A continuous Riemann-Hilbert problem for colliding plane gravitational waves

TL;DR

This paper introduces a novel inverse scattering method using a Riemann-Hilbert problem to solve the characteristic initial value problem for colliding plane gravitational waves, enabling new solution generation and extensions of known spacetimes.

Contribution

It develops a continuous Riemann-Hilbert problem approach for colliding gravitational waves, expanding solution classes and providing a new technique for generating spacetime solutions.

Findings

Extended Szekeres class with 2 additional parameters

Discovered a limiting case of circularly polarized impulsive waves

Presented a new integral equation-based solution method

Abstract

We present the foundations of a new solution technique for the characteristic initial value problem (IVP) of colliding plane gravitational waves. It has extensive similarities to the approach of Alekseev and Griffiths in 2001, but we use an inverse scattering method with a Riemann-Hilbert problem (RHP), which allows for a transformation to a continuous RHP with a solution given in terms of integral equations for non-singular functions. Ambiguities in this procedure lead to the construction of a family of spacetimes containing the solution to the IVP. Therefore the described technique also serves as an interesting solution generating method. The procedure is exemplified by extending the Szekeres class of colliding wave spacetimes with 2 additional real parameters. The obtained solution seems to feature a limiting case of a new type of impulsive waves, which are circularly polarised.