who's afraid of background independence?

21 Pages·2007·0.22 MB·English

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.