Violation of the Equivalence Principle in Modified Theories of Gravity

0.12 MB·English

by  Gonzalo J. Olmo

Violation of the Equivalence Principle in Modified Theories of Gravity
Gonzalo J. Olmo
Physics Department, University of Wisconsin-Milwaukee, P.O.Box 413, Milwaukee, WI 53201, USA
(Dated: December 31st, 2006)
Westudymodifiedtheoriesofgravityofthef(R)typein Palatiniformalism. Forageneric f(R)
lagrangian, weshowthatthemetriccanbesolved astheproductofascalarfunction timesarank-
two tensor (or auxiliary metric). The scalar function is sensitive to the local energy-momentum
density. The auxiliary metric satisfies a set of equations very similar to Einstein’s equations and,
for weak sources, it can be approximated by the Minkowski metric. According to this, the metric
coupled to the matter strongly departs from the Minkowskian one in the neighborhood of any
microscopic physical system. As a consequence, new gravitationally-induced interactions arise and
leadtoobservableeffectsatmicroscopicandmacroscopicscales. Inparticular,testbodytrajectories
experience self-accelerations which depend on the internal structure and composition of the body.
ThesefactsmakeveryunlikelytheviabilityofPalatinif(R)modelsdesignedtochangethelate-time
7 cosmic evolution.
0
PACSnumbers: 98.80.Es,04.50.+h,04.25.Nx
0
2

n Modified theories of gravity have been object of in- a tiny Λ µ2. One is then tempted to conclude that
a ∼
tense study inrecentyearsdue to their ability topredict this model, and any other model with a small Λ, is
J
cosmic acceleration [1] without the need for sources of compatible withsolarsystemobservations[4, 6,7]since,
9
dark energy. In this sense, special attention have re- for sufficiently smallΛ, (1) is virtually undistinguishable

ceivedthose theories of the f(R) type, where f is a non- from the Schwarzschild solution of GR, which passes
2
v linear function of the scalar curvature R, with nonlinear all observational tests. However, though the vacuum
2 terms relevant at low (cosmic) curvatures. For a given solutions are very well known, the equations in the
0 lagrangianf(R), such theories can be formulated in two presence of matter have not been studied in detail yet.
0 inequivalent ways, namely, in metric and in Palatini for- The relation of the integration constant M⊙ with the
2
malism. In the former case, the connection is defined mattersources,forinstance,isstillunknown. Andthisis
1
as the usual Levi-Civita connection. In the latter, the an important aspect, since Palatini theories only depart
6
connectionis regardedasindependentofthe metric and, from GR in the presence of sources, as is well known in
0
/ therefore,mustbe determined by solvingits correspond- cosmology. Some results in this direction were found in
c
ingfieldequations. Accordingtothis, themetric formal- the study of the Newtonian and/or post-Newtonian lim-
q
ism leads to a system of fourth-order partial differential itsin[8,9]. Allthoseworksagree,withsmalldifferences,
-
r equations for the metric, whereas the Palatiniformalism in that in addition to the usual Newtonian potential
g
leads to second-order equations. Only for the linear la- M/r the Newtonian limit has a density-dependent term.
:
v grangian,f(R)=R 2Λ,the twoformalismsleadtothe Such term was considered as irrelevant and negligible
Xi same equations of m−otion, which are those of General in some of those works. However, it was remarked in
Relativity (GR). [9] that it could be potentially dangerous. The reason
r
a Though much has been written about the properties of being that it generates accelerations which depend on
f(R)theoriesin cosmologicalapplications(see [2, 3] and the gradient of the local matter density and which
[4,5]formetricandPalatiniformalism,respectively),we do not decay with distance. Furthermore, when an
do not yethavea clear understandingof their properties atomic/microscopic description of matter is considered,
in other regimes. Let us focus on Palatini theories. It is that density-dependent term seems to break the pertur-
easy to verify (see below for details) that, for arbitrary bative approach itself, which is a disturbing property.
f(R), Palatini theories have the same type of vacuum These facts suggest that further work is needed in or-
equationsas GeneralRelativitywith a cosmologicalcon- dertofullyunderstandthedynamicsofPalatinitheories.
stant. Inparticular,for a sphericallysymmetric star like
the sun, the metric outside the star can be written as a
Schwarzschild-de Sitter solution In this work we study and solve the equations of mo-
tion of Palatini f(R) theories in the presence of matter
dr2 in some cases of interest, namely, spherically symmetric
ds2 =g dxµdxν = A(r)dt2+ +r2dΩ2 (1)
SdS µν − A(r) distributions andalso for veryweak sources without any
particular symmetry. From our analysis it follows that
with A(r)=1 2GM⊙/r Λr2/3, where Λ, G and M⊙ strong gravitational effects arise at microscopic scales
− −
represent a cosmological constant, Newton’s constant which are in conflict with the theoretical foundations
and the mass of the star respectively. The well-known of these theories and with the experimental evidence
model f(R) = R µ4/R of Carroll et al. [2], like any supporting the Einstein equivalence principle and our
−
other f(R) model, admits such solutions and leads to knowledge of elementary-particle physics. These facts
2
strongly suggest that f(R) models designed to modify make our notation clearer, since t and g are con-
µν µν
the gravitationalinteractionatlow curvaturesshould be formally related, it follows that (T) = gµνR (Γ) and
µν
R
ruled out. R(g)=gµνR (g) are related by
µν
3 3
Letus beginbydefiningthe actionofPalatinitheories (T)=R(g)+ ∂ f′(T)∂λf′(T) (cid:3)f′(T)
R 2f′(T) λ − f′(T)
1 (9)
S[g,Γ,ψ ]= d4x√ gf(R)+S [g ,ψ ] (2)
m 2κ2 Z − m µν m where,recall,f′(T) f′( (T))isafunctionofT,which
meansthat the f′(T≡)termRs in(8) actas additionalmat-
Here f(R) is a function of R≡gµνRµν(Γ), with Rµν(Γ) tersources. Thematterterms∂µf′(T)and(cid:3)f′(T)make
given by difficulttheanalisysof(8)toobtaing . Inthissense,we
µν
find very useful to shift the problem and find solutions
R (Γ)= ∂ Γλ +∂ Γλ +Γλ Γρ Γλ Γρ (3)
µν − µ λν λ µν µρ νλ− νρ µλ for the conformally related metric g¯µν = φgµν, where
φ f′(T)/f′(0) is a dimensionless factor which becomes
where Γλ is the connection. The matter action S de- ≡
µν m unityinvacuum,T =0. Theequationsofmotionforg¯µν
pends on the matter fields ψ , the metric g and its
m µν boil down to
derivatives, but not on Γλ . Note that Γλ is regarded
µν µν
as a field of the gravity sector and, therefore, cannot ap- κ2 V(T)
Gν(g¯)= Tν δν (10)
pearinthemattersector,sinceonlythemetricisallowed µ f′φ2(T)(cid:20) µ − 2κ2 µ(cid:21)
0
tocoupletomatteraccordingtothepostulatesofmetric
theories of gravity [11]. Varying (2) with respect to the where f′ f′(0) and we have used the shorthand no-
metric we obtain tation V0(T≡) f′(T) f( ). The above equations
≡ R − R
of motion for g¯ are considerably simpler than those
1 µν
f′(R)Rµν(Γ) f(R)gµν =κ2Tµν (4) for gµν, which makes our task easier. In particular,
− 2 for spherically symmetric systems we can use the ansatz
where f′(R) df/dR. Note that the trace of (4) ds2 =gµνdxµdxν =φ−1g¯µνdxµdxν with
≡
f′(R)R 2f(R)=κ2T, (5) ds2 = 1 A(r)e2Φ(r)dt2+ 1 dr2+r2dΩ2 (11)
− φ(cid:20)− A(r) (cid:21)
implies analgebraicrelationbetweenR andthe trace T.
which leads to
The solution to this algebraic equation will be denoted
mbyusRt v=anRis(hTi)n.dTepheenvdaernitaltyioonfo(4f)(2a)ndwigtihversespect to Γλµν 2dΦ = κ2 Trr−Ttt (12)
r dr f′φ2 (cid:18) A (cid:19)
0
1 1 1 d(r[1 A]) κ2 V(T)
√ g δρf′gµν δµf′gρν δνf′gµρ =0 (6) − = Tt (13)
∇ρ(cid:20) − (cid:18) λ − 2 λ − 2 λ (cid:19)(cid:21) −r2 dr f′φ2 (cid:18) t − 2κ2 (cid:19)
0
where f′ f′( (T)) is also a function of the matter Defining now A(r) = 1 2GM(r)/r in (13), we can
terms. Usi≡ng anRauxiliary tensor t f′g , (6) can be rewrite M(r) and Φ(r) as−
µν µν
≡
readilysolved[12]. The solutionstates the compatibility
betweenΓλµν andthemetrictµν. Inotherwords,Γλµν can M(r) = κ2 rdx x2 Ttt−V(T)/(2κ2) (14)
be written as the Levi-Civita connection of t −2Gf′ Z (cid:20) φ2 (cid:21)
µν 0 0
κ2 r Tr Tt
tλρ Φ(r) = dx x r − t (15)
Γλ = (∂ t +∂ t ∂ t ) (7) 2f′ Z (cid:20) φ2A (cid:21)
µν 2 µ ρν ν ρµ− ρ µν 0 0
Outsideofthesources,(T =0,φ=1),theaboveequa-
Inserting this solution for Γλ , written in terms of g µν
µν µν tions can be readily integrated leading to
and f′( (T)), in (4) we obtain
R
V(0)r3
R (g) 1g R(g) = κ2T Rf′−fg M(r) = M⊙+ 12Gf′ (16)
µν − 2 µν f′ µν − 2f′ µν − 0
Φ(r) = Φ (17)
3 1 0
∂ f′∂ f′ g (∂f′)2 +
− 2(f′)2 (cid:20) µ ν − 2 µν (cid:21) and we recover the Schwarzschild-de Sitter solution (1),
+ 1 [ f′ g (cid:3)f′] (8) with 2Λ=V(0)/f0′ and Φ0 =constant can be eliminated
f′ ∇µ∇ν − µν by a convenient redefinition of the time coordinate.
Note that the constant M⊙ must be fixed by matching
where R (g) and R(g) are computed in terms of the the interior and exterior solutions. Now, since the
µν
Levi-Civita connection of the metric g , i.e., they rep- interior solutions depend on the details of Tν and the
µν µ
resent the usual Ricci tensor and scalar curvature. To particular f(R) lagrangianconsidered throughφ(T) and
3
V(T), the value of M⊙ (and also Φ0) will depend on sity dependent) cosmological constant. For lagrangians
the internal structure and composition of the object ofthe formf(R)=R+ǫg(R),withǫsomesmallparam-
(for a recent example see [10]). In other words, a given eter,we findthat V(T) ǫ. Thus V(T)is verysmall(of
amount of matter-energy, drr2Tt, can lead to different cosmological order) and∼can be neglected in the region
t
external gravitational fieldRs. This contrasts with Gen- of interest. Like in General Relativity, the contribution
eral Relativity, where the external field generated by a of Tν is also negligible for weak sources. Therefore, (10)
µ
spherically symmetric system only depends on the total becomes Gν(g¯) 0 and leads to g¯ η . The metric
µ ≈ µν ≈ µν
matter-energy and not on the structure or composition generated by our test body (atom) is then
(in GR φ = 1,V(T) = 0). This behavior was already
g (x)=φ−1[T(x)]η (18)
reported in [9] within the post-Newtonian regime. µν µν
Note that the factor φ−1[T(x)] is due to the derivatives
We have just seen that outside of the sources (11) of f′(T) on the right hand side of (8) which, unlike T
µν
turns into (1). Let us consider what happens if we and V(T), cannot be neglected. We thus see that the
place a test body with m M⊙ within the external metricintheneighborhoodofanysmallphysicalsystem,
≪
field of M⊙. This test body could be anything from a describedbyacertainprobabilitydensitydistribution,is
laboratory-sized object to a single atom. The metric not in general the Minkowski metric. This result is very
nearby this body is given by (1). However, the metric important and leads to a number of inconsistencies and
inside of the test body, where T = 0, is given by (11) pathological effects, which must be added to those dis-
6
with φ(T) = 1. In the 1/R model, for instance, we cussed above.
find φ = 16 − 2[1+√11+12/σ2], where σ ≡ −T/Tc and Let us first focus on the theoretical inconsistencies. The
T = µ2/κ2 = ρ c with ρ 10−26 g/cm3. When T action (2) was constructed according to the postulates
c µ µ ∼ of metric theories of gravity, namely, i) spacetime is en-
drops below T (outside the body) we have φ 1. For
T above T (icnside the body) we get φ →3/4. We dowedwithasymmetricmetricgµν,ii)theworldlinesof
c → test bodies are geodesics of that metric, and iii) in local
then see that the line element (11) changes suddenly
freely falling frames the nongravitationallaws of physics
from ds2 = ds2 to ds2 = 4ds2 when going from
SdS 3 SdS are those of special relativity. These postulates tell us
the outside to the inside of the test body. Actually, this
how to introduce matter in a theory of gravity based on
happens in general when going from the outside to the
geometry: we must take the Minkowskian theory and
inside of atoms, which leads to strong oscillations in the
change η by g and ∂ by , with g =0. This
metric at microscopic scales both within the test body µν µν µ ∇µ ∇α µν
prescriptionshould guaranteethat the non-gravitational
and within the central object characterized by M⊙.
laws of physics of Minkowski space were recoveredin lo-
This unusual and highly non-perturbative behavior,
cal freely falling frames. In our case, however, even in
which has no precedent in the literature on alternative
local frames in which external gravitational fields have
theories of gravity, is due to the dependence of the
been screened, we do not recover the Minkowskian met-
metric on the factor φ(T), which is a function of the
ric, which violates postulate iii).
local energy-momentumdensity T. Any theory in which
On the other hand, since these postulates assume that
the lagrangian f(R) be sensitive to some low curvature
the Einstein Equivalence Principle (EEP) is valid, it fol-
scale will lead to a φ(T) characterized by two limiting
lows that Palatinif(R)theories must be in conflict with
values, φ 1 for small T and φ φ for large T, as
→ → c the experimental evidence supporting the EEP (see [11]
compared to the characteristic scale T . Therefore, the
c for details). Let us thus focus on the weak equivalence
strong jumps in the metric discussed here for the 1/R
principle (also known as universality of free fall, and a
model occur in all f(R) models aimed at changing the
key piece of the EEP), which states that the trajectory
late-time cosmic expansion.
of a freely falling test body in a local inertial frame is
unaccelerated and independent of the internal structure
To see in more detail the effects of the factor φ(T),
and composition of the body. From (18) we see that the
we will consider a portion of space-time containing the
geodesic equation
above test body. Assume that we can take coordinates
inwhichthelineelement(1)becomesMinkowskianaway d2ξµ dξαdξβ
from the body, i.e., ds2 ηµνdξµdξν. This choice of co- dτ2 +Cαµβ(g) dτ dτ =0, (19)
≈
ordinates eliminates the externalfield generatedby M⊙. where Cµ (g)= gµρ (∂ g +∂ g ∂ g ), leads to
We could also have assumed no external field for sim- αβ 2 α ρβ β ρα− ρ αβ
plicity. The metric g in the region close to our test
µν dvi 1
body can then be computed using again the decomposi- = (1 ~v 2) vi∂ lnφ+∂ilnφ (20)
t
dt 2 −| |
tion g¯µν = φgµν. For a sufficiently light test body, such (cid:2) (cid:3)
as an atom, the metric g¯ can be well approximated by wherevi =dξi/dt. Thisequationshowsthatif∂ φ(T)=
µν µ
6
η . This is so because g¯ is governedby (10), which is 0,thetestbodywillfeelself-accelerationswhich,inaddi-
µν µν
dynamically very similar to General Relativity. In fact, tion, will depend on its own internal structure and com-
the term κ2/(f′φ2) can be seen as an effective (density position. Now, the deep dependence of φ(T) on the mi-
0
dependent) Newton’s constant, and V(T) as some (den- croscopic structure does not allow us to use an averaged
4
descriptionofmatter(perfectfluid)tostudythebehavior otherhand,thefactthatφ(T),inmodelsrelevantforthe
of(20). Oneshoulduseamicroscopicdescriptionforthe late-time cosmology, is sensitive to low energy-density
matter sources, as pointed out in [13]. Let us then con- scales, makes it difficult (or perhaps impossible) an
sider the action of a free Dirac field in Minkowski space expansion of (22) in powers of T to quantitatively study
and use the EEP to construct the corresponding curved the predictions of particular models. This means that
space theory to be put in (2). The result is the non-gravitationallaws ofphysicsofMinkowskispace
are strongly modified and might even lose their predic-
S = d4x√ g ψ¯γµ ψ+imψ¯ψ , (21) tive power. This problem also affects the gravitational
m µ
Z − ∇ laws, since the metric is sensitive to the microscopic
(cid:2) (cid:3)
structure of T , and we must solve the microscopic
µν
where ψ = (∂ Γ )ψ, Γ is the spin connection,
µ µ µ µ matter field equations to get T . This fact, by the
and γ∇µ,γν = 2gµ−ν are the curved-space counterparts µν
way, invalidates all cosmological applications of Palatini
of D{irac’s c}onstant matrices γa,γb = 2ηab. Though
theories considered so far [4, 5], in which an averaged
{ }
the Minkowskian theory should be recovered in a freely
description of matter (perfect fluid) has been assumed.
falling frame, where external gravitational fields have
Insummary,wehaveshownthatthegravitationalsector
been screened, from (18) and (21) we find
inPalatinif(R)theorieshasdramaticeffectsonthemat-
ter sector. When the gravitational interaction is turned
S = d4ξφ−2(T) (22) on, the Minkowskian laws of physics are completely lost
m
Z ×
duetothesensitivityofthespacetimemetrictothelocal
φ1/2(T)ψ¯γa∂ ψ 3∂ φ1/2(T)ψ¯γaψ+imψ¯ψ energy-momentum densities. This fact, besides being a
× (cid:20) a − 2 a (cid:21) theoretical inconsistency, implies observable effects such
as violations of the EEP and new interactions among
where T depends on all the sources present, including ψ elementary particles. Furthermore, if the theory is sen-
itself, and ∂aφ1/2 is due to Γµ [16]. It is quite evident sitive to very low density scales, it might be impossible
that (22) is not the Minkowski space theory we started to usethe theoryto extractquantitativepredictions(see
with and that new gravitationally-induced (first-order) [15]foranexample). AllthissuggeststhatPalatinif(R)
interactions arise [17] via the various φ(T) terms in theories, at least in their current form, should be ruled
(22). This modification of the microscopic interactions out. Only those models in which φ(T) is only sensitive
is what eventually leads to the macroscopic violations of to very high energy densities, above the currently acces-
the universality of free fall observed in (20). Note that sibleexperimentallimits,mighthaveachancetosurvive.
this effect is different in nature from the Nordtvedt effect
[14], which is due to the coupling of the gravitational
self-energy of a massive body to external gravitational This work has been supported by NSF grants PHY-
fields (second order gravity-gravity interaction). On the 0071044and PHY-0503366.
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5
[17] This result is independent of the choice of conformal
frame for thegravitational action