Scalar-Tensor-Vector Gravity, Galaxy Rotation Curves, and Quadrupole Gravitational Polarization

Abstract

The standard cold dark matter model is the best fit to cosmological
observations and to galaxy rotation curves. However, unless there is a direct
detection of dark matter, either in the laboratory or in astronomical
observations, one should allow for modified gravity theories such as MOND or
Scalar-Tensor-Vector Gravity as possible explanations for flatness of galaxy
rotation curves. The STVG theory due to Moffat and collaborators modifies
general relativity by the addition of a massive vector field, and the vector
field coupling constant, its mass, and the gravitational constant, are
dynamical scalar fields. The theory is shown to yield a modified acceleration
law which has a repulsive Yukawa component added to the Newtonian law of
gravitational acceleration, and which can explain the observed flatness of a
large class of galaxy rotation curves by fixing the values of two free
parameters [a mass scale and a length scale]. Here we provide a possible
insight into the success of the STVG theory, by considering the effect of
quadrupole polarization on the averaged gravitational field inside a galaxy,
due to the pull of neighboring galaxies. This effect is analogous to the
polarization induced modification of averaged electromagnetic fields in a
medium, and was studied by Szekeres in the context of propagation of
gravitational waves. The study was generalized to the case of a static
weak-field approximation by Zalaletdinov and collaborators, who showed that the
effect of quadrupole polarization is to modify Poisson's equation for the
gravitational potential to a fourth order [biharmonic] equation. We show that,
remarkably, the biharmonic equation implies a modified Newtonian acceleration
which is of the same repulsive Yukawa form as in the STVG theory, and the
corrections could in principle be large enough to explain flatness of the
rotation curves.

Full text

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arXiv:1108.2375v1 [astro-ph.GA] 11 Aug 2011
Scalar-Tensor-Vector Gravity, Galaxy Rotation
Curves, and Quadrupole Gravitational Polarization
Priti Mishra and T. P. Singh
Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
e-mail : priti@tifr.res.in; tpsingh@tifr.res.in
Abstract
The standard cold dark matter model with a cosmological constant is the best fit to cosmological
observations and to galaxy rotation curves. However, unless there is a direct detection of dark
matter, either in the laboratory or in astronomical observations, one should allow for modified
gravity theories such as MOND or Scalar-Tensor-Vector Gravity [STVG] as possible explanations
for flatness of galaxy rotation curves. The STVG theory due to Moffat and collaborators modifies
general relativity by the addition of a massive vector field, and the vector field coupling constant,
its mass, and the gravitational constant, are dynamical scalar fields. The theory is shown to yield
a modified acceleration law which has a repulsive Yukawa component added to the Newtonian law
of gravitational acceleration, and which can explain the observed flatness of a large class of galaxy
rotation curves by fixing the values of two free parameters [a mass scale and a length scale]. In
the present paper we provide a possible insight into the success of the STVG theory, by staying
within general relativity, and by considering the effect of polarization [quadrupole approximation]
on the averaged gravitational field inside a galaxy, due to the pull of neighboring galaxies. This
effect is analogous to the polarization induced modification of averaged electromagnetic fields in an
electrically charged macroscopic medium, and was studied by Szekeres in the context of propagation
of gravitational waves. The study was generalized to the case of a static weak-field approximation by
Zalaletdinov and collaborators, who showed that the effect of quadrupole polarization is to modify
Poisson’s equation for the gravitational potential to a fourth order [biharmonic] equation. We show
that, remarkably, the biharmonic equation implies a modification of Newtonian acceleration which
is precisely of the same repulsive Yukawa form as in the STVG theory, and the corrections could in
principle be large enough to explain flatness of the rotation curves.
1 Introduction
Dark matter is the most well accepted explanation for the flatness of galaxy rotation curves, and it also fits
very well into the standard ΛCDM paradigm for cosmology. However, unless there is a direct detection of
one or more popular dark matter candidates in the laboratory, or via astronomical observations, alternate
possible explanations should not be excluded. These alternatives include modified theories of gravity
such as MOND (Modified Newtonian Dynamics) and the Scalar-Tensor-Vector Gravity theory developed
by Moffat and collaborators [1], [2], [3]. It is the latter which we will be concerned with, in the present
paper. STVG is a generalization of general relativity which includes a massive vector field, and also
three dynamical scalar fields G(x), µ(x) and ω(x), which are respectively the gravitational ‘constant’,
the mass of the vector field, and its coupling to gravity. It has been shown that STVG possesses a
modified law of acceleration which can explain flat rotation curves for a large class of galaxies after fixing
two parameters, a mass scale and a length scale, to universal values. The choice of the values of these
parameters is dictated by phenomenology, and not determined by theory.
Here we report a modified acceleration law which is essentially identical to that in STVG, but which
comes from an entirely different line of reasoning. As motivation, we begin by stating an analogy
with electrodynamics. If one considers a microscopic [corpuscular] medium of molecules in an external
electromagnetic field, averaging of the Maxwell equations for the microscopic system results in the
inclusion of polarization terms coming from the induction of moments on individual molecules. This is of
course well known from electrodynamics. In an elegant analysis motivated by the study of propagation
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of gravitational waves in a material medium, Szekeres [4] developed an analogous description for general
relativity, via averaging of Bianchi identities. A modification results from quadrupole moments induced
on ‘molecules’ which in the present context are galaxies, whose ‘atoms’ are stars. Zalaletdinov and
collaborators [5], [6] extended this analysis to the static weak-field Newtonian approximation, and showed
that the effect of quadrupole polarization is to modify Poisson’s equation to a fourth order biharmonic
equation for the gravitation potential. In the present paper we obtain the modified acceleration law
implied by this modification of Poisson’s equation, and point out its significance for the observed galaxy
rotation curves.
The fact that our present analysis gives the same acceleration law as STVG is all the more intriguing
because we were not looking for an explanation of STVG. Rather, it was a subsequent realization that
the two laws are the same. Nonetheless, our presentation here emphasizes also the similarity between
the two results - the commonality in spite of apparently different origins suggests the possibility that
this may be a genuine effect, where the quadrupole induced modification can be mimicked by a field
theoretic [fifth force] modification of general relativity. It should be emphasized that at this stage our
analysis is intended to be illustrative, so as to bring out the physics involved, rather than to dwell into
precise fitting of galaxy rotation curves.
The plan of the paper is as follows. In Sections 2 and 3, we recall the acceleration law of STVG,
and the derivation of the biharmonic equation for the gravitational potential. In Section 4 we obtain the
acceleration law which results from this biharmonic equation and in Section 5 we compare it with the
result from STVG. Section 6 briefly discusses the theoretical determination of the free parameters, ad
Conclusions are given in Section 7.
2 Galaxy Rotation Curves in the STVG Theory
Here we briefly recall the form of the acceleration law, as derived in STVG theory. Details can be found
in [1], [2] and [3].
In the STVG theory, the action is given by
S = SGrav+ Sφ+ SS+ SM, (1)
where
SGrav=
1
16π
?
d4x√−g
?1
G(R + 2Λ)
?
, (2)
Sφ= −
?
d4x√−g
?
ω
?1
4BµνBµν+ V (φ)
??
, (3)
and
SS=
?
d4x√−g
?1
G3
?1
2gµν∇µG∇νG − V (G)
?
+1
G
?1
2gµν∇µω∇νω − V (ω)
?
+
1
µ2G
?1
2gµν∇µµ∇νµ − V (µ)
??
. (4)
Here V (φ) is the potential for the massive vector field φµhaving mass µ and coupling ω; V (G),V (ω)
and V (µ) are the potentials associated with the three scalar fields G(x),ω(x) and µ(x), respectively, and
Bµν= ∂µφν− ∂νφµ. (5)
By assuming that Λ = 0, V (φ) = 0, ω ad µ to be constants, and by considering the motion of a test
particle coupled to gravity and to the vector field, it was shown that the law of acceleration is given by
a(r) = −G∞M
r2
+ Kexp(−r/r0)
r2
?
1 +r
r0
?
, (6)
where G∞is defined to be the effective gravitational constant at infinity
G∞= G
?
1 +
?
M0
M
?
(7)
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and r0= 1/µ. Here, M0denotes a parameter that vanishes when ω = 0 and G is Newton’s gravitational
constant. The constant K is assumed to equal
K = G
?
MM0. (8)
The choice of K, which determines the strength of the coupling of Bµνto matter and the magnitude of
the Yukawa force modification of weak Newtonian gravity, is based on phenomenology and M0is a free
parameter of the theory.
By using (7), one can rewrite the acceleration in the form
a(r) = −GM
r2
?
1 +
?
M0
M
?
1 − exp(−r/r0)
?
1 +r
r0
???
. (9)
It is assumed [and justified by further considerations] that one can generalize this to the case of a mass
distribution by replacing the factor GM/r2in (9) by GM(r)/r2. The rotational velocity of a star vcis
obtained from v2
c(r)/r = a(r) and is given by
vc=
?
GM(r)
r
?
1 +
?
M0
M
?
1 − exp(−r/r0)
?
1 +r
r0
???1/2
. (10)
A good fit to a large number of galaxies has been achieved with the parameters:
M0= 9.60 × 1011M⊙,r0= 13.92kpc = 4.30 × 1022cm. (11)
In the fitting of the galaxy rotation curves for both LSB and HSB galaxies, using photometric data to
determine the mass distribution M(r), only the mass-to-light ratio ?M/L? is employed, once the values
of M0and r0are fixed universally for all LSB and HSB galaxies. Dwarf galaxies are also fitted with the
parameters
M0= 2.40 × 1011M⊙,
We will now see how a modified acceleration law of the form Eqn. (9) is obtained without modifying
general relativity, as a result of considering polarization effects on the averaged gravitational field. As
shown in the work of Szekeres and Zalaletdinov et al., which we recall in this next section, such a
consideration leads to the Poisson equation being modified into a fourth order biharmonic equation. In
the subsequent section, we derive the modified acceleration law by solving the biharmonic equation.
r0= 6.96kpc = 2.15 × 1022cm. (12)
3 Quadrupole gravitational polarization and the biharmonic equa-
tion for the potential
This section summarizes the work of Szekeres [4] and Zalaletdinov et al. [5], [6] leading to the derivation
of the biharmonic equation.
Thinking of galaxies as ‘molecules’ made of atoms [the stars], one would like to analyze how the
averaged gravitational field inside a galaxy, is modified by the polarization of the molecules, due to
the external pull of other galaxies. Indeed one has at the back of the mind the polarization induced
modification of electromagnetic fields in a an electrically charged medium.
In vacuum, if the Riemann tensor Rµνρσis regarded as being the gravitational analog of the Maxwell
tensor Fµν, the Bianchi identities
Rµν[ρσ;τ]= 0 ⇔ R;σ
show striking resemblance to the “source-free”Maxwell equations
µνρσ= 0 (13)
F[µν,ρ]= 0, F,ρ
µν= 0. (14)
In the non-vacuum case, the Weyl tensor
Cµνρσ= Rµνρσ− gµ[ρRσ]ν− Rµ[ρgσ]ν+1
3Rgµ[ρgν]σ
(15)
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is the gravitational analog of the Maxwell tensor. The Bianchi identities now take the form,
C;σ
µνρσ= κJµνρ, (16)
where
Jµνρ= J[µν]ρ= −(Tρ[µ;ν]−1
3gρ[µT,ν]). (17)
Eqn. (16) is comparable to the Maxwell equations
F,ν
µν= µ0jµ
(18)
if Cµνρσ is interpreted as representing the free gravitational field and Jµνρis a kind of matter current.
And the corresponding conservation equation is:
J;ρ
µνρ= 0. (19)
The gravitationalfield created by a number of particles represented by a microscopic energy-momentum
tensor tα(micro)
β
is given by Einstein’s equations
gαǫrǫβ−1
2δα
βgµνrµν= −κtα(micro)
β
(20)
where κ = 8πG/c4. The Einstein equations for the microscopic distribution of gravitational molecules
t(molecule)µν(x) = c−1ΣaΣima
i
?
midτa
dτi
dzµ
dτa
i
dzν
dτaδ4[x − yµi
a(τa) − si(τa)]dτa
(21)
are then :
gαǫrǫβ−1
2δα
βgµνrµν= −κtα(molec)
β
. (22)
Averagingthe left-hand side of the Einstein equations using Kauffman’s method of molecular moments
and by taking into account the expression [4] for the tensor of gravitational quadrupole polarization Qµνρσ
in terms of the covariant gravitational quadrupole moment qµνρσ
a
Qµνρσ= c−1?Σa
?
qµνρσ
a
δ4(x − ya)dτa? (23)
brings the averaged Einstein equations in the form:
R(0)
µν−1
2g(0)
µνR(0)= −κ(T(free)
µν
+ T(GW)
µν
+1
2c2Q;ρσ
µρνσ), (24)
where T(GW)
momentum tensor of molecules
µν
is Isaacson’s energy-momentum tensor of gravitational waves and T(free)
µν
is the energy-
T(free)
µν
(x) = c−1Σa
?
madyµ
a
dτi
dyν
dτiδ4[x − yµ
a
a(τa)]dτa.(25)
A model of a static weak-field averagedgravitating medium with quadrupole gravitationalpolarization
is developed next. The model is based on the following three assumptions.
1. The energy-momentum tensor of the molecules T(free)
Newtonian approximation to hold :
µν
satisfies the standard conditions for the
T(free)
00
≫ T(free)
ij
,T
(free)
00
≫ T(free)
0i
,T(free)
00
= T(free)
µν
uµuν= µc2. (26)
2. The macroscopic metric tensor g(0)
µν is static,
∂g(0)
∂t
µν
= 0, (27)
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3. The averaged macroscopic metric obeys the weak field approximation : tensor g(0)
µν,
g(0)
µν= ηµν+ ehµν,(28)
In general relativity the conditions (26)-(28) for a microscopic energy-momentum tensor tα(micro)
?
equations (20) and to result in the Poisson equation for the Newtonian gravitational potential φ = φ(xa)
β
and
for the metric tensor g00= −1 +2φ
c2
?
, g0i= 0, gij = δij lead to the Newtonian limit of the Einstein
∇2φ = 4πGµ.(29)
We are interested in the Newtonian limit of the macroscopic gravity equations (24 ) under conditions
(26)-(28) for the macroscopic tensor g(0)
µν
g(0)
00= −
?
1 +2φ
c2
?
,g(0)
11= 1,g(0)
22= 1,g(0)
33= 1,g(0)
µν= 0,µ ?= ν,(30)
which leads to a generalization of the Poisson equation which incorporates the effect of gravitational
quadrupole polarization.
For the case of the static weak-field macroscopic medium Isaacson’s energy-momentum tensor of
gravitational waves T(GW)
µν
vanishes
T(GW)
µν
= 0. (31)
The matter quadrupole polarization tensor is modelled by assuming, in analogy with the electrostatic
case, that
Qi0j0= ǫgCi0j0
(32)
and the gravitational dielectric constant ǫgis modelled by [4] as
ǫg=1
4
mA2c2
ω2
0
N. (33)
Here, A is the typical linear dimension of a molecule, m is the typical mass of a molecule, ω0is a typical
frequency of harmonically oscillating atoms in the molecule and N is the number density of the molecules
in the medium.
For the metric
g(0)
00= −
?
1 +2φ
c2
?
,g(0)
11= 1,g(0)
22= 1,g(0)
33= 1,g(0)
µν= 0 (for µ ?= ν), (34)
we get
R00
=−1
1
c2
gµνRµν
g00R00+ giiRii
(−1 + 2φ/c2)(−1
2
c2∇2φ.
c2∇2φ,
∂2φ
∂x2
i
(35)
Rii
=,(i is not summed over here.), (36)
R= (37)
= (38)
=
c2∇2φ) + (1
c2
∂2φ
∂x2
i
)(i is summed over here.)(39)
=
(40)
Terms only up to first order in φ have been considered. The last term in equation(24) is Qµρνσ. The
only non-zero components of this tensor are:
Qi0j0
=ǫgRi0j0= −ǫg
−1
c2
∂2φ
∂xi∂xj
(41)
⇒ Q;ij
i0jo
=
c2∇4φ.(42)
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