Linear Waves in the Kerr Geometry: A Mathematical Voyage to Black Hole Physics

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 $s=0,1/2,1,2$, 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.

arXiv comment25 pages, LaTeX, 2 figures, minor changes (published version)

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Cited by in corpus (13)

• A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta
• Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces
• Bound states of the Dirac equation on Kerr spacetime
• Linearized gravity and gauge conditions
• Linear Stability of the Non-Extreme Kerr Black Hole
• The massive Dirac field on a rotating black hole spacetime: Angular solutions
• Global analysis of quasilinear wave equations on asymptotically Kerr-de Sitter spaces
• New Generic Ringdown Frequencies at the Birth of a Kerr Black Hole
• Spin geometry and conservation laws in the Kerr spacetime
• On the linear stability of the extreme Kerr black hole under axially symmetric perturbations
• Monotonic Local Decay Estimates
• Mathematical general relativity: a sampler
• Linear Stability of Rotating Black Holes: Outline of the Proof