A Reexamination of the Canonical Structure of the Einstein-Hilbert Action in First-Order Form

TL;DR

This paper performs a detailed canonical analysis of the Einstein-Hilbert action in first-order form, revealing the structure of constraints and degrees of freedom using Dirac formalism, and explores extensions with matter fields.

Contribution

It provides a comprehensive constraint analysis of the Einstein-Hilbert action in first-order form, identifying novel tertiary constraints and degrees of freedom.

Findings

Primary constraints lead to secondary and tertiary first class constraints.

The phase space has d(d-3) degrees of freedom after gauge fixing.

Adding matter fields alters the constraint structure and degrees of freedom.

Abstract

A canonical analysis of the Einstein-Hilbert action S_d (d>2) is considered, using the first order form with the metric and affine connection as independent fields. We adopt a conservative approach to using the Dirac constraint formalism; we do not use equations of motion which are independent of time derivatives and correspond to first class constraints to eliminate fields. Applying the Dirac procedure, we find that the primary constraints lead to secondary constraints which are equations of motion not involving time derivatives, and that those secondary constraints which are first class imply novel tertiary constraints which are also first class. Once the constraints and their associated gauge conditions are used to eliminate the non-dynamical degrees of freedom in S_d, there are d(d-3) degrees of freedom left in phase space. We also consider the simpler limiting case of the non-interacting graviton in the first order formalism as well as the effect of adding the action for a massless scalar field to the Einstein-Hilbert action.