Investigations of electromagnetic interactions between test-charged and magnetized particles are important in the dynamics of the particles in strong gravitational fields around black holes. Here, we study the dynamics of a particle having an electric charge and a magnetic dipole moment in the spacetime of a Schwarzschild black hole in modified gravity (MOG), called Schwarzschild-MOG black hole. First, we provide a solution of Maxwell equations for the angular component of electromagnetic four potentials in the Schwarzschild-MOG spacetime. Then, we derive equations of motion and effective potential for circular motion of such particles using a hybrid form of the Hamilton-Jacobi equation which includes both interactions of electric charge and magnetic dipole moment with the external magnetic field assumed as asymptotically uniform, and interaction between the particles and the MOG field. Also, we investigate the effects of the three types of interactions on the radius of innermost stable circular orbits (ISCOs) and the energy \& angular momentum of the particles at their corresponding ISCOs. Finally, we provide detailed analyses of the effects of the three interactions mentioned above on the center of mass energy in the collisions between neutral, electrically charged, and magnetized particles.
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 …
Accretion Disks in Schwarzschild-MOG and Kerr-MOG Backgrounds: MOG Parameter in terms of Observational Quantities
Mesoamerican Centre for Theoretical Physics, Universidad Autónoma de Chiapas（中美洲理论物理中心，恰帕斯自治大学）；Instituto de Física, Universidad Autónoma de Santo Domingo（物理学研究所，圣多明各自治大学）；Facultad de Ciencias Físico-Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo（数理科学学院，米却肯圣尼古拉斯德伊达尔戈大学）
This paper explores the dynamics of both neutral as well as charged particles orbiting near a rotating black hole in scalar-tensor-vector gravity. We study the conditions for the particle to escape at the innermost stable circular orbit. We investigate stability of orbits through effective potential and Lyapunov exponent in the presence of magnetic field. The effective force acting on particle is also discussed. We also study the center of mass energy of particle collision near the horizon of this black hole. Finally, we compare our results with the particle motion around Schwarzschild, Kerr a…
This paper aims to investigate the scattering of fermions by spherically symmetric MOG black holes, which are a type of black holes encountered in scalar-tensor-vector modified gravitational theories. After determining the scattering modes in this black hole geometry, we apply the partial wave method to compute analytical expressions for the phase shifts that enter into the definition of scattering amplitudes. An analysis of the influence of the MOG parameter $α$ on the differential scattering cross section and the induced polarization is conducted. Also, a comparison with Schwarzschild scatte…
In this paper, we analyze the thermodynamic stability of Schwarzschild Modified Gravity (MOG) black holes in a non-commutative framework. We show that, unlike a commutative MOG black hole, in the coherent state picture of non-commutativity MOG black holes are thermodynamically stable. At the final stage of evaporation, a stable remnant with zero temperature and finite entropy is left in this noncommutative framework. Also, we consider Parikh-Wilczek tunneling mechanism of massive particles from non-commutative MOG black holes and demonstrate that information leaks out of non-commutative MOG bl…
The Quasinormal modes (QNMs) for gravitational and electromagnetic perturbations are calculated in a Scalar-Tensor-Vector (Modified Gravity) spacetime, which was initially proposed to obtain correct dynamics of galaxies and galaxy clusters without the need for dark matter. It is found that for the increasing model parameter $α$, both the real and imaginary parts of the QNMs decrease compared to those for a standard Schwarzschild black hole. On the other hand, when taking into account the $1/(1+α)$ mass re-scaling factor present in MOG, Im($ω$) matches almost identically that of GR, while Re($ω…
In the work, we have presented detailed analyses of the event horizon and curvature properties of spacetime around a regular black hole in modified gravity so-called regular MOG black hole. The motion of neutral, electrically charged and magnetized particles with magnetic dipole moment, in the close environment of the regular MOG black hole immersed in an external asymptotically uniform magnetic field has been also explored. Through the study on the motion of the neutral test particle, we have obtained that the positive (negative) values of the MOG parameter mimic the spin of a rotating Kerr b…
The field equations for Scalar-Tensor-Vector-Gravity (STVG) or modified gravity (MOG) have a static, spherically symmetric black hole solution determined by the mass $M$ with two horizons. The strength of the gravitational constant is $G=G_N(1+α)$ where $α$ is a parameter. A regular singularity-free MOG solution is derived using a nonlinear field dynamics for the repulsive gravitational field component and a reasonable physical energy-momentum tensor. The Kruskal-Szekeres completion of the MOG black hole solution is obtained. The Kerr-MOG black hole solution is determined by the mass $M$, the …
The shadows cast by non-rotating and rotating modified gravity (MOG) black holes are determined by the two parameters mass $M$ and angular momentum $J=Ma$. The sizes of the shadows cast by the spherically symmetric static Schwarzschild-MOG and Kerr-MOG rotating black holes increase significantly as the free parameter $α$ is increased from zero. The Event Horizon Telescope (EHT) shadow image measurements can determine whether Einstein's general relativity is correct or whether it should be modified in the presence of strong gravitational fields.
Comments on Orbits of particles with magnetic dipole moment around magnetized Schwarzschild black holes: Applications to S2 star orbit, arXiv:2406.03371v2
This paper studies the deflection of charged particles in a dipole magnetic field in Schwarzschild spacetime background in the weak field approximation. To calculate the deflection angle, we use Jacobi metric and Gauss-Bonnet theorem. Since the corresponding Jacobi metric is a Finsler metric of Randers type, we use both the osculating Riemannian metric method and generalized Jacobi metric method. The deflection angle up to fourth order is obtained and the effect of the magnetic field is discussed. It is found that the magnetic dipole will increase (or decrease) the deflection angle of a positi…
We calculate the harmonic-gauge even l=1 mode of the linear metric perturbation (MP) produced by a particle in a weak-field circular orbit around a Schwarzschild black hole (BH). We focus on the Newtonian limit, i.e. the limit in which the mass M of the central BH approaches zero (while fixing the orbital radius and the small-object mass), and obtain explicit expressions for the MP in this limit. We find that the MP are anomalous in this limit, namely, they do not approach their standard, Coulomb-like, flat-space values. Instead, the MP grows on approaching the BH, and this growth becomes wors…
A conformal coupling of the metric in the Jordan frame to the energy-momentum tensor, screens the scalar field gravitational coupling strength $G$ in modified gravity (MOG). The scalar field acquires a mass which depends on the local matter density: the scalar field particle is massive for the Sun and earth, where the density is high compared to low density environments in cosmology and astrophysics. Together with the screening of the vector field $ϕ_μ$, this guarantees that solar system tests of gravity are satisfied. The conformal metric is coupled to the electromagnetic matter field and ene…
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 $a/M\simeq0.93$…
The characteristic features were found in the electromagnetic spectrum of radiation from free falling dipole, when it is fall radially into a Schwarzschild black hole. These features can be used as another method for the black hole mass determinating. Also, these features can be used for the determination some characteristics of the magnetosphere or the accretion disk around the black hole.
Formal solutions of any-order mass, angular-momentum, dipole perturbations on the Schwarzschild background spacetime are derived in a gauge-invariant manner. Once we accept the proposal in [K. Nakamura, Class. Quantum Grav. {\bf 38} (2021), 145010.], we can extend the gauge-invariant linear perturbation theory on the Schwarzschild background spacetime including the monopole ($l=0$) and dipole ($l=1$) modes to any-order perturbations of the same background spacetime through the arguments in [K. Nakamura, Class. Quantum Grav. {\bf 31} (2014), 135013.]. As a result of this resolution, we reached …
We find a new class of exact solutions of the five-dimensional Einstein equations whose corresponding four-dimensional spacetime possesses a Schwarzschild-like behavior. The electromagnetic potential depends on a harmonic function and can be choosen to be of a monopole, dipole, etc. field. The solutions are asymptotically flat and for vanishing magnetic field the four metrics are of the Schwarzschild solution. The spacetime is singular in $r=2m$ for higher multipole moments, but regular for monopoles or vanishing magnetic fields in this point. The scalar field posseses a singular behavior. #(P…
The discontinuities of electromagnetic test fields generated by general layers of electric and magnetic monopoles and dipoles are investigated in general curved spacetimes. The equivalence of electric currents and magnetic dipoles is discussed. The results are used to describe exact "Schwarzschild disk" solutions endowed with such sources. The resulting distributions of charge and dipole densities on the disks are corroborated using the membrane paradigm.
In the Introduction we briefly recall our previous results on stationary electromagnetic fields on black-hole backgrounds and the use of spin-weighted spherical harmonics. We then discuss static electric and magnetic test fields in a Schwarzschild background using some of these results. As sources we do not consider point charges or current loops like in previous works, rather, we analyze spherical shells with smooth electric or magnetic charge distributions as well as electric or magnetic dipole distributions depending on both angular coordinates. Particular attention is paid to the discontin…
In this work we study the effects of non-minimal coupling between electromagnetism and gravity, which are motivated by quantum effects such as vacuum polarization. We investigate the modification of both asymptotically dipole and uniform magnetic fields around a Schwarzschild black hole that come as the result of non-minimal coupling. While in both cases the magnetic field gets enhanced or suppressed with respect to the case of minimal coupling, depending on the sign of non-minimal coupling parameter, in the case of a background uniform magnetic field the direction of the magnetic field also a…
The Kerr solution is generated from the Schwarzschild solution by a simple combination of real global coordinate transformations and of invariance transformations acting on the space of stationary solutions of the Einstein-Maxwell equations. The same transformation can be used to generate a spinning field configuration from any static axisymmetric configuration. We illustrate this by generating from the continuous family of Voorhees--Zipoy vacuum solutions a family of solutions endowed with mass, angular momentum, dipole magnetic moment and quadrupole electric moment.
From Størmer to Schwarzschild: Analytical Dynamics of Charged Particles in a Dipole Magnetosphere
Instituto de Astrofísica e Ciências do Espaço（太空天文与科学研究所）；Faculdade de Ciências da Universidade de Lisboa（里斯本大学理学院）；Departamento de Física, Faculdade de Ciências da Universidade de Lisboa（里斯本大学理学院物理系）；Babeş-Bolyai University（巴比什-博尧伊大学）；Astronomical Observatory（天文台）
We develop an approach to calculate the self-force on a charged particle held in place in a curved spacetime, in which the particle is attached to a massless string and the force is measured by the string's tension. The calculation is based on the Weyl class of static and axially symmetric spacetimes, and the presence of the string is manifested by a conical singularity; the tension is proportional to the angular deficit. A remarkable and appealing aspect of this approach is that the calculation of the self-force requires no renormalization of the particle's field. This is in contract with tra…
We calculate the unregularized monopole and dipole contributions to the self-force acting on a particle of small mass in a circular orbit around a Schwarzschild black hole. From a self-force point of view, these non-radiating modes are as important as the radiating modes with l greater than 2. In fact, we demonstrate how the dipole self-force contributes to the dynamics even at the Newtonian level. The self-acceleration of a particle is an inherently gauge-dependent concept, but the Lorenz gauge is often preferred because of its hyperbolic wave operator. Our results are in the Lorenz gauge and…
Corrected thermodynamics and radiation predictions of modified black bounce compact objects
Kimyo International University in Tashkent（塔什干化学国际大学）；Tashkent State Technical University（塔什干国立技术大学）；School of Physics, Harbin Institute of Technology（哈尔滨工业大学物理学院）；University of Tashkent for Applied Sciences（塔什干应用科学大学）；Farabi University（法拉比大学）；Fesenkov Astrophysical Institute（费森柯夫天体物理研究所）；Center for Theoretical Physics, Khazar University（哈扎尔大学理论物理中心）；Chitkara University Institute of Engineering and Technology, Chitkara University（奇特拉卡大学工程与技术学院，奇特拉卡大学）；National University of Uzbekistan（乌兹别克斯坦国立大学）；Department of Physics, Jizzakh State Pedagogical University（吉扎克国立师范学院物理系）；Tashkent State Medical University（塔什干国立医科大学）
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 …
The electromagnetic bremsstrahlung spectrum for the dipole which falling by a spiral orbit into the Schwarzschild black hole was found. The characteristic features in this electromagnetic spectrum can be used for determine of the black hole mass by the new way. This new way (if implemented) provides higher accuracy in determining of the black hole mass. Also these features in the spectrum can be used for determine of the certain characteristics in the black hole magnetosphere or in the accretion disk characteristics around the black hole. It is also shown that the asymptotic behavior of this s…
The consequences of the vacuum polarization effect in magnetic fields around a Schwarzschild Black Hole on the motion of charged particles are investigated in this work. Using the weak electromagnetic field approximation, we discuss the non-minimal coupling between magnetic fields and gravity caused by the vacuum polarization and study the equations of motion for the case of a magnetic field configuration which asymptotically approaches a dipole magnetic field. It is shown that the presence of non-minimal coupling can significantly influence the motion of charged particles around Black Holes. …
The linear perturbation of the Kerr black hole has been discussed by using the Newman--Penrose and the perturbed Weyl scalars, $ψ_0$ and $ψ_4$ can be obtained from the Teukolsky equation. In order to obtain the other Weyl scalars and the perturbed metric, a formalism was proposed by Chrzanowski and by Cohen and Kegeles (CCK) to construct these quantities in a radiation gauge via the Hertz potential. As a simple example of the construction of the perturbed gravitational field with this formalism, we consider the gravitational field produced by a rotating circular ring around a Schwarzschild bla…
The Mathisson-Papapetrou-Dixon (MPD) equations describe the motion of an extended test body in general relativity. This system of equations, though, is underdetermined and has to be accompanied by constraining supplementary conditions, even in its simplest version, which is the pole-dipole approximation corresponding to a spinning test body. In particular, imposing a spin supplementary condition (SSC) fixes the center of the mass of the spinning body, i.e. the centroid of the body. In the present study, we examine whether characteristic features of the centroid of a spinning test body, moving …
Based on the idea that the vacuum fluctuations of electromagnetic fields can induce instantaneous correlated dipoles, we study the far-zone Casimir-Polder potential between two atoms in the Boulware, Unruh and Hartle-Hawking vacua outside a Schwarzschild black hole. We show that, at spatial infinity, the Casimir-Polder potential in the Boulware vacuum is similar to that in the Minkowski vacuum in flat spacetime with a behavior of $R^{-7}$, so is in the Unruh vacuum as a result of the backscattering of the Hawking radiation from the black hole off the spacetime curvature. However, the interatom…
The free fall of electric charges and dipoles, radial and freely falling into the Schwarzschild black hole event horizon, was considered. Inverse effect of electromagnetic fields on the black hole is neglected. Dipole was considered as a point particle, so the deformation associated with exposure by tidal forces are neglected. According to the theorem, "the lack of hair" of black holes, multipole magnetic fields must be fully emitted by multipole fall into a black hole. The spectrum of electromagnetic radiation power for these multipoles (monopole and dipole) was found. Differences were found …
An analytical solution of Einstein-Maxwell equations with a static fluid as a source is presented. The spacetime is represented by the axially symmetric Weyl metric and the energy-momentum tensor describes a coupling of a fluid with an electromagnetic field. When appropriate limits are performed we recover the well-known solutions of Gutsunaev-Manko and Schwarzschild. Also, using Eckart's thermodynamics, we calculated the temperature, the mechanical pressure, the charge density and the energy density of the system. The analysis of thermodynamic quantities suggests that the solution can be used…
In the present work, our main objective is to investigate the orbits of spinning test particles around a Schwarzschild black hole under the influence of a quintessence matter field (SQBH). We begin with the dynamics of the spinning test particles around SQBH which is governed by the Mathisson-Papapetrou-Dixon (MPD) equations under the pole-dipole approximation, where the gravitational field and the higher multipoles of the particle are neglected. Depending on the types of saddle points,the effective potential are classified and the possibility of chaotic orbits is discussed. The inner most sta…
Spinning particle dynamics, epicyclic frequencies, and transient QPO signatures in Schwarzschild spacetime
School of Physics, Harbin Institute of Technology（哈尔滨工业大学物理学院）；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所）；Tashkent State Technical University（塔什干国立技术大学）；Physics Department, Faculty of Arts and Sciences, Eastern Mediterranean University（东地中海大学文理学院物理系）；Physics Department, School of Foundational Studies and Education, Mapúa University（马普阿大学基础研究与教育学院物理系）；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）
Linearized perturbations of a Schwarzschild black hole are described, for each angular momentum $\ell$, by the well-studied discrete quasinormal modes (QNMs), and in addition a continuum. The latter is characterized by a cut strength $q(γ>0)$ for frequencies $ω= -iγ$. We show that: (a) $q(γ\downarrow0) \propto γ$, (b) $q(Γ) = 0$ at $Γ= (\ell+2)!/[6(\ell-2)!]$, and (c) $q(γ)$ oscillates with period $\sim 1$ ($2M\equiv1$). For $\ell=2$, a pair of QNMs are found beyond the cut on the unphysical sheet very close to $Γ$, leading to a large dipole in the Green's function_near_ $Γ$. For a source near…
This work is devoted to study the effects of Einstein-Æther gravity on the dynamics of magnetized particles orbiting a static, spherically symmetric and uncharged black hole immersed in an external asymptotically uniform magnetic field in both comoving and proper observers frames. The analysis is carried out by varying the free parameters $c_{13}$ and $c_{14}$ of the Einstein-Æther theory and noticing their impacts on the particle trajectories, radii of the innermost stable circular orbits (ISCOs), and the amount of center-of-mass energy produced as a result of the collision. The strength of t…
We find an exact solution on the brane for static black hole in the DGP model. In the appropriate limit we recover the 2 known solutions, the Schwarzschild and the Reissner-Nordstrom solutions with tidal charge. The solution has 2 branches, which correspond asymptotically to de-Sitter or flat Universe. Also we study the linear stability of the solutions. We find that the Regge-Wheeler and the Zerilli potential are positive. Finally the dipole perturbation is derived which corresponds to a linearization of a rotating black hole solution on the brane.
