The imprint of scalar clouds around Kerr black holes : hairy solutions and weak cosmic censorship

In this thesis we explore Einstein gravity coupled to matter, and more specifically, we focus on the Kerr black hole in the presence of a massive scalar field. We investigate the superradiant phenomenon present in this setting - amplified scattering of scalar waves by the black hole given the right conditions on the scalar field's frequency are satisfied. Due to the mass term the spacetime develops a trapping region where the scalar waves can get localised instead of dispersing to infinity. In this way, the energy extracted from the black hole leads to the formation of a quasi bound state of the scalar field around it, known as a scalar cloud. This parallels the situation in Kerr-AdS, where due to the timelike character of the boundary this phenomenon appears more naturally and one does not necessarily need to make the field massive. We explore some of the consequences due to the formation of the scalar clouds in the asymptotically flat context. In the case of a complex scalar field the configuration can actually settle to a stationary solution to the Einstein equation which represents a scalar hairy Kerr black hole in asymptotically flat four dimensions - the first example of such black holes that does not violate energy conditions, discovered recently. These scalar hairy black holes branch off from the Kerr solution at the onset of scalar superradiance, when the scalar cloud starts forming. Since their discovery a number of studies have been carried out, trying to relate their properties to astrophysics. However, prior to our work, no stability analysis was performed. This is particularly important if we want to argue for the astrophysical significance of such solutions. We made the initial step in answering this question by showing that such black holes are unstable towards higher excited superradiant modes than the mode from whose onset they branched off. This required the construction of the black hole backgrounds numerically, utilising the DeTurck method and solving a system of PDEs with extended precision using spectral methods. Afterwards, a clever choice of gauge allowed us to decouple the scalar field perturbations form the gravitational sector and perform the analysis by looking at the quasinormal mode spectrum of the hairy solutions. When the scalar field is real, these stationary hairy solutions do not exist, but scalar clouds still form around the black hole. Moreover, our previous work suggested that for any parameters of the Kerr black hole, an unstable scalar mode will be present. We showed this analytically using a hybrid WKB and matched asymptotic expansion technique and confirmed it with numerical data. This led us to explore an idea for a plausible counterexample to the Weak Cosmic Censorship Conjecture in flat four dimensions, based on the cascading of energies, as seen in AdS, whereby with time even higher superradiant modes become dominant and form clouds around the black hole. In support of our claim we have integrated the sourced Teukolsky equation for gravitational perturbations of Kerr numerically with extended precision in order to obtain the backreaction of the scalar field on the geometry and also analysed the system analytically, as detailed above, in order to confirm our numerics and discuss the endpoint of the evolution of the system.