Abstract
In this paper we argue that classical, asymptotically AdS spacetimes that arise as states in consistent ultraviolet completions of Einstein gravity coupled to matter must satisfy an infinite family of positive energy conditions. To each ball-shaped spatial region $B$ of the boundary spacetime, we can associate a bulk spatial region $\Sigma_B$ between $B$ and the bulk extremal surface $\tilde{B}$ with the same boundary as $B$. We show that there exists a natural notion of a gravitational energy for every such region that is non-negative, and non-increasing as one makes the region smaller. The results follow from identifying this gravitational energy with a quantum relative entropy in the associated dual CFT state. The positivity and monotonicity properties of the gravitational energy are implied by the positivity and monotonicity of relative entropy, which holds universally in all quantum systems.
Highlights
Consider a classical asymptotically AdS spacetime M of (d + 1) dimensions associated with a state in some UV-complete theory of quantum gravity for which the low-energy effective description is Einstein gravity coupled to matter
We find that the relative entropy S(ρΨB||ρvBac) is related to a novel measure of energy associated with the spatial region ΣB between the boundary domain B and the extremal surface B: S(ρΨB||ρvBac) = EnergyM (ΣB) − EnergyAdS(ΣB)
After reviewing the definition of the relative entropy in conformal field theory, we will formulate the holographic dual of the relative entropy in terms of the quasi-local energy associated to the region between the boundary domain B and the extremal surface B (Ryu-Takayanagi surface or its covariant generalization)
Summary
Consider a classical asymptotically AdS spacetime M of (d + 1) dimensions associated with a state in some UV-complete theory of quantum gravity for which the low-energy effective description is Einstein gravity coupled to matter. This same technology is employed in the present paper to derive the result (2) from the expression of [3] involving boundary integrals It was pointed out in [12] that the relative entropy of nearby states in the CFT, in the sense that their gravity duals are different quantum states on the same background geometry, is given by the relative entropy in the bulk.
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