Analytic treatment of the excited instability spectra of the magnetically charged SU(2) Reissner-Nordstr\xf6m 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 {ωn(r+, r−)}n = 0n = ∞ (here r± are the black-hole horizon radii). Based on direct 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 ωn(r+ − r−) = λn, where {λn} 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 analytical proof for the numerically-suggested universal behavior ωn(r+ − r−) = λn in the small frequency ωnr+ ≪ (r+ − r−)/r+ regime. Interestingly, it is shown that the excited black-√hole resonances are characterized by the simple universal relation ωn + 1/ωn = e− 2π/3. Finally, we confirm our analytical results for the black-hole instability spectra with numerical computations.