Holographic Entanglement Entropy for General Higher Derivative Gravity (1310.5713v4)

Abstract: We propose a general formula for calculating the entanglement entropy in theories dual to higher derivative gravity where the Lagrangian is a contraction of Riemann tensors. Our formula consists of Wald's formula for the black hole entropy, as well as corrections involving the extrinsic curvature. We derive these corrections by noting that they arise from naively higher order contributions to the action which are enhanced due to would-be logarithmic divergences. Our formula reproduces the Jacobson-Myers entropy in the context of Lovelock gravity, and agrees with existing results for general four-derivative gravity. We emphasize that the formula should be evaluated on a particular bulk surface whose location can in principle be determined by solving the equations of motion with conical boundary conditions. This may be difficult in practice, and an alternative method is desirable. A natural prescription is simply minimizing our formula, analogous to the Ryu-Takayanagi prescription for Einstein gravity. We show that this is correct in several examples including Lovelock and general four-derivative gravity.

• The paper introduces a novel formula combining Wald's entropy with extrinsic curvature corrections to compute holographic entanglement entropy accurately.
• It extends the Ryu-Takayanagi prescription to include complex frameworks like Gauss-Bonnet and Lovelock gravity theories.
• The findings pave the way for exploring gauge-gravity duality and black hole entropy, raising questions about stability and higher order corrections.

Holographic Entanglement Entropy for Theories with Higher Derivative Gravity

The paper in discussion presents an advancement in the calculation of holographic entanglement entropy for theories dual to higher derivative gravity. Authored by Xi Dong, the paper introduces a general formula for determining entanglement entropy in contexts where the gravitational Lagrangian involves contractions of Riemann tensors. This work extends the approach initially developed by Ryu and Takayanagi for Einstein gravity, encompassing more complex gravitational interactions.

Proposed Formula and Methodology

The pivotal contribution of this paper is the derivation of a holographic entanglement entropy formula that incorporates Wald's entropy along with additional corrections due to extrinsic curvature. This advancement is crucial as Wald's entropy alone does not suffice for obtaining the correct universal terms in theories governed by higher derivative gravity, such as Gauss-Bonnet gravity within the Lovelock framework. The formula is generalized to account for potentially naive higher-order contributions which become significant due to logarithmic divergences.

The proposed entropy formula is expressed as follows:

$$S_{EE}= 2\pi\int d^d y \sqrt{g} \left( \frac{\partial L}{\partial R_{z z}} + \sum_{i} \frac{\partial^2 L}{\partial R_{zizj} \partial R_{kl}} \frac{8 K_{zij} K_{kl}}{q_i+1} \right)$$

The presence of extrinsic curvature terms, which do not vanish in non-Killing horizon settings, necessitates this refinement for accurately capturing the entropy in holographic settings.

Evaluating the Proposed Formula

To employ the formula effectively, it must be evaluated at a specific codimension-2 surface in the bulk. This surface is theoretically determined by solving the equations of motion with conical boundary conditions—a practice, however, recognized as challenging. Therefore, an analogous proposal to the Ryu-Takayanagi prescription is suggested: minimizing the entropy formula itself could identify the appropriate surface location, a conjecture validated in several situations including Lovelock gravity and general four-derivative gravity.

Implications and Future Directions

The implications of this work are profound for the theoretical understanding of entanglement entropy in complex gravity scenarios. It bridges an important gap between classical applications of holography and real-world theories that predict higher derivative terms, like string theory. The proposed framework can also be adapted to compute black hole entropy in these higher derivative theories, evaluated at horizon surfaces.

The results presented open several avenues for further inquiry. A primary question for subsequent paper is the widespread applicability of a minimization principle for other, less straightforward theories of gravity. Additionally, the derivation of embedding equations, which typically involve more complex derivatives for higher order gravity theories, raises further questions about the potential existence and nature of ghosts or other anomalous behaviors.

Lastly, the paper points out the necessity for continued exploration into the connections between entanglement entropy and the stability—both dynamic and statistical—of black hole solutions in such theories.

Conclusion

This paper sets a foundational milestone in the calculation of entanglement entropy for theories with higher derivative gravity, providing a robust formulation that encapsulates both intrinsic and extrinsic geometric features of the bulk space. It marks a significant progression in the paper of gauge-gravity duality and offers insightful perspectives that could guide future research in theoretical and gravitational physics.