Geometrical optics analysis of the short-time stability properties of the Einstein evolution equations

TL;DR

This paper develops geometric optics-based analytical tools to evaluate the short-time stability of Einstein evolution formulations, helping identify ill-behaved formulations by analyzing transient growth of solutions.

Contribution

It introduces a novel geometric optics approach to assess the stability of Einstein evolution equations, focusing on transient amplification to eliminate problematic formulations.

Findings

Identified formulations with unacceptable transient growth in flat space.

Applied techniques to the Kidder-Scheel-Teukolsky family of equations.

Provided constraints on stable formulations for practical scenarios.

Abstract

Many alternative formulations of Einstein's evolution have lately been examined, in an effort to discover one which yields slow growth of constraint-violating errors. In this paper, rather than directly search for well-behaved formulations, we instead develop analytic tools to discover which formulations are particularly ill-behaved. Specifically, we examine the growth of approximate (geometric-optics) solutions, studied only in the future domain of dependence of the initial data slice (e.g. we study transients). By evaluating the amplification of transients a given formulation will produce, we may therefore eliminate from consideration the most pathological formulations (e.g. those with numerically-unacceptable amplification). This technique has the potential to provide surprisingly tight constraints on the set of formulations one can safely apply. To illustrate the application of these techniques to practical examples, we apply our technique to the 2-parameter family of evolution equations proposed by Kidder, Scheel, and Teukolsky, focusing in particular on flat space (in Rindler coordinates) and Schwarzchild (in Painleve-Gullstrand coordinates).