Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole

TL;DR

This paper rigorously defines quasi-normal modes for Kerr-de Sitter black holes, shows their discrete nature under slow rotation, and proves exponential decay of linear wave energy with orthogonality conditions.

Contribution

It provides a rigorous mathematical framework for quasi-normal modes and establishes exponential energy decay for linear waves in Kerr-de Sitter backgrounds.

Findings

Quasi-normal modes are characterized as poles of a meromorphic family of operators.

Under slow rotation, these poles form a discrete set.

Linear wave energy decays exponentially with orthogonality to zero resonance.

Abstract

We provide a rigorous definition of quasi-normal modes for a rotating black hole. They are given by the poles of a certain meromorphic family of operators and agree with the heuristic definition in the physics literature. If the black hole rotates slowly enough, we show that these poles form a discrete subset of the complex plane. As an application we prove that the local energy of linear waves in that background decays exponentially once orthogonality to the zero resonance is imposed.