Causal structure and diffeomorphismsm in Ashtekar's gravity

TL;DR

This paper develops a diffeomorphism invariant extension of Einstein gravity that includes singular metrics, maintaining causal structure and linking Ashtekar's formulation to a covariant theory with local SO(3,C) invariance.

Contribution

It introduces a new extension of Einstein gravity compatible with Ashtekar's formulation, allowing degenerate metrics without disrupting causal structure.

Findings

Degenerate metrics do not destroy causal structure.

Ashtekar's gravity has a covariant extension with local SO(3,C) invariance.

The ADM formulation is shown to be equivalent to the covariant theory.

Abstract

A manifestly diffeomorphism invariant extension of Einstein gravity is constructed, which includes singular metrics, and whose ADM formulation is Ashtekar's gravity. The latter is shown to be locally equivalent to the covariant theory. It turns out that exactly those kinds of degenerate four dimensional metrics are allowed which do not destroy the causal structure of spacetime. It is also shown that Ashtekar's gravity possesses an extension that provides a local SO(3,C) invariance, without complexifying or changing the signature of the metric.