Conformal scattering theories for tensorial wave equations on Schwarzschild spacetime

Abstract

In this paper, we establish the constructions of conformal scattering theories for the tensorial wave equation such as the tensorial Fackerell-Ipser and the spin $\pm 1$ Teukolsky equations on Schwarzschild spacetime. In our strategy, we construct the conformal scattering for the tensorial Fackerell-Ipser equations which are obtained from the Maxwell equation and spin $\pm 1$ Teukolsky equations. Our method combines Penrose's conformal compactification and the energy decay results of the tensorial fields satisfying the tensorial Fackerell-Ipser equation to prove the energy equality of the fields through the conformal boundary $\mathfrak{H}^+\cup \scri^+$ (resp. $\mathfrak{H}^-\cup \scri^-$) and the initial Cauchy hypersurface $\Sigma_0 = \left\{ t=0 \right\}$. We will prove the well-posedness of the Goursat problem by using a generalization of H"ormander's results for the tensorial wave equations. By using the results for the tensorial Fackerell-Ipser equations we will establish the construction of conformal scattering for the spin $\pm 1$ Teukolsky equations.