A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole I: The Teukolsky Equations
Authors
Publication Date
2022-07Journal Title
Communications in Mathematical Physics
ISSN
0010-3616
Publisher
Springer Science and Business Media LLC
Volume
393
Issue
1
Pages
477-581
Language
en
Type
Article
This Version
VoR
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Masaood, H. (2022). A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole I: The Teukolsky Equations. Communications in Mathematical Physics, 393 (1), 477-581. https://doi.org/10.1007/s00220-022-04372-3
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">
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</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">
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</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>
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</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">
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</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>
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</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>
Keywords
Article
Sponsorship
Engineering and Physical Sciences Research Council (EP/L016516/1)
Identifiers
s00220-022-04372-3, 4372
External DOI: https://doi.org/10.1007/s00220-022-04372-3
This record's URL: https://www.repository.cam.ac.uk/handle/1810/338206
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Licence:
http://creativecommons.org/licenses/by/4.0/
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