Quasinormal modes of magnetized black hole

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.