Decay of the Maxwell field on the Schwarzschild manifold

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, $r_{\ast }>ϵt$, we obtain decay for the null components with rates of $\mathrm{\mid }{\varphi }_{+}\mathrm{\mid }\sim \mathrm{\mid }\alpha \mathrm{\mid }<C{r}^{-5\mathrm{/}2}$, $|\phi_0| \sim |\rho| + |\sigma| < C r^{-2} |t-r_|^{-1/2}$,and$|\phi_{-1}| \sim |\underline{\alpha}| < C r^{-1} |t-r_|^{-1}$\mathrm{.}Alongtheeventhorizonandiningoingregions,where$r_*<0$, and when $t+r_{\ast }1$, all components (normalized with respect to an ingoing null basis) decay at a rate of $C{\text{uout}}^{-1}$ with $\text{uout}=t+r_{\ast }$ in the exterior region.