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Linear waves in the Kerr geometry: A mathematical voyage to black hole physics

Authors: Felix Finster, Niky Kamran, Joel Smoller and Shing-Tung Yau
Journal: Bull. Amer. Math. Soc. 46 (2009), 635-659
MSC (2000): Primary 83C57, 35L15, 83C55
Published electronically: May 5, 2009
MathSciNet review: 2525736
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Abstract: This paper gives a survey of wave dynamics in the Kerr space-time geometry, the mathematical model of a rotating black hole in equilibrium. After a brief introduction to the Kerr metric, we review the separability properties of linear wave equations for fields of general spin  $ s=0,\frac{1}{2}, 1, 2$, corresponding to scalar, Dirac, electromagnetic fields and linearized gravitational waves. We give results on the long-time dynamics of Dirac and scalar waves, including decay rates for massive Dirac fields. For scalar waves, we give a rigorous treatment of superradiance and describe rigorously a mechanism of energy extraction from a rotating black hole. Finally, we discuss the open problem of linear stability of the Kerr metric and present partial results.

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Additional Information

Felix Finster
Affiliation: NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Niky Kamran
Affiliation: Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6 Canada

Joel Smoller
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Shing-Tung Yau
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 01238

Received by editor(s): January 9, 2008
Received by editor(s) in revised form: August 11, 2008, and February 18, 2009
Published electronically: May 5, 2009
Additional Notes: The first author’s research was supported in part by the Deutsche Forschungsgemeinschaft.
The second author’s research supported by NSERC grant RGPIN 105490-2004.
The third author’s research was supported in part by the National Science Foundation, Grant No. DMS-0603754.
The fourth author’s research was supported in part by the NSF, Grant No. 33-585-7510-2-30.
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