Dimensional reduction in numerical relativity: Modified Cartoon formalism and regularization

Abstract

We present in detail the Einstein equations in the Baumgarte-Shapiro-Shibata-Nakamura formulation for the case of D-dimensional spacetimes with SO(D - d) isometry based on a method originally introduced in Ref. 1. Regularized expressions are given for a numerical implementation of this method on a vertex centered grid including the origin of the quasi-radial coordinate that covers the extra dimensions with rotational symmetry. Axisymmetry, corresponding to the value d = D - 2, represents a special case with fewer constraints on the vanishing of tensor components and is conveniently implemented in a variation of the general method. The robustness of the scheme is demonstrated for the case of a black-hole head-on collision in D = 7 spacetime dimensions with SO(4) symmetry.