Abstract: |
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 between B and the bulk extremal surface 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. Near the boundary these properties reduce to integrated energy conditions on the bulk matter. In this limit the relative entropy formula can be inverted via the inverse Radon transform, to reconstruct the energy density at a bulk point. |