Einstein's equations in Ashtekar's variables constitute a symmetric hyperbolic system

TL;DR

This paper demonstrates that the vacuum Einstein equations expressed in Ashtekar's variables form a symmetric hyperbolic system when appropriate constraint terms are added, ensuring well-posedness for fixed lapse and shift.

Contribution

It establishes the symmetric hyperbolic structure of Einstein's equations in Ashtekar's variables with fixed gauge conditions, extending the mathematical understanding of these formulations.

Findings

The system is symmetric hyperbolic for arbitrary fixed lapse and shift.

Adding constraint-proportional terms yields a well-posed evolution system.

Supports stability analysis of numerical relativity simulations.

Abstract

We show that the 3+1 vacuum Einstein field equations in Ashtekar's variables constitutes a first order symmetric hyperbolic system for arbitrary but fixed lapse and shift fields, by suitable adding to the system terms proportional to the constraint equations.