Notes on Derivation of 'Generalized Gravitational Entropy'

TL;DR

This work develops a conical-singularity–free derivation of generalized gravitational entropy for co-dimension 2 entangling surfaces ${\cal B}$ by adapting a Jacobson–Myers–like boundary-term approach. The entropy emerges from a boundary contribution ${I}_{\text{str}}[{\cal B}]$ in a small neighbourhood around ${\cal B}$, with $S = -{I}_{\text{str}}[{\cal B}] + \lim_{n\to 1} \partial_n^{\text{pos}} I_{\text{str}}[{\cal B}_n]$, and extremality of ${\cal B}$ ensures the last term vanishes, recovering the familiar Einstein result when ${\cal B}$ is minimal. The framework extends to Lovelock gravities, yielding $S = 4\pi \sum_m m c_m \hat{I}_m[{\cal B}]$ where $\hat{I}_m[{\cal B}] = \int_{\cal B} \hat{L}_{m-1}$ and $\hat{L}_{m-1}$ is a higher–order curvature combination on ${\cal B}$, aligning with known results from other methods. Overall, the approach offers a robust, singularity-free route to generalized gravitational entropy and supports holographic entanglement entropy in higher-derivative theories, subject to regularity and extremality conditions on the entangling surface.

Abstract

An alternative derivation of generalized gravitational entropy associated to co-dimension 2 'entangling' hypersurfaces is given. The approach is similar to the Jacobson-Myers 'Hamiltonian' method and it does not require computations on manifolds with conical singularities. It is demonstrated that the entangling surfaces should be extrema of the entropy functional. When our approach is applied to Lovelock theories of gravity the generalized entropy formula coincides with results derived by other methods.