ProgressinParticleandNuclearPhysics56(2006)340–445 www.elsevier.com/locate/ppnp Review Alternatives to dark matter and dark energy Philip D. Mannheim DepartmentofPhysics,UniversityofConnecticut,Storrs,CT06269,USA Abstract We review the underpinnings of the standard Newton–Einstein theory of gravity, and identify whereitcouldpossiblygowrong.Inparticular,wediscussthelogicalindependencefromeachother ofthegeneralcovarianceprinciple,theequivalenceprincipleandtheEinsteinequations,anddiscuss how to constrain the matter energy–momentum tensor which serves as the source of gravity. We identifytheaprioriassumptionofthevalidityofstandardgravityonalldistancescalesastheroot causeofthedarkmatteranddarkenergyproblems,anddiscusshowthefreedomcurrentlypresent in gravitational theory can enable us to construct candidate alternatives to the standard theory in whichthedarkmatteranddarkenergyproblemscouldthenberesolved.Weidentifythreegeneric aspectsofthesealternateapproaches:thatitisauniversalaccelerationscalewhichdetermineswhen aluminousNewtonianexpectationistofailtofitdata,thatthereisaglobalcosmologicaleffecton local galactic motions which can replace galactic dark matter, and that to solve the cosmological constantproblemitisnotnecessarytoquenchthecosmologicalconstantitself,butonlytheamount bywhichitgravitates. ©2005ElsevierB.V.Allrightsreserved. Keywords:Generalrelativity;Darkmatter;Darkenergy;Cosmology;Cosmologicalconstantproblem;Conformal gravity Contents 1. Introduction...........................................................................................................342 2. Theunderpinningsofthestandardgravitationalpicture...............................................343 2.1. Massivetestparticlemotion.........................................................................343 2.2. Theequivalenceprinciple.............................................................................345 E-mailaddress:[email protected]. 0146-6410/$-seefrontmatter©2005ElsevierB.V.Allrightsreserved. doi:10.1016/j.ppnp.2005.08.001 P.D.Mannheim/ProgressinParticleandNuclearPhysics56(2006)340–445 341 2.3. Masslessfieldmotion..................................................................................346 2.4. Massivefieldmotion...................................................................................348 3. Structureoftheenergy–momentumtensor.................................................................350 3.1. Kinematicperfectfluids...............................................................................350 3.2. PerfectRobertson–Walkerfluids...................................................................352 3.3. Schwarzschildfluidsources..........................................................................353 3.4. Scalarfieldfluidsources..............................................................................356 3.5. Dynamicalfluidsources...............................................................................358 3.6. Implicationsofelementaryparticlephysics....................................................361 4. Newtoniangravity..................................................................................................362 4.1. TheNewtonianpotential..............................................................................362 4.2. ThesecondorderandfourthorderPoissonequations......................................364 4.3. UniversalaccelerationscaleandtheMONDtheory.........................................367 5. Einsteingravity......................................................................................................369 5.1. EinsteingravityforstaticsourcesandtheNewtonianlimit...............................369 5.2. TheEinsteinstaticuniverseandthecosmologicalconstant...............................372 6. Thedarkmatterproblem.........................................................................................374 6.1. Thegalacticrotationcurveproblem...............................................................374 6.2. Thedarkmattersolutionforgalaxies.............................................................377 6.3. Cluesfromthedata.....................................................................................380 7. Thedarkenergyproblem.........................................................................................383 7.1. Thecosmologicaldarkmatterandflatnessproblems.......................................383 7.2. Fine-tuningandtheinflationaryuniverse........................................................386 7.3. Theacceleratinguniverseandthecosmologicaldarkenergysolution................388 7.4. Thefreedominherentinfittingthecurrentsupernovaedata..............................390 7.5. Thecosmiccoincidenceproblem..................................................................391 7.6. Thecosmologicalconstantproblem...............................................................393 7.7. Testingdarkenergybeyondaredshiftofone..................................................394 8. AlternativestoEinsteingravity.................................................................................397 8.1. Puremetrictheorieswithadditionalcurvature-dependentterms........................397 8.2. Additionalfields.........................................................................................398 8.3. Additionalspacetimedimensions..................................................................399 8.4. ThemodifiedFriedmannequationsonthebrane.............................................402 8.5. RelativisticMONDtheory............................................................................404 8.6. Modificationsofthenatureofthespacetimegeometryitself.............................405 8.7. Conformalgravity.......................................................................................408 9. Alternativestodarkmatter.......................................................................................412 9.1. ModifyingNewtoniangravityatlargedistancesorsmallaccelerations..............412 9.2. ModifyingEinsteingravity—theconformalgravityapproach...........................412 9.3. ImpactoftheglobalHubbleflowongalacticrotationcurves............................415 9.4. ComparisonoftheconformalgravityandMONDfits......................................418 10. Alternativestodarkenergy......................................................................................421 10.1. Theconformalgravityalternativetodarkenergy.............................................421 10.2. Conformalgravityandtheacceleratinguniverse.............................................425 10.3. Quenchingthecontributionofthecosmologicalconstanttocosmicevolution....427 11. Futureprospectsandchallenges...............................................................................428 A. Thepotentialofathindisk......................................................................................430 B. Thepotentialofaseparablethickdisk.......................................................................432 C. Thepotentialofasphericalbulge.............................................................................434 342 P.D.Mannheim/ProgressinParticleandNuclearPhysics56(2006)340–445 D. Theghostprobleminfourthordertheories................................................................436 References.............................................................................................................443 1. Introduction Following many years of research in cosmology and astrophysics, a picture of the universehasemerged[1–5]whichis astroublingas itisimpressive.Specifically,a wide varietyofdatacurrentlysupporttheview thatthematter contentoftheuniverseconsists of two primary components, viz. dark matter and dark energy, with ordinary luminous matterbeingrelegatedtoadecidedlyminorrole.Thenatureandcompositionofthesedark matter anddarkenergycomponentsis notatall well-understood,andwhile bothpresent severechallengestothestandardtheory,eachpresentsadifferentkindofchallengetoit.As regardsdarkmatter,thereisnothinginprinciplewrongwiththeexistenceofnon-luminous materialperse(indeedobjectssuchasdeadstars,browndwarfsandmassiveneutrinosare well-establishedinnature).Rather,whatisdisturbingistheadhoc,afterthefact,wayin whichdarkmatterisactuallyintroduced,withitspresenceonlybeinginferredafterknown luminous astrophysical sources are found to fail to account for any given astrophysical observation.Darkmatterthusseemstoknowwhere,andinwhatamount,itistobeneeded, andtoknowwhenitisnotinfactneeded(darkmatterhastoavoidbeingabundantinthe solarsysteminordertonotimpairthesuccessofstandardgravityinaccountingforsolar system observationsusing visible sources alone); and moreover,in the cases where it is needed, what it is actually made of (astrophysical sources (Machos) or new elementary particles(Wimps))isasyettotallyunknownandelusive. Disturbingasthedarkmatterproblemis,thedarkenergyproblemisevenmoresevere, andnotsimplybecauseitscompositionandnatureisasmysteriousasthatofdarkmatter. Rather, for dark energy there actually is a very good, quite clear-cut candidate, viz. a cosmologicalconstant,andtheproblemhereisthatthevalueforthecosmologicalconstant as anticipated from fundamental theory is orders of magnitude larger than the data can possiblypermit.Withdarkmatterthen,weseethatluminoussourcesaloneunderaccount forthedata,whilefordarkenergy,acosmologicalconstantoveraccountsforthedata.Thus, within the standard picture, arbitrary as their introductionmight be, there is nonetheless roomfordarkmattercandidatesshouldtheyultimatelybefound,butfordarkenergythere is a need not to find something which might only momentarilybe missing, but rather to get rid of something which is definitely there. And indeed, if it does not prove possible toquenchthecosmologicalconstantbytherequisiteordersandordersofmagnitude,one wouldhavetoconcludethattheprevailingcosmologicaltheorysimplydoesnotwork. Inarrivingatthepredicamentthatcontemporaryastrophysicalandcosmologicaltheory thus finds itself in, it is important to recognize that the entire case for the existence of dark matter and dark energy is based on just one thing alone, viz. on the validity on all distancescalesofthestandardNewton–Einsteingravitationaltheoryasexpressedthrough theEinsteinequationsofmotion (cid:1) (cid:2) c3 1 −8πG Rµν − 2gµν Rαα = Tµν (1) P.D.Mannheim/ProgressinParticleandNuclearPhysics56(2006)340–445 343 for the gravitational field gµν. Specifically, the standard approach to cosmology and astrophysics is to take the Einstein equations of motion as given, and whenever the theory is found to encounter observational difficulties on any particular distance scale, µν modificationsareto thenbe madeto T throughthe introductionofnew,essentiallyad hoc,gravitationalsourcessothatagreementwithobservationcanthenberestored.While a better understandingofdarkenergyandexplicitobservationaldetectionof darkmatter sourcesmighteventuallybe achievedin the future,at the presenttime the only apparent waytoavoidthedarkmatteranddarkenergyproblemsistomodifyorgeneralizenotthe right-hand side of Eq. (1) but rather its left. In order to see how one might actually do this,itisthusnecessarytocarefullygoovertheentirepackagerepresentedbyEq.(1),to determinewhethersomeofitsingredientsmightnotbeassecureasothers,andinvestigate whethertheweakeronescouldpossiblybereplaced.Todothiswewillthusneedtomake acriticalappraisalofbothsidesofEq.(1). 2. Theunderpinningsofthestandardgravitationalpicture 2.1. Massivetestparticlemotion Followingayearofremarkableachievement,achievementwhosecentennialiscurrently beingcelebrated,attheendof1905Einsteinfoundhimselfinasomewhatparadoxicalsit- uation.While hehadsurmountedanenormoushurdleindevelopingspecialrelativity,its veryestablishmentcreatedanevenbiggerhurdleforhim.Specifically,withthe develop- mentofspecialrelativityEinsteinresolvedtheconflictbetweentheLorentzinvarianceof the MaxwellequationsandtheGalilean invarianceofNewtonianmechanicsbyrealizing that it was Lorentz invariance which was the more basic of the two principles, and that itwasNewtonianmechanicswhichthereforehadtobemodified.Whilespecialrelativity thusascribedprimacytoLorentzinvariancesothatobserversmovingwithlargeorsmall uniformvelocitycouldthenallagreeonthesamephysics,suchobserversstilloccupieda highlyprivilegedpositionsinceuniformvelocityobserversformonlyaverysmallsubset ofallpossibleallowableobservers,observerswhocouldmovewitharbitrarilynon-uniform velocity.Additionally,ifitwasspecialrelativitythatwastobetheall-embracingprinciple, allinteractionswouldthenneedtoobeyit,andyetwhatwasatthetimetheacceptedthe- oryof gravity,viz.Newtoniangravity,in factdidnot. Itwasthe simultaneousresolution ofthesetwoissues(viz.acceleratingobserversandthecompatibilityofgravitywithrela- tivity)atoneandthesametimethroughthespectaculardevelopmentofgeneralrelativity which not only established the Einstein theory of gravity, but which left the impression thattherewasonlyonepossibleresolutiontothetwoissues,viz.thatbasedonEq.(1).To pinpointwhatitisinthestandardtheorywhichleadsustothedarkmatteranddarkenergy problems,wethusneedtounravelthestandardgravitationalpackageintowhatareinfact logicallyindependentcomponents,an exercisewhich is actuallyof value in and of itself regardlessofthedarkmatteranddarkenergyproblems. Inordertomakesuchadissectionofthestandardpicture,webeginwithadiscussion ofastandard,free,spinless,relativistic,classical–mechanicalNewtonianparticleofnon- zero kinematic mass m moving in flat spacetime according to the special relativistic generalizationofNewton’ssecondlawofmotion 344 P.D.Mannheim/ProgressinParticleandNuclearPhysics56(2006)340–445 d2ξα m =0, Rµνστ =0, (2) dτ2 wheredτ =(−ηαβdξαdξβ)1/2isthepropertimeandηαβ istheflatspacetimemetric,and wherewe have indicatedexplicitlythatthe Riemanntensoris (forthe moment)zero.As such,Eq.(2)willbeleftinvariantunderlineartransformationsofthecoordinatesξµ,buton µ makinganarbitrarynon-lineartransformationtocoordinatesx andusingthedefinitions ∂xλ ∂2ξα ∂ξα ∂ξβ Γµλν = ∂ξα ∂xµ∂xν, gµν = ∂xµ ∂xνηαβ, (3) we directly find (see e.g. [6], a reference whose notation we use throughout) that the invariantpropertimeisbroughttotheformdτ =(−gµνdxµdxν)1/2,andthattheequation ofmotionofEq.(2)isrewrittenas (cid:3) (cid:4) (cid:3) (cid:4) D2xλ d2xλ dxµdxν m Dτ2 ≡m dτ2 +Γµλν dτ dτ =0, Rµνστ =0, (4) withEq.(4)servingtodefineD2xλ/Dτ2.Asderived,Eq.(4)sofaronlyholdsinastrictly flatspacetimewithzeroRiemanncurvaturetensor,andindeedEq.(4)isonlyacovariant rewriting of the special relativistic Newtonian second law of motion, i.e. it covariantly describeswhatanobserverwithanon-uniformvelocityinflatspacetimesees,withtheΓµλν term emerging as an inertial, coordinate-dependentforce. The emergence of such a Γµλν termoriginatesinthefactthatevenwhilethefour-velocitydxλ/dτ transformsasageneral contravariantvector,itsordinaryderivatived2xλ/dτ2 (whichsamplesadjacentpointsand not merely the point where the four-velocityitself is calculated) does not, and it is only the four-accelerationD2xλ/Dτ2 which transformsas a generalcontravariantfour-vector, anditisthusonlythisparticularfour-vectoronwhosemeaningall(acceleratingandnon- accelerating)observerscanagree.ThequantityΓµλνisnotitselfageneralcoordinatetensor, andinflatspacetimeonecaneliminateiteverywherebyworkinginCartesiancoordinates. Despite this privileged status for Cartesian coordinate systems, in general it is Eq. (4) ratherthanEq.(2)whichshouldbeused(eveninflatspacetime)sincethisistheformof Newton’ssecondlawofmotionwhichanacceleratingflatspacetimeobserversees. Nowwhilealloftheaboveremarkswheredevelopedpurelyforflatspacetime,Eq.(4) hasanimmediategeneralizationtocurvedspacetimewhereitthentakestheform (cid:3) (cid:4) (cid:3) (cid:4) D2xλ d2xλ dxµdxν m Dτ2 ≡m dτ2 +Γµλν dτ dτ =0, Rµνστ (cid:2)=0. (5) Incurvedspacetime,itisagainonlythequantityD2xλ/Dτ2whichtransformsasageneral contravariantfour-vector,andtheChristoffelsymbolΓµλν isagainnotageneralcoordinate tensor.Consequentlyatanygivenpoint P itcanbemadetovanish,1thoughnocoordinate transformationcanbringittozeroateverypointinaspacetimewhoseRiemanntensoris 1Underthetransformation x(cid:3)λ = xλ+ 12xµxν(Γµλν)P,theprimedcoordinateChristoffelsymbols(Γµ(cid:3)λν)P willvanishatthepointP,regardlessinfactofhowlargetheRiemanntensorintheneighborhoodofthepointP mightactuallybe. P.D.Mannheim/ProgressinParticleandNuclearPhysics56(2006)340–445 345 non-vanishing.Asanequation,Eq.(5)canalsobeobtainedfromanactionprinciple,since itisthestationarityconditionδIT/δxλ =0associatedwithfunctionalvariationofthetest particleaction (cid:5) I =−mc dτ (6) T λ with respect to the coordinate x . To appreciate the ubiquity of the appearance of the covariantaccelerationD2xλ/Dτ2,weconsiderasactionthecurvedspaceelectromagnetic coupling (cid:5) (cid:5) λ I(2) =−mc dτ +e dτdx Aλ, (7) T dτ λ tofindthatitsvariationwithrespecttox leadstothecurvedspaceLorentzforcelaw (cid:3) (cid:4) d2xλ dxµdxν dxα mc dτ2 +Γµλν dτ dτ =eFλα dτ , Rµνστ (cid:2)=0. (8) Similarly,thecouplingofthetestparticletotheRicciscalarvia (cid:5) (cid:5) IT(3) =−mc dτ −κ dτ Rαα, (9) leadsto (cid:6) (cid:7)(cid:3)d2xλ dxµdxν(cid:4) (cid:1) dxλdxβ(cid:2) mc+κ Rαα dτ2 +Γµλν dτ dτ =−κ Rαα;β gλβ + dτ dτ , (10) (Rµνστ necessarilynon-zerohere),whilethecouplingofthetestparticletoascalarfield S(x)via (cid:5) I(4) =−κˆ dτS(x), (11) T leadstothecurvedspace (cid:3) (cid:4) (cid:1) (cid:2) d2xλ dxµdxν dxλdxβ κˆS dτ2 +Γµλν dτ dτ =−κˆS;β gλβ + dτ dτ , Rµνστ (cid:2)=0, (12) an expression incidentally which reduces to Eq. (5) when S(x) is a spacetime constant (withthemassparameterthenbeinggivenbymc = κˆS).Inalloftheabovecasesthenit isthequantityD2xλ/Dτ2 whichmustappear,sinceineachsuchcasetheactionwhichis variedisageneralcoordinatescalar. 2.2. Theequivalenceprinciple Nowassuch,theanalysisgivenaboveisapurelykinematiconewhichdiscussesonly thepropagationoftestparticlesincurvedbackgrounds.Thisanalysismakesnoreference togravityperse,andinparticularmakesnoreferencetoEq.(1)atall,thoughitdoesimply thatinanycurvedspacetimeinwhichthemetricgµν istakentobethegravitationalfield, covariant equations of motion involving four-accelerations would have to based strictly 346 P.D.Mannheim/ProgressinParticleandNuclearPhysics56(2006)340–445 on D2xλ/Dτ2, with neither d2xλ/dτ2 nor Γµλν(dxµ/dτ)(dxν/dτ) having any coordinate independentsignificanceormeaning.Assuch,wethusrecognizetheequivalenceprinciple asthestatementthatthed2xλ/dτ2 andΓµλν(dxµ/dτ)(dxν/dτ)termsmustappearinallof the above propagation equations in precisely the combination indicated,2 and that via a coordinate transformationit is possible to removethe Christoffelsymbols at any chosen point P, with it thereby being possible to simulate the Christoffel symbol contribution to the gravitational field at such a point P by an accelerating coordinate system in flat spacetime. We introduce this particular formulation of the equivalence principle quite guardedly, since in an equation such as the fully covariant Eq. (10), one cannot removethedependenceontheRicciscalarbyanycoordinatetransformationwhatsoever, and we thus define the equivalence principle not as the statement that all gravitational effects at any point P can be simulated by an accelerating coordinate system in flat spacetime (viz. that test particles unambiguously move on geodesics and obey Eq. (5) and none other), but rather that no matter what propagation equation is to be used for test particles, the appropriate acceleration for them is D2xλ/Dτ2.3 With Eq. (4) thus showing the role of coordinate invariance in flat spacetime, with Eqs. (5), (8) and (10) exhibitingthe equivalenceprinciple in curvedspacetime, and with all of these equations being independent of the Einstein equations of Eq. (1), the logical independence of the generalcovarianceprinciple,theequivalenceprincipleand theEinstein equationsis thus established.Hence,anymetrictheoryofgravityinwhichtheactionisageneralcoordinate scalar and the metric is the gravitational field will thus automatically obey both the relativity principle and the equivalence principle, no matter whether or not the Einstein equationsaretobeimposedaswell.4 2.3. Masslessfieldmotion Absentfromtheabovediscussionisthequestionofwhetherreal,asopposedtotest,par- ticlesactuallyobeycurvedspacepropagationequationssuchasEq.(5)atall,andwhether suchadiscussionshouldapplytomasslessparticlesaswellsinceforthembothm andthe 2Thistherebysecurestheequalityoftheinertialandpassivegravitationalmassesofmaterialparticles. 3This particular formulation of the equivalence principle does no violence to observation, since Eotvos experiment typetestingoftheequivalence principle ismadeinRicci-flat Schwarzschild geometries whereall RiccitensororRicciscalardependentterms(suchasforinstancethoseexhibitedinEq.(10))aresimplyabsent, withsuchtests(andinfactanywhichinvolvetheSchwarzschildgeometry)notbeingabletodistinguishbetween Eqs.(5)and(10). 4Tosharpenthispoint,wenotethatitcouldhavebeenthecasethattheresolutionoftheconflictbetween gravityandspecialrelativitycouldhavebeenthroughtheintroductionofthegravitationalforcenotasageometric entityatall,butratherasananalogofthewaytheLorentzforceisintroducedinEq.(8).Insuchacase,inan accelerating coordinate system one would still need to use the acceleration D2xλ/Dτ2 and not the ordinary d2xλ/dτ2.However,ifthegravitationalfieldweretobetreatedthesamewayastheelectromagneticfield,left openwouldthenbetheissueofwhetherphysicsistobeconductedinflatspaceorcurvedspace,i.e.leftopen wouldbethequestionofwhatdoesthenfixtheRiemanntensor.Therewouldthenhavetobesomeadditional equation which would fix it, and curvature would still have to be recognized as having true, non-coordinate artifact effects on particles if the Riemann tensor were then found to be non-zero. Taking such curvature to beassociatedwiththegravitationalfield(ratherthanwithsomefurtherfield)isofcoursethemosteconomical, thoughdoingsowouldnotobligegravitationaleffectstoonlybefeltthroughD2xλ/Dτ2,andwouldnotpreclude somegravitationalLorentzforcetypeterm(suchastheoneexhibitedinEq.(10))fromappearingaswell. P.D.Mannheim/ProgressinParticleandNuclearPhysics56(2006)340–445 347 propertimedτ vanishidentically,withequationssuchasEq.(5) becomingmeaningless. Nowitturnsoutthatthesetwoissuesarenotactuallyindependent,asbothreducetothe questionofhowwavesratherthanparticlescoupletogravity,sincelightisdescribedbya waveequation,andelementaryparticlesareactuallytakentobethequantaassociatedwith quantizedfieldswhicharealsodescribedbywaveequations.Andsincethediscussionwill beofrelevancefortheexplorationoftheenergy–momentumtensortobegivenbelow,we present it in some detail now. For simplicity we look at the standard minimally coupled curvedspacetimemasslessKlein–Gordonscalarfieldwithwaveequation S;µ;µ =0, Rµνστ (cid:2)=0 (13) whereS;µdenotesthecontravariantderivative∂S/∂xµ.Ifforthescalarfieldweintroduce aneikonalphase T(x)via S(x) = exp(iT(x)),thescalar T(x)isthenfoundtoobeythe equation T;µT;µ−iT;µ;µ =0, (14) anequationwhichreducestoT;µT;µ =0intheshortwavelengthlimit.Fromtherelation T;µT;µ;ν = 0 which then ensues, it follows that in the short wavelength limit the phase T(x)obeys T;µT;µ;ν = T;µ[T;ν;µ+∂νT;µ−∂µT;ν] = T;µ[T;ν;µ+∂ν∂µT −∂µ∂νT]= T;µT;ν;µ =0. (15) Sincenormalstowavefrontsobeytheeikonalrelation µ dx T;µ = =kµ (16) dq whereqisaconvenientscalaraffineparameterwhichmeasuresdistancealongthenormals andkµ isthewavevectorofthewave,onnotingthat(dxµ/dq)(∂/∂xµ) = d/dq wethus obtain d2xλ dxµdxν kµkλ;µ = dq2 +Γµλν dq dq =0, (17) a condition which we recognize as being the massless particle geodesic equation, with rays then precisely being found to be geodesic in the eikonallimit. Since the discussion givenearlierofthecoordinatedependenceoftheChristoffelsymbolswaspurelygeomet- ric,wethusseethatoncerayssuchaslightraysaregeodesic,theyimmediatelyobeythe equivalence principle,5 with phenomena such as the gravitational bending of light then immediatelyfollowing. Now while we do obtain strict geodesic motion for rays when we eikonalize the minimally coupled Klein–Gordon equation, the situation becomes somewhat different if weconsideranon-minimallycoupledKlein–Gordonequationinstead.Thus,onreplacing Eq.(13)by 5AccordingtoEq.(17),anobserverinEinstein’s elevator wouldnotbeabletotellifalightrayisfalling downwardsundergravityorwhethertheelevatorisacceleratingupwards. 348 P.D.Mannheim/ProgressinParticleandNuclearPhysics56(2006)340–445 ξ S;µ;µ+ SRαα =0, Rµνστ (cid:2)=0, (18) 6 thecurvedspaceeikonalequationthentakestheform ξ T;µT;µ− Rαα =0, (19) 6 withEq.(17)beingreplacedbytheRicciscalardependent d2xλ dxµdxν ξ +Γµλν = (Rαα);λ (20) dq2 dq dq 12 intheshortwavelengthlimit. A similar dependenceon the Ricci scalar or tensor is also obtained for massless spin one-half and spin one fields. Specifically, even though the massless Dirac equation in curvedspace,viz.iγµ(x)[∂µ+Γµ]ψ(x)=0(whereΓµ(x)isthefermionspinconnection) contains no explicit direct dependenceon the Ricci tensor, nonetheless the second order differential equation which the fermion field then also obeys is found to take the form [∂µ + Γµ][∂µ + Γµ]ψ(x) + (1/4)Rααψ(x) = 0. Likewise, even though the curved space Maxwell equations, viz. Fµν;ν = 0, Fµν;λ + Fλµ;ν + Fνλ;µ = 0 also possess no direct coupling to the Ricci tensor, manipulation of the Maxwell equations leads to the second order gαβFµν;α;β + Fµα Rνα−Fνα Rµα = 0, and thus to the second order equation gαβAµ;α;β − Aα;α;µ+AαRµα = 0 for the vector potential Aµ. Characteristic of all these massless field equationsthen is the emergenceof an explicit dependenceon theRiccitensor,andthusofsomenon-geodesicmotionanalogoustothatexhibitedinEq. (20)whenweeikonalize.6NowequationssuchasEq.(20)willstillobeytheequivalence principle (aswe have definedit above),since the four-accelerationwhich appearsis still onewhichcontainsthenon-tensorChristoffelsymbols.Moreover,despitethefactthatall oftheseparticularcurvedspaceequationsarenon-geodesic,nonetheless,intheflatspace limitallofthemdegenerateintotheflatspacegeodesics,withtheeikonalraystravelling on straight lines. Now in general, what is understood by covariantizingis to replace flat spacetimeexpressionsbytheircurvedspacecounterparts,withEq.(4)forinstancebeing replacedbyEq.(5).However,thisisaveryrestrictiveprocedure,since Eqs.(5) and(10) bothhavethesameflatspacelimit.Thestandardcovariantizingprescriptionwillthusfail togenerateanytermswhichexplicitlydependoncurvature(termswhichinsomecaseswe seemustbethere),andthusconstructingacurvedspaceenergy–momentumtensorpurely bycovariantizingaflatspaceoneisnotatallageneralprescription,anissueweshallreturn tobelowwhenwediscussthecurvedspaceenergy–momentumtensorindetail. 2.4. Massivefieldmotion A situationanalogoustotheabovealsoobtainsformassivefields.Specificallyforthe quantum-mechanical minimally coupled curved space massive Klein–Gordon equation, viz. 6WenotethatnoneoftheseparticularcurvedspacefieldequationsinvolvetheRiemanntensor,butonlythe Riccitensor.ThisisfortunatesinceSchwarzschildgeometrytestsofgravitywouldbesensitivetotheRiemann tensor,atensorwhich,incontrasttotheRiccitensor,doesnotvanishinaSchwarzschildgeometry. P.D.Mannheim/ProgressinParticleandNuclearPhysics56(2006)340–445 349 S;µ;µ−m(cid:1)22c2S =0, Rµνστ (cid:2)=0, (21) thesubstitutionS(x)=exp(iP(x)/(cid:1))yields P;µP;µ+m2c2 =i(cid:1)P;µ;µ. (22) Intheeikonal(orthesmall(cid:1))approximationthei(cid:1)Pµ;µ termcanbedropped,sothatthe phase P(x)isthenseentoobeythepurelyclassicalcondition gµνP;µP;ν +m2c2 =0. (23) WeimmediatelyrecognizeEq.(23)asthecovariantHamilton–Jacobiequationofclassical m(cid:8) echanics, an equation whose solution is none other than the stationary classical action µ pµdx asevaluatedbetweenrelevanten(cid:8)dpoints.Intheeikonalapproximationthenwe canthusidentifythewavephase P(x)as pµdxµ,withthephasederivative P;µ = ∂µP then being given as the particle momentum pµ = mcdxµ/dτ, a four-vectormomentum whichaccordinglyhastoobey gµνpµpν +m2c2 =0, (24) thefamiliarfullycovariantparticleenergy–momentumrelation.Withcovariantdifferenti- ationofEq.(24)immediatelyleadingtotheclassicalmassiveparticlegeodesicequation d2xλ dxµdxν pµ pλ;µ = dτ2 +Γµλν dτ dτ =0 (25) (as obtained here with the proper time dτ appropriate to massive particles and not the affine parameterq), we thusrecoverthe well knownresultthatthe center of a quantum- mechanicalwavepacketfollo(cid:8)wsthestationary(cid:8)classicaltrajectory.Further,sincewemay also reexpressthe stationary pµdxµ as−mc dτ,(cid:8)we see thatwe canalso identifythe quantum-mechanicaleikonalphaseas P(x) = −mc dτ,tothusenableustomakecon- tactwiththe I actiongiveninEq.(6).7 Thoughwehavethusmadecontactwith I ,itis T T importanttorealize thatwe wereonlyableto arriveatEq.(23)after havingstartedwith theequationofmotionofEq.(21),anequationwhoseownvalidityrequiresthatstation- aryvariationoftheKlein–Gordonactionfromwhichitisderivedhadalreadybeenmade, withonlythe stationaryclassicala(cid:8)ctionactuallybeingasolutiontothe Hamilton–Jacobi equation. The action I = −mc dτ as evaluated along the stationary classical path is T thus a part of the solution to the wave equation, i.e. the output, rather than a part of the input.8 Thus in the quantum-mechanicalcase we never need to assume as input the ex- istence of any point particle action such as I at all. Rather, we need only assume the T existence of equations such as the standard Klein–Gordon equation, with eikonalization thenpreciselyputtingparticlesontoclassicalgeodesicsjustasdesired.Toconcludethen, we see that notonly doesthe equivalenceprinciple holdfor light (eventhoughit has no 7WemakecontactwiththeIT ofEq.(6)sincewestartwiththeminimallycoupledEq.(21). 8ThuswecannotappealtoIT toputparticlesongeodesics,sincewealreadyhadtoputthemonthegeodesics whichfollowfromEq.(23)inordertogetto IT inthefirstplace.Whiletheuseofanactionsuchas IT will sufficetoobtaingeodesicmotion,aswethussee,itsuseisnotatallnecessary,withitbeingeikonalizationofthe quantum-mechanicalwaveequationwhichactuallyputsmassiveparticlesongeodesics.