A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole

Abstract

In this thesis we construct a scattering theory for the linearised Einstein equations on a Schwarzschild background in a double null gauge. We perform this construction in two parts. In Part I, we construct a scattering theory for the spin ±2 Teukolsky equations, which govern certain the extremal, gauge-invariant components of the linearised curvature. This is done by exploiting a physical-space version of the Chandrasekhar transformation used by Dafermos, Holzegel and Rodnianski in [17] 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. In Part II, we extend the scattering theory of Part I to the full system of linearised Einstein equations by treating it as a system of transport equations which is sourced by solutions to the Teukolsky equations, leading to Hilbert space-isomorphisms between spaces of finite energy initial data and corresponding spaces of scattering states under suitably chosen gauge conditions on initial and scattering data. As a corollary, we show how the Bondi–Metzner–Sachs group emerges as the group of residual gauge transformations when the double null gauge is normalised to define a Bondi frame.