Solving Einstein's Equation Numerically on Manifolds With Arbitrary Spatial Topologies

Abstract

This paper develops a method for solving Einstein's equation numerically on multi-cube representations of manifolds with arbitrary spatial topologies. This method is designed to provide a set of flexible, easy to use computational procedures that make it possible to explore the never before studied properties of solutions to Einstein's equation on manifolds with arbitrary toplogical structures. A new covariant, first-order symmetric-hyperbolic representation of Einstein's equation is developed for this purpose, along with the needed boundary conditions at the interfaces between adjoining cubic regions. Numerical tests are presented that demonstrate the long-term numerical stability of this method for evolutions of a complicated, time-dependent solution of Einstein's equation coupled to a complex scalar field on a manifold with spatial topology S^3. The accuracy of these numerical test solutions is evaluated by performing convergence studies and by comparing the full non-linear numerical results to the analytical perturbation solutions, which are also derived here.