Generating new gravitational solutions by matrix multiplication

TL;DR

This paper introduces a new method for generating gravitational solutions using matrix multiplication, expanding beyond traditional Riemann-Hilbert approaches, and demonstrates its application to various gravitational wave solutions.

Contribution

It presents a novel solution-generating technique based on matrix multiplication that can produce solutions not accessible by previous Wiener-Hopf factorisation methods.

Findings

New solutions via multiplicative deformation of Wiener-Hopf factorisation

Superposition of solutions is possible with the new method

Applications include Kasner, Einstein-Rosen wave, and gravitational pulse wave solutions

Abstract

Explicit solutions to the non-linear field equations of some gravitational theories can be obtained, by means of a Riemann-Hilbert approach, from a canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices. In this paper we describe other types of factorisation from which solutions can be constructed in a similar way. Our approach is based on an invariance problem, which does not constitute a Riemann-Hilbert problem and allows to construct solutions that could not have been obtained by Wiener-Hopf factorisation of a monodromy matrix. It gives rise to a novel solution generating method based on matrix multiplications. We show, in particular, that new solutions can be obtained by multiplicative deformation of the canonical Wiener-Hopf factorisation, provided the latter exists, and that one can superpose such solutions. Examples of applications include Kasner, Einstein-Rosen wave and gravitational pulse wave solutions.