Summary
In general relativity, the metric, gαβ, is no longer a constant (like ηαβ above) but can vary in space and time, and the equations of electromagnetism in a vacuum become:
where fμ is the density of the Lorentz force, gαβ is the reciprocal of the metric tensor gαβ, and g is the determinant of the metric tensor. Notice that Aα and Fαβ are (ordinary) tensors while, Jν, and fμ are tensor densities of weight +1. Despite the use of partial derivatives, these equations are invariant under arbitrary curvilinear coordinate transformations. Thus if one replaced the partial derivatives with covariant derivatives, the extra terms thereby introduced would cancel out. (Cf. manifest covariance#Example.)
Read more about this topic: Maxwell's Equations In Curved Spacetime
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