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D. RICKLES WHO’S AFRAID OF BACKGROUND INDEPENDENCE? ABSTRACT:Backgroundindependenceisgenerallyconsideredtobe‘themarkofdistinction’ofgen- eralrelativity. However, thereisstillconfusionoverexactly whatbackgroundindependence isand how,ifatall,itservestodistinguishgeneralrelativity fromothertheories. Thereisalsosomecon- fusionoverthephilosophical implications ofbackground independence, stemminginpartfromthe definitional problems. InthispaperIattempt tomakesomeheadway onbothissues. Ineachcase Iarguethataproperaccountoftheobservablesofsuchtheoriesgoesalongwayinclarifyingmat- ters. Further,Iargue,againstcommonclaimstothecontrary,thatthefactthattheseobservablesare relationalhasnobearingonthedebatebetweensubstantivalistsandrelationalists,thoughIdothinkit recommendsastructuralistontology,asIshallendeavourtoexplain. 1 INTRODUCTION Everybodysaystheywantbackgroundindependence,andthenwhentheysee ittheyarescaredtodeathbyhowstrangeitis... Backgroundindependence isabigconceptualjump. Youcannotgetitforcheap... ([Rovelli,2003],p. 1521) In his ‘Who’s Afraid of Absolute Space?’ [1970], John Earman defendedNew- ton’s postulation of absolute, substantival space at a time when it was very un- fashionabletodoso,relationalismbeingalltherage. Later,inhisWorldEnough andSpace-Time, he arguedfora tertium quid, fitting neithersubstantivalismnor relationalism, substantivalism succumbing to the hole argument and relational- ism offering“more promissorynotesthan completedtheories” ([Earman, 1989], p. 195). More recently, in a pair of papers written with Gordon Belot [1999; 2001],substantivalismcomesunderattackagain.Thistimetheculpritistheback- groundindependenceofgeneralrelativity,andthepotentialbackgroundindepen- denceofafuturetheoryofquantumgravity.Theclaimisthatwerethesuccessful futuretheoryofquantumgravityshowntobebackgroundindependent,thensub- stantivalismwouldberendereduntenableforreasonsofphysics—thusprovidinga clear-cutexampleofShimonyan‘experimentalmetaphysics’inaction.1 Stillmore recently, in Volume 1 from this series [Dieks, 2006], Earman returns to his ter- tiumquid idea, defending,againonthebasisof (amanifestationof)background 1Infact,thisisreallyjusttheholeproblemagain. Inanotherpaper,[Rickles,2005b],Iexplicitly translatedtheholeargumentintotheframeworkof(backgroundindependent)loopquantumgravity, thusdemonstratingthat(thisapproachto)quantumgravitydoesnotputthedebatebetweensubstanti- valistsandrelationalistsonbettergroundthanintheclassicaltheory: substantivalistshavenothingto fearfromquantumgravity(notinthecaseofloopquantumgravityatanyrate).Iwillaimtostrengthen thisconclusionfurtherinthischapter. 2 D.RICKLES independence,whatIconsidertobeastructuralistpositionwhichdeniesthefun- damentalexistenceofsubjects(inthesenseof‘bearers’ofproperties),thusruling outbothrelationalismandsubstantivalism[Earman,2006a].2 However, my primarytargetin this paperis the issue of what backgroundin- dependenceis: only when this is resolved can we assess the claim that it might servetosettledebatesovertheontologyofspacetime(mysecondarytarget). Let usbeginbyconsideringsomebasicmetaphysicalaspectsofbackgroundstructure, dependence,andindependence,beforefirmingthediscussionupwiththetechnical anddefinitionalaspects.Ontologicalimplicationsmustwaituntilthefinalsection. 2 METAPHYSICSOFBACKGROUNDS Metaphysiciansliketotellthefollowingstorytodistinguishbetweenphysicalism andothernon-physicalistpositions: WhenGodmadetheworlddidHelayoutallthelocalphysicalmattersoffact(propertiesatspacetime points)andtherest(causation,laws,modality,consciousness,etc.) followed,ordidHethenhaveto addtheseafterorinadditiontodoingthat?3 Physicalists, ofcourse, thinkthatin fixing thelocalphysicalfactsin a world He therebyfixedeverythingthereistothatworld:allthatthereisisphysical.Mutatis mutandis,wecanusethisstrategytodistinguishbetweenpositionsonspacetime ontologytoo: WhenGodmadetheworlddidHefirstcreatespacetimeandthenaddmatter(particles,fields,strings, etc.)toitordidHecreatematterandtherebyfixtheexistenceofspacetime? Or,inotherwords,dospacetime,andspatiotemporalpropertiesandrelations,exist independentlyofphysical,materialobjects(particles,fields,strings,branes,etc...) oristheexistenceofsomesuchobjectsnecessary4 fortheirexistence? Substan- tivalistswillanswerYesto thefirst disjunctandrelationalistswillanswerYesto thesecond.Letusbecleareronexactlywhatismeantbytheseterms. HereIshall followSklar[1974](himselffollowedbyEarmanandagenerationofphilosophers ofphysics)intakingsubstantivalismtobethepositionthatviewsspacetimetobe an entity which exists over and above any material objects it might contain; or, in Earman’s words, “prior to the objects it contains” instead of being “nothing 2Earmangoessofarastosuggestanentirelynewontologicalcategory:a“coincidenceoccurrence” ([Earman,2006a],p. 16). ThisisclosetowhatItaketobeoneofthemainontologicalimplications ofbackground independence; however, asIhavealready intimated, Icouchmatters instructuralist terms—see§4. Seealso[Rickles,2006]forasimilarproposaldrawnfromthefrozenformalismand problemoftimeinclassicalandquantumgravity,and[Rickles,Forthcoming]foramoregeneralde- fenseoftheviewonthebasisof(gauge)symmetriesinphysics. 3Inotherwords,arecausation,laws,modality,consciousness,etc.,supervenientonthelocalphys- icalmattersoffact(butnotviceversa),ordotheyconstitutesomething‘overandabove’thesefacts? 4Orjustsufficientifwewishtowageawaroverontologicalparsimony. WHO’SAFRAIDOFBACKGROUNDINDEPENDENCE? 3 but (might be constituted by, might be reducible to) the mutual relations among coexistentobjects”([Earman,1989],p. 289). Thisalso capturesmuchoftheintuitivedistinctionbetweenbackgroundinde- pendenceanddependence: isspactime(geometry)fixed‘prior’tothedetermina- tionofthestateofmatterintheuniverseordoesoneneedtoknowwhatthestateof matteris‘prior’tothedeterminationofspacetimegeometry? Giventhissuperfi- cialsimilarity,thedistinctionbetweenbackgroundindependenceandbackground dependenceisoftensupposedtolatchontothedistinctionbetweenrelationalism and substantivalism: relationalism being committed to the former; substantival- ismbeingcommittedtothelatter.5 However,the‘dualrole’ofthemetricfieldin generalrelativityrathermuddiesthewatershere. TheschizophrenicnatureofthemetricfieldwasviewedbyEinsteinasaneces- saryconsequenceoftheequivalenceprinciple,identifyinginertiaandweight:“the symmetric‘fundamentaltensor’[g ]determinesthemetricalpropertiesofspace, µν the inertial behaviourof bodies in it, as well as gravitational effects” ([Einstein, 1918c],p. 241).Or,asCarloRovelliputsit:“WhatEinsteinhasdiscoveredisthat Newton had mistaken a physical field for a backgroundentity. The two entities hypostatizedbyNewton, space andtime, arejusta particularlocalconfiguration ofaphysicalentity—thegravitationalfield—verysimilartotheelectricandthe magneticfield” ([Rovelli, 2006], p. 27). In other words: “Newtonian space and timeandthegravitationalfieldarethesameentity”(ibid.). This duality, one expression of background independence in general relativ- ity, has been responsible for much recent debate in the philosophyof spacetime physics. Again,itis seentobe implicatedin thetraditionaldebatebetweensub- stantivalismandrelationalism: InNewtonian physics, ifwetake awaythedynamical entities, what remains isspace andtime. In relativisticphysics,ifwetakeawaythedynamicalentities,nothingremains. AsWhiteheadputit,we cannotsaythatwecanhavespacetimewithoutdynamicalentities, anymorethansayingthatwecan havethecat’sgrinwithoutthecat.([Rovelli,2006],p.28) The reason being that the gravitational field is dynamical and does the work of twoinalsosupplyingthestructuresthatcharacterizespacetime.However,thepro- posed link to the substantivalism-relationalismdebate is problematic. Rovelli is lumpingallofthedynamicalfieldstogether,asbeingontologically‘onallfours’; butthis is a mistake: we can removeall fields with the exceptionof the gravita- tional field and still have a dynamically possible world6—i.e. there are vacuum 5Thereareother,primafaciemoresubstantivereasonsforthealignment; however,thesereasons ultimatelyfailaswell:see[Rickles,2005b;2005a;2006;Forthcoming]forthereasonswhy. 6ThestillMachianEinsteinof1918wouldnotagreewiththisclaim.Hewritesthat“with[Mach’s Principle]accordingtothefieldequationsofgravitation,therecanbenoG-fieldwithoutmatter”([Ein- stein, 1918b], p. 34)—of course, this is where his λ-term appears, precisely inorder to make the fieldequationscompatiblewithMach’sPrinciple.Thefieldequationsloseout,beingtransformedinto Gµν−λgµν =−κ(Tµν− 21gµνT),whichdonotallowforempty(i.e.Tµν =0)spacetimes.The positionIcometodefendisnotamillionmilesawayfromthis: otherfieldsareneededtoformthe gauge-invariantcorrelations(betweenfieldvalues)thatprovidethebasicphysicalcontentofthetheory. 4 D.RICKLES solutions to Einstein’s field equations. But we cannot remove the gravitational field in the same way, leaving the other fields intact. This is an indication that thereissomethingspecialaboutthegravitationalfield: itcan’tbeswitchedoff;it isnotjustonefieldamongmany. Hence,thesubstantivalistwouldbeperfectlywithinherrightstoclaimowner- ship. But,sowouldtherelationalistsincethereisthisambiguityovertheontolog- ical natureof the field7: spacetime ormaterialobject? I take thisstate of affairs (namely‘jointownership’ofthemetricfield)tolendsubstantialsupporttoRobert Rynaseiwicz’s[1996]claimthatthedebatebetweensubstantivalismandrelation- alismis“outmoded”inthiscontext. However,wearedriftingsomewhatfromour brief, whichis to geta griponthe conceptof backgroundindependence(andits companions,backgroundstructureandbackgrounddependence). Backgroundstructuresare contrasted with dynamicalones, and a background independenttheoryonlypossessesthelattertype—obviously,backgrounddepen- denttheoriesare those possessing the formertype in additionto the latter type.8 Philosophers,andsomephysicists, willbe morefamiliarwiththe term‘absolute element’inplaceofbackgroundstructure,andthelatterconceptcertainlysoaksup alargepartoftheformer. Ontheformerconcept,inhis1921PrincetonLectures ontheTheoryofRelativity,Einsteinwrites: JustasitwasconsistentfromtheNewtonianstandpointtomakeboththestatements,tempusestabso- lutum,spatiumestabsolutum,sofromthestandpointofthespecialtheoryofrelativitywemustsay, continuumspatiiettemporisestabsolutm. Inthislatterstatementabsolutummeansnotonly“phys- ically real,” butalso “independent inits physical properties, having aphysical effect, butnotitself influencedbyphysicalconditions.([Einstein,1921],p.315) There are three componentshere: a realist thesis, an independencethesis, and a non-dynamicalthesis. Clearlytherealistthesiscan’tsimplymeanthatspaceand timeexist,forLeibniztoowouldsurelyassenttosuchathesisinsomesense. In- stead,Itakeittomeanthatspaceandtimearefundamentalinthesensethatthey donotsuperveneonanyfurther,underlyingobjects,properties,orfacts. Thein- dependencethesisjustlookslikeadenialofrelationism,whilethenon-dynamical thesisamountsto ‘absolute’in somethinglike the sense of Anderson’snotionof ‘absoluteobject’([Anderson,1967],pp.83-7)—thisitself,ofcourse,corresponds most closely to Newton’s notion of absolute in the sense of immutability, itself followedverycloselybyEinsteinhimself: IfNewtoncalledthespaceofphysics‘absolute’,hewasthinkingofyetanotherpropertyofthatwhich wecall‘ether’. Eachphysicalobjectinfluencesandingeneralisinfluencedinturnsbyothers. The 7Forexample,therelationalistmight,asRovellidoes,drawattentiontothefactthat“astrongburst ofgravitationalwavescouldcomefromtheskyandknockdowntherockofGibraltar,preciselyasa strongburstofelectromagneticradiationcould”([Rovelli,1997],p.193). 8There is often a fair amount of slipping and sliding on this: there are degrees of background structure. Generally, onehasinmindbackgroundfieldsratherthanstructuresperse; thatis,oneis interestedinthefreedom(ornot)fromgeometric-objectfieldsonamanifoldthataredeemed‘back- ground’. Thoughthemanifolditselfappearsasabackgroundstructure,thisisgenerallynotcounted whenassessing atheory’s background independence. This is acontentious point that we return to later—see,especially,footnote10. WHO’SAFRAIDOFBACKGROUNDINDEPENDENCE? 5 latter,however,isnottrueoftheetherofNewtonianmechanics.,Theinertia-producingpropertyofthis ether,inaccordancewithclassicalmechanics,ispreciselynottobeinfluenced,eitherbytheconfigu- rationofmatter,orbyanythingelse. Forthisreason,onemaycallit‘absolute’. ([Einstein,1999],p. 15) Andersonpreferstocallthis“theprincipleofreciprocity”: Itisseenthattheabsoluteelementsofatheoryeffect[sic.] thephysicalbehaviourofasystem. That is,adifferentassignmentofvaluestotheabsoluteelementswouldchangethephysicalbehaviourof thesystem.Forinstance,theassignmentofdifferentvaluestothemetricmightresultinparticlepaths thatarecirclesratherthanstraightlines. Ontheotherhand,thephysicalbehaviourofasystemdoes notaffecttheabsoluteelements.Anabsoluteelementinatheoryindicatesalackofreciprocity;itcan influencethephysicalbehaviourofthesystembutcannot,inturn,beinfluencedbythisbehaviour.This lackofreciprocityseemstobefundamentallyunreasonableandunsatisfactory. Wemayexpressthe converseinwhatmightbecalledageneralprincipleofreciprocity: Eachelementofaphysicaltheory isinfluencedbyeveryotherelement. Inaccordancewiththisprinciple, asatisfactory theoryshould havenoabsoluteelements.([Anderson,1964],p.192] LeeSmolintooadoptsasimilarline,explicitlylinkingthisnotionofabsolutewith thenotionofbackground. Hewritesthat“[t]hebackgroundconsistsofpresumed entities that do not change in time, but which are necessary for the definition of the kinematicalquantitiesand dynamicallaws” ([Smolin, 2006], p. 204). How- ever, matters are not so simple as this. This rough ‘absolute elements’ way of definingbackgroundindependenceandbackgrounddependenceistoovaguetodo anyrealwork,andthevariousmethodsoffirmingthingsupfaceseriousproblems (asweshallsee). Moreover,thepeculiarnatureofgeneralrelativity,repletewith itstreatmentofthemetricasalocaldynamicalvariable,threatenstocollapsethe debatebetweensubstantivalismandrelationalism.Theinterpretationofspacetime physicsappearstobefloundering. Yet both debates, between backgroundindependence and backgrounddepen- denceandbetweensubstantivalismandrelationalism,arebelievedbymanyphysi- cistsandahandfulofphilosopherstoplayavitalroleinthesearchforaquantum theoryofgravity.Forexample,inmuchofhisrecentworkLeeSmolin(e.g.[2004; 2006])defendstheideathatbackgroundindependenceisanecessarypieceofthe quantumgravity puzzle: it is essential to solve the puzzles that quantumgravity raisesthatthegeometryofspacetimeisgivenasasolutionofsomeequationsof motion, rather than placed in the theory ‘by hand.’ But Smolin also argues that background independence uniquely supports relationalism, claiming that physi- cists“oftentakebackgoundindependentandrelationalassynonymous”([Smolin, 2006], p. 204). A big target in this paper is just this claim—a claim also made by Belot and Earman [1999; 2001] in order to prop up the listless body of the substantivalism/relationalism debate. Substantivalists needn’t be afraid of back- groundindependenceanymorethanrelationalists. However,ultimatelybothlose outtoastructuralistposition! 3 6 D.RICKLES DEFINITIONSANDDISPUTATIONS Itisoftenclaimedthatthenoveltyofgeneralrelativityliesinits(manifest)‘back- groundindependence.’ However,backgroundindependenceisaslipperyconcept apparentlymeaningdifferentthingstodifferentpeople.Inthissectionweattempt togainafirmergriponthisslipperycustomerbyconsideringvariouselucidations of background independence that have been suggested. There is a clear core to thenotion,andIarguethatthiscorecanbemadeclearerbyconnectingthecon- ceptofbackgroundindependencetothenatureoftheobservablesinbackground independenttheories. Letusbeginbypresentingageneralwayofmakingsenseofthevariousproposals— here I largely follow [Giulini, Forthcoming]. Let us specify a theory by writing downits laws as a set of equationsof motion representingrelationsbetween the centralobjectsofthetheory.Wegetthefollowingschema: (1) E[D,B]=0 HereDrepresentsthedynamicalstructures(thosethathavetobesolvedtogettheir values,suchastheelectromagneticfieldandthemetricingeneralrelativity—these representthephysicaldegreesoffreedomofthetheory,outwhichtheobservables willbeconstructed)andBthebackgroundstructures(thosewhosevaluesareput in ‘by hand’, such as the topology and, in pre-general relativistic theories, the metric). Now letusrepresentthe spaceofkinematicallypossiblehistoriesbyK. Then E[D,B] = 0 selects a subset P ⊂ K of dynamicallypossible histories (or ‘physically’ possible worlds) relative to B.9 Now, if there are no such Bs (or, rather, no B-fields)then the physicallypossible histories(the dynamics)is given by relationsbetween the Ds (and, atleast fiducially—i.e. in termsof the formal definitionof the fields—the manifold, butthe diffeomorphismsymmetrywashes this dependence away). This impacts on the observables of the theory, for the observables must then make no reference to the Bs, only to the Ds. This is the source of the claim thatgeneralrelativity, and backgroundindependenttheories, arerelational:itsimplymeansthatthestatesandobservablesofthetheorydonot makereferencetobackgroundstructures.10 9As Wheeler puts it, “[k]inematics describes conceivable motions without asking whether they areallowed orforbidden. Dynamicsanalyses thedifference between aphysically reasonable anda disallowedhistory”([Wheeler,1964],p.65). 10Though,again,thisdoesnotincludethemanifoldwhichisrequiredforthe(formal)definitionof thedynamicalfields. Theinescapablepresenceofthemanifold,inwhichdimension,topology,differ- entialstructureandsignaturearefixedindependentlyoftheequationsofmotion,leadsSmolintocall generalrelativityonlya“partlyrelationaltheory”([2006],§7.4).However,theabsenceofbackground fieldscoupledwiththesymmetryofthemanifoldmeansthatadisplacement(viaadiffeomorphism) ofthedynamicalfieldswithrespecttoitsimplyproducesagauge-equivalentrepresentationofoneand thesamephysicalstate. Eliminationoftheseredundantpossibilities(“surplusstructure”inRedhead’s sense[1975])furtherreducesthesizeofP,givingusthereducedspaceP =P/Diff(M).This‘su- perspace’containspointsthatareentireorbitsofthegaugegroup,representingabstract‘delocalized’ structuresknownasa“geometries”—see[Misneretal.,1973],p.522.Thisissupposedtobeaspace WHO’SAFRAIDOFBACKGROUNDINDEPENDENCE? 7 This way of understanding a theory lets us recapitulate in a clearer way our earlier definitions of covariance and invariance. Let G be a group of spacetime symmetries that acts on K as G × K → K—i.e. elements of G map kinemati- callypossiblesolutionsontokinematicallypossiblesolutions. We saythatG isa symmetrygroupofthe theorywhose space ofkinematicallypossible historiesis KjustincaseP isleftinvariantbyitsaction. Alternatively,andmoreusefullyfor what follows, we can express the distinction between covariance and invariance as—thisissummarizedschematicallyinfig. 1:11 [COV]⇒E[D,B]=0 iff E[g·D,g·B]=0 (∀g ∈G) (2) [INV]⇒E[D,B]=0 iff E[g·D,B]=0 (∀g ∈G) (3) Spacetime Model (cid:127)M,B,D Background Fields Dynamical Fields B : M → V D : M → V Diffeomorphism φ : M → M Symmetry Group φ ∈ G ⊆ Diff(M) Equations of Motion E[D,B] = 0 Covariance Invariance E[φ · D,φ · B] = 0 E[φ · D,B] = 0 Figure 1. How to understand covariance and invariance groups in a spacetime theory. Here,thefieldstakevaluesinavectorspace(oramorestructuredspace). Thediffeomorphismsdragfieldsalongtonewpoints.Theequationsofmotionare oftheform‘solveforDgivenB’. fitonlyforrelationalists;however,thereareplentyofgoodargumentsthatshowthatthesubstantivalist hasjustasmuchrighttooccupyit—see[Pooley,Forthcoming]and[Rickles,Forthcoming]formore details. 11Cf. Giulini[Forthcoming],p.6. Irecommendthatallphilosophersofphysicsinterestedinback- groundindependence,andthedifficultiesindefiningabsoluteobjects,readthisarticle: itisanexcep- tionallyclear-headedreview. 8 D.RICKLES As I said, in the context of general relativity [COV] = [INV] since B = ∅ (manifoldaside). Of course, the factthatthemanifoldappearsin the laws—and theabsenceofsymmetry-reducingbackgroundfields(i.e. toreducetheeffective symmetrygroupofthetheory)—meansthatthetherewillbesurplusstructure:the localizationonthemanifoldofthedynamicalfieldsispuregauge.12 All this symmetryaffectsthe dynamicsso that a standardHamiltonianor La- grangianformulationisnotpossible. Respectively,thecanonicalvariablesarenot allindependent(beingrequiredtosatisfyidentitiesknownasconstraints:φ(q,p)= 0)andtheEuler-Lagrangeequationsarenotallindependent.Theseidentitiesserve to‘constrain’thesetofphasespacepointsthatrepresentgenuinephysicalpossi- bilities: onlythosepointssatisfyingtheconstraintsdoso,andtheseformasubset inthefullphasespaceknownasthe‘constraintsurface’. As I also said, this has an impacton the form of the observables—andthis is terriblyimportantforthequantizationofthetheory.Sinceapairofdynamicalvari- ables(notobservables)thatdifferbyagaugetransformationareindistinguishable, correspondingtooneandthesamephysicalstateofaffairs,theobservablesought toregisterthisfacttoo:thatis,theobservablesofagaugetheoryshouldbeinsen- sitivetodifferencesamountingtoagaugetransformation—asshouldthestatesin anyquantizationofsuchatheory: i.e. ifx ∼ y thenΨ(x) = Ψ(y).13 Where‘O’ isadynamicalvariable,‘O’isthesetof(genuine)observables,x,y ∈P,and‘∼’ denotesgaugeequivalence,wecanexpressthisas: (4) O∈O ⇐⇒ (x∼y)⊃(O(x)=O(y)) Or, equivalently, we can say that the genuine observables are those dynamical variablesthatareconstantongaugeorbits‘[x]’(where[x]={x:x∼y}): (5) ∀[x], O∈O ⇐⇒ O[x]=const. Mostoftheworkdoneonfindingtheobservablesofgeneralrelativityisdoneusing the3+1projectionofthespacetimeEinsteinequations.Thatis,theconstraintsare understoodasconditionslaiddownontheinitialdata14hΣ,h,Kiwhenweproject thespacetimesolutionontoaspacelikehypersurfaceΣ—here,hisaRiemannian 12Thisdifferencecorresponds,then,tothatbetween‘passive’and‘active’diffeomorphisminvari- ance.AsRovelliputsit:“Afieldtheoryisformulatedinmannerinvariantunderpassivediffs(orchange ofco-ordinates),ifwecanchangethecoordinatesofthemanifold,re-expressallthegeometricquanti- ties(dynamicalandnon-dynamical)inthenewco-ordinates,andtheformoftheequationsofmotion doesnotchange.Atheoryisinvariantunderactivediffs,whenasmoothdisplacementofthedynamical fields(thedynamicalfieldsalone)overthemanifold,sendssolutionsoftheequationsofmotioninto solutionsoftheequationsofmotion”([Rovelli, 2001],p. 122). Wewillcalltheformergeneralco- varianceandthelatterdiffeomorphisminvariance—Earman[2006b]callsthelattersubstantivegeneral covariance,ontheunderstandingthatitamountstoagaugesymmetry,aswehaveassumed. 13ItseemsthatEinsteinwasawareofthisimplicationsoonaftercompletinghistheoryofgeneral relativity,forhewritesthat“theconnectionbetweenquantitiesinequationsandmeasurablequantities isfarmoreindirectthaninthecustomarytheoriesofold”([Einstein,1918a],p.71). 14NotethatJohnWheelerreferstoconstraintsas“initialvalueequations”([Wheeler,1964],p.83). Thisterminologygetsoneclosertothephysicalmeaningoftheconstraints. WHO’SAFRAIDOFBACKGROUNDINDEPENDENCE? 9 metriconΣandK istheextrinsiccurvatureonΣ. Iwon’tgointothenittygritty detailshere,butitturnsoutthattheHamiltonianofgeneralrelativityisasumof constraintsonthisinitialdata(ofthekindthatgenerategaugemotions,namely1st class)—hence, the dynamicsis entirely generatedby constraintsand is therefore puregauge.15 This formulation allows us to connect the characterization of the observables uptothedynamics(generatedbyconstraintsH )moreexplicitly: i (6) O∈O ⇐⇒ {O,H }≈0 ∀i i Inotherwords,theobservablesofthetheoryarethosefunctionsthathaveweakly vanishing(i.e. ontheconstraintsurface)Poissonbracketswithallofthefirst-class constraints. Thesearethegauge-invariantquantities. Apressingproblemingen- eralrelativity—especiallypressingforquantumgravity—istofindsuitableentities thatsatisfythisdefinition.Thereareatleasttwotypesthatfitthebill:highlynon- local quantities defined over the whole spacetime16 and (differently) non-local, ‘relational’quantitiesbuiltoutofcorrelationsbetweenfieldvalues. Thereseems to be some consensus forming, at least amongst ‘canonical relativists’, that the lattertypearethemostnatural. I return to the interpretation of these correlational observables in §4. Let us nowconsideranumberofstandardtakesonthequestionofwhatbackgroundin- dependenceamountsto.Weassesstwomainwaysofdoingthis—utilizinggeneral covarianceand diffeomorphisminvariance—beforeconsideringJ. L. Anderson’s proposalfor firming up the latter approach and discussing the connectionto ob- servables. Aswesawabove,generalcovariancesimplyreferstothefactthatwhenwehit asolutionwithanarbitrarydiffeomorphism,wegetanothersolutionback.Thatis tosay,theequationsofmotionarecovariantwithrespecttodiffeomorphisms.This amountstoaacarryingalongofallofthefields. Covarianceisnotsorestrictive asinvariance(or“symmetry”asAndersoncallsit). TheformerjustsaysthatifM isasolutionthensoisMwhereM′ =g(M)andgisanelementofthecovariance group, the group that preserves the form of the laws (the equations of motion). Thetheoryissaidtobeg-covariant.Butthisisjustaconstraintontheformofthe theory,notonitsphysicalcontent.Inotherwords,generalcovarianceinthissense issimplyapropertyoftheformulationofthetheory. Thisis,ofcourse,justwhat KretschmanntaughtEinsteinsoonaftergeneralrelativitywaswrittendowninits finalform. Theproblemisthat(general)covariancejustmeansthattheequation 15This turns out to be behind the two worst conceptual problems of general relativity: the hole argumentandthefrozenformalismproblem. Fordetailsontheseconnectionssee[BelotandEarman, 1999;2001;Rickles,2006]. Earman[2003]givesasplendidpresentationoftherelationshipbetween theconstrainedHamiltonianformalismandgauge,includingtheirimplicationsfortimeandchange. 16Thereisaproof(forthecaseofclosedvacuumsolutionsofgeneralrelativity)thattherecanbeno localobservablesatall[Torre,1993]—‘local’heremeansthattheobservableisconstructedasaspatial integraloflocalfunctionsoftheinitialdataandtheirderivatives. 10 D.RICKLES ofmotioniswell-definedinthesenseofdifferentialgeometry: theequationneed simply‘live’onthemanifold. Recallhowcoordinatizationoccurs.Firstly,weassociatepointsofthemanifold withR4sothateachx∈Mgetsfournumbers{xµ}(µ=1,2,3,4)associatedto it.Wecandothisinmanyways,asmentionedabove.Wemightusetheassignment {x′µ} → y ∈ Minstead. Becausethesenumbersareassignedtothesamepoint therewillbesomerelationshipbetweenthecoordinatesystems:17 (7) x′µ =x′µ(xν) Giventhedifferentialstructureofthemanifold,wegetaninfinitelycontinuously differentiablefunctionbetweenthe thusrelated coordinatesystems(with a simi- larlydifferentiableinverse).18 Thisisadiffeomorphismpassivelyconstrued;itis gaugeinaverytrivialsense,asWheelersays:“Howonedrawscoordinatesurfaces throughspace-timeisamatterofpaperworkandbookkeeping,andhasnothingto dowiththerealphysics”([Wheeler,1964], p. 81). Thisgoesforanyreasonable spacetime theory. Hence, generalcovarianceunderstoodin these termsdoesnot havethepowertodistinguishbetweenspacetimetheories,andifbackgroundinde- pendenceissupposedtodistinguishgeneralrelativityfromprevioustheories,then generalcovariancecannotunderwriteit. Hence,anyspacetimetheorywrittenintermsofgeometricobjectsontheman- ifoldwillbegenerallycovariantinthesenseofhavingDiff(M)asitscovariance group. However,invarianceisamuchstrongerrequirementthatpicksoutasub- groupofthecovariancegroup;thissaysthatifMisasolutionthensoisMwhere M′ = g(M)andg andnowgisanelementofasubgroupofthecovariancegroup thatpreservestheabsoluteelements. Thetheoryissaidtobeg-invariant. Hence, inonewemapalloftheobjects,intheotherweonlymapthedynamicalobjects, whilstpreservingthebackgroundstructure.19 Thisis the standardview: ratherthanconsideringthe backgroundfieldsto be transformed along with the dynamical fields, we view the diffeomorphisms in an active way, as shifting the dynamical fields relative to the same background fields. Thus, Lee Smolin writes that “[g]eneral coordinate invariance [general covariance—DR] is not the same thing as diffeomorphism invariance, and it is the latter, and not the former, thatis the key to the physicalinterpretationof the theory”.Hegoesontosaythat 17Itisusefultothinkofthepairofcoordinatesystemsasbeinglikeapairoflanguagesandasthe particularcoordinatesassignedtosomeparticularpointasbeinglikenounsinthelanguagereferring toaparticularobject. Atranslationmanualbetweenthelanguageswouldbeanalogoustothediffer- entiablefunctionsrelatingdistinctcoordinatedescriptionsofoneandthesamepoint;inthiscasewe wouldhavedistinctwordsreferringtothesameobject,thesewordsbeinginter-translatable. 18Generally,becausethemanifoldsingeneralrelativityarecoordinatizedbygluingpatchestogether, thefunctionwillbeevaluatedontheoverlapbetweencoordinatesystems. 19Thestorythengoes: specialrelativity cannotbediffeomorphism invariant—i.e. itcannothave Diff(M)asitssymmetrygroup—becausetheimpositionoftheMinkowskimetricreducestheinvari- ancegrouptoasubgroupofthecovariance group, namelythePoincare` groupofisometries ofthis metric, ofMinkowski spacetime. Thissmaller groupis thelargest thatpreserves the (background) structureofMinkowskispacetime;thereareclearlyelementsofDiff(M)thatwouldnotdoso.

Description:
fusion over the philosophical implications of background independence, it they are scared to death by how strange it is . stantivalism would be rendered untenable for reasons of physics—thus . a large part of the former. On the . As I said, in the context of general relativity [COV] = [INV] sin
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