Maxwell's equations in curved spacetime

Remarks

The second set says that $F$ is a closed differential form, $dF = 0$, with $d$ the exterior derivative; it does not involve the metric. The first set can be written $d \star F = \mu_0 \star J^\flat$, up to a sign that depends on conventions. Here $\star$ is the Hodge star of $g$ and $J^\flat$ is the $1$-form dual to $J$. In coordinates, $\nabla_\mu F^{\mu\nu} = \frac{1}{\sqrt{-\det g}} \partial_\mu \big( \sqrt{-\det g} \, F^{\mu\nu} \big)$, because $F$ is antisymmetric. The same antisymmetry gives $\nabla_\nu J^\nu = 0$, conservation of charge on a curved background.

Example: on Minkowski space, where $g = \operatorname{diag}(1, -1, -1, -1)$ in inertial coordinates, the Christoffel symbols vanish and the equations reduce to the covariant formulation of special relativity.

Non-example: on a Lorentzian manifold, the equation $\partial_\mu F^{\mu\nu} = \mu_0 J^\nu$ with plain partial derivatives is not the right law in general coordinates, since its left side is not a vector field.

When the metric itself is unknown and is coupled to $F$ through the electromagnetic stress–energy tensor, the system becomes the Einstein–Maxwell equations.