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We construct a scattering theory for the spin $\pm2$ Teukolsky equations onthe exterior of the Schwarzschild spacetime, as a first step towards developinga scattering theory for the linearised Einstein equations in double null gauge.This is done by exploiting a physical-space version of the Chandrasekhartransformation used by Dafermos, Holzegel and Rodnianski in [arxiv:1601.06467]to prove the linear stability of the Schwarzschild solution. We also addressthe Teukolsky-Starobinsky correspondence and construct an isomorphism betweenscattering data for the $+2$ and $-2$ Teukolsky equations. This will allow usto state an additional mixed scattering statement for a pair of curvaturecomponents satisfying the spin $+2$ and $-2$ Teukolsky equations and connectedvia the Teukolsky-Starobinsky identities, completely determining the radiatingdegrees of freedom of solutions to the linearised Einstein equations.

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