Towards an understanding of the stability properties of the 3+1 evolution equations in general relativity

TL;DR

This paper investigates the stability issues of the ADM formulation in general relativity by analyzing linear perturbations, identifying zero-speed modes responsible for instabilities, and proposing methods to mitigate these problems.

Contribution

It provides a detailed analysis of zero-speed modes in the ADM formulation and suggests ways to improve stability through gauge decoupling and constraint management.

Findings

Zero-speed gauge modes can be eliminated by conformal rescaling.

Constraint violating modes can be given finite speed via momentum constraints.

Reformulations of the equations show improved stability properties.

Abstract

We study the stability properties of the standard ADM formulation of the 3+1 evolution equations of general relativity through linear perturbations of flat spacetime. We focus attention on modes with zero speed of propagation and conjecture that they are responsible for instabilities encountered in numerical evolutions of the ADM formulation. These zero speed modes are of two kinds: pure gauge modes and constraint violating modes. We show how the decoupling of the gauge by a conformal rescaling can eliminate the problem with the gauge modes. The zero speed constraint violating modes can be dealt with by using the momentum constraints to give them a finite speed of propagation. This analysis sheds some light on the question of why some recent reformulations of the 3+1 evolution equations have better stability properties than the standard ADM formulation.