The information paradox and the infall problem

TL;DR

The paper argues that small quantum-gravity corrections cannot resolve the black hole information paradox due to a strong subadditivity–based inequality, which requires order-unity corrections to horizon physics. It demonstrates this with a solvable model mapped to a 1D Ising chain, showing that entanglement entropy $S_{ent}$ is not appreciably reduced by such corrections. The work then separates the problem into the information paradox (needing horizon hair to imprint information on Hawking quanta) and the infall problem (a coarse-grained effective description for high-energy observers), and connects to string-theory fuzzball microstates that provide the required hair. It further discusses AdS/CFT and the role of collective dynamics, arguing that fuzzball hair resolves the paradox while a hydrodynamic/collective description can address infall. Together, these insights illustrate how order-unity horizon corrections arise in string theory and why they are essential for a unitary evolution of black holes, with broader implications for holography and quantum gravity phenomenology.

Abstract

It is sometimes believed that small quantum gravity corrections to the Hawking radiation process can encode the correlations required to solve the black hole information paradox. Recently an inequality on the entanglement entropy of radiation was derived, which showed that such is not the case; one needs {\it order unity} corrections to low energy modes at the horizon to resolve the problem. In this paper we illustrate this inequality by a simple model where the state of the created Hawking pair at each stage is slightly modified by the state of the previous pair. The model can be mapped onto the 1-dimensional Ising chain and solved explicitly. In agreement with the general inequality we find that very little of the entanglement is removed by the encoded correlations. We then use the general inequality to argue that the black hole puzzles split into two problems: the `information paradox' and the `infall problem'. The former addresses the detailed state of low energy modes at the horizon and asks if these can be modified by order unity, while the latter asks for a coarse grained effective description of the infall of heavy observers into the degrees of freedom of the hole.

Figures (5)

• 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: (a) The traditional black hole geometry: the state at the horizon is the vacuum (b) The simplest microstate (which has the $S^1$ in the shape of an exact circle), and the CFT state that it corresponds to; all 'bits' are in the same state (c) The next simplest geometry; the CFT state has its bits distributed over two different types (d) A generic state, very quantum, but with no traditional horizon; the CFT state has bits distributed over many different types.
• Figure 4: Schematic description of a microstate solution of Einstein's equations. There are 'local ergoregions' with rapidly changing direction of frame dragging near the horizon. The geometry closes off without having an interior horizon or singularity due to its peculiar topological structure. Generic microsates are very quantum, and so this figure just gives a schematic description of the rapid fluctuations near the fuzzball surface.
• Figure 5: The KK monopole times $S^1$ dipole structure of the 2-charge D1D5 fuzzball. Line segments joining points on this $S^1$ correspond to branes wrapping the sphere between KK monopoles.