The instability spectrum of weakly-magnetized SU(2) Reissner-Nordstr\"om black holes

Abstract

It is well known that the U(1) Reissner-Nordstr\"om black hole is stable within the framework of the Einstein-Maxwell theory. However, the SU(2) Reissner-Nordstr\"om black-hole solution of the coupled Einstein-Yang-Mills equations is known to be unstable. In fact, this magnetically charged black hole is characterized by an infinite set of unstable (growing in time) perturbation modes. In the present paper we study {\it analytically} the instability resonance spectrum of weakly-magnetized SU(2) Reissner-Nordstr\"om black holes. In particular, we obtain explicit analytical expressions for the infinite set $\{\omega_n\}_{n=0}^{n=\infty}$ of imaginary eigenvalues that characterize the instability growth rates of the perturbation modes. We discuss the role played by these unstable eigenvalues as critical exponents in the gravitational collapse of the Yang-Mills field. Finally, it is shown that our analytical formulas for the characteristic black-hole instability spectrum agree with new numerical data that recently appeared in the literature.