Conformal scattering theories for tensorial wave equations on Schwarzschild spacetime

TL;DR

This paper develops conformal scattering theories for tensorial wave equations, including Maxwell and Teukolsky equations, on Schwarzschild spacetime, utilizing conformal compactification, energy decay, and well-posedness of the Goursat problem.

Contribution

It introduces a novel approach combining Penrose's compactification and energy decay to establish scattering for tensorial wave equations on Schwarzschild.

Findings

Constructed conformal scattering for tensorial Fackerell-Ipser equations.

Proved energy equality through conformal boundary and initial hypersurface.

Established well-posedness of the Goursat problem for tensorial wave equations.

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.