Constraint damping in the Z4 formulation and harmonic gauge

TL;DR

This paper demonstrates how adding specific lower-order terms to the Z4 formulation of Einstein's equations effectively damps most constraint violations, and shows how to extend this damping to harmonic gauge conditions.

Contribution

It introduces a method to damp constraint violations in the Z4 formulation and extends this approach to harmonic gauge Einstein equations through a variable transformation.

Findings

Constraint violations are damped except for constant modes.

The Z4 formulation acts as a lambda-system for constraint damping.

Harmonic gauge Einstein equations can be derived from Z4 with preserved constraint evolution.

Abstract

We show that by adding suitable lower-order terms to the Z4 formulation of the Einstein equations, all constraint violations except constant modes are damped. This makes the Z4 formulation a particularly simple example of a lambda-system as suggested by Brodbeck et al. We also show that the Einstein equations in harmonic coordinates can be obtained from the Z4 formulation by a change of variables that leaves the implied constraint evolution system unchanged. Therefore the same method can be used to damp all constraints in the Einstein equations in harmonic gauge.