"Separation of Variables and Hamiltonian Formulation for the Ernst Equation"

Abstract

It is shown that the vacuum Einstein equations for an arbitrary stationary axisymmetric space-time can be completely separated by re-formulating the Ernst equation and its associated linear system in terms of a non-autonomous Schlesinger-type dynamical system. The conformal factor of the metric coincides (up to some explicitly computable factor) with the $τ$-function of the Ernst equation in the presence of finitely many regular singularities. We also present a canonical formulation of these results, which is based on a ``two-time" Hamiltonian approach, and which opens new avenues for the quantization of such systems.