Perturbations of global monopoles as a black hole's hair

We study the stability of a spherically symmetric black hole with a global monopole hair. Asymptotically the spacetime is. at but has a deficit solid angle which depends on the vacuum expectation value of the scalar field. When the vacuum expectation value is larger than a certain critical value, this spacetime has a cosmological event horizon. We investigate the stability of these solutions against the spherical and polar perturbations and confirm that the global monopole hair is stable in both cases. Although we consider some particular modes in the polar case, our analysis suggests the conservation of the 'topological charge' in the presence of the event horizons and violation of the black hole no-hair conjecture in asymptotically non-flat spacetime.