Maxwell's equations in curved spacetime I know that we can write Maxwell's equations in the covariant form, and this covariant form can be considered as a generalization of these equations in curved spacetime if we replace ordinary derivatives with covariant derivatives. But I have read somewhere that this generalization is not unique and it is just the simplest one. Can anyone introduce some resources about this subject and how electromagnetism and Maxwell's equations are generalized to curved spacetime?
 A: Here is why I doubt there are other ways to generalize Maxwell's equations to curved spacetime.
Special relativity was obtained from the invariance of the speed of light. In special relativity, the electric field is not a vector field, and the magnetic field is not a pseudovector, but that they transform as the components of a two-form $F_{ab} = \partial_a A_b -\partial_b A_a$, where the four-vector $A_a$ contains the scalar and vector potentials.
Maxwell's equations become
$$d F=0$$
$$d\ast F=J$$
When moving to curved spacetimes, they remain the same, since the Hodge dual $\ast$ is defined at each point $p$ of the manifold, on $\wedge^\bullet T^\ast_p$. When expressed in this form, the covariant derivative is not involved, although the metric is involved in the $\ast$ operator.
While I think the generalization of Maxwell's equations to curved spacetime is very rigid and I see no choice based on simplicity here, it is known that there are modified (nonlinear) versions, like the Born-Infeld theory. But they did not originate because of some freedom of generalizing Maxwell's equations to curved spacetimes.
