Killing vector fields and the Einstein-Maxwell field equations in general relativity

Abstract: A number of theorems concerning non-null electrovac spacetimes, that is space-times whose metric satisfies the source-free Einstein-Maxwell equations for some non-null bivector Fij, are presented. Firstly, we suppose that the metric is invariant under a one-parameter group of isornetries with Killing vector field ξ. It is proved that the electromagnetic field tensor Fij is invariant under the group, in the sense that its Lie derivative with respect to ξ vanishes, if and only if the gradient αij of the complexion scalar is orthogonal to ξ. It is is also proved that if in addition ξ is hypersurface orthogonal, it is necessarily parallel to α,i. These results are used to generalize theorems of Perjes and Majumdar concerning static electrovac space-times. Secondly, we suppose that the metric is invariant under a two-parameter othogonally transitive Abelian group of isometries. It is proved that in this case Fij is necessarily invariant under the group. The above results can be used to simplify many derivations of exact solutions of the Einstein-Maxwell equations.