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 ‘Arnowitt–Deser–Misner’ 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 π ij δh ij , where π ij is the canonical momentum conjugate to the boundary metric h ij . Then, the first-order Lagrangian density is obtained either by integration of π ij over the metric derivative ∂ w h ij normal to the boundary, or by rewriting the Myers term as a bulk term.