Classical Holographic Relations and Alternative Boundary Conditions for Lovelock Gravity

TL;DR

This paper derives classical holographic relations for Lovelock gravity, decomposes the Lagrangian into bulk and surface terms, and explores alternative boundary conditions, showing consistent black hole entropy results.

Contribution

It introduces a foliation-independent and an ADM-based holographic relation for Lovelock gravity, and examines their implications for boundary conditions and black hole entropy.

Findings

Bulk term is non-degenerate in foliation-independent approach.

ADM approach yields a first-order bulk term with alternative boundary conditions.

Black hole entropy remains consistent under different boundary conditions.

Abstract

We obtain the classical holographic relation for the general Lovelock gravity and decompose the full Lagrangian into the bulk term and the surface term, expressed as a total derivative $\partial_\mu J^\mu$. By classical holographic relation, we mean that $J^\mu$ is determined completely by the bulk term. We find that the bulk term is not degenerate, or first-order in this foliation-independent approach. We then consider the Arnowitt-Deser-Misner (ADM) formalism where the foliation coordinate $w$ is treated as special. We obtain the classical holographic-degenerate relation with the first-order bulk term that does not involve higher than one derivative of $w$. For Einstein gravity, the two approaches lead to the same bulk term, but different ones for higher-order Lovelock gravities. The classical holographic-degenerate formulation in the ADM approach allows us to consider alternative boundary conditions in the variation principle with different Myers terms. We show in the semiclassical approximation that the black hole entropy in all cases is the same as the one obtained under the standard Dirichlet boundary condition. We also generalize the formalism to general $f(L_{\rm Lovelock}^{(k)})$-gravity.