First-order Lagrangian and Hamiltonian of Lovelock gravity

TL;DR

This paper derives the first-order Lagrangian and Hamiltonian formulations of Lovelock gravity using an elementary approach, starting from the Lagrangian with Myers boundary term to ensure a proper variational principle.

Contribution

It provides a straightforward derivation of the first-order Lagrangian and Hamiltonian for Lovelock gravity, extending previous insights from Einstein-Hilbert and Gauss-Bonnet theories.

Findings

Derived the first-order Lagrangian density for Lovelock gravity.

Established the Hamiltonian formulation consistent with the variational principle.

Presented an elementary method for obtaining these formulations from the Lagrangian with boundary terms.

Abstract

Based on the insight gained by many authors over the years on the structure of the Einstein-Hilbert, Gauss-Bonnet and Lovelock gravity Lagrangians, we show how to derive -- in an elementary fashion -- their first-order, generalized "ADM" Lagrangian and associated Hamiltonian. To do so, we start from the Lovelock Lagrangian supplemented with the Myers boundary term, which guarantees a Dirichlet variational principle with a surface term of the form $\pi^{ij}\delta h_{ij}$, where $\pi^{ij}$ is the canonical momentum conjugate to the boundary metric $h_{ij}$. Then, the first-order Lagrangian density is obtained either by integration of $\pi^{ij}$ over the metric derivative $\partial_wh_{ij}$ normal to the boundary, or by rewriting the Myers term as a bulk term.