Dynamics of Particles around Time Conformal Schwarzschild Black Hole
In this work, we present the new technique for discussing the dynamical motion of neutral as well as charged particles in the absence/presence of magnetic field around the time conformal Schwarzschild black hole. Initially, we find the numerical solutions of geodesics of Schwarzschild black hole and the time conformal Schwarzschild black hole. We observe that the Schwarzschild spacetime admits the time conformal factor , where is an arbitrary function and is very small which causes the perturbation in the spacetimes. This technique also re-scale the energy content of spacetime. We also investigate the thermal stability, horizons and energy conditions corresponding time conformal Schwarzschild spacetime. Also, we examine the dynamics of neutral and charged particle around time conformal Schwarzschild black hole. We investigate the circumstances under which the particle can escape from vicinity of black hole after collision with another particle. We analyze the effective potential and effective force of particle in the presence of magnetic field with angular momentum graphically.
The dynamics of particles (massless or massive, neutral or charged) in the vicinity of black hole (BH) is the most intriguing problems in the BH astrophysics. The geometrical structure of spacetime in the surrounding of BH could be studied better by these phenomenon [1, 2]. The motion of particles help us to understand gravitational fields of BHs experimentally and to compare them with observational data. The magnetic field in the vicinity of BH is due to the presence of plasma . Magnetic fields can’t change the geometry of BH but its interaction with plasma and charged particles is very important [4, 5]. The transfer of energy to particles moving around BH geometry is due to magnetic field, so that there is a possibility of their escape to spatial infinity . Hence, the high energy may produce by charged particles collision in the presence of magnetic field rather than its absence.
Recent observations have also provided hints about connecting magnetars with very massive progenitor stars, for example an infrared elliptical ring or shell was discovered surrounding the magnetar SGR . The explanation of magnetic white dwarfs was first proposed in the scenario of fossil-field for magnetism of compact objects [8, 9, 10]. Magnetic white dwarfs may be created as a result of rebound shock explosion  and may further give rise to novel magnetic modes of global stellar oscillations . This fossil-field scenario is supported by the statistics for the mass and magnetic field distributions of magnetic white dwarfs. Magnetized massive progenitor stars with a quasi-spherical general polytropic magneto fluid under the self-gravity are modeled by . Over recent years methods are being developed to detect truly cosmologically magnetic fields, not associated with any virilized structure. The spectral energy distribution of some tera electron-volt range (TeV) blazers in the TeV and giga electron volt (GeV) range hint at the presence of a cosmological magnetic field pervading all space [13, 14].
Authors [15, 16, 17] investigated effects on charged particles in the presence of magnetic field near BHs. The motion of charged particles discussed by Jamil et al.  around weakly magnetized BHs in JNW space time. Many aspects of motion of particles around Schwarzschild and Reissner-Nordstrom BH have been studied and the detail review is given the reference [19, 20]. An important question may rise in the studies of these important problems is ”what will happen in the dynamics of particles (neutral or charged) around BHs with respect to time”. Particles moves in circular orbits around BHs in equatorial plane may show drastic change with respect to time. Following the work done by Zaharani et al. , we present at what conditions particle escape to infinity after collision. At some specific time interval, we study the dynamics of neutral and charged particles around time conformal Schwarzschild BH.
The purpose of our work is to investigate under which circumstances the particle moving initially would escape from innermost stable circular orbit (ISCO) or it remain bounded, captured by BH or escape to infinity after collision with another particle. We calculate the escape velocity of particle and investigate some important characteristics such as effective potential and effective force of particle motions around the BH with respect to time. Initially, we will find attraction in particle motion but after some specific time we see the repulsion. The comparison of stability orbits of particle with the help of Lyapunov exponent is also established.
The outline of paper is as follows: In section 2, we discuss the Noether and Approximate Noether Symmetries for Schwarzschild BH and time conformal Schwarzschild BH and find the corresponding conservation laws. In section 3, we investigate the dynamics of neutral particle through effective potential, effective force and escape velocity. In section 4, the motion of charged particle is discussed, the behavior of effective potential, effective force and escape velocity is analyzed in the presence of magnetic field. In section 5, the Lyapunov exponent is discussed. Conclusion and observations are given in the last section.
2 Noether and Approximate Noether Symmetries and Corresponding Conservation Laws
The one to one correspondence between conservation laws and symmetries of Lagrangian (Noether symmetries) has been firstly pointed out by Emmy Noether [21, 22, 23]. She suggested that there exist a conservation law for every Noether symmetry. One can obtain the approximate Noether symmetries of Schwarzschild solution by considering its first order perturbation and investigate the energy-momentum of corresponding spacetime.
2.1 Schwarzschild Black Hole
The line element of the Schwarzschild solution is
the corresponding Lagrangian is
The symmetry generator
is the first order prolongation of
is the Noether symmetry if it satisfies the equation,
where is differential operator of the form
and is Gauge function.
The solution of the system (5) is
The corresponding Noether symmetries generators are
For conservation laws (given in the following table), we use the following equation
For example, let us see the calculation for symmetry . In this symmetry, and . Also from lagrangian, we have . Putting these values in equation (9) we have
Similarly for other symmetries, the conservation laws are given in the table
3 Perturbed Metric
Perturbing the metric given by (1) by using the general time conformal factor , which gives
Picking the first order terms in and neglecting the higher order terms we have
in expanded form
where denotes the differentiation with respect to , is defined in equation (2) and
We define the first order approximate Noether symmetries 
up to the gauge Where
is the approximate Noether symmetry and is the approximate part of the gauge function.
Now X is the first order approximate Noether symmetry if it satisfy the equation
where is the first order prolongation of the first order approximate Noether symmetry X given in equation (13).
The equation (15) split into two parts that is
All and are the functions of and are functions of . From equation (17) we obtained a system of 19 partial differential equations whose solution will provide us the cases where the approximate Noether symmetry(ies) exist(s). By putting the exact solution given in equation (7) in equation (17) we have the following system of 19 PDEs
In symmetry generators form we have
We see that only the symmetry got the non-trivial approximate part which re-scale the energy content of the Schwarzschild spacetime. The corresponding conservation laws are
3.1 Existence and Location of Horizon
Consider the line element of time conformal Schwarzschild BH
The parameter is a dimensionless small parameter that causes the perturbation in the spacetime and is constant of the dimension equal to that of time so that to make the term as dimensionless. is the mass of the black hole. In order to discuss the trapped surfaces and apparent horizon of the above metric, we use the definitions of - and Eqs.(13) and (14) of . Here, we define the mean curvature one-form as
Also, the scalar defining the trapped surface of given spacetime is
In the notation of , the quantities in the above equations are defined as and . The coordinates are defined by and . By using these coordinates, we obtained
Now the surface of the given geometry (22), will be trapped, marginally trapped and absolutely non-trapped if is positive, zero and negative respectively. In order to analyze the nature of trapped surfaces and location of horizons, we solve Eq.(27) for by imposing the restriction on such that , and . In the following we discuss these situations in detail.
, leads to two positive real values of , and such that . Here and corresponds to outer and inner horizons for trapping surface respectively.
, leads to two positive real values of , and such that . Here and corresponds to outer and inner horizons for marginally trapping surface respectively.
, leads to two positive real values of , and such that . Here and corresponds to outer and inner absolutely non-trapping points on the given surface respectively. We would like to mention that we have considered the denominator of Eq.(27) as positive for the arbitrary value of parameters and coordinates.
3.2 Thermal Stability
The time conformal Schwarzschild metric is
where with . Further suppose that . Clearly, for values of this solution is positive definite and coordinate singularity occurs at . The coordinate is identified periodically with period
In the limit of large , the killing vector is normalized to . The temperature measure to infinity may be formally identified with inverse of this period. Hence by Tolman law, for any self gravitating system in thermal equilibrium, a local observer will measure a local temperature T which scales as . The constant of proportionality in present context is
The wall temperature and surface area is defined by York . One topologically regular solution to Einstein equation with these boundary conditions is hot flat space with uniform temperature . Also another solution is Schwarzschild metric. If a BH of horizon does exist then the wall temperature from Tolman Law must satisfy
In terms of and , this equation may be solved for . There is no real positive root for , if .
For any value of and , the entropy of BH solution to Eq. (31) is . The heat capacity of constant surface for any solution is
The heat capacity is positive and equilibrium configuration is locally thermally stable if .
3.3 Energy Conditions
Consider the line element of time conformal Schwarzschild BH of Eq. (22). Let’s suppose that the matter distribution is isotropic in nature, whose energy-momentum tensor is given by
Here the vector is fluid four velocity with , is the matter density and p is the pressure.
Taking , from Einstein field equations one can deduce that
Now, we want to check the energy conditions for time conformal Schwarzschild BH. For Null Energy Condition (NEC), Weak Energy Condition (WEC), Strong Energy Condition (SEC) and Dominant Energy Condition (DEC), the following inequalities must satisfy
WEC: , .
SEC: , .
4 Dynamics of Neutral Particle
We discuss the dynamics of neutral particle around time conformal Schwarzschild BH defined by (1). The approximate energy and the approximate angular momentum are given in Table 2 . The total angular momentum in plane can be calculated from and which has the value
Using the normalization condition given in Eq. (36), we can get the approximate equation of motion of neutral particle
For and , effective potential turns out to be
The effective potential extreme values are obtained by The convolution point of effective potential lies in the inner most circular orbit (ISCO).
The corresponding azimuthal angular momentum and the energy of the particle at the ISCO are
Consider the circular orbit of a particle, where is local minima of effective potential. This orbit exist for . The convolution point of effective potential for ISCO is defined by . Now suppose that the particle collides with another particle which is in ISCO. There are three possibilities after collision (i) bounded around BH, (ii) captured by BH and (iii) escape to . The results depend upon the process of collision. Orbit of particle is slightly change but remains bounded for small changes in energy and momentum. Otherwise, it can be moved from initial position and captured by BH or escape to infinity. After collision, the energy and both angular momentum (total and azimuthal) change . Before simplifying the situation, one can apply some conditions, i.e., azimuthal angular momentum and initial radial velocity do not change but energy can change by which we determine the motion of the particle. Hence the effective potential becomes
This energy is greater then the energy of the particle before collision, because after collision the colliding particle gives some of its energy to the orbiting particle. Simplifying the Eq. (40), we can obtain the escape velocity as follows
4.1 Behavior of Effective Potential of Neutral Particle
We analyze the trajectories of effective potential and explain the conditions on energy required for bound motion or escape to infinity around Schwarzschild BH. Figure 1 represents the behavior of effective potential of particle moving around the Schwarzschild BH for different values of angular momentum. We can see from Figure 1, the maxima of effective potential for at for time interval . Similarly, the minima of effective potential for at for time interval . Hence, it is concluded that maxima and minima is shifting forward for large values of angular momentum. Also the repulsion and attraction of effective force depends upon time. For time interval , there is strong attraction near the BH and it vanishes when radial coordinate approaches to infinity. Moreover, for time interval there is strong repulsion near the BH and it vanishes for approaches to infinity. For , we find the equilibrium i.e. there is no attraction and repulsion in effective potential. Hence, effective potential is shifting from attraction to repulsion when time increases and shifting point of time is . Hence, it is concluded that effective potential is attractive and repulsive with respect to time, maxima and minima of effective potential shifting forward in radial coordinate for large values of angular momentum.
4.2 Behavior of Effective Force of Neutral Particle
The effective force is
We are studying the motion of neutral particle in surrounding of Schwarzschild BH where attractive and repulsive gravitational forces produce by scalar vector tensor field which prevents a particle to fall into singularity . The comparison of effective force on particle around Schwarzschild BH as function of radial coordinate for different values of angular momentum is shown in Figure 2. We can see from figure, at time interval , the attraction of particle to reach singularity is more for as compared to . Similarly, at time interval , the repulsion of particle to reach the singularity is more for as compared to . Also, at there is no effective force. Since acts as a shifting point, i.e., the effective force is shifting from attraction to repulsion at . We can conclude that the attraction and repulsion of particle to reach and escape from singularity is more for large values of angular momentum but it mainly depend upon time.
4.3 Trajectories of Escape Velocity of Neutral Particle
Figure 3 represents the trajectories of escape velocity for different values of as a function of . It is evident that if angular momentum is positive then escape velocity is repulsive while it is attractive for negative values of angular momentum. Also, the repulsion of escape velocity for large values of is strong as compared to small values of . On the other hand, the attraction of escape velocity for is strong as compared to . We can conclude that escape velocity of particle increases as angular momentum increases but it becomes almost constant away from the BH. Also it is interesting that the trajectories of escape velocity do not vary with respect to time.
5 Dynamics of the Charge Particle
Now we consider the dynamics of charged particle around time conformal Schwarzschild BH defined by (1). We assume that a particle has electric charge and its motion is affected by magnetic field in BH exterior. Also, we assume that there exits magnetic field of strength in the neighborhood of BH which is homogenous, static and axisymmetric at spatial infinity. Next, we follow the procedure of  to construct magnetic field. Using metric (1), the general killing vector is 
where is killing vector. Using above equation in Maxwell equation for 4- potential in lorentz gauge , we have 
The killing vectors correspond to 4-potential which is invariant under symmetries as follows
Using magnetic field vector 
where , , , is Levi Civita symbol. The Maxwell tensor is
In metric (1), for local observer at rest, we have
here the magnetic field is directed along z-axis (vertical direction). Since the field is directed upward, so we have .
The Lagrangian of particle moving in curved space time is given by 
where and are the mass and electric charge of the particle respectively. The generalized 4-momentum of particle is
The new constants of motion are defined as
where . Using the constraints in the Lagrangian (12) the dynamical equations for and become
Using the normalization condition, we obtain
The effective potential takes the form
The energy of the particle moving around the BH in orbit at the equatorial plane is
This is a constraint equation i.e. it is always valid if it is satisfied at initial time. Let us discuss the symmetric properties of Eq.(54) which are invariant under the transformation as follows
Therefore, without the loss of generality, we consider and for we should apply transformation (58) because negative and positive both charges are inter related by the above transformation. However, if one choose positive electric charge then both cases of (positive and negative) must be studied. They are physically different: the change of sign of corresponds to the change in direction of Lorentz force acting on the particle .
The system (52) - (54) is also invariant with respect to reflection This transformation preserves the initial position of the particle and changes Therefore it is sufficient to consider the positive value of escape velocity . By differentiating Eq.(57) with respect to , we obtain
The effective potential after the collision when the body in the magnetic field ( and ) is
The escape velocity of the particle takes the form
Differentiating Eq.(59) again with respect to we have
5.1 Behavior of Effective Potential of Charged Particle
Figure 4 represents the behavior of effective potential as a function of radial coordinate for different values of magnetic field B. The minima of effective potential at is approximately at initially but as time increase it shifting near the BH. Hence the presence of magnetic field increases the possibility of particle to move in stable orbit. Similarly, initially the minima of effective potential at is approximately at respectively, as time increases it approaches near BH. One can noticed that in the presence of low magnetic field, the minima of effective potential is shifted away from the horizon and width of ISCO is also decreased as compared to high magnetic field. These results is an agreement with [34, 38]. Therefore we can say that increase in magnetic field act as increase instable orbits of particle. Another aspect from figure is that at initial time, we have possibility of stable orbit but as time increases the possibility of ISCO getting low and at , the stable orbit is lost. Hence it is possible that the particle is captured by the BH or it escape to infinity.
5.2 Behavior of Effective Force of Charged Particle
The effective force on the particle can be defined as as