First-order Lagrangian and Hamiltonian of Lovelock gravity

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.