Decay of solutions to the Maxwell equation on the Schwarzschild background. (English) Zbl 1338.83093
Summary: 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 \(\mathbf{H}_{ab}\) defined in terms of the Maxwell field and its first derivatives. The superenergy tensor, although not conserved, yields a conserved higher order energy current \(\mathbf{H}_{ab}(\partial _{t})^{b}\). The tensor \(\mathbf{H}_{ab}\) 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.
MSC:
83C57 | Black holes |
83C22 | Einstein-Maxwell equations |
83C40 | Gravitational energy and conservation laws; groups of motions |