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, $r_{\ast }>\epsilon t$, we obtain decay for the null components with rates of $|{\varphi }_{+}|\sim |\alpha |<C{r}^{-5/2}$, $|{\varphi }_0|\sim |\rho |+|\sigma |<C{r}^{-2}|t-r_{\ast }{|}^{-1/2}$, and $|{\varphi }_{-1}|\sim |\underline{\alpha }|<C{r}^{-1}|t-r_{\ast }{|}^{-1}$. Along the event horizon and in ingoing regions, where $r_{\ast }<0$, and when $t+r_{\ast }1$, 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

References in corpus (1)

• A note on energy currents and decay for the wave equation on a Schwarzschild background