Spacetime symmetries and the complexion of the electromagnetic field

Considering spacetimes admitting groups of motions, we derive the symmetry imposed on the electromagnetic field for solutions to the Einstein−Maxwell equations. It is found that the scalar products between the gradient of the complexion and the Killing vectors must be constants and that these constants enter into the symmetry condition on the electromagnetic field. We therefore obtain a geometrical interpretation of the complexion in spacetimes having symmetries.