Linear mode stability of Kerr-Newman and its quasinormal modes

TL;DR

This paper demonstrates that Kerr-Newman black holes are linearly stable up to near-extremality by numerically analyzing their quasinormal modes, revealing universal properties and confirming stability in a broad parameter range.

Contribution

It provides the first comprehensive numerical analysis of the QNM spectrum of Kerr-Newman black holes, establishing their linear mode stability and uncovering universal behaviors of QNMs at extremality.

Findings

No unstable modes found up to 99.999% extremality.

QNM frequencies approach the extremal limit with Re(ω)=mΩ_H^{ext} and Im(ω)=0.

QNM spectrum dominated by modes connected to Schwarzschild's ℓ=m=2 mode.

Abstract

We provide strong evidence that, up to $99.999\%$ of extremality, Kerr-Newman black holes (KN BHs) are linear mode stable within Einstein-Maxwell theory. We derive and solve, numerically, a coupled system of two PDEs for two gauge invariant fields that describe the most general linear perturbations of a KN BH (except for trivial modes that shift the parameters of the solution). We determine the quasinormal mode (QNM) spectrum of the KN BH as a function of its three parameters and find no unstable modes. In addition, we find that the QNMs that are connected continuously to the gravitational $\ell=m=2$ Schwarzschild QNM dominate the spectrum for all values of the parameter space ($m$ is the azimuthal number of the wave function and $\ell$ measures the number of nodes along the polar direction). Furthermore, all QNMs with $\ell=m$ approach Re$\,\omega = m \Omega_H^{ext}$ and Im$\,\omega=0$ at extremality; this is a universal property for any field of arbitrary spin $|s|\leq 2$ propagating on a KN BH background ($\omega$ is the wave frequency and $\Omega_H^{ext}$ the BH angular velocity at extremality). We compare our results with available perturbative results in the small charge or small rotation regimes and find good agreement. We also present a simple proof that the Regge-Wheeler (odd) and Zerilli (even) sectors of Schwarzschild perturbations must be isospectral.