First-order quasilinear canonical representation of the characteristic formulation of the Einstein equations

TL;DR

This paper presents a first-order quasilinear canonical form of the characteristic formulation of Einstein's equations, enabling better analytical understanding and numerical simulation of gravitational phenomena.

Contribution

It introduces a new canonical formulation with 18 variables, facilitating solution existence proofs and advanced numerical techniques for Einstein's equations.

Findings

Allows concrete statements about solution existence.

Enables incorporation of advanced numerical methods.

Provides a unified framework for vacuum and matter problems.

Abstract

We prescribe a choice of 18 variables in all that casts the equations of the fully nonlinear characteristic formulation of general relativity in first--order quasi-linear canonical form. At the analytical level, a formulation of this type allows us to make concrete statements about existence of solutions. In addition, it offers concrete advantages for numerical applications as it now becomes possible to incorporate advanced numerical techniques for first order systems, which had thus far not been applicable to the characteristic problem of the Einstein equations, as well as in providing a framework for a unified treatment of the vacuum and matter problems. This is of relevance to the accurate simulation of gravitational waves emitted in astrophysical scenarios such as stellar core collapse.