Entropy localization and extensivity in the semiclassical black hole evaporation

Abstract

We aim to quantify the distribution of information in the Hawking radiation and inside the black hole in the semiclassical evaporation process. The structure quantum field theory forces to consider a shared information between two different regions of space-time. Using this tool, we show that the entropy of a thermal gas at the Unruh temperature underestimates the actual amount of (shared) information present in a region of the Rindler space. Then, we analyze the mutual information between the black hole and the late time radiation region. We show that in the semiclassical picture it is not possible to recover the eventual purity of the initial state in the final Hawking radiation through correlations established during the whole evaporation period, no matter the interactions present in the theory. We find extensivity of the entropy as a consequence of a reduction to a two dimensional conformal problem in a simple approximation. However, this seems not to be guaranteed in a full four dimensional calculation. We also find that a large amount of information is contained in a small approximately flat region of space-time near the point where the horizon begins. This gives place to large violations of the entropy bounds. This problem is not eased by backscattering effects and we argue that a breaking of conformal invariance is necessary to delocalize the entropy. Finally, we indicate that the mutual information could lead to a way to understand the Bekenstein-Hawking black hole entropy which does not require a reduction in degrees of freedom in order to regulate the entanglement entropy. On the contrary a large number of field degrees of freedom at high energies giving place to a Hagedorn transition implements a distance cutoff in the mutual information, which may in consequence turn out to be bounded.