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

TL;DR

This paper performs a canonical analysis of the Einstein-Hilbert action in first order form using Dirac constraints, revealing the structure of constraints and degrees of freedom across different dimensions, especially highlighting differences in 2D.

Contribution

It provides a detailed canonical constraint analysis of the Einstein-Hilbert action in first order form, including the special case of two dimensions and the impact of variable choices.

Findings

For d > 2, tertiary constraints are present and phase space variables match those of a symmetric tensor gauge field.

In 2D, the Hamiltonian is a linear combination of first class constraints with SO(2,1) algebra, indicating no independent degrees of freedom.

The canonical structure differs when using $h^{eta heta}$ instead of $g^{eta heta}$ as a dynamical variable, especially in 2D.

Abstract

Using the Dirac constraint formalism, we examine the canonical structure of the Einstein-Hilbert action $S_d = \frac{1}{16\pi G} \int d^dx \sqrt{-g} R$, treating the metric $g_{\alpha\beta}$ and the symmetric affine connection $\Gamma_{\mu\nu}^\lambda$ as independent variables. For $d > 2$ tertiary constraints naturally arise; if these are all first class, there are $d(d-3)$ independent variables in phase space, the same number that a symmetric tensor gauge field $\phi_{\mu\nu}$ possesses. If $d = 2$, the Hamiltonian becomes a linear combination of first class constraints obeying an SO(2,1) algebra. These constraints ensure that there are no independent degrees of freedom. The transformation associated with the first class constraints is not a diffeomorphism when $d = 2$; it is characterized by a symmetric matrix $\xi_{\mu\nu}$. We also show that the canonical analysis is different if $h^{\alpha\beta} = \sqrt{-g} g^{\alpha\beta}$ is used in place of $g^{\alpha\beta}$ as a dynamical variable when $d = 2$, as in $d$ dimensions, $\det h^{\alpha\beta} = - (\sqrt{-g})^{d-2}$. A comparison with the formalism used in the ADM analysis of the Einstein-Hilbert action in first order form is made by applying this approach in the two dimensional case with $h^{\alpha\beta}$ and $\Gamma_{\mu\nu}^\lambda$ taken to be independent variables.