Analytic treatment of the excited instability spectra of the magnetically charged SU(2) Reissner-Nordstr\"om black holes

TL;DR

This paper analytically confirms the universal behavior of instability spectra in magnetically charged SU(2) Reissner-Nordström black holes, revealing simple relations among eigenvalues and validating numerical findings.

Contribution

It provides the first rigorous analytical proof of the universal instability spectrum behavior in these black holes, complementing previous numerical results.

Findings

Analytical proof of the universal relation mbda_n for eigenvalues

Derivation of the ratio mbda_{n+1}/mbda_n=e^{-2\u03c0/\u221a{3}}

Numerical confirmation of the analytical results

Abstract

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.