Gauge and constraint degrees of freedom: from analytical to numerical approximations in General Relativity

TL;DR

This paper compares analytical and numerical approaches to gauge and constraint degrees of freedom in General Relativity, proposing flexible gauge conditions and analyzing their implications in black hole simulations.

Contribution

It introduces a unified framework for gauge conditions in Einstein's equations, including quasi-stationary gauges and their application to black hole simulations.

Findings

Quasi-stationary gauge conditions improve numerical stability.

The relationship between different formalisms is clarified.

Flexible gauge strategies enhance black hole simulation accuracy.

Abstract

The harmonic formulation of Einstein's field equations is considered, where the gauge conditions are introduced as dynamical constraints. The difference between the fully constrained approach (used in analytical approximations) and the free evolution one (used in most numerical approximations) is pointed out. As a generalization, quasi-stationary gauge conditions are also discussed, including numerical experiments with the gauge-waves testbed. The complementary 3+1 approach is also considered, where constraints are related instead with energy and momentum first integrals and the gauge must be provided separately. The relationship between the two formalisms is discussed in a more general framework (Z4 formalism). Different strategies in black hole simulations follow when introducing singularity avoidance as a requirement. More flexible quasi-stationary gauge conditions are proposed in this context, which can be seen as generalizations of the current 'freezing shift' prescriptions.