General Relativity as a Theory of Two Connections

TL;DR

This paper presents a novel formulation of General Relativity using two $SO(3)$ connections in phase space, offering new insights into the structure of gravity and potential methods for handling degenerate metrics.

Contribution

It introduces a two connection phase space formulation of General Relativity, extending the Ashtekar approach with unique features for constraint representation and degenerate metrics.

Findings

Two connection description for the Husain-Kuchař model

A Hamiltonian formulation close to Ashtekar's with new features

Potential mechanism for managing degenerate metrics

Abstract

We show in this paper that it is possible to formulate General Relativity in a phase space coordinatized by two $SO(3)$ connections. We analyze first the Husain-Kucha\v{r} model and find a two connection description for it. Introducing a suitable scalar constraint in this phase space we get a Hamiltonian formulation of gravity that is close to the Ashtekar one, from which it is derived, but has some interesting features of its own. Among them a possible mechanism for dealing with the degenerate metrics and a neat way of writing the constraints of General Relativity.