Quasinormal modes and stability of spherically symmetric gravitating scalar configurations

Within general relativity the stability of a static spherically symmetric minimally coupled selfgravitating scalar configuration to linear radial\linebreak (monopole) perturbations of the corresponding scalar field is studying. We consider the self-consistently posed problem in which the geometric background is not static and metric perturbations, induced by the scalar field fluctuations, are taken into account. The problem is reduced to a single wave equation and the associated Schrödinger equation for the quasinormal modes. We obtain a general form of the effective potential for an arbitrary selfinteracting scalar field potential and consider the stability of vacuum black holes for scalar field fluctuations.