ShapeDynamics
RelativityandRelationalism
FlavioMercati
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CoverpictureconceptbyJ.Barbour
‘Fromhenceforththeplanetsfollowtheirpathsthroughtheetherlikethebirds intheair.Wemustthereforephilosophizeaboutthesethingsdifferently.’
JohannesKepler(1571–1630)
1Introduction 1
1.1OutlineofChapters1
1.2Notation4
1.3ShapeDynamicsinanutshell4
PartIHistoricalMotivation
2Newton’sBucket 17
2.1Thedefectsofthelawofinertia17
2.2Leibniz’srelationalism18
2.3The Scholium problem19 Furtherreading20
3OriginsoftheMach–PoincaréPrinciple 21
3.1Tait’spartialsolutionoftheScholiumproblem21
3.2Mach’scritiqueofNewton23
3.3Hoffman’sexperiment25
3.4Poincaré’sprinciple26 Furtherreading27
PartIIRelationalParticleDynamics
4Barbour–BertottiBestMatching 31
4.1Bestmatching:intuitiveapproach31 Furtherreading38
5BestMatching:TechnicalDetails 39
5.1MathematicalFormulation39
5.2Temporalrelationalism:Jacobi’sprinciple44
5.3Bestmatching‘inaction’48 Furtherreading55
6HamiltonianFormulation 56
6.1TheHamiltonianconstraint56
6.2AcrashcourseinDirac’sconstraintanalysis57
6.3Applicationtooursystems61
6.4Amatterofunits63 Furtherreading66
PartIIIRelationalFieldTheory
III.1TheOntologyofFields67
III.2TheoriginsofGeometrodynamics69
III.3Anexerciseincounterfactualhistory70
III.4Nuggetsoffunctionalanalysis71
7RelativityWithoutRelativity 73
7.1RigidityofthechoiceoftheLagrangian76
7.2Inclusionofascalarfield:SpecialRelativity77
7.3Inclusionofaone-formfield:GaugeTheory80
7.4Best-matchinggaugetransformations82
7.5Yang–MillsTheory83
7.6Furthergeneralizations84 Furtherreading85
7.7Theproblemofmany-fingeredtime85 8York’sSolutiontotheInitial-ValueProblem 88
8.1Lichnerowicz’spartialsolution89
8.2York’sgeneralsolution92 Furtherreading94
12.1LinkingTheory’sequationofmotion114
12.2ThedegreesoffreedomofShapeDynamics117
12.3ThesolutiontotheproblemoftimeinSD117
12.4Couplingtomatteranduniqueness122
12.5Experiencedspacetime124
12.6‘Symmetrydoubling’:BRSTformulationofSD126
12.7‘Conformalgeometrodynamicsregained’128
13.1Homogeneoussolutions:BianchiIX134
13.2Continuationthroughthesingularity151
13.3Sphericallysymmetricsolutions169
13.4AsymptoticallyflatShapeDynamics209
PartVAppendices
AppendixAArnowitt–Deser–MisnerGravity 223
A.1TheArnowitt–Deser–Misner(ADM)formalism223 Furtherreading225
A.2TheWheeler–DeWittequation226
A.3TheBaierlein–Sharp–Wheeleraction226
A.4AsymptoticallyflatADM227
AppendixBOtherAppendices 230
B.1Thecaseforclosedspacelikehypersurfaces230
B.2Free-end-pointvariation231
B.3Liederivative232
B.4TTdecompositionoftensors233
B.5PointsourcesinADMgravity238
B.6Thepolesofaspherically-symmetricuniverse239 References 243
ListofFigures
1WhatisaConformalTransformation5
2Theconceptofagaugeconstraint6
3Theconceptofagauge-fixing7
4TheiconicdiagramofShapeDynamics9
5SchemeoftheconstraintsofSDandGR10
6Newton’sbucketexperiment19
7Hoffman’sexperiment26
8Bestmatchingillustratedwithpicturesofstars36
9Horizontalstacking36
12StageIofbestmatching44 13StageIIofbestmatching45 14Verticalstacking46
15Faraday’slinesofforce68
16TheissueofMany-FingeredTime86
17GraphicalidentificationoftherootoftheLYpolynomial93 182dquantumharmonicoscillator119
19TheDiracalgebroid129
20ShapepotentialofBianchiIX138
21On-shellsurfaceinKasnerspace139
22IntersectionbetweenKasnersphereandplane140
23DivisionofBianchiIXshapespace142
24Taubtransition144
25ExponentsintheBianchiIXshapepotential147
26BianchiIXshapepotentialvs.shapekineticenergy148
27BianchiIXshapepotentialvs.shapekineticenergy, < 1149
28SphericalBianchiIXshapespace151
29BianchiIsolutionsonBianchiIXshapespace153
xii ListofFigures
30Quiescenceboundindimensionlessvariables—1166
31Quiescenceboundindimensionlessvariables—2167
32Domainofarealradius174
33Thesurface P (z)=0for m > 0, λ< 0179
34Thesurface P (z)=0for m > 0, λ ≥ 0180
35Thesurface P (z)=0for m < 0, λ> 0181
36Thesurface P (z)=0for m =0, λ> 0182
37Singleshelluniversediagram189
38Solutionsoftheconstraintsforthesingle-shelluniverse190
39 σ (R)and V vs. p insingle-shelluniverse194
40Twin-shelluniversediagram197
41On-shellsurfacesfor λ< 0, mB > 0200
42On-shellsurfacesfor λ> 0, mB > 0201
43On-shellsurfacesfor λ> 0, mB < 0202
44On-shellsurfacesfor λ> 0, mB =0203
45On-shellsurfacefor λ< 0, mB < 0204
46On-shellsurfacefor λ< 0, mB =0205
47Determiningthecodomainof σ inthetwin-shelluniverse206
48Determiningthecodomainof σ inthetwin-shelluniverse208
49Theneighboursofthesolarsystem210
50AFsingle-shellon-shellsurfaces217
51RegionsofKruskal’sextensionoccupiedbythe‘wormhole’solution218
52LapseandShift224
Acknowledgements
ThisbookwouldnothaveexistedwithouttheinitialencouragementandcontinuedsupportofJulianBarbour,towhomIamimmenselygrateful.IsendabigthanktoHenrique Gomesforhisextensivehelpatvariousstagesofcompositionofthisbook.Iamindebted alsotomymentorLeeSmolinandthefantasticenvironmentofPerimeterInstitutefor theirsupporttoayoungresearcherthattooktheriskofwritingabookduringapostdoctoralappointment.ThanksalsotoTimKoslowski,DavidSloan,DavidWiltshireand MatteoLostagliofortheirvaluablecommentsanddiscussions.Specialthankstomy formerstudentsAndreaNapoletano,MonicaRinconRamirezandMykolaMurskyjfor beingmyguineapigsandproofreadingearlydraftsofwhatbecamethefirstmonograph onShapeDynamics.
Introduction
ShapeDynamics(SD)isanewtheoryofgravitythatisbasedonfewerandmorefundamentalfirstprinciplesthanGeneralRelativity(GR).ThemostimportantfeatureofSD isthereplacementofGR’srelativityofsimultaneitywithamoretractablegaugesymmetry,namely,invarianceunderspatialconformaltransformations.Thisbookcontains bothaquickintroductionforreaderscuriousaboutSDandadetailedwalkthroughof thehistoricalandconceptualmotivationsforthetheory,itslogicaldevelopmentfrom firstprinciples,andanin-depthdescriptionofitspresentstatus.Thebookissufficiently self-containedforgraduatestudentsandadvancedundergraduatestudentswithsome basicbackgroundinGRandLagrangian/Hamiltonianmechanics.Itisintendedboth asareferencetextforstudentsapproachingthesubjectandasareviewingtoolfor researchersinterestedinthetheory.
1.1OutlineofChapters
ThemainpartoftheintroductionisSection1.3ofChapter1(ShapeDynamicsinanutshell),whereIattempttoofferano-nonsensequickentrytothebasicideasofSD.This servesdualpurposes:ononehand,studentsinterestedinSDwillhaveabriefoverview ofwhatthetheoryisaboutandwhatwehopetoachievewithit;ontheother,researchers curiousaboutSDwillfindashortdescriptionofthetheoryinSec.1.3thatishopefully enoughforthemtodecidewhethertheseideasareworthexaminingindepth.TheminimumofnotionsneededtounderstandthecoreideasofSDareoutlinedwiththeaimof makingthesectionasself-containedaspossible.Alltheconceptsareexplainedindetail intherestofthisbook,whiletakingan‘historico-pedagogical’perspectiveandintroducingthemattheappropriatepointsinthestory.Sec.1.3alsoincludesaquickoutline ofbasicconceptsneededtounderstandSD,whicharenotpartofnormalundergraduatecurricula(likeconstrainedHamiltoniansystemsandgaugetheories).Howeverthe sectionislimitedtojustafewpagestobereadquicklybyexperts,andthementioned outlineisbynomeanssufficienttounderstandthoseconceptsproperly.Itspurposeis togiveundergraduatereadersatasteofthebackgroundknowledgethatisnecessaryto understandSDandgettheoveralldrift.Everythingisexhaustivelyexplainedinthebody ofthetext.
PartIshowswhereSDcomesfrom:weconsideritasthemostadvancedstageof the relationalprogramme, whichseekstoeliminateallabsolutestructuresfromphysics— absolutestructuresmeaninganythingthatdeterminesphysicalphenomenabutisnot determinedbythem.ThechiefexampleisNewton’sabsolutespaceandtime(or,in modernterms,inertialframesofreference).ThebattlefieldofNewton’sabsoluteshas seengiantsofsciencefightingtheabsolute-vs-relativedebate:Galileo,Descartes,Newtonhimself,Leibniz,Mach,PoincaréandEinstein.Anotherexampleisscale,orsize: SDtriestoeliminatepreciselythisabsolutestructurefromphysics.Onecouldimagine pushingthisprogrammefurtherintothefuture:whatdeterminesthetopologyofspace? Arethevaluesofthephysicalconstantsaresultofimmutablelawsorofadynamical evolution?
InChapter2IwillexplainindetailthefundamentalproblemofNewtoniandynamics: everythingisbasedonthelawofinertia,whichinturnreliesontheconceptsofrestand uniformmotion,buttheseconceptsarenotdefinedbyNewton.Chapter3makesitclear whattheproblemwithNewton’sconstructionis.Statinginamathematicallypreciseway thedefectofNewton’stheorywasanincrediblydifficultproblem;HenriPoincarésolved itaftermorethantwocenturies.However,evenPoincaré’sformulation(whichwecall the‘Mach–PoincaréPrinciple’)wasnotrecognizedforitstrueworthuntiltheworkof BarbourandBertottiinthe1960s.
PartIIdealswithrelationaldynamicsinthesimplerframeworkofsystemsofpoint particles.RelationaldynamicsisareformulationofdynamicsthatsatisfiestheMach–PoincaréPrinciple,asformulatedbyBarbourandBertotti.Itusesspecifictechniques thatwereinventedforpurpose,inparticularthatof‘bestmatching’.Thesetechniques turnedouttobeequivalenttothemodernformulation,duetoDirac,ofgaugetheories asconstrainedHamiltoniansystems.InChapter4bestmatchingisintroducedatan intuitivelevel,whileChapter5detailsitusingthelanguageofPrincipalFibreBundles, whichareintroducedtothereader.Chapter6describestheHamiltonianformulationof bestmatchingandlinksittomoderngaugetheory.ThetechniquesdevelopedbyDirac forHamiltonianconstrainedsystemsareneededinthissectionandarethereforebriefly explained.
PartIIIdealswiththemoreadvancedframeworkoffieldtheory.Chapter7details(in modernlanguage)aseriesofresultsduetoBarbour,O’Murchadha,Foster,Anderson andKelleher.Thesearestrikingresults:theyshowthattheprinciplesofrelationalfield theoryalonearesufficienttoderiveGR,thegeneralandspecialrelativityprinciples, theuniversalityofthelightcone,Maxwell’selectromagnetism,thegaugeprincipleand Yang–Millstheory.Chapter8containsmorebackgroundmaterial:itpresentsYork’s methodforthesolutionoftheinitial-valueprobleminGR.Thisprovidesanimportant inputfortheformulationofSD.Chapter9dealswithworkIhavedonetogetherwith E.Anderson,andfinallymakestheconnectionfromrelationalfieldtheorytoSD.The latterisshowntoarisefromtheprinciplesofrelationalfieldtheoryandtheMach–PoincaréPrinciple.Thefinalchapterofthispart,Chapter10,describestheattemptto buildatheorythatincorporatestheprinciplesofrelationalfieldtheoryandassumeslocal scaleinvariance(alsocalled conformal,or Weyl invariance)fromthebeginning(whilein myderivationofSDlocalscaleinvarianceemergesasaconsistencyrequirementinthe
analysisoftheconstraintsofthetheory).Suchatheorywouldimplementbothalocal notionofduration(andthereforeinvarianceunderlocaltimereparametrization)and conformalinvariance.Interestingly,thistheoryprovestobeinconsistent,leavinguswith SDastheonlyviablecandidateforatheoryofevolvingconformalgeometry.AsSDonly hasaglobalreparametrizationinvarianceandimpliesapreferrednotionofsimultaneity, wehavetoconcludethatrefoliationinvarianceandconformalinvariancearedualand alternativetoeachother:theycannotbekeptsimultaneously.
InPartIVSDisfinallyformulatedinitscurrentform.Ibeginwithabriefaccountof thewaytheideasatthebasisofSDweredevelopedinChapter11,theninChapter12 IproceedtoderivetheequationsofSDfromthepointIleftthetheoryinChapter9. InSec.12.1IshalldiscussthephysicaldegreesoffreedomofSD,whicharetheconformallyinvariantpropertiesofathree-dimensional(3D)manifold,andtheirconjugate momenta.InSec.12.2IexplainhowSDrepresentsasimplesolutiontotheproblem oftimeinquantumgravity,andhowonereconstructsthefamiliarfour-dimensional spacetimedescriptionofGRfromasolutionofSD.Sec.12.3dealswiththecoupling ofSDtomatter,whichwasanalizedbyGomesandKoslowski,whoappliedtoSD previousresultsontheconformalmethodbyIsenberg,Nester,ÓMurchadhaandYork. InSec.12.4IsummarizeKoslowski’sworkontheemergenceofthespacetimedescription.Thisworkshowshowthefour-dimensional,CMC-foliatedlineelementthatone deducesfromasolutionofSDisthespacetimethatmatterdegreesoffreedomexperienceinthelimitinwhichbackreactioncanbeignored.InSec.12.5Ibrieflydescribethe resultsbyKoslowskiandGomesontheBRSTformulationofSD,andfinallyinSection12.6IsummarizeGomes’workonaconstructionprincipleforSDalongthelineof rigiditytheoremslikethoseofHojman-Kucha ˇ r-Teitelboim.
Chapter13dealswiththeparticularsolutionsofSDthathavebeenstudiedsofar.In Sec.13.1Istudyhomogeneoussolutionsindetail,withsphericaltopology(theso-called ‘BianchiIX’universes),andshowwhatisperhapsthemoststrikingconsequenceofSD: itssolutionscanbecontinueduniquelythroughtheBigBangsingularity.InSec.13.3 Ianalyzespherically-symmetricsolutions,whicharethebasisfordiscussinggravitational collapseandblackholes,andpresentanotherstrikingresult:theArnowitt–Deser–Misner (ADM)-in-constant-mean-extrinsiccurvature(CMC)-foliationdescriptionofaclosed universewithcollapsingmatterfailsatsomepointduringthecollapse(presumablywhen thesystemgeneratesaneventhorizon),whiletheSDdescriptionseemswell-definedat thatpointandafterwards.InSec.13.4Idiscussthesenseinwhichonecantalkabout asymptoticflatnessinSD(whichisfundamentallyatheoryofcompactuniverses),and Icriticallyevaluatepastresultsobtainedintheasymptoticallyflatcase.
Thefinalpartofthebookcontainstheappendices,whicharedividedintoafirst, majorAppendixA,withabriefaccountoftheHamiltonianformulationofGRdueto Arnowitt,DeserandMisner.ThisisthemaintoolofCanonicalGeneralRelativityandis thetheorywehavetocompareclassicalSDto.InthisAppendix,IgiveastandardderivationofthistheorystartingfromGRandtheEinstein–Hilbertaction.Thesametheory canbededucedfromtheaxiomsofrelationalfieldtheorywithoutpresupposingspacetimeandwithoutstartingfromtheEinstein–Hilbertaction,aswasdoneinChapter7. Thisderivationassumeslessandshouldbeconsideredmorefundamentalthanthatof
Arnowitt,DeserandMisner.However,Ifeltthatthejuniorreadersshouldbeawareof thestandardderivation.Finally,AppendixBcontainsaseriesofresultsandderivations thatareusefulandreferencedtothroughoutthetext,butwhicharemovedtotheendof thebookforthesakeofclarityofexposition.
1.2Notation
InthetextweuseanotationaccordingtowhichtheGreekindices μ, ν , gofrom0 to3,whilethelowercaseLatinindicesfromthemiddleofthealphabet i , j , k, , m ... are spatialandgofrom1to3.WeassumeaLorenziansignature(–,+,+,+).Thelowercase Latinindicesfromthebeginningofthealphabet a, b, c,refertotheparticlenumber andgofrom1to N .Three-dimensionalvectorswillbeindicatedwithLatinorGreek boldletters, q, p, θ , ω , ,whilethree-dimensionalmatriceswillbeuppercaseRomanor Greek , ,U,I, .ThespatialLaplacian gij ∇ i ∇ j willbeindicatedwiththesymbol , whileforthed’Alembertian gμν ∇ μ ∇ ν Iwillusethesymbol .The(spatial)conformal Laplacian8 – R willbeindicatedwiththesymbol .
1.3ShapeDynamicsinanutshell
SDisafieldtheorythatdescribesgravityinadifferentwaytoGR.Howeverthedifferencesbetweenthetwotheoriesaresubtle:inmostsituationstheyareindistinguishable.
1.3.1SDisagaugetheoryofspatialconformal(Weyl) symmetry
SDandGRaretwodifferentgaugetheoriesdefinedinthesamephasespace,bothof whichadmitaparticulargauge-fixinginwhichtheycoincide.Thisdoesnotguarantee completeequivalencebetweenthetwotheories:agauge-fixingis,ingeneral,notcompatiblewitheverysolutionofatheory,inparticularduetoglobalissues.Theequivalence betweenSDandGRthereforefailsinsomesituations.
WhatdistinguishesSDfromGRasafundamentaltheoryofgravityisitsdifferent ontology
Firstly,SDdoeswithoutspacetime:theexistenceofapseudo-RiemannianfourdimensionalmanifoldwithaLorentziansignatureisnotassumedamongtheaxioms ofthetheory.Instead,theprimaryentitiesinSDarethree-dimensionalgeometriesthat arefittedtogetherbyrelationalprinciplesintoa‘stack’whosestructuralpropertiescan beidentifiedinsome,butnotall,caseswiththoseofafour-dimensionalspacetime whichsatisfiesEinstein’sfieldequations.TheclosestagreementwithGRoccursifthe three-geometriesarespatiallyclosedwhentherelationalprinciplesofSDarefullyimplemented.However,thereisalsointerestinthepartialimplementationofSD’srelational principlesincaseswherethethree-geometriesareasymptoticallyflat.
Secondly,thespatialgeometrieswhichmaketheconfigurationspaceofSDarenot Riemannian.Theyare conformalgeometries, definedasequivalenceclassesofmetrics
Figure1 Conformaltransformationofa two-dimensionalsphere.Thetriangledefined bytheintersectionofthethreecurvesistransformedinsuchawaythatitsareaand thelengthsofitsthreeedgesarechanged, butthethreeinternalangles(inred)areleft invariant.
underposition-dependentconformaltransformations(sometimescalled‘Weyltrasformations’;thefourthpowerof φ ischosentosimplifythetransformationlawofthescalar curvature R):
Conformaltransformationschangelengthsandpreserveonlyangles(seeFig.1).Therefore,aconformalgeometrypresupposeslessthanaRiemanniangeometry,forwhich lengthsdeterminedbythemetricareconsideredphysical.WhatisphysicalinSDis the conformalstructure,whichisthe angle-determining partofthemetric.Lengthscan bechangedarbitrarilyandlocallybyaconformaltransformation,whichisagauge transformationforSD.
So,SDassumeslessstructurethanGR,butitisinonesenseaminimalisticlifting ofassumptions:thenextthinginorderofsimplicityafterRiemanniangeometryisconformalgeometry.Someotherapproachestoquantumgravityaredecidedlymoreradical asregardstotheamountofstructuretheyassume:eithermuchmore(e.g. stringtheory) ormuchless(e.g. causalsets).
SDisbasedonfewerandmorebasickinematicalfirstprinciplesthanGR:
Spatialrelationalism: thepositionsandsizesofobjectsaredefinedrelativetoeachother. Thisdetermineswhatthephysicalconfigurationspaceis(seeSec.3.2).Infieldtheorythisprincipletranslatesintoconformalanddiffeomorphisminvariance,andthe requirementofaspatiallyclosedmanifold.
Temporalrelationalism: theflowoftimeissolelyduetophysicalchanges(seeSec.5.2).
The Mach–PoincaréPrinciple: apointandadirection(ortangentvector,initsweak form)inthephysicalconfigurationspacearesufficienttouniquelyspecifythesolution (seeSec.3.4).
Thereisnoneedforgeneralcovariance,therelativityprinciple,theexistenceofspacetimeortheexistenceofmeasuringrodsandclocks.Theseconceptsemergefromthe solutionsofSDascharacteristicbehavioursorusefulapproximations.InthissenseSD ismorefundamentalthanGRbecauseitachievesthesamewithless.SeePartIIIforthe fullconstructionofSDstartingfromitsthreefirstprinciples.
AcommonmistakeistoregardSDjustasagauge-fixingofGR.Itiseasytoseethat thisisnotthecase: therearesolutionsofSDthatarenotsolutionsofGR,andvice-versa.
AsatisfactoryunderstandingoftheGRsolutionswhichSDexcludesandoftheSD solutionswhichGRexcludesisstilllacking. LetusnowhaveabrieflookatwhatexactlySDlookslike.
1.3.2GaugetheoriesareconstrainedHamiltoniansystems
SDismorenaturallyformulatedasagaugetheoryintheHamiltonianlanguage.Gauge theoriesaretheorieswithredundancies:oneusesmoredegreesoffreedomthannecessaryinordertoattainasimplerand local description.IntheHamiltonianpicture,this translatesinto nonholonomicconstraints: functionsofthecanonicalvariables χ = χ (p, q) (withsomedependencyonthemomentaeverywhereinphasespace)whichneedto vanishonthesolutionsofthetheory χ (p, q) ≈ 0.1 Asingleconstraintidentifiesa codimension-1hypersurfaceinphasespace,the constraintsurface,onwhichthesolutionsofthetheoryarelocalized.Forexample,ifagaugeconstraintcanbewrittenas χ = p1 ,where p1 isoneofthecanonicalmomenta(asisalwayspossible,thankstoDarboux’stheorem[1])theconstraintsurfaceisthehyperplane p1 ≈ 0showninFig.2. However, p1 alsoplaystheroleofthe generator ofgaugetransformations,whichhappen tobethetranslationsinthe q1 direction:throughthePoissonbracketitdefinesavector fieldonphasespace{p1 , }= ∂ ∂ q1 ,whichisparalleltothe q1 axis(seeFig.2).Thisvector fieldgeneratesinfinitesimaltransformationsonphasespace(translationsinthe q1 direction),anditsintegralcurvesarethe gaugeorbits ofthetransformations.Allthepoints onthesecurvesaregauge-equivalent(theyarerelatedbygaugetransformations:they havedifferentrepresentationsbutthesamephysicalcontent).Moreover,thevectorfield ∂ ∂ q1 isparalleltotheconstraintsurface p1 ≈ 0byconstruction,anditsintegralcurves
Figure2 Theconstraintsurfaceofagaugeconstraint χ = p1 isrepresented inphasespace,whereIputp1 andq1 ontwoaxes,andalltheotherphasespacevariables (qn , pn ),n =2,3 arerepresentedcollectivelyonthethird axis.Ontheright,arethevectorfieldgeneratedbyp1 throughPoisson-brackets, {p1 , }= ∂ ∂ q1 ,whichpointstowardstheq1 direction.Thevectorfieldisparallel totheconstraintsurface,anditsintegralcurves(thegaugeorbits)lieonit.
1 With‘≈’wemeanthattheequationholdsonthesolutionsoftheconstraintequations,followingDirac’s notation.
Figure3 Theconceptofgauge-fixingsurface:the variableq1 isunphysicalanditsvaluecanbetaken arbitrarily,thereforewemightchooseaconventional valueforq1 ,tobedeterminedbythevalueofallthe otherphase-spacevariables,q1 = q1 (q2 , p2 , ...).One wayofobtainingthisistointersecttheconstraintsurfacep1 ≈ 0 withanothersurface, ξ (q, p) ≈ 0,such thatitisneverparalleltop1 ≈ 0 or,attheintersection,‘runsalong’thegaugeorbits(representedby dashedlinesontheconstraintsurface).
lieonit.Thephysicalmeaningofagaugeconstraint χ = p1 isthatthe q1 coordinateis unphysical,likethenon-gauge-invariantpartoftheelectromagneticpotentials A and ϕ , orlikethecoordinatesofthecentre-of-massofthewholeuniverse.
Sincethe q1 coordinateisnotphysical,wecanassignitanyvaluealongthesolutionwithoutchanginganythingphysical.Itisoftenuseful(andnecessaryinquantum mechanics)tofixthevalueof q1 bysomeconvention.Thestandardwayofdoingit isbychoosinga gauge-fixing: wespecifythevalueof q1 asafunctionoftheother variables, q1 = q1 (q2 , p2 , ...).Thiscorrespondstointersectingtheconstraintsurface p1 ≈ 0withanothersurface ξ (p, q) ≈ 0thatspecifiesanintersectionsubmanifold {p, qs.t. χ ≈ 0, ξ ≈ 0}(seeFig.3).Thegauge-fixingshouldspecifythegaugewithout ambiguity:ithastoformaproperintersectionwith p1 ≈ 0,andthereforecannotbeparalleltoitwheretheyintersect.Moreover,atitsintersectionwiththeconstraintsurface χ ≈ 0,thegauge-fixingsurface ξ ≈ 0cannot‘runalong’(betangentto)anyofthegauge orbits:inthatcasetherewouldbemorethanonevalueof q1 thatwouldcorrespondto thesamevalueof q2 , p2 , ... .Thesetwoconditionsdefineagoodgauge-fixingsurface. FordetailsonconstrainedHamiltoniansystemsandgaugetheories,seeSec.6.2.
1.3.3GRasaconstrainedHamiltoniantheory
Arnowitt,DeserandMisner(ADM)formulatedGRintheHamiltonianlanguage.They foliatedspacetimeintoastackofspatialhypersurfacesandsplitthe4-metric gμν intoa spatialpart gij andfouradditionalcomponents g0i and g00 .Thespatialmetriccomponents gij representthecanonicalvariables,andtheirmomenta pij arerelatedtothe extrinsiccurvatureofthespatialhypersurfacewithrespecttoitsembeddinginspacetime.The g0i and g00 components(orbettersomecombinationsthereof)entertheaction withouttimederivatives,andarethereforeLagrangemultipliers.Theyareassociated withfour local constraints(meaningoneconstraintperspatialpoint).Theseconstraints aretheso-called‘superhamiltonian’ H and‘supermomentum’ Hi constraints.HereIwill
callthemthe‘Hamiltonian’andthe‘diffeomorphism’constraints.Thediffeomorphism constraintadmitsasimplegeometricalinterpretation:itsvectorflowsendsconfiguration variablesintothemselves(onesaysitgenerates‘pointtransformations’ Hi : gij → gij ), andthereisnodoubtaboutitsbeingagaugeconstraint.
FortheHamiltonianconstraintthingsarenotthatsimple:itisquadraticinthemomenta,anditsvectorflowdoesnotadmittheinterpretationofapointtransformation(it sends gij sintoboth gij sand pij s).Thereisalargeliteratureontheproblemofinterpreting H.Ifitisinterpretedasagaugeconstraint,onewouldendupwiththeparadoxical conclusionthatthedynamicalevolutionofGRisjustagaugetransformation.Thereare alsohugeproblemswiththedefinitionofwhatpeoplecall‘Diracobservables’:quantities whosePoissonbracketswithallthefirst-classconstraintsvanishontheconstraintsurface(meaningtheymustbeinvariantundertheassociatedgaugetransformations).In GR’scase,thatdefinitionwouldleadtoobservableswhichareconstantsofmotionand donotevolve(‘perennials’,asKucha ˇ rcalledthem[2]).Kucha ˇ radvocatedadifferent notionofobservables,namelyanideathattheyareonlyrequiredtobeinvariantunderdiffeomorphisms.Thesewouldevolve,buttherearetoomany:theywoulddepend on three polarizationsofgravitationalwaves,whileitiswidelyagreedthatgravitational waveshave two physicalpolarizations.
Thefactthat H isquadraticinthemomentaalsocausesmajorproblemsinitsquantization.Itleadstothenotorious‘Wheeler–DeWittequation’,forwhichtherearemany unsolveddifficulties,aboveallits‘timelsss’nature,butalsoorderingambiguitiesand coincidencelimits.TheADMformulationofGRisdetailedinAppendixA,andthe problemswiththistheorywhichleadtotheintroductionofSDareexplainedattheend ofChapter7andinChapter8.
AsillustratedinFig.4,SDisbasedontheidentificationofthepartof H whichisnot associatedwithagaugeredundancyandtakesitasthegeneratorofthedynamics.The restof H isinterpretedasagauge-fixingforanotherconstraint, C .Thisconstraintislinearinmomentaandgeneratesgenuinegaugetransformations,constrainingthephysical degreesoffreedomtobetwoperpoint.
1.3.4Noteveryconstraintcorrespondstogaugeredundancy
Thatthisisthecaseisprettyobvious:thinkaboutaparticleconstrainedonasphere oraplane, i.e. aholonomicconstraint.Suchaconstraintobviouslyhasnothingtodo withgaugeredundancy.However,thereareconstraintswhichDirac[3,4]arguedcan alwaysberelatedtogaugesymmetries:theyaretheso-called‘first-class’constraints. Beingfirst-classmeansthattheycloseanalgebraunderPoissonbracketswitheachother andwiththeHamiltonianofthesystem.Ifthatisthecase,Diracshowedthatonehas freelyspecifiablevariablesinthesystem,oneforeachfirst-classconstraint,andchanging thesevariablesdoesnotchangethesolutionsofthetheory.ButBarbourandFoster [5]havepointedoutthatthepremisesunderwhichDiracobtainedhisresultdonot holdintheimportantcaseinwhichthecanonicalHamiltonianvanishes.Inthatcase theHamiltonianisjustalinearcombinationofconstraints,butthatdoesnotprevent thetheoryfromhavingsensiblesolutions.Thesolutionswillbecurvesinphasespace,
Figure4 AschematicrepresentationofthephasespaceofGR.Init,twoconstraintscoexist,whichare goodgauge-fixingsforeachotherandarebothfirst-classwithrespecttothediffeomorphismconstraint. OneistheHamiltonianconstraintandtheotheristheconformal(Weyl)constraint.TheHamiltonian constraintiscompletelygauge-fixedbytheconformalconstraintexceptforasingleresidualglobalconstraint.ItPoissoncommuteswiththeconformalconstraintandgeneratesavectorflowontheHamiltonian constraintsurface(representedinthefigure),whichisparalleltotheconformalconstraintsurface.This vectorflowgeneratesthetimeevolutionofthesystemintheintersectionbetweenthetwosurfaces.Any solutioncanthenberepresentedinanarbitraryconformalgaugebyliftingitfromtheintersectiontoan arbitrarycurveontheconformalconstraintsurface.Allsuchliftedcurvesaregauge-equivalentsolutions ofaconformalgaugetheorywithaconformallyinvariantHamiltonian.
andwillstillpossessonefreelyspecifiablevariableforeachconstraint—however,one oftheseredundancieswillnotchangethecurveinphasespace:itwilljustchangeits parametrization.Thereforeoneofthefirst-classconstraintsofthesystemwillnotbe relatedtoanygaugeredundancy:thereisnotanassociatedunphysical‘q1 ’directionasin theexampleabove.ThiscounterexampletoDirac’sstatementisveryimportantbecause itisrealizedinthetheorywecareaboutthemost:GeneralRelativity.Oneofthe(many) constraintsofGR shouldnot beassociatedwithgaugeredundancy.TheBarbour–Foster argumentisexplainedattheendofSec.6.2.
1.3.5SDreinterprets H asagauge-fixingofconformal symmetry
SDidentifiesanotherconstraintsurface C ≈ 0inthephasespaceofGR,whichisa goodgauge-fixingfortheHamiltonianconstraint.Thisgauge-fixingthoughhappensto bealsoagaugesymmetrygenerator.Itgeneratesconformaltransformations(1)ofthe spatialmetric,withtheadditionalconditionthatthesetransformationsmustpreserve thetotalvolumeofspace V = d3 x√g .Theconstraint C ,inaddition,happenstoclose afirst-classsystemwiththediffeomorphismconstraint Hi ,thereforeitisamatterof opinionwhetheritis C thatgauge-fixesthesystem(H, Hi )oritis H whichgauge-fixes
Figure5 SchemeoftheconstraintsofGRandofSD.GR’s Hamiltonianconstraint H hasbeensplitintotheglobalpart Hglobal whichisfirstclasswithrespecttotheconformalconstraint C andthepartthatispurelysecondclass, H(x)– Hglobal .This secondclasssystemadmitstwofirstclasssubalgebras:thelowerlefttriangle,whichconstitutestheconstraintalgebraofGR;and thelower-righttriangle,makingtheconstraintalgebraofSD.
(C , Hi ).Iftherealphysicsonlyliesintheintersectionbetween C ≈ 0and H ≈ 0(which isthebigassumptionatthebasisofSD,anddoesnotholdifspacetimeisassumedasan axiom),thenthelogiccanbereversedandtheHamiltonianconstraintcanbeinterpreted asaspecialgauge-fixingfortheconformalconstraint,seeFig.5.Thengravitycanbe reinterpretedasagaugetheoryofconformaltransformations,whichadmitsagaugefixingthatissingledoutbysomespecialproperties.Theseproperties,asIwillshow, havetodowiththefactthatitgivesa‘natural’notionofscaleandpropertime,which agree(mostofthetime)withthosemeasuredbyphysicalrodsandclocks.
1.3.6SD’sHamiltonianconstraint
H and C donotentirelygauge-fixeachother:thereisasinglelinearcombination of H(x)whichisfirstclasswithrespectto C .Thislinearcombination, Hglobal = d 3 xNCMC (x) H(x),isasingleglobalconstraintwhosevectorflowisparalleltoboth the C ≈ 0andthe H ≈ 0surfacesontheirintersection.Thisvectorflowgeneratesan evolutionintheintersection:ithastobeinterpretedasthegeneratoroftimeevolution. Itisthepartofourconstraintswhichisnotassociatedwithagaugeredundancyandis insteadassociatedwithtimereparametrizationsofthesolutionsofthetheory.
1.3.7The‘LinkingTheory’
SDpaysapriceforitsconceptualsimplicity:thegeneratoroftheevolution Hglobal containsthesolutiontoadifferentialequation, NCMC ,andthereforeisanonlocalexpression.Butonecanrecoveralocaltreatmentbyenlargingthephasespace.SDcan infactbeconsideredasoneofthepossiblegauge-fixingsofafirst-classtheorywhich islocal(itsconstraintsarelocal)andlivesinalargerphasespacethanthatofGR.
ThisphasespaceisobtainedfromthatofGRbyadjoiningascalarfield φ andits conjugatemomentum π .Thelargertheory(calledthe‘LinkingTheory’)isdefined bytheconstraintsofGR, H and Hi ,butexpressedintermsof(volume-preserving) conformally-transformedmetrics e4 ˆ φ gij andmomenta e–4 ˆ φ pij –1 3 (1– e6 ˆ φ )√gg ij p/V , where ˆ φ = φ –1 6 log( d3 x√g exp(6φ )/V )and V = d3 x√g .Inaddition,onehasamodifiedconformalconstraintwhichincludesatermthattransforms φ .Thenewconstraint is Q =4 π – C andgeneratessimultaneoustranslationsof φ and(volume-preserving) conformaltransformationsof π ,sothatthecombination e4 ˆ φ gij ,isleftinvariant.The constraint Q isnowfirstclasswithrespectto H and Hi .Bycompletelygauge-fixing Q, forexamplewiththecondition φ ≈ 0,oneobtainsGR.Ontheotherhand,onecan useadifferentgauge-fixing,namely π ≈ 0,whichisfirstclasswithrespectto Q,but gauge-fixes H almostentirely,leavingonlytheglobalpart Hglobal untouched.
OnecanthenworkwiththeLinkingTheory,wherealltheequationsofmotionand constraintsarelocal(apartfromtheirdependenceonthetotalvolume),andworkoutthe solutionsinthisframework.Aslongasthesolutioniscompatiblewiththegauge-fixing π ≈ 0,itisalegitimateSDsolution.
AllthedetailsoftheSDconstructioncanbefoundinChapter9andinPartIV.
1.3.8ThepresentstatusofSD
ThegreatesthopeofSD’sistoprovideanewapproachtoquantumgravitybasedona sumoverthehistoriesofconformal3-geometries.Suchanapproachissofarunexplored: 3Dconformalsymmetryplaysaroleinsomequantumgravityproposals, e.g. Ho ˇ rava–LifshitzgravityorCausalDynamicalTriangulations,inwhichthereisevidencethatthe theoryadmitsconformally-invariantfixedpoints.However,suchsymmetriesareonly asymptoticanddonotcharacterizethephysicallyrelevantregimesofthesetheories.In particular,itseemsthatacommonfeatureofquantumgravitymodelsisthegeneration ofadditionaldynamicaldegreesoffreedomatthequantumlevel, e.g. thescalarmode inHo ˇ rava–Lifshitzgravity.Aquantumgravitytheorycompatiblewiththeprinciples ofSDshould,presumably,keeponlytwodynamicaldegreesoffreedomalltheway throughitsrenormalizationgrouporbit.Suchaproposalseemstocontradictthefact thatquantummechanicsintroducesascaleinphysicsthrough h,andthereforebreaks conformalinvariance.This,however,shouldnotbetakenasasacredfactofnature:the factthat h isdimensionfulcanbeaconsequenceofthefactthatwemeasureitseffectsin asubsystemoftheuniverse,anditsscalemightadmitarelationalexpressionintermsof thestateofthematterintheuniverse.Indeed,in[6]weformulatedatoymodelofscaleinvariantnon-relativisticquantummechanics,inwhichthefundamentalontologyisthat ofawavefunctionevolvingonshapespace.Theroleof h isplayedbyadimensionless quantitywhichisrelatedtothe‘spreading’ofthewavefunctiononshapespace.Such aquantityisintimatelyrelatedtotheparticleequivalentofYorktime,thedilatational momentum D (inappropriateunits h and D havethesamedimensions).The h which weusetodescribephenomenaweobserveinthelaboratoryemerges,insuchamodel, whenweconcentrateonsubsystemsoftheuniverseandmodelthemsemiclassicallyas
