Entropy of Localized States and Black Hole Evaporation

We call a state \vacuum-bounded" if every measurement performed outside a speci ed 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 S, which for large energies is < 1 6 ln(LinT ), where Lin is the length of the interior region. We treat the state resulting from the evaporation of a black hole as similar to a vacuum-bounded state. Assuming that the similarity between vacuum-bounded and rigid problems extends from 1 to 3 dimensions, we apply our results to the black hole information paradox to con rm that large amounts of information cannot be emitted in the nal explosion of a black hole. 04.70.Dy, 05.30.-d Typeset using REVTEX Email address: [email protected] This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818.