It is possible to create a relativistic theory of gravity more simple than General Relativity via Jefimenko's equations? I've came across the Jefimenko's equations, which are the general solutions of Maxwell's Equations and are compatible with Special Relativity.
They are formulated in terms of the retarded potential:
$$
t_r = t - \frac{|\mathbf r - \mathbf{r'}|}{c} \\
\varphi(\mathbf r, t) = \frac{1}{4\pi\varepsilon_0} \int{\frac{\rho(\mathbf r', t_r)}{|\mathbf r - \mathbf{r'}|} \; \mathrm d^3 \mathbf r'} 
$$
where $t_r$ is the time taken for an observer at $\mathbf{r'}$, to see a change in the potential (or in the field).
It is known that Newton's Theory of Gravity, is not compatible with special relativity and, also, the field propagates throughout space immediately. Would proposing the following be a folly:
$$
\phi(\mathbf r, t) = G \int{\frac{\rho(\mathbf r', t_r)}{|\mathbf r - \mathbf{r'}|} \; \mathrm d^3 \mathbf r'} 
$$
Is there something that doesn't allow you to do this? I know proposing this equation, is pure phenomenology and intuition (because if it describes EM-waves, why not adjust it to "predict" gravity ones?). There it goes my question: would this predict waves of the $\mathbf g$ field? For the Maxwell Equations to predict EM-waves, you need the $\mathbf E$ and $\mathbf B$ fields (two fields), but now, you have only the $\mathbf g$ field.
I did a simulation with only one field that satisfies the potential I proposed and we get the following. It reminds me of  grvitational waves in GR. Here is the simulation of a normal $\mathbf g$ field without retarded potential. We might conclude that a wave equation is hiding in here, no?
My question is: Is it possible to describe relativistic gravity without GR using this potential? Why it haven't been done before? Are there any inconsistencies in the reasoning?
 A: Here is a relevant quote from the Great Master himself [1]:

I came a first step closer to a solution of the problem when I tried
to treat the law of gravitation within the framework of special
relativity theory. Like most authors at that time, I tried to
formulate a field law for gravity, since the introduction of action at
a distance was no longer possible, at least in any natural way, due to
the elimination of the concept of absolute simultaneity.
The simplest and most natural procedure was to retain the scalar
Laplacian gravitational potential and to add a time derivative to the
Poisson equation in such a way that the requirements of the special
theory would be satisfied. In addition, the law of motion for a point
mass in a gravitational field had to be adjusted to the requirements
of special relativity. Just how to do this was not so clear, since the
inertial mass of a body might depend on the gravitational potential.
In fact, this was to be expected in view of the principle of
mass-energy equivalence. However, such considerations led to a result
which made me extremely suspicious. According to classical mechanics,
the vertical motion of a body in a vertical gravitational field is
independent of the horizontal motion. This is connected with the fact
that in such a gravitational field the vertical acceleration of a
mechanical system, or of its center of mass, is independent of its
kinetic energy. Yet, according to the theory which I was
investigating, the gravitational acceleration was not independent of
the horizontal velocity or of the internal energy of the system.
This in turn was not consistent with the well known experimental fact
that all bodies experience the same acceleration in a gravitational
field. This law, which can also be formulated as the law of equality
of inertial and gravitational mass, now struck me in its deep
significance. I wondered to the highest degree about its validity and
supposed it to be the key to a deeper understanding of inertia and
gravitation. I did not seriously doubt its strict validity even
without knowing the result of the beautiful experiment of Eötvös,
which — if I remember correctly — I only heard of later. I now gave up
my previously described attempt to treat gravitation in the framework
of the special theory as inadequate. It obviously did not do justice
to precisely the most fundamental property of gravitation.

[1] "Einstein: The Origins of the General Theory of Relativity", Lecture 1933, quoted in Straumann: General Relativity, pp3-4
A: *

*You could introduce the correction term simply in the Newtonian physics and the topic was an active research area.


*GR was basically the study of (differential) geometry i.e. the curves and the surface, so it might smell similar when you plot the vector fields.


*However, notice that the electromagnetic fields were themself the mass-less field and the integrand the Jefimenko's equations had already taken into account of the 4 space retardation with the Minkowski signature so it's already a simulation of the massless field that satisfied the special relativity.
