On solving the constraints by integrating a strongly hyperbolic system

Abstract

It was shown recently that the constraints on the initial data for Einstein's equations may be posed as an evolutionary problem [9]. In one of the proposed two methods the constraints can be replaced by a first order symmetrizable hyperbolic system and a subsidiary algebraic relation. Here, by assuming that the initial data surface is smoothly foliated by a one-parameter family of topological two-spheres, the basic variables are recast in terms of spin-weighted fields. This allows one to replace all the angular derivatives in the evolutionary system by the Newman-Penrose $\eth$ and $\bar{\eth}$ operators which, in turn, opens up a new avenue to solve the constraints by integrating the resulting system using suitable numerical schemes. In particular, by replacing the $\eth$ and $\bar{\eth}$ operators either by a finite difference or by a pseudo-spectral representation or by applying a spectral decomposition in terms of spin-weighted spherical harmonics, the evolutionary equations may be put into the form of a coupled system of non-linear ordinary differential equations.