Abstract
A gauge- and coordinate-invariant perturbation theory for self-gravitating non-Abelian gauge fields with the gauge group SU(2) is developed and used to analyze local uniqueness and linear stability properties of non-Abelian equilibrium configurations. It is shown that all admissible stationary odd-parity excitations of the static and spherically symmetric Einstein-Yang-Mills soliton and black hole solutions have a total angular momentum number $l=1,$ and are characterized by nonvanishing asymptotic flux integrals. Local uniqueness and stability results with respect to non-Abelian perturbations are also established for the Schwarzschild and Reissner-Nordstr\"om solutions. Finally, unstable modes with $l=1$ are excluded for the static and spherically symmetric non-Abelian solitons and black holes.
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