[Paper Review] Notes on Derivation of 'Generalized Gravitational Entropy'

D. V. Fursaev|arXiv (Cornell University)|Jun 21, 2014

Cosmology and Gravitation Theories15 references3 citations

TL;DR

This paper presents a new derivation of generalized gravitational entropy for co-dimension 2 entangling surfaces without relying on conical singularities, using a Hamiltonian-like boundary term approach. It shows that the entropy functional is extremized on the entangling surface and reproduces the Lovelock gravity entropy formula, confirming consistency with prior results via alternative methods.

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.

Motivation & Objective

• To provide an alternative derivation of generalized gravitational entropy that avoids conical singularities in the computation.
• To establish that the entangling surface must be an extremum of the entropy functional.
• To verify consistency of the derived entropy formula with known results in Lovelock gravity theories.
• To generalize the Jacobson-Myers Hamiltonian method to higher-derivative gravity actions.

Proposed method

• The method isolates a small neighborhood around the entangling surface and extracts the boundary term from the gravitational action.
• It uses the replica trick with $ n $-fold replicated manifolds $ \mathcal{M}_n $, where $ \mathcal{T}_n $ has a $ 2\pi n $-periodic $ \tau $-coordinate.
• The entropy is derived as the limit $ \lim_{n\to1} (n\partial_n - 1) I[\bar{\mathcal{M}}_n] $, avoiding direct use of conical singularities.
• The boundary term is evaluated in the limit $ \epsilon \to 0 $, where $ \epsilon $ is the radial cutoff near the entangling surface $ \mathcal{B} $.
• Complex normal vectors and extrinsic curvatures $ k_{ij}, \bar{k}_{ij} $ are used to project the Riemann tensor onto $ \mathcal{B} $.
• The final entropy formula is derived from the integrated boundary term, leading to a Lovelock-like structure on $ \mathcal{B} $.

Experimental results

Research questions

• RQ1 Can generalized gravitational entropy be derived without relying on conical singularities in the action?
• RQ2 Is the entangling surface $ \mathcal{B} $ an extremum of the entropy functional in this new derivation?
• RQ3 Does the derived entropy formula in higher-derivative gravity (Lovelock) match known results from other methods?
• RQ4 Can the Jacobson-Myers Hamiltonian method be extended to generalized entropy in non-Einstein gravity?
• RQ5 Under what conditions does the entropy functional remain extremal for arbitrary higher-derivative gravity theories?

Key findings

• The generalized gravitational entropy is derived without conical singularities, using a boundary term in the action from a replicated geometry.
• The entangling surface $ \mathcal{B} $ is shown to be an extremum of the entropy functional, a key consistency condition.
• For Lovelock gravity, the derived entropy formula matches known results: $ S = 4\pi \sum_m m c_m \hat{I}_m[\mathcal{B}] $, where $ \hat{I}_m[\mathcal{B}] $ is a Lovelock-like invariant on $ \mathcal{B} $.
• The boundary term evaluation leads to a precise expression involving $ k_{ij} $ and $ \bar{k}_{ij} $ curvatures and the intrinsic Riemann tensor of $ \mathcal{B} $.
• The method confirms that the entropy formula is consistent with holographic entanglement entropy in AdS gravity when $ \mathcal{B} $ is a holographic entangling surface.
• The derivation is self-consistent only if the entangling surface extremizes the entropy functional, which is not guaranteed in all higher-derivative gravities.