Motion of the charged test particles in Kerr-Newman-Taub-NUT spacetime and analytical solutions

Abstract

In this work, we study the motion of charged test particles in
Kerr-Newman-Taub-NUT spacetime. We analyze the angular and the radial parts of
the orbit equations and examine the possible orbit types. We also investigate
the spherical orbits and their stabilities. Furthermore, we obtain the
analytical solutions of the equations of motion and express them in terms of
Jacobian and Weierstrass elliptic functions. Finally, we discuss the
observables of the bound motion and calculate the perihelion shift and
Lense-Thirring effect for three dimensional bound orbits.

Full text

arXiv:1512.08682v1 [gr-qc] 29 Dec 2015
Motion of the charged test particles in
Kerr-Newman-Taub-NUT spacetime and
analytical solutions
Hakan Cebeci∗
Department of Physics, Anadolu University, 26470 Eski¸sehir, Turkey
N¨ulifer ¨
Ozdemir†
Department of Mathematics, Anadolu University, 26470 Eski¸sehir, Turkey
Se¸cil S¸entorun‡
Department of Physics, Anadolu University, 26470 Eski¸sehir, Turkey
Abstract
In this work, we study the motion of charged test particles in Kerr-
Newman-Taub-NUT spacetime. We analyze the angular and the ra-
dial parts of the orbit equations and examine the possible orbit types.
We also investigate the spherical orbits and their stabilities. Further-
more, we obtain the analytical solutions of the equations of motion and
express them in terms of Jacobian and Weierstrass elliptic functions.
Finally, we discuss the observables of the bound motion and calculate
the perihelion shift and Lense-Thirring effect for three dimensional
bound orbits.
PACS numbers: 04.20Jb, 02.30.Gp, 02.30.Hq
∗E.mail: hcebeci@anadolu.edu.tr
†E.mail: nozdemir@anadolu.edu.tr
‡E.mail: secilo@anadolu.edu.tr
1

1 Introduction
The Kerr-Newman-Taub-NUT spacetime is known as a stationary axially
symmetric solution of Einstein-Maxwell field equations. The spacetime rep-
resents a rotating electrically charged source equipped with a gravitomagnetic
monopole moment which is also identified as the NUT charge [1, 2]. The so-
lution contains four physical parameters: The gravitational mass, which is
also called gravitoelectric charge; the gravitomagnetic mass (also known as
the NUT charge); the rotation parameter that is the angular speed per unit
mass and electric charge associated with the Maxwell field. As is well known,
the NUT charge produces an asymptotically non-flat spacetime in contrast to
Kerr geometry that is asymptotically flat [3]. Although the Kerr-Newman-
Taub-NUT spacetime has no curvature singularities, there exist conical sin-
gularities on the axis of symmetry as in its uncharged version (namely the
Kerr-Taub-NUT spacetime) [4]. One can get rid off conical singularities by
taking a periodicity condition over the time coordinate. But, this leads to
the emergence of closed time-like curves in the spacetime as in its uncharged
version. It means that, in contrast to Kerr and Kerr-Newman solutions in-
terpreted as regular rotating black holes, the Kerr-Newman-Taub-NUT solu-
tion cannot be identified as a regular black hole solution due to its singularity
structure. Although charged and uncharged spacetimes with NUT parameter
have unpleasing physical properties, its worth to investigate such spacetimes
in general relativity due to their asymptotically non-flat spacetime struc-
tures. To explore their various physical phenomena the spacetimes with
NUT charge have been vastly studied in the works [5, 6, 7, 8, 9, 10, 11, 12]
where in [5], an alternative physical interpretation of the NUT parameter is
also illustrated.
One way to explore the properties of Kerr-Newman-Taub-NUT spacetime
is to study the motion of (charged) test particles in this background. One
can solve the equations of motion analytically and examine the effect of the
NUT parameter and charge of the test particle. Moreover, by integrating
equations of motion for bound orbits, one can also calculate the precession
of the orbital motion and Lense-Thirring effect.
The geodesic motion of test particles were first examined analytically
in Schwarzchild spacetime in [13]. Later on, the motion of test particles
was extensively investigated in Kerr spacetimes where the circular geodesics
has also been examined [14, 15, 16, 17]. Recently, in [18], analytic solu-
tions of the bound timelike geodesics of test particles in Kerr spacetime
1

have been presented in terms of elliptic integrals using Mino time. In [19]
and [20], the geodesic equations are analytically solved in the background of
Schwarzschild-(anti) de Sitter spacetimes, where the solutions are expressed
in terms of Kleinian sigma functions. In [21], investigation of analytic solu-
tions has been extended to Kerr-(anti) de Sitter spacetimes where in this case
the solutions are presented in terms of Weierstrass elliptic functions. In a sim-
ilar fashion, geodesic equations are solved in the spacetime of Kerr black hole
pierced by a cosmic string [22], where the perihelion shift and Lense-Thirring
effect have also been investigated for bound orbits. In [23], the geodesic
equations are analytically examined in the background of Einstein-Maxwell-
dilaton-axion black hole, where the effect of dilaton charge is investigated.
Similarly, analytic solutions of geodesic equations are given in higher dimen-
sional static spherically symmetric spacetimes [24]. Likewise, in [25] and [26],
the equation of geodesic motion have been examined in singly and doubly
spinning black ring spacetimes respectively. In addition, the orbital motion
of electrically and magnetically charged test particles have been studied in
the background of Reissner-Nordstr¨om [27] and Kerr-Newman spacetimes
[28], where the effect of the charge of the test particle has been observed.
In this work, using Hamilton-Jacobi method, we derive the equations
of motion for a charged test particle in the background of Kerr-Newman-
Taub-NUT spacetime. By making a transformation on the time variable,
we decouple the radial and angular (θ−)part of the equations of motion and
express all the differential equations in terms of Mino time [29]. We analyze
the angular and radial part of the orbit equations and examine the possible
orbit types including the special spherical orbits. Furthermore, we obtain the
energy and the orbital angular momentum of the test particle for a spherical
orbit. We also examine the stability of spherical orbits with respect to NUT
parameter. Next, we present the analytical solutions of the equations of
the motion by expressing them in terms of Weierstrass ℘,σ, and ζfunctions.
Additionally, we calculate the angular frequencies for the bounded radial and
angular motions and examine the perihelion precision and the Lense-Thirring
effect.
Organization of the paper is as follows: In section 2, we introduce Kerr-
Newman-Taub-NUT spacetime. In section 3, we derive the equations of
motion of the test particles by expressing them in terms of Mino time as
well. In section 4, we make a comprehensive analysis of the angular and radial
motion where we also examine the spherical orbits as a subsection. In chapter
5, we give detailed analytical solutions of all the equations that describe the
2

orbital motion. Next, we discuss possible orbit types by obtaining plots of the
different orbits with respect to spacetime parameters, the charge, the energy
and the orbital angular momentum of the test particle. We also calculate
and examine perihellion precision and the Lense-Thirring effect for the bound
orbits. We end up with some comments and conclusions.
2 Kerr-Newman-Taub-NUT spacetime
The Kerr-Newman-Taub-NUT spacetime is a stationary solution of the Einstein-
Maxwell field equations that is asymptotically non-flat. The metric describes
a rotating electrically charged source that also includes a NUT charge which
is also known as gravitomagnetic monopole moment. In Boyer-Lindquist
coordinates, Kerr-Newman-Taub-NUT spacetime can be described by the
metric with asymptotically non-flat structure,
g=−∆
Σ(dt −χdϕ)2+ Σ dr2
∆+dθ2+sin2θ
Σadt −(r2+ℓ2+a2)dϕ2
(2.1)
where
Σ = r2+ (ℓ+acos θ)2,
∆ = r2−2Mr +a2−ℓ2+Q2,(2.2)
χ=asin2θ−2ℓcos θ.
Here, Mis a parameter related to physical mass of the gravitational source.
ais associated with its angular momentum per unit mass and ℓdenotes
gravitomagnetic monopole moment of the source which is also identified as
the NUT charge. Qis the electric charge. For future use, we also include the
inverse metric components:
gtt =−1
∆Σ (r2+a2+ℓ2)2−a2sin2θ∆ + 4ℓ∆cot θ
sin θ(χ+ℓcos θ),(2.3)
grr =∆
Σ, gθθ =1
Σ, gϕϕ =1
∆Σ sin2θ(∆ −a2sin2θ),(2.4)
gtϕ =−1
∆Σ (2Mr + 2ℓ2−Q2)a+ 2ℓ∆cot θ
sin θ.(2.5)
3

The electromagnetic field of the source can be given by the potential 1-form
A=−Qr
Σ(dt −χdϕ) = Atdt +Aϕdϕ, (2.6)
where
At=−Qr
Σ, Aϕ=χQr
Σ.(2.7)
Interestingly, although the spacetime cannot be identified as a black hole, it
has metric singularities at the locations
r±=M±pM2−a2+ℓ2−Q2,(2.8)
where ∆ = 0. It can be seen that, the spacetime allows a family of locally
non-rotating observers which rotate with coordinate angular velocity given
by
Ω = −gtφ
gφφ
=∆χ−asin2θ(r2+ℓ2+a2)
∆χ2−sin2θ(r2+ℓ2+a2)2.(2.9)
Then, at the outermost singularity r+, the angular velocity can be calculated
as
Ω+=−gtφ
gφφ r=r+
=a
r2
++ℓ2+a2·(2.10)
It is also obvious that the Killing vectors ξ(t)and ξ(φ)generate two constants
of motion namely the energy and the angular momentum. It can be shown
that the Killing vector ξ=ξ(t)+Ω+ξ(φ)becomes null at the metric singularity
where r=r+.
3 The motion of charged test particles
In this section, we examine the motion of charged test particle in Kerr-
Newman-Taub-NUT spacetime. We choose the units such that we take the
speed of light c= 1. To this end, we introduce the Hamilton-Jacobi equation
for a charged particle
2∂S
∂τ =gµν ∂S
∂xµ−qAµ∂S
∂xν−qAν(3.1)
where τis an affine parameter and qis the charge of the particle. Since the
spacetime (2.1) admits the timelike Killing vector ξ(t)and spacelike Killing
4

vector ξ(φ), the solution of the Hamilton-Jacobi equation can be written as
S=−1
2m2τ−Et +Lϕ +f(r, θ) (3.2)
where f(r, θ) is a function of the variables rand θ, the constants of motion
m,Eand Ldenote the mass, the energy and the angular momentum of the
particle respectively. Furthermore, the separability of the Hamilton-Jacobi
equation [30, 31, 32] in Kerr-Newman-Tab-NUT spacetime implies that the
function f(r, θ) can be expressed as a sum of two different functions which
only depend on rand θindependently, i.e.
f(r, θ) = Sr(r) + Sθ(θ).(3.3)
The substitution of (3.3) together with (2.3), (2.4), (2.5), (2.7) and (3.2) into
Hamilton-Jacobi equation (3.1) results in two differential equations
dSr
dr 2
=1
∆−K−m2r2+1
∆(r2+a2+ℓ2)E−aL −qQr2(3.4)
and dSθ
dθ 2
=K−m2(ℓ+acos θ)2−χE −L
sin θ2
,(3.5)
where Kcan be identified as the Carter separability constant. Using canon-
ical momenta Pµexpression
Pµ=∂S
∂xµ=mgµν
dxν
dτ +qAµ(3.6)
and identifying
Pt=−E, Pϕ=L(3.7)
we obtain the following equations of motion:
dr
dτ =∓1
Σs(r2+ℓ2+a2)¯
E−a¯
L−¯qQr2−∆K
m2+r2,(3.8)
dθ
dτ =∓1
ΣsK
m2−(ℓ+acos θ)2−χ¯
E−¯
L
sin θ2
,(3.9)
5

dt
dτ =1
Σ∆ sin2θ¯
L∆χ−asin2θ(r2+ℓ2+a2)(3.10)
+¯
E−∆χ2+ sin2θ(r2+ℓ2+a2)2−¯qQr sin2θ(r2+ℓ2+a2),
dϕ
dτ =1
Σ∆ sin2θ¯
E−∆χ+asin2θ(r2+ℓ2+a2)
+¯
L∆−a2sin2θ−¯qQrasin2θ(3.11)
where we define
¯
E:= E
m,¯
L:= L
m,¯q:= q
m·(3.12)
Defining a new time parameter λ(the so-called Mino time) as in [29] such
that dλ
dτ =1
Σ,(3.13)
we can express the equations of motion in terms of the new time parameter:
dr
dλ =∓pPr(r),(3.14)
dθ
dλ =∓pPθ(θ),(3.15)
dt
dλ =χ(¯
L−¯
Eχ)
sin2θpPθ(θ)
dθ
dλ
+ ( ¯
E(r2+a2+ℓ2)−a¯
L−¯qQr)(r2+a2+ℓ2)
∆pPr(r)
dr
dλ,(3.16)
dϕ
dλ =(¯
L−¯
Eχ)
sin2θpPθ(θ)
dθ
dλ
+a(r2+a2+ℓ2)−¯
La −¯qQr
∆pPr(r)
dr
dλ,(3.17)
where
Pr(r) = (r2+ℓ2+a2)¯
E−a¯
L−¯qQr2−∆K
m2+r2(3.18)
and
Pθ(θ) = K
m2−(ℓ+acos θ)2−χ¯
E−¯
L
sin θ2
.(3.19)
6

4 Analysis of the motion:
In this part, we make an analysis or angular and radial motion and examine
the possible orbit types. We also investigate the stability of spherical orbits.
4.1 Analysis of the angular motion (θ-motion)
Writing the angular equation as
dθ
dλ2
=Pθ(θ),(4.1)
we see that Pθ(θ)≥0 for the possibility of the motion where Pθ(θ) can also be
interpreted as the potential associated with the angular motion. Obviously,
it depends on the relation between spacetime parameters a,M,ℓand the
energy and the angular momentum of the test particle. Associated with these
parameters, either the motion is not possible (i.e Pθ(θ)<0 ) or it is confined
to an angular interval where θ1≤θ≤θ2(i.e the angular motion is bound).
We should also comment that depending on the values of the parameters,
θ=π
2can be included in that angular interval or not. It means that, the
angular motion of the test particle is bound such that the particle can cross
or cannot cross the equatorial plane depending on the spacetime parameters.
From the expression of Pθ(θ), one can easily see that if the condition
K
m2≥ℓ2+ (¯
L−a¯
E)2(4.2)
is satisfied, the test particle can cross the equatorial plane (θ=π
2). Other-
wise, the particle cannot pass through the equatorial plane but its motion is
restricted to θ1< θ < θ2,θ=π
2being outside of this interval. We further
remark that, the physical condition Pθ(θ) requires that when the spacetime
parameters (NUT parameter, rotation parameter) are fixed, a constraint re-
lation between the energy and the angular momentum of the test particle
can be developed as is also illustrated in [22]. Nevertheless, for our case, due
to the presence of NUT parameter ℓ, it is very cumbersome to get such a
relation between the energy, angular momentum for non-vanishing spacetime
parameters.
In the graphs (1a), (1b) and (1c), the angular potential Pθ(θ) is plotted
with respect to angular variable θwhere it can be seen that there may exist
7

(a) ¯
L=−0.4, K= 10, ¯
E= 3 (b) ¯
L= 0.5, K= 10, ¯
E= 0.96
(c) ¯
L= 0.5, K= 0.5, ¯
E= 5
Figure 1: Graphs of Pθ(θ) with parameters M= 1, a= 0.9, ℓ= 0.1, Q= 0.4,
¯q= 0.3 and m= 1
one or two angular bound orbits or no motion. It is seen that in the plot
(1a), there exist two angular intervals for which Pθ(θ)≥0. In the plot (1b)
however, only one physical interval seems to exist for the given spacetime
parameters. In both plots, it is also understood that the test particle can
cross the equatorial plane (since when θ=π
2,Pθ(θ)>0). On the other hand,
there exists no angular motion in plot (1c) since the condition Pθ(θ)>0 is
not satisfied for the given spacetime parameters.
In addition, it is also obvious that when
Pθ(θ) = 0 (4.3)
and dPθ(θ)
dθ = 0,(4.4)
the motion is confined to θ=θ0plane. These conditions lead to the following
equations:
K
m2θ=θ0
= (ℓ+acos θ0)2+χ0¯
E−¯
L
sin θ02
=: K0
m2,(4.5)
8

a(ℓ+acos θ0) sin θ0−χ0¯
E−¯
L
sin θ0acos θ0¯
E+2ℓ¯
E+¯
Lcos θ0
sin2θ0= 0,(4.6)
where
χ0=asin2θ0−2ℓcos θ0.(4.7)
By an analytic calculation, it can be seen that θ=π
2is not the solution
of these equations for arbitrary values of the parameters ℓ,a,¯
Eand ¯
L.
This means that for arbitrary values of gravitomagnetic monopole moment ℓ
(ℓ6= 0), there exist no equatorial plane orbits (also called equatorial geodesics
for the motion of uncharged test particle) as is also clearly stated in [6]. On
the other hand, for ℓ= 0, θ=π
2solves the above equations provided that
the Carter constant becomes
K
m2= (a¯
E−¯
L)2.(4.8)
It can be seen that, in the vanishing of gravitomagnetic monopole moment,
equatorial plane orbits can exist for arbitrary values of the parameters a,¯
E
and ¯
L. Interestingly, the equation (4.4) and (4.5) also admit the solution
θ0=π
2, if the constraint relation
¯
L=a(2 ¯
E2−1)
2¯
E(4.9)
is imposed between the angular momentum and the energy of the test particle
for arbitrary NUT parameter, provided that the Carter constant becomes
K0
m2=ℓ2+a2
4¯
E2in that case. It means that if the relation (4.9) is satisfied, the
test particle is confined to equatorial plane. Therefore, equatorial orbits can
also exist for arbitrary NUT parameter if the constraint relation (4.9) holds
between the angular momentum and the energy of the test particle and also
the spacetime rotation parameter. To our knowledge, this is a new result
that has not been mentioned in previous works.
One can also examine the case with vanishing rotation parameter afor ℓ6= 0.
It can be algebraically seen that when a= 0, the motion of the charged
particle is restricted to a cone with opening angle determined by either
cos θ0=−¯
L
2ℓ¯
Eor cos θ0=−2ℓ¯
E
¯
L. It implies that in the vanishing of the
rotation parameter, equatorial orbit may also exist either for the interesting
case ¯
L= 0 (and ℓ6= 0) or for the case ℓ= 0 (and ¯
L6= 0) which is also
discussed in [6] for the uncharged particle motion. As a further remark, we
should also point out that, the presence of the charge Qassociated with the
9

electromagnetic field and the charge qof the test particle does not affect the
angular motion.
4.2 Analysis of the radial motion (r-motion)
First, we express the radial equation (3.8) in the form
dr
dλ2
=Pr(r) (4.10)
where Pr(r) is a fourth order polynomial in rwith real coefficients. As in the
angular case, the possibility of the motion requires that Pr(r)≥0. Then for
r-motion, according to the roots of the polynomial Pr(r), one can identify
the following orbit types:
i. Bound Orbit: If the particle moves in a region r2< r < r1, then the
motion is bound. This can happen if Pr(r) has four positive real roots or two
positive real roots (with two complex roots) or two positive double roots or
one triple positive root and one real positive root. In such a case, there may
exist one or two bound regions.
ii. Flyby Orbit: If the particle starts from ∓∞, and comes to a point r=r1
and goes back to infinity, then the orbit is flyby. Likewise, flyby orbits can be
seen when Pr(r) has four positive real roots or two positive real roots (with
two complex roots) or two positive double roots or one triple positive root
and one real positive root. Similarly, for this case, there may exist one or
two flyby orbits.
iii. Transit Orbit: If the particle starts from ∓∞, crosses r= 0 and goes to
∓∞, then the orbit is transit. This is possible if Pr(r) has no real roots.
iv. Spherical Orbit: This is a special type of orbit, such that Pr(r) has a
real double root at r=rs.
Depending on the value of the energy of the test particle, one can further
examine the possible orbit types:
1. For ¯
E < 1: In that case, for physically acceptable motion Pr(r) can have
two or four real zeros since as r→ ∓∞,Pr(r)→ −∞. Then, there exist
10

either one bound orbit or two bound orbits.
2. For ¯
E > 1: For this case, possible types of orbits can be classified ac-
cording to whether Pr(r) has four real zeros, two real zeros or no real zeros.
When Pr(r) has no real zeros (in other words all the roots are complex), then
only the transit orbit is possible since in that case as r→ ∓∞,Pr(r)→ ∞.
On the other hand, if Pr(r) has two different real zeros (and two complex
conjugate roots), one can get two flyby orbits. Moreover, if Pr(r) has four
different real zeros, there exist either two bound orbits or one bound two
flyby orbits or even two bound two flyby orbits.
3. For ¯
E= 1: For this special case, Pr(r) can possess either three real roots
or one real root (with two complex conjugate roots). When Pr(r) has three
real roots, one can get bound and flyby orbits. In the case that Pr(r) has
only one real root, the possible orbit type is flyby.
In the graphs (2a), (2b) and (2c), some possible orbit types are illustrated.
For the first graph (2a), Pr(r) has no real roots so that the orbit is tran-
sit type. In graphs (2b) and (2c), there exist one and two bound regions
respectively.
Alternatively, one can express the equation (4.10) as
dr
dλ2
= (r2+ℓ2+a2)2¯
E−V+(r)¯
E−V−(r)(4.11)
where
V±(r, ¯
L, a, ℓ, ¯q, Q) =
a¯
L+ ¯qQr ±q∆(r)K
m2+r2
r2+ℓ2+a2(4.12)
can be identified as the effective radial potentials. When Pr(r) = 0, the
r-motion has turning points determined by
¯
E=V±(r, ¯
L, a, ℓ, ¯q, Q)r=r0=
a¯
L+ ¯qQr0±q∆(r0)K
m2+r2
0
r2
0+ℓ2+a2(4.13)
where r0’s denote turning points of the motion. Concentrating on V+(r, ¯
L, a, ℓ, ¯q, Q),
one can analyze the characteristics of the radial motion by plotting V+as a
function of distance r. One can further see that as r→ ∞,V+→1. In
the figure 3, different energy levels determine the turning points of the radial
11

(a) Transit orbit with ℓ= 0.4,
¯
E= 10
(b) One bound orbit with ℓ=
0.4, ¯
E= 0.94
(c) Two bound orbits with ℓ= 0.1, ¯
E= 0.96
Figure 2: Graphs of Pr(r) with parameters M= 1, a= 0.9, K= 10, Q= 0.4,
¯q= 0.3, ¯
L= 0.5 and m= 1.
12

Figure 3: The graph of V+with parameters M= 1, a= 0.9, ℓ= 0.1, Q= 0.4,
¯q= 0.3, ¯
L= 0.5, K= 10, m= 1 which describes possible types of orbits.
motion. As explicitly illustrated in [14], a bound motion is possible if there
exist three or more turning points on the figure of effective potential V+[14].
It is also illustrative to examine the variation of the effective radial potential
for different values of NUT parameter ℓ(the other parameters are fixed)
and the charge ¯qof the test particle. As can be seen from the graphs, the
potentials have some local maxima and minima recalling that Pr(r)→1 as
r→ ∞. From the first graph, for the fixed parameters M= 1, a= 0.9,
Q= 0.4, ¯q= 0.3, ¯
L= 0.5, K= 10 and m= 1, one can realize that as
the value of NUT parameter ℓincreases, local maxima tends to disappear.
On the other hand, one can see from the second graph that for the same
parameter values (except that we take ℓ= 0.1 for this case) as the charge ¯q
of the test particle decreases, the local maxima has a similar behaviour.
It is also of great interest to investigate the existence of the bound orbits
in the region where r > r+(in other words outside the singularity of the
spacetime). This means that a radial bound interval r1≤r≤r2exists such
that for that region, Pr(r)>0 and r+< r1. To make an analysis of bound
motion similar to [14], we can affect a transformation r=: R+r+, where r+
describes the metric singularity (i.e. ∆(r+) = 0), assuming that at least one
region of binding exists where r > r+. To this end, we express Pr(r) in terms
of new variable R. Then in terms of R, we obtain
PR(R) = A4R4+A3R3+A2R2+A1R+A0(4.14)
13

Figure 4: The graph of V+as a function of r, with parameters M= 1,
a= 0.9, Q= 0.4, ¯q= 0.3, ¯
L= 0.5, K= 10, m= 1 for different values of ℓ.
Figure 5: The graph of V+as a function of r, with parameters M= 1,
a= 0.9, ℓ= 0.1, Q= 0.4, ¯
L= 0.5, K= 10, m= 1 for different values of ¯q.
14

where
A0=¯
E(ℓ2+a2+r2
+)−(a¯
L+ ¯qQr+)2,(4.15)
A1= 4 ¯
E2r+(r2
++ℓ2+a2)−2¯
E3¯qQr2
++ 2¯
Lar++ ¯qQ(ℓ2+a2)
+2¯qQ(¯qQr++a¯
L)−2K
m2+r2
+(r+−M) (4.16)
A2= 6( ¯
E2−1)r2
++ 6(M−¯qQ ¯
E)r++ 2 ¯
E2(ℓ2+a2)
−2¯
E¯
La + ¯q2Q2−K
m2−(a2−ℓ2+Q2) (4.17)
A3= 2 2r+(¯
E2−1) + M−¯qQ ¯
E,(4.18)
A4=¯
E2−1.(4.19)
At this stage, let us consider in what conditions this polynomial has positive
roots. We remark that A0>0. According to Descartes’ rule of sign, a
polynomial possessing real coefficients can not have more positive roots than
the number of variations of sign in its coefficients. Then if A4>0 ( ¯
E2>1),
a bound region for r > r+can be realized under the conditions
A1<0, A2>0, A3<0 (4.20)
since we also have A0>0. Then four variations of sign would be possible
and therefore there may exist four real positive roots for PR(R). If the above
conditions are simultaneously satisfied, we have the possibility of having at
most four turning points and therefore a bound motion may exist for ¯
E2>1.
On the other hand, if A4<0 ( ¯
E2<1), there exist at most three variations
of sign since A0>0. For this case, then either of the following inequalities
should be simultaneously fulfilled for the existence of bound region(s):
A1<0, A2<0, A3>0,
A1>0, A2<0, A3>0,(4.21)
A1<0, A2>0, A3>0,
A1<0, A2>0, A3<0.
Similarly, this implies that if any one of the above conditions are simultane-
ously satisfied, there exists at most one region of binding outside the outer
singularity where r > r+.
15

4.3 The spherical orbits and stability
The spherical orbits are special orbits that satisfy
Pr(r=rs) = 0,(4.22)
dPr
dr r=rs
= 0 (4.23)
at r=rs. These require that
(¯
Es
2−1)r4
s+ 2(M−¯qQ ¯
Es)r3
s
+2(ℓ2+a2)¯
Es
2+ ¯q2Q2−2a¯
Es¯
Ls−K
m2−(a2−ℓ2+Q2)r2
s
+2 KM
m2+a¯qQ ¯
Ls−¯qQ(ℓ2+a2)¯
Esrs(4.24)
+(ℓ2+a2)2¯
Es
2+a2¯
Ls
2−2a(ℓ2+a2)¯
Es¯
Ls−K
m2(a2−ℓ2+Q2) = 0
and
4( ¯
Es
2−1)r3
s+ 6(M−¯qQ ¯
Es)r2
s
+2 2(ℓ2+a2)¯
Es
2+ ¯q2Q2−2a¯
Es¯
Ls−K
m2−(a2−ℓ2+Q2)rs
+2 KM
m2+a¯qQ ¯
Ls−¯qQ(ℓ2+a2)¯
Es= 0.(4.25)
Considering that Pr(r) can also be written in the form
Pr(r) = (r−rs)2(¯
Es
2−1)r2+µ1r+µ2,(4.26)
the relations (4.24) and (4.25) can equivalently be expressed in the following
forms:
3r4
s+ 2r2
s(ℓ2+a2)−(ℓ2+a2)2¯
Es
2−a2¯
Ls
2+ 2aℓ2+a2−r2
s¯
Es¯
Ls(4.27)
−4r3
s¯qQ ¯
Es+a2−ℓ2+Q2K
m2−r2
s−r2
s3r2
s−4rsM−¯q2Q2+K
m2= 0
and
2rs(ℓ2+a2+r2
s)¯
Es
2+a¯qQ −2rs¯
Es¯
Ls−3r2
s+ℓ2+a2¯qQ ¯
Es
+3r2
sM−K
m2+a2−ℓ2+Q2−¯q2Q2+ 2r2
srs+KM
m2= 0.(4.28)
16

Now, we find analytical solutions to the energy and angular momentum of
the particle in spherical orbit with radius r=rs. We consider the following
cases:
i. The case a6= 0 and ¯
Ls6= 0:
The equations (4.27) and (4.28) lead to a second order equation in ¯
Esin the
form
ν2¯
Es
2+ν1¯
Es+ν0= 0,(4.29)
where the coefficients read
ν0=−rs¯q2Q2−∆(rs)+ (M−rs)K
m2+r2
s2
(4.30)
+¯q2Q2∆(rs)K
m2−r2
s+ 2rs(M−rs)K
m2+r2
s+r2
s¯q2Q2,
ν1=−4¯qQrsK
m2+r2
s∆(rs) (4.31)
and
ν2= 4r2
sK
m2+r2
s∆(rs).(4.32)
The equation (4.29) can be solved as
¯
Es
±=¯qQ
2rs±√Ds
8r2
s∆(rs)K
m2+r2
s(4.33)
where
Ds= 16r2
s∆(rs)K
m2+r2
s(rs−M)K
m2+r2
s+rs∆(rs)2
.(4.34)
Furthermore, the substitution of ¯
Esin (4.28) leads to
¯
Ls
±=¯qQ
2ars
(a2+ℓ2−r2
s)±(a2+ℓ2+r2
s)√Ds
8K
m2+r2
s∆(rs)ar2
s
(4.35)
∓4rs
a√DsK
m2+r2
s(rs−M)K
m2+r2
s+rs∆(rs)∆(rs).
To get a deeper insight of the analytical expressions of energy and angular
momentum, we plot them as a function of gravitomagnetic monopole moment
17

Figure 6: The graph of angular momentum ¯
Lsas a function of gravitomag-
netic monopole moment ℓwith parameters M= 1, a= 0.4, Q= 0.4, ¯q= 0.3,
K= 20, m= 1, rs= 10.
ℓand the spherical radius rs. In both plots, we concentrate on ¯
L+
sand ¯
E+
s.
First, looking at the graph of ¯
L+
svs ℓfor the parameter values given in
Figure 6, it can be seen that as gravitomagnetic monopole moment increases,
angular momentum also increases becoming zero at some specific value of the
NUT parameter i.e at ℓ=ℓs. It is also interesting to see that ¯
Ls≤0 when
0≤ℓ≤ℓs, while ¯
Ls>0 when ℓ > ℓs. This can be physically interpreted
as such that one obtains retrograde ( ¯
Ls≤0) spherical orbits for 0 ≤ℓ≤ℓs,
while for ℓ > ℓs, the orbits are seen to be direct spherical orbits. In addition,
we should point out that the analytical expression of ¯
Lsrestrict the value of
the NUT parameter ℓsince in the expression (4.35), Ds>0 should also be
imposed.
For the second plot ¯
L+
svs rs, it can be seen that ¯
Lsdecreases to a certain
extremal value and then it again increases. It can also be understood that
¯
Ls= 0 at some particular values of spherical radius rsi.e when rs=rs1and
rs=rs2assuming that rs2< rs1. The graph similarly illustrates that, one
has direct spherical orbits for the radial intervals where rs< rs2and rs> rs1
while one obtains retrograde orbits for the interval rs2< rs< rs1.
For the plot of ¯
E+
svs ℓ, one can see that the energy of the test particle
increases while ℓalso increases. It is also interesting to understand that, the
increase of energy starts from a value where ¯
E < 1 and then it continues to
increase to values where ¯
E > 1, the energy becoming unity at some specific
18

Figure 7: The graph of angular momentum ¯
Lsas a function of spherical
radius rswith parameters M= 1, a= 0.4, ℓ= 0.4, Q= 0.4, ¯q= 0.3,
K= 20, m= 1.
value of NUT parameter ℓ. It should be also added that value of ℓshould
again be restricted since the analytic expression (4.33) of ¯
Esalso suggests
that Ds≥0.
Finally the plot of ¯
Esvs rsshows that, as the value of rsincreases, the energy
of test particle starts to decrease from a certain value having an extremum at
some specific value of the spherical radius and then it continuous to increase
approaching unity (i.e ¯
Es→1) as rs→ ∞.
ii. The case ¯
Ls= 0:
For this special case, one can eliminate the Carter constant K
m2from equation
(4.28) and substitute it into the equation (4.27) to obtain the energy equation
in the form
¯ν2¯
Es
2+ ¯ν1¯
Es+ ¯ν0= 0,(4.36)
where in that case the coefficients become
¯ν0=rs∆2(rs) + ¯q2Q2(−a2+ℓ2−Q2+Mrs),(4.37)
¯ν1= ¯qQ a2(Q2+a2) + ℓ2(Q2−ℓ2) + r2
s(3Q2−2a2−4ℓ2−4Mrs+r2
s)
(4.38)
and
¯ν2=−(a2+ℓ2+r2
s)ℓ2(M−3rs) + a2(M+rs) + rs(2Q2−3Mrs+r2
s).
(4.39)
19

Figure 8: The graph of energy ¯
Esas a function of gravitomagnetic monopole
moment ℓwith parameters M= 1, a= 0.4, Q= 0.4, ¯q= 0.3, K= 20,
m= 1, rs= 10.
Figure 9: The graph of energy ¯
Esas a function of spherical radius rswith
parameters M= 1, a= 0.4, ℓ= 0.4, Q= 0.4, ¯q= 0.3, K= 20, m= 1.
20

Then the energy can be evaluated as
¯
Es
±=−¯ν1±p¯
Ds
2 ¯ν2
,(4.40)
where
¯
Ds= ∆2(rs)4r4
s(2∆(rs) + rs(M−rs))
+¯q2Q2r4
s+ (a2+ℓ2)2(¯q2Q2+ 4M rs) (4.41)
+2(a2+ℓ2)r2
s2∆(rs)−2r2
s−4ℓ2+ 2Q2−¯q2Q2.
iii. The case a= 0:
Remarkably, in the case of vanishing angular momentum a(i.e. a= 0), but
¯q6= 0 which corresponds to spherical orbital motion in Taub-NUT Reissner-
Nordstr¨om spacetime, again by eliminating the Carter constant from (4.28),
one can obtain the following energy equation
˜ν2¯
Es
2+ ˜ν1¯
Es+ ˜ν0= 0 (4.42)
where
˜ν0=rs¯q2Q2(ℓ2−Q2+M rs) + (ℓ2−Q2−rs(rs−2M))2,(4.43)
˜ν1= ¯qQ −ℓ4+ℓ2(Q2−4r2
s) + r2
s(3Q2+rS(−4M+rs)),(4.44)
˜ν2=−(ℓ2+r2
s)ℓ2(M−3rs) + rs(2Q2+rs(−3M+rs))(4.45)
for arbitrary ¯
L. The solution can be given by
¯
Es
±=−˜ν1±p˜
Ds
2 ˜ν2
,(4.46)
where
˜
Ds= ¯q2Q2(ℓ2−r2
s)2+ 4rsℓ4(M−3rs) (4.47)
+r3
s(2Q2+r2
s−3Mrs)−2ℓ2rs(r2
s+rsM−Q2).
Now we examine the stability of spherical orbits for arbitrary spacetime
parameters. The stability of such orbits implies that [14, 33]
d2Pr(r)
dr2r=rs
<0.(4.48)
21

This further leads to the inequality
23r2
s+ℓ2+a2¯
E2+ 2 ¯qQ −a¯
L¯
E(4.49)
−(5rs+ 4M)rs−∆(rs)−K
m2+ ¯q2Q2<0.
This means that the spherical orbits are stable if the energy of the particle
falls into the interval ¯
E−<¯
E < ¯
E+,(4.50)
where
¯
E±=(a¯
L−¯qQ)±√DE
2(3r2
s+ℓ2+a2)(4.51)
and
DE= (a¯
L−¯qQ)2+ 2(3r2
s+ℓ2+a2)5r2
s+ 4Mrs+ ∆(rs) + K
m2−¯q2Q2.
(4.52)
Otherwise, we have an unstable spherical orbit. In Tables 1 and 2, we in-
vestigate the stability with respect to change of the NUT parameter ℓand
the spherical radius rs. For the values of the spacetime parameters given
in tables, as ℓand rsincreases, we obtain that the orbits are all stable. At
this stage, we are unable to see an unstable spherical orbit although it is
clear that there may exit unstable one(s) for the spacetime parameters that
satisfy the inequality (4.49). However looking at the Table 1, stable spherical
orbits change their class from retrograde orbits (a > 0,¯
L < 0) to direct ones
(a > 0,¯
L > 0), as the NUT parameter increases. One can see a similar effect
in Table 2, when the radius of the spherical orbit is increased.
Table 1: a= 0.4M,Q= 0.4M,rs= 10M,M= 1, ¯q= 0.3, m= 1, K= 20.
ℓ¯
E+
s¯
L+
sstability of the orbit
0.1 0.965109 -6.73569 stable
0.2 0.965135 -6.61086 stable
0.4 0.965241 -6.1113 stable
0.8 0.965672 -4.1093 stable
1.2 0.966415 -0.758978 stable
2 0.969014 10.0825 stable
4 0.986713 63.9055 stable
22

Table 2: a= 0.4M,Q= 0.4M,ℓ= 0.4M,M= 1, ¯q= 0.3, m= 1, K= 20.
rs¯
E+
s¯
L+
sstability of the orbit
5 0.991365 -4.04398 stable
10 0.965241 -6.1113 stable
12 0.96844 -5.13413 stable
13 0.970084 -4.49053 stable
15 0.973125 -3.0001 stable
20 0.978958 1.41363 stable
5 Analytical solutions
In this section we investigate and present analytical solutions to r-motion, θ-
motion, t-motion and ϕ-motion. We see that the solutions can be expressed
in terms of Jacobian elliptic functions F(y, k) and Weierstrass ℘, Weierstrass
σand Weierstrass ζfunctions.
5.1 θ-motion
Making a transformation x= cos θ, (3.15) can be cast into the following
form: dx
dλ2
=C0+C1x+C2x2+C3x3+C4x4=: Pθ(x) (5.1)
where
C0=K
m2−ℓ2−¯
L−a¯
E2=: e
K, (5.2)
C1= 2ℓa(2 ¯
E2−1) −4ℓ¯
E¯
L, (5.3)
C2=−K
m2+ℓ21−4¯
E2+a2¯
E2−1+¯
L−a¯
E2−¯
L2
=−e
K−4ℓ2¯
E2+a2¯
E2−1−¯
L2,(5.4)
C3= 2ℓa(1 −2¯
E2),(5.5)
C4=a2(1 −¯
E2).(5.6)
23

We also recall that −1≤x≤1. In general, the solution of the fourth order
polynomial equation (5.1) can be expressed in terms of elliptic functions.
First, we see that the transformation (for ¯
E6= 1)
x=1
ξ+xθ(5.7)
brings the polynomial equation (5.1) into the following third order form in
ξ:dξ
dλ2
=α3ξ3+α2ξ2+α1ξ+C4.(5.8)
Here xθis a real root of Pθ(x) and the coefficients are given by
α1=C3+ 4C4xθ,(5.9)
α2=C2+ 3C3xθ+ 6C4x2
θ,(5.10)
α3=C1+ 2C2xθ+ 3C3x2
θ+ 4C4x3
θ.(5.11)
With the further transformation
ξ=1
α34y−α2
3,(5.12)
we obtain standard Weierstrass form of the differential equation (5.8)
dy
dλ2
= 4y3−g2y−g3(5.13)
where
g2=1
4α2
2
3−α1α3(5.14)
and
g3=1
8α1α2α3
6−C4α2
3
2−α3
2
27 .(5.15)
The solution of equation (5.13) can be given by Weierstrass ℘function
y(λ) = ℘(λ−λ0;g2, g3),(5.16)
where λ0describes the initial Mino time. Hence the solution of θ−equation
can be obtained as
θ(λ) = arccos α3
4℘(λ−λ0;g2, g3)−α2
3
+xθ,(5.17)
24

where the solution is valid for all types of orbits. For that solution, one can
assume that there exist at least one real root (in fact there can exist at least
two real roots for a fourth order polynomial).
Alternatively, there may exist orbits for which the polynomial Pθ(x) has four
real roots. For the orbits ¯
E2<1 (C4>0), one can assume Pθ(x) has real
roots x1,x2,x3,x4ordered as x4< x3< x2< x1. In this case, there exist
three intervals namely x≤x4,x3≤x≤x2and x≥x1where Pθ(x)≥0 [34].
For the intervals x≤x4or x≥x1, one can consider the transformation 1
x−x1
x−x2
=x1−x4
x2−x4
y2
θ.(5.18)
Then with this transformation,
Zx
x1
dx
√C0+C1x+C2x2+C3x3+C4x4
=2
pC4(x1−x3)(x2−x4)Zyθ
0
dyθ
p(1 −k2
θy2
θ)(1 −y2
θ)
=2
pC4(x1−x3)(x2−x4)F(yθ, kθ),
(5.19)
where F(yθ, kθ) describes incomplete Jacobian elliptic function of the first
kind. Furthermore, we also have
Zx
x1
dx
√C0+C1x+C2x2+C3x3+C4x4=Zdλ =λ−λ0.(5.20)
So we obtain
F(yθ, kθ) = pC4(x1−x3)(x2−x4)(λ−λ0)
2(5.21)
with
k2
θ=(x1−x4)(x2−x3)
(x1−x3)(x2−x4).(5.22)
1Equivalently, one can apply a similar transformation on the variables such that x−x3
x−x4=
x2−x3
x2−x4y2
θand obtain the same analytical result for the solution of the angular equation
where in that case the transformation is valid when x3≤x≤x2.
25

For the orbits where ¯
E2>1 (C4<0), as for the case ¯
E2<1, we can still
assume four real roots for the polynomial Pθ(x) ordered as ¯x4<¯x3<¯x2<¯x1.
In this case, there exist two intervals ¯x2≤¯x≤¯x1and ¯x4≤¯x≤¯x3for which
Pθ(x)≥0. Concentrating on the interval ¯x2≤¯x≤¯x1, one may consider the
transformation 2
¯x−¯x2
¯x−¯x3
=¯x1−¯x2
¯x1−¯x3
y2
θ.(5.23)
leading to the solution
F(yθ,˜
kθ) = p−C4(¯x1−¯x3)(¯x2−¯x4)(λ−λ0)
2,(5.24)
with
˜
k2
θ=(¯x1−¯x2)(¯x3−¯x4)
(¯x1−¯x3)(¯x2−¯x4).(5.25)
For ¯
E= 1, Pθ(x) becomes a third order polynomial such that the transfor-
mation
x=1
¯
C34y−¯
C2
3(5.26)
obviously puts the equation (5.1) into standard Weierstrass form
dy
dλ2
= 4y3−¯g2y−¯g3(5.27)
where in that case
¯g2=1
4¯
C2
2
3−¯
C1¯
C3,(5.28)
¯g3=1
8¯
C1¯
C2¯
C3
6−¯
C0¯
C2
3
2−¯
C3
2
27 (5.29)
whose solution can similarly be expressed in terms of Weierstrass ℘function
as
y(λ) = ℘(λ−λ0; ¯g2,¯g3) (5.30)
leading to θ-solution
θ(λ) = arccos 1
¯
C3
(4℘(λ−λ0; ¯g2,¯g3)−¯
C2
3).(5.31)
2Equivalently, one can apply a similar transformation on the variables such that ¯x−¯x4
¯x−¯x1=
¯x3−¯x4
¯x3−¯x1y2
θand obtain the same analytical result for the solution of the angular equation
where the transformation is valid when ¯x4≤¯x≤¯x3.
26

We further remark that, here
¯
C0=C0|¯
E=1 ,¯
C1=C1|¯
E=1 ,¯
C2=C2|¯
E=1 ,¯
C3=C3|¯
E=1 .(5.32)
In addition, one can also consider the special case where ¯
E= 1 and a= 0.
For this special case, the angular equation becomes a second order polynomial
which can be solved as
θ(λ) = arccos "2ℓ¯
L
K
m2+ 3ℓ2+p˜
Csin rK
m2+ 3ℓ2(λ−λ0)!# (5.33)
where we also identify
˜
C=1
K
m2+ 3ℓ22K
m2−ℓ2−¯
L2K
m2+ 3ℓ2+ 4ℓ2¯
L2(5.34)
provided that ˜
C > 0.
5.2 r-motion
On the other hand, the transformation (3.13) on the time variable brings the
radial equation (3.8) into
dr
dλ2
=N0+N1r+N2r2+N3r3+N4r4=: Pr(r),(5.35)
where
N4=¯
E2−1,(5.36)
N3= 2(M−¯
E¯qQ),(5.37)
N2=a2(2 ¯
E2−1) + ℓ2(2 ¯
E2+ 1) + Q2(¯q2−1) −2a¯
E¯
L−K
m2,
=−e
K+a2(¯
E2−1) + 2ℓ2¯
E2+Q2(¯q2−1) −¯
L2,(5.38)
N1= 2¯qQ a¯
L−(ℓ2+a2)¯
E+2MK
m2,(5.39)
and
N0=(ℓ2+a2)¯
E−a¯
L2−a2−ℓ2+Q2K
m2.(5.40)
27

If one performs the transformation (for ¯
E6= 1)
r=β3
4v−β2
3+r1(5.41)
the equation (5.35) can be brought into the standard Weierstrass form
dv
dλ2
= 4v3−h2v−h3,(5.42)
where
β1=N3+ 4N4r1,(5.43)
β2=N2+ 3N3r1+ 6N4r2
1,(5.44)
β3=N1+ 2N2r1+ 3N3r2
1+ 4N4r3
1,(5.45)
with
h2=1
12 β2
2−3β1β3, h3=1
8β1β2β3
6−N4β2
3
2−β3
2
27,(5.46)
whose solution can again be given by Weierstrass ℘function
v(λ) = ℘(λ−λ0;h2, h3),(5.47)
so that the solution for rcan be written as
r=β3
4℘(λ−λ0;h2, h3)−β2
3+r1,(5.48)
where we have assumed that Pr(r) has at least two real roots (Here r1is
assumed to be one real root of Pr(r)).
As in the angular motion, if one considers that the radial polynomial Pr(r)
has 4 distinct real roots r1, r2, r3, r4ordered as r4< r3< r2< r1, one can
alternatively express the solutions in terms of Jacobian Elliptic functions. If
one performs the similar transformations done in the angular case, one can
end up with the following solutions: For the orbits where ¯
E > 1, the solution
reads
F(yr, kr) = pN4(r1−r3) (r2−r4)(λ−λ0)
2,(5.49)
28

where we have affected the transformation
r−r1
r−r2
=r1−r4
r2−r4
y2
r(5.50)
with
k2
r=(r1−r4)(r2−r3)
(r1−r3)(r2−r4)(5.51)
while for the orbits where ¯
E < 1, the solution can similarly be expressed as
F(yr,¯
kr) = p−N4(r1−r3) (r2−r4)(λ−λ0)
2,(5.52)
where in that case, the transformation
r−r2
r−r3
=r1−r2
r1−r3
y2
r(5.53)
is valid with
¯
k2
r=(r1−r2)(r3−r4)
(r1−r3)(r2−r4).(5.54)
Interestingly, for the special case where Pr(r) has a double real root such that
r1=r2=rs, the radial polynomial can be put into the form (with ¯
E6= 1)
[16]
Pr(r) = (r−rs)2(¯
E2−1)r2+ 2rrs¯
E2−1 + M−¯qQ ¯
E
rs+N0
r2
s.
(5.55)
At this stage, we can affect the transformation
ρ=1
r−rs
,(5.56)
to obtain (5.35) in the following form:
dρ
dλ2
=α+βρ +γρ2,(5.57)
where
α=N4=¯
E2−1,(5.58)
β= 4rs(¯
E2−1) + 2(M−¯
E¯qQ),(5.59)
29

γ= 3r2
s(¯
E2−1) + 2rs(M−¯
E¯qQ) + N0
r2
s
.(5.60)
The solution of the equation (5.57) can be obtained for three different cases,
namely for the cases γ > 0, γ= 0 and γ < 0. For γ= 0, the solution can be
expressed as
λ−λ0=∓2√α+βρ
β,(5.61)
for γ > 0, the solution can be given by
λ−λ0=∓1
√γln ρ+β
2γ+sρ2+β
γρ+α
γ,(5.62)
while for γ < 0 the solution reads
λ−λ0=∓1
√−γarcsin 2γρ +β
pβ2−4γα !.(5.63)
Interestingly for ¯
E= 1, Pr(r) turns into a third order polynomial such that
the transformation
r=¯
N3
4v−¯
N2
3(5.64)
brings the radial equation into standard Weierstrass-℘form
dv
dλ2
= 4v3−¯
h2v−¯
h3(5.65)
where in that case we identify
¯
h2=1
4¯
N2
2
3−¯
N1¯
N3,(5.66)
¯
h3=1
8¯
N1¯
N2¯
N3
6−¯
N0¯
N2
3
2−¯
N3
2
27 (5.67)
with
¯
N0=N0|¯
E=1 ,¯
N1=N1|¯
E=1 ,¯
N2=N2|¯
E=1 ,¯
N3=N3|¯
E=1 .(5.68)
Then, for ¯
E= 1, the solution reads
r(λ) = ¯
N3
4℘(λ−λ0;¯
h2,¯
h3)−¯
N2
3
.(5.69)
30

5.3 t-motion
To obtain the solution of the equation (3.16), we recall that it can be written
in differential form as
dt =dI(t)
θ,1+dI(t)
θ,2+dI(t)
r(5.70)
where the integration yields
t−t0=I(t)
θ,1+I(t)
θ,2+I(t)
r.(5.71)
Here
I(t)
θ,1=a¯
LZθ(λ)
θ0
dθ
pPθ(θ)=a¯
L(λ−λ0) (5.72)
where we have integrated (3.15), taking the + sign only. The second expres-
sion I(t)
θ,2can be written as
I(t)
θ,2=−2ℓ¯
LZθ(λ)
θ0
cos θ
sin2θpPθ(θ)dθ −a2¯
EZθ(λ)
θ0
sin2θ
pPθ(θ)dθ (5.73)
+4aℓ ¯
EZθ(λ)
θ0
cos θ
pPθ(θ)dθ −4ℓ2¯
EZθ(λ)
θ0
cos2θ
sin2θpPθ(θ)dθ.
The integration I(t)
θ,2can be accomplished first by taking x= cos θand next
by making further transformation
x=α3
4y−α2
3+xθ,(5.74)
where xθis one real root of the polynomial equation (5.1) and α2and α3are
defined as in (5.11) and (5.12) respectively. With the additional transforma-
31

tion ℘(s) = y, integrations with respect to variable syield
I(t)
θ,2=a2¯
E2(x2
θ−1) + 4aℓ ¯
Exθ−2xθℓ
1−x2
θ2ℓ¯
Exθ+¯
L(λ−λ0)
−2ℓ
2
X
i=1
2
X
j=1
(¯
LGi+ 2ℓ¯
E¯
Gi)
℘′(aij )ζ(aij)(λ−λ0) + ln σ(s−aij )
σ(s0−aij )
+a¯
Eaxθ
2+ℓα32
X
i=1
1
℘′(b1i)ζ(b1i)(λ−λ0) + ln σ(s−b1i)
σ(s0−b1i)
−a2¯
Eα2
3
16
2
X
i=1
1
℘′2(b1i)(λ−λ0)℘(b1i) + ℘′′ (b1i)
℘′(b1i)(5.75)
−a2¯
Eα2
3
16
2
X
i=1
1
℘′2(b1i)ζ(s−b1i) + ℘′′ (b1i)
℘′(b1i)ln σ(s−b1i)
σ(s0−b1i)−ζ(i)
0.
Here we identify ℘(aij) = aiand ℘(b1i) = b1=α2
12 (i= 1,2) with
a1=¯c
4(1 −xθ), a2=−¯c
4(1 + xθ),(5.76)
where
¯c=α3+α2
3(1 −xθ).(5.77)
Also,
G1=α3
8(1 −x2
θ)(1 −3x2
θ)
(1 −xθ)+1
3¯c(3α3xθ−α2),(5.78)
G2=α3
8(1 −x2
θ)(1 −3x2
θ)
(1 + xθ)−1
3¯c(3α3xθ−α2),(5.79)
¯
G1=α3
4(1 −x2
θ)xθ
(1 −xθ)+1
3¯c(xθα2−α3)),(5.80)
¯
G2=α3
4(1 −x2
θ)xθ
(1 + xθ)−1
3¯c(xθα2−α3)).(5.81)
Next, we consider the radial integral
I(t)
r=Zr(λ)
r0¯
E(r2+a2+ℓ2)−a¯
L−¯qQr(r2+a2+ℓ2)
∆pPr(r)dr. (5.82)
32

The radial integrals can be evaluated by a similar transformation such that
r=β3
(4v−β2
3)+r1(5.83)
where r1is again assumed to be one real root of Pr(r). Next taking v=℘(s),
the integration yields
I(t)
r=¯
E(r2
1+ℓ2+a2)2−¯qQ(r2
1+ℓ2+a2)r1−a¯
L(r2
1+ 1)(λ−λ0)
∆(r1)
+
3
X
i=1
2
X
j=1
(¯
Eωi−¯qQ ¯ωi)
℘′(vij )ζ(vij)(λ−λ0) + ln σ(s−vij )
σ(s0−vij)
−¯
Eβ2
3
16
2
X
j=1
1
℘′2(v3j)(λ−λ0)℘(v3j) + ℘′′ (v3j)
℘′(v3j)
+℘′′(v3j)
℘′(v3j)ln σ(s−v3j)
σ(s0−v3j)+ζ(s−v3j)−ζ(j)
0(5.84)
+
2
X
i=1
2
X
j=1
1
℘′(vij )ζ(vij)(λ−λ0) + ln σ(s−vij )
σ(s0−vij )×
2¯
E(ℓ2+a2)−a¯
L˜ωi−¯qQ(ℓ2+a2)ˆωi+ (ℓ2+a2)¯
E(ℓ2+a2)−a¯
Lˇωi.
Here ℘(vij ) = viand ℘(v3j) = v3=β2
12 (i= 1,2) with
v1=1
4∆(r1)∆(r1)β2
3−r1+Mβ3−q(r1−M β3)2−∆(r1)β2
3
(5.85)
and
v2=1
4∆(r1)∆(r1)β2
3−r1+Mβ3+q(r1−M β3)2−∆(r1)β2
3.
(5.86)
We also identify
ω1=−[r1(r1−Mβ3)−β3∆(r1) + 2r1∆(r1)(v2−v1)]4
16∆3(r1)(v2−v1) [r1−Mβ3+ 2∆(r1)(v2−v1)]2,(5.87)
ω2=[r1(r1−Mβ3)−β3∆(r1)−2r1∆(r1)(v2−v1)]4
16∆3(r1)(v2−v1) [r1−Mβ3−2∆(r1)(v2−v1)]2,(5.88)
33

¯ω1=−[r1(r1−Mβ3)−β3∆(r1) + 2r1∆(r1)(v2−v1)]3
16∆3(r1)(v2−v1) [r1−Mβ3+ 2∆(r1)(v2−v1)] ,(5.89)
¯ω2=[r1(r1−Mβ3)−β3∆(r1)−2r1∆(r1)(v2−v1)]3
16∆3(r1)(v2−v1) [r1−Mβ3−2∆(r1)(v2−v1)] ,(5.90)
ω3=β3r1−(r1−Mβ3)
2,¯ω3=β3
4,(5.91)
˜ω1=−[r1(r1−Mβ3)−β3∆(r1) + 2r1∆(r1)(v2−v1)]2
16∆3(r1)(v2−v1),(5.92)
˜ω2=[r1(r1−Mβ3)−β3∆(r1)−2r1∆(r1)(v2−v1)]2
16∆3(r1)(v2−v1),(5.93)
ˆω1=1
∆(r1)(v1−v2)v1−β2
12r1v1−β2
12+β3
4,(5.94)
ˆω2=1
∆(r1)(v2−v1)v2−β2
12r1v2−β2
12+β3
4,(5.95)
ˇω1=1
∆(r1)(v1−v2)v1−β2
122
(5.96)
and
ˇω2=1
∆(r1)(v2−v1)v2−β2
122
.(5.97)
5.4 ϕ-motion
Similarly, the equation (3.17) can be written in differential form as
dϕ =dI(ϕ)
θ+dI(ϕ)
r(5.98)
upon integration which gives
ϕ−ϕ0=I(ϕ)
θ+I(ϕ)
r.(5.99)
The angular integral
I(ϕ)
θ=Z¯
L−¯
E(asin2θ−2ℓcos θ)
sin2θpPθ(θ)dθ (5.100)
34

can be accomplished by making the transformation
cos θ=α3
4℘(s)−α2
3+xθ(5.101)
such that it yields the solution
I(ϕ)
θ=(¯
L+ 2ℓ¯
Exθ)
(1 −x2
θ)−a¯
E(λ−λ0) (5.102)
+
2
X
i=1
2
X
j=1
(¯
L¯
Gi+ 2ℓ¯
EGi)
℘′(aij )ζ(aij)(λ−λ0) + ln σ(s−aij )
σ(s0−aij )
where Giand ¯
Giare defined through (5.78)-(5.81).
On the other hand, the radial integral
I(ϕ)
r=Zr(λ)
r0
a(r2+a2+ℓ2)−a¯
L−¯qQr
∆pPr(r)dr (5.103)
can be calculated by affecting the transformation
r=β3
(4℘(s)−β2
3)+r1(5.104)
that yields
I(ϕ)
r=a(r2
1−¯qQr1+a2+ℓ2−a¯
L)
∆(λ−λ0) (5.105)
+
2
X
i=1
2
X
j=1
a(˜ωi−¯qQˆωi+ (a2+ℓ2−a¯
L)ˇωi)
℘′(vij)ζ(vij)(λ−λ0) + ln σ(s−vij )
σ(s0−vij)
where ˜ωi, ˆωiand ˇωiare identified through (5.92)-(5.97).
6 Discussion of the orbits and observables
In this section, we investigate possible three dimensional orbits by plotting
them for fixed energy, angular momentum and spacetime parameters. In
addition, we also calculate the observables of the bound orbits and express
them in terms of the elliptic functions.
35

We first examine the orbits with respect to value of the energy parameter.
In Figures 10 and 11, orbits are plotted for ¯
E < 1. It can be seen that for
¯
E < 1, there may exist one bound or two bound orbits. In Figure 10, for
the value of NUT parameter ℓ= 0.1, we obtain two bound orbits (Pr(r) has
four real zeros), while in figure 11, for ℓ= 0.3 (i.e NUT parameter is slightly
increased) there exists one bound orbit (Pr(r) has two real zeros).
(a) The graph of Pr(r) (b) The graph of Pθ(θ)
(c) 3D-bound orbit with
¯
E < 1(d) r-θplane
(e) Projection onto xy-
plane
Figure 10: The plots are obtained for the parameters M= 1, a= 0.9,
K= 10, Q= 0.4, ¯q= 0.3, ¯
L= 0.5, ¯
E= 0.96, ℓ= 0.1 and m= 1. Here
ρ2=x2+y2. There exist two bound orbits.
36

(a) The graph of Pr(r) (b) The graph of Pθ(θ)
(c) 3D-bound orbit with
¯
E < 1(d) r-θplane
(e) Projection onto xy-
plane
Figure 11: The plots are obtained for the parameters M= 1, a= 0.9,
K= 10, Q= 0.4, ¯q= 0.3, ¯
L= 0.5, ¯
E= 0.96, ℓ= 0.3 and m= 1. Here
ρ2=x2+y2. There exists one bound orbit.
On the other hand, for ¯
E > 1, one can get one bound and two flyby obits
(Pr(r) has four real zeros) or two flyby orbits (Pr(r) has two real zeros) or
transit orbits (Pr(r) has no real zeros). In Figures 12 and 13, 3D-flyby orbits
are obtained for different values of the NUT parameter, while in Figures 14
and 15, 3D-bound orbits (with ¯
E > 1) are realized similarly for different
values of gravitomagnetic monopole moment ℓ.
37

(a) The graph of Pr(r) (b) The graph of Pθ(θ)
(c) 3D-flyby
orbit for r >
r1with ¯
E >
1.
(d) r-θ
plane (e) Projection onto xy-plane
Figure 12: The plots are obtained for the parameters M= 1, a= 0.9,
K= 50, Q= 0.4, ¯q= 0.3, ¯
L= 1.5, ¯
E= 1.1, ℓ= 0.1 and m= 1. Here
ρ2=x2+y2and r1= 10.768 is one real root of Pr(r). There exists one
bound orbit and two flyby orbits.
38

(a) The graph of Pr(r) (b) The graph of Pθ(θ)
(c) 3D-
flyby
orbit for
r > r1
with
¯
E > 1.
(d) r-θ
plane
(e) Pro-
jection
onto
xy-plane
Figure 13: The plots are obtained for the parameters M= 1, a= 0.9,
K= 100, Q= 0.1, ¯q= 0.3, ¯
L= 1, ¯
E= 1.1, ℓ= 0.2 and m= 1. Here
ρ2=x2+y2and r1= 16.473 is one real root of Pr(r). There exists one
bound orbit and two flyby orbits.
39

(a) The graph of Pr(r) (b) The graph of Pθ(θ)
(c) 3D-bound orbit with
¯
E > 1(d) r-θplane
(e) Projection onto xy-
plane
Figure 14: The plots are obtained for the parameters M= 1, a= 0.9,
K= 10, Q= 0.1, ¯q= 0.3, ¯
L= 4, ¯
E= 1.02, ℓ= 1 and m= 1. Here
ρ2=x2+y2. There exists one bound orbit and two flyby orbits.
40

(a) The graph of Pr(r) (b) The graph of Pθ(θ)
(c) 3D-bound orbit with ¯
E > 1
(d) r-θ
plane
(e) Projection onto xy-
plane
Figure 15: The plots are obtained for the parameters M= 1, a= 0.9,
K= 10, Q= 0.1, ¯q= 0.3, ¯
L= 4, ¯
E= 1.02, ℓ= 0.9 and m= 1. Here
ρ2=x2+y2. There exists one bound orbit and two flyby orbits.
Finally, for ¯
E= 1 the possible orbit types can be one bound and one flyby
(Pr(r) has three real zeros) or one flyby orbit (Pr(r) has only one real zero).
In Figure 16, we give an example of flyby orbit while in Figure 17, we exhibit
3D-bound orbit for ¯
E= 1 (for the fixed value of the NUT parameter but
different values of the test particle charge ¯q).
41

(a) The graph of Pr(r) (b) The graph of Pθ(θ)
(c) 3D-flyby orbit with ¯
E= 1 (d) r-θplane
(e) Projection onto xy-plane
Figure 16: The plots are obtained for the parameters M= 1, a= 0.9, K= 1,
Q= 0.4, ¯q= 0.3, ¯
L= 0.5, ¯
E= 1, ℓ= 0.1 and m= 1. Here ρ2=x2+y2.
There exists one flyby orbit.
42

(a) The graph of Pr(r) (b) The graph of Pθ(θ)
(c) 3D-bound orbit with
¯
E= 1 (d) r-θplane
(e) Projection onto xy-
plane
Figure 17: The plots are obtained for the parameters M= 1, a= 0.9, K= 1,
Q= 0.4, ¯q= 2.6, ¯
L= 0.5, ¯
E= 1, ℓ= 0.1 and m= 1. Here ρ2=x2+y2.
There exists one bound orbit.
Next, to obtain the observables for a charged particle in Kerr-Newman-Taub-
NUT spacetime, we assume that the particle makes a bound motion in rand
θcoordinates. Considering that motion in r-coordinate is bounded in the
interval r2≤r≤r1, one can calculate the fundamental period Λrfor the
radial motion as
Λr= 2 Zr1
r2
dr
pPr(r)= 2 Z∞
v0
dv
pP3(v)(6.1)
where P3(v) = 4v3−h2v−h3with h2and h3introduced in (5.46). The
integral can be accomplished via the transformation
ξr=1
κre2−e3
v−e31/2
(6.2)
43

where e1,e2and e3correspond to the roots of the polynomial P3(v) = 0 with
κ2
r=e2−e3
e1−e3. We also choose v0=e1. Then one obtains the radial period as
Λr=2
√e1−e3
K(κr) (6.3)
where K(κr) denotes the complete elliptic function with modulus κr.
Similarly, if one considers that the angular motion is also bounded in the
angular interval θ2≤θ≤θ1, one can evaluate the fundamental period Λθfor
the angular motion in the form
Λθ= 2 Zθ1
θ2
dθ
pPθ(θ)=2
√¯e1−¯e3
K(κθ),(6.4)
where in this case ¯e1, ¯e2, ¯e3correspond to roots of the polynomial P3(y) =
4y3−g2y−g3. Here, g2and g3are expressed in (5.15) and (5.16) respectively
and κ2
θ=¯e2−¯e3
¯e1−¯e3. Similarly K(κθ) describes the complete elliptic function
with modulus κθ. Then one can also evaluate the corresponding angular
frequencies
Υr=2π
Λr
=π√e1−e3
K(κr)(6.5)
and
Υθ=2π
Λθ
=π√¯e1−¯e3
K(κθ)(6.6)
for the radial and θ-motion respectively.
Furthermore, one can obtain the angular frequencies Υϕand Υtfor the ϕ-
motion and t-motion respectively from the solutions of ϕ(λ) and t(λ). By
using the arguments outlined in [35], one notices that the solutions ϕ(λ) and
t(λ) can be both expressed in the following forms
ϕ(λ) = Υϕ(λ−λ0) + ϕ(r)(λ) + ϕ(θ)(λ) (6.7)
and
t(λ) = Υt(λ−λ0) + r(r)(λ) + t(θ)(λ),(6.8)
where Υϕand Υtcorrespond to frequencies in Mino time for ϕ-motion and
t-motion respectively. From the solutions, one can obtain
Υϕ=(¯
L+ 2ℓ¯
Exθ)
(1 −x2
θ)−a¯
E+a(r2
1−¯qQr1+a2+ℓ2−a¯
L)
∆(6.9)
+
2
X
i=1
2
X
j=1 ζ(aij)(¯
L¯
Gi+ 2ℓ¯
EGi)
℘′(aij )+ζ(vij)a(˜ωi−¯qQˆωi+ (a2+ℓ2−a¯
L)ˇωi)
℘′(vij )
44

and
Υt= Υ(r)
t+ Υ(θ)
t,(6.10)
where
Υ(r)
t=¯
E(r2
1+ℓ2+a2)2−¯qQ(r2
1+ℓ2+a2)r1−a¯
L(r2
1+ 1)1
∆(r1)
+
3
X
i=1
2
X
j=1
(¯
Eωi−¯qQ ¯ωi)
℘′(vij)ζ(vij)−¯
Eβ 2
3
16
2
X
j=1
1
℘′2(v3j)℘(v3j) + ℘′′ (v3j)
℘′(v3j)
+
2
X
i=1
2
X
j=1 2¯
E(ℓ2+a2)−a¯
L˜ωi(6.11)
−¯qQ(ℓ2+a2)ˆωi+ (ℓ2+a2)¯
E(ℓ2+a2)−a¯
Lˇωiζ(vij)
℘′(vij )
and
Υ(θ)
t=a¯
L+a2¯
E2(x2
θ−1) + 4aℓ ¯
Exθ−2xθℓ
1−x2
θ2ℓ¯
Exθ+¯
L
−2ℓ
2
X
i=1
2
X
j=1
(¯
LGi+ 2ℓ¯
E¯
Gi)
℘′(aij )ζ(aij) (6.12)
+a¯
Eaxθ
2+ℓα32
X
i=1
1
℘′(b1i)ζ(b1i)
−a2¯
Eα2
3
16
2
X
i=1
1
℘′2(b1i)℘(b1i) + ℘′′ (b1i)
℘′(b1i).
Finally, as illustrated in [35], the angular frequencies calculated using Mino
time λcan be related to the angular frequencies Ωr, Ωθand Ωϕcalculated
with respect to a distant observer time as
Ωr=Υr
Υt
,Ωθ=Υθ
Υt
,Ωϕ=Υϕ
Υt
.(6.13)
Considering that these frequencies are not equal, it enables us to evaluate the
precession of the orbital ellipse and Lense-Thirring effect for angular motions
ϕand θ. These can be given by
Ωperihelion = Ωϕ−Ωr,ΩLT = Ωϕ−Ωθ.(6.14)
45

Although we couldn’t provide a numerical value for perihelion precision and
Lense-Thirring effect, we can deduce that the NUT parameter and the charge
of the test particle definitely influence these observables.
7 Conclusion
In this work, we have examined the motion of a charged test particle in Kerr-
Newman-Taub-NUT spacetime. We have analyzed the angular and radial
parts of the orbital motion by discussing possible orbit types that may ap-
pear. In the analysis of the angular part, we can see that the NUT parameter
has a significant effect for the existence of equatorial orbits such that unlike
in the background of Kerr and Kerr-Newman spacetimes where equatorial
orbits does exist for any spacetime parameter, one cannot obtain equatorial
orbits in Kerr-Taub-NUT and Kerr-Newman-Taub-NUT spacetimes for ar-
bitrary spacetime parameters, the energy and the angular momentum of the
test particle. Meanwhile, equatorial orbits exist either for vanishing NUT
parameter (ℓ= 0) or if a specific relation is satisfied between the energy and
the orbital angular momentum of the test particle for non-vanishing NUT
parameter. We have also classified the orbit types according to the number
of real roots of the radial polynomial as well as the value of the energy of the
test particle (whether it is smaller, greater than unity or equal to unity). We
can strictly mention that, the existence of the NUT parameter does influence
the radial and the angular part of the motion by causing different types of
orbits to appear as can be seen from three dimensional plots of orbits. As
in [14], we have examined the conditions for a bound orbit outside the outer
singularity (i.e for the region where r > r+) as well. Furthermore, we have
discussed spherical orbits by obtaining the energy and angular momentum
of the test particle in such an orbit by also investigating the stability. We
have observed that stable spherical orbits change their class from retrograde
orbits (where a > 0,¯
L < 0) to direct ones (where a > 0,¯
L > 0) as the
NUT parameter varies. Next, we have obtained the analytical solutions of
the orbit equations in terms of Weierstrass ℘,σ, and ζfunctions. We have
also provided three dimensional plots of the orbits for fixed values of the
spacetime parameters and the energy of the test particle. We have seen that
a bound orbit can exist for all possible values of the energy of the test par-
ticle (i.e for ¯
E < 1, ¯
E > 1 and ¯
E= 1) when the spacetime parameters are
fixed. Moreover, we have also calculated the perihelion precision and Lense-
46

Thirring effect for a bound motion. It can be seen from the relations of the
observables that the NUT parameter and the charge of the test particle do
strictly affect the perihelion precision and Lense-Thirring phenomena.
For future work, it is also of interest to examine the planar orbits of the
charged test particles in Kerr-Newman-Taub-NUT background. It would be
remarkable to examine the orbits over a fixed θ=θ0plane for arbitrary
values of the angular momentum and the energy of the test particle and the
spacetime parameters. One can also study the equatorial orbits with the
relation (4.9) imposed. Furthermore, one can similarly investigate the effect
of the cosmological constant on the orbital motion of a test particle in a
spacetime where both the NUT and rotation parameters exist [8]. These are
devoted to future research.
Acknowledgement
We would like to thank Mehmet Ergen for fruitful discussions and for his
help for obtaining especially three dimensional plots of the orbits.
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