The grand challenges of contemporary fundamental physics—dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem—all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress. This write-up is an initiative taken within the framework of the European Action on ‘Black holes, Gravitational waves and Fundamental Physics’.

We prove the uniqueness theorem for static higher dimensional charged black holes spacetime containing an asymptotically flat spacelike hypersurface with compact interior and with both degenerate and non-degenerate components of the event horizon.

The variational method for Sturm-Liouville eigenvalue problem was employed to study analytically properties of the holographic superconductor with dark matter sector, in which a coupling between Maxwell field and another U (1)-gauge field was considered. The backreation of the dark matter sector on gravitational background in question was also examined. PACS numbers: 11.25.Tq, 98.80.Cq I. INTRODUCTIONThe AdS/CFT correspondence [1] plays a significant role in understanding the strongly coupled field theories. It relates string theory on asymptotically anti-de Sitter spacetime (AdS) to a conformal field theory on the boundary and enables to establish method for obtaining correlation functions in a strongly interacting field theory by means of a dual classical gravity description [2]. The aforementioned correspondence provides a theoretical insight into description of Hall and Nernst effects [3,4] (see also some contemporary reviews and references therein [5]). Moreover, it has been also successful applied to nuclear physics problems like heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) [6].It has been also proposed that AdS/CFT correspondence could be useful in describing superconductor phase transitions, namely when an Abelian symmetry is broken outside an AdS black hole this phenomenon results in the formation of the holographic superconductor (formation of charged scalar condensation) in the dual CFT [7]- [8]. It was indicated that the expectation value of charged operators underwent the U (1)-symmetry breaking and the second order phase transition took place [9]- [11]. Other properties of superconducting materials bounded with the influence of the magnetic field, were also reconstructed. The aforementioned studies covered the topic of critical magnetic fields and Abrikosov vortices [12]- [13]. At the beginning, studies of the holographic superconductivity when no backreations of matter fields were considered. Contrary to the previously elaborated cases, the backreaction caused that even uncharged scalar fields could form a condensation in (2 + 1)-dimensions [11]. In the case of p-wave holographic models it was shown that the phase transition leading to the formation of vector hair, changed from the second order to the first one, depending on the strength of the gravitational coupling [14]-[15]. These facts have triggered studies of the holographic superconductivity including the backreaction effect of the matter fields in question (see, e.g., ). Recently, also the backreaction between holographic insulator and superconductor was considered [18]- [21].The other theories of gravity were also taken into account. Holographic superconductivity in Einstein-Gauss-Bonnet (EGB) gravity were elaborated and among all it was revealed that the higher curvature corrections made the condensations harder to occur [22]- [26]. Realization of holographic superconductivity in Einstein-Maxwell-dilaton gravity being a generalized model built from the most extensive covariant gravity Lagrangian with at most two ...

We elaborated the gravitational collapse of a self-gravitating complex charged scalar field in the context of the low-energy limit of the string theory, the so-called dilaton gravity. We begin with the regular spacetime and follow the evolution through the formation of an apparent horizon and the final central singularity.

We prove the uniqueness theorem for self-gravitating non-linear σ-models in higher dimensional spacetime. Applying the positive mass theorem we show that Schwarzschild-Tagherlini spacetime is the only maximally extended, static asymptotically flat solution with non-rotating regular event horizon with a constant mapping.Nowadays, much effort is being devoted to the study of mathematical topics related to the black hole equilibrium states. The pioneering investigations in this field were attributed to Israel [1], Müller zum Hagen et al.[2] and Robinson [3]. The alternative approach to the problem of the uniqueness of black hole solutions was proposed by and then strengthened to the Einstein-Maxwell (EM) black holes [5,6]. Heusler [7] comprised the magnetically charged RN solution and static Einstein-σ-model case [8]. Recently the classification of static of vacuum black holes was finished [9]. The condition of non-degeneracy of the event horizon was removed and