Analytic treatment of the excited instability spectra of the magnetically charged SU(2) Reissner-Nordström black holes

Abstract

The magnetically charged SU(2) Reissner-Nordström 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-Nordström 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.

7 pages

References in corpus (1)

• Quasinormal resonances of near-extremal Kerr-Newman black holes