Space isotropy and weak equivalence principle in a

arXiv:gr-qc/0412085v330Jan2006SpaceisotropyandweakequivalenceprincipleinascalartheoryofgravityM.ArminjonLaboratoire“Sols,Solides,Structures”(CNRS,Universit´eJosephFourier,&InstitutNationalPolytechniquedeGrenoble)BP53,F-38041Grenoblecedex9,France.CurrentlyatDipartimentodiFisica/Universit`adiBari,&INFN/SezionediBari,ViaAmendola173,I-70126Bari,Italy.AbstractWeconsiderapreferred-framebimetrictheoryinwhichthescalargravitationalfieldbothinfluencesthemetricandhasdirectdynamicaleffects.Amodifiedversion(“v2”)isbuilt,byassumingnowalocally-isotropicdilationofphysicallymeasureddistances,ascomparedwithdistancesevaluatedwiththeEuclideanspacemetric.Thedynamicalequationsstayunchanged:theyarebasedonaconsistentformulationofNewton’ssecondlawinacurvedspace-time.Toobtainalocalconservationequationforenergywiththenewmetric,theequationforthescalarfieldismodified:nowitsl.h.s.istheflatwaveoperator.Fluiddynamicsisformulatedandtheasymptoticschemeofpost-Newtonianapproximationisadaptedtov2.Thelatteralsoexplainsthegravitationaleffectsonlightrays,asdidtheformerversion(v1).Theviolationoftheweakequivalenceprinciplefoundforgravitationally-activebodiesatthepoint-particlelimit,whichdiscardedv1,isprovedtonotexistinv2.Thusthatviolationwasindeedduetotheanisotropyofthespacemetricassumedinv1.1IntroductionTheuniversalityofgravitation,i.e.,thefactthat“allbodiesfallinthesamewayinagravitationalfield,”isadistinctivefeatureofthegravityinteraction.Itisalsoknownasthe“weakequivalenceprinciple”(WEP),theequivalencebeingbetweenthegravita-tionalandinertialeffects:indeed,thefactthatthegravitationalaccelerationofasmallbodyisthesameindependentlyofitscompositionanditsmass,allowstoincorporateinittheaccelerationduetothemotionofthearbitraryreferenceframethroughaninertialframe.FollowingGalileo,Newton,andE¨otv¨os,theempiricalvalidityoftheWEPhasbeencheckedwithincreasingprecisions[1].Therefore,theWEPisinvolvedintheveryconstructionofrelativistictheoriesofgravitation,inparticularinthecon-structionofEinstein’sgeneralrelativity(GR),whichtakesbenefitoftheuniversalityofgravitytogeometrizeit.Moreprecisely,intheconstructionofarelativistictheory,1oneensuresthevalidityoftheWEPfortestparticles,which,bydefinition,donotinfluencethegravitationalfield.ThegeometrizationoperatedbyGRisonewaytodothis.AnotherwayistoformulatethedynamicsbyanextensionofNewton’ssecondlaw,thegravitationalforcebeingmgwithmthe(velocity-dependent)inertialmassandga(theory-dependent)gravityacceleration[2].WewouldliketoemphasizethatEinstein’sgeometrizeddynamics,inwhichtestparticlesfollowgeodesiclinesofthespace-timemetric,canberewritteninthatway[2].Atheoryofgravitationhasbeenbuilt,inwhichonestartsfromasimpleformforthefieldg,suggestedbyaheuristicinterpretationofgravityasArchimedes’thrustduetothespacevariationofthe“etherpressure”pe,andinwhichpealsodeterminesthespace-timemetric.Oneobtainsascalarbimetrictheorywithapreferredreferenceframe.ThistheoryissummarizedinRef.[3],orinmoredetailinRef.[4].Itadmitsa“physicallyobservablepreferredfoliation”[5],inshortadetectableether.Yet,apartfromtheWEPviolationtobediscussedbelow,itseemstoagreewithobservations[3,4].Inparticular,theetherofthattheorycouldbeindeeddetectedbyadjustingonastrodynamicalobservationstheequationsofcelestialmechanicsthatarevalidinthetheory[3].Oneofthemotivationstoinvestigateapreferred-frametheoryisthattheexistenceofapreferredreferenceframe,involvingthatofapreferredtime,mayberegardedasapossiblewaytomakequantumtheoryandgravitationtheorymatchtogether[3].Itisinterestingtonotethat,currently,adetectableetherisadvocatedalreadyintheabsenceofgravitationbySelleri,intheframeworkofatheoryclosetospecialrelativity,butbasedonanabsolutesimultaneity[6].Aswementioned,theWEPisvalidforanytestparticle,byconstruction—intheinvestigatedscalartheoryjustaswellasinGR.Butanyrealbody,howeversmallitmaybe,mustinfluencethegravitationalfield,sothatonecanaprioriexpecttheoc-currenceofself-accelerations.ThelatteronesareexcludedfromNewton’stheorybytheactio-reactioprinciple,butthisprinciplecannotevenbeformulatedinanonlineartheory—asaremostrelativistictheories.Moreover,themass-energyequivalence,whichhastobeaccountedforbyarelativistictheory,impliesthattherest-mass,kinetic,andgravitationalenergiesofthetestbodyallinfluencethegravitationalfield.Thismeansthattheinternalstructureofthetestbodymayaprioribeexpectedtoinfluenceitsmotion,whichwouldbeaviolationoftheWEP.Hence,thevalidityoftheWEPfortestparticlesisfarfromensuringthatthisprincipleappliestorealbodies.InGR,studiesoftheself-forceactingonanextendedbodycanbefound,e.g.[7,8,9,10],buttheyhavebeenbasedonsimplifyingassumptionssuchasthatofablack-holebody[7,8,10],oronlinearizedGR[9].SuchassumptionsdonotseemveryappropriatetocheckwhetheraviolationoftheWEPmightbepredictedbyGRintherealworld,sayforanasteroidoraspacecraftinthesolarsystem.Itiswell-known,sincetheworkofNordtvedt[11,12],thattheWEPmaybeviolatedforextendedbodiesofafinitemassinsomerelativistictheoriesofgravity(seealsoWill[13,14]).However,thekindofviolationoftheWEPwhichwehaveinmindisamoresevereone,thatoccurseveninthelimitinwhichthesize