Geodesics connecting end points of time-like interval in asymptotically AdS spacetime

This paper studies the geodesics connecting two time-like separated boundary points in asymptotically anti-de Sitter (AdS) spacetime. We find that in spherically symmetry Schwarzschild AdS black hole, smooth space-like geodesics can connect timelike-separated points by winding around the horizon multiple times. Similar result will also happen in modified BTZ black hole which contains photon ring in the bulk. According to recent holographic proposal on time-like entanglement entropy, our result suggests that, if there is photon ring/sphere in the bulk, then the time-like entanglement entropy AdS3/CFT2 duality may not have an imaginary part and so further understanding may be necessary.