Darboux Coordinates for the Hamiltonian of First Order Einstein-Cartan Gravity

Based on our 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 that preserve equivalence with the original action in all spacetime dimensions higher than two. Considerable simplification of the Hamiltonian formulation using the Darboux coordinates, as compared with the 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 unique gauge invariances generated by the first class constraints are 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.