Dynamical behavior of black-hole spacetimes

The theory of General Relativity predicts that black holes are a natural constituent of our Universe. Before a black hole binary relaxes to a final state, it undergoes damped oscillations which completely characterize the final object. These oscillations are described by quasinormal modes which are present in many dissipative systems. Quasinormal modes are linked to the way a black hole spacetime responds to small fluctuations. Although the field equations admit a well defined future evolution of spacetime, some black holes, like the ones formed after a binary merger, possess a boundary, beyond which the field equations lose their predictive power. This boundary is called a Cauchy horizon and designates the region beyond which the spacetime can potentially be described in a highly non-unique manner. Strong cosmic censorship states that, generically, suitable initial data should be future inextendible beyond the Cauchy horizon, as a solution to the field equations. Recent studies indicate that the fate of Cauchy horizons, such as those found inside charged and rotating cosmological black holes, is intrinsically connected to the decay of perturbations exterior to the event horizon. As such, the validity of strong cosmic censorship is tied to how effectively the exterior damps fluctuations. In this thesis we will perform a quantitative stability analysis of Cauchy horizons in electrically charged black holes immersed in a de Sitter Universe. We will provide strong numerical evidence which designate that the (linear analog of) strong cosmic censorship is violated by near-extremal Reissner-Nordstr\"om-de Sitter black holes. Moreover, we shall investigate a superradiant instability in $d-$dimensional Reissner-Nordstr\"om-de Sitter black holes and show that the increment of dimensions decreases the timescale of the instability.