Integrals of Motion in the Two Killing Vector Reduction of General Relativity

Abstract

We apply the inverse scattering method to the midi-superspace models that are characterized by a two-parameter Abelian group of motions with two spacelike Killing vectors. We present a formulation that simplifies the construction of the soliton solutions of Belinski\v i and Zakharov. Furthermore, it enables us to obtain the zero curvature formulation for these models. Using this, and imposing periodic boundary conditions corresponding to the Gowdy models when the spatial topology is a three torus $T ^3$, we show that the equation of motion for the monodromy matrix is an evolution equation of the Heisenberg type. Consequently, the eigenvalues of the monodromy matrix are the generating functionals for the integrals of motion. Furthermore, we utilise a suitable formulation of the transition matrix to obtain explicit expressions for the integrals of motion. This involves recursion relations which arise in solving an equation of Riccati type. In the case when the two Killing vectors are hypersurface orthogonal the integrals of motion have a particularly simple form.