Can we construct a logically self-consistent relativistic theory of gravity just by tweaking EM? This question considers a modification of classical E&M where we simply reverse the relative sign in Maxwell's equations and change the "$q$" in the Lorentz force law to an "$m$":
$$\partial_\mu F^{\mu \nu} = -J_m^\nu, \qquad (F_i)^\mu = m_i F^\mu_{\ \ \ \nu}(x_i) (U_i)^\nu,$$
where the $i$'s in the modified Lorentz force law index the source particles. $J_m$ is now interpreted as a mass density current rather than a charge density current.
This theory is certainly completely ruled out experimentally, and as several of the answers point out, it's ambiguous regarding how massless particles should be treated, since the relativistic version of Newton's second law doesn't really apply to massless particles.
The OP asks the rather vague question "What about this approach to gravitation does not work?", so the answers go off in many different directions, none of which I think quite capture the OP's real question. Let me try to make their question more precise:

*

*A priori, is this theory a logically possible self-consistent relativistic generalization of Newtonian gravity (putting aside the treatment of massless particles)?


*(Closely related) Could it have been immediately rejected out of hand in 1915 (the year that Einstein proposed GR) based on qualitative experimental data, without even needing to do any calculations? (I'm not trying to get into the weeds of exactly what was known in the year 1915 with this question. What I really mean is, before there was any experimental evidence for the gravitational aberration of light, could someone immediately rule this theory out, or would they need to do a quantitative calculation of the magnitude of the corrections to Newtonian gravity?)


*If not, did anyone ever consider this theory as a possible relativistic theory of gravity before precision experiments validated GR as a better theory?
(There may be some question of what happens to the ${\bf v} \times {\bf B}$ part of the Lorentz force law. I know that the "nonrelativistic limit of EM" is a subtle subject, but I believe that if you're careful then this term becomes negligible in the limit $v \ll c$; e.g. in CGS units, this term is suppressed by a factor of $c$ relative to the electric term in the Lorentz force law.)
 A: As a comment by @knzhou suggests, $P^μ$ cannot be (proportional) to 4-momentum. But it could be some sort of “matter density” current.
What really would not be consistent is the “Lorentz force law” that is put here by hand only. Force law is not an independent equation (even though Einstein originally thought so) but follows under limiting procedure of self consistent field equations in the presence of point-like sources.
If we flip the sign in for the current in “Maxwell's equation” we would still get the EM theory but with positive charges now being called negative and vice versa. And this means that the field energy density for such putative “Maxwell's gravity theory” in Newtonian approximation would be
$$U'_\text{MGT}=\frac{1} {8\pi G} |\nabla \Phi|^2,$$
Where $\Phi$ is the gravitational potential. This means that the total energy of two point charges held close together would be more than the energy of the same charges held at large distance apart, which in turn means that the like charges repel.
In order to be consistent with the Newtonian gravity limit the energy density of static (or quasi-static) gravitational field must be negative:
$$
U_\text{NG}=- \frac{1} {8\pi G} |\nabla \Phi|^2,
$$
which would ensure attractive force for like charges. This could be achieved with the theory of scalar field (such as Nordström's candidate theory of gravitation) or tensor field (such as GR), but not with vector field.

Update: A proof for a relation between attraction/repulsion between like charges and the spin of the mediating field could be found here:

*

*Jagannathan, K., & Singh, L. P. S. (1986). Attraction/repulsion between like charges and the spin of the classical mediating field. Physical Review D, 33(8), 2475, doi:10.1103/PhysRevD.33.2475.

Under assumptions of

*

*Poincare invariance of the theory,

*existence of Lagrangian formulation,

*KG equation $(\partial^2 +m^2)A_{…}=0$ for the mediating spin–$s$  field $A_{…}$  in the absence  of sources,

*interaction Lagrangian of the form $J^{…} A_{…}$,

*rotational symmetry/absence of intrinsic direction for the source $J^{…}$;

the potential energy between two like charges $e$ and $e'$ has the form:
$$
(-)^{s+1} e e' × (\text{positive number}) × \frac{\exp(-mr)}r.
$$
The proof consists in analyzing the expression for the energy density of the field $A$ after imposing suitable gauge-fixing condition (e.g. variant of the Coulomb gauge). The energy must have the term $(\partial_0 A_{i_1 i_2 …})^2$ with a plus sign (this is the “kinetic” term present for free field oscillations). But this would mean that the term with spatial gradients of Coulomb potential $(\partial_i A_{00…})^2$ (and this is the term contributing to potential enrgy of static point charges) would gain overall $(-1)^{s+1}$ sign because of $s+1$ times raising indices.
A: The insightful answer by A.V.S. gives most of the information here. I wish merely to add that a major shaping factor for a theory of gravity was not just that it be covariant and reduce to Newtonian the right limit, but that it would have the equivalence principle built in. I think if you insist on the latter then you will be led to Einstein's G.R. (or else something more complicated).
A: I've been thinking of this question for quite a while. I posted an answer in the page that you linked where I try to answer in the spirit of OP's question. Although there isn't an a priori contradiction, the theory leads to unphysical pathologies (for example the energy of the field may become not positive definite).
If someone were to propose a theory of gravity in the form of "repackaged Maxwell's equations," I would challenge the person to explain how it evades possibilities #1, #2, and #3 of my post.
I don't know if these exact issues were considered by Einstein and his peers at the time, but I've seen Oliver Heaviside's name attached to the kinds of gravity theories proposed in your post and the post you linked, but I can't give any more detail than that.
