Ashtekar's New Variables and Positive Energy

TL;DR

This paper revisits Ashtekar's New Variables in general relativity, addressing previous challenges in proving positive gravitational energy, and introduces a local positive energy expression using special gauge conditions.

Contribution

It provides a novel local positive energy expression for Ashtekar's variables through specific gauge choices, advancing the understanding of gravitational energy positivity.

Findings

Established a locally positive Hamiltonian expression

Demonstrated a positive energy proof using gauge conditions

Addressed previous difficulties with Witten spinor methods

Abstract

We discuss earlier unsuccessful attempts to formulate a positive gravitational energy proof in terms of the New Variables of Ashtekar. We also point out the difficulties of a Witten spinor type proof. We then use the special orthonormal frame gauge conditions to obtain a locally positive expression for the New Variables Hamiltonian and thereby a ``localization'' of gravitational energy as well as a positive energy proof.