Decay of solutions to the Maxwell equation on the Schwarzschild background

Lars Andersson, Thomas Bäckdahl, Pieter Blue

Published 2015-01-19Version 1

A new Morawetz or integrated local energy decay estimate for Maxwell test fields on the exterior of a Schwarzschild black hole spacetime is proved. The proof makes use of a new superenergy tensor $H_{ab}$ defined in terms of the Maxwell field and its first derivatives. The superenergy tensor, although not conserved, yields a conserved higher order energy current $H_{ab} (\partial_t)^b$. The tensor $H_{ab}$ vanishes for the static Coulomb field, and the Morawetz estimate proved here therefore yields integrated decay for the Maxwell field on the Schwarzschild exterior to the Coulomb solution.