Holographic entanglement entropies for Schwarzschild and   Reisner-Nordstr\"om black holes in asymptotically Minkowski spacetimes
  The holographic entanglement entropies (HEE) associated with four dimensional Schwarzschild and Reisner-Nordstr\"om black holes in asymptotically Minkowski spacetimes are investigated. Unlike the cases of asymptotically AdS spacetimes for which the boundaries are always taken at (timelike) conformal infinities, we take the boundaries at either large but finite radial coordinate (far boundary) or very close to the black hole event horizons (near horizon boundary). The reason for such choices is that such boundaries are similar to the conformal infinity of AdS spacetime in that they are all timelike, so that there may be some hope to define dual systems with ordinary time evolution on such boundaries. Our results indicate that, in the case of far boundaries, the leading order contribution to the HEEs come from the background Minkowski spacetime, however, the next to leading order contribution which arises from the presence of the black holes is always proportional to the black hole mass, which constitutes a version of the first law of the HEE for asymptotically flat spacetimes, and the higher order contributions are always negligibly small. In the case of near horizon boundaries, the leading order contribution to the HEE is always proportional to the area of the black hole event horizon, and the case of extremal RN black hole is distinguished from the cases of non-extremal black holes in that the minimal surface defining the HEE is completely immersed inside the boundary up to the second order in the perturbative expansion. 
Yuan Sun and Liu Zhao
gr-qc hep-th