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

Abstract

We construct a scattering theory for the spin $\pm 2$ 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, 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.

83 pages, 4 figures

References in corpus (5)

• Gravitational instability of an extreme Kerr black hole
• Mode stability for the Teukolsky equation on extremal and subextremal Kerr spacetimes
• The Scattering Map on Oppenheimer--Snyder Spacetime
• Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case $|a|\ll M$
• Conformal scattering theory for the linearized gravity fields on Schwarzschild spacetime

Cited by in corpus (9)

• On the Relation Between Asymptotic Charges, the Failure of Peeling and Late-time Tails
• Soft graviton exchange and the information paradox
• Strong Cosmic Censorship in the presence of matter: the decisive effect of horizon oscillations on the black hole interior geometry
• The non-linear stability of the Schwarzschild family of black holes
• The Case Against Smooth Null Infinity IV: Linearised Gravity Around Schwarzschild -- An Overview
• Peeling for tensorial wave equations on Schwarzschild spacetime
• The Case Against Smooth Null Infinity V: Early-Time Asymptotics of Linearised Gravity Around Schwarzschild for Fixed Spherical Harmonic Modes
• Conformal Scattering of Maxwell Potentials
• Conformal scattering theories for tensorial wave equations on Schwarzschild spacetime