Decay of the Maxwell field on the Schwarzschild manifold

TL;DR

This paper establishes decay rates for solutions of the Maxwell equations on a Schwarzschild black hole background using vector field methods without spherical harmonic decomposition.

Contribution

It provides new decay estimates for Maxwell fields in Schwarzschild spacetime, applicable in stationary, outgoing, and ingoing regions, using vector field techniques.

Findings

Decay rate of t^{-1} for Maxwell components in stationary regions.

Specific decay rates for null components in outgoing regions.

Uniform decay rates along the event horizon and ingoing regions.

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, $r_*>\epsilon t$, we obtain decay for the null components with rates of $|\phi_+| \sim |\alpha| < C r^{-5/2}$, $|\phi_0| \sim |\rho| + |\sigma| < C r^{-2} |t-r_*|^{-1/2}$, and $|\phi_{-1}| \sim |\underline{\alpha}| < C r^{-1} |t-r_*|^{-1}$. 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.