The Einstein equations with two commuting Killing vectors

Abstract

The usual treatment of Einstein's equations in the vacuum with two commuting Killing vectors is based on the restricting assumption of orthogonal transitivity of the metric. The author considers the general problem without making this assumption, and obtain a reasonably simple formulation which is in fact very similar to the equations for the restricted problem. The essential result is the separability of the equations for the diagonal blocks of the metric tensor, using the most natural coordinate system. As a consequence of this separability property, he is able to obtain a non-orthogonally transitive generalization of Weyl's (1917) static metrics. This is described by a fourth-order partial differential system. He has also considered the more obvious reductions to ordinary differential equations; several turn out to be easily integrable in closed form, either by quadratures or by elliptic functions.