The information paradox and the infall problem

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 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 one-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.