Quasinormal modes of magnetized black hole

TL;DR

This paper studies how external magnetic fields affect the quasinormal modes of charged, massive scalar fields around black holes, revealing mode splitting, stability thresholds, and conditions for instability.

Contribution

It introduces a detailed analysis of Zeeman effect-induced mode splitting and stability criteria for charged scalar fields in magnetized black hole backgrounds.

Findings

Mode splitting into 2ℓ+1 modes due to Zeeman effect.

Identification of a threshold effective mass for long-lived quasinormal modes.

Discovery of conditions leading to instability when effective mass squared is negative.

Abstract

We investigate charged, massive scalar field around static, spherically symmetric black hole immersed into an external asymptotically uniform magnetic field $B$. It is shown that for given multipole number $\ell$ there are $2\ell+1$ numbers of modes due to the Zeeman effect appearing by an interaction of the external magnetic and charged scalar fields introducing an effective mass of the scalar field $\mu_{\rm eff}=\sqrt{\mu^2-mqB}$ where $m$ is the azimuthal number and $q$ is the charge coupling constant. We calculate threshold value of effective mass in which quasinormal modes are arbitrarily long lived and beyond that value quasinormal modes vanish. In the case of $m qB<0$ quasinormal modes are longer lived with larger oscillation frequencies. Whenever, magnetic and massive scalar fields satisfies condition $\mu_{\rm eff}^2<0$, an instability appears, i.e., if $qB>0$ or $qB<0$ there is an instability for the values of azimuthal number $m>\mu^2/qB$ or $m<\mu^2/qB$, respectively.