Orthogonal transitivity, invertibility, and null geodesic separability in type D electrovac solutions of Einstein’s field equations with cosmological constant

Abstract

It is shown without explicit integration that all Petrov type D electrovac solutions with cosmological constant for an aligned, nonsingular electromagnetic field which satisfy the generalized Goldberg–Sachs theorem, admit at least a two-parameter, abelian, orthogonally transitive group of local isometries. In the case when the group orbits are non-null the group is invertible, and a symmetric null tetrad is shown to exist in which the principle null congruences defined by the type D Weyl tensor are indistinguishable. An explicit example is given of a solution with null group orbits which contains as a subcase a Kinnersley vacuum solution (with the same property). It is also demonstrated that the Hamilton–Jacobi equation for the null geodesics is always solvable by separation of variables in these solutions, a fact which explains the existence of a conformal Killing tensor therein, and which gives rise to a coordinate system in which the field equations may be integrated in terms of polynomial functions.