Canonical analysis of $n$-dimensional Palatini action without second-class constraints

TL;DR

This paper performs a canonical analysis of the n-dimensional Palatini action, eliminating second-class constraints and expressing the phase space in terms of covariant variables with only first-class constraints, connecting to ADM formalism.

Contribution

It provides a novel canonical formulation of n-dimensional general relativity without second-class constraints, using covariant variables and gauge fixing.

Findings

Phase space described by covariant variables with only first-class constraints

Elimination of nondynamical variables via equations of motion

Connection to ADM formalism through gauge fixing

Abstract

We carry out the canonical analysis of the $n$-dimensional Palatini action with or without a cosmological constant $(n\geq3)$ introducing neither second-class constraints nor resorting to any gauge fixing. This is accomplished by providing an expression for the spatial components of the connection that allows us to isolate the nondynamical variables present among them, which can later be eliminated from the action by using their own equation of motion. As a result, we obtain the description of the phase space of general relativity in terms of manifestly $SO(n-1,1)$ [or $SO(n)$] covariant variables subject to first-class constraints only, with no second-class constraints arising during the process. Afterwards, we perform, at the covariant level, a canonical transformation to a set of variables in terms of which the above constraints take a simpler form. Finally, we impose the time gauge and make contact with the $SO(n-1)$ ADM formalism.