A hole-ographic spacetime
We embed spherical Rindler space – a geometry with a spherical hole in its center – in asymptotically AdS spacetime and show that it carries a gravitational entropy proportional to the area of the hole. Spherical AdS-Rindler space is holographically dual to an ultraviolet sector of the boundary field theory given by restriction to a strip of finite duration in time. Because measurements have finite durations, local observers in the field theory can only access information about bounded spatial regions. We propose a notion of differential entropy that captures uncertainty about the state of a system left by the collection of local, finite-time observables. For two-dimensional conformal field theories we use holography and the strong subadditivity of entanglement to propose a formula for differential entropy and show that it precisely reproduces the areas of circular holes in AdS3. Extending the notion to field theories on strips with variable durations in time, we show more generally that differential entropy computes the areas of all closed, inhomogenous curves on a spatial slice of AdS3. We discuss the extension to higher dimensional field theories, the relation of differential entropy to entanglement between scales, and some implications for the emergence of space from the RG flow of entangled field theories. vijay@physics.upenn.edu,czech@stanford.edu,bdchowdh@asu.edu,m.p.heller,J.deBoer@uva.nl ar X iv :1 31 0. 42 04 v3 [ he pth ] 1 1 Ju n 20 14