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

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Peer-reviewed

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Abstract

jats:titleAbstract</jats:title>jats:pWe construct a scattering theory for the spin jats:inline-formulajats:alternativesjats: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:mn2</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-formulajats:alternativesjats: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:mn2</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and jats:inline-formulajats:alternativesjats: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:mn2</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-formulajats:alternativesjats: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:mn2</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and jats:inline-formulajats:alternativesjats: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:mn2</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>

Description

Keywords

4902 Mathematical Physics, 4904 Pure Mathematics, 49 Mathematical Sciences, 51 Physical Sciences

Journal Title

Communications in Mathematical Physics

Conference Name

Journal ISSN

0010-3616
1432-0916

Volume Title

393

Publisher

Springer Science and Business Media LLC
Sponsorship
Engineering and Physical Sciences Research Council (EP/L016516/1)