Flat-space Limit of Extremal Curves

According to the Ryu-Takayanagi prescription, the entanglement entropy of subsystems in the boundary conformal field theory (CFT) is proportional to the area of extremal surfaces in bulk asymptotically Anti-de Sitter (AdS) spacetimes. The flat-space limit of these surfaces is not well defined in the generic case. We introduce a new curve in the three-dimensional asymptotically AdS spacetimes with a well-defined flat-space limit. We find this curve by using a new vector, which is vanishing on it and is normal to the bulk modular flow of the original interval in the two-dimensional CFT. The flat-space limit of this new vector is well defined and gives rise to the bulk modular flow of the corresponding asymptotically flat spacetime. Moreover, after Rindler transformation, this new vector is the normal Killing vector of the BTZ inner horizon. We reproduce all known results about the holographic entanglement entropy of Bondi-Metzner-Sachs invariant field theories, which are dual to the asymptotically flat spacetimes.