Reality Conditions and Ashtekar Variables: a Different Perspective

TL;DR

This paper introduces a modified self-dual action that simplifies the derivation of the $SO(3)$-ADM formalism and offers new insights into the reality conditions in complex general relativity, leading to a real Lorentzian formulation.

Contribution

It presents a new action principle that avoids second class constraints and provides a real formulation of Lorentzian general relativity in Ashtekar variables.

Findings

Simplified derivation of $SO(3)$-ADM formalism.

New perspective on reality conditions in complex GR.

Real Lorentzian formulation in Ashtekar phase space.

Abstract

We give in this paper a modified self-dual action that leads to the $SO(3)$-ADM formalism without having to face the difficult second class constraints present in other approaches (for example if one starts from the Hilbert-Palatini action). We use the new action principle to gain some new insights into the problem of the reality conditions that must be imposed in order to get real formulations from complex general relativity. We derive also a real formulation for Lorentzian general relativity in the Ashtekar phase space by using the modified action presented in the paper.