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

TL;DR

This paper reformulates the Ernst equation for stationary axisymmetric vacuum Einstein equations as a non-autonomous dynamical system, providing a Hamiltonian framework that could facilitate quantization and deepen understanding of these solutions.

Contribution

It introduces a novel separation of variables and Hamiltonian formulation for the Ernst equation, linking it to a Schlesinger-type system and enabling new analytical and quantization approaches.

Findings

Reformulation of Ernst equation as a Schlesinger-type system

Identification of the conformal factor with the τ-function

Development of a two-time Hamiltonian approach

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 $\tau$-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.