A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole I: The Teukolsky Equations
Hamed Masaood
Abstract
Abstract We construct a scattering theory for the spin $$\pm \,2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>±</mml:mo> <mml:mspace/> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> 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 et al. in (Acta Math 222(1):1–214, 2019. 10.4310/acta.2019.v222.n1.a1 ) 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 $$+\,2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>+</mml:mo> <mml:mspace/> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and $$-\,2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mspace/> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> Teukolsky equations. This will allow us to state an additional mixed scattering statement for a pair of curvature components satisfying the spin $$+\,2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>+</mml:mo> <mml:mspace/> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and $$-\,2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mspace/> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> Teukolsky equations and connected via the Teukolsky–Starobinsky identities, completely determining the radiating degrees of freedom of solutions to the linearised Einstein equations.