Dynamics of particles around time conformal Schwarzschild black hole
Eur. Phys. J. C
Dynamics of particles around time conformal Schwarzschild black hole
Abdul Jawad 2
Farhad Ali 1
M. Umair Shahzad 2
G. Abbas 0
0 Department of Mathematics, The Islamia University of Bahawalpur , Bahawalpur , Pakistan
1 Department of Mathematics, Kohat University of Science and Technology , Kohat , Pakistan
2 Department of Mathematics, COMSATS Institute of Information Technology , Lahore 54000 , Pakistan
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 a magnetic field around the time conformal Schwarzschild black hole. Initially, we find the numerical solutions of geodesics of the Schwarzschild black hole and the time conformal Schwarzschild black hole. We observe that the Schwarzschild spacetime admits the time conformal factor e f (t), where f (t ) is an arbitrary function and is very small, which causes a perturbation in the spacetimes. This technique also rescales the energy content of spacetime. We also investigate the thermal stability, horizons and energy conditions corresponding to time conformal Schwarzschild spacetime. Also, we examine the dynamics of a neutral and charged particle around a time conformal Schwarzschild black hole. We investigate the circumstances under which the particle can escape from the vicinity of a black hole after collision with another particle. We analyze the effective potential and effective force of a particle in the presence of a magnetic field with angular momentum graphically.

The magnetic field in the vicinity of BH is due to the presence
of a plasma [3]. Magnetic fields cannot change the geometry
of a BH but its interaction with plasma and charged particles
is very important [4,5]. The transfer of energy to particles
moving around in a BH geometry is due to the magnetic field,
so that there is a possibility of their escape to spatial infinity
[6]. Hence, a high energy may be produced by a charged
particles collision in the presence of a 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 1900+14 [7]. The
explanation of magnetic white dwarfs was first proposed in the
scenario of a fossilfield for magnetism of compact objects
[8–10]. Magnetic white dwarfs may be created as a result of
a rebound shock explosion [10] and may further give rise to
novel magnetic modes of global stellar oscillations [11]. This
fossilfield scenario is supported by the statistics for the mass
and magnetic field distributions of magnetic white dwarfs.
Magnetized massive progenitor stars with a quasispherical
general polytropic magneto fluid under the selfgravity are
modeled by [12]. Over recent years methods were developed
to detect truly cosmologically magnetic fields, not associated
with any virilized structure. The spectral energy distribution
of some teraelectronvolt range (TeV) blazers in the TeV and
gigaelectronvolt (GeV) range hint at the presence of a
cosmological magnetic field pervading all space [13,14].
The authors [15–17] investigated the effects on charged
particles in the presence of a magnetic field near BHs. The
motion of charged particles was discussed by Jamil et al. [18]
around weakly magnetized BHs in JanisNewmanWinicour
(JNW) spacetime. Many aspects of the motion of particles
around an SH and a Reissner–Nordstrom BH have been
studied and a detailed review is given in Refs. [19,20]. An
important question may rise in the studies of these important
problems: what will happen in the dynamics of a particles
(neutral or charged) around BHs with respect to time? Particles
moving in circular orbits around BHs in an equatorial plane
may show a drastic change with respect to time. Following
the work done by Al Zahrani et al. [20], we show under what
conditions a particle may escape to infinity after collision. At
some specific time interval, we study the dynamics of neutral
and charged particles around a time conformal Schwarzschild
BH.
The purpose of our work is to investigate under which
circumstances the particle moving initially would escape from
an innermost stable circular orbit (ISCO) or would remain
bounded, captured by the BH or escape to infinity after
collision with another particle. We calculate the escape velocity
of a particle and investigate some important characteristics
such as the effective potential and the effective force of
particle motions around the BH with respect to time. Initially,
we will find attraction in the particle motion but after some
specific time we see repulsion. The comparison of the
stability orbits of a particle with the help of a Lyapunov exponent
is also established.
The outline of paper is as follows: in Sect. 2, we
discuss the Noether and approximate Noether symmetries for
a Schwarzschild BH and a time conformal Schwarzschild
BH and find the corresponding conservation laws. In Sect.
3, we investigate the dynamics of a neutral particle through
the effective potential, effective force, and escape velocity.
In Sect. 4, the motion of the charged particle is discussed, the
behavior of the effective potential, effective force and escape
velocity is analyzed in the presence of a magnetic field. In
Sect. 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 a Lagrangian (Noether symmetries) has
been first pointed out by Emmy Noether [21–23]. She
suggested that there exists a conservation law for every Noether
symmetry. One can obtain the approximate Noether
symmetries of a Schwarzschild solution by considering its first
order perturbation and investigating the energymomentum
of corresponding spacetime.
dse2 =
2.1 Schwarzschild black hole
The line element of the Schwarzschild solution is
dt 2 −
× dr 2 − r 2(dθ 2 + sin θ dφ2),
−1
the corresponding Lagrangian is
Le =
t˙2 −
−1
The symmetry generator
Xe1 = ξe ∂∂s + ηei ∂∂xi + ηeis ∂∂x˙i , i = 0, 1, 2, 3,
is the first order prolongation of
Xe = ξe ∂∂s + ηei ∂∂xi .
Xe is the Noether symmetry if it satisfies the equation
For the conservation laws given in the Table 1 we use the
equation
φ = ∂∂xL˙i (ηi − ξ x˙i ) + ξ L − A,
for example, let us show the calculation for the symmetry
X1. In this symmetry ξ = 0 and η0 = 1 and we have the
above Lagrangian from Eq. (9), ∂∂Lt˙ = 2(1 − 2rM )t˙; putting
these values in Eq. (9) we have
2M
t˙(1 − 0) + 0 − 0 = 2 1 − r
Xe1 Le + (Dξe)Le = D Ae,
where D is differential operator of the form
D = ∂∂s + x˙i ∂∂xi ,
and Ae is a gauge function.
The solution of the system (5) is
0
Ae = C2, ηe = C3,
1 2
ηe = 0, ηe = −C5 cos φ + C6 sin φ,
3 cos θ (C5 sin φ + C6 cos φ)
ηe = C4 + sin θ
∂ ∂ ∂
X1 = ∂t , X2 = ∂s , X3 = ∂φ ,
∂ ∂
X4 = cos φ ∂θ − cot θ sin φ ∂φ
∂ ∂
X5 = sin φ ∂θ + cot θ cos φ ∂φ
The corresponding Noether symmetries generators are
Similarly for the other symmetries. The corresponding
conservation laws are
Table 1 First integrals
3 Perturbed metric
Perturbing the metric given by (1) we use the general time
conformal factor e f (t), which gives
ds2 = e f (t)dse2 = (1 + f (t ) +
Picking the first order terms in
order terms we have
and neglecting the higher
or in expanded form
ds2 =
× dr 2 − r 2(dθ 2 + sin θ dφ2)
dt 2 −
dt 2 −
× dr 2 − r 2(dθ 2 + sin θ dφ2) ,
−1
−1
L =
t˙2 −
−1
r˙2 − r 2
× r˙2 − r 2(θ˙2 + sin θ φ˙ 2) ,
t˙2 −
−1
where˙denotes differentiation with respect to s, Le is defined
in Eq. (2), and
La = f (t )
t˙2 −
−1
× r˙2 − r 2(θ˙2 + sin θ φ˙ 2) .
We define the first order approximate Noether symmetries
by [24]
X = Xe +
up to the gauge A = Ae +
Aa . Here
Xa = ξa ∂∂s + ηai ∂∂xi , i = 4, 5, 6, 7,
+ Aar = 0, 2ηs6r 2 + Aaθ = 0,
2M −1r˙2 − r 2 θ˙2 + sin2 θφ˙2
φ2 = − 1 − 2rM t˙2 − 1 − r
φ3 = −2r 2 sin2 θφ˙
φ4 = −2r 2(cos φθ˙ − cot θ sin φφ˙)
φ5 = −2r 2(sin φθ˙ + cot θ cos φφ˙)
is the approximate Noether symmetry and Aa is the
approximate part of the gauge function.
Now X is the first order approximate Noether symmetry
if it satisfies the equation
where X1 is the first order prolongation of the first order
approximate Noether symmetry X given in Eq. (13).
Equation (15) splits into two parts, that is,
Xe Le + (Dξe)Le = D Ae,
Xa1 Le + Xe La + (Dξe)La + (Dξa )Le = D Aa .
1
All of ηei, ηai , ξe, ξa , Ae, and Aa are functions of
s, t, r, θ , φ, and η˙ei, η˙ai are functions of s, t, r, θ , φ, t˙, r˙, θ˙, φ˙ .
From Eq. (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 Eq. (7) in Eq. (17) we have the
following system of 19 PDEs:
ξt1 = ξr1 = ξθ1 = ξφ1 = Aas = 0, 2ηs4 1 − 2rM
− Aat = 0,
ηφ6 − sin2 θηθ7 = 0, C3 ft (t) + r2 η5 + 2ηr5 − ξs1 = 0,
2M 2M η5 + 2ηr5 − ξs1 = 0,
C3 ft (t) − r 2 1 − r
2M 2M η5 + 2ηr5 − ξs1 = 0,
C3 ft (t) + r 2 1 − r
C3 ft (t) + r2 η5 + 2 cot θη6 + 2ηφ7 − ξs1 = 0.
We see that the system (18) have C3 and f (t ) in it. The
solution of this system (18) is
5 6
ηa = 0, ηa = −C5 cos φ + C6 sin φ,
ξ 0 = C1 + C3 αs , f (t ) = αt .
Also, the scalar defining the trapped surface of a given
spacetime is
Combining the solution (7) and (19) we have the following
solution of Eq. (15):
Ae + Aa = C2 + C2, ηe0 + ηa4 = C3 +
ηe1 + ηa5 = 0, ηe2 + ηa6
In symmetry generator form we have
αs ∂∂s , X2 = ∂∂s , X3 = ∂∂φ ,
We see that only the symmetry X1 got the nontrivial
approximate part which rescales the energy content of the
Schwarzschild spacetime. The corresponding conservation
laws are
3.1 Existence and location of horizon Consider the line element of a time conformal Schwarzschild BH
ds2 =
t
1 + α
dt 2 −
−1
× dr 2 − r 2(dθ 2 + sin θ dφ2) .
The parameter is a dimensionless small parameter that
causes the perturbation in the spacetime and α is a constant
with dimension equal to that of time t so as to make the
term αt dimensionless. M 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 [25–28] and Eqs.
(13) and (14) of [28]. Here, we define the mean curvature
oneform by
In the notation of [28], the quantities in the above equations
acroeorddeifinnaetedsaasreGde≡fineeUd =by√{xdae}tg=AB{ta,nrd} ganad={xgAa}Ad=x {Aθ.,Tφh}e.
By using these coordinates, we obtain
gt = 0,
eU =
gr = 0
detgAB =
Hence, using given the metric (22), Eqs. (23)–(26), we get
the following form for the trapping scalar:
Now the surface of the given geometry (22), will be
trapped, marginally trapped, and absolutely nontrapped if
κ is positive, zero, and negative, respectively. In order to
analyze the nature of trapped surfaces and the location of the
horizons, we solve Eq. (27) for r by imposing a restriction
on κ such that κ > 0, κ = 0, and κ < 0. In the following we
discuss these situations in detail.
that α > (4m − t ). Here r+ and r− correspond to outer
and inner horizons for trapping surface, respectively.
• κ = 0 leads to two positive real values of r , r+ =
that α > (4m − t ). Here r+ and r− correspond to
outer and inner horizons for marginally trapping surface,
respectively.
• κ < 0 leads to two positive real values of r , r+ <
that α > (4m − t ). Here r+ and r− correspond to
outer and inner absolutely nontrapping 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
ds2 =
t
1 + α
f (r )dt 2 − ( f (r ))−1dr 2
− r 2(dθ 2 + sin θ dφ2) ,
where f (r ) = 1 − rr∗ with r∗ = 2M . Further suppose that
v(r, t ) = 1+ αt f (r ). Clearly, for values of r > r∗ this
solution is positive definite and a coordinate singularity occurs
at r = r∗. The coordinate t is identified periodically with
period
t
1 − α
In the limit of large r , the Killing vector ∂∂t is normalized
to 1. The temperature measure to infinity may be formally
identified with inverse of this period. Hence by the Tolman
law, for any self gravitating system in thermal equilibrium,
a local observer will measure a local temperature T which
1
scales as g1−12 [29]. The constant of proportionality in the
present context is
t
1 + α
The wall temperature TW and surface area AW = 4πr W2 are
defined by York [30]. One topologically regular solution to
the Einstein equation with these boundary conditions is a hot
flat space with uniform temperature TW . Another solution is
the Schwarzschild metric. If a BH of horizon r∗ < rW does
exist then the wall temperature from the Tolman law must
satisfy
TW =
t
1 + α
In terms of rW and TW , this equation may be solved for r∗.
There is no real positive root for r∗, if rW TW < √8π27 [29].
For any value of rW and TW , the entropy of the BH solution
to Eq. (31) is S = πr∗2. The heat capacity of a constant surface
for any solution is
C A = TW ∂∂TSW AW = −2πra2st 1 − rrW∗
The heat capacity is positive and the equilibrium
configuration is locally thermally stable if r∗ < rW < 32r∗ .
−1
3.3 Energy conditions
Consider the line element of time conformal Schwarzschild
BH of Eq. (22). Let us suppose that the matter distribution is
isotropic in nature, whose energymomentum tensor is given
by
Tμν = (ρ + p)uμuν − pgμν .
Here the vector ui is the fluid fourvelocity with ui =
(√g11, 0, 0, 0), ρ is the matter density, and p is the pressure.
Taking G = 1 = c, from the Einstein field equations one
can deduce that
Now, we want to check the energy conditions for a time
conformal Schwarzschild BH. For the null energy condition
(NEC), the weak energy condition (WEC), the strong energy
condition (SEC), and the dominant energy condition (DEC),
the following inequalities must be satisfied:
NEC: ρ + p ≥ 0.
WEC: ρ + p ≥ 0, ρ ≥ 0.
SEC: ρ + p ≥ 0, ρ + 3 p ≥ 0.
DEC: ρ ≥  p.
From Eqs. (34) and (35), we can see that the NEC, WEC,
and DEC satisfy by time conformal Schwarzschild BH, on
the other hand, SEC violates it.
4 Dynamics of the neutral particle
We discuss the dynamics of a neutral particle around the time
conformal Schwarzschild BH defined by (1). The
approximate energy Eapprox and the approximate angular momentum
L z are given in Table 2. The total angular momentum in the
(θ , φ) plane can be calculated from φ4 and φ5 and it has the
value
Table 2 First integrals
Eapprox = 2 1 − 2rM t˙ + α 2tt˙ 1 − 2rM − s L
Lag = (1 + αt 2M −1r˙2
− r 2 θ˙2 + sin)2 θ1φ˙−2 2rM t˙2 − 1 − r
φ4 = −(1 + αt )r 2(cos φθ˙ − cot θ sin φφ˙)
φ5 = −(1 + αt )r 2(sin φθ˙ + cot θ cos φφ˙)
Eapprox =
L 2 =
t
1 + α
t
1 + α
s
E − α
E =
1 −
Using the normalization condition given in Eq. (36), we can
get the approximate equation of motion of the neutral particle,
r˙2 = E 2
1 −
t
1 − α
E 2 =
1 −
= Ueff . (38)
The effective potential extreme values are obtained by
d Udreff = 0. The convolution point of the effective potential
lies in the inner most circular orbit (ISCO);
t
1 − α
(L 2 + 12M 2) − (L 2 + 9M 2) 4t
α
The corresponding azimuthal angular momentum and the
energy of the particle at the ISCO are
E02 =
t
Mr 1 + α ,
3m
1 − r
Consider the circular orbit r = r0 of a particle, where
r0 is for the local minimum of the effective potential. This
orbit exists for r0 ∈ (4M, ∞). The convolution point of the
effective potential for ISCO is defined by r0 = 4M [31].
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 the BH, and (iii)
escape to ∞. The results depend upon the process of the
collision. The orbit of a particle is slightly changed but remains
bounded for small changes in energy and momentum.
Otherwise, it can be moved from initial position and captured
by the BH or escape to infinity. After collision, the energy
and both angular momenta (total and azimuthal) change [3].
Before simplifying the situation, one can apply some
conditions, i.e., the azimuthal angular momentum and initial radial
velocity do not change but the energy can change, by which
we determine the motion of the particle. Hence the effective
potential becomes
E 2 = f (r )
This energy is greater than the energy of the particle before
collision, because after collision the colliding particle gives
some of its energy to the orbiting particle. Simplifying Eq.
(40), we obtain the escape velocity as follows:
v⊥ =
= Ueff .
4.1 Behavior of the effective potential of the neutral particle
We analyze the trajectories of the effective potential and
explain the conditions on energy required for bound motion
or escape to infinity around the Schwarzschild BH. Figure 1
represents the behavior of the effective potential of a particle
moving around the Schwarzschild BH for different values
of the angular momentum. We can see from Fig. 1 that the
maxima of the effective potential for L z = 5, 10, 15 are at
r ≈ 0.5, 1, 1.5 for the time interval [0, 0.5). Similarly, the
minima of the effective potential for L z = 5, 10, 15 are at
r ≈ 0.5, 1, 1.5 for the time interval (0.5, 1]. Hence, it is
concluded that maxima and minima are shifting forward for large
Fig. 1 The plot of the effective potential (Ueff ) versus r and t for
α = = 1, M = 10−16
= 1, M = 10−16
values of the angular momentum. Also the repulsion and
attraction of effective force depends upon time. For the time
interval [0, 0.5), there is a strong attraction near the BH and
it vanishes when the radial coordinate approaches infinity.
Moreover, for the time interval (0.5, 1] there is strong
repulsion near the BH and it vanishes for r approaching infinity.
For t = 0.5, we find the equilibrium i.e. there is no attraction
and repulsion in the effective potential. Hence, the effective
potential is shifting from attraction to repulsion when time
increases and the shifting point of time is t = 0.5. Hence,
it is concluded that the effective potential is attractive and
repulsive with respect to time, maxima and minima of the
effective potential shifting forward in radial coordinate for
large values of the angular momentum.
4.2 Behavior of effective force of the neutral particle The effective force is
1 −
We are studying the motion of the neutral particle in the
surrounding of a Schwarzschild BH where attractive and
repulsive gravitational forces are produced by a scalar vector
tensor field which prevents a particle to fall into the singularity
[32]. The comparison of the effective force on the particle
around the Schwarzschild BH as a function of the radial
coordinate for different values of the angular momentum is shown
in Fig. 2. We can see from this figure that, at time interval
[0, 0.5), the attraction of a particle to reach the singularity is
higher for L z = 15 as compared to L z = 5, 10. Similarly, at
Fig. 3 Plot of escape velocity versus r and t for E = α =
M = 10−16
= 1,
time interval (0.5, 1], the repulsion of a particle to reach the
singularity is higher for L z = 15 as compared to L z = 5, 10.
Also, at t = 0.5 there is no effective force. t = 0.5 acts
as a shifting point, i.e., the effective force is shifting from
attraction to repulsion at t = 0.5. We can conclude that the
attraction and repulsion of a particle to reach and escape
from the singularity is higher for large values of the angular
momentum but it mainly depends upon time.
4.3 Trajectories of escape velocity of neutral particle
Figure 3 represents the trajectories of the escape velocity for
different values of L z as a function of r . It is evident that if
the angular momentum is positive then the escape velocity is
repulsive while it is attractive for negative values of the
angular momentum. Also, the repulsion of the escape velocity for
large values of L z is strong as compared to the small values
of L z . On the other hand, the attraction of the escape velocity
for L z = −15 is strong as compared to L z = −5, −15. We
can conclude that the escape velocity of a particle increases
as the angular momentum increases, but it becomes almost
constant away from the BH. Also it is interesting that the
trajectories of the escape velocity do not vary with respect to
time.
5 Dynamics of the charge particle
Now we consider the dynamics of the charged particle around
the time conformal Schwarzschild BH defined by (1). We
assume that a particle has electric charge and its motion
is affected by the magnetic field in BH exterior. Also, we
assume that there exists a magnetic field of strength (B) in
the neighborhood of BH which is homogeneous, static, and
The Killing vectors correspond to a 4potential which is
invariant under symmetries as follows:
Lξ Aμ = Aμ,ν ξ ν + Aν ξ,νμ = 0.
Using the magnetic field vector [20]
Bμ = − 21 eμνλσ Fλσ uν ,
where eμνλσ = √μ−νλgσ , 0123 = 1, g = det(gμν ), and μνλσ
is the LeviCivita symbol. The Maxwell tensor is
In the metric (1), for a local observer at rest, we have
The remaining two components are zero at r˙ = 0 (turning
point). From Eqs. (44) and (46), we get
L = 21 gμν uμuν + qmA uμ,
where m and q are the mass and electric charge of the particle
respectively. The generalized 4momentum of a particle is
here the magnetic field is directed along the zaxis (vertical
direction). Since the field is directed upward, we have B > 0.
The Lagrangian of a particle moving in curved spacetime
is given by [36]
axisymmetric at spatial infinity. Next, we follow the
procedure of [33] to construct the magnetic field. Using metric (1),
the general Killing vector is [34]
where ξ μ is the Killing vector. Using the above equation in
the Maxwell equation for the 4potential Aμ in Lorenz gauge
μ
A;μ = 0, we have [35]
r (r − 2M )
− B
The new constants of motion are defined as
t˙ = E1 1−−2rMαt , φ˙ = Lrz 2 1si−n2 θαt
− B,
where B = 2qmB . Using the constraints in the Lagrangian (12)
the dynamical equations for θ and r become
− B 2
− 2 α(r θ−˙E2rM ) − 4 r˙rθ˙ ,
Using the normalization condition, we obtain
E 2 = r˙2 + r 2 1 −
− B
The effective potential takes the form
− B
The energy of the particle moving around the BH in orbit r
at the equatorial plane is
Ueff =
− B
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. (53) which are invariant under the transformation as
follows:
φ → −φ, L z → −L z , B → −B.
Therefore, without the loss of generality, we consider B > 0
and for B < 0 we should apply transformation (58) because
both negative and positive charges are inter related by the
above transformation. However, if one chooses a positive
electric charge (B > 0) then both cases of L z (positive and
negative) must be studied. They are physically different: the
change of sign of L z corresponds to the change in direction
of the Lorentz force acting on the particle [20].
The system (52)–(54) is also invariant with respect to
reflection, θ → π − θ . This transformation preserves the
initial position of the particle and changes v⊥ → −v⊥.
Therefore it is sufficient to consider the positive value of escape
velocity (v⊥) [20]. By differentiating Eq. (57) with respect
to r , we obtain
− 2r B
1 −
− B
− B
The effective potential after the collision when the body
in the magnetic field (θ = π2 and r˙ = 0) is
t
1 − α
− B
v⊥ = ⎣
The escape velocity of the particle takes the form
t
1 + α
Differentiating Eq. (59) again with respect to r we have
t
1 − α
+ (4M − r )
− B
5.1 Behavior of the effective potential of the charged
particle
Figure 4 represents the behavior of the effective potential as a
function of radial coordinate for different values of magnetic
field B. The minimum of the effective potential at B = 0.5
is approximately at r = 2 initially but as time increases it is
shifting near the BH. Hence the presence of a magnetic field
increases the possibility for a particle to move in a stable orbit.
Similarly, initially the minimum of the effective potential at
B = 1, 1.5 is approximately at r = 1.5, 1.25, respectively;
as time increases it approaches near the BH. One may notice
that in the presence of a low magnetic field, the minimum of
the effective potential is shifted away from the horizon and
the width of ISCO is also decreased as compared to a high
Fig. 4 Graph of the effective potential versus r and t given in Eq. (57)
for α = = 1, M = 10−16 and L z = 3
magnetic field. These results are in agreement with [17, 33].
Therefore we can say that an increase in magnetic field acts as
an increase of the instable orbits of a particle. Another aspect
of the figure is that, at initial time, we have the possibility of
a stable orbit but as time increases the possibility of ISCO
gets low and at t = 1, the stable orbit is lost. Hence it is
possible that the particle is captured by the BH or it escapes
to infinity.
5.2 Behavior of effective force of the charged particle
The effective force on the particle can be defined as
− B
1 −
− B .
We have plotted the effective force for different values of
B as a function of r . We see from Fig. 5 that the effective
force is more attractive for large values of a magnetic field as
compared to small values. We can conclude that the effective
force of a particle increases as the strength of a magnetic field
increases near the BH initially but with the passage of time,
the particle moves away from the BH and it becomes almost
constant.
5.3 Trajectories of escape velocity of the charged particle
From Eq. (52), for the angular variable we have
− B.
Fig. 5 Graph of effective force versus r and t given in Eq. (63) for
α = = 1, M = 10−16, and Lz = 3
If the Lorentz force on a particle is attractive then the left hand
side of the above equation is negative [37] and vice versa. The
motion of the charged particles is in the clockwise direction.
Our main focus on the magnetic field acting on a particle, the
large value of the magnetic field deforms the orbital motion
of a particle as compared to small values. Hence, we can
conclude that the possibility to escape for the particle from
the ISCO is greater for large values of the magnetic field. We
also explain the behavior of the escape velocity for different
values of a magnetic field in Fig. 6 graphically. The escape
velocity of a particle increases for large values of the
magnetic field as compared to small values. We can conclude that
the presence of a magnetic field provides more energy to the
particle to escape from the vicinity of BH. Figure 7
represents the escape velocity against r for different values of the
angular momentum. The possibility of the particle to escape
is low for large values of the angular momentum. Also as time
increases the possibility for a particle to escape is decreasing
drastically.
6 Stability orbit The Lyapunov function is defined as [38] (65)
Using Eq. (62) we can get the Lyapunov function as
t
+ B r L z − r 3 B 1 + α
− 2L z(r − 2M)
− B
Fig. 6 Plot of escape velocity versus r and t given in Eq. (61) for
E = α = = 1, M = 10−16, and Lz = 5
Fig. 7 Plot of escape velocity versus r and t given in Eq. (61) for
E = α = = 1, M = 10−16, and B = 0.5
The behavior of λ as a function of α is analyzed Fig. 8.
We can see from the figure that an instability of the circular
orbit increases as α increases but it becomes constant
7 Conclusion and observations
In the present work, we found that one of the Noether
symmetries admits an approximate part. This symmetry is the
translation in time which means that the energy of the
spacetime is rescaling. We also found the conservation laws
corresponding to the exact Schwarzschild spacetime and the
time conformal Schwarzschild spacetime and compare their
numerical solutions. The geodesic deviation has also
preFig. 8 Lyapunov exponent as a function for α for B =
10−16, Lz = 3, r = 1, and t = 1
= 1, M =
sented. The perturbation in the spacetime has perturbed the
geodesic. The numerical solutions showed how much they
deviated from each other.
Using Noether symmetries, we have calculated the
equation of motion. We have found three types of approximate
Noether symmetries which correspond to energy, scaling,
and Lorentz transformation in the conformal plane
symmetric spacetimes. These symmetries approximate the
corresponding quantities in the respective spacetimes. We have
not seen approximate Noether symmetries corresponding to
linear momentum, angular momentum, and galilean
transformation in our calculations. This shows that these quantities
are conserved for plane symmetric spacetimes. The
spacetime section of zero curvature does not admit an
approximate Noether symmetry [39] which shows that the
approximate symmetries disappeared whenever we have a section
of zero curvature in the spacetimes. However, the
approximate Noether symmetry does not exist in flat spacetimes
(Minkowski spacetime).
In addition, we have investigated the motion of neutral
and charged particles in the absence and presence of a
magnetic field around the time conformal Schwarzschild BH.
The behavior of the effective potential, effective force, and
escape velocity of neutral and charged particles with respect
to time are discussed. In the case of neutral particle, for large
values of the angular momentum, we have found attraction
and repulsion with respect to time as shown in Figs. 1, 2, and
3. This effect decreases far away from the BH with respect
to time. The more aggressive attraction and repulsion of a
particle to reach and escape from the BH have been observed
for large values of the angular momentum due to the time
dependence. The escape velocity increases as the angular
momentum increases but it does not vary with respect to
time.
In the case of the charged particle, the presence of a high
magnetic field shifted the minimum of the effective
potential toward the horizon and the width of the stable region is
increased as compared to a low magnetic field. The escape
velocity for different values of the angular momentum and
magnetic field is shown in Figs. 6 and 7. It is found that the
presence of a magnetic field provided more energy to the
particle to escape. The possibility for a particle to escape is
low for a large angular momentum and as time increases it
decreases continuously.
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1. V.P. Frolov , I.D. Novikov , Black Hole Physics, Basic Concepts and New Developments (Springer, Berlin, 1998 )
2. N.A. Sharp , Gen. Relativ. Gravit. 10 , 659 ( 1979 )
3. C.V. Borm , M. Spaans , Astron. Astrophys. 553 , L9 ( 2013 )
4. R. Znajek , Nature 262 , 270 ( 1976 )
5. R.D. Blandford , R.L. Znajek , Mon. Not. R. Astron . Soc. 179 , 433 ( 1977 )
6. S. Koide , K. Shibata , T. Kudoh , D.l. Meier, Science 295 , 1688 ( 2002 )
7. S. Wachter et al., Nature 453 , 626 ( 2008 )
8. J. Braithwaite , H.C. Spruit , Nature 431 , 819 ( 2004 )
9. L. Ferrario , D.T. Wickramasinghe , MNRAS 356 , 615 ( 2005 )
10. Y.Q. Lou , W.G. Wang , MNRAS 378 , L54 ( 2007 )
11. Y.Q. Lou , MNRAS 275 , L11 ( 1995 )
12. W.G. Wang , Y.Q. Lou , ApSS 315 , 135 ( 2008 )
13. A. Neronov , I. Vovk, Science 328 , 73 ( 2010 )
14. I. Vovk , A.M. Taylor , D. Semikoz , A. Neronov , Astrophys . J. 747 , L14 ( 2012 )
15. K.N. Mishra , D.K. Chakraborty , Astrophys. Space Sci . 260 , 441 ( 1999 )
16. E. Teo , Gen. Relativ. Gravit. 35 , 1909 ( 2003 )
17. S. Hussain , I. Hussain , M. Jamil , Eur. Phys. J. C 74 , 3210 ( 2014 )
18. G.Z. Babar , M. Jamil , Y.K. Lim , Int. J. Mod. Phys. D 25 , 1650024 ( 2016 )
19. D. Pugliese , H. Quevedo , R. Ruffni , Phys. Rev. D 83 , 104052 ( 2011 )
20. A.M.A. Zahrani , V.P. Frolov , A.A. Shoom , Phys. Rev. D 87 , 084043 ( 2013 )
21. F. Ali , Appl. Math. Sci. 8 , 4679 ( 2014 )
22. F. Ali , T. Feroze , S. Ali , Theor. Math. Phys. 184 , 92 ( 2015 )
23. F. Ali , Mod. Phys. Lett. A 30 , 1550028 ( 2015 )
24. F. Ali , T. Feroze , Int. J. Theor. Phys . 52 ( 9 ), 3329 ( 2013 )
25. J.M.M. Senovilla, Gen. Relativ. Gravit. 29 , 701 ( 1997 )
26. J.M.M. Senovilla, Int. J. Mod. Phys. Conf. Ser . 7 , 1 ( 2012 )
27. I. Bengtsson , J.M.M. Senovilla , Phys. Rev. D 83 , 044012 ( 2011 )
28. J.M.M. Senovilla, Class. Quantum Grav . 19 , L113 ( 2002 )
29. T. Prestidge , Phys. Rev. D 61 , 084002 ( 2000 )
30. J.W. York , Phys. Rev. D 33 , 2092 ( 1986 )
31. S. Chandrasekher , The Mathematical Theory of Black Holes (Oxford University Press, Oxford, 1983 )
32. J.W. Moffat , JCAP 0603 , 3004 ( 2006 )
33. A.N. Aliev , D.V. Gal 'tsov, Sov. Phys. Usp . 32 ( 1 ), 75 ( 1989 )
34. R.M. Wald, Phys. Rev. D 10 , 1680 ( 1974 )
35. A.N. Aliev , N. Ozdemir , Mon. Not. R. Astron . Soc. 336 , 241 ( 1978 )
36. L.D. Landau , E.M. Lifshitz , The Classical Theory of Fields (Pergamon Press, Oxford, 1975 )
37. V.P. Frolov , A.A. Shoom , Phys. Rev. D 82 , 084034 ( 2010 )
38. V. Cardoso , A.S. Miranda , E. Berti , H. Witech , V.T. Zanchin , Phys. Rev. D 79 , 064016 ( 2009 )
39. T. Feroze , F. Ali , J. Geom. Phys . 80 , 88 ( 2014 )