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In this paper, I present a few results devoted to variational methods in General Relativity Theory. I will focus on the equivalence between three different variational formulations. Moreover, I analyze how the dependence on covariant derivatives affects the affine connection.

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We apply the Dirac procedure for constrained systems to the Arnowitt–Deser–Misner (ADM) formalism linearized around a Bianchi I universe with a single minimally coupled scalar field. We discuss and employ basic concepts such as Dirac observables, Dirac brackets, gauge-fixing conditions, reduced phase space, physical Hamiltonian, canonical isomorphism between different gauge-fixing surfaces and spacetime reconstruction. We show that the definition of a gravitational wave as a traceless-transverse mode of the metric perturbation needs to be revised. Moreover, there exist coordinate systems in which a polarization mode of the gravitational wave is given entirely in terms of a scalar metric perturbation. The obtained fully canonical formalism will serve as a starting point for a complete quantization of the cosmological perturbations and the cosmological background.

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We consider curvature-based gravity depending on non-local terms as \(\Box ^{-1} R\), with \(\Box ^{-1}\) being the inverse of the d’Alembert operator and \(R\) the Ricci curvature scalar. Specifically, we select the functional form of the effective Lagrangians by the Noether symmetries and then find out exact solutions in cosmological and spherically symmetric backgrounds. A comparison with experimental data is provided.

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Complex, 4-dimensional spaces which are equipped with congruences of self-dual and anti-self-dual null strings are considered. Criteria of a classification of such spaces are given. Some interesting classes of two-sided Walker and para-Kähler spaces are presented.

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The so-called BSW effect is an idealised scenario for high-energy test particle collisions in the vicinity of black holes; if the black hole is extremal and one of the particles fine-tuned, the centre-of-mass collision energy can be arbitrarily high. It has been recently shown that the energy of escaping particles produced in this process can also be arbitrarily high in the given approximation, as long as both the black hole and the escaping particles are charged, regardless of how small the black-hole charge might be. We revisit these results and show that they are also compatible with properties of microscopic particles for the case of motion in the equatorial plane of an extremal Kerr–Newman black hole.

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We investigate the properties of the magnetically dominated jets from accreting black hole sources. We run the numerical simulation in a 3D general relativistic magneto-hydrodynamic setup and we study the connection between the properties of the jet and the accretion disk. We focus on the formation of magnetically arrested disk state.

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New developments related to our recent study of the accretion of the Vlasov gas onto a moving Schwarzschild black hole are presented. We discuss the low-temperature limit of the mass accretion rate and a simple Monte Carlo simulation used to check the results obtained in this limit. We also comment on several numerical aspects related with momentum integrals expressing the particle density current and the particle density.

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We summarize recent results on \(D\)-dimensional Robinson–Trautman solutions of Einstein’s gravity in the presence of a conformally invariant non-linear electromagnetic field and a cosmological constant. These spacetimes contain static dyonic black holes with various horizon geometries and their time-dependent radiating generalizations, as well as a class of stealth solutions. Extensions to \(f(R)\) and the Gauss–Bonnet gravity are mentioned.

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We discuss an integrable model of string on AdS\(_3 \times \)S\(^3\times \)T\(^4\) in a thermodynamical bath. We show that scattering of the excitations above equilibrium states has some novel features. Thermodynamics points to interesting deformation of the original model for which we discuss finite size effect through mirror TBA equations.

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We describe stationary and axisymmetric gas configurations surrounding black holes. They consist of a collisionless relativistic kinetic gas of identical massive particles following bound orbits in a Schwarzschild exterior spacetime and are modeled by a one-particle distribution function which is the product of a function of the energy and a function of the orbital inclination associated with the particle’s trajectory. The morphology of the resulting configuration is analyzed.

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We analyse the Hamiltonian charges for Maxwell and scalar field theory on light-cones on a de Sitter background.

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In this talk, I gave an account of my article arXiv:1909.07756[gr-qc] which considered the following question: Penrose gave a construction which associates a plane-wave space-time \(P(M,{\mit \Gamma })\) with any pair \((M,{\mit \Gamma })\) where \(M\) is a space-time and \({\mit \Gamma }\) is a null geodesic in \(M\); what condition must \(M\) satisfy if \(P\) is diagonalisable for every \({\mit \Gamma }\) in \(M\)?

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A perturbed black hole emits radiation at certain characteristic frequencies, the quasinormal frequencies, similar to the spectrum of frequencies produced by a struck guitar string. The normal modes of a guitar string are complete, in the sense that any oscillation of the string may be written as a superposition of these modes. In the case of quasinormal modes, this is not the case in general. We present here a simple proof of this fact.