Testing a theory of gravity in celestial mechanics

arXiv:gr-qc/0305078v26Jun2003Testingatheoryofgravityincelestialmechanics:anewmethodanditsfirstresultsforascalartheoryM.Arminjon1Laboratoire“Sols,Solides,Structures”[Unit´eMixtedeRechercheoftheCNRS],BP53,F-38041Grenoblecedex9,Franceemail:[e-mail]st-Newtonianapproximation(PNA)forweakgravitationalfieldsispresentedtogetherwithitsapplicationtotestanalternative,scalartheoryofgravitation.Thenewmethodconsistsindefiningaone-parameterfamilyofsystems,byapplyingaNewtoniansimilaritytransformationtotheinitialdatathatdefinesthesystemofinterest.Thismethodisrigorous.ItsdifferencewiththestandardPNAisemphasized.Inparticular,thenewmethodpredictsthattheinternalstructureofthebodiesdoeshaveaninfluenceonthemotionofthemasscenters.Thetranslationalequationsofmotionobtainedwiththismethodinthescalartheoryareadjustedinthesolarsystem,andcomparedwithanephemerisbasedonthestandardPNAofGR.1Thisisarevisedandexpandedversionofthetextofalecturegivenatthe8thConf.“PhysicalInterpretationsofRelativityTheory”(London,September2002),organizedbytheBritishSocietyforthePhilosophyofSciences.TheinitialversionwillappearintheProceedingsofthatConference(M.C.Duffy,ed.).11IntroductionOnefirstexpectsfromatheoryofgravitythatitshouldprovideanac-curatecelestialmechanics.Inotherwords,thetheoryshouldtellushowmassivecelestialbodiespreciselymovewithrespecttoeachotherundertheeffectofthegravitationalfieldproducedbythemall.Thus,Einstein’sgen-eralrelativity(GR)wonitsfirstadvantageovertheoldertheoryofNewtonwhenitgaveanexplanationtoMercury’sresidualadvanceinperihelion.In1972,Weinbergstatedaboutthisexplanation([1],p.198):“Thisisbyfarthemostimportantexperimentalverificationofgeneralrelativity.”Itishenceextremelyimportantforatheoryofgravitation,notonlythatitproducesaccurateephemerides,butevenmorethatoneissurethatitreallyproducesthoseephemerides,i.e.,thatthesolutionoftheapproximateequationsusedinthecomputationdoesapproachaccuratelyenoughtherelevantsolutionoftheexactequations.TheworksofFock[2]andChandrasekhar[3]aimedatansweringthelatterquestionforGR.Thelaterworkoncelestialmechan-icsinGRreliesonessentiallythesameapproximationschemeasthesetwoworks,whichareequivalentinthisregard.Yetin1966,Synge,whodidknowFock’swork(whichisquotedinRef.[4])andmostprobablyknewalsothatofChandrasekhar,wrote[5]:“IamstillwaitingforarationaltreatmentofthedynamicsofthesolarsystemaccordingtoEinstein’stheory.Intheverynatureofthecase,anyargumentmustbeofanapproximatenature;anassessmentoftheerrorisaprimarydesideratum.”Comparinghissuccessivesentences,wemayinferthatSyngewasnotsatisfiedwiththeapproximationmethodusedintheworks[2,3]norwiththeonehehimselfproposedwithcoworkers[6,4],andwhichwaslimitedtostationaryfields—thisrestrictionisindeedinappropriatetodescribethesolarsysteminarealisticway.Theaimofthispaperistosummarizetheprinciples,thedevelopmentandthenumericalimplementationofanewapproximationmethodforce-lestialmechanicsinrelativistictheoriesofgravitation.ThisapproximationmethodmighthavesatisfiedSynge,perhaps,becauseitismathematicallysoundandgeneral,andbecauseittoopredictsasalientresultwhichhefoundinhisworkforGR[6,4],namelythefactthat,insuchtheories,theinternalstructureofabodydoesinfluencethegravitationalfieldproducedbyit,hencealsothemotionofexternalbodies[9,10].Thenewmethodconsistsbasicallyinassociatingaone-parameterfamily(Sλ)ofgravitatingsystemswiththephysicallygivensystemS,bydefiningafamilyofinitial2conditions.ItwasinitiatedbyFutamase&Schutz[7]forGR,withfur-thermathematicaldevelopmentsgivenbyRendall[8].However,Futamase&Schutz[7]assumedaveryrestrictiveinitialconditionforthespatialmetric.AstoRendall[8],heconsideredanapriorigivenone-parameterfamilyofsolutionsofthefieldequations,withoutinvestigatingthedefinitionofasuchfamilyfromthegivensystemS.Moreover,thesetwoworkswerelimitedtothelocalequationsandsomeoftheirmathematicalproperties.Inparticular,theydidnotprovideequationsofmotionforthemasscentersofasystemofextendedbodies,asoneneedstocomputeanephemeris.Wecametothenewmethodindependently[11,12],totestanalternativetheoryofgravita-tionbasedonjustascalarfield[13,14],andwedidobtainsuchequationsofmotion[15,9,10].ThatscalartheorygivesthesamepredictionsasGRforlightrays[11].Therefore,itisworthtestingthistheoryfurther.Moreover,sincethattheoryismuchsimplerthanGR,itiseasiertoimplementthenewmethodforthattheory,aswellastodiscussthedifferencebetweenthenewmethodandthestandardPNA.2Generalframework:themethodofasymp-toticexpansionsAsiswell-known,anasymptoticexpansionofarealfunctionϕoftherealvariableλintheneighborhoodofsomevalueΛisanexpressionϕ(λ)=a0ψ0(λ)+...+anψn(λ)+R(λ),(1)theknownfunctionsψ0,...,ψnbeingpositiveandbelongingtoadefinitecom-parisonsetE,endowedwithcertainproperties,andwithψ0≫...≫ψn≫Rasλ→Λ[16].Inphysics,therelevantvalueisusuallyΛ=0,andonespeaksofthe“smallparameter”λ.Inparticular,ifthebehaviourasλ→0isregu-larenough,aTaylorexpansionmayapply,sothatψk(λ)=λk(k=0,...,n).However,itmaybethattheTaylorexpansioncanbepushedonlytosomeordern=nmax,beyondwhichamoreaccurateexpansioncanbeobtainedonlyifoneacceptstoconsidermoregeneralfu