Vacuum - Bounded States and the Entropy of Black Hole Evaporation

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. For large energies we show that a vacuum-bounded system with length Lin and a given energy has entropy no more than S + 1 6 lnS, where S is the entropy in a rigid box with the same size and energy. 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 box problems extends from 1 to 3 dimensions, we apply these results to the black hole information paradox. Under these assumptions we conclude that large amounts of information cannot be emitted in the final explosion of a black hole. We also consider vacuum-bounded states at very low energies and come to the surprising conclusion that the entropy of such a state can be much higher than that of a rigid box state with the same energy. For a fixed E we let Lin ′ be the length of a rigid box which gives the same entropy as a vacuum-bounded state of length Lin. In the E → 0 limit we conjecture that the ratio Lin ′ /Lin grows without bound and support this conjecture with numerical computations. Thesis Supervisor: Alan H. Guth Title: Weisskopf Professor of Physics To Judy and Valerie, with love.