Dimensional reduction in numerical relativity: Modified cartoon formalism and regularization

TL;DR

This paper develops a detailed formulation of Einstein's equations for higher-dimensional spacetimes with symmetry, enabling efficient numerical simulations of phenomena like black hole collisions in such contexts.

Contribution

It introduces a modified cartoon formalism for dimensional reduction in numerical relativity, including regularization techniques for implementation on a vertex-centered grid.

Findings

Successfully simulated a black-hole head-on collision in 7D spacetime.

Demonstrated robustness of the scheme for higher-dimensional black hole scenarios.

Provided regularized expressions suitable for numerical implementation with symmetry considerations.

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.