Darboux coordinates for the Hamiltonian of first order Einstein-Cartan gravity

TL;DR

This paper derives Darboux coordinates for the Hamiltonian formulation of first order Einstein-Cartan gravity, simplifying the analysis and clarifying the gauge invariances as translations and rotations in tangent space, distinct from diffeomorphisms.

Contribution

It introduces a unique set of Darboux coordinates for the Hamiltonian of Einstein-Cartan gravity, simplifying the formulation and clarifying the nature of gauge invariances.

Findings

Darboux coordinates preserve equivalence with the original action in all spacetime dimensions higher than two.

Simplification of the Hamiltonian formulation using Darboux coordinates is explicitly demonstrated.

The gauge invariances are identified as translation and rotation in tangent space, not diffeomorphisms.

Abstract

Based on preliminary analysis of the Hamiltonian formulation of the first order Einstein-Cartan action (arXiv:0902.0856 [gr-qc] and arXiv:0907.1553 [gr-qc]) we derive the Darboux coordinates, which are a unique and uniform change of variables preserving equivalence with the original action in all spacetime dimensions higher than two. Considerable simplification of the Hamiltonian formulation using the Darboux coordinates, compared with direct analysis, is explicitly demonstrated. Even an incomplete Hamiltonian analysis in combination with known symmetries of the Einstein-Cartan action and the equivalence of Hamiltonian and Lagrangian formulations allows us to unambiguously conclude that the \textit{unique} \textit{gauge} invariances generated by the first class constraints of the Einstein-Cartan action and the corresponding Hamiltonian are \textit{translation and rotation in the tangent space}. Diffeomorphism invariance, though a manifest invariance of the action, is not generated by the first class constraints of the theory.