Particle Dynamics Around Weakly Magnetized Reissner-Nordstr\"{o}m Black Hole

Considering the geometry of Reissner-Nordstr\"{o}m (RN) black hole immersed in magnetic field we have studied the dynamics of neutral and charged particles. A collision of particles in the inner stable circular orbit is considered and the conditions for the escape of colliding particles from the vicinity of black hole are given. The trajectories of the escaping particle are discussed. Also the velocity required for this escape is calculated. It is observed that there are more than one stable regions if magnetic field is present in the accretion disk of black hole so the stability of ISCO increases in the presence of magnetic field. Effect of magnetic field on the angular motion of neutral and charged particles is observed graphically.


I. INTRODUCTION
Studying the dynamics of a particle around the gravitational source such as a black hole (BH) is important because it is responsible for understanding the geometrical structure of spacetime near the BH. Geodesics may display a rich structure and they can convey a very reliable information to understand the geometry of the BH. There are many types of geodesic motion but the circular geodesics are specially important [1,2]. The motion of test particles helps to study the gravitational fields of objects experimentally and to compare the observations with the predictions about observable effects (lightlike deflection, gravitational timedelay and perihelion shift). In the surrounding of the BH, a magnetic field is present [3], due to existence of plasma in the vicinity of the BH. The accretion disk or a charged gas cloud [4,5] is responsible for this field. The magnetic field is stronger in the vicinity of BH's event horizon yet, it does not effect the geometry of the BH but the motion of the charged particle moving around a BH is effected [6,7]. The magnetic field plays an important role in transferring sufficient energy to the surrounding particles for escaping to spatial infinity [8,9].
In literature many aspects of the particles motion in the vicinity of RN-BH have been studied, e.g. in [12] authors have studied the high energy collisions phenomenon between the particles.
Geodesics of the neutral and charged particles have been studied in [14]. In [13] authors have studied in detail the spatial regions for circular motion of charged test particles around RN-BH and naked singularities.
We consider the Reissner-Nordström (RN) BH is surrounded by an axially-symmetric magnetic field which is homogeneous at infinity. Particles in the accretion disc moves in circular orbits in the equatorial plane. Here we will focus on the timelike geodesics in the presence of magnetic field present in the vicinity of RN-BH. In this paper, collision of a neutral and a charged particle with another neutral particle is studied in the vicinity of magnetized RN-BH. We focus under what circumstances the particle can escape to infinity after collision? To evade the complication in the modeling the particle's motion around a BH under the influence of both gravitational and magnetic forces, we first consider the motion of a neutral particle in the absence of magnetic field [10,11].
The outline of the paper is as follows: In section II we describe the basics of RN-BH and then derive an expression for escape velocity of the neutral particle. In section III the equations of motion of the charged particle moving around weakly magnetized RN-BH are derived. Geodesics of the particles moving around the extremal RN-BH are plotted in section IV. Lyapunov exponent is explained in section V. In section VI the dimensionless form of the equations are given. In section the motion of particle in the equatorial plane to simplify the calculations. Throughout this work we use metric signature as (+, −, −, −) and also c = 1, G = 1.

II. ESCAPE VELOCITY FOR A NEUTRAL PARTICLE
We start with the simple case of calculating the escape velocity when the particle is neutral and magnetic field is absent. The RN-BH metric is given by where here M is the mass and Q is electric charge of the BH and we assume that M and Q, both are positive. The horizon of the RN-BH is given by the solution of the equation f (r) = 0. The roots are If M > Q the equation has two real positive roots, the larger root corresponds to the event horizon and the smaller root corresponds to the Cauchy horizon which is associated with the timelike singularity at r = 0. BH is known to be extremal BH if M = Q. Extremal BH has only one event horizon at r = M = Q. If M < Q, there is no real root of the equation f (r) = 0 and there is no event horizon. This case is known as naked singularity of RN spacetime. We will not discuss this case over here. Our main results are interpreted for the extremal RN-BH. The BH metric is invariant under time translation and rotation around symmetry axis so the Killing vectors will give the constants of motion, where ξ µ (t) = (1, 0, 0, 0) and ξ µ (φ) = (0, 0, 0, 1). The corresponding conserved quantities are the total energy E and azimuthal angular momentum L z of the moving particleṫ the calculations. Considering the planar motion of the particle i.e. for θ = π/2 the normalization condition u µ u µ = 1, givesṙ The extreme values of the effective potential are correspond to dU eff dr = 0, which are given by The ISCO is at r = 4M for extreme RN-BH (Q = M ). It occurs at r = 6M for Q = 0 as we have for the Schwarzschild black hole [11]. The point where the ISCO exists, r = 4M = r o , is the convolution point of the effective potential [15]. The energy and the azimuthal angular momentum of the particle corresponding to ISCO are respectively For extremal BH case, i.e. at Q = M , Eqs. (10) and (11) become We consider the situation when near the ISCO a particle collides with another particle which is coming from infinity initially at rest. After collision, there are three possibilities depending on the progression of the collision: (i) bounded motion (ii) captured by BH (iii) escape to infinity. For small changes in energy and angular momentum, orbit of the particle alters very slightly, while large changes result to take the particle away from the original path which may result in capture by BH or escape to infinity. After collision, particle will no longer remain in the same equatorial plane, so further discussion would be completed with respect to new equatorial plane (θ = π/2). But note tat due to spherical symmetry all equatorial planes are equivalent. For simplification of our problem we consider the case of collision when (i) the azimuthal angular momentum is invariant (ii) initial radial velocity remains same. These condition imply that only energy of the particle will change hence its motion would be determined by considering only the change in the energy. After collision particle acquires an escape velocity (v esc ) = v ⊥ in orthogonal direction of the equatorial plane as explained in [19]. The momentum and energy of the particle becomes (at the equatorial Here v esc ≡ −rθ o andθ o is the initial polar angular velocity of the particle. Further or for extremal BH where E o is defined in Eq. (12). It is clear that these values of angular momentum and energy are greater than the values before collision. Also Eq. (16) shows that as r → ∞, E new → E o = 1. So for E ≡ E new ≥ 1 particle will have unbounded motion. In other words the particle can not escape to infinity if E new < 1. Hence for escape to infinity it is required that The last expression for velocity v esc is obtained by solving equation (16) at E new = 1.

III. CHARGED PARTICLE AROUND A RN-BH SURROUNDED BY MAGNETIC FIELD
In this section we explore how the presence of a magnetic field in the BH exterior stimulate the motion of a charged particle. For this case the Lagrangian of the moving particle of mass m and charge q modifies as where ξ µ is a Killing vector. Eq. (20) corresponds to the Maxwell equation for 4-potential A µ in Here B is the magnetic field strength defined as [11] where here ǫ µνλσ is the Levi Civita symbol, and F µν is the Maxwell tensor, defined as For the local observer in the RN geometry, The only two components of F µν will survive which are F 23 = −F 32 and F 13 = −F 31 . By using these components of F µν and u µ 0 in Eq. (22) the magnetic field expression becomes Using Euler-Lagrange equations for Eq. (19) we geṫ Note that Eq. (26) is similar to Eq. (5), obtained in the case of neutral particle and The normalization condition (u µ u µ = 1) gives where A charged particle moving in an external electromagnetic field F µν obeys the equation of motion: Using Eqs. (31) and (1), we get the dynamical equations of θ and r respectively as where γ = M r 2 − Q 2 r 3 .

IV. TIME-LIKE GEODESICS OF THE PARTICLES IN THE VICINITY OF RN-BH
Time-like Geodesics of a Neutral Particle Moving Around a RN-BH: For a particle approaching the RN-BH time-like geodesics could be obtained by using Eqs. (5) and (7) together: where positive root gives the path of the particle going away from the BH, and negative root gives the path of an ingoing particle. For the particle which is approaching the BH, coming from infinity and initially at rest, the time-like geodesics in (r, t) coordinates are shown in figure 1. It is observed that the particle goes closer to the extremal RN-BH as compared to the case when it is moving around RN-BH.
Time-like geodesics of particles approaching the magnetized RN-BH could be obtained by using Eqs. (26) and (35) together: FIG. 1: Time-like geodesics of a particle coming from rest at infinity. We chose L 2 = 9, M = 1 and E = 1.
Thick line shows the path of a particle around RN-BH with Q = 0.9, motion is bounded at r = r b . Thin line is path around an extremal BH (Q = M = 1) and there is a boundary at r = r c .
FIG. 2: Geodesics of an ingoing particle starting its motion from some finite distance boundary (r = r c ).
Thin line shows the behavior of particle moving around an Extremal BH (Q = M = 1) and thick line is for the RN-BH when Q = 0.9 < M . Particles go back before reaching the horizons r = r h and r = r outer for the both cases respectively.
where positive and negative signs give the path of the outgoing and ingoing particle. Setting E 2 = 1, Note that time-like geodesics of a neutral particle start from infinity and reach to the BH bounded by inner boundary defined as the horizon of the BH but the geodesics of a charged particle in the presence of magnetic field are not only bounded by inner boundary but there is an outer boundary on the trajectory of the particle. In the later case a particle could reach near the horizon of the BH only if it starts its motion from some finite distance, on the other hand if motion starts from infinity, then the presence of magnetic field around BH helps the particle to escape and it never reaches near the horizons, this escaping phenomenon is explained in sec VIII in details.

V. STABILITY OF CIRCULAR ORBITS
Lyapunov exponents λ are the measurements of the rate of convergence or divergence of the nearby trajectories in the phase space. It is highly sensitive to initial conditions. Greater the value of λ more the nearby trajectories diverge. Therefore we can check the stability of orbits by the λ [18]. It is given by where r o is the ISCO of the particle moving around RN-BH. With the use of Eqs.
(2) and (30) the expression for λ defined in Eq. (37) becomes We have plotted the λ in the absence and presence of magnetic field B. One can see from figure 9 that the orbits are more stable for B = 0.5 compared to the case when B = 0. The stable region for B = 0.5 is represented by line α and in this case there are two local maxima while for B = 0 there is only one. The local minima might be related to the change in behavior of the effective potential, corresponds to BH (black brane transition) as observed in [20]. In figure 10 we are comparing the Lyapunov exponent of RN-BH with Schwarzschild-BH (Q=0). Results show that the circular orbits are more stable around RN-BH compared to the Schwarzschild-BH. Hence charge of the BH increases the stability of the orbits of moving particles.

VI. DYNAMICAL EQUATIONS IN DIMENSIONLESS FORM
For the sake of convenience we can rewrite the dynamical equations of r and θ in dimensionless form by introducing the following dimensionless quantities [11]: here r d = 2M . Using the quantities defined in Eq. (39), Eqs. (30)-(32) take the form For extremal black hole case, it becomes For the particle moving in equatorial plane, Eq. (40) is obviously satisfied and Eqs. (41) and (42) become and and for extreme black hole Eq (30) for effective potential becomes for extreme black hole it become For the particle moving around RN-BH in the equatorial plane ,θ = π/2, at radius ρ o , Eqs. (46) and (48) become and for extreme black hole Again considering the ideal scenario of collision which does not change the azimuthal angular momentum of the particle except its energy i.e. E o → E, defined as for extremal BH where E o is the energy defined in Eq. (49). As already mentioned that when ρ → ∞ the energy E → 1. So for the unbound motion the energy of the particle should be E ≥ 1. Solving equation (52) at E = 1, for escape velocity of the particle, we get following expression and for extremal RN-BH For simplicity we are considering the particle to be initially in ISCO, further we discuss the behavior of the particle when it escapes to asymptotic infinity. The only parameters required for describing the motion of the particle are the parameters ℓ and b defined in term of ρ o and the energy of the particle. The expression for the parameters ℓ and b in term of ρ could be obtained by dealing with Eq. (51). The first and second derivatives of the effective potential defined in Eq. (51) are and U ′′ eff = 2bℓρ 2 (−3 + 4ρ) + 2ℓ 2 (5 + 6(−2 + ρ)ρ) + ρ 2 (3 − 4ρ + 4b 2 ρ 4 ) 2ρ 6 .
The values of ℓ and b can be found by solving simultaneously U ′ eff = 0 and U ′′ eff = 0 and are obtained as given below It can be seen from Eqs. (59) and (58) that these Eqs. have real values only when ρ ∈ (0.5, 1.5).
In section VII the parameters ℓ and b are plotted against ρ.

VII. BEHAVIOR OF EFFECTIVE POTENTIAL
In this section we discuss the behaviors of effective potentials graphically and explain the conditions on the energy of the particle required for escape to infinity or for bounded motion around In figure 5 we have plotted the effective potential as a function of ρ. This figure shows that there are two minima u min and u min1 , in the presence of magnetic field while in the absence of magnetic field there is only one minima, u min . Hence we can say that the presence of magnetic filed increases the possibility of the particle to move in a stable orbit. A comparison of effective potential of Schwarzschild-BH with that of RN-BH is established in figure 6. It is clear from figure that the maxima for the effective potential of RN-BH has greater value in comparison with the maxima of effective potential Schwarzschild-BH. Since a particle moving around BH could be captured if it has energy greater then maxima of its effective potential otherwise it will move back to infinity or may reside in some stable orbit. Therefore, we can say that the possibility for a particle to escape from the vicinity of RN-BH or to stay in some stable orbit is more as compared to its behavior while moving around Schwarzschild-BH.
In figure 7 different regions of effective potential which are linked to escape and bounded motion of the particle are shown. Here α and β are the regions which correspond to stable orbits for b = 0.5.
For b = 0 there is only one stable region represented by γ which is related to a stable orbits. Dotted line represents the minimum energy required to escape from the vicinity of BH. If the particle has energy E ≥ 1 and move toward the BH it will bounce back to infinity which is represented by κ.
In figure 8 we are comparing the effective potentials of extremal BH in the presence of magnetic field and without magnetic field. One can notice that for b = 0.5, the effective potential has two local minima which corresponds to two stable regions while for b = 0 it has only one minima. We use the notation U max and U min for unstable and stable circular orbits of the particle respectively.
Here U 1min corresponds to ISCO which coincides with ISCO of the case when b = 0 (dotted curve) and U 2min correspond to stable circular orbits which occur due to presence of magnetic field.
Therefore we can say that magnetic field contributes to increase the stability of the orbits.
In figure 9 Lyapunov exponent is plotted as a function of radial coordinate, it can be seen that there is no stable region in case of B = 0. But it can be seen that there is a stable region for B = 0.5 represented by line α and two Local maxima instead of one as in the case of B = 0. It can be seen from the figures 8 and 9 that stability of ISCO increases due to the presence of magnetic field. This is in agrement with the results of [17]. In figure 10 we are comparing the Lyapunov exponent of a charged BH with the uncharged BH. One can see that the λ of uncharged black hole sharply goes to zero as compared to charged BH. Therefore, stability is more for charged black hole.
In figure 11 we have plotted the magnetic field as a function of ρ. The magnetic field to be real, imposes a restriction on ρ that ρ ∈ (0.5, 1.5). One can notice that magnetic field decreases abruptly as particle moves away from the source (BH). We have plotted the angular momentum ℓ as a function of ρ in figure 12, it is clear from the figure that ℓ → ∞ for ρ = 0.5. Angular momentum ℓ ± for ISCO as a function of magnetic field b are plotted in figures 13 and 14. Lorentz force is repulsive for ℓ > 0 and attractive for ℓ < 0. These figures show that for b → +∞ we have ℓ → ±∞.

VIII. TRAJECTORIES OF ESCAPE VELOCITY
Escape velocity of the charged particle is plotted in figure 16, the shaded region correspond to escape of particle from ISCO to ∞ and −∞ respectively. The solid curve represents the minimum velocity which is required to escape from the ISCO. The unshaded region represents that if the value of velocity lies in this region then particle will remain in the ISCO or some other stable orbit.
In figure 15 we are comparing the escape velocity of a particle moving around the Schwarzschild-BH with the particle moving around the RN-BH. It can be seen from this figure that the difference between the velocities is large near the black hole (initially) and it become almost same as we move away from the black hole. Therefore, we can conclude that the effect of the charge of black hole on the motion of the particle is large and it is reducing as particle moves away from it. For the angular variable we have, If the left hand side of equation (60) is negative then the Lorentz force on the particle moving around the RN-BH is attractive [19]. The motion of the charged particle is in clockwise direction.
Motion is kind of oscillation in the radial direction which is deformed smoothly by magnetic field present in the vicinity of BH. Lorentz force is repulsive if Left hand side is positive. We are not going to the detail here because it is already discussed in [15,19]. Our concern is only about the effect of magnetic field on the charged particle. Therefore, magnetic field create a deformation in the oscillatory motion, so, greater the strength magnetic field greater will be the deformation.
Hence we can conclude that larger the strength of magnetic field, it is easy for a particle to escape from the ISCO. We have also plotted in figure 17 escape velocity as a function of radius for different values of magnetic field b. It can be seen that the escape velocity of the particle increases as the magnetic field strength increases and, just like magnetic field, away from the BH it becomes almost constant. As the magnetic field is strong near the BH, therefore, we can conclude that presence of magnetic field will provide more energy to particle, so that it might easily escape from the vicinity of BH. Figure 18 represents the escape velocity against radius ρ for different values of angular momentum ℓ. From figure 18 we can say that the possibility for a particle to escape having large ℓ is more in comparison with the particle with smaller value of ℓ. The behavior of escape velocity for extreme RN-BH is shown in figure 19.
The motion of particles in the RN geometry in the presence of magnetic field is investigated in this work. The problem is simplified a little by using some assumption as mentioned in sec II.
We first studied the neutral particle moving around RN-BH and derived the expressions for the energy and azimuthal angular momentum of the particle corresponding to ISCO. We obtained the expressions for escape velocity of the particle, after its collision with some other particle. Then analysis for a charged particle is done, and dynamical equations of θ and r are obtained. The time-like geodesics of a particle moving around RN-BH are plotted in figure (1) and geodesics of a charged particle in the vicinity of a magnetized RN-BH are shown in figure (2). We see that when a charged particle is moving around a magnetized RN-BH there are two boundaries on the geodesics, r = r h and r = r c unlike the former case in which a neutral particle comes from infinity and goes back to infinity before reaching the horizon, r = r h , of the BH. A comparison of the time-like geodesics around RN and extremal RN-BH is established. It is observed that when Q = M the particle goes more close to the BH as compared to the case when Q < M . We find out the condition on the energy of the particle required to escape or to remain bounded in orbit. Expressions for escape velocity of a charged particle moving around RN-BH, in the presence of magnetic field in the vicinity of BH due to plasma, are derived in this paper.
Behavior of effective potentials is studied in details and its effect on the stability of the orbits is explained graphically. More importantly a comparison of the expressions for effective potentials, obtained in the presence and absence of magnetic field, is established. It is seen that presence of magnetic field increase the stability of the orbits of the moving particles, in fact two stable region (local minima) are obtained in contrast to the only stable region obtained in the case when magnetic field is absent. Further we did comparison of Lyapunov exponent for both cases B = 0 and B = 0.5. It is noticed that existence of magnetic field in the accretion disc might increase the probability of the particle to escape from the ISCO. Effects of magnetic field on the dimensionless form of angular momentum ℓ and on escape velocity are also observed graphically. Results show that the parameter ℓ asymptotically goes to ±∞ as magnetic field strength increases.