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Lectures on Linear Stability of Rotating Black Holes

  • Felix FinsterEmail author
Chapter
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)

Abstract

These lecture notes are concerned with linear stability of the non-extreme Kerr geometry under perturbations of general spin. After a brief review of the Kerr black hole and its symmetries, we describe these symmetries by Killing fields and work out the connection to conservation laws. The Penrose process and superradiance effects are discussed. Decay results on the long-time behavior of Dirac waves are outlined. It is explained schematically how the Maxwell equations and the equations for linearized gravitational waves can be decoupled to obtain the Teukolsky equation. It is shown how the Teukolsky equation can be fully separated to a system of coupled ordinary differential equations. Linear stability of the non-extreme Kerr black hole is stated as a pointwise decay result for solutions of the Cauchy problem for the Teukolsky equation. The stability proof is outlined, with an emphasis on the underlying ideas and methods.

Keywords

Black holes Linear stability Kerr geometry Gravitational waves Linear hyperbolic PDEs Teukolsky Equation 

Mathematics Subject Classification (2000)

Primary 83C57 83C35; Secondary 58J45 83C20 83C60 35Q75 35L15 35L52 

Notes

Acknowledgements

I would like to thank the organizers of the first “Domoschool—International Alpine School of Mathematics and Physics” held in Domodossola, 16–20 July 2018, for the kind invitation. This article is based on my lectures delivered at this summer school. I am grateful to Niky Kamran and Igor Khavkine for helpful comments on the manuscript.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany

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