Linear derivative Cartan formulation of General Relativity

TL;DR

This paper presents a new formulation of Einstein gravity as a gauge theory with manifest local Lorentz invariance, auxiliary fields, and a bi-complex algebra, providing a systematic and potentially superior approach to the theory.

Contribution

It introduces a linear derivative Cartan formulation of General Relativity incorporating auxiliary fields and a bi-complex algebra, offering a systematic embedding and improved constraint handling.

Findings

Provides a gauge theory formulation of Einstein gravity with local Poincare symmetry.

Clarifies the role of auxiliary two-form fields in the Hamiltonian structure.

Shows relations between constraints and how to determine Lagrange multipliers.

Abstract

Beside diffeomorphism invariance also manifest SO(3,1) local Lorentz invariance is implemented in a formulation of Einstein Gravity (with or without cosmological term) in terms of initially completely independent vielbein and spin connection variables and auxiliary two-form fields. In the systematic study of all possible embeddings of Einstein gravity into that formulation with auxiliary fields, the introduction of a ``bi-complex'' algebra possesses crucial technical advantages. Certain components of the new two-form fields directly provide canonical momenta for spatial components of all Cartan variables, whereas the remaining ones act as Lagrange multipliers for a large number of constraints, some of which have been proposed already in different, less radical approaches. The time-like components of the Cartan variables play that role for the Lorentz constraints and others associated to the vierbein fields. Although also some ternary ones appear, we show that relations exist between these constraints, and how the Lagrange multipliers are to be determined to take care of second class ones. We believe that our formulation of standard Einstein gravity as a gauge theory with consistent local Poincare algebra is superior to earlier similar attempts.