Circumventing the black hole hair-loss problem

We provide a possible way of avoiding the hair-loss problem by studying gauge fluctuations on a classical Schwarzschild black hole background. The hair-loss problem arises due to the small fidelity of reconstructing the interior operator at the end stage of the evaporation, which is general in most schemes trying to decouple the early and late radiation to avoid firewall. To circumvent the problem, we put the black hole in a cavity as a toy model and study the entropy behavior of the system with different temperatures. By analyzing gauge fluctuations with nontrivial boundary conditions, we find that the entropy of the black hole system increases at the end stage, rather than directly dropping to zero. Besides the end stage, the entropy is the same as Page's original argument. The hair-loss problem can be avoided by the proposed model and we can gain important insights into the end stage of the evaporation and quantum effects of gravity.