A constrained scheme for Einstein equations based on Dirac gauge and 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.