Entanglement entropy of asymptotically flat non-extremal and extremal black holes with an island

The island rule for the entanglement entropy is applied to an eternal Reissner-Nordstr\"om black hole. key ingredient that hole assumed be in thermal equilibrium with a heat bath of arbitrary temperature and so generalized treated as being off-shell. Taking on-shell condition off-shell entropy, we find then obtain following rule. For non-extremal hole, grows linearly time can saturated after Page expected. also has well-defined Schwarzschild limit. In extremal prescription provides logarithmically growing constant time. boundary hits curvature singularity where semi-classical approximations appear invalid. To avoid encountering singularity, apply this procedure Hayward regular at origin. Consequently, presence holes provide finite which might imply non-trivial vacuum configurations holes.