Abstract
In the duality of 3-dimensional anti de-Sitter (AdS) spacetime and 2-dimensional conformal field theory (CFT), it was argued that the holographic entanglement entropy between two timelike separated events needs both timelike and spacelike geodesics. This paper studies the possibility of using smooth spacelike geodesics to connect two timelike separated events of boundary. We find that in a spherically symmetry Schwarzschild anti-de Sitter spacetime black hole, smooth spacelike geodesics can connect timelike-separated points by winding around the horizon multiple times. A similar result will also happen in a three-dimensional asymptotically AdS black hole that contains a photon ring in the bulk. Our result suggests that, if there is a photon ring, then the timelike entanglement entropy in the AdS3/CFT2 duality may not have an imaginary part. The result implies that the bulk photon ring may play an important role in understanding analytical behaviors of the entanglement spectrum for AdS3/CFT2 duality. Published by the American Physical Society 2025
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