Entropy of Localized States and Black Hole Evaporation

TL;DR

This paper introduces the concept of vacuum-bounded states, computes their maximum entropy in a 1D model, and discusses implications for black hole evaporation and the information paradox.

Contribution

It defines vacuum-bounded states, calculates their maximum entropy numerically, and applies these findings to black hole evaporation scenarios.

Findings

Maximum entropy of vacuum-bounded states exceeds that of rigid-wall states by a calculable amount.

The entropy difference scales with the interior region length and temperature.

Implications suggest limited information release during black hole final evaporation.

Abstract

We call a state "vacuum-bounded" if every measurement performed outside a specified interior region gives the same result as in the vacuum. We compute the maximum entropy of a vacuum-bounded state with a given energy for a one-dimensional model, with the aid of numerical calculations on a lattice. The maximum entropy is larger than it would be for rigid wall boundary conditions by an amount $\delta S$ which for large energies is less than or approximately $(1/6) ln (L_in T)$, where L_in is the length of the interior region. Assuming that the state resulting from the evaporation of a black hole is similar to a vacuum-bounded state, and that the similarity between vacuum-bounded and rigid-wall-bounded problems extends from 1 to 3 dimensions, we apply these results to the black hole information paradox. We conclude that large amounts of information cannot be emitted in the final explosion of a black hole.