The magnetically charged SU(2) Reissner-Nordstr\"om black-hole solutions of the coupled nonlinear Einstein-Yang-Mills field equations are known to be characterized by infinite spectra of unstable (imaginary) resonances $\{ \omega_n(r_+,r_-)\}_{ n=0}^{ n=\infty}$ (here $r_{ \pm}$ are the black-hole horizon radii). Based on direct { \it numerical} computations of the black-hole instability spectra, it has recently been observed that the excited instability eigenvalues of the magnetically charged black holes exhibit a simple universal behavior. In particular, it was shown that the numerically computed instability eigenvalues of the magnetically charged black holes are characterized by the small frequency universal relation $\omega_n(r_+-r_-)=\lambda_n$, where $\{ \lambda_n\}$ are dimensionless constants which are independent of the black-hole parameters. In the present paper we study analytically the instability spectra of the magnetically charged SU(2) Reissner-Nordstr\"om black holes. In particular, we provide a rigorous { \it analytical} proof for the { \it numerically}-suggested universal behavior $\omega_n(r_+-r_-)=\lambda_n$ in the small frequency $\omega_n r_+\ll (r_+-r_-)/r_+$ regime. Interestingly, it is shown that the excited black-hole resonances are characterized by the simple universal relation $\omega_{ n+1}/\omega_n=e^{ -2\pi/\sqrt{ 3}}$. Finally, we confirm our analytical results for the black-hole instability spectra with numerical computations.
Categories: hep-th astro-ph.HE gr-qc