A symmetric hyperbolic formulation of the vacuum Einstein equations in affine-null coordinates

TL;DR

This paper introduces a symmetric hyperbolic formulation of the vacuum Einstein equations in affine-null coordinates, addressing hyperbolicity issues in previous formulations and enabling better numerical stability for gravitational wave studies.

Contribution

It presents a new tetrad-based symmetric hyperbolic formulation that overcomes weak hyperbolicity problems in existing affine-null coordinate formulations.

Findings

Formulation is strongly hyperbolic and numerically stable.

Avoids hyperbolicity issues of previous affine-null formulations.

Potential application in gravitational wave scattering studies.

Abstract

We present a symmetric hyperbolic formulation of the Einstein equations in affine-null coordinates. Giannakopoulos et. al. (arXiv:2007.06419) recently showed that the most commonly numerically implemented formulations of the Einstein equations in affine-null coordinates (and other single-null coordinate systems) are only weakly-but not strongly-hyperbolic. By making use of the tetrad-based Newman-Penrose formalism, our formulation avoids the hyperbolicity problems of the formulations investigated by Giannakopoulos et. al. We discuss a potential application of our formulation for studying gravitational wave scattering.