Hamiltonian analysis of the double null 2+2 decomposition of General Relativity expressed in terms of self-dual bivectors

TL;DR

This paper develops a Hamiltonian formulation of General Relativity using a 2+2 double null decomposition with self-dual bivectors, revealing a Lie algebra structure of the constraints.

Contribution

It introduces a novel Hamiltonian framework for GR based solely on self-dual variables in a double null decomposition, with a clear geometric interpretation.

Findings

First class constraint algebra forms a Lie algebra

Constraints have simple geometric interpretation

Complete canonical analysis in self-dual variables

Abstract

In this paper we obtain a 2+2 double null Hamiltonian description of General Relativity using only the (complex) SO(3) connection and the components of the complex densitised self-dual bivectors. We carry out the general canonical analysis of this system and obtain the first class constraint algebra entirely in terms of the self-dual variables. The first class algebra forms a Lie algebra and all the first class constraints have a simple geometrical interpretation.