Abstract
We explore several aspects of the relation between gravity and entanglement in the context of AdS/CFT, in the simple setting of 3 bulk dimensions. Specifically, we consider small perturbations of the AdS metric and the CFT vacuum state and study what can be learnt about the metric perturbation from the Ryu-Takayanagi (RT) formula alone. It is well-known that, if the RT formula holds for all boundary spacelike segments, then the metric perturbation satisfies the linearized Einstein equations throughout the bulk. We generalize this result by showing that, if the RT formula holds for all spacelike segments contained in a certain boundary region, then the metric perturbation satisfies the linearized Einstein equations in a corresponding bulk region (in fact, it is completely determined in that region). We also argue that the same is true for small perturbations of the planar BTZ black hole and the CFT thermal state. We discuss the relation between our results and the ideas of subregion-subregion duality, and we point out that our argument also serves as a holographic proof of the linearized RT formula for boundary segments.
Highlights
Beyond its power as a tool for computing entanglement entropies, the RT formula provides deep insight into the workings of Anti de Sitter (AdS)/conformal field theory (CFT)
We will show that the linearized RT formula for spacelike segments contained in U implies the linearized Einstein equations in W (U ), but is equivalent to those equations supplemented with a boundary condition, which relates the metric perturbation to the state perturbation via the usual holographic formula for the expectation value of the CFT stress tensor
We prove the equivalence announced above for perturbations of the zero-temperature background. We do it in three steps: in subsection 2.3 we find a pair of local equations which is equivalent to the condition that the linearized RT formula hold for all segments contained in some boundary spacelike segment; this result is used in subsection 2.4 to show that the linearized RT formula for spacelike segments contained in a generic boundary open region U is equivalent to the linearized Einstein equations in a corresponding bulk region G(U ) plus a standard holographic formula, which plays the role of a boundary condition; in subsection 2.5 we show that, in the case where U is the domain of dependence of a spacelike segment, G(U ) = W (U ), which completes the argument
Summary
Consider the Poincare patch of the 3-dimensional Anti de Sitter (AdS) space and, on its boundary, a conformal field theory (CFT) in the vacuum state. In these circumstances the Ryu-Takayanagi (RT) formula [2, 3], S. is known to hold for any boundary spacelike segment. Itn [14,15,16] it was shown that, if the RT formula (2.1) continues to hold to first order for all boundary spacelike segments, the metric perturbation satisfies the linearized Einstein equations. If the RT formula (2.1) continues to hold to first order for all spacelike segments contained in U , the metric perturbation satisfies the linearxized Einstein equations in. U implies the linearized Einstein equations in W (U ),zbut is equivalent to those equations supplemented with a boundary condition, which relates the metric perturbation to the state perturbation via the usual holograzphic formula for the expectation value of the CFT stress tensor
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