Quasinormal modes on Kerr spacetimes

TL;DR

This paper develops a rigorous mathematical framework for defining and analyzing quasinormal modes on Kerr spacetimes, connecting various existing notions and proving stability and convergence results.

Contribution

It introduces a new framework for quasinormal modes on Kerr spacetimes, including resolvent estimates and meromorphic continuation, unifying different approaches in the literature.

Findings

Established a rigorous definition of quasinormal modes as eigenvalues.

Proved stability of quasinormal frequencies under small perturbations.

Demonstrated convergence of Kerr--de Sitter frequencies as the cosmological constant tends to zero.

Abstract

We introduce a rigorous framework for defining quasinormal modes on stationary, asymptotically flat spacetimes as isolated eigenvalues of the infinitesimal generator of time translations. We consider time functions corresponding to a foliation of asymptotically hyperboloidal hypersurfaces and restrict to suitable Hilbert spaces of functions. These functions have finite Sobolev regularity in bounded regions, but need to be Gevrey-regular at null infinity. This framework is developed in the context of sub-extremal Kerr spacetimes, but also gives uniform-in-$\Lambda$ resolvent estimates on Kerr--de Sitter spacetimes with a small cosmological constant $\Lambda$. As a corollary, we also construct the meromorphic continuation (in a sector of the complex plane) of the cut-off resolvent in Kerr that is associated to the standard Boyer--Lindquist time function. The framework introduced in this paper bridges different notions of quasinormal modes found in the literature. As further applications of our methods, we prove stability of quasinormal frequencies in a sector of the complex plane, with respect to suitably small perturbations and establish convergence properties for Kerr--de Sitter quasinormal frequencies when the cosmological constant approaches zero.