Einstein Metrics with Two-Dimensional Killing Leaves and Their Applications in Physics

Abstract

Solutions of vacuum Einstein’s field equations, for the class of pseudo-Riemannian four-metrics admitting a non Abelian two dimensional Lie algebra of Killing fields, are explicitly described. When the distribution orthogonal to the orbits is completely integrable and the metric is not degenerate along the orbits, these solutions are parameterized either by solutions of a transcendental equation (the tortoise equation), or by solutions of a linear second order differential equation in two independent variables. Metrics, corresponding to solutions of the tortoise equation, are characterized as those that admit a three dimensional Lie algebra of Killing fields with two dimensional leaves. Metrics, corresponding to the case in which the commutator of the two Killing fields is isotropic, represent nonlinear gravitational waves.