What’s wrong with this Nordström-like scalar theory of gravity? I got very perplexed while reading a few papers on the old Nordström theory of relativistic scalar gravity.  I would like to know what's wrong with the following, which isn't exactly the same as Nordström old theory (AFAIK, since there appears to be several inconsistencies with the description on the Wikipedia's page: https://en.wikipedia.org/wiki/Nordstr%C3%B6m%27s_theory_of_gravitation).
I consider gravity as a pure scalar field in Minkowski spacetime.  The test-particle equation of motion is the following (I use units so $c \equiv 1$ and metric signature $\eta = (1, -1, -1, -1)$):
$$\tag{1}
\frac{d u^a}{d \sigma} = (u^a \, u^b - \eta^{ab}) \, \partial_b \, \phi,
$$
where $\phi$ represents the gravitational potential, and $u_a \, u^a \equiv 1$ is the usual four-velocity norm in Minkowski spacetime.  The four-force on the right side is orthogonal to the four-velocity, so $u_a \, \dot{u}^a = 0$ as it should.  The relativistic Poisson equation is assumed to be this:
$$\tag{2}
\square \, \phi = -\, 4 \pi G T,
$$
where $T \equiv \eta^{ab} \, T_{ab}$ is the trace of the energy-momentum tensor (including a possible non-linear contribution from the scalar field itself).
I'm not interested in the experimental failure of this theory, which predicts that light would not produce any gravity (since the trace of the electromagnetic contribution vanishes).
So what are the theoretical issues with these equations, in a special-relativity classical field context?  What are the contradictions, or inconsistencies?  What are the non-experimental concerns with this theory?  As an example, maybe these equations couldn't be found from an action, and this could be raised as an objection (even in a classical field context), since it would be hard to find the scalar field energy-momentum without the lagrangian density (unless the energy-momentum is already God-given...).
 A: An interesting analysis could be found in a following paper:

*

*Giulini, D. (2008). What is (not) wrong with scalar gravity? Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 39(1), 154-180, doi:10.1016/j.shpsb.2007.09.001, arXiv:gr-qc/0611100.

Abstract:

On his way to General Relativity, Einstein gave several arguments as to why a special-relativistic theory of gravity based on a massless scalar field could be ruled out merely on grounds of theoretical considerations. We re-investigate his two main arguments, which relate to energy conservation and some form of the principle of the universality of free fall. We find that such a theory-based a priori abandonment not to be justified. Rather, the theory seems formally perfectly viable, though in clear contradiction with (later) experiments.

OP's model is discussed in this paper as a “naive theory”. The trouble with it, is that the equation of motion for test particle is incompatible with the coupling of matter to scalar field implied by the field equation on $\phi$. This requirement is referred to as Principle of universal coupling:

All forms of matter (including test particles) couple to the gravitational field in a universal fashion.

The theory could be remedied by deriving equations of motion from the joint  action of scalar field and matter (including test point particles) satisfying this principle of universal coupling taking, taking into account that point particle has the following stress-energy tensor: \begin{equation}
\label{eq:T-PointParticle}
T^{\mu\nu}_p(x)=mc\,\int 
{\dot x}^\mu(\tau){\dot x}^\nu(\tau)\ \delta^{(4)}(x-x(\tau))\ d\tau\,.
\end{equation}
The resulting equation for particle motion in this improved theory of scalar gravity:
\begin{align}
\ddot x^\mu(\tau)&=
P^{\mu\nu}(\tau)\partial_\nu φ(x(\tau))\,,&\\
\text{where}\quad
P^{\mu\nu}(\tau)&=
\eta^{\mu\nu}-{\dot x}^\mu(\tau){\dot x}^\nu(\tau)/c^2&\\
\text{and}\qquad
φ&:=c^2\ln(1+\phi/c^2)\,,&
\end{align}
differs from that of the naive theory in that it is now in $φ$, nonlinear function of the scalar field $\phi$.
This improved model is actually internally consistent at least as a classical field theory, and in particular, the arguments by Einstein used to dismiss scalar theories of gravity do not work against it. So it is only experiments, such as the absence of light deflection by a gravitating body and the wrong prediction for the perihelion precession (which is different by a factor of $-1/6$ from value predicted by GR) that rule this theory out.
