The hyperbolic Ernst equation in a triangular domain

TL;DR

This paper solves the hyperbolic Ernst equation in a triangular domain using a Riemann-Hilbert approach, enabling analysis of gravitational wave collisions with singular boundary data.

Contribution

It develops a Riemann-Hilbert framework for the Ernst equation with unbounded boundary derivatives, applicable to gravitational wave scenarios.

Findings

Solution via Riemann-Hilbert problem for boundary data

Explicit inversion of singular integral operator at boundary

Characterization of solution behavior near singular boundary points

Abstract

The collision of two plane gravitational waves in Einstein's theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain. We use the integrable structure of the Ernst equation to present the solution of this problem via the solution of a Riemann--Hilbert problem. The formulation of the Riemann--Hilbert problem involves only the prescribed boundary data, thus the solution is as effective as the solution of a pure initial value problem via the inverse scattering transform. Our results are valid also for boundary data whose derivatives are unbounded at the triangle's corners---this level of generality is crucial for the application to colliding gravitational waves. Remarkably, for data with a singular behavior of the form relevant for gravitational waves, it turns out that the singular integral operator underlying the Riemann--Hilbert formalism can be explicitly inverted at the boundary. In this way, we are able to show exactly how the behavior of the given data at the origin transfers into a singular behavior of the solution near the boundary.