Entanglement entropy and Page curve of black holes with island in massive gravity

By applying the island rule proposed recently, we compute the entanglement entropy of Hawking radiation and study the Page curve for the eternal black holes in massive gravity. We investigate for both the neutral and charged black holes which the corresponding results of Schwarzschild and Reissner-Nordstr\"{o}m black holes are restored in the limit of massless graviton. We show for the neutral and non-extremal charged black holes that the island is not formed at the early times of the evaporation and hence the entanglement entropy increases linearly in time. However, for the extremal charged black hole, the calculation of the entanglement entropy at the early times without the island is ill-defined because the metric is divergent at the curvature singularity. This implies that new physics in the UV region must be taken into account to make the metric behaving smoothly at the very short distances. At the late times, with the emergence of one island near the event horizon, the entanglement entropy is saturated by the Bekenstein-Hawking entropy of black holes. In addition, we analyze the impact of massive gravity parameters on the size of island, the entanglement entropy, the Page time, and the scrambling time in detail.