Almost-stationary motions and gauge conditions in General Relativity

TL;DR

This paper introduces an almost-stationary gauge condition for Numerical Relativity, enabling hyperbolic formulations of Einstein's field equations by defining time lines via harmonic almost-Killing vectors.

Contribution

It proposes a new gauge condition based on an almost-Killing equation, facilitating hyperbolic evolution in Numerical Relativity.

Findings

Enables hyperbolic form of Einstein equations in ADM and Z4 formalisms.

Defines time lines as integral curves of harmonic almost-Killing vectors.

Provides a variational derivation for the gauge condition.

Abstract

An almost-stationary gauge condition is proposed with a view to Numerical Relativity applications. The time lines are defined as the integral curves of the timelike solutions of the harmonic almost-Killing equation. This vector equation is derived by a variational principle, by minimizing the deviations from isometry. The corresponding almost-stationary gauge condition allows one to put the field equations in hyperbolic form, both in the free-evolution ADM and in the Z4 formalisms.