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 coordinate $r$ ranges over $2M < r_1 < r < r_2$, we obtain a decay rate of $t^{-1}$ for all components of the Maxwell field. We use vector field methods and do not require a spherical harmonic decomposition. In outgoing regions, where the Regge-Wheeler tortoise coordinate is large, $r_*>εt$, we obtain decay for the null components with rates of $|ϕ_+| \sim |α| < C r^{-5/2}$, $|ϕ_0| \sim |ρ| + |σ| < C r^{-2} |t-r_*|^{-1/2}$, and $|ϕ_{-1}| \sim |\underlineα| < C r^{-1} |t-r_*|^{-1}$. Along the event horizon and in ingoing regions, where $r_*<0$, and when $t+r_*1$, all components (normalized with respect to an ingoing null basis) decay at a rate of $C \uout^{-1}$ with $\uout=t+r_*$ in the exterior region.

37 pages, 5 figures