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A review of the original thermodynamic formulation of the Tolman-Ehrenfest effect prescribing the temperature profile of uncharged fluid at thermal equilibrium forming stationary configurations in curved space-time is proposed. A statistical description based on relativistic kinetic theory is implemented. In this context the Tolman-Ehrenfest relation arises in the Schwarzschild space-time for collisionless uncharged particles at Maxwellian kinetic equilibrium. However, the result changes considerably when non-ideal fluids, i.e., non-Maxwellian distributions, are treated, whose statistical temperature becomes non-isotropic and gives rise to a tensor pressure. This is associated with phase-space anisotropies in the distribution function, occurring both for diagonal and non-diagonal metric tensors, exemplified by the Schwarzschild and Kerr metrics respectively. As a consequence, it is shown that for these systems it is not possible to define a Tolman-Ehrenfest relation in terms of an isotropic scalar temperature. Qualitative properties of the novel solution are discussed., Comment: 7 pages. arXiv admin note: text overlap with arXiv:2306.10434

Non-ideal fluids are generally subject to the occurrence of non-isotropic pressure tensors, whose determination is fundamental in order to characterize their dynamical and thermodynamical properties. This requires the implementation of theoretical frameworks provided by appropriate microscopic and statistical kinetic approaches in terms of which continuum fluid fields are obtained. In this paper the case of non-relativistic magnetized fluids forming equilibrium toroidal structures in external gravitational fields is considered. Analytical solutions for the kinetic distribution function are explicitly constructed, to be represented by a Chapman-Enskog expansion around a Maxwellian equilibrium. In this way, different physical mechanisms responsible for the generation of non-isotropic pressures are identified and proved to be associated with the kinetic constraints imposed on single and collective particle dynamics by phase-space symmetries and magnetic field. As a major outcome, the validity of a polytropic representation for the kinetic pressure tensors corresponding to each source of anisotropy is established, whereby directional pressures exhibit a specific power-law functional dependence on fluid density. The astrophysical relevance of the solution for the understanding of fluid plasma properties in accretion-disc environments is discussed., Comment: 16 pages

Charged fluids rotating around compact objects can form unique equilibrium structures when ambient large-scale electromagnetic fields combine with strong gravity. Equatorial as well as off-equatorial toroidal structures are among such figures of equilibrium with a direct relevance for astrophysics. To investigate their geometrical shapes and physical properties in the near-horizon regime, where effects of general relativity play a significant role, we commonly employ a scheme based on the energy-momentum conservation written in a standard representation. Here, we develop its interesting alternatives in terms of two covariant force representations, both based on a hypersurface projection of the energy-momentum conservation. In a proper hypersurface, space-like forces can be defined, following from a decomposition of the fluid four-acceleration. Each of the representations provides us with an insight into properties of the fluid flow, being well reflected in related conformal hypersurface geometries; we find behaviour of centrifugal forces directly related to geodesics of these conformal hypersurfaces and their embedding diagrams. We also reveal correspondence between the charged fluid flow world-lines from an ordinary spacetime, and world-lines determined by a charged test particles equation of motion in a conformal spacetime., Comment: 23 pages, 3 figures

We discuss the effects of electric charging on the equilibrium configurations of magnetized, rotating fluid tori around black holes of different mass. In the context of gaseous/dusty tori in galactic nuclei, the central black hole dominates the gravitational field and it remains electrically neutral, while the surrounding material acquires some electric charge and exhibits non-negligible self-gravitational effect on the torus structure. The structure of the torus is influenced by the balance between the gravitational and electromagnetic forces. A cusp may develop even in Newtonian tori due to the charge distribution., Comment: 5 pages, 1 figure; to appear in Proceedings of the 15th Marcel Grossman Meeting on General Relativity - the session AC3 on "Accretion Discs and Jets" by Eva Hackmann & Audrey Trova (Rome, 1-7 July 2018), edited by Elia Battistelli, Robert T. Jantzen, and Remo Ruffini, in preparation

We present a detailed comparison of several integration schemes applied to the dynamic system consisting of a charged particle on the Kerr background endowed with the axisymmetric electromagnetic test field. In particular, we compare the performance of the symplectic integrator with several non-symplectic routines and discuss under which circumstances we should choose the symplectic one and when we should switch to some other scheme. We are basically concerned with two crucial, yet opposing aspects - accuracy of the integration and CPU time consumption. The latter is generally less critical in our application while the highest possible accuracy is strongly demanded., Comment: 11 pages, 5 figures, 2 tables, prepared in 2011 for the Proceedings of RAGtime 10-13: Workshops on black holes and neutron stars (Opava, September 15 - 17, 2010)

We study charged-fluid toroidal structures surrounding a non-rotating charged black hole immersed in a large-scale, asymptotically uniform magnetic field. In continuation of our former study on electrically charged matter in approximation of zero conductivity, we demonstrate existence of orbiting structures in permanent rigid rotation in the equatorial plane, and charged clouds hovering near the symmetry axis. We constrain the range of parameters that allow stable configurations and derive the geometrical shape of equi-pressure surfaces. Our simplified analytical study suggests that these regions of stability may be relevant for trapping electrically charged particles and dust grains in some areas of the black hole magnetosphere, being thus important in some astrophysical situations., Comment: 14 pages, 5 figures

Circular motion of particles, dust grains and fluids in the vicinity of compact objects has been investigated as a model for accretion of gaseous and dusty environment. Here we further discuss, within the framework of general relativity, figures of equilibrium of matter under the influence of combined gravitational and large-scale magnetic fields, assuming that the accreted material acquires a small electric charge due to interplay of plasma processes and photoionization. In particular, we employ an exact solution describing the massive magnetic dipole and we identify the regions of stable motion. We also investigate situations when the particle dynamics exhibits the onset of chaos. In order to characterize the measure of chaoticness we employ techniques of Poincar\'e surfaces of section and of recurrence plots., Comment: 11 pages, 6 figures, published in the proceedings of the conference "Relativity and Gravitation: 100 Years after Einstein in Prague" (25. - 29. 6. 2012, Prague)

The possible occurrence of equilibrium off-equatorial tori in the gravitational and electromagnetic fields of astrophysical compact objects has been recently proved based on non-ideal MHD theory. These stationary structures can represent plausible candidates for the modelling of coronal plasmas expected to arise in association with accretion discs. However, accretion disc coronae are formed by a highly diluted environment, and so the fluid description may be inappropriate. The question is posed of whether similar off-equatorial solutions can be determined also in the case of collisionless plasmas for which treatment based on kinetic theory, rather than fluid one, is demanded. In this paper the issue is addressed in the framework of the Vlasov-Maxwell description for non-relativistic multi-species axisymmetric plasmas subject to an external dominant spherical gravitational and dipolar magnetic field. Equilibrium configurations are investigated and explicit solutions for the species kinetic distribution function are constructed, which are expressed in terms of generalized Maxwellian functions characterized by isotropic temperature and non-uniform fluid fields. The conditions for the existence of off-equatorial tori are investigated. It is proved that these levitating systems are admitted under general conditions when both gravitational and magnetic fields contribute to shaping the spatial profiles of equilibrium plasma fluid fields. Then, specifically-kinetic effects carried by the equilibrium solution are explicitly provided and identified here with diamagnetic, energy-correction and electrostatic contributions. It is shown that these kinetic terms characterize the plasma equation of state by introducing non-vanishing deviations from the assumption of thermal pressure., Comment: Accepted for publication in The Astrophysical Journal - Supplement Series

Within the framework of Bonnor's exact solution describing a massive magnetic dipole, we study the motion of neutral and electrically charged test particles. In dependence on the Bonnor spacetime parameters, we determine regions enabling the existence of stable circular orbits confined to the equatorial plane and of those levitating above the equatorial plane. Constructing Poincar\'e surfaces of section and recurrence plots, we also investigate the dynamics of particles moving along general off-equatorial trajectories bound in effective potential wells forming around the stable circular orbits. We demonstrate that the motion in the Bonnor spacetime is not integrable. This extends previous investigations of generalized St\"ormer's problem into the realm of exact solutions of Einstein-Maxwell equations, where the gravitational and electromagnetic effects play a comparable role on the particle motion., Comment: 27 pages, 14 figures

Astrophysical fluids may acquire non-zero electrical charge because of strong irradiation or charge separation in a magnetic field. In this case, electromagnetic and gravitational forces may act together and produce new equilibrium configurations, which are different from the uncharged ones. Following our previous studies of charged test particles and uncharged perfect fluid tori encircling compact objects, we introduce here a simple test model of a charged perfect fluid torus in strong gravitational and electromagnetic fields. In contrast to ideal magnetohydrodynamic models, we consider here the opposite limit of negligible conductivity, where the charges are tied completely to the moving matter. This is an extreme limiting case which can provide a useful reference against which to compare subsequent more complicated astrophysically-motivated calculations. To clearly demonstrate the features of our model, we construct three-dimensional axisymmetric charged toroidal configurations around Reissner-Nordstr\"om black holes and compare them with equivalent configurations of electrically neutral tori., Comment: 14 pages, 7 figures

We study the motion of charged test particles around a Kerr black hole immersed in the asymptotically uniform magnetic field, concluding that off-equatorial stable orbits are allowed in this system. Being interested in dynamical properties of these astrophysically relevant orbits we employ rather novel approach based on the analysis of recurrences of the system to the vicinity of its previous states. We use recurrence plots (RPs) as a tool to visualize recurrences of the trajectory in the phase space. Construction of RPs is simple and straightforward regardless of the dimension of the phase space, which is a major advantage of this approach when compared to the "traditional" methods of the numerical analysis of dynamical systems (for instance the visual survey of Poincar\'{e} surfaces of section, evaluation of the Lyapunov spectra etc.). We show that RPs and their quantitative measures (obtained from recurrence quantification analysis -- RQA) are powerful tools to detect dynamical regime of motion (regular vs. chaotic) and precisely locate the transitions between these regimes., Comment: 10 pages, 10 figures, published

Accretion onto black holes and compact stars brings material in a zone of strong gravitational and electromagnetic fields. We study dynamical properties of motion of electrically charged particles forming a highly diluted medium (a corona) in the regime of strong gravity and large-scale (ordered) magnetic field. We start our work from a system that allows regular motion, then we focus on the onset of chaos. To this end, we investigate the case of a rotating black hole immersed in a weak, asymptotically uniform magnetic field. We also consider a magnetic star, approximated by the Schwarzschild metric and a test magnetic field of a rotating dipole. These are two model examples of systems permitting energetically bound, off-equatorial motion of matter confined to the halo lobes that encircle the central body. Our approach allows us to address the question of whether the spin parameter of the black hole plays any major role in determining the degree of the chaoticness. To characterize the motion, we construct the Recurrence Plots (RP) and we compare them with Poincar\'e surfaces of section. We describe the Recurrence Plots in terms of the Recurrence Quantification Analysis (RQA), which allows us to identify the transition between different dynamical regimes. We demonstrate that this new technique is able to detect the chaos onset very efficiently, and to provide its quantitative measure. The chaos typically occurs when the conserved energy is raised to a sufficiently high level that allows the particles to traverse the equatorial plane. We find that the role of the black-hole spin in setting the chaos is more complicated than initially thought., Comment: 21 pages, 20 figures, accepted to ApJ

Off-equatorial circular orbits with constant latitudes (halo orbits) of electrically charged particles exist near compact objects. In the previous paper, we discussed this kind of motion and demonstrated the existence of minima of the two-dimensional effective potential which correspond to the stable halo orbits. Here, we relax previous assumptions of the pseudo-Newtonian approach for the gravitational field of the central body and study properties of the halo orbits in detail. Within the general relativistic approach, we carry out our calculations in two cases. Firstly, we examine the case of a rotating magnetic compact star. Assuming that the magnetic field axis and the rotation axis are aligned with each other, we study the orientation of motion along the stable halo orbits. In the poloidal plane, we also discuss shapes of the related effective potential halo lobes where the general off-equatorial motion can be bound. Then we focus on the halo orbits near a Kerr black hole immersed in an asymptotically uniform magnetic field of external origin. We demonstrate that, in both the cases considered, the lobes exhibit two different regimes, namely, one where completely disjoint lobes occur symmetrically above and below the equatorial plane, and another where the lobes are joined across the plane. A possible application of the model concerns the structure of putative circumpulsar discs consisting of dust particles. We suggest that the particles can acquire a small (but non-zero) net electric charge, and this drives them to form the halo lobes., Comment: 24 pages, 7 figures

Pseudo-Newtonian gravitational potential describing the gravitational field of static and spherically symmetric black holes in the universe with a repulsive cosmological constant is introduced. In order to demonstrate the accuracy of the pseudo-Newtonian approach, the related effective potential for test-particle motion is constructed and compared with its general relativistic counterpart given by the Schwarzschild-de Sitter geometry. The results indicate that such an approach could be useful in applications of developed Newtonian theories of accretion discs in astrophysically interesting situations in large galactic structures for the Schwarzschild-de Sitter spacetimes with the cosmological parameter y=(1/3)\Lambda M^2<10^{-6}., Comment: 17 pages, 11 figures, accepted by International Journal of Modern Physics D (2008)

Near a black hole or an ultracompact star, motion of particles is governed by strong gravitational field. Electrically charged particles feel also electromagnetic force arising due to currents inside the star or plasma circling around. We study a possibility that the interplay between gravitational and electromagnetic action may allow for stable, energetically bound off-equatorial motion of charged particles. This would represent well-known generalized Stormer's 'halo' orbits, which have been discussed in connection with the motion of dust grains in planetary magnetospheres. We demonstrate that such orbits exist and can be astrophysically relevant when a compact star or a black hole is endowed with a dipole-type magnetic field. In the case of Kerr-Newman solution, numerical analysis shows that the mutually connected gravitational and electromagnetic fields do not allow existence of stable halo orbits above the outer horizon of black holes. Such orbits are either hidden under the inner black-hole horizon, or they require the presence of a naked singularity., Comment: 16 pages, 7 figures, accepted in Class. Quantum Grav. (2008)

Optical reference geometry and related concept of inertial forces are investigated in Kerr-de Sitter spacetimes. Properties of the inertial forces are summarized and their typical behaviour is illustrated. The intuitive 'Newtonian' application of the forces in the relativistic dynamics is demonstrated in the case of the test particle circular motion, static equilibrium positions and perfect fluid toroidal configurations. Features of the optical geometry are illustrated by the embedding diagrams of its equatorial plane. The embedding diagrams do not cover whole the stationary regions of the spacetimes, therefore the limits of embeddability are established. A shape of the embedding diagrams is related to the behaviour of the centrifugal force and it is characterized by the number of turning points of the diagrams. Discussion of the number of embeddable photon circular orbits is also included and the typical embedding diagrams are constructed. The Kerr-de Sitter spacetimes are classified according to the properties of the inertial forces and embedding diagrams., Comment: 31 pages, 10 figures

Equilibrium conditions and spin dynamics of spinning test particles are discussed in the stationary and axially symmetric Kerr-de Sitter black-hole or naked-singularity spacetimes. The general equilibrium conditions are established, but due to their great complexity, the detailed discussion of the equilibrium conditions and spin dynamics is presented only in the simple and most relevant cases of equilibrium positions in the equatorial plane and on the symmetry axis of the spacetimes. It is shown that due to the combined effect of the rotation of the source and the cosmic repulsion the equilibrium is spin dependent in contrast to the spherically symmetric spacetimes. In the equatorial plane, it is possible at the so-called static radius, where the gravitational attraction is balanced by the cosmic repulsion, for the spinless particles as well as for spinning particles with arbitrarily large azimuthal-oriented spin or at any radius outside the ergosphere with a specifically given spin orthogonal to the equatorial plane. On the symmetry axis, the equilibrium is possible at any radius in the stationary region and is given by an appropriately tuned spin directed along the axis. At the static radii on the axis the spin of particles in equilibrium must vanish.

Forces defined in the framework of optical reference geometry are introduced in the case of stationary and axially symmetric Kerr black-hole and naked-singularity spacetimes with a repulsive cosmological constant. Properties of the forces acting on test particles moving along circular orbits in the equatorial plane are discussed, whereas it is shown where the gravitational force vanishes and changes its orientation and where the centrifugal force vanishes and changes its orientation independently of the velocity of test particles related to the optical geometry; the Coriolis force does not vanish for the velocity being non-zero. The spacetimes are classified according to the number of circular orbits where the gravitational and centrifugal forces vanish.