It is possible, apparently, to describe gravitational lensing as if gravitational potential induces an effective refractive index change in the vacuum, and spacetime is flat.
As pointed out by @AndrewSteane, this has nothing really to do with electromagnetism; it is merely a useful fiction that can be helpful in calculating null geodesics- which correspond, of course, to light rays in the vicinity of a gravitating mass.
I would like to know if something analogous can be done to describe the electric field in the vicinity of a charged mass, as if space were flat. I imagine this would have the form of an "effective permittivity field" around the mass, as a function of gravitational potential.
Edit 1/4/19: After a bit more thought -- there would likely be an "effective magnetic permeability field" as well; and both effective fields might well be anisotropic. This paper, "Eﬀective refractive index tensor for weak-ﬁeld gravity" appears to point in this direction.
Edit 1/16/19: After a lot more searching, this paper turned up. The authors, Isabel Fernandez-Nunez and Oleg Bulashenko, show that the trajectory of an electromagnetic wave is accurately described by specifying electric permittivity and magnetic permeability, dependent respectively on gravitational time dilation and spatial curvature. Moreover, the authors state:
Then, it can be shown that the covariant Maxwell’s equations written in curved coordinates can be transformed into their standard form for flat space but in the presence of an effective medium.
The "effective medium" is described by tensors dependent on the components of the space time metric.
The statement is not about electromagnetic waves; it is about Maxwell's equations. So-- if the authors are right -- my conjecture that something analogous can be done to describe the electric field in the vicinity of a charged mass, as if space were flat must be correct.
I'll welcome any informed comments. "Informed" means that the commenter has read and understood the two papers I've referenced.