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A new Morawetz or integrated local energy decay estimate for Maxwell testfields on the exterior of a Schwarzschild black hole spacetime is proved. Theproof makes use of a new superenergy tensor $H_{ab}$ defined in terms of theMaxwell field and its first derivatives. The superenergy tensor, although notconserved, yields a conserved higher order energy current $H_{ab}(\partial_t)^b$. The tensor $H_{ab}$ vanishes for the static Coulomb field, andthe Morawetz estimate proved here therefore yields integrated decay for theMaxwell field to the Coulomb solution on the Schwarzschild exterior.

Decay of solutions to the Maxwell equation on the Schwarzschild background