Parametrized Post-Newtonian Expansion of Chern-Simons Gravity
Stephon Alexander1 and Nicolás Yunes1
Center for Gravitational Wave Physics, Institute for Gravitational Physics and Geometry and Department of Physics,
The Pennsylvania State University, University Park, PA 16802, USA
(Dated: November 4, 2018)
We investigate the weak-field, post-Newtonian expansion to the solution of the field equations
in Chern-Simons gravity with a perfect fluid source. In particular, we study the mapping of this
solution to the parameterized post-Newtonian formalism to 1 PN order in the metric. We find
that the PPN parameters of Chern-Simons gravity are identical to those of general relativity, with
the exception of the inclusion of a new term that is proportional to the Chern-Simons coupling
parameter and the curl of the PPN vector potentials. We also find that the new term is naturally
enhanced by the non-linearity of spacetime and we provide a physical interpretation for it. By
mapping this correction to the gravito-electro-magnetic framework, we study the corrections that
this new term introduces to the acceleration of point particles and the frame-dragging effect in
gyroscopic precession. We find that the Chern-Simons correction to these classical predictions could
be used by current and future experiments to place bounds on intrinsic parameters of Chern-Simons
gravity and, thus, string theory.
PACS numbers: 11.25.Wx, 95.55.Ym, 04.60.-m, 04.80.Cc
I. INTRODUCTION
Tests of alternative theories of gravity that modify gen-
eral relativity (GR) at a fundamental level are essential to
the advancement of physics. One formalism that has had
incredible success in this task is the parameterized post-
Newtonian (PPN) framework [1, 2, 3, 4, 5, 6]. In this
formalism, the metric of the alternative theory is solved
for in the weak-field limit and its deviations from GR are
expressed in terms of PPN parameters. Once a metric
has been obtained, one can calculate predictions of the
alternative theory, such as light deflection and the perihe-
lion shift of Mercury, which shall depend on these PPN
parameters. Therefore, experimental measurements of
such physical effects directly lead to constraints on the
parameters of the alternative theory. This framework,
together with the relevant experiments, have already
been successfully employed to constrain scalar-tensor the-
ories (Brans-Dicke, Bekenstein) [7], vector-tensor the-
ories (Will-Nordtvedt [8], Hellings-Nordtvedt [9]), bi-
metric theories (Rosen [10, 11]) and stratified theories
(Ni [12]) (see [13] for definitions and an updated review.)
Only recently has this framework been used to study
quantum gravitational and string-theoretical inspired
ideas. On the string theoretical side, Kalyana [14] investi-
gated the PPN parameters associated with the graviton-
dilaton system in low-energy string theory. More re-
cently, Ivashchuk, et. al. [15] studied PPN parameters
in the context of general black holes and p-brane spher-
ically symmetric solutions, while Bezerra, et. al. [16]
considered domain wall spacetimes for low energy effec-
tive string theories and derived the corresponding PPN
parameters for the metric of a wall. On the quan-
tum gravitational side, Gleiser and Kozameh [17] and
more recently Fan, et. al. [18] studied the possibility
of testing gravitational birefringence induced by quan-
tum gravity, which was proposed by Amelino-Camelia,
el. al. [19] and Gambini and Pullin [20]. Other non-
PPN proposals have been also put forth to test quan-
tum gravity, for example through gravitational waves
[21, 22, 23, 24, 25, 26, 27, 28], but we shall not discuss
those tests here.
Chern-Simons (CS) gravity [29, 30] is one such ex-
tension of GR, where the gravitational action is mod-
ified by the addition of a parity-violating term. This
extension is promising because it is required by all 4-
dimensional compactifications of string theory [31] for
mathematical consistency because it cancels the Green-
Schwarz anomaly [32]. CS gravity, however, is not unique
to string theory and in fact has its roots in the standard
model, where it arises as a gravitational anomaly pro-
vided that there are more flavours of left handed leptons
than right handed ones. Moreover the CS extension to
GR can arise via the embedding of the three dimensional
Chern-Simons topological current into a 4D space-time
manifold, decsribed by Jackiw and Pi [30]
Chern-Simons gravity has been recently studied in
the cosmological context. In particular, this framework
was used to shed light on the anisotropies of the cos-
mic microwave background (CMB) [33, 34, 35] and the
leptogenesis problem [34, 36, 37]. Parity violation has
also been shown to produce birefringent gravitational
waves [28, 29], where different polarizations modes ac-
quire varying amplitudes. These modes obey different
propagation equations because the imaginary sector of
the classical dispersion relation is CS corrected. Different
from [20], in CS birefringence the velocity of the gravita-
tional wave remains that of light.
In this paper we study CS gravity in the PPN frame-
work, extending the analysis of [38] and providing some
missing details. In particular, we shall consider the ef-
fect of the CS correction to the gravitational field of,
for instance, a pulsar, a binary system or a star in the
weak-field limit. These corrections are obtained by solv-
http://arxiv.org/abs/0704.0299v1
ing the modified field equations in the weak-field limit for
post-Newtonian (PN) sources, defined as those that are
weakly-gravitating and slowly-moving [39]. Such an ex-
pansion requires the calculation of the Ricci and Cotton
tensors to second order in the metric perturbation. We
then find that CS gravity leads to the same gravitational
field as that of classical GR and, thus, the same PPN
parameters, except for the inclusion of a new term in the
vectorial sector of the metric, namely
0i = 2ḟ (∇× V )i , (1)
where ḟ acts as a coupling parameter of CS theory and
Vi is a PPN potential. We also show that this solution
can be alternatively obtained by finding a formal solu-
tion to the modified field equations and performing a PN
expansion, as is done in PN theory. The full solution
is further shown to satisfy the additional CS constraint,
which leads to equations of motion given only by the di-
vergence of the stress-energy tensor.
The CS correction to the metric found here leads to
an interesting interpretation of CS gravity and forces us
to consider a new type of coupling. The interpretation
consists of thinking of the field that sources the CS cor-
rection as a fluid that permeates all of spacetime. Then
the CS correction in the metric is due to the “dragging” of
such a fluid by the motion of the source. Until now, cou-
plings of the CS correction to the angular momentum of
the source had been neglected by the string theory com-
munity. Similarly, curl-type terms had also been consid-
ered unnecessary in the traditional PPN framework, since
previous alternative gravity theories had not required it.
As we shall show, in CS gravity and thus in string the-
ory, such a coupling is naturally occurring. Therefore, a
proper PPN mapping requires the introduction of a new
curl-type term with a corresponding new PPN parameter
of the type of Eq. (1).
A modification to the gravitational field leads natu-
rally to corrections of the standard predictions of GR.
In order to illustrate such a correction, we consider the
CS term in the gravito-electro-magnetic analogy [40, 41],
where we find that the CS correction accounts for a mod-
ification of gravitomagnetism. Furthermore, we calculate
the modification to the acceleration of point particles and
the frame dragging effect in the precession of gyroscopes.
We find that these corrections are given by
δai = −
c2 r2
δΩi = − ḟ
c3 r2
ni − v
, (2)
where m and v are the mass and velocities of the source,
while r is the distance to the source and ni = xi/r is a
unit vector, with · and × the flat-space scalar and cross
products. Both corrections are found to be naturally en-
hanced in regions of high spacetime curvature. We then
conclude that experiments that measure the gravitomag-
netic sector of the metric either in the weak-field (such as
Gravity Probe B [42]) and particularly in the non-linear
regime, will lead to a direct constraint on the CS cou-
pling parameter ḟ . In this paper we develop the details
of how to calculate these corrections, while the specifics
of how to actually impose a constraint, which depend
on the experimental setup, are beyond the scope of this
paper.
The remainder of this paper deals with the details of
the calculations discussed in the previous paragraphs.
We have divided the paper as follows: Sec. II describes
the basics of the PPN framework; Sec. III discusses CS
modified gravity, the modified field equations and com-
putes a formal solution; Sec. IV expands the field equa-
tions to second order in the metric perturbation; Sec. V
iteratively solves the field equations in the PN approx-
imation and finds the PPN parameters of CS gravity;
Sec. VI discusses the correction to the acceleration of
point particles and the frame dragging effect; Sec. VII
concludes and points to future research.
The conventions that we use throughout this work are
the following: Greek letters represent spacetime indices,
while Latin letters stand for spatial indices only; semi-
colons stand for covariant derivatives, while colons stand
for partial derivatives; overhead dots stand for deriva-
tives with respects to time. We denote uncontrolled re-
mainders with the symbol O(A), which stands for terms
of order A. We also use the Einstein summation con-
vention unless otherwise specified. Finally, we use ge-
ometrized units, where G = c = 1, and the metric signa-
ture (−,+,+,+).
II. THE ABC OF PPN
In this section we summarize the basics of the PPN
framework, following [6]. This framework was first
developed by Eddington, Robertson and Schiff [1, 6],
but it came to maturity through the seminal papers of
Nordtvedt and Will [2, 3, 4, 5]. In this section, we de-
scribe the latter formulation, since it is the most widely
used in experimental tests of gravitational theories.
The goal of the PPN formalism is to allow for com-
parisons of different metric theories of gravity with each
other and with experiment. Such comparisons become
manageable through a slow-motion, weak-field expansion
of the metric and the equations of motion, the so-called
PN expansion. When such an expansion is carried out
to sufficiently high but finite order, the resultant solu-
tion is an accurate approximation to the exact solution
in most of the spacetime. This approximation, however,
does break down for systems that are not slowly-moving,
such as merging binary systems, or weakly gravitating,
such as near the apparent horizons of black hole binaries.
Nonetheless, as far as solar system tests are concerned,
the PN expansion is not only valid but also highly accu-
rate.
The PPN framework employs an order counting-
scheme that is similar to that used in multiple-scale anal-
ysis [43, 44, 45, 46]. The symbol O(A) stands for terms of
order ǫA, where ǫ ≪ 1 is a PN expansion parameter. For
convenience, it is customary to associate this parameter
with the orbital velocity of the system v/c = O(1), which
embodies the slow-motion approximation. By the Virial
theorem, this velocity is related to the Newtonian poten-
tial U via U ∼ v2, which then implies that U = O(2) and
embodies the weak-gravity approximation. These expan-
sions can be thought of as two independent series: one
in inverse powers of the speed of light c and the other in
positive powers of Newton’s gravitational constant.
Other quantities, such as matter densities and deriva-
tives, can and should also be classified within this order-
counting scheme. Matter density ρ, pressure p and spe-
cific energy density Π, however, are slightly more com-
plicated to classify because they are not dimensionless.
Dimensionlessness can be obtained by comparing the
pressure and the energy density to the matter density,
which we assume is the largest component of the stress-
energy tensor, namely p/ρ ∼ Π/ρ = O(2). Derivatives
can also be classified in this fashion, where we find that
∂t/∂x = O(1). Such a relation can be derived by noting
that ∂t ∼ vi∇i, which comes from the Euler equations of
hydrodynamics to Newtonian order.
With such an order-counting scheme developed, it is
instructive to study the action of a single neutral particle.
The Lagrangian of this system is given by
L = (gµνu
−g00 − 2g0ivi − gijvivj
where uµ = dxµ/dt = (1, vi) is the 4-velocity of the parti-
cle and vi is its 3-velocity. From Eq. (3), note that knowl-
edge of L to O(A) implies knowledge of g00 to O(A), g0i
to O(A − 1) and gij to O(A − 2). Therefore, since the
Lagrangian is already known to O(2) (the Newtonian so-
lution), the first PN correction to the equations of motion
requires g00 to O(4), g0i to O(3) and gij to O(2). Such
order counting is the reason for calculating different sec-
tors of the metric perturbation to different PN orders.
A PPN analysis is usually performed in a particular
background, which defines a particular coordinate sys-
tem, and in an specific gauge, called the standard PPN
gauge. The background is usually taken to be Minkowski
because for solar system experiments deviations due to
cosmological effects are negligible and can, in principle,
be treated as adiabatic corrections. Moreover, one usu-
ally chooses a standard PPN frame, whose outer regions
are at rest with respect to the rest frame of the universe.
Such a frame, for example, forces the spatial sector of the
metric to be diagonal and isotropic [6]. The gauge em-
ployed is very similar to the PN expansion of the Lorentz
gauge of linearized gravitational wave theory. The differ-
ences between the standard PPN and Lorentz gauge are
of O(3) and they allow for the presence of certain PPN
potentials in the vectorial sector of the metric perturba-
tion.
The last ingredient in the PPN recipe is the choice of
a stress-energy tensor. The standard choice is that of a
perfect fluid, given by
T µν = (ρ+ ρΠ+ p)uµuν + pgµν . (4)
Such a stress-energy density suffices to obtain the PN
expansion of the gravitational field outside a fluid body,
like the Sun, or of compact binary system. One can show
that the internal structure of the fluid bodies can be ne-
glected to 1 PN order by the effacement principle [39] in
GR. Such effacement principle might actually not hold
in modified field theories, but we shall study this subject
elsewhere [47].
With all these machinery, on can write down a super-
metric [6], namely
g00 = −1 + 2U − 2βU2 − 2ξΦW + (2γ + 2 + α3 + ζ1
− 2ξ)Φ1 + 2 (3γ − 2β + 1 + ζ2 + ξ)Φ2
+ 2 (1 + ζ3)Φ3 + 2 (3γ + 3ζ4 − 2ξ)Φ4 − (ζ1 − 2ξ)A,
g0i = −
(4γ + 3 + α1 − α2 + ζ1 − 2ξ)Vi
(1 + α2 − ζ1 + 2ξ)Wi,
gij = (1 + 2γU) δij , (5)
where δij is the Kronecker delta and where the PPN
potentials (U,ΦW ,Φ1,Φ2,Φ3,Φ4,A, Vi,Wi) are defined
in Appendix A. Equation (5) describes a super-metric
theory of gravity, because it reduces to different met-
ric theories, such as GR or other alternative theo-
ries [6], through the appropriate choice of PPN param-
eters (γ, β, ξ, α1, α2, α3, ζ1, ζ2, ζ3, ζ4). One could obtain
a more general form of the PPN metric by performing
a post-Galilean transformation on Eq. (5), but such a
procedure shall not be necessary in this paper.
The super-metric of Eq. (5) is parameterized in terms
of a specific number of PPN potentials, where one usually
employs certain criteria to narrow the space of possible
potentials to consider. Some of these restriction include
the following: the potentials tend to zero as an inverse
power of the distance to the source; the origin of the co-
ordinate system is chosen to coincide with the source,
such that the metric does not contain constant terms;
and the metric perturbations h00, h0i and hij transform
as a scalar, vector and tensor. The above restrictions
are reasonable, but, in general, an additional subjective
condition is usually imposed that is based purely on sim-
plicity: the metric perturbations are not generated by
gradients or curls of velocity vectors or other generalized
vector functions. As of yet, no reason had arisen for re-
laxing such a condition, but as we shall see in this paper,
such terms are indeed needed for CS modified theories.
What is the physical meaning of all these parame-
ters? One can understand what these parameters mean
by calculating the generalized geodesic equations of mo-
tion and conservation laws [6]. For example, the param-
eter γ measures how much space-curvature is produced
by a unit rest mass, while the parameter β determines
how much “non-linearity” is there in the superposition
law of gravity. Similarly, the parameter ξ determines
whether there are preferred-location effects, while αi rep-
resent preferred-frame effects. Finally, the parameters ζi
measure the amount of violation of conservation of to-
tal momentum. In terms of conservation laws, one can
interpret these parameters as measuring whether a the-
ory is fully conservative, with linear and angular momen-
tum conserved (ζi and αi vanish), semi-conservative, with
linear momentum conserved (ζi and α3 vanish), or non-
conservative, where only the energy is conserved through
lowest Newtonian order. One can verify that in GR,
γ = β = 1 and all other parameters vanish, which implies
that there are no preferred-location or frame effects and
that the theory is fully conservative.
A PPN analysis of an alternative theory of gravity then
reduces to mapping its solutions to Eq. (5) and then de-
termining the PPN parameters in terms of intrinsic pa-
rameters of the theory. The procedure is simply as fol-
lows: expand the modified field equations in the metric
perturbation and in the PN approximation; iteratively
solve for the metric perturbation to O(4) in h00, to O(3)
in h0i and to O(2) in hij ; compare the solution to the
PPN metric of Eq. (5) and read off the PPN parameters
of the alternative theory. We shall employ this procedure
in Sec. V to obtain the PPN parameters of CS gravity.
III. CS GRAVITY IN A NUTSHELL
In this section, we describe the basics of CS gravity,
following mainly [29, 30]. In the standard CS formalism,
GR is modified by adding a new term to the gravitational
action. This term is given by [30]
SCS =
d4xf (⋆R R) , (6)
where mpl is the Planck mass, f is a prescribed external
quantity with units of squared mass (or squared length
in geometrized units), R is the Ricci scalar and the star
stands the dual operation, such that
R⋆R =
Rαβγδǫ
αβµνRγδµν , (7)
with ǫµνδγ the totally-antisymmetric Levi-Civita tensor
and Rµνδγ the Riemann tensor.
Such a correction to the gravitational action is inter-
esting because of the unavoidable parity violation that
is introduced. Such parity violation is inspired from CP
violation in the standard model, where such corrections
act as anomaly-canceling terms. A similar scenario oc-
curs in string theory, where the Green-Schwarz anomaly
is canceled by precisely such a CS correction [32], al-
though CS gravity is not exclusively tied to string the-
ory. Parity violation in CS gravity inexorably leads to
birefringence in gravitational propagation, where here
we mean that different polarization modes obey differ-
ent propagation equations but travel at the same speed,
that of light [29, 30, 36, 47]. If CS gravity were to
lead to polarization modes that travel at different speeds,
then one could use recently proposed experiments [17]
to test this effect, but such is not the case in CS grav-
ity. Birefringent gravitational waves, and thus CS grav-
ity, have been proposed as possible explanations to the
cosmic-microwave-background (CMB) anisotropies [36],
as well as the baryogenesis problem during the inflation-
ary epoch [33].
The magnitude of the CS correction is controlled by the
externally-prescribed quantity f , which depends on the
specific theory under consideration. When we consider
CS gravity as an effective quantum theory, then the cor-
rection is suppressed by some mass scale M , which could
be the electro-weak scale or some other scale, since it is
unconstrained. In the context of string theory, the quan-
tity f has been calculated only in conservative scenar-
ios, where it was found to be suppressed by the Planck
mass. In other scenarios, however, enhancements have
been proposed, such as in cosmologies where the string
coupling vanishes at late times [48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58], or where the field that generates f cou-
ples to spacetime regions with large curvature [59, 60] or
stress-energy density [28, 47]. For simplicity, we here as-
sume that this quantity is spatially homogeneous and its
magnitude is small but non-negligible, so that we work
to first order in the string-theoretical correction. There-
fore, we treat ḟ as an independent perturbation parame-
ter, [70] unrelated to ǫ, the PN perturbation parameter.
The field equations of CS modified gravity can be ob-
tained by varying the action with respect to the metric.
Doing so, one obtains
Gµν + Cµν = 8πTµν , (8)
where Gµν is the Einstein tensor, Tµν is a stress-energy
tensor and Cµν is the Cotton tensor. The latter tensor is
defined via
Cµν = −
(µDαRν)β + (Dσf,τ )
⋆Rτ (µ
where parenthesis stand for symmetrization, g is the de-
terminant of the metric, Da stands for covariant differ-
entiation and colon subscripts stand for partial differen-
tiation.
Formally, the introduction of such a modification to
the field equations leads to a new constraint, which is
compensated by the introduction of the new scalar field
degree of freedom f . This constraint originates by re-
quiring that the divergence of the field equations vanish,
namely
DµCµν =
Dνf (
⋆RR) = 0, (10)
where the divergence of the Einstein tensor vanished by
the Bianchi identities. If this constraint is satisfied, then
the equations of motion for the stress-energy DµT
µν are
unaffected by CS gravity. A common source of confusion
is that Eq. (10) is sometimes interpreted as requiring that
R⋆R also vanish, which would then force the correction
to the action to vanish. However, this is not the case be-
cause, in general, f is an exact form (d2f = 0) and, thus,
Eq. (10) only implies an additional constraint that forces
all solutions to the field equations to have a vanishing
The previous success of CS gravity in proposing plau-
sible explanations to important cosmological problems
prompts us to consider this extension of GR in the weak-
field regime. For this purpose, it is convenient to rewrite
the field equations in trace-reversed form, since this form
is most amenable to a PN expansion. Doing so, we find,
Rµν + Cµν = 8π
Tµν −
, (11)
where the trace of the Cotton tensor vanishes identically
and T = gµνT
µν is the four dimensional trace of the
stress-energy tensor. To linear order, the Ricci and Cot-
ton tensors are given by [30]
Rµν = −
�hµν +O(h)2,
Cµν = −
ǫ̃0αβ(µ�ηhν)β,α +O(h)2, (12)
where ǫ̃αβγδ is the Levi-Civita symbol, with conven-
tion ǫ̃0123 = +1, and �η = −∂2t + ηij∂i∂j is the flat
space D’Alambertian, with ηµν the Minkowski metric. In
Eq. (12), we have employed the Lorentz gauge condition
α = h,µ/2, where h = g
µνhµν is the four dimensional
trace of the metric perturbation.
The Cotton tensor changes the characteristic behav-
ior of the Einstein equations by forcing them to become
third order instead of second order. Third-order par-
tial differential equations are common in boundary layer
theory [43]. However, in CS gravity, the third-order con-
tributions are multiplied by a factor of f and we shall
treat this function as a small independent expansion pa-
rameter. Therefore, the change in characteristics in the
modified field equations can also be treated perturba-
tively, which is justified because eventhough ḟ might be
enhanced by standard model currents, extra dimensions
or a vanishing string coupling, it must still carry some
type of mass suppression.
The trace-reversed form of the field equations is useful
because it allows us to immediately find a formal solution.
Inverting the D’Alambertian operator we obtain
Hµν = −16π �−1η
Tµν −
+O(h)2, (13)
where we have defined an effective metric perturbation
Hµν ≡ hµν + ḟ ǫ̃0αβ(µhν)β,α. (14)
Note that this formal solution is identical to the formal
PN solution to the field equations in the limit ḟ → 0.
Also note that the second term in Eq. (14) is in essence
a curl operator acting on the metric. This antisymmetric
operator naturally forces the trace of the CS correction
to vanish, as well as the 00 component and the symmetric
spatial part.
From the formal solution to the modified field equa-
tions, we immediately identify the only two possible non-
zero CS contributions : a coupling to the vector compo-
nent of the metric h0i; and coupling to the transverse-
traceless part of the spatial metric hTTij . The latter
has already been studied in the gravitational wave con-
text [29, 30, 47] and it vanishes identically if we require
the spatial sector of the metric perturbation to be confor-
mally flat. The former coupling is a new curl-type contri-
bution to the metric perturbation that, to our knowledge,
had so far been neglected both by the string theory and
PPN communities. In fact, as we shall see in later sec-
tions, terms of this type will force us to introduce a new
PPN parameter that is proportional to the curl of certain
PPN potentials.
Let us conclude this section by pushing the formal so-
lution to the modified field equations further to obtain a
formal solution in terms of the actual metric perturba-
tion hµν . Combining Eqs. (13) and (14) we arrive at the
differential equation
hµν+ḟ ǫ̃
(µhν)β,α = −16π �−1η
Tµν −
+O(h)2.
Since we are searching for perturbations about the gen-
eral relativistic solution, we shall make the ansatz
hµν = h
µν + ḟζµν +O(h)2, (16)
where h
µν is the solution predicted by general relativity
h(GR)µν ≡ −16π �−1η
Tµν −
, (17)
and where ζµν is an unknown function we are solving for.
Inserting this ansatz into Eq. (15) we obtain
ζµν+ḟ ǫ̃
(µζν)β,α = 16πǫ̃
(µ∂α�
Tν)β −
gν)βT
We shall neglect the second term on the left-hand side be-
cause it would produce a second order correction. Such
conclusion was also reached when studying parity viola-
tion in GR to explain certain features of the CMB [35].
We thus obtain the formal solution
ζµν = 16πǫ̃
(µ∂α�
Tν)β −
gν)βT
and the actual metric perturbation to linear order be-
comes
hµν = −16π �−1η
Tµν −
+ 16πḟ ǫ̃kℓi�−1η
δi(µTν)ℓ,k −
δi(µην)ℓT,k
+O(h)2,
where we have used some properties of the Levi-Civita
symbol to simplify this expression. The procedure pre-
sented here is general enough that it can be directly ap-
plied to study CS gravity in the PPN framework, as well
as possibly find PN solutions to CS gravity.
IV. PN EXPANSION OF CS GRAVITY
In this section, we perform a PN expansion of the field
equations and obtain a solution in the form of a PN se-
ries. This solution then allows us to read off the PPN
parameters by comparing it to the standard PPN super-
metric [Eq. (5)]. In this section we shall follow closely the
methods of [6] and [61] and indices shall be manipulated
with the Minkowski metric, unless otherwise specified.
Let us begin by expanding the field equations to second
order in the metric perturbation. Doing so we find that
the Ricci and Cotton tensors are given to second order
Rµν = −
�ηhµν − 2hσ(µ,ν)σ + h,µν
2hρ(µ,ν)λ − hµν,ρλ − hρλ,µν
hρλ,µhρλ,ν + h
ν,λ (21)
− hρµ,λhρν,λ +
h,λ − 2hλρ,ρ
hµν,λ − 2hλ(µ,ν)
+O(h)3,
Cµν = −
ǫ̃0αβ(µ
�ηhν)β,α − hσβ,αν)σ
ǫ̃0αβ(µ
�ηhν)β,α − hσβ,αν)σ
2hν)(λ,α) − hλα,ν)
β − 2hσ(λ,β)σ + h,βλ
− 2Q̂Rν)β,α
ǫ̃σαβ(µ
2h0(σ,τ) − hστ,0
hτ [β,α]ν) − hν)[β,α]τ
hµλǫ̃
0αβ(λ
�ηhν)β,α − hσβ,αν)σ
ǫ̃0αβ(µ
β,α − hσβ,ασλ)
hνλ +O(h)3. (22)
where index contraction is carried out with the
Minkowski metric and where we have not assumed any
gauge condition. The operator Q̂(·) takes the quadratic
part of its operand [of O(h)2] and it is explained in more
detail in Appendix B, where the derivation of the expan-
sion of the Cotton tensor is presented in more detail. In
this derivation, we have used the definition of the Levi-
Civita tensor
ǫαβγδ = (−g)1/2ǫ̃αβγδ =
ǫ̃αβγδ +O(h)2, (23)
ǫαβγδ = −(−g)−1/2ǫ̃αβγδ = −
ǫ̃αβγδ +O(h)2.
Note that the PN expanded version of the linearized Ricci
tensor of Eq. (21) agrees with previous results [6]. Also
note that if the Lorentz condition is enforced, several
terms in both expressions vanish identically and the Cot-
ton tensor to first order reduces to Eq. (12), which agrees
with previous results [30].
Let us now specialize the analysis to the standard PPN
gauge. For this purpose, we shall impose the following
gauge conditions
k − 1
h,j = O(4),
k − 1
hkk,0 = O(5), (24)
where hkk is the spatial trace of the metric perturbation.
Note that the first equation is the PN expansion of one of
the Lorentz gauge conditions, while the second equation
is not. This is the reason why the previous equations
where not expanded in the Lorentz gauge. Nonetheless,
such a gauge condition does not uniquely fix the coordi-
nate system, since we can still perform an infinitesimal
gauge transformation that leaves the modified field equa-
tions invariant. One can show that the Lorentz and PPN
gauge are related to each other by such a gauge transfor-
mation. In the PPN gauge, then, the Ricci tensor takes
the usual form
R00 = −
∇2h00 −
h00,ih00,
hijh00,ij +O(6),
R0i = −
∇2h0i −
h00,0i +O(5),
Rij = −
∇2hij +O(4), (25)
which agrees with previous results [6], while the Cotton
tensor reduces to
C00 = O(6),
C0i = −
ḟ ǫ̃0kli∇2h0l,k +O(5),
Cij = −
ḟ ǫ̃0kl(i∇2hj)l,k +O(4), (26)
where ∇ = ηij∂i∂j is the Laplacian of flat space [see Ap-
pendix B for the derivation of Eq. (26).] Note again the
explicit appearance of two coupling terms of the Cotton
tensor to the metric perturbation: one to the transverse-
traceless part of the spatial metric and the other to the
vector metric perturbation. The PN expansions of the
linearized Ricci and Cotton tensor then allow us to solve
the modified field equations in the PPN framework.
V. PPN SOLUTION OF CS GRAVITY
In this section we shall proceed to systematically solve
the modified field equation following the standard PPN
iterative procedure [6]. We shall begin with the 00 and
ij components of the metric to O(2), and then proceed
with the 0i components to O(3) and the 00 component
to O(4). Once all these components have been solved for
in terms of PPN potentials, we shall be able to read off
the PPN parameters adequate to CS gravity.
A. h00 and hij to O(2)
Let us begin with the modified field equations for the
scalar sector of the metric perturbation. These equations
are given to O(2) by
∇2h00 = −8πρ, (27)
because T = −ρ. Eq. (27) is the Poisson equation, whose
solution in terms of PPN potentials is
h00 = 2U +O(4). (28)
Let us now proceed with the solution to the field equa-
tion for the spatial sector of the metric perturbation.
This equation to O(2) is given by
∇2hij + ḟ ǫ̃0kl(i∇2hj)l,k = −8πρδij , (29)
where note that this is the first appearance of a Cotton
tensor contribution. Since the Levi-Civita symbol is a
constant and ḟ is only time-dependent, we can factor
out the Laplacian and rewrite this equation in terms of
the effective metric Hij as
∇2Hij = −8πρδij , (30)
where, as defined in Sec. III,
Hij = hij + ḟ ǫ̃0kl(ihj)l,k. (31)
The solution of Eq. (30) can be immediately found in
terms of PPN potentials as
Hij = 2Uδij +O(4), (32)
which is nothing but Eq. (13). Recall, however, that in
Sec. III we explicitly used the Lorentz gauge to simplify
the field equations, whereas here we are using the PPN
gauge. The reason why the solutions are the same is that
the PPN and Lorentz gauge are indistinguishable to this
order.
Once the effective metric has been solved for, we can
obtain the actual metric perturbation following the pro-
cedure described in Sec. III. Combining Eq. (31) with
Eq. (32), we arrive at the following differential equation
hij + ḟ ǫ̃
(ihj)l,k = 2Uδij. (33)
We look for solutions whose zeroth-order result is that
predicted by GR and the CS term is a perturbative cor-
rection, namely
hij = 2Uδij + ḟζij , (34)
where ζ is assumed to be of O(ḟ)0. Inserting this ansatz
into Eq. (33) we arrive at
ζij + ḟ ǫ̃
(iζj)l,k = 0, (35)
where the contraction of the Levi-Civita symbol and the
Kronecker delta vanished. As in Sec. III, note that the
second term on the left hand side is a second order cor-
rection and can thus be neglected to discover that ζij
vanishes to this order.
The spatial metric perturbation to O(2) is then simply
given by the GR prediction without any CS correction,
namely
hij = 2Uδij +O(4). (36)
Physically, the reason why the spatial metric is unaf-
fected by the CS correction is related to the use of a
perfect fluid stress-energy tensor, which, together with
the PPN gauge condition, forces the metric to be spa-
tially conformally flat. In fact, if the spatial metric were
not flat, then the spatial sector of the metric perturba-
tion would be corrected by the CS term. Such would
be the case if we had pursued a solution to 2 PN or-
der, where the Landau-Lifshitz pseudo-tensor sources a
non-conformal correction to the spatial metric [39], or if
we had searched for gravitational wave solutions, whose
stress-energy tensor vanishes [29, 36]. In fact, one can
check that, in such a scenario, Eq. (30) reduces to that
found by [29, 30, 36, 47] as ρ → 0. We have then found
that the weak-field expansion of the gravitational field
outside a fluid body, like the Sun or a compact binary, is
unaffected by the CS correction to O(2).
B. h0i to O(3)
Let us now look for solutions to the field equations for
the vector sector of the metric perturbation. The field
equations to O(3) become
∇2h0i +
h00,0i +
ḟ ǫ̃0kli∇2h0l,k = 16πρvi, (37)
where we have used that T 0i = −T0i. Using the lower
order solutions and the effective metric, as in Sec. III, we
obtain
∇2H0i + U,0i = 16πρvi, (38)
where the vectorial sector of the effective metric is
H0i = h0i +
ḟ ǫ̃0klih0l,k. (39)
We recognize Eq. (38) as the standard GR field equation
to O(3), except that the dependent function is the effec-
tive metric instead of the metric perturbation. We can
thus solve this equation in terms of PPN potentials to
obtain
H0i = −
Wi, (40)
where we have used that the superpotential X satisfies
X,0j = Vj − Wj (see Appendix A for the definitions.)
Combining Eq. (39) with Eq. (40) we arrive at a differ-
ential equation for the metric perturbation, namely
h0i +
ḟ ǫ̃0klih0l,k = −
Wi. (41)
Once more, let us look for solutions that are perturbation
about the GR prediction, namely
h0i = −
Wi + ḟζi, (42)
where we again assume that ζi is of O(ḟ)0. The field
equation becomes
ḟ (∇× ζ)i =
(∇× V )i +
(∇×W )i
where (∇×A)i = ǫijk∂jAk is the standard curl operator
of flat space. As in Sec. III, note once more that the
second term on the left-hand side is again a second order
correction and we shall thus neglect it. Also note that
the curl of the Vi potential happens to be equal to the
curl of the Wi potential. The solution for the vectorial
sector of the actual gravitational field then simplifies to
h0i = −
Wi + 2ḟ (∇× V )i +O(5). (44)
We have arrived at the first contribution of CS mod-
ified gravity to the metric for a perfect fluid source.
Chern-Simons gravity was previously seen to couple to
the transverse-traceless sector of the metric perturbation
for gravitational wave solutions [29, 30, 36, 47]. The CS
correction is also believed to couple to Noether vector
currents, such as neutron currents, which partially fueled
the idea that this correction could be enhanced. However,
to our knowledge, this correction was never thought to
couple to vector metric perturbations. From the analy-
sis presented here, we see that in fact CS gravity does
couple to such terms, even if the matter source is neu-
trally charged. The only requirement for such couplings
is that the source is not static, ie. that the object is ei-
ther moving or spinning relative to the PPN rest frame
so that the PPN vector potential does not vanish. The
latter is suppressed by a relative O(1) because in the far
field the velocity of a compact object produces a term of
O(3) in Vi, while the spin produces a term of O(4). In
a later section, we shall discuss some of the physical and
observational implications of such a modification to the
metric.
C. h00 to O(4)
A full analysis of the PPN structure of a modified the-
ory of gravity requires that we solve for the 00 component
of the metric perturbation to O(4). The field equations
to this order are
∇2h00 −
h00,ih00,i +
hijh00,ij = 4πρ [1
v2 − U + 1
, (45)
where the CS correction does not contribute at this order
(see Appendix B.) Note that the h0i sector of the metric
perturbation to O(3) does not feed back into the field
equations at this order either. The terms that do come
into play are the h00 and hij sectors of the metric, which
are not modified to lowest order by the CS correction.
The field equation, thus, reduce to the standard one of
GR, whose solution in terms of PPN potentials is
h00 = 2U − 2U2 + 4Φ1 + 4Φ2 +2Φ3 +6Φ4 +O(6). (46)
We have thus solved for all components of the metric per-
turbation to 1 PN order beyond the Newtonian answer,
namely g00 to O(4), g0i to O(3) and gij to O(2).
D. PPN Parameters for CS Gravity
We now have all the necessary ingredients to read off
the PPN parameters of CS modified gravity. Let us begin
by writing the full metric with the solutions found in the
previous subsections:
g00 = −1 + 2U − 2U2 + 4Φ1 + 4Φ2 + 2Φ3 + 6Φ4 +O(6),
g0i = −
Wi + 2ḟ (∇× V )i +O(5),
gij = (1 + 2U) δij +O(4). (47)
One can verify that this metric is indeed a solution of
Eqs. (27), (29), (37) and (45) to the appropriate PN or-
der and to first order in the CS coupling parameter. Also
note that the solution of Eq. (47) automatically satis-
fies the constraint ⋆RR = 0 to linear order because the
contraction of the Levi-Civita symbol with two partial
derivatives vanishes. Such a solution is then allowed in
CS gravity, just as other classical solutions are [62], and
the equations of motion for the fluid can be obtained di-
rectly from the covariant derivative of the stress-energy
tensor.
We can now read off the PPN parameters of the CS
modified theory by comparing Eq. (5) to Eq. (47). A vi-
sual inspection reveals that the CS solution is identical
to the classical GR one, which implies that γ = β = 1,
ζ = 0 and α1 = α2 = α3 = ξ1 = ξ2 = ξ3 = ξ4 = 0 and
there are no preferred frame effects. However, Eq. (5)
contains an extra term that cannot be modeled by the
standard PPN metric of Eq. (5), namely the curl contri-
bution to g0i. We then see that the PPN metric must be
enhanced by the addition of a curl-type term to the 0i
components of the metric, namely
g0i ≡ −
(4γ + 3 + α1 − α2 + ζ1 − 2ξ)Vi
(1 + α2 − ζ1 + 2ξ)Wi + χ (r∇× V )i , (48)
where χ is a new PPN parameter and where we have
multiplied the curl operator by the radial distance to
the source, r, in order to make χ a proper dimension-
less parameter. Note that there is no need to introduce
any additional PPN parameters because the curl of Wi
equals the curl of Vi. In fact, we could have equally pa-
rameterized the new contribution to the PPN metric in
terms of the curl of Wi, but we chose not to because Vi
appears more frequently in PN theory. For the case of
CS modified gravity, the new χ parameter is simply
χ = 2
, (49)
which is dimensionless since ḟ has units of length. If an
experiment could measure or place bounds on the value
of χ, then ḟ could also be bounded, thus placing a con-
straint on the CS coupling parameter.
VI. ASTROPHYSICAL IMPLICATIONS
In this section we shall propose a physical interpreta-
tion to the CS modification to the PPN metric and we
shall calculate some GR predictions that are modified by
this correction. This section, however, is by no means a
complete study of all the possible consequences of the CS
correction, which is beyond the scope of this paper.
Let us begin by considering a system of A nearly spher-
ical bodies, for which the gravitational vector potentials
are simply [6]
V i =
viA +
, (50)
W i =
(vA · nA)niA +
where mA is the mass of the Ath body, rA is the field
point distance to the Ath body, niA = x
A/rA is a unit
vector pointing to the Ath body, vA is the velocity of the
Ath body and J iA is the spin-angular momentum of the
Ath body. For example, the spin angular momentum for
a Kerr spacetime is given by J i = m2ai, where a is the
dimensionless Kerr spin parameter. Note that if A = 2
then the system being modeled could be a binary of spin-
ning compact objects, while if A = 1 it could represent
the field of the sun or that of a rapidly spinning neutron
star or pulsar.
In obtaining Eq. (50), we have implicitly assumed a
point-particle approximation, which in classical GR is
justified by the effacement principle. This principle pos-
tulates that the internal structure of bodies contributes
to the solution of the field equations to higher PN order.
One can verify that this is indeed the case in classical
GR, where internal structure contributions appear at 5
PN order. In CS gravity, however, it is a priori unclear
whether an analogous effacement principle holds because
the CS term is expected to couple with matter current via
standard model-like interactions. If such is the case, it is
possible that a “mountain” on the surface of a neutron
star [63] or an r-mode instability [64, 65, 66] enhances
the CS contribution. In this paper, however, we shall
neglect these interactions, and relegate such possibilities
to future work [47].
With such a vector potential, we can calculate the CS
correction to the metric. For this purpose, we define the
correction δg0i ≡ g0i − g(GR)0i , where g
0i is the GR
prediction without CS gravity. We then find that the CS
corrections is given by
δg0i = 2
(vA × nA)i −
(JA · nA)
where the · operator is the flat-space inner product and
where we have used the identities ǫ̃ijk ǫ̃klm = δilδjm −
δimδjl and ǫ̃ilk ǫ̃jlm = 2δij . Note that the first term of
Eq. (51) is of O(3), while the second and third terms
are of O(4) as previously anticipated. Also note that ḟ
couples both to the spin and orbital angular momentum.
Therefore, whether the system under consideration is the
Solar system (vi of the Sun is zero while J i is small), the
Crab pulsar (vi is again zero but J i is large) or a binary
system of compact objects (neither vi nor J i vanish),
there will in general be a non-vanishing coupling between
the CS correction and the vector potential of the system.
From the above analysis, it is also clear that the CS
correction increases with the non-linearity of the space-
time. In other words, the CS term is larger not only for
systems with large velocities and spins, but also in re-
gions near the source. For a binary system, this fact
implies that the CS correction is naturally enhanced
in the last stages of inspiral and during merger. Note
that this enhancement is different from all previous en-
hancements proposed, since it does not require the pres-
ence of charge [28, 47], a fifth dimension with warped
compactifications [59, 60], or a vanishing string cou-
pling [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. Un-
fortunately, the end of the inspiral stage coincides with
the edge of the PN region of validity and, thus, a com-
plete analysis of such a natural enhancement will have to
be carried out through numerical simulations.
In the presence of a source with the vector potentials of
Eq. (50), we can write the vectorial sector of the metric
perturbation in a suggestive way, namely
g0i =
viA −
vA − v(eff)A
(eff)
A · nA − 2
(eff)
where we have defined an effective velocity and angular
momentum vector via
viA(eff) = v
A − 6ḟ
J iA(eff) = J
A − ḟmAviA, (53)
or in terms of the Newtonian orbital angular momentum
= rA × pA and linear momentum piA(N) = mAv
LiA(eff) = L
A(N) − 6ḟ (nA × JA)
J iA(eff) = J
A − ḟpiA. (54)
From this analysis, it is clear that the CS corrections
seems to couple to both a quantity that resembles the or-
bital and the spin angular momentum vector. Note that
when the spin angular momentum vanishes the vectorial
metric perturbation is identical to that of a spinning mov-
ing fluid, but where the spin is induced by the coupling
of the orbital angular momentum to the CS term.
The presence of an effective CS spin angular momen-
tum in non-spinning sources leads to an interesting phys-
ical interpretation. Let us model the field that sources
ḟ as a fluid that permeates all of spacetime. This field
could be, for example, a model-independent axion, in-
spired by the quantity introduced in the standard model
to resolve the strong CP problem [67]. In this scenario,
then the fluid is naturally “dragged” by the motion of any
source and the CS modification to the metric is nothing
but such dragging. This analogy is inspired by the er-
gosphere of the Kerr solution, where inertial frames are
dragged with the rotation of the black hole. In fact, one
could push this analogy further and try to construct the
shear and bulk viscosity of such a fluid, but we shall not
attempt this here. Of course, this interpretation is to be
understood only qualitatively, since its purpose is only
to allow the reader to picture the CS modification to the
metric in physical terms.
An alternative interpretation can be given to the CS
modification in terms of the gravito-electro-magnetic
(GED) analogy [40, 41], which shall allow us to eas-
ily construct the predictions of the modified theory. In
this analogy, one realizes that the PN solution to the
linearized field equations can be written in terms of a
potential and vector potential, namely
ds2 = − (1− 2Φ)dt2 − 4 (A · dx) dt+ (1 + 2Φ) δijdxidxj ,
where Φ reduces to the Newtonian potential U in the
Newtonian limit [41] and Ai is a vector potential related
to the metric via Ai = −g0i/4. One can then construct
GED fields in analogy to Maxwell’s electromagnetic the-
ory via
Ei = − (∇Φ)i − ∂t
Bi = (∇×A)i , (56)
which in terms of the vectorial sector of the metric per-
turbation becomes
Ei = − (∇Φ)i + 1
Bi = −1
(∇× g)i , (57)
where we have defined the vector gi = g0i. The geodesic
equations for a test particle then reduce to the Lorentz
force law, namely
F i = −mEi − 2m (v ×B)i . (58)
We can now work out the effect of the CS correction
on the GED fields and equations of motion. First note
that the CS correction only affects g. We can then write
the CS modification to the Lorentz force law by defining
δai = ai − ai
, where ai
is the acceleration vector
predicted by GR, to obtain,
δai =
δġi +
(v × δΩ)i , (59)
where we have defined the angular velocity
δΩi = (∇× δg)i . (60)
The time derivative of the vector gi is of O(5) and can
thus be neglected, but the angular velocity cannot and it
is given by
δΩi = −
3 (vA · nA)niA − viA
, (61)
which is clearly of O(3). Note that although the first
term between square brackets cancels for circular orbits
because niA is perpendicular to v
A to Newtonian order,
the second term does not. The angular velocity adds a
correction to the acceleration of O(4), namely
δai = −3
(vA · nA) (vA × nA)i , (62)
which for a system in circular orbit vanishes to Newto-
nian order. One could use this formalism to find the
perturbations in the motion of moving objects by inte-
grating Eq. (62) twice. However, for systems in a circular
orbit, such as the Earth-Moon system or compact bina-
ries, this correction vanishes to leading order. Therefore,
lunar ranging experiments [68] might not be able to con-
straint ḟ .
Another correction to the predictions of GR is that
of the precession of gyroscopes by the so-called Lense-
Thirring or frame-dragging effect. In this process, the
spin angular momentum of a source twists spacetime in
such a way that gyroscopes are dragged with it. The
precession angular velocity depends on the vector sector
of the metric perturbation via Eq. (61). Thus, the full
Lense-Thirring term in the precession angular velocity of
precessing gyroscopes is
ΩiLT = −
J iA(eff) − 3n
JA(eff) · nA
. (63)
Note that this angular velocity is identical to the GR
prediction, except for the replacement J iA → J iA(eff).
In CS modified gravity, then, the Lense-Thirring effect
is not only produced by the spin angular momentum of
the gyroscope but also by the orbital angular momentum
that couples to the CS correction. Therefore, if an ex-
periment were to measure the precession of gyroscopes
by the curvature of spacetime (see, for example, Gravity
Probe B [42]) one could constraint ḟ and thus some in-
trinsic parameters of string theory. Note, however, that
the CS correction depends on the velocity of the bodies
with respect to the inertial PPN rest-frame. In order to
relate these predictions to the quantities that are actually
measured in the experiment, one would have to transform
to the experiment’s frame, or perhaps to a basis aligned
with the direction of distant stars [6].
Are there other experiments that could be performed
to measure such a deviation from GR? Any experiment
that samples the vectorial sector of the metric would in
effect be measuring such a deviation. In this paper, we
have only discussed modifications to the frame-dragging
effect and the acceleration of bodies through the GED
analogy, but this need not be the only corrections to
classical GR predictions. In fact, any predictions that de-
pends on g0i indirectly, for example via Christoffel sym-
bols, will probably also be modified unless the corrections
is fortuitously canceled. In this paper, we have laid the
theoretical foundations of the weak-field correction to the
metric due to CS gravity and studied some possible cor-
rections to classical predictions. A detailed study of other
corrections is beyond the scope of this paper.
VII. CONCLUSION
We have studied the weak-field expansion of the solu-
tion to the CS modified field equations in the presence
of a perfect fluid PN source in the point particle limit.
Such an expansion required that we linearize the Ricci
and Cotton tensor to second order in the metric pertur-
bation without any gauge assumption. An iterative PPN
formalism was then employed to solve for the metric per-
turbation in this modified theory of gravity. We have
found that CS gravity possesses the same PPN parame-
ters as those of GR, but it also requires the introduction
of a new term and PPN parameter that is proportional
to the curl of the PPN vector potentials. Such a term
is enhanced in non-linear scenarios without requiring the
presence of standard model currents, large extra dimen-
sions or a vanishing string coupling.
We have proposed an interpretation for the new term
in the metric produced by CS gravity and studied some
of the possible consequences it might have on GR predic-
tions. The interpretation consists of picturing the field
that sources the CS term as a fluid that permeates all of
spacetime. In this scenario, the CS term is nothing but
the “dragging” of the fluid by the motion of the source.
Irrespective of the validity of such an interpretation, the
inclusion of a new term to the weak-field expansion of
the metric naturally leads to corrections to the standard
GR predictions. We have studied the acceleration of
point particles and the Lense-Thirring contribution to
the precession of gyroscopes. We have found that both
corrections are proportional to the CS coupling parame-
ter and, therefore, experimental measurement of these ef-
fects might be used to constraint CS and, possibly, string
theory.
Future work could concentrate on studying further the
non-linear enhancement of the CS correction and the
modifications to the predictions of GR. The PPN analysis
performed here breaks down very close to the source due
to the use of a point particle approximation in the stress
energy tensor. One possible research route could consists
of studying the CS correction in a perturbed Kerr back-
ground [69]. Another possible route could be to analyze
other predictions of the theory, such as the perihelion
shift of Mercury or the Nordtvedt effect. Furthermore,
in light of the imminent highly-accurate measurement of
the Lense-Thirring effect by Gravity Probe B, it might be
useful to revisit this correction in a frame better-adapted
to the experimental setup. Finally, the CS modification
to the weak-field metric might lead to non-conservative
effects and the breaking of the effacement principle [47],
which could be studied through the evaluation of the
gravitational pseudo stress-energy tensor. Ultimately, it
will be experiments that will determine the viability of
CS modified gravity and string theory.
Acknowledgments
The authors acknowledge the support of the Center
for Gravitational Wave Physics funded by the National
Science Foundation under Cooperative Agreement PHY-
01-14375, and support from NSF grants PHY-05-55-628.
We would also like to thank Cliff Will for encouraging
one of us to study the PPN formalism and Pablo Laguna
for suggesting one of us to look into the PPN expansion
of CS gravity. We would also like to thank R. Jackiw,
R. Wagoner and Ben Owen for enlightening discussions
and comments.
APPENDIX A: PPN POTENTIALS
In this appendix, we present explicit expressions for
the PPN potentials used to parameterize the metric in
Eq. (5). These potentials are the following:
|x− x′|
d3x′,
ρ′v′i
|x− x′|
d3x′,
ρ′v′j(x− x′)j(x− x′)i
|x− x′|3
d3x′,
ρ′ρ′′
(x− x′)i
|x− x′|3
(x′ − x′′)i
|x− x′′| −
(x− x′′)i
|x′ − x′′|
d3x′d3x′′,
ρ′v′2
|x− x′|
d3x′, Φ2 ≡
ρ′U ′
|x− x′|
d3x′,
|x− x′|
d3x′, Φ4 ≡
|x− x′|
d3x′,
v′i (x− x′)
|x− x′| d
ρ′|x− x′|d3x′. (A1)
These potentials satisfy the following relations
∇2U = −4πρ, ∇2Vi = −4πρvi,
∇2Φ1 = −4πρv2, ∇2Φ2 = −4πρU,
∇2Φ3 = −4πρΠ, ∇2Φ4 = −4πp,
∇2X = −2U (A2)
The potential X is sometimes referred to as the super-
potential because it acts as a potential for the Newtonian
potential.
APPENDIX B: LINEARIZATION OF THE
COTTON TENSOR
In this appendix, we present some more details on the
derivation of the linearized Cotton tensor to second order.
We begin with the definition of the Cotton tensor [30] in
terms of the symmetrization operator, namely
Cµν = − 1√
(Dσf) ǫ
σαβ(µDαR
β + (Dστf)
Rτ(µ|σ|ν)
Using the symmetries of the Levi-Civita and Riemann
tensor, as well as the fact that f depends only on time,
we can simplify the Cotton tensor to
Cµν = (−g)−1ḟ
ǫ̃0αβ(µRν)β,α + ǫ̃
0αβ(µΓ
Γ0στ ǫ̃
σαβ(µRν)τ αβ
. (B2)
Noting that the determinant of the metric is simply g =
−1 + h, so that (−g)−1 = 1 + h, we can identify four
terms in the Cotton tensor
A = ḟ ǫ̃
0αβ(µ
L̂Rν)β,α
B = ḟ ǫ̃
0αβ(µhρρ
L̂Rν)β,α
C = ḟ ǫ̃
0αβ(µ
L̂Rλβ
ǫ̃σαβ(µ
L̂Γ0στ
L̂Rν)ταβ
E = ḟ ǫ̃
0αβ(µ
Q̂Rν)β,α
, (B3)
where the L̂ operator stands for the linear part of its
operand, while the Q̂ operator isolates the quadratic part
of its operand. For example, if we act L̂ and Q̂ on (1+h)n,
where n is some integer, we obtain
L̂(1 + h)n
= nh,
Q̂(1 + h)n
n(n− 1)
h2.(B4)
Let us now compute each of these terms separately.
The first four terms are given by
A = −
ǫ̃0αβ(µ
β,α − hσβ,νασ
B = −
hǫ̃0αβ(µ
β,α − hσβ,νασ
C = −
ǫ̃0αβ(µ
hν)λ,α + h
α,λ − hλα,ν)
β − hσλ,βσ − hσβ,λσ + h,λβ
ǫ̃σαβ(µ
2h0(σ,τ) − hστ,0
hτ [β,α]
ν − hν [β,α]τ
The last term of the Cotton tensor is simply the deriva-
tive of the Ricci tensor which we already calculated to
second order in Eq. (21). In order to avoid notation clut-
ter, we shall not present it again here, but instead we
combine all the Cotton tensor pieces to obtain
Cµν = − ḟ
ǫ̃0αβ(µ
β,α − hσβ,ασν)
ǫ̃0αβ(µ
β,α − hσβ,ασν)
2hν)(λ,α) − hλα,ν)
β − 2hσ(λ,β)σ + h,βλ
− 2Q̂Rν)β,α
ǫ̃σαβ(µ
2h0(σ,τ) − hστ,0
hτ [β,α]
ν) − hν)[β,α]τ
+O(h)3
where its covariant form is
Cµν = −
ǫ̃0αβ(µ
�ηhν)β,α − hσβ,αν)σ
ǫ̃0αβ(µ
�ηhν)β,α − hσβ,αν)σ
2hν)(λ,α) − hλα,ν)
β − 2hσ(λ,β)σ + h,βλ
− 2Q̂Rν)β,α + hνλ
β,α − hσβ,ασλ)
ǫ̃σαβ(µ
2h0(σ,τ) − hστ,0
hτ [β,α]ν) − hν)[β,α]τ
hµλǫ̃
0αβ(λ
�ηhν)β,α − hσβ,αν)σ
+O(h)3. (B7)
For the PPN mapping of CS modified gravity, only the
00 component of the metric is needed to second order,
which implies we only need C00 to O(h)2. This compo-
nent is given by
C00 =
ǫ̃ijk0
2h0(i,ℓ) − hiℓ,0
hℓ[k,j]0 − h0[k,j]ℓ
h0ℓǫ̃
0jk(ℓ
�ηh0k,j − hik,j0i
+O(h)3, (B8)
where in fact the last term vanishes due to the PPN gauge
condition. Note that this term is automatically of O(6),
which is well beyond the required order we need in h00.
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http://einstein.standfod.edu

  We investigate the weak-field, post-Newtonian expansion to the solution of
the field equations in Chern-Simons gravity with a perfect fluid source. In
particular, we study the mapping of this solution to the parameterized
post-Newtonian formalism to 1 PN order in the metric. We find that the PPN
parameters of Chern-Simons gravity are identical to those of general
relativity, with the exception of the inclusion of a new term that is
proportional to the Chern-Simons coupling parameter and the curl of the PPN
vector potentials. We also find that the new term is naturally enhanced by the
non-linearity of spacetime and we provide a physical interpretation for it. By
mapping this correction to the gravito-electro-magnetic framework, we study the
corrections that this new term introduces to the acceleration of point
particles and the frame-dragging effect in gyroscopic precession. We find that
the Chern-Simons correction to these classical predictions could be used by
current and future experiments to place bounds on intrinsic parameters of
Chern-Simons gravity and, thus, string theory.

Parametrized Post-Newtonian Expansion of Chern-Simons Gravity
Stephon Alexander1 and Nicolás Yunes1
Center for Gravitational Wave Physics, Institute for Gravitational Physics and Geometry and Department of Physics,
The Pennsylvania State University, University Park, PA 16802, USA
(Dated: November 4, 2018)
We investigate the weak-field, post-Newtonian expansion to the solution of the field equations
in Chern-Simons gravity with a perfect fluid source. In particular, we study the mapping of this
solution to the parameterized post-Newtonian formalism to 1 PN order in the metric. We find
that the PPN parameters of Chern-Simons gravity are identical to those of general relativity, with
the exception of the inclusion of a new term that is proportional to the Chern-Simons coupling
parameter and the curl of the PPN vector potentials. We also find that the new term is naturally
enhanced by the non-linearity of spacetime and we provide a physical interpretation for it. By
mapping this correction to the gravito-electro-magnetic framework, we study the corrections that
this new term introduces to the acceleration of point particles and the frame-dragging effect in
gyroscopic precession. We find that the Chern-Simons correction to these classical predictions could
be used by current and future experiments to place bounds on intrinsic parameters of Chern-Simons
gravity and, thus, string theory.
PACS numbers: 11.25.Wx, 95.55.Ym, 04.60.-m, 04.80.Cc
I. INTRODUCTION
Tests of alternative theories of gravity that modify gen-
eral relativity (GR) at a fundamental level are essential to
the advancement of physics. One formalism that has had
incredible success in this task is the parameterized post-
Newtonian (PPN) framework [1, 2, 3, 4, 5, 6]. In this
formalism, the metric of the alternative theory is solved
for in the weak-field limit and its deviations from GR are
expressed in terms of PPN parameters. Once a metric
has been obtained, one can calculate predictions of the
alternative theory, such as light deflection and the perihe-
lion shift of Mercury, which shall depend on these PPN
parameters. Therefore, experimental measurements of
such physical effects directly lead to constraints on the
parameters of the alternative theory. This framework,
together with the relevant experiments, have already
been successfully employed to constrain scalar-tensor the-
ories (Brans-Dicke, Bekenstein) [7], vector-tensor the-
ories (Will-Nordtvedt [8], Hellings-Nordtvedt [9]), bi-
metric theories (Rosen [10, 11]) and stratified theories
(Ni [12]) (see [13] for definitions and an updated review.)
Only recently has this framework been used to study
quantum gravitational and string-theoretical inspired
ideas. On the string theoretical side, Kalyana [14] investi-
gated the PPN parameters associated with the graviton-
dilaton system in low-energy string theory. More re-
cently, Ivashchuk, et. al. [15] studied PPN parameters
in the context of general black holes and p-brane spher-
ically symmetric solutions, while Bezerra, et. al. [16]
considered domain wall spacetimes for low energy effec-
tive string theories and derived the corresponding PPN
parameters for the metric of a wall. On the quan-
tum gravitational side, Gleiser and Kozameh [17] and
more recently Fan, et. al. [18] studied the possibility
of testing gravitational birefringence induced by quan-
tum gravity, which was proposed by Amelino-Camelia,
el. al. [19] and Gambini and Pullin [20]. Other non-
PPN proposals have been also put forth to test quan-
tum gravity, for example through gravitational waves
[21, 22, 23, 24, 25, 26, 27, 28], but we shall not discuss
those tests here.
Chern-Simons (CS) gravity [29, 30] is one such ex-
tension of GR, where the gravitational action is mod-
ified by the addition of a parity-violating term. This
extension is promising because it is required by all 4-
dimensional compactifications of string theory [31] for
mathematical consistency because it cancels the Green-
Schwarz anomaly [32]. CS gravity, however, is not unique
to string theory and in fact has its roots in the standard
model, where it arises as a gravitational anomaly pro-
vided that there are more flavours of left handed leptons
than right handed ones. Moreover the CS extension to
GR can arise via the embedding of the three dimensional
Chern-Simons topological current into a 4D space-time
manifold, decsribed by Jackiw and Pi [30]
Chern-Simons gravity has been recently studied in
the cosmological context. In particular, this framework
was used to shed light on the anisotropies of the cos-
mic microwave background (CMB) [33, 34, 35] and the
leptogenesis problem [34, 36, 37]. Parity violation has
also been shown to produce birefringent gravitational
waves [28, 29], where different polarizations modes ac-
quire varying amplitudes. These modes obey different
propagation equations because the imaginary sector of
the classical dispersion relation is CS corrected. Different
from [20], in CS birefringence the velocity of the gravita-
tional wave remains that of light.
In this paper we study CS gravity in the PPN frame-
work, extending the analysis of [38] and providing some
missing details. In particular, we shall consider the ef-
fect of the CS correction to the gravitational field of,
for instance, a pulsar, a binary system or a star in the
weak-field limit. These corrections are obtained by solv-
http://arxiv.org/abs/0704.0299v1
ing the modified field equations in the weak-field limit for
post-Newtonian (PN) sources, defined as those that are
weakly-gravitating and slowly-moving [39]. Such an ex-
pansion requires the calculation of the Ricci and Cotton
tensors to second order in the metric perturbation. We
then find that CS gravity leads to the same gravitational
field as that of classical GR and, thus, the same PPN
parameters, except for the inclusion of a new term in the
vectorial sector of the metric, namely
0i = 2ḟ (∇× V )i , (1)
where ḟ acts as a coupling parameter of CS theory and
Vi is a PPN potential. We also show that this solution
can be alternatively obtained by finding a formal solu-
tion to the modified field equations and performing a PN
expansion, as is done in PN theory. The full solution
is further shown to satisfy the additional CS constraint,
which leads to equations of motion given only by the di-
vergence of the stress-energy tensor.
The CS correction to the metric found here leads to
an interesting interpretation of CS gravity and forces us
to consider a new type of coupling. The interpretation
consists of thinking of the field that sources the CS cor-
rection as a fluid that permeates all of spacetime. Then
the CS correction in the metric is due to the “dragging” of
such a fluid by the motion of the source. Until now, cou-
plings of the CS correction to the angular momentum of
the source had been neglected by the string theory com-
munity. Similarly, curl-type terms had also been consid-
ered unnecessary in the traditional PPN framework, since
previous alternative gravity theories had not required it.
As we shall show, in CS gravity and thus in string the-
ory, such a coupling is naturally occurring. Therefore, a
proper PPN mapping requires the introduction of a new
curl-type term with a corresponding new PPN parameter
of the type of Eq. (1).
A modification to the gravitational field leads natu-
rally to corrections of the standard predictions of GR.
In order to illustrate such a correction, we consider the
CS term in the gravito-electro-magnetic analogy [40, 41],
where we find that the CS correction accounts for a mod-
ification of gravitomagnetism. Furthermore, we calculate
the modification to the acceleration of point particles and
the frame dragging effect in the precession of gyroscopes.
We find that these corrections are given by
δai = −
c2 r2
δΩi = − ḟ
c3 r2
ni − v
, (2)
where m and v are the mass and velocities of the source,
while r is the distance to the source and ni = xi/r is a
unit vector, with · and × the flat-space scalar and cross
products. Both corrections are found to be naturally en-
hanced in regions of high spacetime curvature. We then
conclude that experiments that measure the gravitomag-
netic sector of the metric either in the weak-field (such as
Gravity Probe B [42]) and particularly in the non-linear
regime, will lead to a direct constraint on the CS cou-
pling parameter ḟ . In this paper we develop the details
of how to calculate these corrections, while the specifics
of how to actually impose a constraint, which depend
on the experimental setup, are beyond the scope of this
paper.
The remainder of this paper deals with the details of
the calculations discussed in the previous paragraphs.
We have divided the paper as follows: Sec. II describes
the basics of the PPN framework; Sec. III discusses CS
modified gravity, the modified field equations and com-
putes a formal solution; Sec. IV expands the field equa-
tions to second order in the metric perturbation; Sec. V
iteratively solves the field equations in the PN approx-
imation and finds the PPN parameters of CS gravity;
Sec. VI discusses the correction to the acceleration of
point particles and the frame dragging effect; Sec. VII
concludes and points to future research.
The conventions that we use throughout this work are
the following: Greek letters represent spacetime indices,
while Latin letters stand for spatial indices only; semi-
colons stand for covariant derivatives, while colons stand
for partial derivatives; overhead dots stand for deriva-
tives with respects to time. We denote uncontrolled re-
mainders with the symbol O(A), which stands for terms
of order A. We also use the Einstein summation con-
vention unless otherwise specified. Finally, we use ge-
ometrized units, where G = c = 1, and the metric signa-
ture (−,+,+,+).
II. THE ABC OF PPN
In this section we summarize the basics of the PPN
framework, following [6]. This framework was first
developed by Eddington, Robertson and Schiff [1, 6],
but it came to maturity through the seminal papers of
Nordtvedt and Will [2, 3, 4, 5]. In this section, we de-
scribe the latter formulation, since it is the most widely
used in experimental tests of gravitational theories.
The goal of the PPN formalism is to allow for com-
parisons of different metric theories of gravity with each
other and with experiment. Such comparisons become
manageable through a slow-motion, weak-field expansion
of the metric and the equations of motion, the so-called
PN expansion. When such an expansion is carried out
to sufficiently high but finite order, the resultant solu-
tion is an accurate approximation to the exact solution
in most of the spacetime. This approximation, however,
does break down for systems that are not slowly-moving,
such as merging binary systems, or weakly gravitating,
such as near the apparent horizons of black hole binaries.
Nonetheless, as far as solar system tests are concerned,
the PN expansion is not only valid but also highly accu-
rate.
The PPN framework employs an order counting-
scheme that is similar to that used in multiple-scale anal-
ysis [43, 44, 45, 46]. The symbol O(A) stands for terms of
order ǫA, where ǫ ≪ 1 is a PN expansion parameter. For
convenience, it is customary to associate this parameter
with the orbital velocity of the system v/c = O(1), which
embodies the slow-motion approximation. By the Virial
theorem, this velocity is related to the Newtonian poten-
tial U via U ∼ v2, which then implies that U = O(2) and
embodies the weak-gravity approximation. These expan-
sions can be thought of as two independent series: one
in inverse powers of the speed of light c and the other in
positive powers of Newton’s gravitational constant.
Other quantities, such as matter densities and deriva-
tives, can and should also be classified within this order-
counting scheme. Matter density ρ, pressure p and spe-
cific energy density Π, however, are slightly more com-
plicated to classify because they are not dimensionless.
Dimensionlessness can be obtained by comparing the
pressure and the energy density to the matter density,
which we assume is the largest component of the stress-
energy tensor, namely p/ρ ∼ Π/ρ = O(2). Derivatives
can also be classified in this fashion, where we find that
∂t/∂x = O(1). Such a relation can be derived by noting
that ∂t ∼ vi∇i, which comes from the Euler equations of
hydrodynamics to Newtonian order.
With such an order-counting scheme developed, it is
instructive to study the action of a single neutral particle.
The Lagrangian of this system is given by
L = (gµνu
−g00 − 2g0ivi − gijvivj
where uµ = dxµ/dt = (1, vi) is the 4-velocity of the parti-
cle and vi is its 3-velocity. From Eq. (3), note that knowl-
edge of L to O(A) implies knowledge of g00 to O(A), g0i
to O(A − 1) and gij to O(A − 2). Therefore, since the
Lagrangian is already known to O(2) (the Newtonian so-
lution), the first PN correction to the equations of motion
requires g00 to O(4), g0i to O(3) and gij to O(2). Such
order counting is the reason for calculating different sec-
tors of the metric perturbation to different PN orders.
A PPN analysis is usually performed in a particular
background, which defines a particular coordinate sys-
tem, and in an specific gauge, called the standard PPN
gauge. The background is usually taken to be Minkowski
because for solar system experiments deviations due to
cosmological effects are negligible and can, in principle,
be treated as adiabatic corrections. Moreover, one usu-
ally chooses a standard PPN frame, whose outer regions
are at rest with respect to the rest frame of the universe.
Such a frame, for example, forces the spatial sector of the
metric to be diagonal and isotropic [6]. The gauge em-
ployed is very similar to the PN expansion of the Lorentz
gauge of linearized gravitational wave theory. The differ-
ences between the standard PPN and Lorentz gauge are
of O(3) and they allow for the presence of certain PPN
potentials in the vectorial sector of the metric perturba-
tion.
The last ingredient in the PPN recipe is the choice of
a stress-energy tensor. The standard choice is that of a
perfect fluid, given by
T µν = (ρ+ ρΠ+ p)uµuν + pgµν . (4)
Such a stress-energy density suffices to obtain the PN
expansion of the gravitational field outside a fluid body,
like the Sun, or of compact binary system. One can show
that the internal structure of the fluid bodies can be ne-
glected to 1 PN order by the effacement principle [39] in
GR. Such effacement principle might actually not hold
in modified field theories, but we shall study this subject
elsewhere [47].
With all these machinery, on can write down a super-
metric [6], namely
g00 = −1 + 2U − 2βU2 − 2ξΦW + (2γ + 2 + α3 + ζ1
− 2ξ)Φ1 + 2 (3γ − 2β + 1 + ζ2 + ξ)Φ2
+ 2 (1 + ζ3)Φ3 + 2 (3γ + 3ζ4 − 2ξ)Φ4 − (ζ1 − 2ξ)A,
g0i = −
(4γ + 3 + α1 − α2 + ζ1 − 2ξ)Vi
(1 + α2 − ζ1 + 2ξ)Wi,
gij = (1 + 2γU) δij , (5)
where δij is the Kronecker delta and where the PPN
potentials (U,ΦW ,Φ1,Φ2,Φ3,Φ4,A, Vi,Wi) are defined
in Appendix A. Equation (5) describes a super-metric
theory of gravity, because it reduces to different met-
ric theories, such as GR or other alternative theo-
ries [6], through the appropriate choice of PPN param-
eters (γ, β, ξ, α1, α2, α3, ζ1, ζ2, ζ3, ζ4). One could obtain
a more general form of the PPN metric by performing
a post-Galilean transformation on Eq. (5), but such a
procedure shall not be necessary in this paper.
The super-metric of Eq. (5) is parameterized in terms
of a specific number of PPN potentials, where one usually
employs certain criteria to narrow the space of possible
potentials to consider. Some of these restriction include
the following: the potentials tend to zero as an inverse
power of the distance to the source; the origin of the co-
ordinate system is chosen to coincide with the source,
such that the metric does not contain constant terms;
and the metric perturbations h00, h0i and hij transform
as a scalar, vector and tensor. The above restrictions
are reasonable, but, in general, an additional subjective
condition is usually imposed that is based purely on sim-
plicity: the metric perturbations are not generated by
gradients or curls of velocity vectors or other generalized
vector functions. As of yet, no reason had arisen for re-
laxing such a condition, but as we shall see in this paper,
such terms are indeed needed for CS modified theories.
What is the physical meaning of all these parame-
ters? One can understand what these parameters mean
by calculating the generalized geodesic equations of mo-
tion and conservation laws [6]. For example, the param-
eter γ measures how much space-curvature is produced
by a unit rest mass, while the parameter β determines
how much “non-linearity” is there in the superposition
law of gravity. Similarly, the parameter ξ determines
whether there are preferred-location effects, while αi rep-
resent preferred-frame effects. Finally, the parameters ζi
measure the amount of violation of conservation of to-
tal momentum. In terms of conservation laws, one can
interpret these parameters as measuring whether a the-
ory is fully conservative, with linear and angular momen-
tum conserved (ζi and αi vanish), semi-conservative, with
linear momentum conserved (ζi and α3 vanish), or non-
conservative, where only the energy is conserved through
lowest Newtonian order. One can verify that in GR,
γ = β = 1 and all other parameters vanish, which implies
that there are no preferred-location or frame effects and
that the theory is fully conservative.
A PPN analysis of an alternative theory of gravity then
reduces to mapping its solutions to Eq. (5) and then de-
termining the PPN parameters in terms of intrinsic pa-
rameters of the theory. The procedure is simply as fol-
lows: expand the modified field equations in the metric
perturbation and in the PN approximation; iteratively
solve for the metric perturbation to O(4) in h00, to O(3)
in h0i and to O(2) in hij ; compare the solution to the
PPN metric of Eq. (5) and read off the PPN parameters
of the alternative theory. We shall employ this procedure
in Sec. V to obtain the PPN parameters of CS gravity.
III. CS GRAVITY IN A NUTSHELL
In this section, we describe the basics of CS gravity,
following mainly [29, 30]. In the standard CS formalism,
GR is modified by adding a new term to the gravitational
action. This term is given by [30]
SCS =
d4xf (⋆R R) , (6)
where mpl is the Planck mass, f is a prescribed external
quantity with units of squared mass (or squared length
in geometrized units), R is the Ricci scalar and the star
stands the dual operation, such that
R⋆R =
Rαβγδǫ
αβµνRγδµν , (7)
with ǫµνδγ the totally-antisymmetric Levi-Civita tensor
and Rµνδγ the Riemann tensor.
Such a correction to the gravitational action is inter-
esting because of the unavoidable parity violation that
is introduced. Such parity violation is inspired from CP
violation in the standard model, where such corrections
act as anomaly-canceling terms. A similar scenario oc-
curs in string theory, where the Green-Schwarz anomaly
is canceled by precisely such a CS correction [32], al-
though CS gravity is not exclusively tied to string the-
ory. Parity violation in CS gravity inexorably leads to
birefringence in gravitational propagation, where here
we mean that different polarization modes obey differ-
ent propagation equations but travel at the same speed,
that of light [29, 30, 36, 47]. If CS gravity were to
lead to polarization modes that travel at different speeds,
then one could use recently proposed experiments [17]
to test this effect, but such is not the case in CS grav-
ity. Birefringent gravitational waves, and thus CS grav-
ity, have been proposed as possible explanations to the
cosmic-microwave-background (CMB) anisotropies [36],
as well as the baryogenesis problem during the inflation-
ary epoch [33].
The magnitude of the CS correction is controlled by the
externally-prescribed quantity f , which depends on the
specific theory under consideration. When we consider
CS gravity as an effective quantum theory, then the cor-
rection is suppressed by some mass scale M , which could
be the electro-weak scale or some other scale, since it is
unconstrained. In the context of string theory, the quan-
tity f has been calculated only in conservative scenar-
ios, where it was found to be suppressed by the Planck
mass. In other scenarios, however, enhancements have
been proposed, such as in cosmologies where the string
coupling vanishes at late times [48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58], or where the field that generates f cou-
ples to spacetime regions with large curvature [59, 60] or
stress-energy density [28, 47]. For simplicity, we here as-
sume that this quantity is spatially homogeneous and its
magnitude is small but non-negligible, so that we work
to first order in the string-theoretical correction. There-
fore, we treat ḟ as an independent perturbation parame-
ter, [70] unrelated to ǫ, the PN perturbation parameter.
The field equations of CS modified gravity can be ob-
tained by varying the action with respect to the metric.
Doing so, one obtains
Gµν + Cµν = 8πTµν , (8)
where Gµν is the Einstein tensor, Tµν is a stress-energy
tensor and Cµν is the Cotton tensor. The latter tensor is
defined via
Cµν = −
(µDαRν)β + (Dσf,τ )
⋆Rτ (µ
where parenthesis stand for symmetrization, g is the de-
terminant of the metric, Da stands for covariant differ-
entiation and colon subscripts stand for partial differen-
tiation.
Formally, the introduction of such a modification to
the field equations leads to a new constraint, which is
compensated by the introduction of the new scalar field
degree of freedom f . This constraint originates by re-
quiring that the divergence of the field equations vanish,
namely
DµCµν =
Dνf (
⋆RR) = 0, (10)
where the divergence of the Einstein tensor vanished by
the Bianchi identities. If this constraint is satisfied, then
the equations of motion for the stress-energy DµT
µν are
unaffected by CS gravity. A common source of confusion
is that Eq. (10) is sometimes interpreted as requiring that
R⋆R also vanish, which would then force the correction
to the action to vanish. However, this is not the case be-
cause, in general, f is an exact form (d2f = 0) and, thus,
Eq. (10) only implies an additional constraint that forces
all solutions to the field equations to have a vanishing
The previous success of CS gravity in proposing plau-
sible explanations to important cosmological problems
prompts us to consider this extension of GR in the weak-
field regime. For this purpose, it is convenient to rewrite
the field equations in trace-reversed form, since this form
is most amenable to a PN expansion. Doing so, we find,
Rµν + Cµν = 8π
Tµν −
, (11)
where the trace of the Cotton tensor vanishes identically
and T = gµνT
µν is the four dimensional trace of the
stress-energy tensor. To linear order, the Ricci and Cot-
ton tensors are given by [30]
Rµν = −
�hµν +O(h)2,
Cµν = −
ǫ̃0αβ(µ�ηhν)β,α +O(h)2, (12)
where ǫ̃αβγδ is the Levi-Civita symbol, with conven-
tion ǫ̃0123 = +1, and �η = −∂2t + ηij∂i∂j is the flat
space D’Alambertian, with ηµν the Minkowski metric. In
Eq. (12), we have employed the Lorentz gauge condition
α = h,µ/2, where h = g
µνhµν is the four dimensional
trace of the metric perturbation.
The Cotton tensor changes the characteristic behav-
ior of the Einstein equations by forcing them to become
third order instead of second order. Third-order par-
tial differential equations are common in boundary layer
theory [43]. However, in CS gravity, the third-order con-
tributions are multiplied by a factor of f and we shall
treat this function as a small independent expansion pa-
rameter. Therefore, the change in characteristics in the
modified field equations can also be treated perturba-
tively, which is justified because eventhough ḟ might be
enhanced by standard model currents, extra dimensions
or a vanishing string coupling, it must still carry some
type of mass suppression.
The trace-reversed form of the field equations is useful
because it allows us to immediately find a formal solution.
Inverting the D’Alambertian operator we obtain
Hµν = −16π �−1η
Tµν −
+O(h)2, (13)
where we have defined an effective metric perturbation
Hµν ≡ hµν + ḟ ǫ̃0αβ(µhν)β,α. (14)
Note that this formal solution is identical to the formal
PN solution to the field equations in the limit ḟ → 0.
Also note that the second term in Eq. (14) is in essence
a curl operator acting on the metric. This antisymmetric
operator naturally forces the trace of the CS correction
to vanish, as well as the 00 component and the symmetric
spatial part.
From the formal solution to the modified field equa-
tions, we immediately identify the only two possible non-
zero CS contributions : a coupling to the vector compo-
nent of the metric h0i; and coupling to the transverse-
traceless part of the spatial metric hTTij . The latter
has already been studied in the gravitational wave con-
text [29, 30, 47] and it vanishes identically if we require
the spatial sector of the metric perturbation to be confor-
mally flat. The former coupling is a new curl-type contri-
bution to the metric perturbation that, to our knowledge,
had so far been neglected both by the string theory and
PPN communities. In fact, as we shall see in later sec-
tions, terms of this type will force us to introduce a new
PPN parameter that is proportional to the curl of certain
PPN potentials.
Let us conclude this section by pushing the formal so-
lution to the modified field equations further to obtain a
formal solution in terms of the actual metric perturba-
tion hµν . Combining Eqs. (13) and (14) we arrive at the
differential equation
hµν+ḟ ǫ̃
(µhν)β,α = −16π �−1η
Tµν −
+O(h)2.
Since we are searching for perturbations about the gen-
eral relativistic solution, we shall make the ansatz
hµν = h
µν + ḟζµν +O(h)2, (16)
where h
µν is the solution predicted by general relativity
h(GR)µν ≡ −16π �−1η
Tµν −
, (17)
and where ζµν is an unknown function we are solving for.
Inserting this ansatz into Eq. (15) we obtain
ζµν+ḟ ǫ̃
(µζν)β,α = 16πǫ̃
(µ∂α�
Tν)β −
gν)βT
We shall neglect the second term on the left-hand side be-
cause it would produce a second order correction. Such
conclusion was also reached when studying parity viola-
tion in GR to explain certain features of the CMB [35].
We thus obtain the formal solution
ζµν = 16πǫ̃
(µ∂α�
Tν)β −
gν)βT
and the actual metric perturbation to linear order be-
comes
hµν = −16π �−1η
Tµν −
+ 16πḟ ǫ̃kℓi�−1η
δi(µTν)ℓ,k −
δi(µην)ℓT,k
+O(h)2,
where we have used some properties of the Levi-Civita
symbol to simplify this expression. The procedure pre-
sented here is general enough that it can be directly ap-
plied to study CS gravity in the PPN framework, as well
as possibly find PN solutions to CS gravity.
IV. PN EXPANSION OF CS GRAVITY
In this section, we perform a PN expansion of the field
equations and obtain a solution in the form of a PN se-
ries. This solution then allows us to read off the PPN
parameters by comparing it to the standard PPN super-
metric [Eq. (5)]. In this section we shall follow closely the
methods of [6] and [61] and indices shall be manipulated
with the Minkowski metric, unless otherwise specified.
Let us begin by expanding the field equations to second
order in the metric perturbation. Doing so we find that
the Ricci and Cotton tensors are given to second order
Rµν = −
�ηhµν − 2hσ(µ,ν)σ + h,µν
2hρ(µ,ν)λ − hµν,ρλ − hρλ,µν
hρλ,µhρλ,ν + h
ν,λ (21)
− hρµ,λhρν,λ +
h,λ − 2hλρ,ρ
hµν,λ − 2hλ(µ,ν)
+O(h)3,
Cµν = −
ǫ̃0αβ(µ
�ηhν)β,α − hσβ,αν)σ
ǫ̃0αβ(µ
�ηhν)β,α − hσβ,αν)σ
2hν)(λ,α) − hλα,ν)
β − 2hσ(λ,β)σ + h,βλ
− 2Q̂Rν)β,α
ǫ̃σαβ(µ
2h0(σ,τ) − hστ,0
hτ [β,α]ν) − hν)[β,α]τ
hµλǫ̃
0αβ(λ
�ηhν)β,α − hσβ,αν)σ
ǫ̃0αβ(µ
β,α − hσβ,ασλ)
hνλ +O(h)3. (22)
where index contraction is carried out with the
Minkowski metric and where we have not assumed any
gauge condition. The operator Q̂(·) takes the quadratic
part of its operand [of O(h)2] and it is explained in more
detail in Appendix B, where the derivation of the expan-
sion of the Cotton tensor is presented in more detail. In
this derivation, we have used the definition of the Levi-
Civita tensor
ǫαβγδ = (−g)1/2ǫ̃αβγδ =
ǫ̃αβγδ +O(h)2, (23)
ǫαβγδ = −(−g)−1/2ǫ̃αβγδ = −
ǫ̃αβγδ +O(h)2.
Note that the PN expanded version of the linearized Ricci
tensor of Eq. (21) agrees with previous results [6]. Also
note that if the Lorentz condition is enforced, several
terms in both expressions vanish identically and the Cot-
ton tensor to first order reduces to Eq. (12), which agrees
with previous results [30].
Let us now specialize the analysis to the standard PPN
gauge. For this purpose, we shall impose the following
gauge conditions
k − 1
h,j = O(4),
k − 1
hkk,0 = O(5), (24)
where hkk is the spatial trace of the metric perturbation.
Note that the first equation is the PN expansion of one of
the Lorentz gauge conditions, while the second equation
is not. This is the reason why the previous equations
where not expanded in the Lorentz gauge. Nonetheless,
such a gauge condition does not uniquely fix the coordi-
nate system, since we can still perform an infinitesimal
gauge transformation that leaves the modified field equa-
tions invariant. One can show that the Lorentz and PPN
gauge are related to each other by such a gauge transfor-
mation. In the PPN gauge, then, the Ricci tensor takes
the usual form
R00 = −
∇2h00 −
h00,ih00,
hijh00,ij +O(6),
R0i = −
∇2h0i −
h00,0i +O(5),
Rij = −
∇2hij +O(4), (25)
which agrees with previous results [6], while the Cotton
tensor reduces to
C00 = O(6),
C0i = −
ḟ ǫ̃0kli∇2h0l,k +O(5),
Cij = −
ḟ ǫ̃0kl(i∇2hj)l,k +O(4), (26)
where ∇ = ηij∂i∂j is the Laplacian of flat space [see Ap-
pendix B for the derivation of Eq. (26).] Note again the
explicit appearance of two coupling terms of the Cotton
tensor to the metric perturbation: one to the transverse-
traceless part of the spatial metric and the other to the
vector metric perturbation. The PN expansions of the
linearized Ricci and Cotton tensor then allow us to solve
the modified field equations in the PPN framework.
V. PPN SOLUTION OF CS GRAVITY
In this section we shall proceed to systematically solve
the modified field equation following the standard PPN
iterative procedure [6]. We shall begin with the 00 and
ij components of the metric to O(2), and then proceed
with the 0i components to O(3) and the 00 component
to O(4). Once all these components have been solved for
in terms of PPN potentials, we shall be able to read off
the PPN parameters adequate to CS gravity.
A. h00 and hij to O(2)
Let us begin with the modified field equations for the
scalar sector of the metric perturbation. These equations
are given to O(2) by
∇2h00 = −8πρ, (27)
because T = −ρ. Eq. (27) is the Poisson equation, whose
solution in terms of PPN potentials is
h00 = 2U +O(4). (28)
Let us now proceed with the solution to the field equa-
tion for the spatial sector of the metric perturbation.
This equation to O(2) is given by
∇2hij + ḟ ǫ̃0kl(i∇2hj)l,k = −8πρδij , (29)
where note that this is the first appearance of a Cotton
tensor contribution. Since the Levi-Civita symbol is a
constant and ḟ is only time-dependent, we can factor
out the Laplacian and rewrite this equation in terms of
the effective metric Hij as
∇2Hij = −8πρδij , (30)
where, as defined in Sec. III,
Hij = hij + ḟ ǫ̃0kl(ihj)l,k. (31)
The solution of Eq. (30) can be immediately found in
terms of PPN potentials as
Hij = 2Uδij +O(4), (32)
which is nothing but Eq. (13). Recall, however, that in
Sec. III we explicitly used the Lorentz gauge to simplify
the field equations, whereas here we are using the PPN
gauge. The reason why the solutions are the same is that
the PPN and Lorentz gauge are indistinguishable to this
order.
Once the effective metric has been solved for, we can
obtain the actual metric perturbation following the pro-
cedure described in Sec. III. Combining Eq. (31) with
Eq. (32), we arrive at the following differential equation
hij + ḟ ǫ̃
(ihj)l,k = 2Uδij. (33)
We look for solutions whose zeroth-order result is that
predicted by GR and the CS term is a perturbative cor-
rection, namely
hij = 2Uδij + ḟζij , (34)
where ζ is assumed to be of O(ḟ)0. Inserting this ansatz
into Eq. (33) we arrive at
ζij + ḟ ǫ̃
(iζj)l,k = 0, (35)
where the contraction of the Levi-Civita symbol and the
Kronecker delta vanished. As in Sec. III, note that the
second term on the left hand side is a second order cor-
rection and can thus be neglected to discover that ζij
vanishes to this order.
The spatial metric perturbation to O(2) is then simply
given by the GR prediction without any CS correction,
namely
hij = 2Uδij +O(4). (36)
Physically, the reason why the spatial metric is unaf-
fected by the CS correction is related to the use of a
perfect fluid stress-energy tensor, which, together with
the PPN gauge condition, forces the metric to be spa-
tially conformally flat. In fact, if the spatial metric were
not flat, then the spatial sector of the metric perturba-
tion would be corrected by the CS term. Such would
be the case if we had pursued a solution to 2 PN or-
der, where the Landau-Lifshitz pseudo-tensor sources a
non-conformal correction to the spatial metric [39], or if
we had searched for gravitational wave solutions, whose
stress-energy tensor vanishes [29, 36]. In fact, one can
check that, in such a scenario, Eq. (30) reduces to that
found by [29, 30, 36, 47] as ρ → 0. We have then found
that the weak-field expansion of the gravitational field
outside a fluid body, like the Sun or a compact binary, is
unaffected by the CS correction to O(2).
B. h0i to O(3)
Let us now look for solutions to the field equations for
the vector sector of the metric perturbation. The field
equations to O(3) become
∇2h0i +
h00,0i +
ḟ ǫ̃0kli∇2h0l,k = 16πρvi, (37)
where we have used that T 0i = −T0i. Using the lower
order solutions and the effective metric, as in Sec. III, we
obtain
∇2H0i + U,0i = 16πρvi, (38)
where the vectorial sector of the effective metric is
H0i = h0i +
ḟ ǫ̃0klih0l,k. (39)
We recognize Eq. (38) as the standard GR field equation
to O(3), except that the dependent function is the effec-
tive metric instead of the metric perturbation. We can
thus solve this equation in terms of PPN potentials to
obtain
H0i = −
Wi, (40)
where we have used that the superpotential X satisfies
X,0j = Vj − Wj (see Appendix A for the definitions.)
Combining Eq. (39) with Eq. (40) we arrive at a differ-
ential equation for the metric perturbation, namely
h0i +
ḟ ǫ̃0klih0l,k = −
Wi. (41)
Once more, let us look for solutions that are perturbation
about the GR prediction, namely
h0i = −
Wi + ḟζi, (42)
where we again assume that ζi is of O(ḟ)0. The field
equation becomes
ḟ (∇× ζ)i =
(∇× V )i +
(∇×W )i
where (∇×A)i = ǫijk∂jAk is the standard curl operator
of flat space. As in Sec. III, note once more that the
second term on the left-hand side is again a second order
correction and we shall thus neglect it. Also note that
the curl of the Vi potential happens to be equal to the
curl of the Wi potential. The solution for the vectorial
sector of the actual gravitational field then simplifies to
h0i = −
Wi + 2ḟ (∇× V )i +O(5). (44)
We have arrived at the first contribution of CS mod-
ified gravity to the metric for a perfect fluid source.
Chern-Simons gravity was previously seen to couple to
the transverse-traceless sector of the metric perturbation
for gravitational wave solutions [29, 30, 36, 47]. The CS
correction is also believed to couple to Noether vector
currents, such as neutron currents, which partially fueled
the idea that this correction could be enhanced. However,
to our knowledge, this correction was never thought to
couple to vector metric perturbations. From the analy-
sis presented here, we see that in fact CS gravity does
couple to such terms, even if the matter source is neu-
trally charged. The only requirement for such couplings
is that the source is not static, ie. that the object is ei-
ther moving or spinning relative to the PPN rest frame
so that the PPN vector potential does not vanish. The
latter is suppressed by a relative O(1) because in the far
field the velocity of a compact object produces a term of
O(3) in Vi, while the spin produces a term of O(4). In
a later section, we shall discuss some of the physical and
observational implications of such a modification to the
metric.
C. h00 to O(4)
A full analysis of the PPN structure of a modified the-
ory of gravity requires that we solve for the 00 component
of the metric perturbation to O(4). The field equations
to this order are
∇2h00 −
h00,ih00,i +
hijh00,ij = 4πρ [1
v2 − U + 1
, (45)
where the CS correction does not contribute at this order
(see Appendix B.) Note that the h0i sector of the metric
perturbation to O(3) does not feed back into the field
equations at this order either. The terms that do come
into play are the h00 and hij sectors of the metric, which
are not modified to lowest order by the CS correction.
The field equation, thus, reduce to the standard one of
GR, whose solution in terms of PPN potentials is
h00 = 2U − 2U2 + 4Φ1 + 4Φ2 +2Φ3 +6Φ4 +O(6). (46)
We have thus solved for all components of the metric per-
turbation to 1 PN order beyond the Newtonian answer,
namely g00 to O(4), g0i to O(3) and gij to O(2).
D. PPN Parameters for CS Gravity
We now have all the necessary ingredients to read off
the PPN parameters of CS modified gravity. Let us begin
by writing the full metric with the solutions found in the
previous subsections:
g00 = −1 + 2U − 2U2 + 4Φ1 + 4Φ2 + 2Φ3 + 6Φ4 +O(6),
g0i = −
Wi + 2ḟ (∇× V )i +O(5),
gij = (1 + 2U) δij +O(4). (47)
One can verify that this metric is indeed a solution of
Eqs. (27), (29), (37) and (45) to the appropriate PN or-
der and to first order in the CS coupling parameter. Also
note that the solution of Eq. (47) automatically satis-
fies the constraint ⋆RR = 0 to linear order because the
contraction of the Levi-Civita symbol with two partial
derivatives vanishes. Such a solution is then allowed in
CS gravity, just as other classical solutions are [62], and
the equations of motion for the fluid can be obtained di-
rectly from the covariant derivative of the stress-energy
tensor.
We can now read off the PPN parameters of the CS
modified theory by comparing Eq. (5) to Eq. (47). A vi-
sual inspection reveals that the CS solution is identical
to the classical GR one, which implies that γ = β = 1,
ζ = 0 and α1 = α2 = α3 = ξ1 = ξ2 = ξ3 = ξ4 = 0 and
there are no preferred frame effects. However, Eq. (5)
contains an extra term that cannot be modeled by the
standard PPN metric of Eq. (5), namely the curl contri-
bution to g0i. We then see that the PPN metric must be
enhanced by the addition of a curl-type term to the 0i
components of the metric, namely
g0i ≡ −
(4γ + 3 + α1 − α2 + ζ1 − 2ξ)Vi
(1 + α2 − ζ1 + 2ξ)Wi + χ (r∇× V )i , (48)
where χ is a new PPN parameter and where we have
multiplied the curl operator by the radial distance to
the source, r, in order to make χ a proper dimension-
less parameter. Note that there is no need to introduce
any additional PPN parameters because the curl of Wi
equals the curl of Vi. In fact, we could have equally pa-
rameterized the new contribution to the PPN metric in
terms of the curl of Wi, but we chose not to because Vi
appears more frequently in PN theory. For the case of
CS modified gravity, the new χ parameter is simply
χ = 2
, (49)
which is dimensionless since ḟ has units of length. If an
experiment could measure or place bounds on the value
of χ, then ḟ could also be bounded, thus placing a con-
straint on the CS coupling parameter.
VI. ASTROPHYSICAL IMPLICATIONS
In this section we shall propose a physical interpreta-
tion to the CS modification to the PPN metric and we
shall calculate some GR predictions that are modified by
this correction. This section, however, is by no means a
complete study of all the possible consequences of the CS
correction, which is beyond the scope of this paper.
Let us begin by considering a system of A nearly spher-
ical bodies, for which the gravitational vector potentials
are simply [6]
V i =
viA +
, (50)
W i =
(vA · nA)niA +
where mA is the mass of the Ath body, rA is the field
point distance to the Ath body, niA = x
A/rA is a unit
vector pointing to the Ath body, vA is the velocity of the
Ath body and J iA is the spin-angular momentum of the
Ath body. For example, the spin angular momentum for
a Kerr spacetime is given by J i = m2ai, where a is the
dimensionless Kerr spin parameter. Note that if A = 2
then the system being modeled could be a binary of spin-
ning compact objects, while if A = 1 it could represent
the field of the sun or that of a rapidly spinning neutron
star or pulsar.
In obtaining Eq. (50), we have implicitly assumed a
point-particle approximation, which in classical GR is
justified by the effacement principle. This principle pos-
tulates that the internal structure of bodies contributes
to the solution of the field equations to higher PN order.
One can verify that this is indeed the case in classical
GR, where internal structure contributions appear at 5
PN order. In CS gravity, however, it is a priori unclear
whether an analogous effacement principle holds because
the CS term is expected to couple with matter current via
standard model-like interactions. If such is the case, it is
possible that a “mountain” on the surface of a neutron
star [63] or an r-mode instability [64, 65, 66] enhances
the CS contribution. In this paper, however, we shall
neglect these interactions, and relegate such possibilities
to future work [47].
With such a vector potential, we can calculate the CS
correction to the metric. For this purpose, we define the
correction δg0i ≡ g0i − g(GR)0i , where g
0i is the GR
prediction without CS gravity. We then find that the CS
corrections is given by
δg0i = 2
(vA × nA)i −
(JA · nA)
where the · operator is the flat-space inner product and
where we have used the identities ǫ̃ijk ǫ̃klm = δilδjm −
δimδjl and ǫ̃ilk ǫ̃jlm = 2δij . Note that the first term of
Eq. (51) is of O(3), while the second and third terms
are of O(4) as previously anticipated. Also note that ḟ
couples both to the spin and orbital angular momentum.
Therefore, whether the system under consideration is the
Solar system (vi of the Sun is zero while J i is small), the
Crab pulsar (vi is again zero but J i is large) or a binary
system of compact objects (neither vi nor J i vanish),
there will in general be a non-vanishing coupling between
the CS correction and the vector potential of the system.
From the above analysis, it is also clear that the CS
correction increases with the non-linearity of the space-
time. In other words, the CS term is larger not only for
systems with large velocities and spins, but also in re-
gions near the source. For a binary system, this fact
implies that the CS correction is naturally enhanced
in the last stages of inspiral and during merger. Note
that this enhancement is different from all previous en-
hancements proposed, since it does not require the pres-
ence of charge [28, 47], a fifth dimension with warped
compactifications [59, 60], or a vanishing string cou-
pling [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. Un-
fortunately, the end of the inspiral stage coincides with
the edge of the PN region of validity and, thus, a com-
plete analysis of such a natural enhancement will have to
be carried out through numerical simulations.
In the presence of a source with the vector potentials of
Eq. (50), we can write the vectorial sector of the metric
perturbation in a suggestive way, namely
g0i =
viA −
vA − v(eff)A
(eff)
A · nA − 2
(eff)
where we have defined an effective velocity and angular
momentum vector via
viA(eff) = v
A − 6ḟ
J iA(eff) = J
A − ḟmAviA, (53)
or in terms of the Newtonian orbital angular momentum
= rA × pA and linear momentum piA(N) = mAv
LiA(eff) = L
A(N) − 6ḟ (nA × JA)
J iA(eff) = J
A − ḟpiA. (54)
From this analysis, it is clear that the CS corrections
seems to couple to both a quantity that resembles the or-
bital and the spin angular momentum vector. Note that
when the spin angular momentum vanishes the vectorial
metric perturbation is identical to that of a spinning mov-
ing fluid, but where the spin is induced by the coupling
of the orbital angular momentum to the CS term.
The presence of an effective CS spin angular momen-
tum in non-spinning sources leads to an interesting phys-
ical interpretation. Let us model the field that sources
ḟ as a fluid that permeates all of spacetime. This field
could be, for example, a model-independent axion, in-
spired by the quantity introduced in the standard model
to resolve the strong CP problem [67]. In this scenario,
then the fluid is naturally “dragged” by the motion of any
source and the CS modification to the metric is nothing
but such dragging. This analogy is inspired by the er-
gosphere of the Kerr solution, where inertial frames are
dragged with the rotation of the black hole. In fact, one
could push this analogy further and try to construct the
shear and bulk viscosity of such a fluid, but we shall not
attempt this here. Of course, this interpretation is to be
understood only qualitatively, since its purpose is only
to allow the reader to picture the CS modification to the
metric in physical terms.
An alternative interpretation can be given to the CS
modification in terms of the gravito-electro-magnetic
(GED) analogy [40, 41], which shall allow us to eas-
ily construct the predictions of the modified theory. In
this analogy, one realizes that the PN solution to the
linearized field equations can be written in terms of a
potential and vector potential, namely
ds2 = − (1− 2Φ)dt2 − 4 (A · dx) dt+ (1 + 2Φ) δijdxidxj ,
where Φ reduces to the Newtonian potential U in the
Newtonian limit [41] and Ai is a vector potential related
to the metric via Ai = −g0i/4. One can then construct
GED fields in analogy to Maxwell’s electromagnetic the-
ory via
Ei = − (∇Φ)i − ∂t
Bi = (∇×A)i , (56)
which in terms of the vectorial sector of the metric per-
turbation becomes
Ei = − (∇Φ)i + 1
Bi = −1
(∇× g)i , (57)
where we have defined the vector gi = g0i. The geodesic
equations for a test particle then reduce to the Lorentz
force law, namely
F i = −mEi − 2m (v ×B)i . (58)
We can now work out the effect of the CS correction
on the GED fields and equations of motion. First note
that the CS correction only affects g. We can then write
the CS modification to the Lorentz force law by defining
δai = ai − ai
, where ai
is the acceleration vector
predicted by GR, to obtain,
δai =
δġi +
(v × δΩ)i , (59)
where we have defined the angular velocity
δΩi = (∇× δg)i . (60)
The time derivative of the vector gi is of O(5) and can
thus be neglected, but the angular velocity cannot and it
is given by
δΩi = −
3 (vA · nA)niA − viA
, (61)
which is clearly of O(3). Note that although the first
term between square brackets cancels for circular orbits
because niA is perpendicular to v
A to Newtonian order,
the second term does not. The angular velocity adds a
correction to the acceleration of O(4), namely
δai = −3
(vA · nA) (vA × nA)i , (62)
which for a system in circular orbit vanishes to Newto-
nian order. One could use this formalism to find the
perturbations in the motion of moving objects by inte-
grating Eq. (62) twice. However, for systems in a circular
orbit, such as the Earth-Moon system or compact bina-
ries, this correction vanishes to leading order. Therefore,
lunar ranging experiments [68] might not be able to con-
straint ḟ .
Another correction to the predictions of GR is that
of the precession of gyroscopes by the so-called Lense-
Thirring or frame-dragging effect. In this process, the
spin angular momentum of a source twists spacetime in
such a way that gyroscopes are dragged with it. The
precession angular velocity depends on the vector sector
of the metric perturbation via Eq. (61). Thus, the full
Lense-Thirring term in the precession angular velocity of
precessing gyroscopes is
ΩiLT = −
J iA(eff) − 3n
JA(eff) · nA
. (63)
Note that this angular velocity is identical to the GR
prediction, except for the replacement J iA → J iA(eff).
In CS modified gravity, then, the Lense-Thirring effect
is not only produced by the spin angular momentum of
the gyroscope but also by the orbital angular momentum
that couples to the CS correction. Therefore, if an ex-
periment were to measure the precession of gyroscopes
by the curvature of spacetime (see, for example, Gravity
Probe B [42]) one could constraint ḟ and thus some in-
trinsic parameters of string theory. Note, however, that
the CS correction depends on the velocity of the bodies
with respect to the inertial PPN rest-frame. In order to
relate these predictions to the quantities that are actually
measured in the experiment, one would have to transform
to the experiment’s frame, or perhaps to a basis aligned
with the direction of distant stars [6].
Are there other experiments that could be performed
to measure such a deviation from GR? Any experiment
that samples the vectorial sector of the metric would in
effect be measuring such a deviation. In this paper, we
have only discussed modifications to the frame-dragging
effect and the acceleration of bodies through the GED
analogy, but this need not be the only corrections to
classical GR predictions. In fact, any predictions that de-
pends on g0i indirectly, for example via Christoffel sym-
bols, will probably also be modified unless the corrections
is fortuitously canceled. In this paper, we have laid the
theoretical foundations of the weak-field correction to the
metric due to CS gravity and studied some possible cor-
rections to classical predictions. A detailed study of other
corrections is beyond the scope of this paper.
VII. CONCLUSION
We have studied the weak-field expansion of the solu-
tion to the CS modified field equations in the presence
of a perfect fluid PN source in the point particle limit.
Such an expansion required that we linearize the Ricci
and Cotton tensor to second order in the metric pertur-
bation without any gauge assumption. An iterative PPN
formalism was then employed to solve for the metric per-
turbation in this modified theory of gravity. We have
found that CS gravity possesses the same PPN parame-
ters as those of GR, but it also requires the introduction
of a new term and PPN parameter that is proportional
to the curl of the PPN vector potentials. Such a term
is enhanced in non-linear scenarios without requiring the
presence of standard model currents, large extra dimen-
sions or a vanishing string coupling.
We have proposed an interpretation for the new term
in the metric produced by CS gravity and studied some
of the possible consequences it might have on GR predic-
tions. The interpretation consists of picturing the field
that sources the CS term as a fluid that permeates all of
spacetime. In this scenario, the CS term is nothing but
the “dragging” of the fluid by the motion of the source.
Irrespective of the validity of such an interpretation, the
inclusion of a new term to the weak-field expansion of
the metric naturally leads to corrections to the standard
GR predictions. We have studied the acceleration of
point particles and the Lense-Thirring contribution to
the precession of gyroscopes. We have found that both
corrections are proportional to the CS coupling parame-
ter and, therefore, experimental measurement of these ef-
fects might be used to constraint CS and, possibly, string
theory.
Future work could concentrate on studying further the
non-linear enhancement of the CS correction and the
modifications to the predictions of GR. The PPN analysis
performed here breaks down very close to the source due
to the use of a point particle approximation in the stress
energy tensor. One possible research route could consists
of studying the CS correction in a perturbed Kerr back-
ground [69]. Another possible route could be to analyze
other predictions of the theory, such as the perihelion
shift of Mercury or the Nordtvedt effect. Furthermore,
in light of the imminent highly-accurate measurement of
the Lense-Thirring effect by Gravity Probe B, it might be
useful to revisit this correction in a frame better-adapted
to the experimental setup. Finally, the CS modification
to the weak-field metric might lead to non-conservative
effects and the breaking of the effacement principle [47],
which could be studied through the evaluation of the
gravitational pseudo stress-energy tensor. Ultimately, it
will be experiments that will determine the viability of
CS modified gravity and string theory.
Acknowledgments
The authors acknowledge the support of the Center
for Gravitational Wave Physics funded by the National
Science Foundation under Cooperative Agreement PHY-
01-14375, and support from NSF grants PHY-05-55-628.
We would also like to thank Cliff Will for encouraging
one of us to study the PPN formalism and Pablo Laguna
for suggesting one of us to look into the PPN expansion
of CS gravity. We would also like to thank R. Jackiw,
R. Wagoner and Ben Owen for enlightening discussions
and comments.
APPENDIX A: PPN POTENTIALS
In this appendix, we present explicit expressions for
the PPN potentials used to parameterize the metric in
Eq. (5). These potentials are the following:
|x− x′|
d3x′,
ρ′v′i
|x− x′|
d3x′,
ρ′v′j(x− x′)j(x− x′)i
|x− x′|3
d3x′,
ρ′ρ′′
(x− x′)i
|x− x′|3
(x′ − x′′)i
|x− x′′| −
(x− x′′)i
|x′ − x′′|
d3x′d3x′′,
ρ′v′2
|x− x′|
d3x′, Φ2 ≡
ρ′U ′
|x− x′|
d3x′,
|x− x′|
d3x′, Φ4 ≡
|x− x′|
d3x′,
v′i (x− x′)
|x− x′| d
ρ′|x− x′|d3x′. (A1)
These potentials satisfy the following relations
∇2U = −4πρ, ∇2Vi = −4πρvi,
∇2Φ1 = −4πρv2, ∇2Φ2 = −4πρU,
∇2Φ3 = −4πρΠ, ∇2Φ4 = −4πp,
∇2X = −2U (A2)
The potential X is sometimes referred to as the super-
potential because it acts as a potential for the Newtonian
potential.
APPENDIX B: LINEARIZATION OF THE
COTTON TENSOR
In this appendix, we present some more details on the
derivation of the linearized Cotton tensor to second order.
We begin with the definition of the Cotton tensor [30] in
terms of the symmetrization operator, namely
Cµν = − 1√
(Dσf) ǫ
σαβ(µDαR
β + (Dστf)
Rτ(µ|σ|ν)
Using the symmetries of the Levi-Civita and Riemann
tensor, as well as the fact that f depends only on time,
we can simplify the Cotton tensor to
Cµν = (−g)−1ḟ
ǫ̃0αβ(µRν)β,α + ǫ̃
0αβ(µΓ
Γ0στ ǫ̃
σαβ(µRν)τ αβ
. (B2)
Noting that the determinant of the metric is simply g =
−1 + h, so that (−g)−1 = 1 + h, we can identify four
terms in the Cotton tensor
A = ḟ ǫ̃
0αβ(µ
L̂Rν)β,α
B = ḟ ǫ̃
0αβ(µhρρ
L̂Rν)β,α
C = ḟ ǫ̃
0αβ(µ
L̂Rλβ
ǫ̃σαβ(µ
L̂Γ0στ
L̂Rν)ταβ
E = ḟ ǫ̃
0αβ(µ
Q̂Rν)β,α
, (B3)
where the L̂ operator stands for the linear part of its
operand, while the Q̂ operator isolates the quadratic part
of its operand. For example, if we act L̂ and Q̂ on (1+h)n,
where n is some integer, we obtain
L̂(1 + h)n
= nh,
Q̂(1 + h)n
n(n− 1)
h2.(B4)
Let us now compute each of these terms separately.
The first four terms are given by
A = −
ǫ̃0αβ(µ
β,α − hσβ,νασ
B = −
hǫ̃0αβ(µ
β,α − hσβ,νασ
C = −
ǫ̃0αβ(µ
hν)λ,α + h
α,λ − hλα,ν)
β − hσλ,βσ − hσβ,λσ + h,λβ
ǫ̃σαβ(µ
2h0(σ,τ) − hστ,0
hτ [β,α]
ν − hν [β,α]τ
The last term of the Cotton tensor is simply the deriva-
tive of the Ricci tensor which we already calculated to
second order in Eq. (21). In order to avoid notation clut-
ter, we shall not present it again here, but instead we
combine all the Cotton tensor pieces to obtain
Cµν = − ḟ
ǫ̃0αβ(µ
β,α − hσβ,ασν)
ǫ̃0αβ(µ
β,α − hσβ,ασν)
2hν)(λ,α) − hλα,ν)
β − 2hσ(λ,β)σ + h,βλ
− 2Q̂Rν)β,α
ǫ̃σαβ(µ
2h0(σ,τ) − hστ,0
hτ [β,α]
ν) − hν)[β,α]τ
+O(h)3
where its covariant form is
Cµν = −
ǫ̃0αβ(µ
�ηhν)β,α − hσβ,αν)σ
ǫ̃0αβ(µ
�ηhν)β,α − hσβ,αν)σ
2hν)(λ,α) − hλα,ν)
β − 2hσ(λ,β)σ + h,βλ
− 2Q̂Rν)β,α + hνλ
β,α − hσβ,ασλ)
ǫ̃σαβ(µ
2h0(σ,τ) − hστ,0
hτ [β,α]ν) − hν)[β,α]τ
hµλǫ̃
0αβ(λ
�ηhν)β,α − hσβ,αν)σ
+O(h)3. (B7)
For the PPN mapping of CS modified gravity, only the
00 component of the metric is needed to second order,
which implies we only need C00 to O(h)2. This compo-
nent is given by
C00 =
ǫ̃ijk0
2h0(i,ℓ) − hiℓ,0
hℓ[k,j]0 − h0[k,j]ℓ
h0ℓǫ̃
0jk(ℓ
�ηh0k,j − hik,j0i
+O(h)3, (B8)
where in fact the last term vanishes due to the PPN gauge
condition. Note that this term is automatically of O(6),
which is well beyond the required order we need in h00.
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[70] Formally, ḟ by itself is dimensional, so it cannot be
treated as an expansion parameter. A dimensionless pa-
rameter can, however, be constructed by dividing ḟ by
some length scale squared.
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