Correlations in Hawking radiation and the infall problem

TL;DR

This paper shows that small quantum gravity corrections at the black-hole horizon cannot restore unitarity, proving that order‑unity corrections (hair) are needed via fuzzball microstates. By constructing several numerical toy models and a burning-paper analogy, the authors demonstrate that Hawking-like entanglement entropy $S_{ent}$ grows monotonically under small corrections, whereas Page’s burning-paper scenario yields a rise followed by a fall to zero. They argue that fuzzball structure provides the required horizon corrections and discuss a complementary framework in which high-energy infalling observers can be described by ensemble-averaged correlators, effectively reproducing a horizon-crossing geometry. The work unifies a microscopic fuzzball picture with coarse-grained, ensemble-based descriptions to address both the information paradox and the infall problem, with implications for complementarity and the role of microstate ensembles in quantum gravity.

Abstract

It is sometimes believed that small quantum gravity effects can encode information as `delicate correlations' in Hawking radiation, thus saving unitarity while maintaining a semiclassical horizon. A recently derived inequality showed that this belief is incorrect: one must have order unity corrections to low energy evolution at the horizon (i.e. fuzzballs) to remove entanglement between radiation and the hole. In this paper we take several models of `small corrections' and compute the entanglement entropy numerically; in each case this entanglement is seen to monotonically grow, in agreement with the general inequality. We also construct a model of `burning paper', where the entanglement is found to rise and then return to zero, in agreement with the general arguments of Page. We then note that the fuzzball structure of string microstates offers a version of `complementarity'. Low energy evolution is modified by order unity, resolving the information problem, while for high energy infalling modes ($E>> kT$) we may be able to replace correlators by their ensemble averaged values. Israel (and others) have suggested that this ensemble sum can be represented in the thermo-field-dynamics language as an entangled sum over two copies of the system, giving the two sides of the extended black hole diagram. Thus high energy correlators in a microstate may be approximated by correlators in a spacetime with horizons, with the ensemble sum over microstates acting like the `sewing' prescription of conformal field theory.

Figures (16)

• Figure 1: The Penrose diagram of a black hole formed by collapse of the 'infalling matter'. The spacelike slices satisfy all the niceness conditions required for semiclassical evolution in a gravity theory.
• Figure 2: A schematic set of coordinates for the Schwarzschild hole. Spacelike slices are $t=const$ outside the horizon and $r=const$ inside. Infalling matter is very far from the place where pairs are created.
• Figure 3: Entanglement entropy as a function of timestep in model A, for two different values of the smallness parameter $\epsilon$.
• Figure 4: Entanglement entropy as a function of timestep in model B, for two different values of the smallness parameter $\epsilon$. In the first case the matter is given by $M= \{100111010110\}$ and in the second by $M=\{ 110100010010\}$.
• Figure 5: Entanglement entropy as a function of timestep in model C, for two different values of the smallness parameter $\epsilon$.