A trick for passing degenerate points in Ashtekar formulation

TL;DR

This paper proposes a method involving complexification of variables to handle degenerate points in Ashtekar's formulation of general relativity, demonstrating its effectiveness through numerical integration of a degenerate solution.

Contribution

It introduces a novel trick of complexifying variables and applying reality conditions to manage degenerate points in Ashtekar's formulation, supported by numerical evidence.

Findings

Complexification enables continuous evolution through degenerate points.

Numerical integration confirms the effectiveness of the proposed method.

The approach offers new insights into the dynamics of degenerate spacetimes.

Abstract

We examine one of the advantages of Ashtekar's formulation of general relativity: a tractability of degenerate points from the point of view of following the dynamics of classical spacetime. Assuming that all dynamical variables are finite, we conclude that an essential trick for such a continuous evolution is in complexifying variables. In order to restrict the complex region locally, we propose some `reality recovering' conditions on spacetime. Using a degenerate solution derived by pull-back technique, and integrating the dynamical equations numerically, we show that this idea works in an actual dynamical problem. We also discuss some features of these applications.