Phase space analysis on some black hole manifolds

The Schwarzschild and Reissner–Nordstrøm 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 L6 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 loss of angular derivatives.