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

TL;DR

This paper develops a scattering theory for the spin ±2 Teukolsky equations on Schwarzschild spacetime, advancing understanding of linearised gravity and the Einstein equations in a black hole exterior.

Contribution

It constructs a scattering framework for the Teukolsky equations, including the Teukolsky-Starobinsky correspondence, as a step towards linearised Einstein equations analysis.

Findings

Established a scattering theory for spin ±2 Teukolsky equations.

Constructed an isomorphism between scattering data for +2 and -2 equations.

Connected curvature components via Teukolsky-Starobinsky identities.

Abstract

We construct a scattering theory for the spin $\pm2$ 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.