Constraint propagation in N+1-dimensional space-time

TL;DR

This paper investigates how constraint propagation behaves in higher-dimensional space-times, highlighting potential numerical stability issues and emphasizing the importance of constraint analysis for reformulating Einstein equations.

Contribution

It analyzes the dimensional dependence of constraint propagation in N+1-dimensional space-time and discusses implications for numerical relativity formulations.

Findings

Constraint propagation equations are independent of dimension N.

Potential accuracy and stability issues in N+1-dimensional numerical simulations.

Reformulation efforts based on constraint analysis remain applicable.

Abstract

Higher dimensional space-time models provide us an alternative interpretation of nature, and give us different dynamical aspects than the traditional four-dimensional space-time models. Motivated by such recent interests, especially for future numerical research of higher-dimensional space-time, we study the dimensional dependence of constraint propagation behavior. The $N+1$ Arnowitt-Deser-Misner evolution equation has matter terms which depend on $N$, but the constraints and constraint propagation equations remain the same. This indicates that there would be problems with accuracy and stability when we directly apply the $N+1$ ADM formulation to numerical simulations as we have experienced in four-dimensional cases. However, we also conclude that previous efforts in re-formulating the Einstein equations can be applied if they are based on constraint propagation analysis.