We present stationary solutions of magnetized, viscous thick accretion disks around a Schwarzschild black hole. We assume that the tori are not self-gravitating, are endowed with a toroidal magnetic field, and obey a constant angular momentum law. Our study focuses on the role of the black hole curvature in the shear viscosity tensor and in their potential combined effect on the stationary solutions. Those are built in the framework of a causality-preserving, second-order gradient expansion scheme of relativistic hydrodynamics in the Eckart frame description which gives rise to hyperbolic equations of motion. The stationary models are constructed by numerically solving the general relativistic momentum conservation equation using the method of characteristics. We place constraints in the range of validity of the second-order transport coefficients of the theory. Our results reveal that the effects of the shear viscosity and curvature are particularly noticeable only close to the cusp of the disks. The surfaces of constant pressure are affected by viscosity and curvature and the self-intersecting isocontour - the cusp - moves to smaller radii (i.e. towards the black hole horizon) as the effects become more significant. For highly magnetized disks the shift in the cusp location is smaller. Our findings might have implications on the dynamical stability of constant angular momentum tori which, in the inviscid case, are affected by the runaway instability.
Geodesics for a 5D magnetized Schwarzschild-like solution are analyzed by reducing the problem to the motion of a test particle in an effective potential. In absence of magnetic field comparison is established with Schwarzschild's geometry. Embedding diagrams are constructed in order to visualize the geometry of the metric. The study performed here is also valid, when the electromagnetic interactions are neglected, for the low energy superstring theory and the Brans-Dicke theory.
Neutral test particles around a Schwarzschild black hole immersed in an external uniform magnetic field have no interactions of electromagnetic forces, but their motions can be chaotic. This chaotic behavior is induced owing to the gravitational effect of the magnetic field leading to the nonintegrability of the magnetized Ernst-Schwarzschild spacetime geometry. In fact, chaos is strengthened typically with an increase of the energy or the magnetic field under appropriate circumstances. When these test particles have charges, the electromagnetic forces are included. As a result, the electromag…
We study the circular orbits of charged particles around a weakly charged Schwarzschild black hole immersed in a weak, axisymmetric magnetic field. We start by reviewing the circular orbits of neutral particles and charged particles around only weakly charged and only weakly magnetized black holes. The case of a weakly magnetized and charged black hole is investigated then. In particular, we study the effect of the electromagnetic forces on the charged particles innermost stable circular orbits. We show that negative energy stable circular orbits are possible and that two bands of charged part…
Comments on Orbits of particles with magnetic dipole moment around magnetized Schwarzschild black holes: Applications to S2 star orbit, arXiv:2406.03371v2
In this paper, we have presented the studies of the motion of magnetized particles and energetic processes around Schwarzschild black holes in modified gravity (MOG). The study of circular stable orbits shows that orbits of magnetized particles can not be stable for the values of magnetic coupling parameter $β\geq 1$. It was also shown that the range of stable circular orbits increases with the increase of both MOG and magnetic coupling parameters, while the effects of magnetic interaction stronger than the gravity. It was obtained that the increase of the MOG parameter causes the increase of …
This paper investigates wave properties of hot plasma in a Veselago medium. For the Schwarzschild black hole, the $3+1$ GRMHD equations are re-formulated which are linearly perturbed and then Fourier analyzed for rotating (non-magnetized and magnetized) plasmas. The graphs of wave vector, refractive index and change in refractive are used to discuss the wave properties. The results obtained confirm the presence of Veselago medium for both rotating (non-magnetized and magnetized) plasmas. This work generalized the isothermal plasma waves in the Veselago medium to hot plasma case.
We re-formulate the 3+1 GRMHD equations for the Schwarzschild black hole in a Veselago medium. Linear perturbation in rotating (non-magnetized and magnetized) plasma is introduced and their Fourier analysis is considered. We discuss wave properties with the help of wave vector, refractive index and change in refractive index in the form of graphs. It is concluded that some waves move away from the event horizon in this unusual medium. We conclude that for the rotating non-magnetized plasma, our results confirm the presence of Veselago medium while the rotating magnetized plasma does not provid…
This paper is dedicated to the investigation of the positivity of the Hayward quasilocal energy of tori. Marginally trapped tori have nonnegative Hayward energy. We consider a scenario of a spherically symmetric constant density star matched to an exterior Schwarzschild solution. We show that any generic tori within the star, distorted or not, trapped or not, have strictly positive Hayward energy. Surprisingly we find analytic examples of `thin' tori with negative Hayward energy in the outer neighborhood of the Schwarzschild horizon. These tori are swept out by rotating the standard round circ…
In this paper, we study the influence of the axion-plasmon, as proposed in (Phys. Rev. Lett. 120, 181803 (2018)) on the optical properties of the Schwarzschild black hole. Our aim is to provide a test to detect the effects of a fixed axion background using black holes. To accomplish our goal, we explore the effect of the axion-plasmon coupling on the motion of photons around the Schwarzschild black hole and check the possibility of observing those effects upon the black hole shadow, the gravitational deflection angle, Einstein rings and shadow images obtained by radially infalling gas on a bla…
We consider a synchrotron radiation from a charged particle moving in a bound orbit around a weakly magnetized Schwarzschild black hole (a static black hole immersed into a constant uniform magnetic field) in its equatorial plane, perpendicular to the magnetic field. In particular, we study the case when the Lorentz force acting on the charged particle is directed outward from the black hole. The particle is initially moving in a nongeodesic bound orbit which due to synchrotron radiation decays to a nongeodesic circular orbit. We study this transition and calculate the radiated power and energ…
In this paper we consider magnetized black holes and black rings in the higher dimensional dilaton gravity. Our study is based on exact solutions generated by applying a Harrison transformation to known asymptotically flat black hole and black ring solutions in higher dimensional spacetimes. The explicit solutions include the magnetized version of the higher dimensional Schwarzschild-Tangherlini black holes, Myers-Perry black holes and five dimensional (dipole) black rings. The basic physical quantities of the magnetized objects are calculated. We also discuss some properties of the solutions …
In this paper we examine the wave properties of hot plasma living in Schwarzschild magnetosphere. The 3+1 GRMHD perturbation equations are formulated for this scenario. These equations are Fourier analyzed and then solved numerically to obtain the dispersion relations for non-rotating, rotating non-magnetized and rotating magnetized plasma. The wave vector is evaluated which is used to calculate refractive index. These quantities are shown in graphs which are helpful to discuss the dispersive properties of the medium near the event horizon.
We discuss charged particles motion in a spacetime of a weakly magnetized static non-rotating black hole. We study under which conditions a charged particle originally revolving around the black hole at a circular orbit after being kicked by another particle or photon can escape to infinity. We determine the escape velocity for particles at the innermost stable circular orbits and discuss the properties of particles moving with near-critical velocity. We show that in a general case such a motion is chaotic.
We present the first family of magnetically polarized equilibrium tori around a Kerr black hole. The models were obtained in the test fluid approximation by assuming that the tori is a linear media, making it is possible to characterize the magnetic polarization of the fluid through the magnetic susceptibility $χ_{m}$. The magnetohydrodynamic (MHD) structure of the models was solved by following the Komissarov approach, but with the aim of including the magnetic polarization of the fluid, the integrability condition for the magnetic counterpart was modified. We build two kinds of magnetized to…
Chaotic motion and power spectral density in Schwarzschild Bertotti-Robinson black hole spacetime
Harbin Institute of Technology（哈尔滨工业大学）；University of Tashkent for Applied Sciences（塔什干应用科学大学）；Institute for Advanced Studies, New Uzbekistan University（新乌兹别克斯坦大学高级研究院）；Institute of Theoretical Physics, National University of Uzbekistan（乌兹别克斯坦国立大学理论物理研究所）；Institute of Fundamental and Applied Research, National Research University TIIAME（TIIAME国立研究型大学基础与应用研究所）；Tashkent State Technical University（塔什干国立技术大学）
We present a numerical study of the dynamics of magnetized, relativistic, non-self-gravitating, axisymmetric tori orbiting in the background spacetimes of Schwarzschild and Kerr black holes. The initial models have a constant specific angular momentum and are built with a non-zero toroidal magnetic field component, for which equilibrium configurations have recently been obtained. In this work we extend our previous investigations which dealt with purely hydrodynamical thick discs, and study the dynamics of magnetized tori subject to perturbations which, for the values of the magnetic field str…
Charged particle bound orbits around magnetized Schwarzschild black holes: S2 star and hotspot applications
Harbin Institute of Technology（哈尔滨工业大学）；Tashkent University of Applied Sciences（塔什干应用科学大学）；Institute for Advanced Studies, New Uzbekistan University（新乌兹别克斯坦大学高级研究院）；Universidad Central de Chile（智利中央大学）；National University of Uzbekistan（乌兹别克斯坦国立大学）；Tashkent State Technical University（塔什干国立技术大学）；Urgench State University（乌尔根奇国立大学）
A charged particle kicked from an initial circular orbit around a weakly magnetized Schwarzschild black hole undergoes transient chaotic motion before either getting captured by the black hole or escaping upstream or downstream with respect to the direction of the magnetic field. These final states form basins of attraction in the space of initial states. We provide a detailed numerical study of the basin structure of this initial state space. We find it to possess the peculiar Wada property: each of its basin boundaries is shared by all three basins. Using basin entropy as a measure, we show …
In the present paper, we have investigated the motion of charged particles together with magnetic dipoles to determine how well the spacetime deviation parameter $ε$ and external uniform magnetic field can mimic the spin of a rotating Kerr black hole. Investigation of charged particle motion has shown that the deviation parameter $ε$ in the absence of external magnetic fields can mimic the rotation parameter of Kerr spacetime up to $a/M \approx0.5$. The combination of external magnetic field and deviation parameter can do even a better job mimicking the rotation parameter up to $a/M\simeq0.93$…
We study motion of a charged particle in the vicinity of a weakly magnetized Schwarzschild black hole and focus on its bounded trajectories lying in the black hole equatorial plane. If the Lorentz force, acting on the particle, is directed outward from the black hole, there exist two qualitatively different types of trajectories, one is a curly motion and another one is a trajectory without curls. We calculated the critical value of the magnetic field for the transition between these two types. If the magnetic field is greater than the critical one, for fixed values of the particle energy and …
Pseudo-Newtonian gravitational potential introduced in spherically symmetric black-hole spacetimes with a repulsive cosmological constant is tested for equilibrium toroidal configurations of barotropic perfect fluid orbiting the black holes. Shapes and potential depths are determined for the marginally stable barotropic tori with uniform distribution of specific angular momentum, using both the pseudo-Newtonian and fully relativistic approach. For the adiabatic (isoentropic) perfect fluid, temperature profiles, mass-density and pressure profiles and total masses of pseudo-Newtonian and relativ…
We investigate the distribution of gravitational energy on the spacetime of a Schwarzschild black hole immersed in a cosmic magnetic field. This is done in the context of the {\it Teleparallel Equivalent of General Relativity}, which is an alternative geometrical formulation of General Relativity, where gravity is describe by a spacetime endowed with torsion, rather than curvature, with the fundamental field variables being tetrads. We calculate the energy enclosed by a two-surface of constant radius - in particular, the energy enclosed by the event horizon of the black hole. In this case we f…
We consider the Klein-Gordon equation on the exterior of the toric anti de-Sitter Schwarzschild black hole with Dirichlet, Neumann and Robin boundary conditions at $\mathcal{I}$. We define a non-degenerate energy for the equation which controls the renormalised $H^1$ norm of the field. We then establish both decay and integrated decay of this energy through vector field methods. Finally we demonstrate the necessity of `losing a derivative' in the integrated energy estimate through the construction of a Gaussian beam staying in the exterior of the event horizon for arbitrary long co-ordinate ti…
We study geometrically thick perfect-fluid tori with constant specific angular momentum, so-called "Polish doughnuts", orbiting deformed compact objects with a quadrupole moment. More specifically, we consider two different asymptotically flat, static and axisymmetric vacuum solutions to Einstein's field equation with a non-zero quadrupole moment, the q-metric and the Erez-Rosen spacetime. It is our main goal to find features of Polish doughnuts in these two spacetimes which qualitatively distinguish them from Polish doughnuts in the Schwarzschild spacetime. As a main result we find that, for …
We study behaviour of ionized region of a Keplerian disk orbiting a Schwarzschild black hole immersed in an asymptotically uniform magnetic field. In dependence on the magnetic parameter ${\cal B}$, and inclination angle $θ$ of the disk plane with respect to the magnetic field direction, the charged particles of the ionized disk can enter three regimes: a) regular oscillatory motion, b) destruction due to capture by the magnetized black hole, c) chaotic regime of the motion. In order to study transition between the regular and chaotic type of the charged particle motion, we generate time serie…
It has recently been demonstrated that magnetized black holes in composed Einstein-Maxwell-scalar-Gauss-Bonnet field theories with a non-minimal negative coupling of the scalar field to the Gauss-Bonnet curvature invariant may support spatially regular scalar hairy configurations. In particular, it has been revealed that, for Schwarzschild-Melvin black-hole spacetimes, the onset of the near-horizon spontaneous scalarization phenomenon is marked by the numerically computed dimensionless critical relation $(BM)_{\text{crit}}\simeq0.971$, where $\{M,B\}$ are respectively the mass and the magnetic…
In this paper we have investigated the dynamics of neutral, electrically charged and magnetized particles around renormalized group improved (RGI) Schwarzschild black hole in the presence of external asymptotically uniform magnetic field. We have analyzed the spacetime structure around RGI black hole by investigating Ricci, the square of Ricci tensor and Kretschmann curvature scalars and shown that only in the case when the parameter $γ=0$ the curvature becomes infinite at the center of the black hole, while for non-zero values of $γ$ parameter the black hole curvature reflects the properties …
This paper studied the relativistic accretion thick disc model raised by a deformed compact object that slightly deviated from spherical up to the quadrupole moment by utilising $\rm q$-metric. This metric is the simplest asymptotically flat solution of Einstein's equation with quadrupole moment. This work aims to study the effects of quadrupole moments in combination with the parameters of the thick magnetised disc model via studying the properties of these equilibrium sequences of magnetised, non-self-gravitating discs in this space-time. We employed different angular momentum distributions …
Based on the scalar-tensor-vector modified gravitational theory, a modified gravity Schwarzschild black hole solution has been given in the existing literature. Such a black hole spacetime is obtained through the inclusion of a modified gravity coupling parameter, which corresponds to the modified gravitational constant and the black hole charge. In this sense, the modified gravity parameter acts as not only an enhanced gravitational effect but also a gravitational repulsive force contribution to a test particle moving around the black hole. Because the modified Schwarzschild spacetime is stat…
We analyze the dispersion modes of hot plasma around Schwarzschild de-Sitter event horizon in the presence of Veselago medium. For this purpose, we apply ADM 3+1 formalism to develop GRMHD equations for the Schwarzschild de-Sitter spacetime. Implementation of linear perturbation yields perturbed GRMHD equations that are used for the Fourier analysis of rotating (non-magnetized and magnetized) plasma. Wave analysis is described by the graphical representation of the wave vector, refractive index, change in refractive index, phase and group velocities. The outcome of this work supports the valid…
We apply the ADM 3+1 formalism to derive the general relativistic magnetohydrodynamic equations for cold plasma in spatially flat Schwarzschild metric. Respective perturbed equations are linearized for non-magnetized and magnetized plasmas both in non-rotating and rotating backgrounds. These are then Fourier analyzed and the corresponding dispersion relations are obtained. These relations are discussed for the existence of waves with positive angular frequency in the region near the horizon. Our results support the fact that no information can be extracted from the Schwarzschild black hole. It…
This is the first of a series of papers investigating the oscillation properties of relativistic, non-selfgravitating tori orbiting around a black hole. In this initial paper we consider the axisymmetric oscillation modes of a torus constructed in a Schwarzschild spacetime. To simplify the treatment and make it as analytical as possible, we build our tori with vertically integrated and vertically averaged quantities. Our investigation progresses by steps. We first consider a local analysis in Newtonian gravity and determine the properties of acoustic wave propagation within these objects, as w…
We have studied the wave properties of the cold and isothermal plasma in the vicinity of the Schwarzschild black hole event horizon. The Fourier analyzed perturbed 3+1 GRMHD equations are taken on the basis of which the complex dispersion relations are obtained for non-rotating, rotating non-magnetized and rotating magnetized backgrounds. The propagation and attenuation vectors along with the refractive index are obtained (shown in graphs) to study the dispersive properties of the medium near the event horizon. The results show that no information can be obtained from the Schwarzschild magneto…
In many astrophysically relevant situations radiation reaction force acting upon a charge can not be neglected and the question arises about the location and stability of circular orbits in such regime. Motion of point charge with radiation reaction in flat spacetime is described by Lorenz-Dirac (LD) equation, while in curved spacetime -- by DeWitt-Brehme (DWB) equation containing the Ricci term and the tail term. We show that for the motion of elementary particles in vacuum metrics the DWB equation can be reduced to the covariant form of the LD equation which we use here. Generically, the LD …
The 3+1 GRMHD equations for Schwarzschild spacetime in Rindler coordinates with isothermal state of plasma are formulated. We consider the cases of non-rotating and rotating backgrounds with non-magnetized and magnetized plasmas. For these cases, the perturbed form of these equations are linearized and Fourier analyzed by introducing plane wave type solutions. The determinant of these equations in each case leads to two dispersion relations which give value of the wave number $k$. Using the wave number, we obtain information like phase and group velocities etc. which help to discuss the nature…
The capacity to model magnetohydrodynamical (MHD) flows in dynamical, strongly curved spacetimes significantly extends the reach of numerical relativity in addressing many problems at the forefront of theoretical astrophysics. We have developed and tested an evolution code for the coupled Einstein-Maxwell-MHD equations which combines a BSSN solver with a high resolution shock capturing scheme. As one application, we evolve magnetized, differentially rotating neutron stars under the influence of a small seed magnetic field. Of particular significance is the behavior found for hypermassive neutr…
In order to test the role of large-scale magnetic fields in quasiperiodic oscillation phenomena observed in microquasars, we study oscillatory motion of charged particles in vicinity of a Schwarzschild black hole immersed into an external asymptotically uniform magnetic field. We determine the fundamental frequencies of small harmonic oscillations of charged test particles around stable circular orbits in the equatorial plane of a magnetized black hole, and discuss the radial profiles of frequencies of the radial and latitudinal harmonic oscillations in dependence on the mass of the black hole…
In this paper, we examine the weak deflection angle and greybody bound for magnetized regular black hole.$~$ For this purpose, we apply the Gauss-Bonnet theorem on the black hole and obtain the deflection angle in plasma and non-plasma mediums.$~$ Moreover, we investigate graphically the effect of impact parameter on the deflection angle for regular black hole in both mediums. We examine that deflection angle goes to infinity when the impact parameter approaches to zero. We also observe that deflection angle shows negative behaviour at $q=0.6$ and $q=2.09$ but at $0.6<q<2.09$, angle shows the …
I present results of two-dimensional general relativistic hydrodynamical simulations of constant specific angular momentum tori orbiting around a Schwarzschild black hole. After introducing axisymmetric perturbations, these objects either become unstable to the runaway instability or respond with regular oscillations. The latter, in particular, are responsible for quasi-periodic bursts of accretion onto the black hole as well as for the emission of intense gravitational radiation, with signal-to-noise ratios at the detector which are comparable or even larger than the typical ones expected in …
We present new equilibrium solutions of stationary models of magnetized thick disks (or tori) around Kerr black holes with synchronised scalar hair. The models reported here largely extend our previous results based on constant radial distributions of the specific angular momentum along the equatorial plane. We introduce a new way to prescribe the distribution of the disk's angular momentum based on a combination of two previous proposals and compute the angular momentum distribution outside the equatorial plane by resorting to the construction of von Zeipel cylinders. We find that the effect …
This is the second of a series of papers investigating the oscillation properties of relativistic, non-selfgravitating tori orbiting around black holes. Extending the work done in a Schwarzschild background, we here consider the axisymmetric oscillations of vertically integrated tori in a Kerr spacetime. The tori are modeled with a number of different non-Keplerian distributions of specific angular momentum and we discuss how the oscillation properties depend on these and on the rotation of the central black hole. We first consider a local analysis to highlight the relations between acoustic a…
We use the Ernst-Schwarzschild solution for a black hole immersed in a uniform magnetic field to estimate corrections to the bending angle and time delay due-to presence of weak magnetic fields in galaxies and between galaxies, and also due-to influence of strong magnetic field near supermassive black holes. The magnetic field creates a kind of confinement in space, that leads to increasing of the bending angle and time delay for a ray of light propagating in the equatorial plane.
We present general relativistic hydrodynamics simulations of constant specific angular momentum tori orbiting a Schwarzschild black hole. These tori are expected to form as a result of stellar gravitational collapse, binary neutron star merger or disruption, can reach very high rest-mass densities and behave effectively as neutron stars but with a toroidal topology (i.e. ``toroidal neutron stars''). Our attention is here focussed on the dynamical response of these objects to axisymmetric perturbations. We show that, upon the introduction of perturbations, these systems either become unstable t…
Testing the true nature of black holes - the no-hair hypothesis - will become increasingly more precise in the next few years as new observational data is collected in both the gravitational wave channel and the electromagnetic channel. In this paper we consider numerically generated spacetimes of Kerr black holes with synchronised scalar hair and build stationary models of magnetized thick disks (or tori) around them. Our approach assumes that the disks are not self-gravitating, they obey a polytropic equation of state, the distribution of their specific angular momentum is constant, and they…
We discuss the conditions for the existence of extended matter configurations orbiting in the ergoregion or close to the outer ergosurface of the Kerr black hole ("dragged" configurations). The corotating tori under consideration are perfect fluid configurations with barotropic equation of state, orbiting on the equatorial plane of the central Kerr black hole. The possibility of magnetized tori with a toroidal magnetic field is also discussed. Indications on the attractors where dragged tori can be observed are provided with the analysis of the fluid characteristics and geometrical features, r…
We present stationary solutions of geometrically thick discs (or tori) endowed with a self-consistent toroidal magnetic field distribution surrounding a non-rotating black hole in an analytical, static, spherically-symmetric $f(R)$-gravity background. These $f(R)$-gravity models introduce a Yukawa-like modification to the Newtonian potential, encoded in a single parameter $δ$ which controls the strength of the modified potential and whose specific values affect the disc configurations when compared to the general relativistic case. Our models span different magnetic field strengths, from purel…
