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It seems difficult to track down a clear explanation of this statement:

So although the Coulomb law was discovered in a supporting frame, general relativity tells us that the field of such a charge is not precisely $1 / r^2$.

Papers I've found seem to say either that the inverse square law for the Coulomb field of a massive point charge remains exactly true in general relativity, or that it has corrections on the order of $1/r^4$ and higher.

I suspect that I may be getting lost in the coordinate transformations and the in-context meaning of $r$. What I would like to know is if general relativity predicts any deviations from the inverse square law for the electric field surrounding a point massive charged particle, as seen by a distant observer.

Presumably the measurement method should be specified, so here's a possibility: Attach a charge $Q$ to an extremely massive particle, then probe the electric field of the charge by first measuring the distribution of electrons, then of positrons, shot at various energies toward the massive particle a la the Rutherford experiment. The difference between the two would be used to subtract out the purely gravitational attraction between the massive particle and the probe particles. I realize that this approach would be impractical for measuring extremely small deviations from Coulomb's law, but it should at least provide a way to dodge some of the difficulties associated with defining $r$ near a point mass. I also realize that quantum corrections would totally change things in real experiment. I'm just looking for a clear classical explanation of what happens to Coulomb's law due to the effects of gravity per general relativity.

EDIT 12/24/18: Specifically: in the scattering measurement proposed above, does it appear to a distant observer that there is a deviation from the inverse square law - a deviation that is a bit different from that due strictly to gravity?

The Reissner-Nordstrom metric describes a massive point charge in General Relativity. The covariant component of the electrostatic four-potential, $A_t$, is exactly $Q/r$, but the contravariant component $A^t$ is more complicated. Similarly, the purely covariant component $F_{rt}$ of the electrostatic field, and also the purely contravariant component $F^{rt}$, is exactly $Q/r^2$ but the mixed components are more complicated.

This is in Schwarzschild-style coordinates where the area of a sphere around the charge is $4pi r^2$ as $rrightarrowinfty$.

There is no meaningful way to say which components are the potential and the field. The potential is a four-vector and the field is a four tensor, and they have both covariant and contravariant components which are equally valid.

Regardless of which components you use, the electric flux through a sphere at infinity is $4pi Q$.

When you think about Coulomb’s Law, you should not think of it in the “high-school” form that the field is inverse-square. Instead you should think of it in the “university” form that $nablacdotmathbf{E}=4pirho$. This equation, generalized to take curved spacetime into account, remains true when the effect of gravity is considered.