Maxwell's equations in curved spacetime

Summary

Maxwell's equations in curved spacetime describe the electromagnetic field in the presence of gravity, where spacetime is a Lorentzian manifold. They are obtained from the covariant formulation by replacing partial derivatives with the covariant derivatives of the Levi-Civita connection of the metric. In the language of differential forms they need no metric for one half and only the Hodge star for the other. They describe the bending of light by gravity and, coupled to the Einstein equations, charged black holes.