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Résumé |
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/(4GN), 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.
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