Hamiltonian analysis of $n$-dimensional Palatini gravity with matter

TL;DR

This paper performs a Hamiltonian analysis of $n$-dimensional Palatini gravity coupled with a scalar field, showing it yields the same dynamics as the Hilbert action but does not directly produce a connection dynamics formalism.

Contribution

It provides a detailed Hamiltonian analysis of $n$-dimensional Palatini gravity with matter, highlighting its equivalence to the Hilbert action and limitations in connection formalism.

Findings

Palatini and Hilbert actions produce the same geometric dynamics.

The Palatini formalism does not directly lead to a connection dynamics formalism.

The analysis applies for dimensions greater than two.

Abstract

We consider the Palatini formalism of gravity with cosmological constant $\Lambda$ coupled to a scalar field $\phi$ in $n$-dimensions. The $n$-dimensional Einstein equations with $\Lambda$ can be derived by the variation of the coupled Palatini action provided $n>2$. The Hamiltonian analysis of the coupled action is carried out by a $1+(n-1)$ decomposition of the spacetime. It turns out that both Palatini action and Hilbert action lead to the same geometric dynamics in the presence of $\Lambda$ and $\phi$. While, the $n$-dimensional Palatini action could not give a connection dynamics formalism directly.