Planar black holes in holographic axion gravity: Islands, Page times, and scrambling times

The present work investigates the entanglement entropies of the Hawking radiations, Page times, and scrambling times for the eternal planar black holes in the holographic axion gravity. The solutions correspond to a new class of charged black holes, because the boundary diffeomorphism is broken due to the graviton mass induced by the axion fields in the bulk. The information theoretical aspects of these black hole solutions are determined upon applying the island rule for the entanglement entropy. Like nonextremal charged black holes, the radiation entropy grows linearly in the no-island configurations, while it is saturated at late times by asymptotic values set by the Bekenstein-Hawking entropy in the island configurations, with the boundary being located slightly outside the outer horizon. In particular, for the extremal black planes of holographic axion gravity, we find that (a) the entanglement entropy of the Hawking radiation is ill-defined at the early times when the island is absent, (b) it tends to a distinctive constant at the late times, and (c) the late-time location of the island is indeed universal. Moreover, we investigate how the Page time is affected by the holographic massive gravity deformation. For neutral solutions at the small deformation parameter, and for charged solutions with almost-extremal deformation parameter, we find that the Page transition happens at earlier times.