Gauge stability of 3+1 formulations of General Relativity

TL;DR

This paper develops a general method to analyze the gauge stability of 3+1 formulations of General Relativity, revealing that most fixed gauges are ill-posed or unstable, with implications for choosing stable gauges in numerical relativity.

Contribution

It introduces a universal approach to gauge stability analysis in 3+1 GR formulations, applicable regardless of specific system forms, and evaluates the stability of various gauges.

Findings

Most fixed gauges are ill-posed except synchronous gauge.

Maximal slicing and its parabolic extension are ill-posed.

Metric-dependent algebraic gauges are necessary but generally unstable.

Abstract

We present a general approach to the analysis of gauge stability of 3+1 formulations of General Relativity (GR). Evolution of coordinate perturbations and the corresponding perturbations of lapse and shift can be described by a system of eight quasi-linear partial differential equations. Stability with respect to gauge perturbations depends on a choice of gauge and a background metric, but it does not depend on a particular form of a 3+1 system if its constrained solutions are equivalent to those of the Einstein equations. Stability of a number of known gauges is investigated in the limit of short-wavelength perturbations. All fixed gauges except a synchronous gauge are found to be ill-posed. A maximal slicing gauge and its parabolic extension are shown to be ill-posed as well. A necessary condition is derived for well-posedness of metric-dependent algebraic gauges. Well-posed metric-dependent gauges are found, however, to be generally unstable. Both instability and ill-posedness are associated with perturbations of physical accelerations of reference frames.