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Dynamics of a Charged Particle Around a Slowly Rotating Kerr Black Hole Immersed in Magnetic Field

The dynamics of a charged particle moving around a slowly rotating Kerr black hole in the presence of an external magnetic field is investigated. We are interested to explore the conditions under which the charged particle can escape from the gravitational field of the black hole after colliding with another particle. The escape velocity of the charged particle in the innermost stable circular orbit is calculated. The effective potential and escape velocity of the charged particle with angular momentum in the presence of magnetic field is analyzed. This work serves as an extension of a preceding paper dealing with the Schwarzschild black hole [Zahrani {\it et al}, Phys. Rev. D 87, 084043 (2013)].


I. INTRODUCTION
The motion of a particle around a black hole is among the important problems of black hole astrophysics. It helps in understanding the geometrical structure of spacetime near the black hole.
There are observational and theoretical evidences that magnetic field is present in the nearby surrounding of the black hole [1]. The origin of this magnetic field is the probable existence of plasma in the vicinity of the black hole such as the accretion disk or a charged gas cloud [2,3].
The relativistic motion of particles in the conducting matter in the accretion disk can generate the regular magnetic field inside the disk. Therefore near the event horizon of the black hole, there exists much strong magnetic field. This field does not effect the geometry of the black hole but it does effect the motion of the charged particle moving around it [4,5]. More importantly, a rotating black hole may provide sufficient energy to the particle due to which it may escape to spatial infinity. This physical effect appears to play crucial role in the ejection of high energy particles from accretion disks around black holes. The main role is played by magnetic field, along with black hole's rotation in the transfer of energy to the particle [6,7]. Other interesting processes around black holes include evaporation and phantom energy accretion onto black holes [8].
Modeling of the charged particle's motion around a magnetized black hole is a complicated problem as during its motion around the black hole, particle is under the influence of both gravitational and magnetic forces [9,10]. In the present paper we consider a slowly rotating Kerr black hole which is surrounded by a axially-symmetric magnetic field. Magnetic field is homogeneous at infinity.
Same problem was studied for weakly charged rotating black holes in [11]. The effect of magnetic field on innermost stable circular orbit (ISCO) is studied in [12] and their main conclusion is that, if the magnetic field is present then the (ISCO) is located closer to the black hole horizon. The effect of the black hole rotation on the motion of neutral particle which is moving around it is same as the effect of the magnetic field on the motion of the charged particle which is discussed in [13,14].
In this paper first we consider a charged particle moving around magnetized black hole in equatorial plane which collide with another particle. We focus under what circumstances the particle can escape to infinity. Magnetic field is homogeneous far from the black hole and gravitational field is ignorable. Therefore, far from the black hole charged particle moves in a homogeneous magnetic field. This is the simple case. If the magnetic field is not consider then the equations of motion are simple a little. This simplicity to solve equations of motion is also their for the motion of the particle around a black hole immersed in magnetic field, provided the particle moves in the equatorial plan perpendicular to the magnetic field. However, before reaching the region where the magnetic field is homogeneous and gravitational field is ignorable, particle passes through a region where both fields effect its motion. At that time particle's motion is totally unpredictable (chaotic) and we can not solve the equations. When a particle moving in a non uniform magnetic field in the absence of black hole its motion is also unpredictable [15,16]. We are extending a previous work [10] by choosing the slowly rotating Kerr Black hole. We calculate the escape velocity of a particle needed to escape from the vicinity of black hole to infinity and investigated its characteristics.
The outline of the paper is as follows: In section II we explain our model and derive an expression for escape velocity of the neutral particle. In section III we derive the equations of motion of the charged particle moving around slowly rotating weakly magnetized Kerr black hole. In section IV we give the dimensionless form of the equations. Trajectories for escape energy and escape velocity of the particle are discussed in section V and VI respectively and their graphs are given in the appendix. Summery and conclusion are presented in section VII. Throughout we use sign convention (+, −, −, −) and units where 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 Kerr metric is given by [17] where m is the mass and a is the spin of the black hole and interpreted as the angular momentum per unit mass of black hole a = L m . The Kerr metric diverges when From the above equation we get two values of r: ∆ > 0 for r + < r < r − and ∆ < 0 for r > r − and r < r + . The region r = r + represents the event horizon. Further r = 0 and θ = π 2 is the location of a curvature singularity in the Kerr metric. Due to computational and analytical problems in dealing with the analysis for Kerr metric, we consider the slowly rotating black hole and neglect the terms involving a 2 , then equation (1) becomes Here r g = 2m is the gravitational radius of a slowly rotating Kerr black hole just like Schwarzschild black hole (Note that for a slowly rotating Kerr and Schwarzschild black hole the horizon occur at r = r g ). The black hole metric has the following symmetries These symmetries imply the following transformation [10] There are three constants of motion in which two of them arise as a result of two Killing vectors The black hole metric is invariant under time translation and rotation around symmetry axis. The corresponding conserved quantities are the energy E and azimuthal angular momentum Throughout in this paper the over dot represents the differentiation with respect to proper time τ . The third integral of motion is the sum of the square of total angular momentum of the particle and the black hole [11] Here we denote v ⊥ ≡ −rθ o . Using the normalization condition u µ u µ = 1, we geṫ In the above equations (8), (9) and (11) the upper sign represents the co-rotation (direction of rotation of particle and black hole is same) and the lower sign represents the contra-rotation (direction of rotation of particle and black hole is opposite) and we are considering here only the upper signs.
At the turning points (ṙ = 0), the equation (11) becomes quadratic in E whose solution is which gives E = V eff , effective potential.
If we solve equation (11) without putting (ṙ = 0) then we get the same expression (12) with an additional term in the square root In case of a Schwarzschild black hole we discard the negative values of the energy by the condition that the energy of the particle is positive in the exterior region of black hole. However the negative energies are allowed in Kerr geometry because in case of Kerr black hole if r 2 << L 2 z (here L z is the angular momentum per unit mass) this condition can always be satisfied by taking the mass of the particle to be small. Inside static limit surface which is defined as (r = r st = 2m = r g ) the term r 4 L 2 always contributes less as compared to r 2 g r 2 L 2 , (as inside static limit surface r g > r) and hence its square root always contributes less than r g a|L|r to E. Thus If L < 0 then E < 0. At the static limit surface of Kerr black hole in the equatorial plane r = r st = 2m = r g , there are orbits inside r st in which L < 0 then energy is also negative E < 0, or we can say that within the static limit surface there are retrograde orbits which have negative energy. It means that energy required to remove a particle from its orbit to infinity is greater than its rest mass.
Consider a particle at the circular orbit r = r o , where r is the local minima of the effective potential. This orbit exists for r o ∈ (3r g , ∞). The corresponding energy and azimuthal angular momentum are The ISCO is defined by r o = 3r g , which is the convolution point of the effective potential [25].
Now consider the particle is in a ISCO and collides with another particle, so after the collision it will move within a new plane with respect to the new equatorial plane. After collision between particles, three cases are possible for the particles: (i) bounded motion (ii) captured by black hole (iii) escape to infinity. The result depends on the collision process. For small change in energy and momentum, orbit of the particle is slightly perturbed. While for large change in energy and momentum, it can go away from initial path and captured by black hole or escape to infinity.
After the collision particle should have new values of energy and momentum E, L z and the total angular momentum L 2 . We simplify the problem by applying the following conditions (i) the azimuthal angular momentum is fixed (ii) initial radial velocity remains same after the collision (ṙ = 0). Under these condition only energy can change by which we can determine the motion of the particle. After collision particle acquires an escape velocity (v ⊥ ) in orthogonal direction of the equatorial plane as explained in [1].
After the collision the momentum and energy of the particle from equations (12) and (10) becomes (at the equatorial plan θ = π 2 ) These values of momentum and energy are greater then the values of momentum and energy before collision.
We can see from the equation (18) that as r → ∞, E new → 1. So for the unbound motion E new ≥ 1. Physically it means that the energy of the particle exceeds its rest mass energy. Therefore, all the orbits with E new ≥ 1 are unbounded in the sense that particle escape to infinity. Conversely for E new < 1, particle cannot escape to infinity (the orbits are always bounded).
Therefore particle escapes to infinity if We get the above expression for velocity v from equation (18) by putting E = 1 and then solve it for v. Here we put E = 1 because as r → ∞ energy E → 1, this is the minimum energy for the particle to escape from the vicinity of black hole. Particle escape condition is |v| ≥ v ⊥ .

III. CHARGED PARTICLE AROUND THE SLOWLY ROTATING KERR BLACK HOLE
We investigate the motion of a charged particle q (electric charge) is in the presence of magnetic field in the exterior of the black hole. The Killing vector equation is [22] ξ µ ;ν where ξ µ is a Killing vector. Eq. (20) where B is the magnetic field strength. The 4-potential is invariant under the symmetries which correspond to the Killing vectors, i . e., A magnetic field vector is defined as [10] where In equation (24) ǫ µνλσ is the Levi Civita symbol and the Maxwell tensor is defined as For a local observer at rest we have (8) and the other two components u µ 1 = 0, u µ 2 = 0 at the turning point (ṙ = 0). From equations (23) − (26) we can obtain the components of magnetic field The Lagrangian of the particle of mass m and charge q moving in an external magnetic field of a curved space-time is given by [23] and generalized 4-momentum of the particle P µ = mu µ + qA µ and the new constants of motion Here we denote Using these constant of motion in the Lagrangian we get the dynamical equations of θ and r respectively as Following the procedure of section II, we get the expression of effective potential If the equation (34) is satisfied initially (at the time of collision), then it is always valid (throughout the motion), provided that θ(τ ) and r(τ ) are controlled by equation (32) Therefore, without losing the generality, we consider the particle of positive electric charge (and hence B > 0). To consider the particle of negative charge one should apply the transformation (35). So, the trajectory of a negatively charge particle is related to positive charge's trajectory by transformation (35). If we make a choice B > 0 then we will have to study both cases when L z > 0, L z < 0. They are physically different: the change of sign of L z means the change of direction of the Lorentz force on the particle.
The system of equations (28) − (34) is invariant with respect to reflection (θ → π − θ). This transformation retain the initial position of the particle and changes So, that is why it sufficient to consider only the positive value of (v ⊥ ).

IV. DIMENSIONLESS FORM OF THE DYNAMICAL EQUATIONS
We have to integrate our dynamical equations of r and θ numerically. First we make these equation dimensionless by introducing the following dimensionless quantities [11]: Eqs. (32), (33) and (34) take the form +2aE(−1 + 4b + cos 2θ) sin θ + 4bqρ 4 sin 2θ and The energy of the particle moving around the Kerr black hole of radius ρ o at the equatorial plane is azimuthal angular momentum of the particle but it change the transverse velocity v ⊥ > 0. Due to this, the particle energy changes E o → E and is given by the equation We explained before as theρ → ∞ then the energy E → 1. For the unbound motion the energy of the particle should be E ≥ 1. By solving equation (41) and putting E = 1, we get four values of escape velocity of the particle as given below We now discuss the be behavior of the particle when it escapes to asymptotic infinity. For simplicity we consider the particle initially in ISCO. The parameter ℓ and b defined in term of ρ o and only one parameter (the energy of the particle) specify the motion of the particle. We can express the parameters ℓ and b in term of ρ o by calculating the first and second derivative equation (40). The first derivative of the effective potential of equation (40) is and the second derivative is Here k 1 and k 2 are defined as we cannot solve these two equations to get explicit expression for ℓ and b. We can determine the range of ℓ from the escape velocity formula, for this range of ℓ the escape velocity is real by fixing the other parameter b and a. We have done this as we have four expression for escape velocity (two positive and two negative) and all the 4 expressions for escape velocity are different so they also have different range for ℓ. We have mentioned the range of ℓ for every expression of escape velocity in section VI and also plot the trajectory for escape velocity against the parameter (ℓ, ρ) and b which are given in the appendix.

V. TRAJECTORIES FOR ESCAPE ENERGY
We plot only the positive energy of the particle E + . Here we cannot consider the negative value of energy because it only exists within the static limit surface (r st = 2m). There are retrograde orbits which have negative energy. Here we ignore the (a 2 ) term so, we have only one horizon (r = 2m = r g ). That is way there is no orbit inside static limit surface and we cannot consider the negative energy. For the rotational (angular) variable If (ℓ < 0) and (ℓ + b > E) then the right hand side of equation (47) is always negative. For ℓ < 0 the Lorentz force on the charged particle is attractive (charged particle rotates around the black hole in clockwise direction). This motion is like the oscillation in the radial direction. This motion is very similar to the bounded motion of the test particles rotating near the Schwarzschild black hole as explained in [25]. Presence of magnetic field might shift the trajectory of the particle away from the black hole. So it is easy to escape when the Lorentz force is attractive. If (ℓ > 0) and (ℓ + E > b) then right hand side of equation (47) is always positive. For (ℓ > 0) the Lorentz force on the particle is repulsive means the rotation of the charged particle is in anti-clock wise direction. The repulsive Lorentz force and the magnetic field might shift the orbit of the particle very close to innermost stable circular orbit as discussed in [11,27]. All the figures (1-6) correspond to equation (40) and given in the appendix. In figure-1 the shaded region corresponds to escape value of energy (unbound motion) and the region below the line or not shaded corresponds to the bounded trajectories of the particle. The curved line represents the minimum energy required to escape form the vicinity of black hole for the particle. In figure-2 we have plotted effective potential and ρ as we increase the value of angular momentum the square root term of the equation (40) is dominating and plot is like Schwarzschild plot for effective potential as given in [13]. Here we plot the effective potential against ρ (ℓ = 10), (b = .5), (a = .5). Here (E max ) corresponds to stable circular orbit and (E min ) refers to unstable circular orbit.
We can see from the figures 3 and 4 that the energy have same trajectories for both ℓ + and (ℓ − ). As we have explained before that the expression of the energy become (E → 1) as (r → ∞) so , for unbound motion or for particle to escape the energy should be (E ≥ 1). Figures 5 and 6 shows how the energy changes as magnetic field and angular momentum is changing. For figure 5 we have plotted b and E. It is shown by the figure 5 as the magnetic field increases the energy also increases. So stronger the magnetic field means the particle have higher escape energy by that it can easily escape. Figure 6 shows the variation in energy with the change in angular momentum ℓ i. e. initially it is decreasing then it is increasing. So initially the Lorentz force on the particle is not enough that it may escape from the black hole vicinity as the Lorentz force increases on the particle the angular momentum (ℓ) also increases and particle will escape to infinity.  figure 7, the shaded region corresponds to escape velocity of the particle and the solid curved line represents the minimum velocity required to escape from the vicinity of the black hole to +∞ and the unshaded region is for bounded motion around the black hole. In figure 8 the shaded region corresponds to escape velocity of the particle and the solid curved line represents the minimum velocity required to escape from the vicinity of the black hole to −∞ and the unshaded region is for bounded motion around the black hole.  figure 11 we shows how the escape velocity is changed with the change in magnetic field. It can be seen initially it is increasing exponentially near the source (black hole surrounding by conducting matter (plasma)) because the magnetic field is strong near the black hole and then it increase linearly until it become constant because the magnetic field is become homogeneous as we move away from the black hole (as we have explained before).

VI. TRAJECTORIES FOR ESCAPE VELOCITY
So, magnetic field provides sufficient amount of energy to particle to escape from the black hole's vicinity. Figure 12 shows the variation in escape velocity v + by the change of angular momentum is parabolic. Initially as the angular momentum is zero the particle is at the convolution point either it goes to clockwise direction or in anti-clockwise direction around the black hole. Here in figure 15 the velocity increases as the magnetic field increases linearly. Figures 16-19 corresponds to escape velocity as given by equation (43) for the negative value of it. Figures 16 and 17 represents the same trajectories of the particle escape velocity for ℓ + and ℓ − .
The range for angular momentum for which the escape velocity to be real is −4.291 ≤ ℓ ≤ 4.291. Figure 18 shows that as the magnetic field is increasing the escape velocity also increases. Figure   19 shows the variation by the change of angular momentum. The trajectories of the escape velocity is parabolic. Initially as the angular momentum is zero the particle is at the convolution point either it goes to clockwise direction or in anti-clockwise direction around the black hole. The range for angular momentum for which the escape velocity to be real is −1.605 ≤ ℓ ≤ 1.605.
It can be seen from figure 22 escape velocity is increasing as magnetic field increases.

VII. SUMMERY AND CONCLUSION
The study of motion of charged particle moving around the Kerr black hole is complicated.
Here we considered a particle under the influence of both gravitational and magnetic force and we simplify it by using some assumptions explained below. In this paper we studied the motion of the charged particle in a space-time of slowly rotating magnetized Kerr black hole. We have obtained equations of motion by using Lagrangian formalism and expression for magnetic field also derived in section III . It is assumed that the particle is initially in the ISCO. We have discussed the conditions for the energy of the particle that how it change when particle collide with any other particle during their motion in the ISCO. It is studied what is the minimum energy for a particle to escape or its motion remain bounded around a black hole. By taking above mentioned assumptions our problem is simplified a little and we have calculated the expression for escape velocity of the particle moving around the black hole. We have discussed the approximations that when our results reduced to the case of Schwarzschild's black hole as studied in [10]. We have also discussed the behaviour of energy and escape velocity with angular momentum in the presence of magnetic field. It is observed that negative energy is also possible for the Kerr metric inside the static limit case (r st = 2m = r g ) (called ergo-sphere). Here we have considered the slowly rotating case, therefore we have only one horizon and we did not have ergo sphere. Event horizon are same for both Schwarzschild and slowly rotating Kerr black hole (r = r g ). We have shown all the aspects of the particle's motion by the trajectories in the appendix.
The motion of the charged particle around the black hole after the collision with any other particle is mainly unpredictable. By using the assumptions that particle moving in the equatorial plane and after the collision with any other particle its angular momentum and the initial radial velocity did not change then we predicted the motion of the particle. In this paper we found the escape conditions for a charged particle from the surrounding of the black hole. Magnetic field is present in the vicinity of the black hole and particle's motion strongly effected by it. It is shown by figures under what conditions particle escape from the black hole vicinity or remain bounded around the black hole. It is seen that particle's energy will remain same for both positive (ℓ > 0) and negative (ℓ < 0) value of angular momentum.
It can be seen from the equations (13) and (18) for large value of angular momentum L z the term in the square root will dominate and effective potential reduced to Schwarzschild effective potential which is discussed in [13] as shown by the figure 2. We have obtained convolution point by plotting the effective potential against angular momentum. It is shown that magnetic field largely effect the motion of the particle near the black hole as the magnetic field is very strong in the vicinity of black hole and decreases far away from it.
As discussed before trajectories for escape velocity are same for both positive (ℓ > 0) and negative (ℓ < 0) value of angular momentum like the energy of particle. Particle can escape to both +∞ and −∞ depending on its energy. Due to the presence of magnetic field in the vicinity of black hole escape velocity of the particle increases exponentially and it becomes almost constant as it move away from the black hole. It is shown for the change of angular momentum escape velocity potential as given in [13]. Here we plot the effective potential against ρ (ℓ = 10), (b = .5), (a = .5). In this graph (E max ) corresponds to stable circular orbit and (E min ) corresponds to unstable circular orbit.