General-covariant evolution formalism for Numerical Relativity

TL;DR

This paper introduces a covariant extension of Einstein's equations with an extra four-vector, enabling the formulation of hyperbolic evolution systems for Numerical Relativity, potentially improving stability and coordinate handling.

Contribution

It proposes a covariant formalism with an additional four-vector to reformulate Einstein's equations for better numerical evolution in relativity.

Findings

Recover Einstein's solutions when the four-vector vanishes

Formulate symmetric hyperbolic evolution systems with harmonic coordinates

Applicable to Numerical Relativity simulations

Abstract

A general covariant extension of Einstein\'{}s field equations is considered with a view to Numerical Relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector $Z_\mu$. Einstein's solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints amount to the covariant algebraic condition $Z_\mu=0$. The extended field equations can be supplemented by suitable coordinate conditions in order to provide symmetric hyperbolic evolution systems: this is actually the case for either harmonic coordinates or normal coordinates with harmonic slicing.