Boundedness and Decay for the Teukolsky Equation on Kerr Spacetimes I: The Case | a| ≪ M

Mihalis Dafermos; Gustav Holzegel; Igor Rodnianski

doi:10.1007/s40818-018-0058-8

Boundedness and Decay for the Teukolsky Equation on Kerr Spacetimes I: The Case | a| ≪ M

Mihalis Dafermos, Gustav Holzegel, Igor Rodnianski

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Abstract

We prove boundedness and polynomial decay statements for solutions of the spin ±2 Teukolsky equation on a Kerr exterior background with parameters satisfying | a| ≪ M. The bounds are obtained by introducing generalisations of the higher order quantities P and P̲ used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters | a| < M. As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.

Original language	English (US)
Article number	2
Journal	Annals of PDE
Volume	5
Issue number	1
DOIs	https://doi.org/10.1007/s40818-018-0058-8
State	Published - Jun 1 2019

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics
Geometry and Topology
General Physics and Astronomy
Mathematical Physics

Keywords

General relativity
Kerr black hole
Teukolsky equation

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10.1007/s40818-018-0058-8

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abstract = "We prove boundedness and polynomial decay statements for solutions of the spin ±2 Teukolsky equation on a Kerr exterior background with parameters satisfying | a| ≪ M. The bounds are obtained by introducing generalisations of the higher order quantities P and P̲ used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters | a| < M. As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.",

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T1 - Boundedness and Decay for the Teukolsky Equation on Kerr Spacetimes I

T2 - The Case | a| ≪ M

AU - Dafermos, Mihalis

AU - Holzegel, Gustav

AU - Rodnianski, Igor

PY - 2019/6/1

Y1 - 2019/6/1

N2 - We prove boundedness and polynomial decay statements for solutions of the spin ±2 Teukolsky equation on a Kerr exterior background with parameters satisfying | a| ≪ M. The bounds are obtained by introducing generalisations of the higher order quantities P and P̲ used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters | a| < M. As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.

AB - We prove boundedness and polynomial decay statements for solutions of the spin ±2 Teukolsky equation on a Kerr exterior background with parameters satisfying | a| ≪ M. The bounds are obtained by introducing generalisations of the higher order quantities P and P̲ used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters | a| < M. As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.

KW - General relativity

KW - Kerr black hole

KW - Teukolsky equation

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Boundedness and Decay for the Teukolsky Equation on Kerr Spacetimes I: The Case | a| ≪ M

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