Hyperbolic formulations and numerical relativity: Experiments using Ashtekar's connection variables

TL;DR

This paper compares different hyperbolic formulations of Einstein's equations using Ashtekar's connection variables, demonstrating that strongly and symmetric hyperbolic systems improve numerical stability and accuracy in gravitational wave simulations.

Contribution

First to perform full numerical simulations with Ashtekar's variables comparing weak, strong, and symmetric hyperbolic systems, highlighting their numerical performance differences.

Findings

Strong and symmetric hyperbolic systems reduce constraint violations.

Symmetric hyperbolic system is not always the best in numerical performance.

Using Ashtekar's variables allows consistent comparison across hyperbolicity levels.

Abstract

In order to perform accurate and stable long-time numerical integration of the Einstein equation, several hyperbolic systems have been proposed. We here present numerical comparisons between weakly hyperbolic, strongly hyperbolic, and symmetric hyperbolic systems based on Ashtekar's connection variables. The primary advantage for using this connection formulation in this experiment is that we can keep using the same dynamical variables for all levels of hyperbolicity. Our numerical code demonstrates gravitational wave propagation in plane symmetric spacetimes, and we compare the accuracy of the simulation by monitoring the violation of the constraints. By comparing with results obtained from the weakly hyperbolic system, we observe the strongly and symmetric hyperbolic system show better numerical performance (yield less constraint violation), but not so much difference between the latter two. Rather, we find that the symmetric hyperbolic system is not always the best in numerical performances. This study is the premier to present full numerical simulations using Ashtekar's variables. We also describe our procedures in detail.