Lectures on Linear Stability of Rotating Black Holes

Abstract

These lecture notes are concerned with linear stability of the nonextreme 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.

Slow decay of waves in gravitational solitons

“…Though (1.2) is the simplest version of the gravitational perturbation equations, it still preserves many geometric features of the spacetime through the metric, g. Hence understanding the properties of solutions to (1.2) is a useful precursor to the problem of nonlinear stability in the spacetime. The study of linear scalar waves in spacetimes has a well established history; [2,29,30,39,41] are essential reviews on the subject. The study of linear wave equations on explicit stationary solutions has also seen remarkable advancements for spacetimes with other asymptotics and dimensions greater than four.…”