DECAY OF THE MAXWELL FIELD ON THE SCHWARZSCHILD MANIFOLD

Abstract

We study solutions of the decoupled Maxwell equations in the exterior region of a Schwarzschild black hole. In stationary regions, where the Schwarzschild radial coordinate takes values in a bounded interval away from the event horizon, we obtain decay for all components of the Maxwell field at a rate which is bounded by the inverse of the standard time coordinate. We use vector field methods and no not require a spherical harmonic decomposition. In outgoing regions, where the Regge–Wheeler tortoise coordinate grows at least linearly with the time coordinate, we obtain decay rates for each of the null components. These rates are similar to the rates in flat space but weaker. Along the event horizon and in ingoing regions, where the Regge–Wheeler coordinate is negative and the outgoing, Eddington–Finkelstein null-coordinate is positive, all components (normalized with respect to an ingoing null basis) decay at a rate which is bounded by the inverse of the outgoing null coordinate.