Maxwell equations in curved spacetime

In curved spacetime, Maxwell's equations can be expressed in forms valid in Minkowski background, with the effect of the metric (gravity) appearing as effective polarizations and magnetizations. The electric and magnetic (EM) fields depend on the observer's frame four-vector. We derive Maxwell's equations valid in general curved spacetime using the fields defined in the normal frame, the coordinate frame, and two other non-covariant methods used in the literature. By analyzing the case in the generic frame we show that the EM fields, as well as the charge and current densities, defined in non-covariant ways do not correspond to physical ones measured by an observer. We show that modification of the homogeneous part is inevitable to any observer, and such a modification is difficult to interpret as the effective medium property. The normal frame is the relevant one to use as it gives the EM fields measured by an Eulerian observer.