Charged particle motion around non-singular black holes in conformal gravity in the presence of external magnetic field

We consider electromagnetic fields and charged particle dynamics around non-singular black holes in conformal gravity immersed in an external, asymptotically uniform magnetic field. First, we obtain analytic solutions of the electromagnetic field equation around rotating non-singular black holes in conformal gravity. We show that the radial components of the electric and magnetic fields increase with the increase of the parameters $L$ and $N$ of the black hole solution. Second, we study the dynamics of charged particles. We show that the increase of the values of the parameters $L$ and $N$ and of magnetic field causes a decrease in the radius of the innermost stable circular orbits (ISCO) and the magnetic coupling parameter can mimic the effect of conformal gravity giving the same ISCO radius up to $\omega_{\rm B}\leq 0.07$ when $N=3$.

One of the examples to resolve the singularity problem has been proposed in Refs. [19], where, within a large class of conformally invariant theories of gravity, singularity-free black hole solutions have been proposed. The proposed spacetimes are geodetically complete and the curvature invariants do not diverge at r = 0.
Ref. [24] has been devoted to study the electromagnetic field around compact starts in conformal gravity. Using X-ray observations of supermassive black holes (SMBHs), it was proposed a test of conformal gravity in [25]. The energy condition and scalar perturbations of the spacetime in conformal gravity have been studied in [26,27]. Recently, magnetized particle motion around black holes in conformal gravity in the presence of external magnetic * Electronic address: nbakhtiyor18@fudan.edu.cn † Electronic address: javlon@astrin.uz ‡ Electronic address: ahmadjon@astrin.uz § Electronic address: bambi@fudan.edu.cn fields has been studied in [28]. In this work, we plan to investigate charged particle motion acceleration around singularity-free black holes immersed in an external magnetic field. A black hole itself cannot have its own magnetic field. However the curved spacetime will change the structure of electromagnetic field surrounding the black hole. The pioneer work of Wald has been devoted to study the electromagnetic field around a Kerr black hole immersed in an external, asymptotically uniform magnetic field [29]. A number of works have been devoted to study the electromagnetic field and charged particle motion around compact objects in external magnetic fields .
The energetic process around black holes can be used to model the observational features of astrophysical objects (Relativistic jets, Soft gamma ray repeaters -SGR and etc.). Different mechanisms of energy extraction from black holes have been proposed: Penrose process [55], Blandford-Znajeck mechanism [56], Magnetic Penrose process [57][58][59][60], and particle acceleration mechanism (BSW) [61]. Particularly, in Ref. [61] it was shown that the center of mass energy of colliding particles near an extreme rotating Kerr black hole may diverge for the fine tuned values of the angular momentum of the particles. Magnetic fields may play an important role in the charged particle acceleration near black holes [33,34]. Thus we are also interested to study the effects of conformal gravity and magnetic fields on the charged particle acceleration process.
In this paper, we study the dynamics of charged particles around black holes in conformal gravity in the presence of magnetic fields. The paper is organized as follows: Sect. II is devoted to study the electromagnetic field around black holes in conformal gravity immersed in an external, asymptotically uniform magnetic field. The charged particle motion around non-rotating and rotating black holes in conformal gravity in the presence of an external magnetic field has been studied in Sects. III and IV, respectively. In Sect. V, we study the collision of charged particles near a black hole in conformal gravity. We summarize our results in Sect. VI. Throughout this work, we use (−, +, +, +) for the space-time signature and a system of units where G = c = 1 .

II. ELECTROMAGNETIC FIELD AROUND A SINGULARITY FREE BLACK HOLE
In this section, we study the electromagnetic field around a singularity-free black hole in conformal gravity described by the line element where with where a is the spin parameter of the black hole with total mass M , and L and N are conformal parameters; that is, parameters related to the conformal rescaling. We start with considering the electromagnetic field around a singularity-free black hole immersed in an external, asymptotically uniform magnetic field aligned along the direction of the axis of symmetry of the space-time. The energy momentum tensor of the electromagnetic field is assumed to be negligibly small, does not change the spacetime metric, and is of the following order of magnitude Using a Killing vector ξ µ being responsible for the symmetry of spacetime geometry of the black hole, one may recall the equation Expressions (6) can be used to have the following equation where the Riemann curvature tensor R λαβγ can be transformed to the Ricci one in the following way For the spacetime of the singularity free rotating black hole in the conformal gravity, the right hand side of Eq. (8) can be chosen as R α γ ξ γ = η α . The Maxwell equations then can be expressed in the following form where F αβ is electromagnetic field tensor and can be selected as Here C 0 is an integration constant and the 4-vector a α is a correction to the potential due to the presence of the conformal gravity parameters responsible for the Ricci non flatness of the spacetime. The 4-vector a α can be found from the equation a α = η α . The electromagnetic potential then can be written as the sum of two contributions whereÃ α is the potential being proportional to the Killing vectors. To find the solution forÃ α , one can use the ansatz for the vector potentialÃ α of the electromagnetic field in the Lorentz gauge in the formÃ α = C 1 ξ α (t) + C 2 ξ α (ϕ) [29]. The constant C 2 = B/2, where the gravitational source is immersed in a uniform magnetic field B being parallel to its axis of rotation. The value of the remaining constant C 1 = aB can be easily calculated from the asymptotic properties of the spacetime (2) at the asymptotical infinity.
The second part a α of the total vector potential of the electromagnetic field is produced due to the contribution of conformal gravity and has the following solution a α = kBL 2 /r, 0, 0, 0 , where the expression for the constant k can be easily found from the asymptotic properties of the spacetime (2) at infinity [41,44]. However, since the effect of conformal gravity is negligibly small at large distances, one might exclude this part tending k → 0 and use the traditional contravariant expression for the 4-vector potential as in the Kerr case.
Finally, the components of the 4-vector potential A α of the electromagnetic field will take a form where, for simplicity, we take N = 1. The components of the electric and magnetic fields in the frame moving with four velocity u α read where the electromagnetic field tensor, F αβ , in terms of the four potential can be expressed as Using the the expressions above for the zero angular momentum observers (ZAMO) with the four-velocity components (u α ) ZAMO ≡ −g 00 1, 0, 0, g 03 g 00 , (u α ) ZAMO ≡ −1 −g 00 1, 0, 0, 0 , one can easily find the components of the electromagnetic field using the expressions in (16)- (18). The nonvanishing orthonormal components of the electromagnetic field measured by zero angular momentum observers with the four-velocity that has the form (19) are given by the expressions in (A1)-(A4) and linear approximation of conformal gravity parameter L 2 is presented by the expressions in (A5)-(A8) in appendix A. Fig. 1 shows the radial dependence of the components of electromagnetic fields in the cases of different values of the angle θ and conformal gravity parameter L 2 . Since the expressions in (A1)-(A4) have a complex form, one might be interesting to see the structure of the electromagnetic fields around BHs in the ZAMO frame, which is presented in Fig. 2.
In the case of slow rotation (a M ) and far distance (M/r 1), expressions (A1)-(A4) reduce to From the expressions in (20)- (23), one can see that in such limits only the components Er and Bθ are affected by conformal gravity while other components (Eθ and Br) do not include any contribution from L 2 (for the chosen approximation). The latter is especially clearly seen for Eθ in the graphs presented in Fig. 1. In the limit of flat spacetime, i.e. for M/r → 0, M a/r 2 → 0 and L 2 /r 2 → 0, expressions (A1)-(A4) give the following limiting expressions: Br = B cos θ, Bθ = B sin θ, Er = Eθ = 0, being consistent with the solutions for the homogeneous magnetic field in Newtonian limit. Figure 3 presents the profiles of magnetic fields around black holes for different values of spin and conformal parameter. One can see from the right top and bottom panels that the conformal parameter forces the parallel field line to have a dipole-like structure. Finally, bottom panels give the possibilities of the comparison of the nature of conformal and spin parameters of the black hole.

III. CHARGED PARTICLE MOTION AROUND A CONFORMAL NON-ROTATING BLACK HOLE IMMERSED IN A UNIFORM MAGNETIC FIELD
Here we study charged particle motion around a black hole in conformal gravity in the presence of an external, uniform magnetic field. The Hamilton-Jacobi equation for a test particle with mass m and charge q can be written as The solution of equation (24) can be sought in the following form where E = E/m and L = l/m are the specific energy and the specific angular momentum of the test particle, respectively. It is convenient to consider the particle motion on the equatorial plane plane, whereθ = 0 (p θ = 0). Then we can writeṙ  where the effective potential has a form and ω B = eB/2m is the magnetic coupling parameter, or the so-called cyclotron frequency, which characterizes the interaction between the charged particle and the magnetic field. The effective potential is invariant under the following transformations: where the Lorentz force acting on the charged particle is repulsive and has the same direction as the centrifugal force, and (L, ω B ) ←→ (−L, −ω B ), where the Lorentz force is attractive and has the same direction as the gravitational force. Below we analyze the effective potential (27) for positive angular momenta of the particle and either positive or negative magnetic coupling parameter. Fig. 4 shows the radial dependence of the effective potential on the equatorial plane. One can see from Fig. 4 that when ω B > 0 the effective potential is higher than in the case ω B < 0 and increases with the increase of the values of parameters L and N . It is worth to note that at large distances the effect of the magnetic field plays a more important role than the effect of conformal gravity.
a. b.

A. Stable Circular orbits
Now we will consider the innermost stable circular orbits of charged particles using following the standard conditions In fact, circular orbits can be stable when the second derivative of the effective potential with respect to both coordinates (∂ r V eff ≥ 0 and ∂ θ V eff ≥ 0) is positive and the ISCO on the equatorial plane corresponds to the zero value of this derivative ∂ r V eff = 0. The angular momentum of circular orbits can be found in the following form In order to ensure that we obtain a real solution of equation (29), we require the function under the square root to be always positive. Since the second part of the equation under the square root is always positive, it implies must be satisfied for any values of L and N . Now we will analyze the distance where L cr is always positive i.e. positions where circular motion can occurs in the equatorial plane. Fig. 5 illustrates the dependence of the minimum distance where circular orbits are allowed by the conformal parameter L for the values of the parameter N . One can see from this figure that in caseL = 0, such a distance is r = 3M and is the same as in the Schwarzschild spacetime. The distance decreases with the increase of the conformal parameter L and the decreasing rate increases as the parameter N increases. Fig. 6 shows the radial dependence of the critical value of the angular momentum for circular orbits. One can see from this figure that the value of the critical angular momentum of the charged particle increases in the presence of an external magnetic field.
The energy of the charged particle at circular orbits can be obtained substituting Eq. (29) into Eq. (27) Here we will study radial dependence of the energy in the equatorial plane, where sin θ = 1. Fig. 7 illustrates the radial profiles of the charged particle energy in circular orbits on the equatorial plane. One can see that the magnetic field increases the energy of the charged particle. One more thing is that the rate of energy increase in the case of ω B > 0 is higher than the case of ω B < 0. In Fig. 8, we present the effect of conformal gravity and magnetic field on the ISCO radius of charged particles. One can see from the figure that increasing the conformal parameters L and N and the magnetic coupling parameter decreases of the ISCO radius.

B. Charged particles trajectories
Now we will study the effects of the parameters of conformal gravity on the trajectories of charged particles.
Trajectories of positive and negative charged particles for absolute values of the magnetic coupling parameter |ω B | = 0.1 are shown in Fig. 9. All plots in this figure are taken at the value of the conformal parameter L = M and trajectories of the charged particles in red lines on first and fourth rows correspond to the values of the conformal parameters N = 1, black and blue lines correspond to the conformal parameter N = 4 and N = 8, respectively for the fixed value of the specific angular momentum L = 5M . One can see from the figures that the specific energy of the charged particles increases with increasing of the conformal parameter N . Moreover, the orbits of charged particles started at equatorial plane θ 0 = π/2 become unstable and the particle leaves the central object at higher values of N . It can be explained by the magnetic field structure around the black hole in conformal gravity which is shown in the third column of the plots in grey lines that becomes dipol like structure at the higher values of the parameter N .

C. Conformal non-rotating black hole versus Schwarzschild black hole in a uniform magnetic field
In this subsection, we consider two different cases: the motion of a charged particle around a Schwarzschild black hole and a non-rotating black hole in conformal gravity, in a magnetic field, with the same ISCO radius for the charged particle. Fig. 11 show that the impact of the conformal parameter L and of the magnetic coupling parameter ω B on the ISCO radius is the same. Fig. 12 shows the relation between the conformal and magnetic coupling parameters L and ω B , for the same ISCO radius. One can see from this figure that a magnetic field can mimic conformal gravity for the values of the parameter L = 1 and the other conformal parameter N = 1, 2, 3 at ω B ≤ 0.003068, ω B ≤ 0.021015 and ω B ≤ 0.06771, respectively. Since the spacetime of a rotating black hole in conformal gravity admits separation of variables on the equato-rial plane (θ = π/2) we will study the motion around the source described by the metric (1) using the Hamilton-Jacobi equation where the action S can be decomposed in the form as Eq. (25). Finally we obtain the equation of motion in the following form (for N = 1): where FIG. 10: As in Fig.9, but for negatively charged particles.
One may define the effective potential in the following form Radial dependence of the effective potential is presented in Fig. 14, where graphs are plotted in the case in which the values of the angular momentum L and energy E of the particle are chosen to be equal to the values of a particle moving around the innermost circular orbit; so, turning points of the lines on the graphs represent the ISCO radius.
The stability of the equatorial orbits can be checked by plotting the trajectories of charged particles for given values of the external magnetic field and the conformal parameter L as illustrated in Fig. 13 (the z axis is assumed to be parallel to the symmetry axis and the origin coincides with the centre of the gravitating object). It is clearly seen from the second row of figures that for fixed values of the parameters mentioned the trajectory remains stable.
The value of the ISCO radius can be obtained from the following standard conditions with R(r) taking the form (35). The results of equations (37)-(39) are expressed in Fig. 15, panel a for a vanishing magnetic field and a non-vanishing spin parameter and panel b for a non-vanishing magnetic field and a vanishing spin parameter. We can clearly see from the figures that in the absence of an external magnetic field but nonvanishing spin parameter the ISCO radius first slightly increases for small values of the conformal parameter and then goes down for higher values of L 2 . In the case of a vanishing spin parameter but in the presence of an external magnetic field, it always monotonically decreases with the increase of the conformal parameter L 2 .

V. CENTER-OF-MASS ENERGY OF CHARGED PARTICLES COLLISIONS
In this section, we investigate the center-of-mass energy from collisions of two charged particles near rotating magnetized black holes in conformal gravity. The general expression for the center-of-mass energy for two particles coming from infinity with masses m 1 and m 2 and fourvelocities u α 1 and u β 2 , respectively, can be found as the sum of their four-momenta [62,63] {E cm , 0, 0, 0} = m 1 u µ Square of the center-of-mass energy can be defined in (40) and we have after algebraic substitutions, we have the expression in a dimensionless form Using the expression for the four velocities of the charged particles around magnetized rotating black holes in conformal gravity and considering the collision of the particles with equal mass m 1 = m 2 and initial energy E 1 = E 2 = 1, one may get the expression for the centerof-mass energy in the following form We forces. This implies that the collision does not occur at the point where the energy disappears. The distance where the center-of-mass energy disappears increases (decreases) increasing the conformal (spin) parameter.

VI. CONCLUSION
In this work we have considered dynamics of charged particles and electromagnetic fields in the vicinity of rotating black holes in conformal gravity immersed in an external, asymptotically uniform magnetic field. The study of electromagnetic fields shows that the angular (radial) component of the magnetic field and the absolute value of the external magnetic field decrease (increases) with the increase of both parameters of conformal gravity, L and N , and the increase of the parameters of conformal gravity forces the external uniform magnetic field to have a dipole-like structure.
One may see from the studies of particle dynamics of charged particles around conformal non-rotating black holes in the presence of a magnetic field that the minimum circular orbits and ISCO radius decrease as with the increase of the conformal gravity and magnetic cou-pling parameters and in the case of rotating black holes the ISCO decreases faster. Moreover, it is shown that the particle orbits become unstable at higher values of both conformal parameter as a result of the fact that the magnetic field gets a dipole structure.
We have studied the effect of conformal gravity on the ISCO radius of charged particles around non-rotating black holes in the presence of an external magnetic field. We have shown that the conformal parameters can mimic the magnetic coupling parameters when ω B ≤ 0.003068 (ω B ≤ 0.021015, ω B ≤ 0.06771 ) at the values of conformal parameters N = 1 (N = 2, N = 3) while L ∈ (0, 1) and with increasing of the value of the conformal parameter N the mimic value of the coupling parameter ω B increases.