Mutual information of subsystems and the Page curve for the Schwarzschild–de Sitter black hole

Abstract

In this work, we show that the two proposals associated with the mutual information of matter fields can be given for an eternal Schwarzschild black hole in de Sitter spacetime. These proposals also depict the status of associated entanglement wedges and their role-play in obtaining the correct Page curve of radiation. The first proposal has been given for the before Page time scenario, which shows that the mutual information $I({R}_{H}^{+}:{R}_{H}^{\ensuremath{-}})$ vanishes at a certain value of the observer's time ${t}_{{b}_{H}}={t}_{H}$ (where ${t}_{H}\ensuremath{\ll}{\ensuremath{\beta}}_{H}$). We claim that this is the Hartman-Maldacena time at which the entanglement wedge associated with ${R}_{H}^{+}\ensuremath{\cup}{R}_{H}^{\ensuremath{-}}$ gets disconnected and the fine-grained radiation entropy has the form $S({R}_{H})\ensuremath{\sim}\mathrm{log}({\ensuremath{\beta}}_{H})$. The second proposal depicts the fact that just after the Page time, when the replica wormholes are the dominating saddle points, the mutual information $I({B}_{H}^{+}:{B}_{H}^{\ensuremath{-}})$ vanishes as soon as the time difference ${t}_{{a}_{H}}\ensuremath{-}{t}_{{b}_{H}}$ equals the scrambling time. Holographically, this reflects that the entanglement wedge associated with ${B}_{H}^{+}\ensuremath{\cup}{B}_{H}^{\ensuremath{-}}$ jumps to the disconnected phase at this particular timescale. Furthermore, these two proposals lead us to the correct time evolution of the fine-grained entropy of radiation as portrayed by the Page curve. We have also shown that similar observations can be obtained for the radiation associated with the cosmological horizon.