Exact solutions to Ernst-like equation in (2+2) Hamiltonian reduction

Abstract

We apply the method of Hamiltonian reduction without isometry as a way to find exact solutions to Einstein’s equations. To find exact solutions, we introduce two spatial Killing vector fields to the Einstein’s equations obtained through the Hamiltonian reduction, and derive the Ernst-like equation in the privileged coordinates. By solving the Ernst-like equation, we found a four-parameter family of exact solutions, one of which is interpreted as a deformation of the general Kasner spacetime. We extend our method to spacetimes where two independent gravitational degrees of freedom co-exist and interact with each other, and obtain a set of two partial differential equations satisfied by them. If we substitute a pre-fixed diagonal mode into these equations, and then the equations reduce to a single non-linear partial differential equation, which is interpreted as the equation of non-diagonal mode of gravitational waves propagating on the “background” spacetime determined by the diagonal mode. We choose three simplest “background” spacetimes, and discuss the corresponding non-diagonal modes in each case.