A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole I: The Teukolsky Equations

Abstract

We construct a scattering theory for the spin $\pm 2$ Teukolsky equations on the exterior of the Schwarzschild spacetime, as a first step towards developing a scattering theory for the linearised Einstein equations in double null gauge. This is done by exploiting a physical-space version of the Chandrasekhar transformation used by Dafermos, Holzegel and Rodnianski in [arxiv:1601.06467] to prove the linear stability of the Schwarzschild solution. We also address the Teukolsky-Starobinsky correspondence and construct an isomorphism between scattering data for the $+2$ and $-2$ Teukolsky equations. This will allow us to state an additional mixed scattering statement for a pair of curvature components satisfying the spin $+2$ and $-2$ Teukolsky equations and connected via the Teukolsky-Starobinsky identities, completely determining the radiating degrees of freedom of solutions to the linearised Einstein equations.

arXiv comment83 pages, 4 figures

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