Symmetric hyperbolic system in the Ashtekar formulation

TL;DR

This paper develops a symmetric hyperbolic formulation of vacuum general relativity using Ashtekar variables, emphasizing Hermiticity, reality conditions, and characteristic speeds for improved mathematical and numerical analysis.

Contribution

It introduces a new symmetric hyperbolic system in Ashtekar variables, utilizing Hermiticity and constraint addition, differing from previous approaches by focusing on characteristic matrix properties.

Findings

System is symmetric hyperbolic with Hermitian characteristic matrix.

Ensures consistency with reality conditions.

Analyzes characteristic speeds of the system.

Abstract

We present a first-order symmetric hyperbolic system in the Ashtekar formulation of general relativity for vacuum spacetime. We add terms from constraint equations to the evolution equations with appropriate combinations, which is the same technique used by Iriondo, Leguizam\'on and Reula [Phys. Rev. Lett. 79, 4732 (1997)]. However our system is different from theirs in the points that we primarily use Hermiticity of a characteristic matrix of the system to characterize our system "symmetric", discuss the consistency of this system with reality condition, and show the characteristic speeds of the system.