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

DOI10.1007/S00220-022-04372-3arXiv2007.13658MaRDI QIDQ6345931

Hamed Masaood

Publication date: 27 July 2020

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.

Mathematics Subject Classification ID

Black holes (83C57) Scattering theory for PDEs (35P25) Linearizations (93B18) Applications of differential geometry to physics (53Z05) Einstein's equations (general structure, canonical formalism, Cauchy problems) (83C05) Yang-Mills and other gauge theories in mechanics of particles and systems (70S15)

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