Note on generalized gravitational entropy in Lovelock gravity

TL;DR

The note extends the generalized gravitational entropy framework to Lovelock gravity using the replica trick and analyzes the resulting constraint on the codimension-two entangling surface. It finds that, for maximally symmetric bulks, the surface must be minimal, but the bulk-derived constraint does not generally coincide with the variation of the proposed holographic entanglement entropy functionals (Wald or Jacobson-Myers), signaling a tension between generalized entropy and holographic entropy in higher-curvature theories. The authors show this mismatch persists in Gauss-Bonnet and general Lovelock gravity, though certain cases (e.g., sphere/cylinder in AdS_5) may approximate agreement when higher-order extrinsic-curvature terms are small. This work highlights the need to reassess or modify the entropy functionals in Lovelock theories to consistently capture holographic entanglement, or to reinterpret generalized gravitational entropy beyond its holographic entanglement interpretation.

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.