Misner,C.W.,Thorne,K.S.,andWheeler,J.A.(1973)GravitationNewYork:Freeman.253\nNOTES24Gödel,K.(1949)‘Anexampleofanewtypeofcosmologicalsolution’ReviewofModernPhysics21pp.447–50;Oszvàth,I.andSchücking,E.(1969)‘Thefiniterotatinguniverse’AnnalsofPhysicsNewYork55pp.166–204.25Weyl,H.‘Gravitationandelectricity’inEinstein,A.etal.(1923)op.cit.,pp.200–16.26Davies,P.C.W.(1982)TheAccidentalUniverseCambridge:CambridgeUniversityPressandHawking,S.W.(1983)‘Thecosmologicalconstant’inMcCrea,W.H.andRees,MJ.(eds)TheConstantsofPhysicsLondon:RoyalSociety.27Blau,S.K.andGuth,A.H.‘Inflationarycosmology’inHawking,S.W.andIsrael,W.(eds)(1987)300YearsofGravitationCambridge:CambridgeUniversityPress;seealsotheexcellentreviewbyAbbott,L.(1988)‘Themysteryofthecosmologicalconstant’ScientificAmerican2585pp.106–13.28SeeChapter10andalsoRay,C.(1987)op.cit.,ch.5.29Cartwright,N.(1982)HowtheLawsofPhysicsLieOxford:OxfordUniversityPress.Wemighttrytoarguethatafundamentallawmightbetruewhenitbelongstoaunifiedtheory;butsadlythehistoryofsciencerevealslittleevidencethatweareconvergingonsuchatheory:ifanything,thelastonehundredyearshaveshownamarkeddivergenceaswehavemovedfromacontextwithtwobasicforces(gravityandelectromagnetism)toonewiththreeforces(gravity,strongandelectro-weak).Peoplekeepdreamingofunifiedtheories,butnaturekeepsthrowingupsurprises.And,asCartwrightpointsout,althoughwewantlawsthatunify…whathappensmaywellbevariedanddiverse.Weareluckywecanorganisephenomenaatall.Thereisnoreasontothinkthattheprinciplesthatbestorganizewillbetrue,northattheprinciplesthataretruewillorganizemuch.(Cartwright1982op.cit.:53)SeeHawking,S.W.(1988)ABriefHistoryofTimeLondon:BantamPress;andDavies,P.C.W.andBrown,J.R.(eds)(1988)Superstrings:aTheoryofEverything?Cambridge:CambridgeUniversityPress—noteespeciallySheldonGlashow’sscepticismaboutunificationandhisremark,directedatsuperstringtheory,thatheis‘waitingforthesuperstringtobreak’:p.180.30Hawking,S.W.andEllis,G.F.R.(1973)TheLarge-ScaleStructureofSpace-TimeCambridge:CambridgeUniversityPress.31Barrow,J.D.andTipler,F.J.(1986)TheAnthropicCosmologicalPrincipleOxford:OxfordUniversityPress;Hawking,S.W.(1982)op.cit.;and.Davies,P.C.W.(1982)op.cit.32Carter,B.(1974)‘Largenumbercoincidencesandtheanthropicprincipleincosmology’inLongair,M.S.(ed.)ConfrontationofCosmologicalTheorieswithObservationalDataDordrecht:Reidelpp.291–8.254\nNOTES33Davies,P.C.W.andBrown,J.R.(eds)(1986)TheGhostintheAtomCambridge:CambridgeUniversityPressdiscussavarietyofviewsontheideathatquantumconsiderationsshowthattheobserverissomehowresponsiblefortheexistenceofuniverseincludingWheeler’sclaimthatobserversarerequiredtobringtheuniverseintoexistenceinlinewithsomequantumcosmologicalfeedbackprocess.Thelinkbetweenanthropocentricityanddimensionality,mentionedbrieflyinthethirdchapterofthisbook,isdiscussedbyBarrow,J.D.(1983)‘Dimensionality’inMcCrea,W.H.andRees,M.J.(eds.)(1983)op.cit.,pp.337–46.34Theotherfourparametersare:theHubbleconstantH,whichcharacterisestherateatwhichgalaxiesaremovingapart;thephoton/protonratioS,whichplaysanessentialroleintheformationofgalaxies;thenumberofprotonsNwithintheobservableuniverse;theparameterq,whichdescribestherateatwhichtheexpansionoftheuniverseisslowingdown.Weretheyallnotapproximatelytheircurrentlyobservedvalues,thentheuniversewouldbeinconsistentwiththepossibilityoflife;seeBarrow,J.D.andTipler,F.J.(1986)op.cit.forafulldiscussion.Rees,M.J.(1983)‘Largenumbersinastrophysics’,inMcCrea,W.H.andRees,M.J.(eds)(1983)op.cit.,pp.311–22,carriesthediscussionfurtherwithareviewofthe‘largenumber’coincidencesincosmology.35Davies,P.C.W.(1982)op.cit.,p.121.36Earman,J.S.(1987)‘TheSAPalsorises:acriticalexaminationoftheanthropicprinciple’AmericanPhilosophicalQuarterly24pp.307–17.37SeeChapter10foradiscussionofinflationarycosmology.NotealsothediscussionofSAPandcosmologicalperspectivesinHacking,I.(1987)‘TheInverseGambler’sfallacy;theargumentfromdesign.TheanthropicprincipleappliedtoWheeleruniverses’Mind96pp.331–40.38Hawking,S.W.(1982)op.cit.,p.425.39Carter,B.(1974)op.cit.40Carter,B.(1983)‘Theanthropicprincipleanditsimplicationsforbiologicalevolution’inMcCrea,W.H.andRees,M.J.(eds)(1983)op.cit.,pp.347–63.10COSMOLOGICALCONUNDRUMS1Leibniz’sargumentisalmostexactlythatgiven(earlier)bytheEnglishphilosopherThomasHobbes(1588–1679)inhisessay‘Onlibertyandnecessity’—seethereferencetoHobbes’argumentinAnscombe,G.E.M.(1986)‘Times,beginnings,andcauses’inKenny,A.(ed.)Rationalism,Empiricism,andIdealismOxford:OxfordUniversityPressp.93.ForthequotationfromLeibnizseeAlexander,H.G.(ed.)(1956)TheLeibniz—ClarkeCorrespondenceManchester:ManchesterUniversityPresspp.36–8.2PresumablyareferencetoDescartes’sviews.3SeeAlexander,H.G.(ed.)(1956)op.cit.p.49.255\nNOTES4DetailsoftheinitialdiscoveryaregiveninPenzias,A.A.andWilson,R.W.(1965)‘Ameasurementofexcessantennatemperatureat4080Mc/s’AstrophysicalJournal142pp.419–21;afulldiscussionofitsimplicationsmaybefoundinRaychaudhuri,A.K.(1979)TheoreticalCosmologyOxford:OxfordUniversityPressch.6.5TheproofsthatsingularitiesmustexistinGTRwereprovidedbyRogerPenroseandStephenHawkinginthe1960s.SeePenrose,R.(1965)‘Gravitationalcollapseandspacetimesingularities’PhysicalReviewLetters14pp.57–9;andHawking,S.W.(1966)Theoccurrenceofsingularitiesincosmology’ProceedingsoftheRoyalSocietyA294pp.511–21.6SeeGeroch,R.P.andHorowitz,G.T.(1979)‘Globalstructuresofspacetimes’inHawking,S.W.andIsrael,W.(eds)GeneralRelativityCambridge:CambridgeUniversityPress.7SeetheoriginalideainBondi,H.andGold,T.(1948)‘Thesteady-statetheoryoftheexpandinguniverse’MonthlyNoticesoftheRoyalAstronomicalSociety108pp.252–70andHoyle,F.(1948)‘Anewmodelfortheexpandinguniverse’MonthlyNoticesoftheRoyalAstronomicalSociety108pp.372–82;andalsothe(supportive)commentsonthetheorybyNarlikar,J.V.(1988)ThePrimevalUniverseOxford:OxfordUniversityPresspp.217–28.ForalesspartisanaccountseeBarrow,J.D.(1988)TheWorldwithintheWorldOxford:OxfordUniversityPresspp.212–18.8Thispointismadebymanywhorefusetograntanyforcetothe‘cosmologicalargument’fortheexistenceofGod;thisargumentstartsfromthepremissthatnothinghappenswithoutacause(orreason),andleadsustotheconclusionthatGodexistsonthegroundsthattheremustbesomecause(orreason)fortheexistenceoftheworldasawhole;see,forexample,thetextofaradiodebatebetweenBertrandRussellandFrederickCoplestoninVesey,G.(ed.)(1974)PhilosophyintheOpenMiltonKeynes:OpenUniversityPresspp.115–20.9Hawking,S.W.(1987)‘Quantumcosmology’inHawking,S.W.andIsrael,W.(eds)300YearsofGravitationCambridge:CambridgeUniversityPress;seealsotheintroductiontoHawking,S.W.andIsrael,W.(eds)(1979)op.cit.10SeeHawking,S.W.(1987)op.cit.,especiallypp.633–6.11Hawking,S.W.(1987)op.cit.,p.635;seealsothecommentsonthisproposalbyGrunbaum,A.(1989)‘Thepseudo-problemofcreationinphysicalcosmology’PhilosophyofScience56p.393;thisarticlebyGrunbaumisreprintedinthevaluablecollectionbyLeslie,J.(ed.)(1989)PhysicalCosmologyandPhilosophyNewYork:Macmillan.12Similarly,iftheuniversedoescollapse,theremaybenodefiniteend-pointtothefuture.ButHawkingarguesthatevenuniverseswhicharenon-singularatthestartmayrecollapsetoasingularityattheend.SeeHawking,S.W.(1987)op.cit.,p.650.13Thereareseveralexcellentgeneralreviewsoftheimportantfeaturesofinflationarycosmology;non-technicalaccountsoftheoriginaltheorymaybefoundinGuth,A.H.andSteinhardt,P.J.(1984)‘Theinflationaryuniverse’ScientificAmericanMay1984,reprintedinDavies,P.C.W.(ed.)(1989)TheNewPhysicsCambridge:CambridgeUniversityPress;Narlikar,J.V.(1988)op.cit.,ch.5;and(inbrief)inGribbin,J.andRees,M.(1989)Cosmic256\nNOTESCoincidencesLondon:BantamPresspp.277–83;acomprehensivebuttechnicalaccountoftheoriginalandrevisedtheoriesisgiveninBlau,S.K.andGuth,A.H.(1987)‘Inflationarycosmology’inHawking,S.W.andIsrael,W.(1987)(eds)op.cit.,pp.524–604.Acomprehensivenon-technicalaccountoftherefinedtheoryisgiveninGuth,A.H.(1989)‘Startingtheuniverse’inCornell,J.(ed.)Bubbles,VoidsandBumpsinTime:theNewCosmologyCambridge:CambridgeUniversityPress.14Ellis,G.F.R.andWilliams,R.M.(1988)FlatandCurvedSpace-TimesOxford:OxfordUniversityPresspp.277–85givesamoredetaileddiscussionofthisproblem.15Blau,S.K.andGuth,A.H.(1987)op.cit.,p.542.16Guth,A.H.(1989)op.cit.,p.136;notethatzerodegreesKelvinequalsroughlyminus273degreesCelsiusorcentigrade.17Withoutanyrecognisable,distinctiveparticlesandantiparticles,theideaofsuchahightemperatureisspelledoutintermsofexcitationsinthematterfieldinsteadofthemoreusualtermsofaverageparticlespeeds.18Blau,S.K.andGuth,A.H.(1987)op.cit.19Blau,S.K.andGuth,A.H.(1987)op.cit.,p.541.20SeethediscussionofthisanalogyinHawking,S.W.(1988)ABriefHistoryofTimeLondon:BantamPresspp.127–8.21Guth,A.H.andSteinhardt,P.J.(1984)op.cit.22ThispointismadewithsomeforcebyGrunbaum,A.(1989)op.cit.,pp.389–93,whoisreactingtoaclaim,bytheastronomerBernardLovell,thatinflationarytheoryallowsustosaythattheuniversewascreatedoutofnothing.23Hawking,S.W.(1988)op.cit.,p.128.24Linde,A.(1985)‘Theuniverse:inflationoutofchaos’NewScientist105,1446pp.14–18;reprintedinLeslie,J.(ed.)(1989)op.cit.25AtranslationofLaplace’s(1799)paperdiscussingthepossibilityof‘blackholes’appearsasAppendixAinHawking,S.W.andEllis,G.F.R.(1973)TheLargeScaleStructureofSpace-TimeCambridge:CambridgeUniversityPress.PaulDavies,inanarticleintheSundayCorrespondent,April1990,notesthattheearliestmentionoftheideaofablackholemayhavebeenin1783,bytheclergymanJohnMichell,whoaddressedhimselftothesameproblemsinNewtoniantheoryasLaplace.26SeeChandrasekhar’sremarksonthisincidentinMehra,J.(ed.)(1973)ThePhysicist’sConceptionofNatureDordrecht:Reidel,pp.36–7.27Hawking,S.W.andIsrael,W.(eds)(1979)op.cit.,seetheir‘Introduction’.28Penrose,R.(1979)‘Singularitiesandtime-asymmetry’inHawking,S.W.andIsrael,W.(eds)(1979)op.cit.,p.618.29SeeEarman,J.S.(1971)‘Laplaciandeterminism,oristhisanywaytorunauniverse?’JournalofPhilosophy68pp.729–44;andMalament,D.B.(1977)‘Observationallyindistinguishablespacetimes’inEarman,J.S.,Glymour,C.N.,andStachel,J.J.(eds)FoundationsofSpace-timeTheoriesMinneapolis:UniversityofMinnesotaPress.30SeeGeroch,R.P.andHorowitz,G.T.(1979)op.cit.31SeeClarke,C.J.S.(1976)‘Spacetimesingularities’CommunicationsinMathematicalPhysics49pp.17–23;andClarke,C.J.S.(1982)‘Singularspacetimes’CommunicationsinMathematicalPhysics84pp.329–31.Seealsothe257\nNOTEScommentsbyEarman,J.S.(1989)WorldEnoughandSpace-TimeCambridge,MA:MITPress,chs5,8and9.32SeeRay,C.(1987)TheEvolutionofRelativityBrisol:AdamHilger,ch.4;andEarman,J.S.(1989)op.cit.,chs8and9.33ForamoredetailedgeneralreviewofthephilosophicalissuesconnectedwithblackholesandcosmiccensorshipseeWeingard,R.(1979)‘Somephilosophicalaspectsofblackholes’Synthèse42pp.191–219.AnexcellentgeneraldiscussionofthephysicsofblackholesappearsinWald,R.M.(1977)Space,Time,andGravityChicago:ChicagoUniversityPress.CONCLUSION:RELATIVITY—JUSTANOTHERBRICKINTHEWALL?1Hawking,S.W.(1980)IstheEndofTheoreticalPhysicsinSight?Cambridge:CambridgeUniversityPress;thisisareprintofHawking’slectureonbeingmadeLucasianProfessorofMathematicsatCambridgeUniversity.2Cartwright,N.(1982)HowtheLawsofPhysicsLieOxford:OxfordUniversityPress;Hacking,I.(1983)RepresentingandInterveningCambridge:CambridgeUniversityPress;andGalison,P.(1987)HowExperimentsEndChicago:ChicagoUniversityPress.3Kuhn,T.S.(1970)TheStructureofScientificRevolutionssecondeditionChicago:ChicagoUniversityPress:seepostscript;andLakatos,I.(1970)‘Falsificationandthemethodologyofscientificresearchprogrammes’inLakatos,I.andMusgrave,A.(eds)CriticismandtheGrowthofKnowledgeCambridge:CambridgeUniversityPress.4Kuhn,T.S.(1970)op.cit,p.182.5Kuhn,T.S.(1970)op.cit.andKuhn,T.S.(1974)‘Secondthoughtsonparadigms’inSuppe,F.(ed.)TheStructureofScientificTheoriesUrbana,IL:UniversityofIllinoisPress;Hesse,M.B.(1974)TheStructureofScientificInferenceLondon:MacmillanandHesse,M.B.(1980)RevolutionsandReconstructionsBloomington,IN:IndianaUniversityPress;Lakatos,I.(1970)op.cit.;andLaudan,L.(1986)ScienceandValuesBerkeley:UniversityofCaliforniaPress.6Kuhn,T.S.(1970)op.cit,pp.181–7.7Galison,P.(1988)‘Philosophyinthelaboratory’JournalofPhilosophy85p.10.8SeeDavies,P.C.W.andBrown,J.(eds)(1988)Superstrings:aTheoryofEverything?Cambridge:CambridgeUniversityPress;formoretechnicaldetailseetheopeningreviewofstringtheoryinGreen,M.B,Schwarz,J.H.,andWitten,E.(1987)SuperstringTheoryvol.1Cambridge:CambridgeUniversityPress.9SeeBlau,S.K.andGuth,A.H.(1987)‘Inflationarycosmology’inHawking,S.W.andIsrael,W.(eds)300YearsofGravitationCambridge:CambridgeUniversityPresspp.524–603.10Wald,R.M.(1984)GeneralRelativityChicago:ChicagoUniversityPress;seepp.450–69.11Damour,T.(1987)‘TheproblemofmotioninNewtonianandEinsteiniangravity’inHawking,S.W.andIsrael,W.(eds)(1987)op.cit.,pp.128–98.258\nNOTES12Zahar,E.(1989)Einstein’sRevolution:aStudyinHeuristicLaSalle,IL:OpenCourt.13SeetheinterestingessaybyKuhn,T.S.(1981)‘Afunctionforthoughtexperiments’inHacking,I.(ed.)ScientificRevolutionsOxford:OxfordUniversityPress.14SeeLaudan,L.(1986)op.cit.,andalsoHesse,M.B.(1980)op.cit.forinvestigationsintotheroleofvaluesinscience.Seealsothe‘sociologically’basedanalysisofthoselikeBloor,Latour,andWoolgar,whoarguethatscientificknowledgeisafunctionofsociology—thatscienceisnomorethanasocialconstruction.ThisprogrammeisanextremeextensionofKuhn’sthinking.Thereisaparticularfocusintheseaccountsonthewaythatscientists,withalltheirpettyattitudesandrivalriesandvaluesandsoon,andwithalltheconstraintswhichsocietyimposesonthescientificcommunity,literallynegotiateamongstthemselveswhatideasshouldemergefromtheirresearch—itisasiftheycansaywhattheywillabouttheworld.Thisstrikesmeasacuriousattitude.Yes,scienceisundoubtedlyinfluencedbysociety.Butitisalsoinfluencedbythewaytheworldis.Fordetailsofthosewhopursuethesociologyofscience,seeRichards,S.(1987)PhilosophyandSociologyofSciencesecondeditionOxford:BasilBlackwell.15See,foradiscussionoftheimportanceofsymmetriesandinvarianceforphysics,vanFraassen,B.(1989)LawsandSymmetriesOxford:OxfordUniversityPress.16Cartwright,N.(1982)op.cit.259\nSELECTBIBLIOGRAPHYFIVEKEYBOOKSFORFURTHERSTUDYAlexander,H.G.(ed.)(1956)TheLeibniz-ClarkeCorrespondenceManchester:ManchesterUniversityPressFlood,R.andLockwood,M.(eds)TheNatureofTimeOxford:BasilBlackwellPais,A.(1982)SubtleistheLordOxford:OxfordUniversityPress(abrilliantbiographyofEinstein)Reichenbach,H.(1957)ThePhilosophyofSpaceandTimeNewYork:DoverSklar,L.(1974)Space,Time,andSpacetimeBerkeley,CA:UniversityofCaliforniaPressTENUSEFULBOOKSWITHAPHILOSOPHICALSLANTAngel,R.B.(1980)Relativity:theTheoryandItsPhilosophyOxford:PergamonEarman,J.S.(1989)WorldEnoughandSpace-TimeCambridge,MA:MITPressvanFraassen,B.C.(1980)AnIntroductiontothePhilosophyofTimeandSpacesecondeditionNewYork:RandomHouseFriedman,M.(1983)FoundationsofSpace-TimeTheoriesPrinceton,NJ:PrincetonUniversityPressHorwich,P.(1987)AsymmetriesinTimeCambridge,MA:MITPressMellor,D.H.(1981)RealTimeCambridge:CambridgeUniversityPressMunitz,M.K.(1986)CosmicUnderstanding:PhilosophyandtheScienceoftheUniversePrinceton,NJ:PrincetonUniversityPressNewton-Smith,W.H.(1980)TheStructureofTimeLondon:RoutledgeRay,C.(1987)TheEvolutionofRelativityBristol:AdamHilgerSalmon,W.C.(1980)Space,Time,andMotionsecondeditionMinneapolis:UniversityofMinnesotaPressTENUSEFULBOOKSWITHASCIENTIFICORMATHEMATICALSLANTBarbour,J.B.(1989)AbsoluteorRelativeMotion?vol.ICambridge:CambridgeUniversityPress(lookoutforvol.II)Cohen,I.B.(1987)TheBirthoftheNewPhysicsOxford:OxfordUniversityPress260\nSELECTBIBLIOGRAPHYEddingtonA.S.(1920)Space,Time,andGravitationCambridge:CambridgeUniversityPressEllis,G.F.R.andWilliams,R.M.(1988)FlatandCurvedSpace-TimesOxford:OxfordUniversityPressFrench,A.P.(1968)SpecialRelativityLondon:VanNostrandReinholdGribbin,J.andRees,M.(1989)CosmicCoincidencesLondon:BantamHawking,S.W.(1988)ABriefHistoryofTimeLondon:BantamStewart,I.(1987)TheProblemsofMathematicsOxford:OxfordUniversityPressToretti,R.(1983)RelativityandGeometryOxford:PergamonWald,R.M.(1984)GeneralRelativityChicago:ChicagoUniversityPressTENUSEFULCOLLECTIONSOFARTICLESANDCLASSICPAPERSBernstein,J.andFeinberg,G.(eds)(1986)CosmologicalConstants:PapersinModernCosmologyNewYork:ColumbiaUniversityPressCapek,M.(ed.)(1976)TheConceptsofSpaceandTimeDordrecht:ReidelCornell,J.(ed.)(1989)Bubbles,VoidsandBumpsinTime:theNewCosmologyCambridge:CambridgeUniversityPressEarman,J.S.,Glymour,C.N.,andStachel,J.J.(eds)(1977)FoundationsofSpace-TimeTheoriesMinneapolis:UniversityofMinnesotaPressEinstein,A.,Lorentz,H.A.,Weyl,H.,andMinkowski,H.(1923)ThePrincipleofRelativityLondon:MethuenHawking,S.W.andIsrael,W.(eds)(1987)300YearsofGravitationCambridge:CambridgeUniversityPressHealey,R.A.(ed.)(1981)Reduction,Time,andRealityCambridge:CambridgeUniversityPressLeslie,J.(ed.)(1989)PhysicalCosmologyandPhilosophyNewYork:MacmillanSalmon,W.C.(ed.)(1970)Zeno’sParadoxesIndianapolis,IN:Bobbs-MerrillSchilpp,P.(ed.)(1969)AlbertEinstein:Philosopher-ScientistLaSalle,IL:OpenCourt.FIVEUSEFULINTRODUCTIONSTOPROBLEMSINTHEPHILOSOPHYOFSCIENCEChalmers,A.F.(1982)WhatisThisThingCalledScience?secondeditionMiltonKeynes:OpenUniversityPressHacking,I.(1983)RepresentingandInterveningCambridge:CambridgeUniversityPressKuhn,T.S.(1970)TheStructureofScientificRevolutionssecondeditionChicago:ChicagoUniversityPressNewton-Smith,W.H.(1981)TheRationalityofScienceLondon:RoutledgeO’Hear,A.(1989)AnIntroductiontothePhilosophyofScienceOxford:OxfordUniversityPress261\nSELECTBIBLIOGRAPHYTheliteratureonspaceandtimecontainsmanyfinebooksinadditiontotheonesmentionedabove.Furtherreferencesmaybefoundinthenotestoeachchapter.Inevitablymanyhavenotbeenmentionedinthispersonalselection.ButIbelievethatyouwillfindeverybookinthisbibliographyinterestingandrewarding,thoughsomewillbehardgoing.Unfortunately,noteverybookmentionedaboveisstillinprint;however,inter-libraryloanfacilitiesmayhelpinlocatingcopies,shouldanybehardtofind.262\nINDEXabsolutespace99–103,134–7,146–Berresford,G.C.230n.50,179–84,215–16Bertotti,B.244n.,248n.absolutespacetime134–9,146–50,bigbang84–6,146,196–9,204–9215–16blackholes195,201–3,209–15,224absolutetime102Black,M.20,239n.Achinstein,P.242n.Blau,S.K.206,254n.,256n.257n.,Adams,J.C.90258n.adhochypotheses91,184Bloor,D.259n.affinegeometry54,134,139–42,197Bohm,D.42,61,233n.,235n.Alexander,H.G.241n,244n.,245n.,Bolyai,J.71246n,255n.Bondi,H.145–6,200,232n.,249n.,Anscombe,G.E.M.255n.256n.AnthropicPrinciple:83,226–7;Bowman,P.A.57,234n.Strong189–92,222;Weak191–2Boyle,R.104,113Aristotle5,109,229n.,249n.Brans,C.244n.,248n.Armstrong,D.M.241n.Brown,H.R.234n.Asquith,P.A.239n.,251n.Brown,J.238n.,254n.,258n.Ayer,A.J.241n.Butterfield,J.249n.backwardscausation153,171–5Calder,N.236n.Barbour,J.B.240n.,241n.,244n.,Capek,M.230n.248n.Carnap,R.244n.Barnes,J.229n.Cartan’sspacetimereformulationofBarrow,J.D.83,238n.,254n.,255n.Newtoniangravity247n.Bastin,E.229n.Carter,B.189,192,254n.,255n.Beck,A.230n.Cartwright,N.188,218,226–7,beginningoftime199–203254n.,258n.,259n.Beltrami,E.71Cauchy,A.11,212–13,216,229n.,Benacerraf,P.14,230n.249n.Bentley,R.177Chandler,M.237n.Bergson,H.25–6,44–5,230n.Chandrasekhar210,257n.Berkeley,G.116,122,126,243n.Chiu,H.Y.247n.Bernstein,J.248n.,253n.Clarke,C.J.S.213,257n.263\nINDEXClarke,S.99–100,103,106–12,194,empiricalequivalence92–5207,242n.Euclid69,70Cleomedes242n.Euclideangeometry69–78clockparadox24–6expansionofspacetime84–90,196–closedcausalloops79–82,153–719,204–9conformalgeometry53–4,97experimentation127–30,224–5conventionalism48,90–8Cornell,J.257n.Feigl,H.245n.cosmiccensorship195,211–15Feinberg,G.248n.,253n.cosmologicalconstant176–92,223Feyerabend,P.126,128,245n.cosmologicalmodels84–90,133,Feynman,R.64–5,83,91,164–5,143,186,196–209235n,238n,239n.,250n.CosmologicalPrinciple86–90Feynmandiagrams64–5,164–5Cottingham,J.234n.fieldequationsofGTR78,131–2,covariance147139–46,176–89Fine,A.240n.,252n.Damour,T.258n.Fizeau,A.46Davies,P.C.W.190–1,238n.,250n.,Flood,R.250n.254n.,256n.,257n,258n.Foucault,J.46Descartes,R.104,113,240n.,241n.,French,A.P.42,230n.,232n.,233n.255n.Friedman,M.97,114–15,122,determinism148–50,215–16234n.,235n,237n.,240n.,243n.,Devitt,M.234n.244n.,246n.,247n.,248n.,249n.Dicke,R.244n,248n.Friedmann,A.85–6,133,143–4,dimensionality5,82–3186,196,212,225,248n.,253n.Dopplereffect45Frisch,D.H.230n.Duff,M.J.252n.Duhem,P.90,230n.,239n.Galileo120Duhem-Quinethesis22,90,92Galison,P.218,220–1,245n.,258n.Dummett,M.156–7,250n.,251n.Gamov,G.251n.Gardner,M.R.246n.Earman,J.S.115,146,148–50,190,Gauss,C.F.71216,235n.,241n.,242n.,243n.,Geroch,R.210,256n.,257n.246n.,247n.,249n.,251n.,255n.,Giannoni,C.234n.257n.,258n.Giere,R.239n.Eddington,A.S.1,74,210,236n.Gilman,R.C.244n.,248n.Ehrenfest,P.82,238n.Glashow,S.254n.Einstein,A.1–4,24–5,82,119,131–Glymour,C.N.246n,249n.,257n.3,139,146–8,176–7,179–87,Gödel,K.142,186,248n.,250n.,221–3,225,230n.,236n.,243n.,251n.,253n.244n.,247n,249n.,252n.,253n.,Gold,T.200,256n.254n.Graves,J.C.243n.Einstein’sstaticuniverse180–4Green,M.B.70,82–3,224,258n.Elliot,R.66–8,235n.Greenstreet,W.J.236n.Ellis,B.57,234n.,235n.Gribbin,J.238n.,243n.,256n.Ellis,G.F.R.88–9,145,188,233n.,Grunbaum,A.52–55,59,234n.,235n.,238n.,239n.,248n.,249n.,235n.,256n.250n.,254n.,257n.264\nINDEXGuth,A.204–8,254n.,256n.,257n.,infinitespeed66–8258n.inflationarycosmology70–1,83,195,203–9Hacking,I.127–30,218,245n.,instrumcntalism150259n.intervals,spacetime36–7Halfele-Keatingexperiment26,invariance36,61,226231n.Isham,C.J.252n.Hall,A.R.241n,243n.isotropy86–90Hall,M.B.241n,243n.Israel,W.211,229n.,236n.,238n.,Halley,E.253n.250n.,254n.,255n.,256n.,257n.,Hanfling,O.244n.258n.Hanson,N.R.95,240n.,244n.Harré,R.233n.Kagon,R.242n.Harrison,J.168,251n.Kenny,A.255n.Hartle,J.202Kerrmetric:rotatingblackhole142,Hawking,S.W.145,176,188,191,248n.201–2,208,210,217,221,224,Kerszberg,P.252n.229n.,235n,236n.,238n.,248n.,Kitcher,P.251n.249n.,250n,253n.,254n.,255n.,Koyré,A.241n.256n.,257n,258n.Kretschmann,E.249n.Healey,R.A.243n,244n.Kuhn,T.S.95,124–5,219,220,Hesse,M.B.219,230n.,239n.,225–6,240n.,245n.,258n.,259n.240n.,244n,258n.,259n.Hobbes,T.255n.Lakatos,I.219–20,226,258n.Hoefer,C.248n.Laplace,P.S.209,212,257n.Hoffman,B.233n.,234n.Latour,B.259n.Hoffman,W.247n.Laudan,L.219,258n.holeargument146–50,213–16lawsofphysics187–9,221,226–7Holton,G.252n.Leibniz,G.3,67,99–100,105–13,homogeneity86–90118,131,148,193–5,203,241n.,Hookway,C.230n.242n.,243n.,244n.,245n.,246n.,Horowitz,G.T.256n.,257n.255n.Horwich,P.170,250n.,251n.Leibniz-Clarkecorrespondence108–Hoyle,F.200,256n.13,193–5,203Hubble,E.146,184Leplin,J.233n.,245n.Hume,D.122Leslie,J.256n.,257n.Huygens,C.99,110–11,246n.LeVerrier,J.90–1,185Lewis,D.251n.identityofindiscernibles67,107–8,lightcone60148limitingvalues11–14Iltis,C.243n.Linde,A.70,83,204,209,236n.,indeterminism148–50,215–16238n.,257n.inertialforces27–32,99–103,113–Lobatschewskii,N.I.7115,118–20Locke,J.104–5,113,223,241n.inertialframes27–32Lockwood,M.250n.inertialmotion27–32,99–103,118–Longair,M.S.254n.20,134–9Lorentz,H.A.58,131–2,230n.,infinitesimals10236n.,252n.265\nINDEXLorentztransformations27,33–6Newton’sthoughtexperiments100–3,110–12,118–20,133,225McCrea,W.H.238n.,254n.,255n.non-Euclideangeometry2,69–78Mach,E.2–3,102,115–23,125,North,J.D.252n.130–1,133–4,138,144–6,177,Norton,J.146,249n.,253n.179–83,222,227,243n.,244n.,247n.,252n.Occam’srazor97,116,122,240n.Machamer,P.K.243n.Occam,Williamof240n.Mach’sPrinciple133–4,138–9,143–Olber’sparadox253n.6,181–4,222Oszvàth,I.142,144,186,254n.Malament,D.B.53,97,234n.,247n.,Owen,G.E.L.10,229n.251n.Manders,K.243n.Pais,A.176,181,184,229n.,249n.,manifold,spacetime54,131,148–9251n.,252n.,253n.Marder,L.232n.Papineau,D.240n.Mates,B.241n.paradoxes:clock24–6;lampsuper-Maudlin,T.249n.task14–15;parallelsuper-taskMaxwell,G.245n.15–19;timetravel166–71;tripletsMaxwell,J.C.2441–4;twins36–41,134;Zeno’sMehra.J.257n.5–11Mellor,D.H.152,165,172–4,249n.,Parfit,D.235n.250n.,251n.Peirce,C.S.229n.metricalgeometry54,131,139,140–Penrose,R.6,150,155,210–11,213,2221,229n,238n.,250n.,255n.,Meyerson,E.253n.257n.Michell,J.257n.Penzias,A.196,255n.microwavebackground196Planck,H.223Minkowski,H.73,97,132,138,140,Poincaré,H.70–1,75,79,90,92,144,212,230n.,247n.,250n.,235n.,236n.252n.Poisson’sequation178Misner,C.246n.,247n.,248n.,Popper,K.R.91,183,239n,253n.249n.,253n.positivism40,117,122–30,190–1,Mundy,B.243n.222Munitz,M.251n.primaryqualities104Murdoch,D.234n.Putnam,H.245n.Musgrave,A.258n.Quine,W.V.O.90–1,234n.,239n.nakedsingularities214Narlikar,J.V.256n.Raine,D.143–4,244n.,248n.Nerlich,G.235n.,237n.Ray,C.235n.,240n.,241n.,244n.,Neumann,C.177–8,180,182–3,248n.,251n.,253n.,254n.,258n.252n.Raychaudhuri,A.K.255n.Newton,I.2,34,99–105,107,109,realism48110–12,117–20,123,132,135–7,Recami,E.66,235n.180–2,184–5,203,240n.,241n.,Redmount,I.250n.242n.,243n.,246n.,247n.,252n.Rees,M.238n.,254n.,255n.,256n.Newton-Smith,W.H.22,230n.,232n.,233n.,238n.,244n.,245n.266\nINDEXReichenbach,H.52,70–1,75,78–9,Stewart,I.10,229n.80–1,91,95–6,131–2,234n.,Stoothoff,R.234n.236n,237n.,238n.,239n.,244n.,substantivalism103,146–50,215–16246n.SufficientReason,Principleof107–8,Reichenbach,M.236n.193–5relationism105–8,118–20,138–9,superstrings70–1,82–3,224215–16super-tasks14–19Richards,S.259n.Suppe,F.258n.Riemann,G.72,82,236n.Swinburne,R.250n.Riggs,P.J.251n.Rindler,W.248n.tachyons60–6,153Rømer,O.57–8,234n.Taylor,E.F.232n.Russell,B.256n.Teller,P.146,148,241n.,242n.,249n.Sainsbury,M.14–17,19,29,230n.tensors72,147Salmon,W.6,20,46,49,50,53–4,theories218–28229n.,230n.,233n.,234n.,235n.theory-dependenceofobservationSchilpp,P.244n.124–30Schücking,E.142,144,186,254n.Thomson,J.6,14–15,17,19,229n.Schwarz,J.H.70,82–3,224,236n.,Thorne,K.246n.,247n.,248n.,238n.,258n.249n.,253n.Schwarzschildsolution,the132,141,Thurston,W.P.239n.144,222time,beginningof199–203Sciama,D.W.143,244n.,248n.timemachines151–71secondaryqualities104timetravel64–6,151–75Seeliger,H.177–8,180,182–3Tipler,F.J.254n.,255n.simplicity120–2,222Tolman,R.C.61,235n.Simplicius5,10,229n.topologicalholes198–9,213–16simultaneity:absolute53–7,136;topology20–3,79–90,139,141,metrical51–3,57,97,147;146–51,154–6topological51–3,57Turnbull,R.G.243n.singularities193–216two-slitexperiment49–50Sitter,W.de133,176,182,200,251n.,252n.underdeterminationoftheorybySklar,L.103,110,113–15,149,215,data49,90–8236n.,238n.,240n.,242n.,243n.,246n.,247n.,248n.,251n.vanFraassen,B.126–7,129,236n.,slowclocktransport57–9245n.,259n.Smith,J.H.230n.Vesey,G.256n.Sorabji,R.242n.vonLeyden,W.241n.spacetime:diagrams27–33;expansionof84–90,196–9,204–Wald,R.M.238n.,258n.9Waylen,P.C.244n.,248n.Stachel,J.J.230n.,242n.,246n.,Weekes,J.R.239n.257n.Weierstrass,K.11standardsignalsynchrony47–56Weinberg,S.249n.Stein,H.241n.,242n.,243n.,246n.,Weingard,R.258n.247n.Wells,H.G.151,249n.Steinhardt,P.J.204,256n.,257n.267\nINDEXWeyl,H.82,143,186,230n,236n,Witten,E.258n.238n.,252n.Wittgenstein,L.124259n.Wheeler,J.A.73,134,209,232n.,Woolgar,S.259n.246n.,247n.,248n.,249n.,253n.,wormholes160–1229n.255n.Williams,R.M.88–9,233n.,238n.,Young,T.49–50257n.Winnie,J.53,234n.Zahar,E.225,245n.,253n.,Wilson,M.243n.Zeno2,5–12,16,20–2,79,Wilson,R.W.196,255n.Zeno’sparadoxes5,7–11Winch,P.241n.268