Note on generalized gravitational entropy in Lovelock gravity

TL;DR

This paper extends the concept of gravitational entropy to Lovelock gravity in arbitrary dimensions, deriving constraints on minimal surfaces via the replica trick, and highlighting differences from known holographic entanglement entropy functionals.

Contribution

It provides a generalized framework for gravitational entropy in Lovelock gravity and derives new constraint equations for minimal surfaces in this context.

Findings

Constraints show surfaces are minimal when spacetime is maximally symmetric.

The derived constraints differ from known holographic entanglement entropy functionals.

The study extends gravitational entropy concepts beyond Einstein gravity.

Abstract

The recently proposed gravitational entropy generalize the usual black hole entropy to Euclidean solutions without U(1) symmetry in the framework of Einstein gravity. The entropy of such smooth configuration is given by the area of minimal surface, therefore explaining the Ryu-Takayanagi formula of holographic entanglement entropy. In this note we investigate the generalized gravitational entropy for general Lovelock gravity in arbitrary dimensions. We use the replica trick and consider the Euclidean bulk spacetime with conical singularity localized at a codimension two surface. We obtain a constraint equation for the surface by requiring the bulk equation of motion to be of good behavior. When the bulk spacetime is maximally symmetric, the constraints show that the traces of the extrinsic curvatures of the surface are vanishing, i.e. the surface has to be geometrically a minimal surface. However the constraint equation cannot be obtained by the variation of the known functional for holographic entanglement entropy in Lovelock gravity.