Hamiltonian analysis of the double null 2+2 decomposition of Ashtekar variables

TL;DR

This paper presents a Hamiltonian analysis of General Relativity using a double null 2+2 decomposition with Ashtekar variables, revealing the algebra of constraints and their geometric interpretations.

Contribution

It introduces a canonical formulation of GR in double null 2+2 variables with complex self-dual forms, detailing the structure of the first class constraints and their Lie algebra.

Findings

Derived the constraint algebra as a Lie algebra.

Identified constraints generating surface diffeomorphisms and null generator transformations.

Clarified the role of self-dual spin and boost constraints.

Abstract

We derive a canonical analysis of a double null 2+2 Hamiltonian description of General Relativity in terms of complex self-dual 2-forms and the associated SO(3) connection variables. The algebra of first class constraints is obtained and forms a Lie algebra that consists of two constraints that generate diffeomorphisms in the two surface, a constraint that generates diffeomorphisms along the null generators and a constraint that generates self-dual spin and boost transformations.