Trick for passing degenerate points in the Ashtekar formulation

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 a pullback 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.