The Spin $\varvec{\pm }$ 1 Teukolsky Equations and the Maxwell System on Schwarzschild

Abstract

In this note, we prove decay for the spin ± 1 Teukolsky equations on the Schwarzschild spacetime. These equations are those satisfied by the extreme components ($\alpha$ and ${\underline{\alpha }}$) of the Maxwell field, when expressed with respect to a null frame. The subject has already been addressed in the literature, and the interest in the present approach lies in the connection with the recent work by Dafermos, Holzegel and Rodnianski on linearized gravity (Dafermos et al. in The linear stability of the Schwarzschild solution to gravitational perturbations, 2016. arXiv:1601.06467). In analogy with the spin $\pm 2$ case, it seems difficult to directly prove Morawetz estimates for solutions to the spin $\pm 1$ Teukolsky equations. By performing a differential transformation on the extreme components $\alpha$ and ${\underline{\alpha }}$, we obtain quantities which satisfy a Fackerell–Ipser Equation, which does admit a straightforward Morawetz estimate and is the key to the decay estimates. This approach is exactly analogous to the strategy appearing in the aforementioned work on linearized gravity. We achieve inverse polynomial decay estimates by a streamlined version of the physical space $r^p$ method of Dafermos and Rodnianski. Furthermore, we are also able to prove decay for all the components of the Maxwell system. The transformation that we use is a physical space version of a fixed-frequency transformation which appeared in the work of Chandrasekhar (Proc R Soc Lond Ser A 348(1652):39–55, 1976). The present note is a version of the author’s master thesis and also serves the “pedagogical” purpose to be as complete as possible in the presentation.