Classical Holographic Relations and Alternative Boundary Conditions for Lovelock Gravity

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 $f(L_{\mathrm{Lovelock}}^{(k)})$-gravity.

Latex, 27 pages, title slightly modified, typos corrected, new comments added, references added and updated, to appear in PRD

References in corpus (6)

• Holography of Gravitational Action Functionals
• Counterterms in Dimensionally Continued AdS Gravity
• Holographic Heat Current as Noether Current
• Role of spacetime boundaries in Einstein's other gravity
• First-order Lagrangian and Hamiltonian of Lovelock gravity
• Black Hole Entropy and Boundary Conditions

Cited by in corpus (5)

• On the Lagrangian Holographic Relation at $D\to 2$ and $4$ Limits of Gravity
• Holographic dark energy satisfying the energy conditions in Lovelock gravity
• Observational Feasibility of 4D Einstein-Gauss-Bonnet Cosmology: Bouncing and Non-Bouncing Universes
• General Solutions of Einstein Gravity at $D\to 2$
• Cosmology with Higher-Derivative Gravities