Asymptotically constrained and real-valued system based on Ashtekar's variables

TL;DR

This paper introduces a dynamical system based on Ashtekar's variables that naturally evolves towards solutions satisfying Einstein's constraints and reality conditions, potentially improving numerical relativity methods.

Contribution

It develops a new set of evolution equations that ensure the system asymptotically satisfies constraints and reality conditions, extending previous approaches with dissipative forces.

Findings

System evolves towards constraint-satisfying manifold

Ensures stability against perturbative errors

Potentially useful for numerical relativity simulations

Abstract

We present a set of dynamical equations based on Ashtekar's extension of the Einstein equation. The system forces the space-time to evolve to the manifold that satisfies the constraint equations or the reality conditions or both as the attractor against perturbative errors. This is an application of the idea by Brodbeck, Frittelli, Huebner and Reula who constructed an asymptotically stable (i.e., constrained) system for the Einstein equation, adding dissipative forces in the extended space. The obtained systems may be useful for future numerical studies using Ashtekar's variables.