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

Abstract

AbstractWe construct a scattering theory for the spin $$\pm \,2$$ ± 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 et al. in (Acta Math 222(1):1–214, 2019. 10.4310/acta.2019.v222.n1.a1) 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 $$+\,2$$ + 2 and $$-\,2$$ - 2 Teukolsky equations. This will allow us to state an additional mixed scattering statement for a pair of curvature components satisfying the spin $$+\,2$$ + 2 and $$-\,2$$ - 2 Teukolsky equations and connected via the Teukolsky–Starobinsky identities, completely determining the radiating degrees of freedom of solutions to the linearised Einstein equations.