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

Abstract

<jats:title>Abstract</jats:title><jats:p>We construct a scattering theory for the spin <jats:inline-formula><jats:alternatives><jats:tex-math>$$\pm \,2$$</jats:tex-math><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></jats:alternatives></jats:inline-formula> 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. <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" ext-link-type="doi" xlink:href="https://doi.org/10.4310/acta.2019.v222.n1.a1">10.4310/acta.2019.v222.n1.a1</jats:ext-link>) 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$+\,2$$</jats:tex-math><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></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$-\,2$$</jats:tex-math><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></jats:alternatives></jats:inline-formula> Teukolsky equations. This will allow us to state an additional mixed scattering statement for a pair of curvature components satisfying the spin <jats:inline-formula><jats:alternatives><jats:tex-math>$$+\,2$$</jats:tex-math><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></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$-\,2$$</jats:tex-math><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></jats:alternatives></jats:inline-formula> Teukolsky equations and connected via the Teukolsky–Starobinsky identities, completely determining the radiating degrees of freedom of solutions to the linearised Einstein equations.</jats:p>