The reality conditions for the new canonical variables of General Relativity

TL;DR

This paper investigates the reality conditions necessary for the Ashtekar formulation of 4-dimensional general relativity, establishing their consistency with constraints and clarifying the conditions under which the theory remains equivalent to Einstein's.

Contribution

It demonstrates the polynomial reality conditions' consistency with constraints and clarifies their applicability depending on the non-degeneracy of the inverse metric.

Findings

Reality conditions are consistent with constraints when the inverse metric is non-degenerate.

The theory is equivalent to Einstein's general relativity under these conditions.

Degenerate metrics lead to complex general relativity, where reality conditions cannot be imposed generally.

Abstract

We examine the constraints and the reality conditions that have to be imposed in the canonical theory of 4--d gravity formulated in terms of Ashtekar variables. We find that the polynomial reality conditions are consistent with the constraints, and make the theory equivalent to Einstein's, as long as the inverse metric is not degenerate; when it is degenerate, reality conditions cannot be consistently imposed in general, and the theory describes complex general relativity.