Comment by raattgift

Not sure what you mean; you can't have any mass in a flat spacetime and obey the Einstein Field Equations for a Lorentzian spacetime (because T_{\mu\nu} doesn't vanish everywhere).

There are a variety of types of variable speed of light. If we foliate to 3+1 the usual picture is that c is constant on all spatial slices. Some VSL theories have the same c at all points on a given slice, but introduce a time variation of c. Other VSL theories introduce spatial variation as well (or instead). These families of theories all have significantly different equations of motion or actions from one another (cf. <https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_actio...>). There's no obvious reason why c couldn't relate in a more complicated way to the stress-energy tensor than the Einstein gravitational constant does, but there's also no obvious reason to think such an alternative theory should produce free-fall trajectories similar to those from GR.

In any case, I think you have to choose your function on c, obtain the field equations, decide which energy conditions and constraint equations you want to impose, set appropriate boundary conditions, choose a curve along which to foliate, and run with enough different initial-value surfaces (each of which must satisfy the constraints initially), that eventually an intuition develops. A Will-like parameterized post-Newtonian formalism approach would also be a good idea (<https://en.wikipedia.org/wiki/Parameterized_post-Newtonian_f...>).

Unfortunately I'm unable to guess your choice of "c(m) formula".