Phase space analysis on some black hole manifolds

The Schwarzschild and Reissner-Nordstrom solutions to Einstein's equations describe space-times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space-time of this type. We show that for solutions with initial data which decay at infinity and at the bifurcation sphere, a weighted L-6 norm in space decays like t(-1/3). This weight vanishes at the event horizon, but not at infinity. To obtain this control, we require only an is an element of loss of angular derivatives. (C) 2008 Elsevier Inc. All rights reserved.