533 Page 2 of 11 Eur. Phys. J. C (2018) 78:533
itational field. The theories predict specific values for the
classical gravitational tests, see in addition [53–57]. To com-
pare 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 com-
pare them using the Parametrized Post Newtonian (PPN) for-
malism [58,59]. The PPN formalism was devised for con-
fronting 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 cos-
mological 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 alter-
natives 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 the-
ories 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 draw-
back 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 gravita-
tional 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 com-
pare between scalar theories of gravitation.
This paper is organized as follows. In Sect. 2, we define
the general stratified scalar theory by postulating a gravita-
tional 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 theo-
ries of gravitation. In Sect. 5 we conclude.
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 nec-
essary 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 ds
M
given by
ds
2
M
= γ
ab
dx
a
dx
b
, (1)
where γ
ab
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 γ
ab
need not
be η
ab
= 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, (2)
(∇
a
t)(∇
b
t)γ
ab
=−1, (3)
where ∇
a
is the covariant derivative with respect to the metric
γ
ab
.
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
γ
ab
∇
a
Φ∇
b
Φ
c
2
, (4)
where is the d’Alembertian, ∇
a
is the covariant derivative,
both with respect to the metric γ
ab
, G is Newton’s gravita-
tional 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 =−T
ab
u
a
u
b
where u
a
is the four-velocity of the source with respect to γ
ab
,
or ρ =−γ
ab
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 =−T
ab
u
a
u
b
[50,61]. This entails that ρ =−γ
ab
T
ab
is the most realistic
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