Abelian Ashtekar formulation from the ADM action

TL;DR

This paper derives the Ashtekar formulation of linear gravity directly from the ADM action by linearization and canonical transformation, confirming consistency with the standard linearization approach.

Contribution

It presents a novel derivation of the linearized Ashtekar variables directly from the ADM action, providing an alternative to the standard linearization method.

Findings

The derived Ashtekar variables match those obtained via the standard linearization.

The approach confirms the consistency of the Ashtekar formulation with linearized gravity.

Provides a new perspective on connecting ADM and Ashtekar formulations for linear theories.

Abstract

We study the Ashtekar formulation of linear gravity starting from the ADM first order action for the non linear theory, linearizing it, and performing a canonical transformation that coordinatizes the phase space in terms of the already linearized Ashtekar variables. The results obtained in this way are in accordance with those obtained through the standard method, in which, after introducing the Ashtekar variables for the full theory, a linearization around the flat Abelian connection and its conjugate momentum is performed.