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

TL;DR

This survey reviews the mathematical understanding of wave behavior in Kerr black hole spacetimes, including decay, superradiance, and stability issues, highlighting recent advances and open problems.

Contribution

It provides a comprehensive overview of wave dynamics in Kerr geometry, including new results on decay rates, superradiance, and the linear stability problem.

Findings

Decay rates for scalar and Dirac waves established.

Rigorous analysis of superradiance and energy extraction.

Partial results on the linear stability of Kerr black holes.

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.