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

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  • 17 July 2003
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, Einstein equations are reduced to a system of five quasi-linear elliptic equations (including the Hamiltonian and momentum constraints) coupled to two quasi-linear wave equations. The remaining three degrees of freedom are fixed quasi-algebraically by the Dirac gauge. Within this framework, we also derive a slow evolution approximation for Einstein equations which consists in neglecting the gravitational waves and is analogous to the anelastic approximation in hydrodynamics.

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