The spectrum of known black-hole solutions to the stationary Einstein equations has been steadily increasing, sometimes in unexpected ways. In particular, it has turned out that not all black-hole-equilibrium configurations are characterized by their mass, angular momentum and global charges. Moreover, the high degree of symmetry displayed by vacuum and electro-vacuum black-hole spacetimes ceases to exist in self-gravitating non-linear field theories. This text aims to review some developments in the subject and to discuss them in light of the uniqueness theorem for the Einstein-Maxwell system.

The fate of Cauchy horizons, such as those found inside charged black holes, is intrinsically connected to the decay of small perturbations exterior to the event horizon. As such, the validity of the strong cosmic censorship (SCC) conjecture is tied to how effectively the exterior damps fluctuations. Here, we study massless scalar fields in the exterior of Reissner-Nordström-de Sitter black holes. Their decay rates are governed by quasinormal modes of the black hole. We identify three families of modes in these spacetimes: one directly linked to the photon sphere, well described by standard WKB-type tools; another family whose existence and timescale is closely related to the de Sitter horizon. Finally, a third family which dominates for near-extremally-charged black holes and which is also present in asymptotically flat spacetimes. The last two families of modes seem to have gone unnoticed in the literature. We give a detailed description of linear scalar perturbations of such black holes, and conjecture that SCC is violated in the near extremal regime.

It was recently shown that Strong Cosmic Censorship may be violated in highly charged black-hole spacetimes living in a universe with a positive cosmological constant. Several follow-up works have since suggested that such result, while conceptually interesting, cannot be upheld in practice. We focus here on the claim that the presence of charged massive scalars suffices to save Strong Cosmic Censorship. To the contrary, we show that there still exists a finite region in parameter space where Strong Cosmic Censorship is expected to be violated.

This paper is the second part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the EinsteinMaxwell-scalar field system with a cosmological constant Λ, with the data on the outgoing initial null hypersurface given by a subextremal ReissnerNordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a "suitably regular" Lorentzian manifold.In the first paper of this sequence [4], we established well posedness of the characteristic problem with general initial data.In this second paper, we generalize the results of Dafermos [6] on the stability of the radius function at the Cauchy horizon by including a cosmological constant. This requires a considerable deviation from the strategy followed in [6], focusing on the level sets of the radius function instead of the red-shift and blue-shift regions. We also present new results on the global structure of the solution when the free data is not identically zero in a neighborhood of the origin.In the third and final paper [5], we will consider the issue of mass inflation and extendibility of solutions beyond the Cauchy horizon.

We present the key elements of the proof of an upper bound for angular-momentum and charge in terms of the mass for electro-vacuum asymptotically flat axisymmetric initial data sets with simply connected orbit space.

This paper is the third part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the EinsteinMaxwell-scalar field system with a cosmological constant Λ, with the data on the outgoing initial null hypersurface given by a subextremal ReissnerNordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a "suitably regular" Lorentzian manifold.In the first part [7] of this series we established the well posedness of the characteristic problem, whereas in the second part [8] we studied the stability of the radius function at the Cauchy horizon.In this third and final paper we show that, depending on the decay rate of the initial data, mass inflation may or may not occur. When the mass is controlled, it is possible to obtain continuous extensions of the metric across the Cauchy horizon with square integrable Christoffel symbols. Under slightly stronger conditions, we can bound the gradient of the scalar field. This allows the construction of (non-isometric) extensions of the maximal development which are classical solutions of the Einstein equations. Our results provide evidence against the validity of the strong cosmic censorship conjecture when Λ > 0.

This paper is the first part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the EinsteinMaxwell-scalar field system with a cosmological constant Λ, with the data on the outgoing initial null hypersurface given by a subextremal ReissnerNordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development (MGHD) as a "suitably regular" Lorentzian manifold.In this first part we establish well posedness of the Einstein equations for characteristic data satisfying the minimal regularity conditions leading to classical solutions. We also identify the appropriate notion of maximal solution, from which the construction of the corresponding MGHD follows, and determine breakdown criteria. This is the unavoidable starting point of the analysis; our main results will depend on the detailed understanding of these fundamentals.In the second part of this series [12] we study the stability of the radius function at the Cauchy horizon. In the third and final paper [13] we show that, depending on the decay rate of the initial data, mass inflation may or may not occur; in fact, it is even possible to have (non-isometric) extensions of the spacetime across the Cauchy horizon as classical solutions of the Einstein equations.

We prove an upper bound for angular momentum and charge in terms of the mass for electro-vacuum asymptotically flat axisymmetric initial data sets with simply connected orbit space. This completes the work started in (Chruściel and Costa 2009 Class. Quantum Grav. 26 235013 (arXiv:gr-qc/0909.5625)) where this charged Dain inequality was first presented but where the proof of the main result, based on the methods of Chruściel et al (Ann. Phys. 2008 323 2591-613 (arXiv:gr-qc/0712.4064v2)), was only sketched. Here we present a complete proof while simplifying the methods suggested by Chruściel and Costa (2009 Class. Quantum Grav. 26 235013 (arXiv:gr-qc/0909.5625)).