Factorisation of symmetric matrices and applications in gravitational theories

TL;DR

This paper presents a method for Wiener-Hopf factorisation of symmetric 2x2 matrices with rational quotient, with applications to solving Einstein's equations via Riemann-Hilbert problems.

Contribution

It introduces a technique to determine second columns of matrix factors in symmetric Wiener-Hopf factorisation when the quotient is rational, simplifying the process.

Findings

The second column in each factor is determined by the first column via a rational matrix.

The method is illustrated with examples related to Einstein field equations.

Applicable in Riemann-Hilbert approaches for gravitational solutions.

Abstract

We consider the canonical Wiener-Hopf factorisation of $2 \times 2$ symmetric matrices $\mathcal M$ with respect to a contour $\Gamma$. For the case that the quotient $q$ of the two diagonal elements of $\mathcal M$ is a rational function, we show that due to the symmetric nature of the matrix $\mathcal M$, the second column in each of the two matrix factors that arise in the factorisation is determined in terms of the first column in each of these matrix factors, by multiplication by a rational matrix, and we give a method for determining the second columns of these factors. We illustrate our method with two examples in the context of a Riemann-Hilbert approach to obtaining solutions to the Einstein field equations.