Constrained scheme for the Einstein equations based on the 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 $t=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, 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 quasilinear elliptic equations (including the Hamiltonian and momentum constraints) coupled to two quasilinear scalar wave equations. The remaining 3 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 3D gravitational wave spacetimes which demonstrates the stability of the proposed scheme.