Instability of Gravitating Sphalerons

We prove the instability of the gravitating regular sphaleron solutions of the $SU(2)$ Einstein-Yang-Mills-Higgs system with a Higgs doublet, by studying the frequency spectrum of a class of radial perturbations. With the help of a variational principle we show that there exist always unstable modes. Our method has the advantage that no detailed knowledge of the equilibrium solution is required. It does, however, not directly apply to black holes.