Decay of solutions to the Maxwell equation on the Schwarzschild background

Abstract

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 defined in terms of the Maxwell field and its first derivatives. The superenergy tensor, although not conserved, yields a conserved higher order energy current . The tensor vanishes for the static Coulomb field, and the Morawetz estimate proved here therefore yields integrated decay for the Maxwell field to the Coulomb solution on the Schwarzschild exterior.