Aspects of self-gravitating solitons and hairy black holes

This thesis considers two particular systems of gravity coupled to matter: Einstein-non-Abelian-Proca (ENAP) theory with gauge group $SU(2)$ in asymptotically anti-de Sitter (AdS) spacetime and Einstein-charged-scalar theory in a cavity. The first part of this thesis is devoted to the ENAP-AdS model. For a purely magnetic gauge field we obtain spherically symmetric solitons and black holes with non-Abelian Proca hair. This is achieved by solving the corresponding field equations numerically. We prove that the equilibrium gauge field must have at least one node. Then we turn to dyons and dyonic black holes which carry both electric and magnetic charge. We show that no non-trivial dyons or dyonic black holes exist in this model. We perturb the equilibrium solutions under linear, spherically symmetric perturbations of the metric and gauge field. We find numerical evidence which reveals that the solitons and hairy black holes are linearly unstable. These hairy black holes violate the generalized no-hair conjecture in the sense that they look identical to the Schwarzschild-AdS metric when observed from infinity. In the second part of this thesis, we investigate a plausible end-point of the charged superradiant instability. We study the Einstein-Maxwell-Klein-Gordon (EMKG) equations with a mirror-like boundary condition. We construct numerical solitons and black holes with charged scalar hair. Then we study the stability of the equilibrium solutions under linear, spherically symmetric perturbations of the metric, electromagnetic and scalar fields. When the mirror is located at the first zero of the static scalar field, we find stable solitons if the mirror radius is sufficiently large. However when the mirror radius is sufficiently small, some solitons are found to be unstable. In the black hole case, we find no evidence of instability when the mirror is located at the first zero of the static scalar field. In contrast, numerical evidence shows that the hairy black holes are unstable if the mirror is located at the second zero of the static scalar field. We conclude that these stable hairy black holes could represent an end-point of the charged superradiant instability.