Gravitational Fields with 2-Dimensional Killing Leaves and the Gravitational Interaction of Light

Abstract

Gravitational fields invariant for a non Abelian Lie algebra generating a 2-dimensional distribution, are explicitly described. When the orthogonal distribution is 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 Darboux equation. 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. It is shown that the remaining metrics represent nonlinear gravitational waves obeying to two nonlinearsuperposition laws. The energy and the polarization of this family of waves are explicitly evaluated; it is shown that they have spin$-1$ and their possible sources are also described. Old results by Tolman, Ehrenfest, Podolsky and Wheeler on the gravitational interaction of photons are naturally reinterpreted.