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Decay of solutions to the Maxwell equation on the Schwarzschild background

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Andersson,  Lars
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Bäckdahl,  Thomas
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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1501.04641.pdf
(Preprint), 254KB

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Citation

Andersson, L., Bäckdahl, T., & Blue, P. (2016). Decay of solutions to the Maxwell equation on the Schwarzschild background. Classical and Quantum Gravity, 33(8): 085010. doi:10.1088/0264-9381/33/8/085010.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0029-0037-1
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 $H_{ab}$ 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.