A constrained scheme for Einstein equations based on Dirac gauge and spherical coordinates

TL;DR

This paper introduces a new constrained formulation for 3+1 numerical relativity using Dirac gauge and spherical coordinates, simplifying Einstein equations into elliptic and wave equations, and demonstrating stable numerical evolution of gravitational waves.

Contribution

It extends Dirac gauge to spherical coordinates and reduces Einstein equations to a coupled system of elliptic and wave equations for improved numerical relativity simulations.

Findings

Stable numerical evolution of 3D gravitational waves

Reduction of Einstein equations to elliptic and wave equations

Effective use of Dirac gauge in spherical coordinates

Abstract

We propose a new formulation for 3+1 numerical relativity, based on a constrained scheme and a generalization of Dirac gauge to spherical coordinates. This is made possible thanks to the introduction of a flat 3-metric on the spatial hypersurfaces t=const, which corresponds to the asymptotic structure of the physical 3-metric induced by the spacetime metric. Thanks to the joint use of Dirac gauge, maximal slicing and spherical components of tensor fields, the ten Einstein equations are reduced to a system of five quasi-linear elliptic equations (including the Hamiltonian and momentum constraints) coupled to two quasi-linear scalar wave equations. The remaining three degrees of freedom are fixed by the Dirac gauge. Indeed this gauge allows a direct computation of the spherical components of the conformal metric from the two scalar potentials which obey the wave equations. We present some numerical evolution of 3-D gravitational wave spacetimes which demonstrates the stability of the proposed scheme.