Stability properties of a formulation of Einstein's equations

TL;DR

This paper analyzes the stability of the KST formulation of Einstein's equations for weak gravitational waves, showing that stability depends on hyperbolicity and illustrating instabilities through numerical simulations.

Contribution

It provides a combined continuum and numerical stability analysis of the KST formulation, highlighting the role of hyperbolicity and parameter choices.

Findings

Stability depends on hyperbolicity of the system.

Weak field limit simplifies stability analysis.

Numerical simulations reveal parameter-dependent instabilities.

Abstract

We study the stability properties of the Kidder-Scheel-Teukolsky (KST) many-parameter formulation of Einstein's equations for weak gravitational waves on flat space-time from a continuum and numerical point of view. At the continuum, performing a linearized analysis of the equations around flat spacetime, it turns out that they have, essentially, no non-principal terms. As a consequence, in the weak field limit the stability properties of this formulation depend only on the level of hyperbolicity of the system. At the discrete level we present some simple one-dimensional simulations using the KST family. The goal is to analyze the type of instabilities that appear as one changes parameter values in the formulation. Lessons learnt in this analysis can be applied in other formulations with similar properties.