In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime (where the metric may deviate from the Minkowski metric) or where one uses an arbitrary (not necessarily Cartesian) coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields (or energy/matter in general) induce curvature in spacetime, Maxwell's equations in flat spacetime should be viewed as a convenient approximation. When working in the presence of bulk matter, distinguishing between free and bound electric charges may facilitate analysis. When the distinction is made, they are called the macroscopic Maxwell's equations. Without this distinction, they are sometimes called the "microscopic" Maxwell's equations for contrast. The electromagnetic field admits a coordinate-independent geometric description, and Maxwell's equations expressed in terms of these geometric objects are the same in any spacetime, curved or not. Also, the same modifications are made to the equations of flat Minkowski space when using local coordinates that are not rectilinear. For example, the equations in this article can be used to write Maxwell's equations in spherical coordinates. For these reasons, it may be useful to think of Maxwell's equations in Minkowski space as a special case of the general formulation.
Summary In general relativity, the metric tensor g α β {\displaystyle g_{\alpha \beta }} is no longer a constant (like η α β {\displaystyle \eta _{\alpha \beta }} as in Examples of metric tensor) but can vary in space and time, and the equations of electromagnetism in vacuum become
F α β = ∂ α A β − ∂ β A α , D μ ν = 1 μ 0 g μ α F α β g β ν − g c , J μ = ∂ ν D μ ν , f μ = F μ ν J ν , {\displaystyle {\begin{aligned}F_{\alpha \beta }&=\partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha },\\{\mathcal {D}}^{\mu \nu }&={\frac {1}{\mu _{0}}}\,g^{\mu \alpha }\,F_{\alpha \beta }\,g^{\beta \nu }\,{\frac {\sqrt {-g}}{c}},\\J^{\mu }&=\partial _{\nu }{\mathcal {D}}^{\mu \nu },\\f_{\mu }&=F_{\mu \nu }\,J^{\nu },\end{aligned}}}
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