Characteristic formulations of general relativity and applications

TL;DR

This paper investigates the hyperbolicity and well-posedness of characteristic formulations of general relativity, revealing weak hyperbolicity issues in Bondi-like systems and exploring implications for gravitational wave modeling and holography.

Contribution

It provides a detailed analysis of the hyperbolic properties of Bondi-like formulations, identifies their limitations, and discusses potential alternative approaches for better numerical stability.

Findings

Several Bondi-like formulations are only weakly hyperbolic.

Weak hyperbolicity affects the accuracy of gravitational waveform modeling.

Numerical tests confirm weak hyperbolicity in practice.

Abstract

General relativity can describe various gravitational systems of astrophysical relevance, like black holes and neutron stars, or even strongly coupled systems through the holographic duality. The characteristic initial (boundary) value problem has numerous applications in general relativity involving numerical studies and is often formulated using Bondi-like coordinates. Well-posedness of the resulting systems of partial differential equations, however, remains an open question. The answer to this question affects the accuracy, and potentially the reliability of conclusions drawn from numerical studies based on such formulations. In the first part of this thesis, we expand our understanding of the hyperbolicity and well-posedness of Bondi-like free evolution systems. We show that several prototype Bondi-like formulations are only weakly hyperbolic and examine the root cause of this result. Consequently, the characteristic initial (boundary) value problem of general relativity in these gauges is rendered ill-posed in the simplest norms one would like to employ. We discuss the implications of this result in accurate gravitational waveform modeling methods and work towards the construction of alternative norms that might be more appropriate. We also present numerical tests that demonstrate weak hyperbolicity in practice and highlight important features to perform them effectively. In the second part, we turn our attention to applications of these formulations to the qualitative behavior of strongly coupled systems via holography.