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

Abstract

It is shown without explicit integration that all Petrov type-D vacuum solutions with cosmological constant admit at least a two-parameter, abelian, orthogonally transitive group of local isometries. In the case when the orbits are non-null the group is invertible and a symmetric null tetrad is shown to exist in which it is manifest that the principal null congruences defined by the type-D Weyl tensor are indistinguishable. General forms for the metrics are given for both the case of non-null and null group orbits. It is also demonstrated that there always exists a system of coordinates for these solutions in which the Hamilton–Jacobi equation for the null geodesics is solvable by separation of variables, a fact that explains the existence of a conformal Killing tensor therein. A partial invariant characterization of Kinnersley’s type-D vacuum solutions is given from which it follows that all his metrics except those of Case III admit a (0,2) Yano–Killing tensor and hence a full (0,2) Killing tensor.