Local and global gravity

arXiv:gr-qc/9611038v113Nov1996LocalandGlobalGravity∗PhilipD.MannheimDepartmentofPhysics,UniversityofConnecticut,Storrs,[e-mail]ueofFoundationsofPhysicshonoringProfessorLawrenceHorwitzontheoccasionofhis65thbirthday;A.vanderMerweandS.Raby,Editors,PlenumPublishingCompany,N.Y.,1997.AbstractOurlongexperiencewithNewtonianpotentialshasinuredustotheviewthatgravityonlyproduceslocaleffects.Inthispaperwechallengethisquitedeeplyin-grainednotionandexplicitlyidentifysomeintrinsicallyglobalgravitationaleffects.InparticularweshowthattheglobalcosmologicalHubbleflowcanactuallymodifythemotionsofstarsandgaswithinindividualgalaxies,andevendosoinawaywhichcanapparentlyeliminatetheneedforgalacticdarkmatter.Alsoweshowthataclassicallightwaveacquiresanobservable,global,pathdependentphaseintraversingagravi-tationalfield.Bothoftheseeffectsservetounderscoretheintrinsicdifferencebetweennon-relativisticandrelativisticgravity.1IntroductionSinceitsveryinceptiongravitationaltheoryhasalwaysbeenregardedasbeingintrinsicallylocal.Specifically,withtheNewtoniangravitationalpotentialfallingas1/r,themotionofanygivensystemisthenprimarilydeterminedbythemostnearbygravitationalsources,withstatic,sphericallysymmetricallydistributeddistantmatteractuallymakingnonetcontributionatall.Thusinastatic,sphericallysymmetricNewtonianuniverseonlynearbysourcesarerelevant.Moreover,theadventofEinstein’sgeneraltheoryofrelativityactuallyonlyservedtoreinforcethisviewpoint,sincethemetricinsideastatic(orevenaradiallyvibrating)sphericallysymmetricshellisfoundtobeflatinEinsteingravity,tothusagainpermitustolocallyignoretherestofauniverseofthesamesymmetry.Thusfirstlywithpotentialswhichfalltozeroatlargedistancesandthensubsequentlywiththeanalogousrelativisticnotionofasymptoticflatness,aviewofgravityhasemergedthatonlylocalsourcesarerelevant.Inthispaperweexamineandchallengethislongstandingandbynowquiteingrainednotionbyidentifyingsomeexplicitlyglobalaspectsofgravity,bothinstandard∗UCONN96-09,November19961Einsteingravityitself,aswellasintherecentlyadvancedfourthorderconformalinvariantalternatetheoryofgravity(seee.g.[1]),atheoryinwhichindividualgravitationalsourcesarefoundtoactuallymakeasymptoticallynon-vanishingcontributionstothegeometry,andinwhichtheglobalcosmologicalHubbleflowactuallyhaslocalconsequences,tothusgivegravityaquiteMachianflavor.SinceagreatdealofourgravitationalintuitioncomesfromNewtoniancelestialme-chanics,webegininSec.(2)withananalysisoftheobservationalbasisforinferringthatnon-relativisticpotentialsactuallyfallatlargedistances;andshowthatrisingonesarenotmerelynotexcludedbyavailabledata,butthataphenomenologicalanalysisofthesystem-aticsofgalacticrotationcurvesindicatesthat(linearly)risingpotentialsmayinfactevenbefavored.Moreover,ourspecificstudyrevealsthatnotonebutinfacttwolinearpotentialsappeartobeoperativeingalaxies,onewhichscaleslinearlywiththeamountofmatterinagivengalaxy,andasecondonewhichintriguinglyhasauniversal,galaxyindependentnormalizationofordertheHubblescale.InSec.(3)weshowthatrelativisticcovariancedoesnotinfactleadusunambiguouslytostandardEinsteingravity,andthatitispossibletoconsiderother,equallycovariant,puremetrictheoriesofgravity,theorieswhichcouldthenhavetotallydifferentasymptoticproperties.Inparticularweshowthatthetwophe-nomenologicallyfoundlinearlyrisinggalacticpotentialsofSec.(2)evenhaveanaturalhomeintheconformalgravityalternativewheretheycanbothbeproduced,onelocallybytheluminousmatterwithinagivengalaxyandtheothergloballybytheuniversalHubbleflowduetotherestofthegalaxiesintheuniverse.InSec.(4)weshowthattheCOW(Colella,OverhauserandWerner)quantum-mechanicalneutronbeaminterferometryexperimentinabackgroundgravitationalfieldhasapurelyclassicalcounterpart,withclassicallightwavesalsoexperiencingglobal,pathdependentinterferencephaseshiftsinbackgroundgravity,tothusyieldanotherglobal,expresslyrelativisticeffect;andwiththeinterferencebeingfoundtobeaccreditabletothegravitationalbendingoflight,thiseffectisthenonewhichwilloc-curinanycovariant,puremetricbasedtheoryofgravity.Fromthevariousspecificstudiesofthispapertherethusemergesaglobalcharactertogravityofasurprisinglyfarreachingnature.2Localandglobalnon-relativisticgravityInNewtoniangravitythe−MG/rpotentialhasfourveryremarkableproperties.Firstly,itpermitsthereplacementofastatic,sphericallysymmetricsourceofnon-zeroradiusbyapointsourcelocatedatitscenter.Secondly,itprovidesforthemutualcancellationofthepotentialsofdistant,static,sphericallysymmetricallydistributedmattersources,sothatsuchmatterhasnonetlocaleffectatall.Thirdly,since−1/4πristheGreen’sfunctionofthe∇2operator,the−MG/rpotentialalsoemergesastheexteriorsolutiontothesecondorderPoissonequation,viz.∇2φ(r)=4πGρ(r),φ(rR0)=−4πGrZR00dr′ρ(r′)r′2(1)forastatic,sphericallysymmetricsourceofradiusR0andmassdensityρ(r),tothusconnecttheNewtonianpotentialtoadynamicalequationofmotion;anduniquelysoinfactwithin2theframeworkofsecondorderequations,sincenootherpowerbehavedfunctionr−nissimultaneouslyboththeinputGreen’sfunctionandtheoutputpotentialtoasecondor