Entropy localization and extensivity in the semiclassical black hole evaporation

TL;DR

The paper tackles how information is distributed and shared during semiclassical black hole evaporation. It proposes mutual information $I(A,B)$ as a regulator-independent measure to quantify entanglement between regions, applying a two-dimensional reduction to model the evaporation and Rindler spacetimes. The results indicate that the information carried by Hawking radiation can be extensive in the radiation region and that a substantial amount of information localizes near the horizon (Aleph), with the mutual information potentially exceeding the semiclassical Bekenstein bound in some setups, yet remaining finite and well-defined. These findings offer a framework to reinterpret black hole entropy and the generalized second law without requiring a fundamental reduction of degrees of freedom, though a complete four-dimensional treatment and the role of nonconformal dynamics remain open questions.

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.

Figures (5)

• Figure 1: The spatial surfaces $A$ and $A^\prime$ correspond to the same causally complete region $\hat{A}$ (diamond shaped set), while $B$ or $B^\prime$ are Cauchy surfaces for the causally complete region $\hat{B}$.
• Figure 2: The wedge $A$ and diamond set $B$ in the exterior region. The curves of constant $\nu$ are lines through the origin.
• Figure 3: (a) Two different spatial surfaces $\Sigma_1$ and $\Sigma_2$ on the space-time of a black hole which completely evaporates. (b) The set $A$ is placed inside the black hole and $B$ is in the radiation region.
• Figure 4: Penrose diagram for the black hole formation and evaporation. In the text the mutual information $I(A,B)$ between $A$ and $B$ is calculated in a two dimensional approximation.
• Figure 5: Black hole formed by the collapse of a null shell. The region I below the shell and above the horizon is a diamond shaped subset (intersection of the past of a point and the future of another point) of Minkowski space. Region II contains all points space-like separated from region I. The arrows show the trajectories of the modular group. It acts geometrically in region I for conformal fields and in the late time region II if thermal equilibrium is achieved.