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 ${\partial }_w{h}_{ij}$ normal to the boundary, or by rewriting the Myers term as a bulk term.

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Cited by in corpus (3)

• On the Lagrangian Holographic Relation at $D\to 2$ and $4$ Limits of Gravity
• Classical Holographic Relations and Alternative Boundary Conditions for Lovelock Gravity
• Thin shell dynamics in Lovelock gravity