Einstein metrics with 2-dimensional Killing leaves and their physical interpretation

Abstract

Solutions of vacuum Einstein’s field equations, for the class of pseudo-Riemannian 4-metrics admitting a non Abelian 2-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 3-dimensional Lie algebra of Killing fields with 2-dimensional leaves. A formalism (ζ-complex analysis), allowing one to construct global Einstein metrics by matching together these solutions, is also described. It is shown that the remaining metrics, corresponding to the case in which the commutator of the two Killing fields is isotropic, represent nonlinear gravitational waves obeying to two nonlinear superposition laws. The energy and the polarization of this family of waves are explicitly evaluated; it is shown that they have spin-1 and their sources are also described.