A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole II: The Full System

TL;DR

This paper develops a comprehensive scattering theory for the full linearised Einstein equations on Schwarzschild spacetime, extending previous work on the Teukolsky equations and revealing relations between past and future memories.

Contribution

It extends the scattering theory from Teukolsky equations to the full linearised Einstein system on Schwarzschild background, using a transport equation approach.

Findings

Constructed a Hilbert space isomorphism between initial data and scattering states.

Linked past and future linear memories via an antipodal map.

Extended the scattering framework to the full Einstein system.

Abstract

We construct a scattering theory for the linearised Einstein equations on a Schwarzschild background in a double null gauge. We build on the results of Part I \cite{Mas20}, where we used the energy conservation enjoyed by the Regge--Wheeler equation associated with the stationarity of the Schwarzschild background to construct a scattering theory for the Teukolsky equations of spin $\pm2$. We now 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 that for a solution which is Bondi-normalised at both past and future null infinity, past and future linear memories are related by an antipodal map.