Electromagnetic Displacement
The electric displacement field, D and the auxiliary magnetic field, H form an antisymmetric contravariant rank 2 tensor density of weight +1. In a vacuum, this is given by
Notice that this equation is the only place where the metric (and thus gravity) enters into the theory of electromagnetism. Furthermore even here, the equation is invariant under a change of scale, that is, multiplying the metric by a constant has no effect on this equation. Consequently, gravity can only affect electromagnetism by changing the speed of light relative to the global coordinate system being used. Light is only deflected by gravity because it is slower when near to massive bodies. So it is as if gravity increased the index of refraction of space near massive bodies.
More generally, in materials where the magnetization-polarization tensor is non-zero, we have
The transformation law for electromagnetic displacement is
where the Jacobian determinant is used. If the magnetization-polarization tensor is used, it has the same transformation law as the electromagnetic displacement.
Read more about this topic: Maxwell's Equations In Curved Spacetime