Continuity equations for general matter: applications in numerical relativity

TL;DR

This paper derives and discusses the use of continuity equations for matter in curved spacetime, enabling better numerical relativity simulations despite the coordinate dependence of energy and momentum.

Contribution

It provides explicit expressions for matter source terms in the ADM formalism applicable to numerical relativity.

Findings

Formulated general matter source terms for ADM decomposition.

Applied equations to simulations of compact objects.

Discussed implications for cosmological models.

Abstract

Due to the absence of symmetries under time and spatial translations in a general curved spacetime, the energy and momentum of matter is not conserved as it is in flat space. This means, for example, that the flux of matter energy through a surface is in general not balanced by an equal increase in the energy of the matter contained within the enclosed volume - there is an additional "source" resulting from the curvature of spacetime acting on the matter (and vice versa). One can calculate this source term and reconcile the flux and energy accumulation over time in an arbitrary volume, although a foliation of the spacetime must be chosen, making the quantities inherently coordinate dependent. Despite this dependence, these quantities are practically useful in numerical relativity simulations for a number of reasons. We provide expressions for general matter sources in a form appropriate for implementation in the ADM decomposition, and discuss several applications in simulations of compact object dynamics and cosmology.