Magnetized black hole as an accelerator of charged particle

: Astrophysical accretion processes near the black hole candidates, such as active galactic nuclei (AGN), X-ray binary (XRB), and other astrophysical sources, are associated with high- 2 energetic emission of radiation of relativistic particles and outﬂows (winds and/or jets). It is widely believed that the magnetic ﬁeld plays a very important role to explain such high energetic 4 processes in the vicinity of those astrophysical sources. In the present research note, we propose 5 that the black hole is embedded in an asymptotically uniform magnetic ﬁeld. We investigate the dynamical motion of charged particles in the vicinity of a weakly magnetized black hole. We show 7 that in the presence of the magnetic ﬁeld, the radius of the innermost stable circular orbits (ISCO) for a charged particle is located close to the black hole’s horizon. The fundamental frequencies, such as Keplerian and epicyclic frequencies of the charged particle are split into two parts due to 10 the magnetic ﬁeld, as an analog of the Zeeman effect. The orbital velocity of the charged particle 11 measured by a local observer has been computed in the presence of the external magnetic ﬁeld. 12 We also present an analytical expression for the four-acceleration of the charged particle orbiting 13 around black holes. Finally, we determine the intensity of the radiating charged accelerating 14 relativistic particle orbiting around the magnetized black hole. 15


Introduction
Observation evidence of astrophysical black holes, such supermassive black hole 18 (SMBH) and stellar black hole provides new motivation to investigate charged particle 19 dynamics around black hole in the presence of the external electromagnetic field. It is 20 generally accepted that a magnetic field is considered one of main sources of the most 21 energetic processes around supermassive black holes at the center of galaxies, playing 22 the role of "feeder" of the supermassive black hole by trapping dust near the galaxy's 23 center [1]. 24 Synchrotron radiation is, a relativistic case of cyclotron radiation, characterized by emitting photons due to the acceleration of charged particles in the external magnetic field. In a flat space, radiation from a rapidly moving charge and synchrotron radiation (magnetic bremsstahlung) from charged particle moving along circular trajectory arbitrary relativistic velocity in uniform magnetic field has been investigated in [2]. These facts provide new motivations for investigating of radiation from charged particles in the framework of general relativity (GR). It is worth notice that according to "no-hair theorem", the black hole can not possess magnetic field. However, the external magnetic field around the black hole can be generated by its accretion disc, or a sourunded rotating matter, or a companion in binary systems containing neutron star (NS) or/and magnetar with strong magnetic field. One of simple model of magnetized black hole has performed by Wald [3], and similar physical scenarios on magnetized black hole have been considered later, e.g., in [4][5][6][7][8][9][10][11][12][13][14]. According to this model, the black hole immersed in an asymptotically uniform external magnetic field that is small enough to change its spacetime grometry. In the Ref. [15], it is shown that the external magnetic field B is negligibally small than the critical magnetic field B M that can infulience the spacetime of the black hole and satisfies the following condition: According to the Ref. [16] the magnetic field strength around supermassive black hole (SMBH) is order of ∼ 10 2 G, while in the vicinity of the stellar black hole (SBH), it is about ∼ 10 4 G. The energy of the emitted photon through the cyclotron frequency around SMBH then estimated as while around SBH, it has A complete detailed analysis of interaction between black hole and magnetic field 25 generated by the accretion disc or companion object (it can be neutron star or magnetar 26 with strong magnetic field) is complicated problem which requires numerical magneto-27 hydrodynamic (MHD) simulations [17]. However, approximative methods are also very 28 useful to draw picture of this phenomenon, by considering stationary magnetized black 29 hole solutions in general relativity as was done by Wald [3] and Ernst [18].

30
The discussing of an interaction between charged particle and electromagnetic 31 fields is very interesting topic from theoretical and observational point of view. A com-32 prehensive physical aspects of the theory of black holes in an external electromagnetic 33 field are reviewed in [19][20][21][22]. In the papers [23,24] propagation of scalar field in the 34 background of strongly magnetized black hole (or Ernst spacetime) has been studied and 35 later it is considered for the massive scalar field in [25]. It is shown that in the presence 36 of the strong magnetic field the quasinormal modes are longer lived and have larger 37 oscillation frequencies in both massless and massive scalar fields [26,27]. The effect of 38 the magnetic field in optical properties of black hole has been discussed in Refs. [28,29].

39
The paper is organized as follows. In Sec. 2, we provide basic necessary equa-40 tions related to charged test particle motion around the Schwarzschild black hole in the 41 presence of the electromagnetic field. In Sec. 3, we investigate a general description to 42 derive the fundamental frequencies for charged particle orbiting around static black Sec. 5, we summarize found results and give a future outlook related to this work. 49 2. Charged particle dynamics 50 In this section, we provide equations of motion for charged particle around the black hole immersed in the uniform magnetic field. In Boyer-Lindquist coordinates x α = (t, r, θ, φ), the Schwarzschild metric is given by 1 where M is total mass of the black hole.

51
The configuration of the electromagnetic field near the black hole has explicitly shown in [3], the non-zero component of the vector potential is given by where B is the uniform magnetic field strength.

52
The dynamical motion of charged particle of the mass m and charge q is governed by the following non-geodesic equation where U α = dx α /dλ is the four-velocity of the test particle, λ is an affine parameter, Γ α µν are the Christoffel symbols and F αβ = A β,α − A α,β is the electromagnetic field tensor. The conserved quantities, namely, the specific energy E , and specific angular momentum L of charged particle measured at the infinity, can be easily found as Using the normalization of four-velocity of the test particle, one can have the following expression: and hereafter introducing the spatial components of the velocity of particle measured by a local observer with the total velocity v 2 = v 2 r + v 2 θ + v 2 φ , the expression for the specific energy of charged particle can be expressed as Note that the energy expression (10) is obviously independent of the external magnetic 53 field, however, the radius r and velocity v of charged particle depend on the external 54 magnetic field. One can easily see from the expression (10) that absence the black hole's 55 mass, i.e. M = 0, the classical expression for the energy of relativistic particle can be 56 obtained as follows, 3. Fundamental frequency of charged particle 58 Hereafter using normalization of the four-velocity of the test particle i.e. U α U α = −1, taking into account the expressions (7), one can obtain where As one can see from the expression for the potential (12) for a charged particle one needs 59 the explicit form of the vector potential, while for magnetized particle depends on the 60 components of the magnetic field which means that if we wish to consider both the 61 charged and magnetized particle in the presence of external magnetic field then we need 62 the expressions for the vector potential and components of the magnetic field.

63
It is interesting to consider the periodic motion of the charged particle orbiting around the black hole which allows determining the fundamental frequencies such as Keplerian and Larmor frequencies. The simple way of deriving the expressions for thus frequencies is to consider motion in the stable circular orbit with, U α = (U t , 0, 0, U φ ), which allows writing where Ω = dφ/dt is the angular velocity of the orbital motion measured by a distant 64 observer.

65
It is important to determine radius of the innermost stable circular orbit (ISCO) for charged particle. The ISCO radius can be easily determined from the following conditions: Considering charged particle motion in the vicinity of the Schwarzschild black in the 66 presence of the uniform magnetic field, we found that the ICSO radii for both positively 67 and negatively charged particles decrease due to the external uniform magnetic field.

68
Similarly, careful numerical analyses show that the radius of the marginally bound orbit, 69 where the energy of the particle in circular orbit will be the same as its rest energy, or 70 E = 1, for charged particle also decrease due to the effect of the external magnetic field. It is also interesting to produce the trajectories of the charged particle orbiting 80 around the magnetized black hole. As we mentioned before that charged particle moves   in the gravitational and magnetic field and motion is governed by the four second order 82 equations as shown in (6). In order to construct particle trajectory one needs eight 83 initial conditions, two of them can be eliminated using the conserved quantities, due to 84 symmetry another two condition can be written as t(0) = φ(0) = 0, while normalization 85 of the four-velocity eliminates one of conditions. Then, we only need to give initial 86 position of particle with random velocity. Figure 3 illustrates the trajectories of charged 87 particle orbiting around magnetized black hole. Now we focus on the derivation of the expression for the orbital angular frequencies, such as Keplerian, and Larmor frequencies, of the charged particle orbiting around the black hole. To do this, let us again consider motion in circular orbit with, U α = U t (1, 0, 0, Ω). In the case, from (6), equations for radial and vertical motion can be written Note that the physical meaning of the quantity Ω above equations are different and this difference can be easily shown by considering neutral particle motion i.e., q = 0. In this case the solution of the first equation in (17) becomes Ω 0 = −g tt,r /g φφ,r which represents Keplerian frequency for neutral particle. On the other hand solution of the second equation of (17) vanishes Ω = 0 for neutral particle, so that non-trivial solution can be found only for charged particle in the presence of external electromagnetic field. In order to find the explicit expressions for Keplerian and Larmor frequencies, one has to eliminate U t by inserting the expression (13) into (17), and after performing simple algebraic manipulations one can obtain

The epicyclic frequencies 90
It is also interesting to determine the epicyclic frequencies (Ω r , Ω θ ) produced by oscillatory motion of charged particle along radial and vertical direction at stable circular Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 June 2021 doi:10.20944/preprints202106.0302.v1 orbit around black hole in the presence of external magnetic field. Here we study quasiperiodic oscillation of charged particle around given stable circular orbit. Before move on further that we expand of the function V(r, θ) in the form where x 0 = (r 0 , θ 0 ) are the stationary points. Here we have used the conditions (??)-(??). Now inserting the expression (12) into (11), using the expression (20) one can obtain equation of harmonic oscillatory motion for charged particle, around the stationary orbit (r 0 , θ 0 ), for the displacement δ r = r − r 0 , δ θ = θ − θ 0 in the form: where the epicyclic frequencies in (21) can be calculated by Finally, using the equations (12)-(15) the explicit form of the epicyclic frequencies (Ω r , Ω θ ) of charged particle orbiting around black hole can be expressed as where i = (r, θ). From equation (23) one can see that the radial and vertical frequencies

91
(Ω r , Ω θ ) depend on the the background geometry, the external magnetic field and also 92 parameters of the test particle. Once background spacetime geometry and external 93 magnetic field are given then one can immediately determine Ω r and Ω θ , however 94 keep in mind that they still depend Keplerian frequency which is most important in the 95 calculation of the fundamental frequencies.

96
Using the general expressions (18), (19), and (23) for the fundamental frequencies such as Keplerian, epicyclic and Larmor frequencies of charged particle orbiting around Schwarzschild blach immersed in uniform magnetic field can be expressed as where ω B = qB/m is the cyclotron frequency for charged particle and Ω 0 = √ M/r 3 is Keplerian frequency for neutral particle in Schwarzschild space. Absence of the external uniform magnetic field, i.e. B = 0 or ω B = 0, the expressions for the fundamental frequencies take the form: Notice that angular frequencies, Ω i , is related to the frequencies, ν i , as follows Ω i = 2πν i . To have an idea about order of magnitude of these frequencies, one can estimate the orbital frequency of test particle for a Schwarzschild black hole in the form: Hz .
It is worth noting that these frequencies can be observed distance greater that the 97 ISCO position of test particle around the black hole. Figure 4 draws radial dependence

105
Now we focus on investigating of synchrotron radiation from relativistic charged particle in the vicinity of magnetized Schwarzschild black hole. According to the Ref. [2], the expression for the four-momentum loss of the accelarating test particle can be written as It is well-known that accelerating relativistic charged particle emits radiation. Now we concentrate on the radiation of the accelerating charged particle orbiting around the black hole. The radiation spectrum of the relativistic charged particle in curved spacetime can be expressed as [2] where w α is the four-acceleration of particle in a curved space defined as w α = U β ∇ β U α , on the other hand taking account non-geodesic equation (6), one can write For simplicity, we consider the motion of charged particle in the stable circular orbit with U α = U t (1, 0, 0, Ω) and to see the behavior of the radiation spectrum. Since the velocity and acceleration of particle are orthogonal to each other, i.e., w α U α ≡ 0, we can immediately express the four-acceleration of particle in the form, w α = (0, w r , w θ , 0), where the components of the acceleration can be defined as Finally, the expressions for the intensity (31) of the radiating accelerated charged particle orbiting around magnetized black hole is Similarly, one can also consider more realistic situation that the charged particle falling into black hole with the four-velocity, U α = U t (1, u, 0, ω), where u = dr/dt is radial velocity and ω = dφ/dt is angular velocity of particle. In this case, from the condition w α U α = 0, one can argue that the radial acceleration of charged particle vanishes w r = 0, however, vertical acceleration still should be exist, i.e., w θ = 0. Finally, the intensity of charged particle can be expressed as which concludes that accreting charged particle onto magnetized black hole emits the 106 electromagnetic radiation.

108
The study of black holes analyzing the observed data on the accretion disc may be 109 helpful to investigate the electromagnetic radiation in the vicinity of compact objects. In 110 the present research work, we have investigated the motion of charged particle and the 111 energetic process, namely, the fundamental frequencies and synchrotron radiation by the other, resulting in other two particles with the same spin but less charge, then both of 132 these two particles will have a smaller ISCO orbit after the instantaneous event. We 133 suspect that there must be an astrophysically observable phenomenon corresponding to 134 this interesting collisional event.

135
Finally, we investigate synchrotron radiation from the acceleration of charged parti-

143
It is also interesting to study charged particle acceleration around rotating magne-