Alternatives to dark matter and dark energy

106 Pages·2006·1.59 MB·English

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.