Particle Dynamics Around Weakly Magnetized Riessner-Nordström Black Hole
Abstract: Considering the geometry of Reissner-Nordström (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.
The geometrical structure of the spacetime around black hole (BH) could be understood better by studying the dynamics of particles in the vicinity of black holes frolov1998 (); sh5979 (). Circular geodesics give information about geometry of spacetime po9064 (); am5968 (). 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 (light like deflection, gravitational time delay and perihelion shift). Presence of plasma is responsible for magnetic field bo2013 (), in the the surrounding of the black hole mc2307 (); do2008 (). Near the event horizon effect of magnetic field is strong but not enough to disturb the geometry of the black hole. Motion of a charged around black hole in the presence of magnetic field gets influenced zn7076 (); bl3377 (). Black holes with such scenario are known as ‘weakly magnetized’ fr2012 () i.e. the magnetic field strength lies between Gauss. Magnetic field is responsible for transferring energy to the particles moving in the geometry around black hole, so that their escape to spatial infinity is possible ko8802 (). Hence the collision of charged particles near the black hole may produce much higher energy in the presence of magnetic field than in its absence. In mi4199 (); te0903 (); saqib (); ja2415 () the effects of magnetic fields on the charged particles around black holes were investigated. Timelike geodesics in modified gravity black hole in the presence of axially symmetric magnetic field are studied in saqib-mog (). In gulmina () authors have studied the dynamics of a charged particle around a weakly magnetized naked singularity, in the Janis-Newman-Winicour (JNW) spacetime. Kaya ka1107 () studied the motion of charged particles around a five dimensional rotating black hole in a uniform magnetic field and found stable circular orbits around the black hole. In literature many aspects of the particles motion in the vicinity of RN-black hole have been studied. In za7110 () authors have studied the high energy collisions phenomenon between the particles, currently termed as BSW mechanism. In pu5211 () the spatial regions for circular motion of neutral and charged test particles around RN-BH and naked singularities have been studied. Critical escape velocity for a charged particle moving around a weakly magnetized Schwarzschild black hole has been studied in za4313 (). We consider Reissner-Nordström (RN) black hole 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. Following the work done by Zahrani et. al. za4313 (), 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 modeling the particle’s motion around a black hole under the influence of both gravitational and magnetic forces, we first consider the motion of a neutral particle in the absence of magnetic field. The study of particle dynamics around RN spacetime is also relevant as this metric represents the extreme RN-BH and a naked singularity as special cases.
Outline of the paper is as follows: In section II metric of RN-BH is discussed and escape velocity of a neutral particle is calculated. In section III the equations of motion of a charged particle moving around weakly magnetized RN-BH are derived. Trajectories of the particles moving around the extremal RN-BH are plotted in section IV. In section V the dimensionless form of the equations are given. In section VI trajectories for escape energy and escape velocity of the particle are presented. Motion of the particle is initially considered in the equatorial plane for the sake of simplicity. The metric signature is and .
Ii Escape Velocity For a Neutral Particle
We first work for the escape velocity of a neutral particle in the absence of magnetic field. The RN-BH metric is given by
Here is the mass and is electric charge of the black hole. Horizon of the RN-BH are located at:
If , there are two real positive roots; the larger root corresponds to the event horizon and the smaller root refers to the Cauchy horizon which is associated with the timelike singularity at . Black hole is known to be extremal black hole if and it has only one event horizon at . If , there is no real root of the equation and there is no event horizon. This case is known as naked singularity of RN spacetime. Symmetries of the black hole metric are along the time translation and rotation around symmetry axis. The corresponding constants of motion can be calculated using the Killing vectors
here and . The corresponding conserved quantities are the total energy of the moving particle and its azimuthal angular momentum
Over dot is the differentiation with respect to proper time and . From the normalization condition , we have
we considered , i.e. the planar motion of the particle. Solving , we get the value of corresponding to the extreme values of the effective potential (the convolution point) chandrasekher1983 ():
The ISCO is at for extreme RN-BH . For it reduces to (Schwarzschild black hole) za4313 (). The corresponding energy and the azimuthal angular momentum of the particle (in ISCO) are respectively
For extremal black hole case, i.e. at , Eq. (8) becomes
Now consider the collision of a particle, moving in the ISCO, with another particle which is coming from infinity (initially at rest). This collision may result in three possibilities (depending on the progression of the collision): (i) a bounded motion (ii) particle captured by black hole (iii) particle escape to infinity. Orbit of the particle alters slightly if the energy and angular momentum of the particle do not undergo a major change, otherwise the particle can move away from the original path resulting in captured by black hole or an escape to infinity may occur. Collision of the particles changes the equatorial plane of the moving particle but since the metric is spherical symmetric so all the equatorial planes are similar. We consider the collision occurring in such a way that the azimuthal angular momentum remains invariant and initial radial velocity also remains same. Hence only the change in energy will be considered for determining the motion of particle after collision. These condition are imposed for simplification only. Particles gains an escape velocity in orthogonal direction of the equatorial plane after collision frolov3410 () and its momentum and energy (in the new equatorial plan) become
here and denotes the particles’s initial polar angular velocity. Energy of the particle is
with , given in Eq. (9). After collision, particle gains greater angular momentum and energy as compared to before collision. From Eq. (11) it is clear that in the asymptotic limit (), . So for unbounded motion (escape) particle requires . Hence for escape to infinity the necessary condition is
we have solved equation taking and the quantities with subscript denotes the extremal black hole case.
Iii Charged Particle Around RN-BH surrounded by Magnetic Field
The presence of magnetic field interrupts the motion of a charged particle around black hole. To know the aftermath of this perturbation let us start with the Lagrangian of the moving particle as
here is mass of the particle and is the charge of particle. The Killing vector equation , resembles to the Maxwell equation for in the Lorentz gauge wald8074 () here denotes the Killing vector and is the 4-potential defined as wald8074 ()
with as the magnetic field strength given as
and using Levi Civita symbol, , one can write
The Maxwell tensor, , is defined as
For a local observer in RN geometry, . The only two components of will survive which are and .
Using the Euler-Lagrange equations for the Lagrangian defined in Eq. (13) one can get easily
here . The normalization condition gives
Equation of motion of a charged particle moving in an external electromagnetic field satisfies:
Iv Dynamical Equations in Dimensionless Form
For the sake of convenience we can rewrite the dynamical equations of and in dimensionless form by introducing the following dimensionless quantities frolov3410 ():
For extremal black hole case, it becomes
For the particle moving in equatorial plane, Eqs. and (27) become
Using the built in command NDSolve in Mathematica , Eq. can be solved and the behavior of the obtained interpolating function can be better understood by plotting it against . Using Eq. (23) for Eqs. we obtain
and for extreme black hole
The effective potential given in Eq. (19) becomes
for extreme black hole it becomes
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. , defined as
for extremal black hole
where is the energy defined in Eq. (34). As already mentioned that when the energy . So for the unbound motion the energy of the particle should be . Solving equation at , 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 and defined in term of and the energy of the particle. The expression for the parameters and in term of could be obtained by dealing with Eq. (36). The first and second derivatives of the effective potential defined in Eq. (36) are
The values of and can be found by solving simultaneously and , as given below
In section VII the parameters and are plotted against .
V Effect of Magnetic Field on Motion of Particles
Consider the neutral particle moving around RN-BH. Writing the equations associated with the constants of motion, in dimensionless form we have
where positive sign is for the particle going away from the black hole, and negative sign is for the path of an ingoing particle, also
For extremal black hole it becomes
It is observed graphically that a particle having less angular momentum approaches the black hole more closely as compared to the one having large angular momentum. This shows that when a particle does not move radially, its chances for approaching the black hole event horizon are very less, Fig. (2). For a charged particle moving around RN-BH in the presence of magnetic field, we can write
For extremal black hole
Change in during the motion of particle, around RN-BH, starting its motion from some finite distance is shown in Fig. (2). It is observed that behavior of angular motion of particle is linked with the strength of magnetic field.
Vi Behavior of Effective Potential
In this section the behaviors of effective potentials of particle are studied graphically and the energy required for its escape to infinity or for bounded motion is discussed. In Fig. (4) we have plotted the effective potential as a function of . There are two minima and , in the presence of magnetic field while in the absence of magnetic field there is only one minima, . 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 black hole with that of RN-BH is established in Fig. (4). It is clear that the maxima for the effective potential of particle moving around RN-BH has greater value in comparison with the maxima of effective potential for Schwarzschild black hole. Since a particle moving around black hole 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 black hole. In Fig. (6) different regions of effective potential which are linked to escape and bounded motion of the particle are shown. Here and are the regions which correspond to stable orbits for . For there is only one stable region represented by which is related to a stable orbits. Dotted line represents the minimum energy required to escape from the vicinity of black hole. If the particle has energy and move toward the black hole it will bounce back to infinity which is represented by . In Fig. (6) we are comparing the effective potentials of extremal black hole in the presence of magnetic field and without magnetic field. One can notice that for , the effective potential has two local minima which corresponds to two stable regions while for it has only one minima. We use the notation and for unstable and stable circular orbits of the particle respectively. Here corresponds to ISCO which coincides with ISCO of the case when (dotted curve) and 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 Fig. (8) we have plotted the magnetic field as a function of . One can notice that magnetic field decreases abruptly as particle moves away from the source (black hole). We have plotted the angular momentum as a function of in Fig. (8), it is clear that for . Angular momentum for ISCO as a function of magnetic field are plotted in Fig. (8).
Vii Trajectories of Escape Velocity
Escape velocity of the charged particle is plotted in Fig. (9), 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 Fig. (11) we are comparing the escape velocity of a particle moving around the Schwarzschild black hole with the particle moving around the RN-BH. It is clear that the difference between the velocities is large near the black hole 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. The behavior of escape velocity for extreme RN-BH is shown in Fig. (11). The paths followed by the kicked (escaping) particle, moving initially in the ISCO, are shown in Figs. (13), (13), and (14), which are obtained by solving Eqs. (24), (25) and (26) numerically, we have taken initial radial velocity after collision as zero. We are interested to know the effect of magnetic field on the motion of charged particle (the magnitude of deformation produced in oscillatory motion). This effect increases as strength of magnetic field increases. Escape velocity is plotted in Fig. (16) for different values of magnetic field . Escape of the particle from the vicinity of black hole becomes easier in the presence of stronger magnetic field. As particle goes away from the black hole its escape velocity becomes almost constant, just like the strength of magnetic field. Hence presence of magnetic field will provide more energy to particle, so that it might easily escape from the vicinity of black hole. In Fig. (16) connection of escape velocity with angular momentum is shown. It is clear that escape velocity of a particle with larger value of is greater compared to the particle with smaller value of .
Viii Summary and Conclusion
Motion of particles in the RN geometry in the presence of magnetic
field is investigated in this paper. To avoid complications in the
analysis some assumption are made, 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 and are obtained. Effect of
angular momentum and magnetic field on motion of neutral and charged
particles is observed graphically. It is noticed that a particle
with less angular momentum will approach the black hole more
closely, as compared to the case when angular momentum is large. We
find out the condition on 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 black hole, are also obtained. Trajectories
of the escaping charged particle are also shown graphically. It is
observed that a slight change in the initial condition of the
colliding particle affects the escaping behavior. Presence of
magnetic field also disturbs the escaping trajectories. 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 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 regions (local minima) are obtained in
contrast to the only stable region obtained in the case when
magnetic field is absent. Such analysis helps us to understand the
effect of black hole on its surrounding matter. We intend to extend
the similar analysis for RN-de-Sitter black hole.
Acknowledgement: The authors would like to thank the referees for their useful comments to improve this work.
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