#### Stratified scalar field theories of gravitation with self-energy term and effective particle Lagrangian

Eur. Phys. J. C
Stratified scalar field theories of gravitation with self-energy term and effective particle Lagrangian
Diogo P. L. Bragança 0 1
José P. S. Lemos 1
0 Present Address: Department of Physics, Stanford Institute of Theoretical Physics-SITP, Stanford University , Stanford, CA 94305 , USA
1 Centro de Astrofísica e Gravitação-CENTRA, Departamento de Física, Instituto Superior Técnico-IST, Universidade de Lisboa-UL , Av. Rovisco Pais 1, 1049-001 Lisbon , Portugal
We construct a general stratified scalar theory of gravitation from a field equation that accounts for the self-interaction of the field and a particle Lagrangian, and calculate its post-Newtonian parameters. Using this general framework, we analyze several specific scalar theories of gravitation and check their predictions for the solar system post-Newtonian effects.
1 Introduction
Newtonian gravitation is a scalar theory in which the
gravitational interaction is described by a gravitational scalar field
or potential Φ, that satisfies a Poisson equation. It is
complemented by Newton’s second law of mechanics for the
trajectories of particles moving in the gravitational potential Φ.
With the advent of special relativity it became clear that
energy and mass are equivalent and so any form of energy,
like the energy contained in any physical field, produces
gravitational field and also gravitates. The gravitational field,
possessing itself gravitational energy, should thus produce
additional gravitational field and thus the Poisson law should
be modified to contain this field self-energy. A consistent
generalization of Newtonian’s gravitation accounting for the
weight of gravitational self-energy has been performed by
several authors, see e.g. [1–4], see also [5] for a discussion
of gravitational self-energy terms.
Special relativity also implied that any proposed theory of
gravitation should be relativistic. The simplest way is to put
Newtonian’s gravitation in a relativistic form. Scalar
gravitational theories were initiated by Nordström with the
gravitational potential being treated as a scalar field on a Minkowski
background [6] and then modifying it into a scalar theory in
a conformal background [7], with its full structure displayed
by Einstein and Fokker [8] who showed that it is a covariant
scalar theory in a conformally flat space-time, i.e.,
gravitational effects can be seen as a consequence of having a curved
metric generated by a scalar gravitational potential, see also
the review by Laue [9]. The idea of conformal theories of
gravitation were resurrected by Littlewood [10], whose
theory arose the interest of Pirani [11], and further developed by
Gürsey [12], Bergmann [13], and Dowker [14]. Several other
studies analyzed their properties [15–30]. Reviews, analyses,
and modifications of these scalar theories can also be seen in
[31–39]. Scalar conformal theories of gravitation are simple,
interesting, didactic, but suffer from the problem that due
to the conformal character and the coupling of
electromagnetism with gravitation, they yield a zero light deflection in
the presence of a gravitational field.
An additional interesting set of theories of gravitation that
involve a scalar field are the stratified scalar theories. These
theories are not purely scalar, they also possess a universal
reference frame, where a universal time t , and thus a universal
vector field, is defined with the space slices composing a
stratum that is conformally flat. Einstein was the first to compose
such a theory in which the velocity of light is a variable
quantity that plays the role of the gravitational potential [40] and
was also developed by Abraham [41]. Other stratified
theories were composed by Papapetrou [42–44], Yilmaz [45,46],
Whitrow and Morduch [47], Page and Tupper [48], Rosen
[49], Ni [50,51], and Broekaert [52]. In Ni [50] a review of
stratified scalar theories is given. Stratified scalar theories can
bypass the light deflection problem. Further, these theories,
due to the existence of a preferred vector field, break Lorentz
symmetry and thus could be candidates to a fundamental
theory where Lorentz symmetry is not essential.
Each of these particular scalar theories, be they conformal
or stratified, have a field equation and a particle Lagrangian
that together give the particle trajectories in the external
gravitational field. The theories predict specific values for the
classical gravitational tests, see in addition [53–57]. To
compare the different theories in these tests, it should then be
enough to compare just a few specific parameters of the field
equation and of the particle Lagrangian of each theory. They
also have different post-Newtonian effects, and we can
compare them using the Parametrized Post Newtonian (PPN)
formalism [58,59]. The PPN formalism was devised for
confronting general relativity and other theories of gravitation
with observational data [58–61]. In order to easily compare
theories defined by a field equation and particle Lagrangian,
a systematic formalism to get PPN parameters from some
parameters of the field equation and of particle Lagrangian
would be of great help. General relativity, so far the most
successful theory, is a tensorial theory of gravitation. But, it
is believed that both at the quantum gravity level and at
cosmological scales there are corrections to general relativity.
On one hand, these corrections bring complexity and give
rise to new fields that enter the scene in the same footing as
the tensor field of general relativity. Indeed, extensions of
general relativity admitting, in addition to the metric tensor
field, vector and scalar fields, have been proposed as
alternatives theories of gravitation [58,59]. On the other hand,
general relativity could be emergent from some underlying
simpler phenomena, such as atoms of spacetime, perhaps in
the form of simple scalar or vectorial fields. Thus, scalar
theories of the type just mentioned, or even vectorial theories,
can be sought for, although vectorial theories of gravitation
modeled in Maxwell electromagnetism suffer from the
drawback of admitting negative field energy not admissible for the
gravitational field.
In this paper we propose a general stratified scalar theory
of gravitation postulating first a field equation that accounts
for the self-interaction of the gravitational field and second
a Lagrangian for describing particle motion in the
gravitational field. Conformal scalar theories of gravitation are also
obtained as a specific case of this general stratified scalar
theory. We also give a direct method to compute the two
PPN parameters that affect the classical solar system tests,
namely, Mercury’s perihelion precession, light deflection,
gravitational redshift, and the Shapiro effect. This method
reads the PPN parameters directly from the field equation
and the particle Lagrangian. Using specific scalar theories of
gravitation we then confront them with experimental data.
This method provides a simple, straightforward way to
compare between scalar theories of gravitation.
This paper is organized as follows. In Sect. 2, we define
the general stratified scalar theory by postulating a
gravitational field equation that accounts for self-interaction and a
particle Lagrangian that gives the particle motion. In Sect. 3,
we calculate the weak field limit and two post-Newtonian
parameters of the theory. In Sect. 4, we use the general scalar
theory to compute PPN parameters for specific scalar
theories of gravitation. In Sect. 5 we conclude.
(
1
)
(
2
)
(
3
)
(
4
)
2 Building a general stratified scalar field theory of gravitation with self-interaction
2.1 Postulates and equations of the theory
In a general stratified scalar gravitational theory it is
necessary to define an existing prior background structure [50]
(see also [58]). Moreover, to build a gravitational theory, one
needs to know how a particle moves in a gravitational field
and how gravitation is generated by matter. We thus have to
(
1
) define an existing prior background structure, (
2
) give a
general field equation, and (
3
) give a particle Lagrangian for
the particle’s trajectories.
The stratified scalar gravitational theory we are going to
work with is defined on a background Minkowski spacetime,
with line element dsM given by
dsM2 = γabd x a d x b,
where γab is the flat spacetime metric, and a and b run from
0 to 3. Note that the coordinates defining Eq. (
1
) need not
be Minkowskian coordinates and so in general γab need not
be ηab = diag (
−1, 1, 1, 1
). We stratify the theory using a
universal time parameter t . This parameter is a scalar field
satisfying the following equations
∇b∇a t = 0,
(∇a t )(∇bt ) γ ab = −1,
where ∇a is the covariant derivative with respect to the metric
γab.
To define the field equation for the gravitational scalar field
Φ, we generalize the Poisson equation in order to account for
a self-interaction of the gravitational field. We then assume
that the field equation is given by
Φ = 4π Gρ − k
where is the d’Alembertian, ∇a is the covariant derivative,
both with respect to the metric γab, G is Newton’s
gravitational constant, ρ is the gravitational source density and k is
a dimensionless constant.
The gravitational source density ρ is a scalar and thus can
be defined in two different ways, namely, ρ = −Tabua ub
where ua is the four-velocity of the source with respect to γab,
or ρ = −γab T ab. Although in our work we do not need to
specify ρ, it is important to note that the Kreuzer experiment
is not compatible with the first possibility ρ = −Tabua ub
[50,61]. This entails that ρ = −γab T ab is the most realistic
choice to be used in Eq. (
4
). In the case of electromagnetic
radiation, we then have ρ = 0, which means that light does
not generate gravitational field. As we shall see, this does not
necessarily imply that light is not bent by gravity; this is only
the case for conformal scalar theories of gravitation.
Note also that we explicitly include a self-interaction term
with coupling constant k, but even though different scalar
theories of gravity yield a particular k, we will let k have
a priori any value. See below a derivation of the modified
Poisson equation Eq. (
4
).
We now formulate how test particles behave in the theory.
For this, we impose that the particle’s trajectories are the
geodesics of a metric gab which itself is generated by the
scalar gravitational potential Φ and the universal time t . Thus,
we write quite generally [50].
ds2 = gabd xa d x b,
= −(g(Φ) − f (Φ))c2dt 2 + f (Φ)γabd xa d x b,
(
5
)
where c is the velocity of light, dt is the differential of the
universal time t , Φ is the gravitational potential, and g and f
are two scalar functions of Φ. Since gab in Eq. (
5
) defines the
geodesics it must be considered the physical metric. In
general, this physical metric breaks Lorentz symmetries (even
though these are preserved in the background metric). This
Lorentz symmetry breaking arises also in standard
cosmology (e.g. the Cosmic Microwave Background indicates a
preferred set of reference frames), and that may indicate that
at a fundamental level there must be a breaking of Lorentz
symmetries, and as such these stratified theories should not
be discarded a priori. Note that light propagates in the null
geodesics of this metric, and therefore, in the general case,
does not follow straight lines. For the physical metric, the
action S for a particle trajectory is then
where dτ = 1c −gabd xa d x b, and m is the mass of the
particle. From the standard definition of a Lagrangian L,
namely,
S = −mc2
dτ,
S =
L d x 0,
in a given coordinate system xa , with x 0 being some time
coordinate, we get, from Eqs. (
5
), (
6
), and (
7
), the following
particle Lagrangian,
L =
− mc (g(Φ) − f (Φ))c2
dt
d x 0
2
d xa d x b
− f (Φ)γab d x 0 d x 0 .
(
6
)
(
7
)
(
8
)
In a stratified theory, without loss of generality, it is clearly
convenient, from (
8
), that x 0 should be identified with t ,
x 0 = t . In this case there are further simplifications, namely,
γ00 = −c2, and imposing further that the metric is static one
can set γ0α = 0, where α is a spatial index running from 1
to 3. Using these facts, the Lagrangian (
8
) becomes
(
9
)
(
10
)
(
11
)
L = −mc g(Φ)c2 − f (Φ)v2,
where the particle’s 3-velocity v is defined through the
relation
v2 = γαβ ddxxα0 ddxxβ0 .
Since in special relativity the Lagrangian for a particle is [58]
L = −mc√c2 − v2, we see that to get the correct special
relativity limit, we have to impose the following condition on the
functions f and g, f (0) = g(0) = 1, where without loss of
generality we are assuming that no gravitational field means
Φ = 0 rather than Φ = constant. See below a derivation of
the particle Lagrangian given in Eq. (
9
).
It is important to remark that this formalism can be used to
study conformally flat theories as well. In fact, to cancel the
influence of the universal time t , it is sufficient to set g = f
in Eq. (
8
). Choosing a reference frame in which γ0α = 0, the
Lagrangian becomes
L = −mc f (Φ) c2 − v2,
which is simply Eq. (
9
) with g = f , as expected, i.e., it
represents a conformal scalar field theory. We will use this fact
throughout the paper. Finally, note that, in this case, Eq. (
5
)
shows that light propagates in straight lines.
2.2 Derivation of the Poisson equation with a self-energy
term and of the particle Lagrangian
2.2.1 Derivation of the Poisson equation with a self-energy
term
We can motivate our field equation (
4
) through the following
scheme. In electrodynamics, the electromagnetic energy is
stored in the field with a positive energy density ρEM given
by ρEM = 21 | E|2 + | B|2 , where E and B are the electric
and magnetic fields, respectively. A similar expression can be
obtained for the gravitational field in Newtonian gravitation.
Indeed, using Poisson’s equation,
∇2Φ = 4π Gρ ,
one can show that the total gravitational potential energy
Egrav in a given volume V can be written as (see e.g. [1])
(
12
)
energy of the particle measured in the local observer’s frame
is E = − pbub. Identifying the particle’s energy E with
the Hamiltonian H , we can then find the expression for the
Lagrangian L.
To start we assume that gαβ (x α) is a flat spatial metric, not
necessarily Euclidean, afterwards we will relax this
assumption. Thus, calling γαβ a general flat spatial metric, we put
(
13
)
gαβ = γαβ .
Egrav = 21 V ρΦ dV = V − |∇8πΦG|2 dV , where ρ is the
matter density, Φ is the gravitational potential, G is Newton’s
gravitational constant, ∇ is the gradient operator, and an
integration by parts has been performed. Therefore, in the
Newtonian theory of gravitation, one may define a gravitational
field energy density as
|∇Φ|2 .
ρgrav ≡ − 8π G
It is interesting to note that this is a negative definite energy
density. This stems from the fact that gravity in the Newtonian
theory is exclusively an attractive force.
Now, we assume that the energy of the gravitational
field can also gravitate. Thus, Eq. (
12
) with Eq. (
13
) yields
1
∇2Φ = 4π Gρ − 2 |∇Φ|2. Notice that this approach is not
self-consistent (for self-consistent constructions see [4,16]);
nonetheless, this modified Poisson equation is valid to the
first post-Newtonian order, which is the order we are
interested in for PPN formalism purposes. Moreover, to put in
a relativistic setting, and in order to get a Lorentz scalar,
we replace the Laplacian in Poisson’s equation (
12
) by the
d’Alembertian and generalize the gradient ∇ to the covariant
derivative ∇a . Then we get Φ = 4π Gρ − 21 γ ab∇acΦ2∇bΦ ,
where γab is the Minkowski metric. In order to consider
a more general theory, we let the factor 21 that multiplies
γ ab∇aΦ∇bΦ be undetermined, call it k, obtaining thus the
c2
sought for equation
We can motivate the definition of our particle Lagrangian (
9
)
from an expression for the energy of a particle in a curved
static spacetime.
A static spacetime with metric gab as can be written
ds2 = gab d xa d x b = g00dt 2 + gαβ d x αd x β ,
(
15
)
where Latin indices a, b run from 0 to 3, Greek indices α, β
correspond to the spatial part of the metric and run from 1 to
3, g00 = g00(x α), and in general gαβ = gαβ (x α). We assume
asymptotic flatness.
We now calculate the appropriate expression for the
observed energy E of a particle measured by a static observer
in the metric given by Eq. (
15
). The four-velocity va of the
particle is va = ddxτa , where τ is the proper time of the
particle, and its four-momentum pa is pa = m gabvb. If an
observer has four-velocity ub, say, then the total observed
Consider now the metric given in Eq. (
15
) and consider
a coordinate system in which the metric (
16
) is diagonal,
for instance in static spherical coordinates. Then, we have
tdhdxatat ddtτthe=pardtdixctalve’0s, fwohuerrevevlo0cit=y cdadτnt. bTehewrsipttaetnialascovma
po=nents of the velocity vα defined with respect to the time
t are vα = ddxtα and the square of the spatial velocity
is v2 = γαβ vαvβ . Then, since gabva vb = −c2, we get
gabva vb = g00 v0 2 + v0 2 v2 = −c2. We can then solve
this equation for v0, getting v0 = c g00−+1v2 . We want
the particle energy measured by an inertial static observer
at infinity. Such a static observer has spatial velocity zero,
uα = 0. Since such an observer is also a test particle the
above derivation for v0 holds, but now we have to replace v0
by u0 and vα by uα = 0, an so u2 = 0, and from
asymptotically flatness g00 = −c2. So, u0 = c g−010 = 1. Since
the metric is diagonal and uα = 0, the energy expression
E = − pbub simplifies to E = −mg00 v0u0. Then using the
expressions for v0 and u0 just found we find for the energy
the expression
E = −m g00 c
−1
g00 + v2
.
We proceed by identifying this energy with the Hamiltonian
H of the particle, i.e., H = E . Writing the Hamiltonian as a
function of the spatial components of the momentum of the
particle pα = mgαbvb, we obtain
H =
−g00 p2 − g00 m2c2 ,
where here p2 = γ αβ pα pβ . This reduces to special
relativity when g00 = −c2. Now, we want the
corresponding Lagrangian L. Using the Legendre transformation
relating H and L, namely, L = ddxtα pα − H , together with the
Hamilton equation ddxtα = ∂∂pHα , we can verify that the particle
Lagrangian is
(
17
)
(
18
)
(
19
)
We generalize this approach for a non-flat static metric
whose spatial part gαβ can be put in isotropic coordinates.
We then write
where A and B are dimensionless constants. Since f (Φ)
appears already multiplied by v2 in Eq. (
9
), it suffices to
expand to order Φ/c2, so we write
(
21
)
(
22
)
(
20
)
Φ
f (Φ) = 1 − 2C c2 ,
(
24
)
gαβ = f γαβ ,
where γαβ is a flat metric, not necessarily Euclidean metric,
and f is a conformal factor. With this spatial metric, the
particle Lagrangian given in Eq. (
19
) becomes
where v2 = γαβ vαvβ . Writing −g00 ≡ g(Φ)c2 and f =
f (Φ), for some Φ and identifying this Φ with the
gravitational potential, we see that the particle Lagrangian in
Eq. (
21
) is the same as
L = −mc g(Φ)c2 − f (Φ)v2 ,
i.e., we recover the postulated particle Lagrangian given in
Eq. (
9
).
It is interesting to notice that this particle Lagrangian
Eq. (
22
), or Eq. (
9
), is the same as the one that comes
from the requirement that the trajectory in spacetime is a
geodesic of the metric gab. In fact, the particle action in
general relativity SGR is given by Sparticle GR = −mc dτ ,
which corresponds to the Lagrangian Lparticle GR = −mc ddτt .
It is immediate to verify that Lparticle GR = L. Therefore,
the particle trajectory in our theory is a geodesic of the
metric ds2 = −g(Φ)c2dt 2 + f (Φ)γαβ d x αd x β , γαβ being the
spatial flat metric. However, we have derived the Lagrangian
Eq. (
22
), or Eq. (
9
), without the requirement that a
particle follows a geodesic of spacetime and without requiring
the covariant divergence of the particle’s energy-momentum
tensor to vanish. This result is certainly very interesting.
3 Weak field limit of the theory
3.1 Choices of g and f
In order to compare the stratified scalar theory given in
Eqs. (
4
) and (
9
) with experiment, we take advantage of the
Parametrized Post Newtonian (PPN) formalism. To use it, we
Taylor expand f and g to second order in 1/c2. We define
second order terms as having the following factors, vc4 , Φcv42 ,
4
or Φc42 . The function g(Φ) is expanded as
Φ Φ2
g(Φ) = 1 + 2 A c2 + 2B c4 ,
(
23
)
G M
Φ = − r
(G M )2
− k 2c2r 2 ,
with the condition that at infinity Φ = 0.
where C is a dimensionless constant. To have the correct
Newtonian limit in the Lagrangian (
9
), we must have A = 1.
From now on, we therefore consider A = 1.
In order to further understand the physical meaning of
B and C , we interpret the Lagrangian Eq. (
9
) as containing
an interaction term. For that we should expand the
particle Lagrangian given in Eq. (
9
) to second order and
analyze how these parameters affect the interaction part of the
Lagrangian. We assume that to second order
approximation, we can write our Lagrangian as the sum of a free
special relativistic part Lparticle SR plus an interaction part
Lint, L = Lparticle SR + Lint = −mc2 1 − vc22 + Lint.
Expanding the special relativistic term to second order, we
get L = −mc2 + 21 mv2 + 81 m vc24 + Lint . On the other
hand, expanding the particle Lagrangian given by Eq. (
9
),
4
we get L = −mc2 + 21 mv2 1 − (2C + 1) cΦ2 + 81 m vc2 −
m Φ + B − 21 Φc22 . Comparing these two equations for L
gives the interaction Lagrangian to the relevant order, Lint =
L − LSR = −m Φ + B − 21 Φc22 − 21 mv2(2C + 1) cΦ2 .
Analyzing this equation, we can say first that B affects the
effective gravitational mass of the particle. This is in line with
Nordström’s interpretation that gravitational mass is affected
by the gravitational field itself [7]. Second, that C affects
the interaction between the particle’s kinetic energy and the
gravitational field.
3.2 Spherical symmetric solution of the field equation
We have to solve the field equation to obtain the gravitational
potential to second order. We solve Eq. (
4
) in a spherically
symmetric vacuum, i.e., ρ = 0, using spherical coordinates
(r, θ , φ), obtaining ddΦr = r2+ kGcMr , where M is the
gravitaG M
2
tional mass of the central body. Up to order 1/c2 this yields
dΦ
dr =
G M
r 2 + k
(G M )2
c2r 3 .
Integrating Eq. (
25
) we obtain This corresponds to a potential
(
25
)
(
26
)
3.3 PPN parameters
We may now substitute Eq. (
26
) into Eqs. (
23
) and (
24
) to
get
(
27
)
(
28
)
(
29
)
G M (G M )2
g(Φ) = 1 − 2 c2r + (2B − k) c4r 2 ,
G M
f (Φ) = 1 + 2C c2r ,
where we kept only the terms to the order previously
mentioned.
We want to verify how the parameters B, C , and k are
related with PPN parameters β and γ defined in the standard
PPN metric on a static, spherically symmetric spacetime in
isotropic coordinates by
ds2 = −
c2 − 2 GrM
G M
+ 1 + 2γ c2r
+ 2β
(G M )2
c2r 2
dt 2
γαβ d x αd x β ,
where γαβ is a flat 3-metric.
Using the correspondence between g00 and f , and g(Φ)
and f (Φ) and comparing with the standard PPN metric
Eq. (
29
), we immediately identify the following relations
ity, whether stratified scalar field theories or conformally flat
scalar theories with only one metric potential.
In the next section, we study consistent theories of
gravitation with only one metric potential and calculate for each
theory the parameters B, C and k to verify if they predict the
correct solar system effects.
4 Application to scalar field theories of gravitation:
Stratified and conformal theories
4.1 Page and Tupper theory
The Page and Tupper theory [48] is a stratified scalar field
theory of gravitation and as such is an instance of the set
of equations given in Eqs. (
4
), (
5
), and (
9
). In the preferred
reference frame, the theory has the following field equation
and particle Lagrangian
Φ = 4π G F (Φ/c2)4ρ ,
∗
L = −mc F 2(Φ/c∗2)c∗(Φ)2 − F 2(Φ/c2)v2,
∗
respectively, where is the d’Alembertian in the Minkowski
metric, ρ is the gravitational source density, e.g. ρ =
Tabua ub where ua is the four-velocity of the source, or
ρ = Taa , and the functions F (Φ/c2) and c∗ are given by
∗
Φ
F = 1 − a1 c2 + (a1 Q + a2)
Φ 2
c2
where a1, a2, Q, and R, are dimensionless constant
parameters in the theory. Here, c∗ is interpreted as a variable speed of
light. Using Eqs. (
5
), (
9
), and (
32
), it is immediate to recover
the physical metric of the Page and Tupper theory, and we get
for the line element ds2 = F 2(Φ/c2)(−c∗2(Φ)dt 2 + d x 2 +
d y2 + d z2). ∗
The field equation Eq. (
31
) of the Page and Tupper
theory corresponds to k = 0 in Eq. (
4
). The Lagrangian in
Eq. (
32
) of the Page and Tupper theory corresponds to
g(Φ)c2 = F 2(Φ/c∗2)c∗(Φ)2 and to f (Φ) = F 2(Φ/c∗2) in
Eq. (
9
). Expanding to the post-Newtonian order, we get for
g(Φ) and f (Φ)
Φ
g(Φ) = c2 1 + (Q − 2a1) c2
Φ
f (Φ) = 1 − 2a1 c2 ,
+(a12 + 2a2 + Q2 − R)
Φ 2
c2
To be compatible with every solar system test, it is enough
to replace β and γ by the values of general relativity, that
is β = γ = 1, and guarantee that our theory satisfies the
system (
30
).
Therefore, a stratified scalar field theory of gravitation that
has
– A field equation given in this approximation by Eq. (
4
),
– A particle Lagrangian given in this approximation by
Eq. (
9
),
– Parameters that satisfy Eqs. (
30
) for β = γ = 1.
predicts correctly every solar system effect predicted by
general relativity.
Moreover, a conformally flat scalar theory of
gravitation that has a field equation given in this approximation by
Eq. (
4
), and parameters A, B, C , and k, that satisfy Eqs. (
30
)
for β = γ = 1 would also correctly predict every relativistic
solar system effect.
This approach also provides a clean and fast way to
compute the PPN parameters β and γ for most theories of
gravrespectively. Looking at Eqs. (
23
) and (
24
) this corresponds
to
1
A = 2 Q − a1,
1
B = 2 (a12 + Q2 − R) + a2,
C = a1.
Therefore, although in general there is a self-interaction term,
for the static case (which is the case we are interested in) that
term vanishes and Eq. (
42
) becomes ∇2Φ = 0. This implies
that Eq. (
42
) corresponds to k = 0 in Eq. (
4
).
The Lagrangian in Eq. (
43
) corresponds to g(Φ) = e2Φ/c2
and to f (Φ) = e−2Φ/c2 in Eq. (
9
). Expanding to the
postNewtonian order, we get for g(Φ) and f (Φ)
2Φ
g(Φ) = 1 + c2 +
2Φ
f (Φ) = 1 − c2 ,
In order to have the correct Newtonian limit, we must have
A = 1, that is Q = 2a1 + 2. Therefore, the PPN parameters
defined in Eq. (
29
), obeying the relation in Eq. (
30
), are given
by
1
β = 2 (a12 + Q2 − R) + a2,
1
= 2 (5a12 + 8a1 + 4 − R) + a2,
γ = a1.
(
37
)
(
38
)
(
39
)
(
40
)
(
41
)
(
42
)
(
43
)
(
44
)
(
45
)
The PPN parameter β has to be one to account for the solar
system tests. We see from Eq. (
40
) that the Page and Tupper
parameters R and a2 provide two degrees of freedom, so there
are many possible combinations in the theory that give the
correct value β = 1. In addition, if in the Page and Tupper
theory a1 = 1, then from Eq. (
41
) the theory has the correct
value for γ , γ = 1.
4.2 Ni’s Lagrangian-based stratified theory
This Ni’s theory [50] is a stratified scalar field theory of
gravitation and as such is an instance of the set of equations given
in Eqs. (
4
), (
5
), and (
9
). This theory has a field Lagrangian
density which yields the following field equation
∂
∂ xa
√−ggab ∂Φ
∂ x b
+ 2π(−g)1/2T ab ∂gab
∂Φ
1 ∂√−ggab ∂Φ ∂Φ
− 2 ∂Φ ∂ xa ∂ x b = 0,
L = −mc e2Φ/c2 c2 − e−2Φ/c2 v2.
Using Eqs. (
5
), (
9
), and (
43
), it is immediate to recover the
physical metric gab of this theory, and we get for the line
element ds2 = −e2Φ/c2 dt 2 + e−2Φ/c2 (d x 2 + d y2 + d z2).
In a vacuum, this equation simplifies to
√−ggab ∂2Φ 1 ∂(√−ggab) ∂Φ ∂Φ
∂ xa ∂ x b + 2 ∂Φ ∂ xa ∂ x b = 0.
Using the fact that √−g = e−2Φ/c2 , we obtain
√−ggab = diag(−e−4Φ , 1, 1, 1).
A = 1,
B = 1,
Therefore, the PPN parameters β and γ defined in Eq. (
29
),
obeying the relation in Eq. (
30
), are given by
We can conclude that this theory correctly predicts the
classical solar system tests.
4.3 Ni’s general conformally flat theory
This Ni’s theory [50] is a conformally flat theory of
gravitation and as such is an instance of the set of equations
given in Eqs. (
4
), (
5
), and (
9
). The field equation and
particle Lagrangian in this theory are given by
Φ = 4π G K (Φ)ρ ,
L = −mc e−2F(Φ)c2 − e−2F(Φ)v2,
Φ
respectively, where K (Φ) = 1 − pΦ and F (Φ) = − c2 +
q Φc42 + . . ., with p and q being two dimensionless constants.
Note that Nordström’s theory [7] is a particular case of this
theory. Using Eqs. (
5
), (
9
), and (
52
), it is immediate to recover
the physical metric of this theory, and we get for the line
element, i.e., the metric, gab = e−2F(Φ)γab, where γab is a
flat metric in Minkowski spacetime, as in Eq. (
1
).
The field equation Eq. (
51
) of the theory corresponds to
k = 0.
in Eq. (
4
). The particle Lagrangian in Eq. (
52
) corresponds
to g(Φ) = e−2F(Φ) and to f (Φ) = e−2F(Φ) in Eq. (
9
).
Expanding to the post-Newtonian order, we get for g(Φ)
(
46
)
(
47
)
(
48
)
(
49
)
(
50
)
(
51
)
(
52
)
and f (Φ)
2Φ 2Φ2
g(Φ) = 1 + c2 + (1 − q) c4 ,
2Φ
f (Φ) = 1 + c2 ,
respectively. Note, that f (Φ) also gets a (1 − q) 2cΦ42 in the
expansion, but we do not need it in the calculations. Looking
at Eqs. (
23
) and (
24
) this corresponds to
B = 1 − q,
C = −1.
β = 1 − q,
γ = −1.
Therefore, the PPN parameters defined in Eq. (
29
), obeying
the relation in Eq. (
30
), are given by
Therefore, since γ = 1, this theory does not have the
correct post-Newtonian form that could explain solar system
phenomena.
4.4 A new conformal scalar theory of gravitation in flat
spacetime
Following Freund and Nambu [16] (see also [4] for the
static case), one can build a general self-consistent relativistic
scalar theory of gravitation in flat spacetime (see Appendix
A).
In this new theory, the vacuum field equation and particle
Lagrangian are given by (see Appendix A)
1 1
Φ = − 1 − 2cΦ2 c2 (∇cΦ)(∇cΦ),
L = −mc2 1 − c2
v2
1 +
,
where h2 is a dimensionless function satisfying h2(0) = 1.
Using Eqs. (
5
), (
9
), and (
56
), it is immediate to recover the
physical metric of this theory, and we get for the line element,
i.e., the metric, gab = 1 + Φhc22(Φ) 2 γab, where γab is a flat
metric in Minkowski spacetime, as in Eq. (
1
).
The field equation Eq. (
55
) of the theory corresponds k =
1 in Eq. (
4
). The particle Lagrangian in Eq. (
56
) corresponds
to g(Φ) = f (Φ) = 1 + Φh2(Φ)/c2 2 in Eq. (
9
).
Expanding to the post-Newtonian order, and assuming that we can
expand h2 to first order in Φ, we get for g(Φ) and f (Φ)
(
57
)
(
58
)
(
53
)
(
54
)
where means differentiation with respect to Φ. Note again,
that f (Φ) also gets a h2(0) + 21 2cΦ42 in the expansion, but
we do not need it in the calculations. Looking at Eqs. (
23
)
and (
24
) this corresponds to
2Φ
f (Φ) = 1 + c2 ,
Therefore, the PPN parameters defined in Eq. (
29
), obeying
the relation in Eq. (
30
), are given by
Even though we have a free parameter h2(0) which could
be adjusted to one, this theory always yields γ = −1, and
therefore does not explain all the solar system tests. In order
to solve this problem, one could think about relaxing
equation Eq. (
55
) for self-coupling and allow a general k. This
would only change the expression of β, which would become
1
β = h2(0) + 2 − 2k ; γ would still be −1. This is why it is
so challenging to build a relativistic scalar theory of
gravitation that respects Lorentz symmetries. Since this theory is
very general, the only way to obtain the correct PPN
parameters from a scalar field theory in flat spacetime would be to
modify the field Lagrangian density by adding for instance a
term proportional to T ab(∂a Φ)(∂bΦ). The consequences of
this modification cannot be straightforwardly derived with
the formalism developed in this paper, since in this case the
effective physical metric also depends on ∂a Φ (that is f and
g are functions of Φ and ∂a Φ) and here we assumed that f
and g are only functions of Φ, because this is what was used
in the literature. The extension of this model to account for
derivative couplings is then left for future work.
5 Conclusion
In this paper, we presented a general stratified scalar field
theory of gravitation in a Minkowski background. Then, we
calculated two post-Newtonian parameters from three
general parameters of the theory B, C and k, concluding that it
is perfectly possible for such a scalar theory to explain the
four solar system tests. Finally, we used this general theory
to rapidly compute the PPN parameters β and γ for a set of
scalar theories of gravitation to verify if they agree with the
experimental tests of gravitation in the solar system.
Therefore, with this formalism, one can directly find those two
PPN parameters only from the field equation and the particle
Lagrangian of a given scalar theory of gravitation. Although
this is a very efficient method to calculate β and γ for a given
theory, it does not allow one to compute the other PPN
parameters. It would be interesting to generalize this approach to
efficiently calculate the remaining PPN parameters for scalar
theories and verify if it is possible for such a theory to explain
every phenomenon predicted by general relativity.
The stratified theories that were analyzed (Page and
Tupper’s, and Ni’s) yielded the correct PPN parameters relevant
for solar system tests. One could wonder whether this
indicates that they are valid theories, and the answer to that relies
in analyzing the remaining PPN parameters. This analysis
was done by Nordtvedt and Will [60] and Ni [50] and the
conclusion was that stratified theories cannot account for
Earthtide measurements due to the motion of the solar system
relative to the preferred frame (defined by the distant stars).
The conformal theories that were analyzed did not yield
the correct γ parameter even in very general cases. This
motivates future work on the analysis of a relativistic scalar
theory including a derivative coupling in the Lagrangian, of the
type T ab(∂a Φ)(∂bΦ). Such a theory would not have
preferred frame effects (it would respect Lorentz symmetries),
so if it predicted the correct parameters β and γ it would not
have the problem of Earth-tide measurements.
If such a scalar theory correctly predicts the outcome of
every weak field gravity experiment, then we can only rule it
out using strong gravity experiment results (e.g. LIGO,
neutron star binaries, cosmology). Note also that a scalar theory
of gravity is much simpler than general relativity, since it
describes gravity with one function instead of ten. In such
theories, unlike general relativity, it is generally possible to
define a local gravitational energy-momentum tensor, which
is always an attractive feature, and is still a problem in general
relativity.
Acknowledgements DPLB thanks a grant from Fundação para a
Ciência e Tecnologia (FCT), Portugal, through project No. UID/FIS/00099/
2013. JPSL thanks FCT for the Grant No. SFRH/BSAB/128455/2017,
Coordenação de Aperfeiçoamento do Pessoal de Nível Superior
(CAPES), Brazil, for support within the Programa CSF-PVE, Grant
No. 88887.068694/2014-00.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: A new conformal scalar field theory of gravitation in flat spacetime: criteria and equations
In this section, following the approach of Freund and Nambu
[16] (see also Franklin [4] for the static case) we build a
1
LNewt = −ρΦ − 8π G |∇Φ|2.
3. The general form of the Lagrangian is
L = Lfree + Lm + Lint,
where
Lfree =
h1(Φ)
8π G
(∂a Φ)(∂ a Φ),
with h1 being a dimensionless function to be determined
which accounts for self-interaction and satisfies h1(0) =
−1, Lm is the matter Lagrangian density, and
Φ
Lint = c2 h2(Φ)Tm,
with h2 a dimensionless free function satisfying h2(0) =
1, and Tm is the trace of the matter energy-momentum
tensor Tmab defined as √−γ Tmab = δδLγamb , γ being the
determinant of γab, and δ denotes functional variation.
4. The energy-momentum tensor for the gravitational field
is given by Noether’s expression
Tgarbav = ∂∂(L∂afrΦee) ∂ bΦ − γ abLfree,
where Lfree is the free field Lagrangian of Eq. (A.4).
5. The exact free field equation should be of the form
new conformal scalar theory of gravitation in flat Minkowski
spacetime from a set of criteria, see Sect. 4.4. These criteria
are
1. The spacetime metric is given by
ds2 = γab d x a d x b,
where γab is a flat metric not necessarily of Minkowski
form, i.e., not necessarily γab = ηab = diag (
−1, 1, 1, 1
).
2. In the Newtonian limit, the field Lagrangian density
should be equal to
Φ = κ Tgrav,
∇2Φ = −
|∇Φ|2 .
c2
in order to account explicitly for the self interaction of the
field, where Tgrav is the trace of Tgarbav and α is a coupling
constant to be determined.
6. In a static vacuum, the field equation (A.7) should
simplify to
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
With these requirements in hand, we begin by calculating
the expression for Tgarbav using Eqs. (A.4) and (A.6). We then
obtain
h1(Φ)
Tgarbav = 4π G
(∂a Φ)(∂bΦ) − 21 γ ab(∂cΦ)(∂cΦ) .
Calculating the trace yields
Tgrav = − h41π(ΦG) (∂cΦ)(∂cΦ).
First we fix the constant of proportionality κ in Eq. (A.7).
In the static, Newtonian limit, where h1(Φ) = −1, we
1
have Tgrav = 4π G |∇Φ|2. Therefore, in order to account for
Eq. (A.8) one has κ = − 4πc2G and the field equation in
vacuum is
4π G
Φ = − c2 Tgrav.
Φ = − 2hh11((ΦΦ)) (∂cΦ)(∂cΦ).
Second we determine h1. The Euler-Lagrange equation give
for the free Lagrangian Eq. (A.4),
Replacing Eqs. (A.10) and (A.12) in Eq. (A.11) yields the
following differential equation for h1, h1 = − c22 h12, which
upon integration gives, considering that h1(0) = −1,
h1(Φ) = −
1
2Φ .
1 − c2
Thus, using Eq. (A.13) in Eq. (A.10) together with (A.11),
or directly in (A.12), we obtain the field equation for the
gravitational field Φ in vacuum,
1
1
Φ = − c2 1 − c2
2Φ
Φ = − c2 1 − c2
2Φ
4π G
− c2
2Φ
1 − c2
(∂a Φ)(∂a Φ).
(∂a Φ)(∂a Φ),
The full field equation, i.e., the equation derived taking into
account Lfree and Lint in Eq. (A.3) is then
(h2 + Φh2)Tm.
(A.15)
Finally, we want to find an expression for the matter
Lagrangian from Lm and Lint in Eq. (A.3). Since we want
to compute how particles behave in the theory our matter
is represented by a point particle. To simplify the
analysis we use Minkowski coordinates, i.e., γab = ηab, ηab =
diag(
−1, 1, 1, 1
). In this case the matter Lagrangian density
is the Lagrangian density for a point particle
Lm = ρ0ηabua ub,
v2
= mδ3(x − x0) 1 − c2 ηabua ub,
where ρ0 is the scalar proper mass density, x represents
spatial coordinates and x0 the spatial position of the
particle, and ua is the particle’s four-velocity with respect to
the metric ηab. The matter energy-momentum tensor is the
energy-momentum tensor for a point particle determined
from T ab δLm [37].
m = δηab
(A.16)
(A.17)
(A.18)
(A.19)
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)
T ab
m = ρ0ua ub,
= mδ3(x − x0) 1 − vc22 ua ub.
This yields the trace
Tm = −mc2δ3(x − x0) 1 − c2 .
v2
Thus, using Eqs. (A.16) and (A.17) for the sum Lm +
Lint that appears in Eq. (A.3) we have Lm + Lint =
v2
−mc2δ3(x − x0) 1 − c2 1 + cΦ2 h2(Φ) , where we have
used ηabua ub = −c2. Integrating over all space, we get
the matter plus the interaction Lagrangian for the
particle which we simply call the particle Lagrangian L, L =
d3x (Lm + Lint), i.e.,
The field equation for the gravitational field (A.15)
together with the particle Lagrangian (A.19) are the
equations of this theory and this is all we need to know in order
to calculate the trajectory of particles.
One could make the theory even more interesting by
modifying the field Lagrangian density through the addition of a
term proportional to T ab(∂a Φ)(∂bΦ).
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