Island rules for the noncommutative black hole

Abstract: In the context of noncommutative black holes, we reconsider the island rule and reproduce the Page curve. Since the radiation entropy of the noncommutative black hole will eventually diverge in the absence of islands, it is crucial to include the contribution of islands in the radiation process to recover the Page curve. When cutoff surface is far from event horizon, we find that the existence of islands helps to improve the radiation entropy and avoid its divergence at the late stage, while the radiation entropy for the commutative black hole with islands is still divergent in this case. When cutoff surface is close to event horizon, the existence of islands can lead to the radiation process consistent with the Page curves for both the commutative and noncommutative black holes, but it is interesting to obtain a significantly delay Page time for the noncommutative case, allowing us to avoid considering the contribution of islands for a sufficiently long period of time.