Fluctuations in the entropy of Hawking radiation

Abstract

We use the gravitational path integral (GPI) to compute the fluctuations of the Hawking radiation entropy around the Page curve in a two-dimensional model introduced by Penington Before the Page time, we find that δS=e−S/2, where S is the black hole entropy. This result agrees with the Haar-averaged entropy fluctuations in a bipartite system. After the Page time, we find that δS∼e−S, up to a prefactor that depends logarithmically on the width of the microcanonical energy window. This is not symmetric under exchange of subsystem sizes and so does not agree with the Haar average for a subsystem of fixed Hilbert space dimension. The discrepancy can be attributed to the fact that the black hole Hilbert space dimension is not fixed by the state preparation: even in a microcanonical ensemble with a top-hat smearing function, the GPI yields an additive fluctuation in the number of black hole states. This result, and the fact that the Page curve computed by the GPI is smooth, all point towards an ensemble interpretation of the GPI. Published by the American Physical Society 2024