Hyperbolicity of General Relativity in Bondi-like gauges

TL;DR

This paper investigates the hyperbolicity and well-posedness of Bondi-like formulations of general relativity, revealing they are only weakly hyperbolic and exploring ways to recover convergence and stability in numerical simulations.

Contribution

It identifies the weak hyperbolicity of Bondi-like formulations and demonstrates how alternative norms can restore convergence in numerical models.

Findings

Bondi-like systems are only weakly hyperbolic due to shared pathological structures.

Alternative norms can recover convergence in toy model tests.

Incompatibility of norms prevents energy estimates in Cauchy-Characteristic-Matching models.

Abstract

Bondi-like (single-null) characteristic formulations of general relativity are used for numerical work in both asymptotically flat and anti-de Sitter spacetimes. Well-posedness of the resulting systems of partial differential equations, however, remains an open question. The answer to this question affects accuracy, and potentially the reliability of conclusions drawn from numerical studies based on such formulations. A numerical approximation can converge to the continuum limit only for well-posed systems; for the initial value problem in the $L^2$ norm this is characterized by strong hyperbolicity. We find that, due to a shared pathological structure, the systems arising from the aforementioned formulations are however only weakly hyperbolic. We present numerical tests for toy models that demonstrate the consequence of this shortcoming in practice for the characteristic initial boundary value problem. Working with alternative norms in which our model problems may be well-posed we show that convergence may be recovered. Finally we examine well-posedness of a model for Cauchy-Characteristic-Matching in which model symmetric and weakly hyperbolic systems communicate through an interface, with the latter playing the role of GR in Bondi gauge on characteristic slices. We find that, due to the incompatibility of the norms associated with the two systems, the composite problem does not naturally admit energy estimates.