Instability of hairy black holes in spontaneously-broken Einstein-Yang-Mills-Higgs systems

The stability of a new class of hairy black hole solutions in the coupled system of Einstein-Yang-Mills-Higgs is examined, generalising a method suggested by Brodbeck and Straumann and collaborators, and Volkov and Gal'tsov. The method maps the algebraic system of linearised radial perturbations of the various field modes around the black hole solution into a coupled system of radial equations of Schr\"odinger type. No detailed knowledge of the black hole solution is required, except from the fact that the boundary conditions at the physical space-time boundaries (horizons) must be such so as to guarantee the {\it finiteness} of the various expressions involved. In this way, it is demonstrated that the above Schr\"odinger equations have bound states, which implies the instability of the associated black hole solution.