Relativistic gravitation theory for the MOND parad

arXiv:astro-ph/0403694v623Aug2005RelativisticgravitationtheoryfortheMONDparadigmJacobD.Bekenstein∗RacahInstituteofPhysics,HebrewUniversityofJerusalemGivatRam,Jerusalem91904ISRAEL(Dated:February2,2008)Themodifiednewtoniandynamics(MOND)paradigmofMilgromcanboastofanumberofsuccessfulpredictionsregardinggalacticdynamics;thesearemadewithouttheassumptionthatdarkmatterplaysasignificantrole.MONDrequiresgravitationtodepartfromNewtoniantheoryintheextragalacticregimewheredynamicalaccelerationsaresmall.SofarrelativisticgravitationtheoriesproposedtounderpinMONDhaveeitherclashedwiththepost-Newtoniantestsofgeneralrelativity,orfailedtoprovidesignificantgravitationallensing,orviolatedhallowedprinciplesbyexhibitingsuperluminalscalarwavesoranapriorivectorfield.WedeveloparelativisticMONDinspiredtheorywhichresolvestheseproblems.Initgravitationismediatedbymetric,ascalarfieldanda4-vectorfield,allthreedynamical.Forasimplechoiceofitsfreefunction,thetheoryhasaNewtonianlimitfornonrelativisticdynamicswithsignificantacceleration,butaMONDlimitwhenaccelerationsaresmall.WecalculatetheβandγPPNcoefficientsshowingthemtoagreewithsolarsystemmeasurements.Thegravitationallightdeflectionbynonrelativisticsystemsisgovernedbythesamepotentialresponsiblefordynamicsofparticles.TotheextentthatMONDsuccessfullydescribesdynamicsofasystem,thenewtheory’spredictionsforlensingbythatsystem’svisiblematterwillagreeaswellwithobservationsasgeneralrelativity’spredictionsmadewithadynamicallysuccessfuldarkhalomodel.Cosmologicalmodelsbasedonthetheoryarequitesimilartothosebasedongeneralrelativity;theypredictslowevolutionofthescalarfield.Forarangeofinitialconditions,thislastresultmakesiteasytoruleoutsuperluminalpropagationofmetric,scalarandvectorwaves.PACSnumbers:95.35.+d,95.30.Sf,98.62.Sb,04.80.CcI.INTRODUCTIONIntheextragalacticregime,whereNewtoniangravitationaltheorywouldhavebeenexpectedtobeanexcellentdescription,accelerationsofstarsandgas,asestimatedfromDopplervelocitiesandgeometricconsiderations,areasarulemuchlargerthanthoseduetotheNewtonianfieldgeneratedbythevisiblematterinthesystem[1,2].Thisisthe“missingmass”problem[3]or“accelerationdiscrepancy”[4].Itisfashionabletoinferfromittheexistenceofmuchdarkmatterinsystemsrangingfromdwarfspheroidalgalaxieswithmasses∼106M⊙togreatclustersofgalaxiesinthe1013M⊙regime[3,5].Andagain,galaxiesandclustersofgalaxiesarefoundtogravitationallylensebackgroundsources.Wheninterpretedwithingeneralrelativity(GR),thislensingisanomalouslylargeunlessoneassumesthepresenceofdarkmatterinquantitiesandwithdistributionsimilartothoserequiredtoexplaintheaccelerationsofstarsandgas.Thusextragalacticlensinghasnaturallybeenregardedasconfirmingthepresenceofthedarkmattersuggestedbythedynamics.Buttheputativedarkmatterhasneverbeenidentifieddespitemuchexperimentalandobservationaleffort[6].ThisraisesthepossibilitythattheaccelerationdiscrepancyaswellasthegravitationallensinganomalymayreflectdeparturesfromNewtoniangravityandGRongalacticandlargerscales.NowalternativestoGRaretraditionallyrequiredtopossessaNewtonianlimitforsmallvelocitiesandpotentials;thustheaccelerationdiscrepancyalsoraisesthepossibilitythatthecorrectrelativisticgravitationaltheorymaybeofakindnotgenerallyconsideredhitherto.InthelasttwodecadesMilgrom’smodifiedNewtoniandynamics(MOND)paradigm[7,8,9]hasgainedrecognitionasasuccessfulschemeforunifyingmuchofextragalacticdynamicsphenomenologywithoutinvoking“darkmatter”.IncontrastwithearliersuggestedmodificationsofNewton’slawofuniversalgravitation[10,11,12,13],MONDischaracterizedbyanaccelerationscalea0,notadistancescale,anditsdeparturefromNewtonianpredictionsisaccelerationdependent:˜μ(|a|/a0)a=−∇ΦN.(1)HereΦNistheusualNewtonianpotentialofthevisiblematter,while˜μ(x)≈xforx≪1and˜μ(x)→1forx≫1.∗Electronicaddress:[e-mail];URL:~bekenste/2Milgromestimateda0≈1×10−8cms−2fromtheempiricaldata.Inthelaboratoryandthesolarsystemwhereaccelerationsarestrongcomparedtoa0,formula(1)goesovertotheNewtonianlawa=−∇ΦN.Milgromconstructedformula(1)toagreewiththefactthatrotationcurvesofdiskgalaxiesbecomeflatoutsidetheircentralparts.ThatfaroutagalaxyofmassMexhibitsanapproximatelysphericalNewtonianpotential.Thescalesaresuchthat|∇ΦN|≈GMr−2≪a0inthisregion,andsoEq.(1)with˜μ(x)≈xgives|a|≈(GMa0)1/2r−1whichhastherdependenceappropriateforthecentripetalaccelerationv2c/rofaradiusindependentrotationalvelocityvc—anasymptoticallyflatrotationcurve.InfactoneobtainstherelationM=(Ga0)−1vc4whichleadstothepredictionthatforanyclassofgalaxieswithaconstantmasstoluminosityratioΥinaspecifiedspectralband,theluminosityinthatbandshouldscaleasvc4.Andindeed,thereexistsanempiricallawofjustthisform:theTully–Fisherlaw[14](TFL)relatingnearinfrared(H–band)luminosityLHofaspiraldiskgalaxytoitsrotationvelocity,LH∝vc4,withtheproportionalityfactorbeingconstantwithineachgalacticmorphologyclass.ThisversionoftheTFLwasestablishedonlyafterMONDwasenunciated[15].ItisinharmonywiththeMONDpredictionintwoways.First,theinfraredlightofagalaxycomesmostlyfromcooldwarfstarswhichmakeupmostofitsmass(hencegivingatightcorrelati