Trapping of waves and null geodesics for rotating black holes

TL;DR

This paper investigates the behavior of linear waves and null geodesics around rotating black holes, revealing their dynamical properties and stability features through geometric optics and hyperbolicity analysis.

Contribution

It establishes the link between wave dynamics and geodesic flow in Kerr black holes, highlighting the role of hyperbolic trapping and quasi-normal mode bounds in perturbed spacetimes.

Findings

Null geodesic flow exhibits r-normal hyperbolicity of the trapped set.

Distribution of quasi-normal modes provides bounds on decay rates.

Perturbations do not alter the fundamental geometric optics approximation.

Abstract

We present dynamical properties of linear waves and null geodesics valid for Kerr and Kerr-de Sitter black holes and their stationary perturbations. The two are intimately linked by the geometric optics approximation. For the nullgeodesic flow the key property is the r-normal hyperbolicity of the trapped set and for linear waves it is the distribution of quasi-normal modes: the exact quantization conditions do not hold for perturbations but the bounds on decay rates and the statistics of frequencies are still valid.