electromagnetism - Coulomb's Law in the presence of a strong gravitational field

I was under the impression that the $1/r^2$ falloff of various forces were because of the way the area of a expanding sphere scales. But that strict $1/r^2$ falloff would only be globally true in a strictly Euclidean geometry, yes? So, if you had an electron and a positron in a warped space due to gravity, wouldn't that $1/r^2$ term in Coulomb's Law end up needing to be adjusted because of the warping of space leading to an increase or decrease in effect due to gravitational lensing or scattering?

I found one related question here: Does Gravity Warp Coulomb's force? But it only has one down-voted and unclear answer.

Answer

Of course Coulomb's law has to be adapted! And it is therefore fortunate that there exist manifestly covariant formulations of electromagnetism that do not care how spacetime is curved. However, we should first briefly observe that Coulomb's law is not one of the fundamental laws of electromagnetism, though it has played a great role in its inception:

Coulomb's law is just the solution of Maxwell's equations for a point charge and no current in flat Minkowski space. Maxwell's equations can jointly be generalized to arbitrary spacetimes:

The electric field strength is a 2-form $F = \frac{1}{2}F_{\mu\nu}\mathrm{d}x^\mu \wedge \mathrm{d}x^\nu$ on spacetime, and electric current is a 3-form $J = \frac{1}{6}J_{\mu\nu\rho}\mathrm{d}x^\mu \wedge \mathrm{d}x^\nu \wedge \mathrm{d}x^\rho$, as the Hodge dual of the usual vector current. Maxwell's equations now simply read

$$ \mathrm{d}F = 0 \; \text{and} \; \mathrm{d}\star F = J$$

where, since the Hodge star depends on the metric, the curvature of spacetime indeed influences the form of our laws.

It has to be noted that, on arbitrary spacetimes, the notion of having "separate laws" for electric on magnetic fields doesn't really make sense anymore, since they mix in (almost) arbitrary ways, depending on the metric. You can still get the electric and magnetic fields as components $F^{0i}$ and $F^{ij}$ of the field strength, but you won't be writing any nice, frame-independent laws for them. Maxwell's equations do not dissolve nicely into "Gauss' law", "Faraday's law" or such things in a general setting.