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 {{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 to the Coulomb solution on the Schwarzschild exterior.
 Publication:

Classical and Quantum Gravity
 Pub Date:
 April 2016
 DOI:
 10.1088/02649381/33/8/085010
 arXiv:
 arXiv:1501.04641
 Bibcode:
 2016CQGra..33h5010A
 Keywords:

 Mathematics  Analysis of PDEs;
 General Relativity and Quantum Cosmology;
 Mathematical Physics;
 35Q61;
 83C60;
 83C57
 EPrint:
 15 pages, updated references