Solving gravitational field equations by Wiener-Hopf matrix factorisation, and beyond

TL;DR

This paper reviews how Wiener-Hopf matrix factorisation techniques can be used to find exact solutions to two-dimensional Einstein's field equations, connecting integrable systems, Riemann-Hilbert problems, and gravitational theories.

Contribution

It highlights recent advances in Wiener-Hopf methods applied to gravitational equations and introduces a new solution-generating approach via $ au$-invariance.

Findings

Explicit solutions to gravitational field equations derived from Wiener-Hopf factorisation.

Demonstration of the $ au$-invariance property as a novel solution-generating technique.

Interdisciplinary approach linking General Relativity, Complex Analysis, and Operator Theory.

Abstract

By viewing Einstein's field equations -- reduced to two dimensions -- as an integrable system, one can simultaneously obtain exact solutions to both the equations themselves and their associated Lax pair via a canonical Wiener-Hopf factorisation of a so-called monodromy matrix. In this article, we review this remarkable interplay between gravitational field equations, integrable systems, Riemann-Hilbert problems, and Wiener-Hopf factorisation theory, with particular emphasis on developments from the past decade enabled by advances in Wiener-Hopf factorisation techniques arising from the study of singular integral equations and Toeplitz operators. Through a variety of concrete examples, we illustrate how Wiener-Hopf factorisation yields explicit, exact solutions to the field equations of gravitational theories, and how its generalisation through a so-called $\tau$-invariance property provides a new solution-generating method. Along the way, we aim to demonstrate the importance of an interdisciplinary approach -- grounded in General Relativity, Complex Analysis, and Operator Theory -- for the study of gravitational field equations.