Gravitation from Entanglement in Holographic CFTs (1312.7856v2)

Abstract: Entanglement entropy obeys a 'first law', an exact quantum generalization of the ordinary first law of thermodynamics. In any CFT with a semiclassical holographic dual, this first law has an interpretation in the dual gravitational theory as a constraint on the spacetimes dual to CFT states. For small perturbations around the CFT vacuum state, we show that the set of such constraints for all ball-shaped spatial regions in the CFT is exactly equivalent to the requirement that the dual geometry satisfy the gravitational equations of motion, linearized about pure AdS. For theories with entanglement entropy computed by the Ryu-Takayanagi formula $S=A/(4G_N)$, we obtain the linearized Einstein equations. For theories in which the vacuum entanglement entropy for a ball is computed by more general Wald functionals, we obtain the linearized equations for the associated higher-curvature theories. Using the first law, we also derive the holographic dictionary for the stress tensor, given the holographic formula for entanglement entropy. This method provides a simple alternative to holographic renormalization for computing the stress tensor expectation value in arbitrary higher derivative gravitational theories.

• The paper demonstrates that quantum entanglement enforces spacetime geometry by translating the first law of entanglement into linearized gravitational equations via the Ryu-Takayanagi formula.
• It introduces a streamlined approach to calculating the holographic stress tensor, enhancing evaluations in higher-derivative gravitational theories.
• The findings imply that the universal entropic law underlies black hole thermodynamics, paving the way for unifying quantum mechanics and gravity.

Summary of "Gravitation from Entanglement in Holographic CFTs"

The paper, "Gravitation from Entanglement in Holographic CFTs," underscores the pivotal role of quantum entanglement in the context of the AdS/CFT correspondence, proposing a substantive equivalence between entanglement entropy and spacetime geometry through gravitational laws. The research focuses on understanding this relation in the framework of conformal field theories (CFTs) with semiclassical holographic duals.

Core Contributions and Analytical Framework

The authors investigate the 'first law' of entanglement entropy, which resembles the first law of thermodynamics but applies to entropies of quantum systems. Specifically, they demonstrate that for small perturbations around the CFT vacuum state, this entropic law translates into linearized gravitational equations in the holographic dual, supporting the thesis with rigorous mathematical detail. The argument’s backbone is the Ryu-Takayanagi formula for entanglement entropy, which associates it with the area of extremal surfaces in the AdS space, yielding the linearized Einstein equations.

Key Results

1. Linearized Einstein Equations: The paper asserts that for CFTs wherein the Ryu-Takayanagi prescription is applicable, the perturbations of the vacuum state necessitate solutions that satisfy Einstein's equations linearized around pure AdS spacetime. The results are generalized to theories with more complex entanglement entropies computed by Wald functionals, deriving equivalent higher-curvature gravitational equations.
2. Holographic Stress Tensor: A significant methodological innovation is the introduction of a simplified alternative to holographic renormalization for calculating expectations of the CFT stress tensor. This is particularly effective for higher derivative gravitational theories, providing a streamlined approach for computational evaluations.
3. Microscopic Interpretation and Universality: The first law of entanglement entropy is postulated to be the underlying microscopic principle related to the first law of black hole thermodynamics for AdS-Rindler horizons. This showcases the universal nature of these constraints as they imply the linearized field equations in any classical gravitational theory where this microscopic entropic law holds.

Theoretical and Practical Implications

• Theoretical: The derivation reinforces the robustness of holographic duals of CFTs, specifically illustrating how quantum entanglement contributes to defining spacetime metrics and dynamics. This approach enhances the conceptual framework linking quantum information theory with gravitational physics.
• Practical: By enabling the extraction of gravitational equations in arbitrary higher-derivative theories, the methods developed may streamline computational aspects of exploring quantum gravity scenarios, potentially leading to new insights into both cosmological models and black hole physics.

Future Directions

The results provoke several intriguing directions for further research, such as extending the relationship between entanglement entropy and gravitational dynamics beyond linear perturbations to encompass nonlinear solutions. Additionally, the consideration of quantum corrections and their contributions to bulk gravitational theories offers fertile ground for new endeavors informed by holographic principles.

In essence, this paper substantiates the conjecture that spacetime geometry—or more explicitly, gravitational dynamics—could emerge from the quantum entanglement properties intrinsic to the dual CFTs, thus providing a promising pathway toward unifying quantum mechanics and gravity under the ambit of holographic duality.