The constraints as evolution equations for numerical relativity

TL;DR

This paper introduces a novel method to reformulate Einstein's constraint equations as evolution equations, ensuring their satisfaction in numerical relativity simulations and simplifying their conservation.

Contribution

The authors present a new approach to rewriting constraints as first-order evolution equations, improving constraint preservation in numerical solutions of Einstein's equations.

Findings

Reformulation guarantees constraints are satisfied to chosen accuracy.

Simplifies the subsidiary constraints compared to standard equations.

Easily integrates into existing Einstein equation formulations.

Abstract

The Einstein equations have proven surprisingly difficult to solve numerically. A standard diagnostic of the problems which plague the field is the failure of computational schemes to satisfy the constraints, which are known to be mathematically conserved by the evolution equations. We describe a new approach to rewriting the constraints as first-order evolution equations, thereby guaranteeing that they are satisfied to a chosen accuracy by any discretization scheme. This introduces a set of four subsidiary constraints which are far simpler than the standard constraint equations, and which should be more easily conserved in computational applications. We explore the manner in which the momentum constraints are already incorporated in several existing formulations of the Einstein equations, and demonstrate the ease with which our new constraint-conserving approach can be incorporated into these schemes.