# Geodesics motion of test particles around SK-wormhole with topological
defects and gravitational lensing

Faizuddin Ahmed  0000-0003-2196-9622

###### Abstract

This study investigates the geodesic motion of test particles, both massless
and massive, within a Schwarzschild-Klinkhamer (SK) wormhole space-time. We
specifically consider the influence of cosmic strings on the system and
analyze the effective potential, and observing that the presence of a cosmic
string parameter alters it for null and time-like geodesics. Moreover, we
calculate the deflection angle for null geodesics, and demonstrate that the
cosmic string modifies this angle and induces a shift in the results.
Additionally, we extend our investigation in this SK-wormhole space-time but
with a global monopole. We explore the geodesic motion of test particles in
this scenario and find that the effective potential is affected by the global
monopole. Similarly, we determine the deflection angle for null geodesics and
show that the global monopole parameter introduces modifications to this
angle. Lastly, we present several known solutions for space-times involving
cosmic strings and global monopoles within the framework of this SK-wormhole.

##  1 Introduction

Geodesic motion plays a crucial role in understanding the fundamental aspects
of black hole (BH) and wormhole space-times. It provides valuable insights
into the underlying structure of the space-time and offers significant
information pertaining to these astrophysical phenomena. The behavior of
geodesics in the background space-time exhibits a remarkable complexity,
carrying essential implications for BHs or wormholes. Among the various types
of geodesic motions, circular geodesic motion holds particular interest due to
its connection with gravitational binding energy. These motions provide
insights into the dynamics of particles within the gravitational field.
Notably, null geodesics are highly effective in explaining characteristic
modes of BHs, such as quasinormal modes, as well as the behavior of wormholes.
In the case of an extreme Schwarzschild black hole, there exist a simple set
of unstable circular orbits, which arise as a consequence of the non-linear
nature of GR. These orbits represent an important outcome of the theory,
shedding light on the intricate dynamics of gravitational systems.

Gravitational lensing [1, 2] has been a prominent area of research in
cosmology and gravitation ever since its initial experimental observation by
A. Eddington following the proposal of the general theory of relativity in
1915. This phenomenon manifests in both weak field scenarios, where light rays
pass at a considerable distance from the source, and strong field situations,
where light rays come into close proximity with massive objects. Extensive
investigations have been undertaken to explore gravitational lensing,
encompassing a wide array of space-time configurations. These studies
encompass diverse subjects such as gravitational lensing by black holes [5, 4,
3, 8, 12, 10, 6, 9, 13, 14, 15, 7, 11, 16], naked singularities [17, 18, 19],
spherically symmetric space-time [20], rotating global monopole space-times
[21, 22], cosmic string space-time under LSV [23], including the cosmological
constant in rotating cosmic string space-time [24], Kerr-Newman-Kasuya space-
time [25], black-bounce space-time [26, 27], stationary axisymmetric space-
time [28], and more. Furthermore, gravitational lensing has been investigated
in topologically charged space-times within the framework of Eddington-
inspired Born-Infeld gravity [29], massive gravity [30], spherically symmetric
and static space-time [31], topologically charged Eddington-inspired Born-
Infeld gravity space-time [32], and a space-time with cosmic string in
Eddington-inspired Born-Infeld gravity space-time [33].

Wormholes are hypothetical passageways that connect different regions of the
same universe or even distinct universes through a narrow throat. The study of
wormholes originated from the manipulation of the vacuum solution of the
Schwarzschild black hole. However, in 1973, the exact solution for a wormhole
was first provided by Ellis [34], with a similar solution independently
proposed by Bronnikov [35] in the same year. These wormholes, collectively
known as the Ellis-Bronnikov wormhole, have been extensively explored in
scientific literature. The fascination with wormholes grew significantly
following the groundbreaking works of Morris and Thorne on a similar type of
wormhole referred to as the EB-type wormhole [36]. This particular wormhole
concept introduced a novel perspective on the possibility of time travel
within the framework of general relativity theory. The origins of this concept
can be traced back to 1949 when K. Gödel presented his seminal work on a
rotating universe [37]. Numerous wormhole solutions, both with and without a
cosmological constant, have been constructed and investigated in scientific
literature, contributing to our understanding of these hypothetical structures
(for a comprehensive review, see Ref. [38]). Gravitational lensing in various
wormholes geometries has been extensively studied by numerous authors [39, 58,
59, 50, 51, 52, 41, 40, 48, 44, 53, 45, 46, 49, 47, 42, 43, 54, 62, 60, 55,
61, 56, 57]. These investigations have shed light on the gravitational lensing
effects and properties associated with wormholes.

In Ref. [39], author presented a new type of wormhole space-time, a non-vacuum
solution of the field equations which is a degenerate one given by

$ds^{2}=-dt^{2}+\Big{(}\frac{x^{2}}{x^{2}+b^{2}}\Big{)}\,dx^{2}+(x^{2}+a^{2})\,(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}).$
(1.1)

Here $a,b$ are nonzero real constants and the coordinates
$-\infty<(t,x)<+\infty$, $\theta\in[0,\pi)$, and $\phi\in[0,2\,\pi]$. For
$b^{2}=a^{2}$, this wormhole space-time reduces to a vacuum solution. Author
discussed there that the matter-energy distribution of the metric (1.1)
satisfies the weak energy condition (WEC) and the null energy condition (NEC)
provided $b^{2}>a^{2}$.

In Ref. [63], author discussed the thermodynamics of Schwarzschild black hole
with a cosmic string. In the coordinate chat $(t,r,\theta,\phi)$ with $0\leq
r$ and $0\leq\phi<2\,\pi$, the line-element is described by

$ds^{2}=-A(r)\,dt^{2}+\frac{dr^{2}}{A(r)}+r^{2}\,(d\theta^{2}+\alpha^{2}\,\sin^{2}\theta\,d\phi^{2}),\quad
A(r)=\Big{(}1-\frac{2\,M}{r}\Big{)}.$ (1.2)

In this study, our objective is to examine the behavior of light-like and
time-like geodesics within the framework of a Schwarzschild-Klinkhamer-type
wormhole space-time, taking into account the presence of topological defects
caused by a cosmic string and a global monopole. Specifically, we investigate
the effective potential of the system in both geometries and analyze how the
parameters of cosmic string and global monopole influences this potential,
leading to a shift in the results obtained. Furthermore, we focus on
determining the deflection angle for light-like geodesics within these
geometries. We observe that the topological defects arising from the cosmic
string and global monopole introduce modifications to this deflection angle,
indicating that the presence of these defects has an impact on the bending of
light within these wormhole space-times. Additionally, we discuss various
well-known space-time geometries that can be obtained as special cases from
the SK-wormhole with a cosmic string and a global monopole, further
highlighting the significance of these topological defects in shaping the
overall properties of these wormholes.

##  2 Schwarzschild-Klinkhamer wormhole with a cosmic string

In this section, we consider a generalized version of the Klinkhamer wormhole
space-time (1.1) by incorporating a factor $A(x)$ analogue to the
Schwarzschild solution in the presence of a cosmic string passing through it.
This new metric we called Schwarzschild-Klinkhamer wormhole space-time with a
cosmic string, a non-vacuum solution of the field equations. This new metric
is described by the following line-element in the chat $(t,x,\theta,\phi)$
as111We thank professor F. R. Klinkhamer who pointed out Ref. [64] where a
similar metric was presented without a cosmic string, $\alpha\to 1$.

$\displaystyle
ds^{2}=-A(x)\,dt^{2}+\frac{1}{\Big{(}1+\frac{b^{2}}{x^{2}}\Big{)}}\,\frac{dx^{2}}{A(x)}+(x^{2}+a^{2})\,(d\theta^{2}+\alpha^{2}\,\sin^{2}\theta\,d\phi^{2}),$
$\displaystyle A(x)=\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}.$ (2.1)

Here $\alpha<1$ is a cosmic string parameter related with linear mass density
of the string. We choose the parameters $a,b$, and $M$ are all positive
integer with $b>2\,M$, $b^{2}>a^{2}$, and $a>0$. The above metric (2.1) of
Schwarzschild-Klinkhamer wormhole with a cosmic string can reduce to
Schwarzschild solution with different spatial structure and the original
Klinkhamer wormhole metric (1.1) [39]. Note that the condition $b>2\,M$ in
(2.1) is required to prevent the formation of event horizon. The introduction
of a cosmic string into the space-time brings about significant alterations to
its geometrical and curvature characteristics.

The nonzero components of the Einstein tensor $G_{\mu\nu}$ for the space-time
(2.1) are

$\displaystyle-G^{t}_{t}=\frac{(b^{2}-a^{2})}{(x^{2}+a^{2})^{2}}\,\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}=G^{x}_{x},\quad
G^{\theta}_{\theta}=G^{\phi}_{\phi}=-\frac{R}{2}.$ (2.2)

The Ricci scalar $R$ and the Kretschmann scalar curvatures of the above space-
time are given by

$\displaystyle
R=\frac{4\,M}{(x^{2}+b^{2})^{3/2}}-\frac{2\,(a^{2}-b^{2}+2\,M\,\sqrt{x^{2}+b^{2}})}{(x^{2}+a^{2})^{2}},$
$\displaystyle\mathcal{K}=\frac{4}{(x^{2}+a^{2})^{4}\,(x^{2}+b^{2})^{3}}\,\Big{[}4\,M^{2}\,(a^{2}+x^{2})^{4}-4\,M\,(a^{2}+x^{2})^{2}\,(b^{2}+x^{2})^{5/2}-12\,b^{2}\,M\,(b^{2}+x^{2})^{7/2}$
$\displaystyle-12\,M\,x^{2}(b^{2}+x^{2})^{7/2}+16\,M\,(a^{2}+x^{2})\,(b^{2}+x^{2})^{7/2}+3\,(b^{2}+x^{2})^{4}\,(b^{2}+4M^{2}+x^{2})$
$\displaystyle+(a^{2}+x^{2})^{2}\,(b^{2}+x^{2})^{2}\,(3b^{2}+4M^{2}+3x^{2})-2\,(a^{2}+x^{2})\,(b^{2}+x^{2})^{3}\,(3b^{2}+4M^{2}+3x^{2})\Big{]},$
(2.3)

where are finite at $x=0$ even for the case $b^{2}=a^{2}$ and vanishes for
$x\to\pm\,\infty$. Note that the curvature components of
$R_{\mu\nu\alpha\beta}$ are influenced by the cosmic string parameter
$\alpha$, and hence, changes the curvature characteristics.

For the energy-momentum tensor
$T^{\mu}_{\nu}=\mbox{diag}\,(-\rho,p_{x},p_{t},p_{t})$, where $\rho$ is the
energy-density, $p_{x},p_{t}$ are the pressure components along $x$-direction
and tangential. We have

$\displaystyle\rho=p_{x}=\frac{(b^{2}-a^{2})}{(x^{2}+a^{2})^{2}}\,\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}>0,$
$\displaystyle
p_{t}=-\frac{2\,M}{(x^{2}+b^{2})^{3/2}}+\frac{(a^{2}-b^{2}+2\,M\,\sqrt{x^{2}+b^{2}})}{(x^{2}+a^{2})^{2}},$
(2.4)

provided $b^{2}>a^{2}$, where $b>2\,M$ to satisfies the weak energy condition.
We see that these physical quantities as well as the scalar curvatures (2.3)
are finite at $x=0$ and vanishes for $x\to\pm\,\infty$.

To check the validate of the energy conditions, we consider the following
vector fields, such as the null vector $k^{\mu}$, the time-like four-vector
$U^{\mu}$ and a spacelike unit vector $\eta^{\mu}$ along $x$-direction for the
space-time (2.1) defined by

$\displaystyle
k^{\mu}=\frac{1}{\sqrt{2}}\,\Bigg{[}-\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}^{-1/2},\sqrt{1+\frac{b^{2}}{x^{2}}}\,\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}^{1/2},0,0\Bigg{]},$
$\displaystyle
U^{\mu}=\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}^{-1/2}\,\delta^{\mu}_{0},$
$\displaystyle\eta^{\mu}=\Bigg{[}\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}^{1/2}\,\sqrt{1+\frac{b^{2}}{x^{2}}}\Bigg{]}\,\delta^{\mu}_{x},$
$\displaystyle-U^{\mu}\,U_{\mu}=1=\eta^{\mu}\,\eta_{\mu},\quad
U_{\mu}\,\eta^{\mu}=0=k^{\mu}\,k_{\mu},\quad
k_{\mu}\,\eta^{\mu}=\frac{1}{\sqrt{2}}=U_{\mu}\,k^{\mu}.$ (2.5)

We choose the energy-momentum tensor as follows:

$\displaystyle
T_{\mu\nu}=(\rho+p_{t})\,U_{\mu}\,U_{\nu}+p_{t}\,g_{\mu\nu}+(p_{x}-p_{t})\,\eta_{\mu}\,\eta_{\nu}.$
(2.6)

With the help of (2.5), we have the following energy conditions

$T_{\mu\nu}\,U^{\mu}\,U^{\nu}=\rho>0\quad,\quad
T_{\mu\nu}\,k^{\mu}\,k^{\nu}=\frac{1}{2}(\rho+p_{x})>0$ (2.7)

which satisfies the null energy condition provided $b^{2}>a^{2}$. Thus, we see
that the new wormhole space-time (4.5) which is a non-vacuum solution of the
field equation satisfies the weak energy condition (WEC) as well as the null
energy condition (NEC).

We discuss below a few special cases of the above Schwarzschild-Klinkhamer
wormhole metric with a cosmic string.

CASE A: If one choose $M=0$, then the space-time (2.1) reduces to the
following form

$ds^{2}=-dt^{2}+\frac{1}{\Big{(}1+\frac{b^{2}}{x^{2}}\Big{)}}\,dx^{2}+(x^{2}+a^{2})\,(d\theta^{2}+\alpha^{2}\,\sin^{2}\theta\,d\phi^{2}),$
(2.8)

the Klinkhamer wormhole non-vacuum space-time with a cosmic string [39].
However, for $\alpha\to 1$, we will get back the original Klinkhamer wormhole
space-time (1.1) (see Eq. (4) in [39]), which is a non-vacuum and degenerate
metric for $b^{2}>a^{2}$.

For the case $b^{2}=a^{2}$, the space-time (2.8) reduces to the following form

$ds^{2}=-dt^{2}+\Big{(}1+\frac{b^{2}}{x^{2}}\Big{)}^{-1}\,dx^{2}+(x^{2}+b^{2})\,(d\theta^{2}+\alpha^{2}\,\sin^{2}\theta\,d\phi^{2})$
(2.9)

the well-known Klinkhamer wormhole vacuum space-time with a cosmic string [39]
(see Eq. (4) in [39], where $\lambda=b_{0}$). Transforming to a coordinate via
$r=\sqrt{x^{2}+b^{2}}$ into the space-time (2.9), we obtain

$ds^{2}=-dt^{2}+dr^{2}+r^{2}\,(d\theta^{2}+\alpha^{2}\,\sin^{2}\theta\,d\phi^{2}),$
(2.10)

which looks similar to a cosmic string space-time in the spherical system [65,
66] with $r\in[b,\infty)$ instead of $r\in[0,\infty)$. It is worthwhile
mentioning that the gravitational lensing effects of vacuum strings was
investigated in Ref. [67].

CASE B: If one choose $b^{2}=a^{2}$, where $2\,M<b<3\,M$, then the space-time
(2.1) reduces to the following form

$\displaystyle
ds^{2}=-\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}\,dt^{2}+\Bigg{[}\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}\Big{(}1+\frac{b^{2}}{x^{2}}\Big{)}\Bigg{]}^{-1}\,dx^{2}$
$\displaystyle+(x^{2}+b^{2})\,(d\theta^{2}+\alpha^{2}\,\sin^{2}\theta\,d\phi^{2})$
(2.11)

which is a vacuum space-time of the Schwarzschild wormhole or Schwarzschild-
Klinkhamer wormhole with a cosmic string. The Kretschmann scalar curvature of
the metric will be given by $\mathcal{K}=\frac{48\,M}{(x^{2}+b^{2})^{2}}$
which is finite at $x=0$ and vanishes for $x\to\pm\,\infty$. Transforming to a
new coordinate via $r=\sqrt{x^{2}+b^{2}}$ into the space-time (2.11), we
obtain

$\displaystyle
ds^{2}=-\Big{(}1-\frac{2\,M}{r}\Big{)}\,dt^{2}+\Big{(}1-\frac{2\,M}{r}\Big{)}^{-1}\,dr^{2}+r^{2}\,(d\theta^{2}+\alpha^{2}\,\sin^{2}\theta\,d\phi^{2}),$
(2.12)

which is similar to Schwarzschild vacuum space-time with a cosmic string [63]
(see Eq. (5) in [63]) with $r\in[b>2\,M,\infty)$.

CASE C: If one choose $M=0$ and $b=0$, then the space-time (2.1) reduces to
the following form

$ds^{2}=-dt^{2}+dx^{2}+(x^{2}+a^{2})\,(d\theta^{2}+\alpha^{2}\,\sin^{2}\theta\,d\phi^{2}),$
(2.13)

an example of Morris-Thorne-type wormhole metric with a cosmic string [62], a
non-vacuum solution of the Einstein’s field equations. The Ricci scalar given
by $R=-\frac{2\,a^{2}}{(x^{2}+a^{2})^{2}}$ and the Kretschmann scalar
curvature by $\mathcal{K}=\frac{12\,a^{4}}{(x^{2}+a^{2})^{4}}$ both are finite
at $x=0$ and vanishes for $x\to\pm\,\infty$. The matter energy-density and the
pressure components given by
$\rho=-p_{r}=p_{t}=\frac{a^{2}}{(x^{2}+a^{2})^{2}}$ [62] are finite at $x=0$
and vanishes for $x\to\pm\,\infty$.

CASE D: If one choose $M=0$ and $a=0$, then the space-time (2.1) reduces to
the following form

$ds^{2}=-dt^{2}+\Big{(}1+\frac{b^{2}}{x^{2}}\Big{)}^{-1}\,dx^{2}+x^{2}\,(d\theta^{2}+\alpha^{2}\,\sin^{2}\theta\,d\phi^{2}),$
(2.14)

another example of Morris-Thorne-type wormhole space-time with a cosmic string
in Eddington-inspired Born-Infeld (EiBI) gravity background. This metric is a
non-vacuum solution of the field equations with the Ricci scalar
$R=\frac{2\,b^{2}}{x^{4}}$ and the Kretschmann scalar curvature
$\mathcal{K}=\frac{12\,b^{4}}{x^{8}}$ both diverges at $x\to 0$ and vanishes
for $x\to\pm\,\infty$. Recently, a space-time with cosmic string in Eddington-
inspired Born-Infeld (EiBI) gravity background has been discussed in Ref.
[33].

In this study, our objective is to explore the influence of a cosmic string on
the deflection angle of photon light and the corresponding effective potential
of the system within the framework of wormhole space-time described by (2.1).
We derive the deflection angle for null geodesics and carefully analyze the
results.

##  3 Geodesics motion of test particles by wormhole with a cosmic string

In this section, we analyze the motion of both massless and massive particles
along geodesics in a wormhole space-time described by the metric (2.1).
Notably, this space-time exhibits independence with respect to the coordinates
$t$ and $\phi$, allowing for the existence of two Killing vectors. These
Killing vectors are denoted as $\eta^{\mu}_{(t)}=\delta^{\mu}_{t}$, associated
with time-translational invariance, and
$\eta^{\mu}_{(\phi)}=\delta^{\mu}_{\phi}$, associated with rotational
invariance. During the motion of a photon or a time-like geodesic, the
projection of the four-momentum of the particles along these Killing vectors
remains conserved. This conservation gives rise to two fundamental quantities:
the energy $E=-p_{\mu}\,\eta^{\mu}_{(t)}$ and the angular momentum
$J=p_{\mu}\,\eta^{\mu}_{(\phi)}$.

We begin this section by writing the Lagrangian of a system defined by [18,
19, 41, 44, 60, 62]

$\mathcal{L}=\frac{1}{2}\,g_{\mu\nu}\,\left(\frac{dx^{\mu}}{d\tau}\right)\,\left(\frac{dx^{\nu}}{d\tau}\right),$
(3.1)

where $\tau$ is the affine parameter along the geodesics, and $g_{\mu\nu}$ is
the metric tensor. Using the line-element (2.1), the Lagrangian (3.1) in the
equatorial plane $\theta=\frac{\pi}{2}$ (or photon or massive particles
confined to the equatorial plane) is given by

$\mathcal{L}=\frac{1}{2}\,\Bigg{[}-A(x)\,\dot{t}^{2}+\frac{1}{\Big{(}1+\frac{b^{2}}{x^{2}}\Big{)}}\,\frac{\dot{x}^{2}}{A(x)}+\alpha^{2}\,(x^{2}+a^{2})\,\dot{\phi}^{2}\Bigg{]}.$
(3.2)

As stated earlier, there are two constant of motions and these are given by

$\displaystyle
A(x)\,\dot{t}=E\Rightarrow\dot{t}=\frac{E}{A(x)}\quad,\quad\alpha^{2}\,(x^{2}+a^{2})\,\dot{\phi}=J\Rightarrow\dot{\phi}=\frac{J}{\alpha^{2}\,(x^{2}+a^{2})}.$
(3.3)

For null or time-like geodesics, the Lagrangian (3.2) after transforming
$r=\sqrt{x^{2}+b^{2}}$ becomes

$\frac{1}{2}\,\Big{(}\frac{dr}{d\tau}\Big{)}^{2}+V_{eff}(r)=\frac{E^{2}}{2}$
(3.4)

which has the same form of the equation for a unit mass particle with energy
$E^{2}/2$ moving in one-dimensional effective potential given by

$V_{eff}(r)=\Big{(}\frac{1}{2}-\frac{M}{r}\Big{)}\,\Bigg{[}\frac{J^{2}}{\alpha^{2}\,(r^{2}+a^{2}-b^{2})}-\varepsilon\Bigg{]},$
(3.5)

where $\varepsilon=0$ for null geodesics and $\varepsilon=-1$ for time-like
geodesics.

Figure 1:  Effective potential for null geodesics without ($\alpha=1$) and
with cosmic string ($\alpha=0.5$) parameter. $a=1.5\,M$, $b=2.3\,M$ (left);
$a=2\,M$, $b=2.3\,M$ (right)

Figure 2:  Effective potential for null geodesics for different values of
cosmic string parameter $\alpha$. $a=1.5\,M$, $b=2.3\,M$ (left); $a=2\,M$,
$b=2.3\,M$ (right)

Figure 3:  Effective potential for time-like geodesics without ($\alpha=1$)
and with cosmic string ($\alpha=0.5$) parameter. $a=1.5\,M$, $b=2.3\,M$,
$J=2\,M$ (left); $a=2\,M$, $b=2.3\,M$, $J=2\,M$ (right)

Figure 4: Effective potential for time-like geodesics for different values of
cosmic string parameter $\alpha$. Here, $a=1.5\,M$, $b=2.3\,M$, $J=2\,M$
(left); $a=2\,M$, $b=2.3\,M$, $J=2\,M$ (right).

The effective potential of the system described by equation (3.5) is
influenced not only by the parameters $a$ and $b$, but also by the cosmic
string parameter $\alpha$. In gravitational and cosmological contexts, the
cosmic string parameter typically lies within the interval $0<\alpha<1$.
Consequently, the effective potential is increased due to the presence of the
cosmic string.

In Figure 1, we present plots of the effective potential for null geodesics
with different values of the cosmic string parameter: $\alpha=1$ and
$\alpha=0.5$. On the left, we set $b=2.3\,M>2\,M$ and $a=1.5\,M$, while on the
right, we use $b=2.3\,M>2\,M$ and $a=2\,M$. It is evident that the effective
potential for null geodesics is significantly influenced by the cosmic string,
showing a higher magnitude compared to the case without the cosmic string.

Moving on to Figure 2, we plot the effective potential for various values of
the cosmic string parameter $\alpha$. Similarly, on the left side, we have
$b=2.3\,M>2\,M$ and $a=1.5\,M$, and on the right side, we have $b=2.3\,M$ and
$a=2\,M$. Here as well, we observe that the effective potential increases as
the value of the cosmic string parameter decreases. Furthermore, in Figures 3
and 4, we depict the effective potential for time-like geodesics, which
correspond to massive objects. On the left side, $b=2.3\,M>2\,M$ and
$a=1.5\,M$, and on the right side, we have $b=2.3\,M>2\,M$ and $a=2\,M$. We
observe that the same effects seen in the case of null geodesics apply here as
well, with the presence of the cosmic string leading to an increased effective
potential.

Now, we discuss circular photon orbits which satisfies the conditions
$\dot{r}=0$ and $\ddot{r}=0$. Unstable photon orbits, which constitute the so-
called photon sphere, additionally satisfy $\dddot{r}>0$. Generally, for
unstable photon orbit (photon sphere), the following conditions in terms of
the effective potential satisfies

$\frac{dV_{eff}}{dr}|_{r=r_{ph}}=0,\quad\frac{d^{2}V_{eff}}{dr^{2}}|_{r=r_{ph}}<0,$
(3.6)

where $r_{ph}$ is the radius of a photon sphere. Note that a photon sphere
corresponds to the maximum potential. For the space-time (2.1), using the
first condition, we obtain

$r^{3}_{ph}-M\,r^{2}_{ph}-2\,M\,r_{ph}+M\,(b^{2}-a^{2})=0$ (3.7)

whose real valued solution gives us the radius of the photon sphere that
exists for $b^{2}\geq a^{2}$ and $(b^{2}-a^{2})<4\,M^{2}$ with $2\,M<b<3\,M$,
and $a\geq 0$. Thus, the exact location of the photon sphere depends on the
parameters $a,b$ and $M$ which is not located at the wormhole throat. For the
case $b^{2}=a^{2}$, from above equation (3.7), we find this photon sphere
radius $r_{ph}=\frac{M}{2}+\sqrt{M\,\Big{(}\frac{M}{4}+2\Big{)}}$ that solely
depends on the mass of the object $M$.

To find the path equation for photon ray, from Eqs. (3.4) and (3.5), we obtain

$\Big{(}\frac{dr}{d\tau}\Big{)}^{2}=E^{2}-\Big{(}1-\frac{2\,M}{r}\Big{)}\,\frac{J^{2}/\alpha^{2}}{(r^{2}-f^{2})}.$
(3.8)

Transforming $r=\frac{1}{u(\phi)}$ into the above equation and then
differentiating w. r. t. $\phi$, one will find the following path equation by
neglecting the terms $\alpha^{4}$ (assuming $\alpha$ has finite value not very
very small)

$\frac{1}{\alpha^{2}}\,\Big{(}\frac{d^{2}u}{d\phi^{2}}\Big{)}+u=3\,M\,u^{2}+2\,(b^{2}-a^{2})\,u^{3}-5\,M\,(b^{2}-a^{2})\,u^{4}.$
(3.9)

For $b^{2}=a^{2}$, one will find similar path equation for photon ray as in
the case of Schwarzschild solution [68].

Overall, our analysis demonstrates that the effective potential in a
Schwarzschild-Klinkhamer wormhole space-time is influenced by the cosmic
string parameter $\alpha$. The effective potential increases with the presence
of the cosmic string, as confirmed by our plots for both null and time-like
geodesics.

###  3.1 Gravitational lensing by wormhole with a cosmic string for $M=0$

In this part, we focus on analyzing the deflection angle of photon light
within the space-time geometry given by (2.1). Specifically, we consider here
the case where $M=0$. By evaluating Eqs. (3.3) and (3.4) for null geodesics,
we derive the following expression:

$\displaystyle\frac{d\phi}{dr}=\frac{\dot{\phi}}{\dot{r}}$ $\displaystyle=$
$\displaystyle\frac{J}{\alpha^{2}}\,\frac{1}{(r^{2}-f^{2})\,\sqrt{E^{2}-2\,V_{eff}(r)}}=\frac{\beta}{\alpha^{2}}\,\frac{1}{\sqrt{(r^{2}-f^{2})^{2}-\frac{\beta^{2}}{\alpha^{2}}\,(r^{2}-f^{2})}}$
(3.10)

where $\beta=\frac{J}{E}$ is the impact parameter characterizing a particular
null geodesic with the conserved energy parameter $E$ and the angular momentum
$J$, and we have defined $f^{2}=(b^{2}-a^{2})\geq 0$ for $b^{2}\geq a^{2}$.
Hence, we can obtain the deflection angle [18, 19, 44, 62, 33, 32]

$\displaystyle\delta\phi=\Delta\phi-\pi,$ (3.11)

where

$\displaystyle\Delta\phi=\frac{2\,\beta}{\alpha^{2}}\,\int^{\infty}_{r_{0}}\,\frac{dr}{\sqrt{(r^{2}-f^{2})^{2}-\frac{\beta^{2}}{\alpha^{2}}\,(r^{2}-f^{2})}}=\frac{2\,\beta}{\alpha^{2}}\,\int^{\infty}_{r_{0}}\,\frac{dr}{\sqrt{(r^{2}-f^{2})(r^{2}-r^{2}_{0})}}.$
(3.12)

Here $r=r_{0}$ is the turning point which can be obtained by setting
$\Big{(}\frac{dr}{d\phi}\Big{)}|_{r=r_{0}}=0$ for null geodesics, and in terms
of the impact parameter $\beta$, we obtain

$r_{0}=\sqrt{f^{2}+\frac{\beta^{2}}{\alpha^{2}}}=\sqrt{b^{2}-a^{2}+\frac{\beta^{2}}{\alpha^{2}}}.$
(3.13)

Defining a new coordinate via $y=r_{0}/r$ and $g=f/r_{0}$ into the integral
(3.12), we obtain

$\displaystyle\Delta\phi=\frac{2\,\beta}{\alpha^{2}\,r_{0}}\,K(g),$ (3.14)

where the complete elliptic integral of the first kind is given by [69]

$K(g)=\int^{1}_{0}\,\frac{dy}{\sqrt{(1-g^{2}\,y^{2})(1-y^{2})}},\quad 0<g<1.$
(3.15)

The expression of this complete elliptic integral becomes

$\displaystyle
K(g)=\frac{\pi}{2}\,\sum^{\infty}_{n=0}\,\Bigg{(}\frac{(2\,n)!}{2^{2\,n}\,(n!)^{2}}\Bigg{)}^{2}\,g^{2\,n}=\frac{\pi}{2}\,\Big{[}1+\frac{g^{2}}{4}+\frac{9\,g^{4}}{64}+\frac{2\,5}{256}\,g^{6}+...\Big{]},$
(3.16)

for $g<1$ since $f^{2}<r^{2}_{0}$, that is, in the weak field limit.

Therefore, the deflection angle from (3.11) using (3.14) becomes

$\displaystyle\delta\phi(g)=\frac{2\,\beta}{\alpha^{2}\,\sqrt{b^{2}-a^{2}+\frac{\beta^{2}}{\alpha^{2}}}}\,K(g)-\pi.$
(3.17)

We illustrate the deflection angle given by equation (3.17) as a function of
$g$ in Figure 5, considering both cases: without the effects of a cosmic
string and with the effects of a cosmic string. It is observed that the
deflection angle is significantly increased in the presence of a cosmic
string, indicating the substantial influence of the cosmic string on the
deflection of light.

Figure 5:  Deflection angle without ($\alpha=1$, left one) and with
($\alpha=1/2$, right one) cosmic string parameter. Here, $a=2.2$, $b=2.3$,
$\beta=1.5$.

As the parameter $g$ always falls within the range $0<g<1$, in the weak field
limit, we can express the deflection angle from equation (3.17) as follows:

$\displaystyle\delta\phi=\frac{\pi\,\beta}{\alpha^{2}\,\sqrt{b^{2}-a^{2}+\frac{\beta^{2}}{\alpha^{2}}}}\,\Bigg{[}1+\frac{(b^{2}-a^{2})/4}{\Big{(}b^{2}-a^{2}+\frac{\beta^{2}}{\alpha^{2}}\Big{)}}+\frac{9(b^{2}-a^{2})^{2}/64}{\Big{(}b^{2}-a^{2}+\frac{\beta^{2}}{\alpha^{2}}\Big{)}^{2}}+\mathcal{O}(g)^{6}\Bigg{]}-\pi.$
(3.18)

It is evident that the deflection angle of photon light is influenced by the
cosmic string parameter $\alpha$ and undergoes modifications. When
$b^{2}=a^{2}$, the space-time corresponds to a Schwarzschild-Klinkhamer
wormhole solution with cosmic string, as given by the metric (2.9). In that
case, the expression of the deflection angle for null geodesics in the weak
field limit, derived from equation (3.18), can be written as
$\delta\phi=\pi\,\left(\frac{1}{\alpha}-1\right)>0$. This implies that the
deflection angle depends on the cosmic string parameter $\alpha$.

###  3.2 Gravitational lensing by wormhole with a cosmic string for $M\neq 0$

In this part, we study the deflection of photon ray for the case $M\neq 0$.
Therefore, from equation (3.4) using (3.5) for null geodesics, we can write

$\dot{r}^{2}=\Big{(}\frac{dr}{d\tau}\Big{)}^{2}=E^{2}-\Big{(}1-\frac{2\,M}{r}\Big{)}\,\frac{J^{2}/\alpha^{2}}{(r^{2}-f^{2})}.$
(3.19)

From (3.3) and (3.19), we derive the following expression

$\displaystyle\frac{dr}{d\phi}=\frac{\alpha^{2}}{\beta}\,\sqrt{(r^{2}-f^{2})-\frac{\beta^{2}}{\alpha^{2}}\,\Big{(}1-\frac{2\,M}{r}\Big{)}\,(r^{2}-f^{2})}.$
(3.20)

Therefore, the deflection angle of photon ray is given by

$\delta\phi=\Delta\phi-\pi,$ (3.21)

where

$\Delta\phi=\frac{2}{\alpha}\,\int^{\infty}_{\bar{r}}\,\frac{dr}{\sqrt{r^{2}-f^{2}}\,\sqrt{\alpha^{2}\,\Big{(}\frac{r^{2}-f^{2}}{\beta^{2}}\Big{)}-\Big{(}1-\frac{2\,M}{r}\Big{)}}}.$
(3.22)

Here $r=\bar{r}$ is the turning point and can be obtained by setting
$\frac{dr}{d\phi}|_{r=\bar{r}}=0$ for null geodesics, and one will find impact
parameter in terms of this turning point by
$\beta(\bar{r})=\alpha\,\sqrt{\bar{r}^{2}-f^{2}}\,\Big{(}1-\frac{2\,M}{\bar{r}}\Big{)}^{-1/2}$.
Thus, equation (3.22) can be expressed by replacing the impact parameter as
follows:

$\Delta\phi=\frac{2}{\alpha}\,\int^{\infty}_{\bar{r}}\,\frac{dr}{\sqrt{r^{2}-f^{2}}\,\sqrt{\Big{(}\frac{r^{2}-f^{2}}{\bar{r}^{2}-f^{2}}\Big{)}\Big{(}1-\frac{2\,M}{\bar{r}}\Big{)}-\Big{(}1-\frac{2\,M}{r}\Big{)}}}.$
(3.23)

If one consider $b^{2}=a^{2}$, as discussed in CASE B of section 2, the space-
time (2.1) under this condition becomes a vacuum Schwarzschild-Klikhamer
wormhole space-time with a cosmic string. Therefore, for $b^{2}=a^{2}$, we
have $f\to 0$, and the impact parameter in terms of the turning point becomes
$\beta(\bar{r}_{0})=\alpha\,\bar{r}_{0}\,\Big{(}1-\frac{2\,M}{\bar{r}_{0}}\Big{)}^{-1/2}$,
where $\bar{r}_{0}>2\,M$. In that case, from equation (3.23), we find

$\Delta\phi=\frac{2}{\alpha}\,\int^{\infty}_{\bar{r}_{0}}\,\frac{dr}{r\,\sqrt{\Big{(}\frac{r}{\bar{r}_{0}}\Big{)}^{2}\Big{(}1-\frac{2\,M}{\bar{r}_{0}}\Big{)}-\Big{(}1-\frac{2\,M}{r}\Big{)}}}$
(3.24)

which is equals to $\frac{1}{\alpha}$ times the expression obtained in the
background of a Schwarzschild solution [68]. Here, $r$ coordinate lies in the
interval $b\leq r<\infty$, where $2\,M<b<3\,M$ instead of $r\in[0,\infty)$. We
see that the presence of a cosmic string in the space-time geometry increases
the deflection angle compared to the known expression in Schwarzschild
solution.

##  4 Schwarzschild-Klinkhamer wormhole with a global monopole

In this section, we consider a generalized version of the Klinkhamer wormhole
space-time (1.1) by incorporating a factor $A(x)$ analogue to the
Schwarzschild solution in the presence of a global monopole. Therefore, the
wormhole space-time with a global monopole which we are interested here is
described by the following line-element

$\displaystyle
ds^{2}=-A(x)\,dt^{2}+\frac{1}{\Big{(}1+\frac{b^{2}}{x^{2}}\Big{)}}\,\frac{dx^{2}}{\alpha^{2}_{0}\,A(x)}+(x^{2}+a^{2})(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),$
$\displaystyle A(x)=\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)},$ (4.1)

where $0<\alpha_{0}<1$ is the global monopole parameter. The introduction of a
global monople into the space-time brings about significant alterations to its
geometry and curvature characteristics. This wormhole space-time (4.1)
possesses the following nonzero components of the Ricci tensor $R_{\mu\nu}$
given by

$\displaystyle
R_{tt}=\frac{2\,M\,\alpha^{2}_{0}\,(b^{2}-a^{2})}{(x^{2}+a^{2})\,(x^{2}+b^{2})^{3/2}}\,\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)},$
$\displaystyle
R_{xx}=\frac{2\,M\,x^{2}}{(x^{2}+b^{2})^{5/2}}\,\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}^{-1}+\frac{2}{(x^{2}+a^{2})^{2}}\,\Bigg{[}-a^{2}+\frac{x^{2}+a^{2}}{x^{2}+b^{2}}\,\Bigg{\\{}b^{2}+\frac{M\,x^{2}}{2\,M-\sqrt{x^{2}+b^{2}}}\Bigg{\\}}\Bigg{]},$
$\displaystyle R_{\theta\theta}=(1-\alpha^{2}_{0}),\quad
R_{\phi\phi}=(1-\alpha^{2}_{0})\,\sin^{2}\theta.$ (4.2)

The Ricci $R$ and Kretschmann scalar curvatures $\mathcal{K}$ for the metric
(4.1) are given by

$\displaystyle
R=2\,\Bigg{[}\frac{1}{x^{2}+a^{2}}+\alpha^{2}_{0}\,\Bigg{\\{}\frac{2\,M}{(x^{2}+b^{2})^{3/2}}+\frac{-2\,a^{2}+b^{2}-x^{2}-2\,M\,\sqrt{x^{2}+b^{2}}}{(x^{2}+a^{2})^{2}}\Bigg{\\}}\Bigg{]},$
$\displaystyle\mathcal{K}=\frac{4}{(x^{2}+a^{2})^{4}\,(x^{2}+b^{2})^{3}}\,\Big{[}4\,M\,(a^{2}+x^{2})\,(b^{2}+x^{2})^{7/2}\,\alpha^{2}_{0}+4\,M^{2}\,(a^{2}+x^{2})^{4}\,\alpha^{4}_{0}$
$\displaystyle-4\,M\,(a^{2}+x^{2})^{2}\,(b^{2}+x^{2})^{5/2}\,\alpha^{4}_{0}-12\,b^{2}\,M\,(b^{2}+x^{2})^{7/2}\,\alpha^{4}_{0}-12\,M\,x^{2}\,(b^{2}+x^{2})^{7/2}\,\alpha^{4}_{0}$
$\displaystyle+12\,M\,(a^{2}+x^{2})\,(b^{2}+x^{2})^{7/2}\,\alpha^{4}_{0}+3\,(b^{2}+x^{2})^{4}\,(b^{2}+4M^{2}+x^{2})\,\alpha^{4}_{0}$
$\displaystyle-2\,(a^{2}+x^{2})\,(b^{2}+x^{2})^{3}\,\alpha^{2}_{0}\,\Big{(}b^{2}+x^{2}+2\,(b^{2}+2M^{2}+x^{2})\,\alpha^{2}_{0}\Big{)}$
$\displaystyle+(a^{2}+x^{2})^{2}\,(b^{2}+x^{2})^{2}\,\Big{(}b^{2}+x^{2}+2\,(b^{2}+2M^{2}+x^{2})\,\alpha^{4}_{0}\Big{)}\Big{]},$
(4.3)

which are finite at $x=0$ even for the case $b^{2}=a^{2}$ and vanishes for
$x\pm\,\infty$.

Considering the energy-momentum tensor
$T^{\mu}_{\nu}=\mbox{diag}(-\rho,p_{x},p_{t},p_{t})$, we obtain the following
physical parameters for the space-time (4.1) as

$\displaystyle\rho=-T^{t}_{t}=-G^{t}_{t}=\frac{2\,M\,\alpha^{2}_{0}}{(x^{2}+a^{2})\,\sqrt{x^{2}+b^{2}}}+\frac{1}{x^{2}+a^{2}}+\frac{\alpha^{2}_{0}\Big{(}-2\,a^{2}+b^{2}-x^{2}-2\,M\,\sqrt{x^{2}+b^{2}}\Big{)}}{(x^{2}+a^{2})^{2}},$
$\displaystyle
p_{x}=T^{x}_{x}=G^{x}_{x}=\frac{2\,\alpha^{2}_{0}\,(1+b^{2}/x^{2})}{(x^{2}+a^{2})^{2}}\,\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}\,\Bigg{[}-a^{2}+\Big{(}\frac{x^{2}+a^{2}}{x^{2}+b^{2}}\Big{)}\,\Bigg{\\{}b^{2}+\frac{M\,x^{2}}{2\,M-\sqrt{x^{2}+b^{2}}}\Bigg{\\}}\Bigg{]}$
$\displaystyle-\frac{1}{x^{2}+a^{2}}-\frac{\alpha^{2}_{0}\,\Big{(}-2\,a^{2}+b^{2}-x^{2}-2\,M\,\sqrt{x^{2}+b^{2}}\Big{)}}{(x^{2}+a^{2})^{2}},$
$\displaystyle
p_{t}=T^{\theta}_{\theta}=G^{\theta}_{\theta}=G^{\phi}_{\phi}=-\alpha^{2}_{0}\,\Bigg{[}\frac{2\,M}{(x^{2}+b^{2})^{3/2}}+\frac{b^{2}-a^{2}-2\,M\,\sqrt{x^{2}+b^{2}}}{(x^{2}+a^{2})^{2}}\Bigg{]}.$
(4.4)

From above we see that the wormhole space-time considered by (4.1) is a non-
vacuum solution of the field equations. The energy-density given by
$\rho=\Bigg{[}\frac{2\,M\,\alpha^{2}_{0}\,(b^{2}-a^{2})}{(x^{2}+a^{2})\,(x^{2}+b^{2})^{3/2}}+\frac{R}{2}\Bigg{]}>0$
is positive provided $b^{2}\geq a^{2}$, where $b>2\,M$. All these physical
quantities, such as the energy-density ($\rho$), the tangential pressure
($p_{t}$), and the scalar curvatures given in (4.3) are finite at $x=0$ and
vanishes for $x\pm\,\infty$. Note that the pressure component $p_{x}$ is
undefined at $x=0$ under the case $b^{2}>a^{2}$. However, for $b^{2}=a^{2}$ in
the subsequent discussion, we show that this pressure component is finite at
$x=0$ and vanishes for $x\pm\,\infty$.

CASE A: If one choose $b^{2}=a^{2}$, then the space-time (4.1) reduces to the
following form

$\displaystyle
ds^{2}=-\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}\,dt^{2}+\Bigg{[}\alpha^{2}_{0}\,\Bigg{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Bigg{)}\,\Bigg{(}1+\frac{b^{2}}{x^{2}}\Bigg{)}\Bigg{]}^{-1}dx^{2}$
$\displaystyle+(x^{2}+b^{2})(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}).$ (4.5)

This space-time (4.5) is a new non-vacuum wormhole solution and a degenerate
one. This wormhole space-time possesses the following nonzero components of
the Einstein tensor,
$G^{\mu}_{\,\nu}=R^{\mu}_{\,\nu}-\frac{1}{2}\,\delta^{\mu}_{\,\nu}\,R$, given
by

$G^{t}_{t}=G^{x}_{x}=-\frac{(1-\alpha^{2}_{0})}{x^{2}+b^{2}}=-R/2$ (4.6)

and $G^{\theta}_{\,\theta}=0=G^{\phi}_{\,\phi}$.

The Ricci scalar and the Kretschmann scalar curvatures of this new wormhole
space-time (4.5) are given by

$R=\frac{2\,(1-\alpha^{2}_{0})}{x^{2}+b^{2}}\quad,\quad\mathcal{K}=\frac{2\,\Big{[}289\,(x^{2}+b^{2})-1248\,M\,\sqrt{x^{2}+b^{2}}+2124\,M^{2}\Big{]}}{(x^{2}+b^{2})^{3/2}}.$
(4.7)

One can see from above that both the scalar curvatures are finite at $x=0$ and
vanishes for $x\to\pm\,\infty$. Assuming that this new non-vacuum wormhole
space-time (4.5) is a solution of the Einstein’s field equations,
$G^{\mu}_{\,\nu}=T^{\mu}_{\,\nu}$, where $\frac{8\,\pi\,G}{c^{4}}=1$, then the
energy density of the stress-energy fluid is given by

$\rho=-T^{t}_{t}=-G^{t}_{t}=R/2=\frac{1-\alpha^{2}_{0}}{x^{2}+b^{2}}>0$ (4.8)

which is always positive since global monopole parameter lies $0<\alpha_{0}<1$
in gravitation and cosmology. The only pressure component along the
$x$-direction is given by

$p_{x}=-R/2=\frac{\alpha^{2}_{0}-1}{x^{2}+b^{2}}<0$ (4.9)

with the tangential component $p_{t}=0$. One can see that $\rho+p_{x}=0$ which
implies that the null energy condition (NEC) is automatically satisfied.

To check the validate of the energy conditions, we consider the following
vector fields, such as the null vector $k^{\mu}$, the time-like four-vector
$U^{\mu}$ and a spacelike unit vector $\eta^{\mu}$ along $x$-direction for the
space-time (4.5) defined by

$\displaystyle
k^{\mu}=\frac{1}{\sqrt{2}}\,\Bigg{[}-\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}^{-1/2},\alpha_{0}\,\sqrt{1+\frac{b^{2}}{x^{2}}}\,\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}^{1/2},0,0\Bigg{]},$
$\displaystyle
U^{\mu}=\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}^{-1/2}\,\delta^{\mu}_{0},$
$\displaystyle\eta^{\mu}=\Bigg{[}\alpha_{0}\,\Big{(}1-\frac{2\,M}{\sqrt{x^{2}+b^{2}}}\Big{)}^{1/2}\,\sqrt{1+\frac{b^{2}}{x^{2}}}\Bigg{]}\,\delta^{\mu}_{x},$
$\displaystyle-U^{\mu}\,U_{\mu}=1=\eta^{\mu}\,\eta_{\mu},\quad
U_{\mu}\,\eta^{\mu}=0=k^{\mu}\,k_{\mu},\quad
k_{\mu}\,\eta^{\mu}=\frac{1}{\sqrt{2}}=U_{\mu}\,k^{\mu}.$ (4.10)

For zero tangential pressure, $p_{t}=0$, we choose the following energy-
momentum tensor

$\displaystyle
T_{\mu\nu}=(\rho+p_{t})\,U_{\mu}\,U_{\nu}+p_{t}\,g_{\mu\nu}+(p_{x}-p_{t})\,\eta_{\mu}\,\eta_{\nu}=\rho\,U_{\mu}\,U_{\nu}+p_{x}\,\eta_{\mu}\,\eta_{\nu}.$
(4.11)

With the help of (4.10), we have the following energy conditions

$T_{\mu\nu}\,U^{\mu}\,U^{\nu}=\rho>0\quad,\quad
T_{\mu\nu}\,k^{\mu}\,k^{\nu}=0.$ (4.12)

Thus, we see that the new wormhole space-time (4.5) which is a non-vacuum
solution of the field equation satisfies the weak energy condition (WEC) as
well as the null energy condition (NEC). We called this new non-vacuum space-
time (4.5) as Schwarzschild-Klinkhamer (SK) wormhole metric with a global
monopole. Since the space-time described by the line-element (4.1) is a
generalization of the non-vacuum space-time (4.5) under the case $b^{2}\neq
a^{2}$, we called this new metric (4.1) the generalized SK-wormhole space-time
with a global monopole which satisfies both the weak energy condition (WEC)
and the null energy condition (NEC).

If one perform a transformation via $r=\sqrt{x^{2}+b^{2}}$ into the space-time
(4.5), we obtain

$\displaystyle
ds^{2}=-\Big{(}1-\frac{2\,M}{r}\Big{)}\,dt^{2}+\alpha^{-2}_{0}\,\Big{(}1-\frac{2\,M}{r}\Big{)}^{-1}\,dr^{2}+r^{2}\,(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),\,r\in[b>2\,M,\infty)$
(4.13)

the Schwarzschild non-vacuum wormhole space-time with a global monopole.

CASE B: If one choose $M=0$, then the space-time (4.1) reduces to the
following form

$ds^{2}=-dt^{2}+\Big{(}1+\frac{b^{2}}{x^{2}}\Big{)}^{-1}\,\alpha^{-2}_{0}\,dx^{2}+(x^{2}+a^{2})\,(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),$
(4.14)

the Klinkhamer wormhole space-time with a global monopole [39], which is a
non-vacuum and degenerate metric. The Ricci scalar and the Kretschamnn scalar
curvatures are given by

$\displaystyle
R=\frac{2\,[x^{2}+a^{2}+(-2\,a^{2}+b^{2}-x^{2})\alpha^{2}_{0}]}{(x^{2}+a^{2})^{2}},$
$\displaystyle\mathcal{K}=\frac{2\,[371\,a^{4}+706\,b^{4}+624\,b^{2}\,x^{2}+289\,x^{4}-2\,a^{2}\,(394\,b^{2}+23\,x^{2})]}{(a^{2}+x^{2})^{4}}.$
(4.15)

Both the scalar curvatures are finite at $x=0$ and vanishes for
$x\to\pm\,\infty$.

Even for $b^{2}=a^{2}$, the metric (4.14) is still a non-vacuum solution of
the field equations with the energy density $\rho=R/2$, and the radial
pressure $p_{r}=-R/2<0$, where the Ricci scalar $R$ is given by (4.7) and the
Kretschmann scalar will be $\mathcal{K}=\frac{578}{(x^{2}+b^{2})}$. All these
physical quantities are finite at $x=0$ and vanishes for $x\to\pm\,\infty$.
For $\alpha_{0}\to 1$ and , the space-time (4.14) reduces to the original
Klinkhamer non-vacuum wormhole solution (see Eq. (4) in [39]) and becomes a
vacuum one for $b^{2}=a^{2}$.

For the case, $b^{2}=a^{2}$, and $\alpha_{0}\neq 1$, one can do a
transformation via $r=\sqrt{x^{2}+b^{2}}$ into the space-time (4.14), we
obtain

$ds^{2}=-dt^{2}+\frac{dr^{2}}{\alpha^{2}_{0}}+r^{2}\,(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),\quad
r\in[b>0,\infty)$ (4.16)

the global monopole-like space-time [70, 71].

CASE C: If one choose $M=0$ and $a=0$, then the space-time (4.1) reduces to
the following form

$ds^{2}=-dt^{2}+\alpha^{-2}_{0}\,\Big{(}1+\frac{b^{2}}{x^{2}}\Big{)}^{-1}\,dx^{2}+x^{2}\,(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),$
(4.17)

an example of a topologically charged Eddington-inspired Born-Infeld (EiBI)
gravity background space-time, which is a non-vacuum solution. The Ricci
scalar and the Kretschmann scalar curvatures are given by

$R=\frac{2\,[(1-\alpha^{2}_{0})\,x^{2}+b^{2}\,\alpha^{2}_{0}]}{x^{4}}\quad,\quad\mathcal{K}=\frac{2\,[706\,b^{4}+624\,b^{2}\,x^{2}+289\,x^{4}]}{x^{8}}.$
(4.18)

Both the scalar curvatures vanishes at $x\to\pm\,\infty$. Recently, examples
of a topologically charged Eddington-inspired Born-Infeld gravity background
space-time has been discussed in Ref. [32, 72].

CASE D: If one choose $M=0$ and $b=0$, then the space-time (4.1) reduces to
the following form

$ds^{2}=-dt^{2}+\frac{dx^{2}}{\alpha^{2}_{0}}+(x^{2}+a^{2})\,(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),$
(4.19)

the topologically charged Ellis-Bronnikov-type wormhole [73], and becomes an
Ellis-Bronnikov-type or Morris-Thorne-type wormhole for $\alpha_{0}\to 1$ [34,
35, 36].

##  5 Geodesics motion of test particles by wormhole with a global monopole

In this part, we study the geodesics motions of test particles in the space-
time background (4.1). Therefore, the Lagrangian of the system on the
equatorial plane $\theta=\frac{\pi}{2}$ is given by

$\mathcal{L}=\frac{1}{2}\,\Bigg{[}-A(x)\,\dot{t}^{2}+\frac{1}{\alpha^{2}_{0}\,\Big{(}1+\frac{b^{2}}{x^{2}}\Big{)}}\,\frac{\dot{x}^{2}}{A(x)}+(x^{2}+a^{2})\,\dot{\phi}^{2}\Bigg{]}.$
(5.1)

Here also, two constant of motions corresponding to the coordinate $(t,\phi)$
which are as follows:

$\displaystyle
A(x)\,\dot{t}=E\Rightarrow\dot{t}=\frac{E}{A(x)}\quad,\quad(x^{2}+a^{2})\,\dot{\phi}=J_{0}\Rightarrow\dot{\phi}=\frac{J_{0}}{(x^{2}+a^{2})},$
(5.2)

where $E$ is the conserved energy parameter, and $J_{0}$ is the conserved
angular momentum.

Henre, the Lagrangian (5.1) for null or time-like geodesics after transforming
$r=\sqrt{x^{2}+b^{2}}$ becomes

$\frac{1}{2}\,\Big{(}\frac{dr}{d\tau}\Big{)}^{2}+V_{eff}(r)=\frac{\alpha^{2}_{0}\,E^{2}}{2}=\frac{E^{2}_{eff}}{2},$
(5.3)

which has the same form of the equation for a unit mass particle with energy
$E^{2}_{eff}/2$ moving in one-dimensional effective potential given by

$V_{eff}(r)=\alpha^{2}_{0}\,\Big{(}\frac{1}{2}-\frac{M}{r}\Big{)}\,\Bigg{[}\frac{J^{2}_{0}}{r^{2}+a^{2}-b^{2}}-\varepsilon\Bigg{]},$
(5.4)

where $\varepsilon=0$ for null geodesics and $\varepsilon=-1$ for time-like
geodesics.

The effective potential (5.7) of the system for both null and time-like
geodesics is influenced not only by the parameters $a,b$ and $M$, but also by
the global monopole parameter $\alpha_{0}$. One can see that the effective
potential of the system decreases compared to the result without the effects
of global monopole, since in the context of gravitation and cosmology, the
values of the global monopole parameter $\alpha_{0}$ typically lies
$0<\alpha_{0}<1$.

In Figure 6, we present plots of the effective potential for null geodesics
without and with the effects of the global monopole parameter: $\alpha_{0}=1$
and $\alpha_{0}=0.5$. On the left side, we set $b=2.3\,M>2\,M$ and $a=1.5\,M$,
while on the right side, we use $b=2.3\,M>2\,M$ and $a=2\,M$. It can be
observed that the effective potential for null geodesics is diminished under
the influence of the global monopole, as compared to the case without the
presence of global monopole parameter ($\alpha_{0}\to 1$).

Moving on to Figure 7, we plot the effective potential for null geodesics
considering various values of the global monopole parameter $\alpha_{0}$.
Similarly, on the left side, we have $b=2.3\,M>2\,M$ and $a=1.5\,M$, and on
the right side, we have $b=2.3\,M>2\,M$ and $a=2\,M$. Here as well, we observe
that the effective potential increases as the value of the global monopole
parameter $\alpha_{0}$ increases.

To find the path equation for photon ray, from Eqs. (5.3) and (5.4), we obtain

$\Big{(}\frac{dr}{d\tau}\Big{)}^{2}=\alpha^{2}_{0}\,\Bigg{[}E^{2}-\Big{(}1-\frac{2\,M}{r}\Big{)}\,\frac{J_{0}^{2}}{(r^{2}-f^{2})}\Bigg{]}.$
(5.5)

Transforming $r=\frac{1}{u(\phi)}$ into the above equation and then
differentiating w. r. t. $\phi$, one will find the following path equation

$\frac{1}{\alpha^{2}_{0}}\,\Big{(}\frac{d^{2}u}{d\phi^{2}}\Big{)}+\Bigg{[}1+\frac{2\,(b^{2}-a^{2})}{\beta^{\prime
2}}\Bigg{]}\,u=3\,M\,u^{2}+2\,(b^{2}-a^{2})\,\Bigg{[}1+\frac{b^{2}-a^{2}}{\beta^{\prime
2}}\Bigg{]}\,u^{3}-5\,M\,(b^{2}-a^{2})\,u^{4}.$ (5.6)

For $b^{2}=a^{2}$, one will find similar path equation for photon ray as in
the case of Schwarzschild solution [68].

In summary, the plots of the effective potential for both null geodesics
demonstrate the significant influence of the global monopole parameter. The
effective potential decreases compared to previous results, indicating the
impact of the global monopole on the dynamics of particles moving along
geodesics in this system.

Figure 6:  Effective potential for light-like geodesics without
($\alpha_{0}=1$) and with global monopole ($\alpha_{0}=0.5$) parameter. Here,
$a=1.5\,M$, $b=2.3\,M$ (left); $a=2\,M$, $b=2.3\,M$ (right)

Figure 7:  Effective potential for light-like geodesics for different values
of global monopole parameter $\alpha_{0}$. Here, $a=1.5\,M$, $b=2.3\,M$
(left); $a=2\,M$, $b=2.3\,M$ (right)

###  5.1 Gravitational lensing by wormhole with a global monopole for $M=0$

In this part, we calculate the deflection angle of photon rays within the
space-time geometry described by equation (4.1) for the specific case where
$M=0$. By considering Eqs. (5.2) and (5.3) for the null geodesics with $M=0$,
we obtain the following expression:

$\displaystyle\frac{d\phi}{dr}=\frac{\dot{\phi}}{\dot{r}}=\frac{\beta^{\prime}}{\alpha_{0}}\,\frac{1}{\sqrt{(r^{2}-f^{2})^{2}-\beta^{\prime
2}\,(r^{2}-f^{2})}},$ (5.7)

where $\beta^{\prime}=\frac{J_{0}}{E}$ is the impact parameter characterizing
a particular null geodesic with the conserved energy parameter $E$ and the
angular momentum $J_{0}$. Hence, we can obtain the deflection angle [44]

$\displaystyle\delta\phi=\Delta\phi-\pi,$ (5.8)

where

$\displaystyle\Delta\phi=\frac{2\,\beta^{\prime}}{\alpha_{0}}\,\int^{\infty}_{r_{0}}\,\frac{dr}{\sqrt{(r^{2}-f^{2})(r^{2}-r^{2}_{0})}}$
(5.9)

where $r=r_{0}$ is the turning point obtained by setting
$\Big{(}\frac{dr}{d\phi}\Big{)}=0$ in equation (5.3) for the null geodesics
corresponding to the case $M=0$, and in terms of impact parameter
$\beta^{\prime}$, it is given by

$r_{0}=\sqrt{f^{2}+\beta^{\prime 2}}=\sqrt{b^{2}-a^{2}+\beta^{\prime 2}}.$
(5.10)

Defining $z=r_{0}/r$ and $g^{\prime}=f/r_{0}$ into the integral (5.9), we
obtain

$\displaystyle\Delta\phi=\frac{2\,\beta^{\prime}}{\alpha_{0}\,r_{0}}\,K(g^{\prime}),$
(5.11)

where the complete elliptic integral of the first kind is given by [69]

$K(g^{\prime})=\int^{1}_{0}\,\frac{dy}{\sqrt{(1-g^{\prime
2}\,z^{2})(1-z^{2})}},\quad 0<g^{\prime}<1.$ (5.12)

The expression of this complete elliptic integral becomes

$\displaystyle
K(g^{\prime})=\frac{\pi}{2}\,\sum^{\infty}_{n=0}\,\Bigg{(}\frac{(2\,n)!}{2^{2\,n}\,(n!)^{2}}\Bigg{)}^{2}\,g^{\prime
2\,n}=\frac{\pi}{2}\,\Big{[}1+\frac{g^{\prime 2}}{4}+\frac{9\,g^{\prime
4}}{64}+\frac{2\,5}{256}\,g^{\prime 6}+...\Big{]},$ (5.13)

for $g^{\prime}<1$ since $f<r_{0}$.

Figure 8:  Deflection angle for $\alpha_{0}=1$ (left one) and $\alpha_{0}=1/2$
(right one). Here, $a=2.2$, $b=2.3$, $\beta^{\prime}=1.5$.

Therefore, the deflection angle from (5.10) using (5.13) becomes

$\displaystyle\delta\phi(g^{\prime})=\frac{2\,\beta^{\prime}}{\alpha_{0}\,\sqrt{b^{2}-a^{2}+\beta^{\prime
2}}}\,K(g^{\prime})-\pi.$ (5.14)

We plot the deflection angle given by equation (5.14) as a function of
$g^{\prime}$ in Figure 8, considering both cases: without the effects of the
global monopole parameter $\alpha_{0}=1$, and with the effects of the global
monopole parameter $\alpha_{0}\neq 1$. It is observed that the deflection
angle is significantly increased in the presence of the global monopole,
indicating the substantial influence of the global monopole on the deflection
of light.

As the parameter $g^{\prime}$ always falls within the range $0<g^{\prime}<1$,
the deflection angle given by equation (5.14) can be expressed as follows:

$\displaystyle\delta\phi=\frac{\pi\,\beta^{\prime}}{\alpha_{0}\,\sqrt{b^{2}-a^{2}+\beta^{\prime
2}}}\Bigg{[}1+\frac{(b^{2}-a^{2})/4}{\Big{(}b^{2}-a^{2}+\beta^{\prime
2}\Big{)}}+\frac{9(b^{2}-a^{2})^{2}/64}{\Big{(}b^{2}-a^{2}+\beta^{\prime
2}\Big{)}^{2}}+\mathcal{O}(g^{\prime})^{6}\Bigg{]}-\pi.$ (5.15)

It is evident that the deflection angle for null geodesics is influenced by
the global monopole parameter $\alpha_{0}$ and undergoes modifications. When
$b^{2}=a^{2}$, the space-time (4.14) corresponds to a non-vacuum solution of
the field equations known as Klinkhamer wormhole with a global monopole. In
that case, the expression for the deflection angle, derived from equation
(5.15), can be written as
$\delta\phi=\pi\,\left(\frac{1}{\alpha_{0}}-1\right)>0$. This implies that the
deflection angle is positive and dependent on the global monopole parameter
$\alpha_{0}$.

###  5.2 Gravitational lensing by wormhole with a global monopole for $M\neq
0$

In this part, we study the deflection of photon ray for the case $M\neq 0$ in
the space-time geometry (4.1). By considering Eqs. (5.2) and (5.3) for the
null geodesics with the case $M\neq 0$, we obtain the following expression:

$\displaystyle\frac{d\phi}{dr}=\frac{\dot{\phi}}{\dot{r}}=\frac{\beta^{\prime}}{\alpha_{0}}\,\frac{1}{\sqrt{(r^{2}-f^{2})^{2}-\beta^{\prime
2}\,\Big{(}1-\frac{2\,M}{r}\Big{)}\,(r^{2}-f^{2})}}.$ (5.16)

Therefore, the deflection angle is given by

$\delta\phi=\Delta\phi-\pi,$ (5.17)

where

$\Delta\phi=\frac{2}{\alpha_{0}}\,\int^{\infty}_{\tilde{r}}\,\frac{dr}{\sqrt{r^{2}-f^{2}}\,\sqrt{\Big{(}\frac{r^{2}-f^{2}}{\beta^{\prime
2}}\Big{)}-\Big{(}1-\frac{2\,M}{r}\Big{)}}}.$ (5.18)

Here $\tilde{r}$ is obtained by setting $\frac{dr}{d\tau}|_{r=\tilde{r}}=0$
for null geodesics, and one will find that the impact parameter in terms of
this point will be
$\beta^{\prime}(\tilde{r})=\sqrt{\tilde{r}^{2}-f^{2}}\,\Big{(}1-\frac{2\,M}{\tilde{r}}\Big{)}^{-1/2}$.
From equation (5.18), we obtain

$\Delta\phi=\frac{2}{\alpha_{0}}\,\int^{\infty}_{\tilde{r}}\,\frac{dr}{\sqrt{r^{2}-f^{2}}\,\sqrt{\Big{(}\frac{r^{2}-f^{2}}{\tilde{r}^{2}-f^{2}}\Big{)}-\Big{(}1-\frac{2\,M}{r}\Big{)}}}.$
(5.19)

If we consider $b^{2}=a^{2}$, as discussed in case A of section 4, the space-
time (4.1) becomes non-vacuum solution of the field equations. Therefore, for
$b^{2}=a^{2}$, we have $f\to 0$, and setting the impact parameter
$\beta^{\prime}(\tilde{r}_{0})=\tilde{r}_{0}\,\Big{(}1-\frac{2\,M}{\tilde{r}_{0}}\Big{)}^{-1/2}$,
where $\tilde{r}_{0}>2\,M$. In that case, we obtain from (5.19) the following
expression

$\displaystyle\Delta\phi$ $\displaystyle=$
$\displaystyle\frac{2}{\alpha_{0}}\,\int^{\infty}_{\tilde{r}_{0}}\,\frac{dr}{r\,\sqrt{\Big{(}\frac{r}{\tilde{r}_{0}}\Big{)}^{2}\Big{(}1-\frac{2\,M}{\tilde{r}_{0}}\Big{)}-\Big{(}1-\frac{2\,M}{r}\Big{)}}}=\frac{1}{\alpha_{0}}\,\Delta\phi_{Sch}$
(5.20)

which is equals to $\frac{1}{\alpha_{0}}$ times the value obtained in the
background of Schwarzschild-type black hole solution [68]. We see that the
presence of a global monopole increases the deflection angle for null
geodesics compared to the known expression in the literature.

##  6 Conclusions

This study investigates the impact of topological defects on the effective
potential of massless and/or massive particles in the context of generalized
SK-wormhole space-time, a new non-vacuum wormhole solution that satisfies the
weak and null energy conditions. By deriving the effective potential in the
presence of cosmic strings and global monopoles, we demonstrated that these
topological defects modify the effective potential and induced shifts in the
results. Specifically, the effective potential increases under the influence
of cosmic strings and decreases due to the presence of global monopoles.

We visually illustrate the effects of cosmic strings on the effective
potential for null and time-like geodesics through graphical representations
(figures 1-4). Also, we visually illustrate the effects of global monopole on
the effective potential for null geodesic by figures 6-7. These plots provide
a clear understanding of how the presence of these topological defects alters
the behavior of particles moving along geodesics in the generalized SK-
wormhole space-time.

Furthermore, we determine the deflection angle for null geodesics in both the
SK-wormhole with cosmic string and with global monopole backgrounds, focusing
on the special case where $M=0$ and later on $M\neq 0$. Our analysis reveals
that the deflection angle is influenced by the cosmic string and global
monopole, resulting in an increased deflection compared to the case without
these topological defects. We present graphical representations of the
deflection angle for $b^{2}>a^{2}$, and $a>0$ (figures 5 & 8) and observed the
significant impact of cosmic strings and global monopoles on this angle of
deflection for null geodesics.

In summary, this comprehensive analysis investigates the behavior of geodesics
in the generalized Schwarzschild-Klinkhamer-type wormhole space-time, taking
into account the presence of topological defects such as cosmic strings and
global monopoles. By deriving and analyzing the effective potential, we gain
valuable insights into the modifications and effects induced by these
topological defects within the wormhole geometry. Additionally, we determined
the deflection angle for photon rays in both scenarios, highlighting the
impact of cosmic strings and global monopoles on this angle. Overall, this
study provides a deeper understanding of how topological defects can modify
the behavior of geodesics and the deflection of light within the considered
wormhole space-time.

## References

  * [1] P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses, Springer-Verlag, Berlin, Germany (1992).
  * [2] P. Schneider, C. Kochanek, and J. Wambsganss, Gravitational Lensing: Strong, Weak and Micro, Springer-Verlag, Berlin, Germany (2006).
  * [3] K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000).
  * [4] E. F. Eiroa, G. E. Romero and D. F. Torres, Phys. Rev. D 66, 024010 (2002).
  * [5] R. Whisker, Phys. Rev. D 71, 064004 (2005).
  * [6] V. Bozza, Gen. Relativ. Gravit. 42, 2269 (2010).
  * [7] E. F. Eiroa and C. M. Sendra, Classical Quantum Gravity 28, 085008 (2011).
  * [8] M. C. Werner, Gen. Relativ. Grav. 44, 3047 (2012).
  * [9] I. Sakalli and A. Övgün, EPL 118, 60006 (2017).
  * [10] K. Jusufi, A. Övgün, J. Saavedra, Y. Vasquez and P. A. Gonzalez, Phys. Rev. D 97, 124024 (2018).
  * [11] A. Övgün, Phys. Rev. D 99, 104075 (2019).
  * [12] N. Tsukamoto, Phys. Rev. D 103, 024033 (2021).
  * [13] F. Atamurotov et al., Phys. Rev. D 104, 064053 (2021).
  * [14] J. Chagoya, C. Ortiz, B. Rodriguez, and A. A. Roque, Classical Quantum Gravity 38, 075026 (2021).
  * [15] M. A. Alawadi, D. Batic, and M. Nowakowski, Classical Quantum Gravity 38, 045003 (2021).
  * [16] S. Ul Islam1, J. Kumar and S. G. Ghosh, JCAP 10 (2021) 013.
  * [17] K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 65, 103004 (2002).
  * [18] F. Ahmed, Adv. High Energy Phys. 2017, 3587018 (2017); Prog. Theor. Exp. Phys. 2017, 083E03 (2017).
  * [19] F. Ahmed and F. Rahaman, Adv. High Energy Phys. 2018, 7839619 (2018); Eur. Phys. J A 54, 52 (2018).
  * [20] N. Tsukamoto, Phys. Rev. D 95, 064035 (2017).
  * [21] K. Jusufi, M. C. Werner, A. Banerjee and A. Övgün, Phys. Rev. D 95, 104012 (2017).
  * [22] T. Ono, A. Ishihara and H. Asada, Phys. Rev. D 99, 124030 (2019).
  * [23] K. Jusufi, I. Sakalli and A. Övgün, Phys. Rev. D 96, 024040 (2017).
  * [24] K. Jusufi and A. Övgün,, Phys. Rev. D 97, 064030 (2018).
  * [25] A. Övgün, I. Sakalli and J. Saavedra, JCAP 10 (2018) 041.
  * [26] J. R. Nascimento, A. Yu. Petrov, P. J. Porfirio, A. R. Soares, Phys. Rev. D 102, 044021 (2020).
  * [27] N. Tsukamoto, Phys. Rev. D 104, 064022 (2021).
  * [28] T. Ono, A. Ishihara and H. Asada, Phys. Rev. D 96, 104037 (2017).
  * [29] C. Furtado, J. R. Nascimento, A. Yu. Petrov, P. J. Porfírio, A. R. Soares, Phys. Rev. D 103, 044047 (2021).
  * [30] K. Jusufi, N. Sarkar, F. Rahaman, A. Banerjee, and S. Hansraj, Eur. Phys. J. C 78, 349 (2018).
  * [31] V. Perlick, Phys. Rev. D 69, 064017 (2004).
  * [32] A. R. Soares, R. L. L. Vitoria and C. F. S. Pereira, arXiv: 2305.11105 [gr-qc].
  * [33] F. Ahmed, arXiv: 2306. 12455 [gr-qc].
  * [34] H. G. Ellis, J. Math. Phys. 14, 104 (1973).
  * [35] K. A. Bronnikov, Acta Phys. Pol. B 4, 251 (1973).
  * [36] M. S. Morris and K. S. Thorne, Amer. J. Phys. 56, 395 (1988).
  * [37] K. Gödel, Rev. Mod. Phys. 21, 447 (1949).
  * [38] M. Visser, Lorentzian Wormholes. From Einstein to Hawking, Springer-Verlag, New York, USA (1995).
  * [39] F. R. Klinkhamer, Acta Phys. Polon. B 54, 5-A3 (2023) [arXiv:2301.00724 [gr-qc]].
  * [40] K. K. Nandi, Y.-Z. Zhang and A. V. Zakharov, Phys. Rev. D 74, 024020 (2006).
  * [41] T. Müller, Phys. Rev. D 77, 044043 (2008).
  * [42] F. Abe, Astrophys. J. 725, 787 (2010).
  * [43] Y. Toki, T. Kitamura, H. Asada and F. Abe, Astrophys. J. 740, 121 (2011).
  * [44] K. Nakajima and H. Asada, Phys. Rev. D 85, 107501 (2012).
  * [45] G. W. Gibbons and M. Vyska, Class. Quantum Grav. 29, 065016 (2012).
  * [46] N. Tsukamoto, T. Harada, and K. Yajima, Phys. Rev. D 86, 104062 (2012).
  * [47] C.-M. Yoo, T. Harada and N. Tsukamoto, Phys. Rev. D 87, 084045 (2013).
  * [48] N. Tsukamoto, Phys. Rev. D 94, 124001 (2016).
  * [49] N. Tsukamoto and T. Harada, Phys. Rev. D 95, 024030 (2017).
  * [50] A. Övgün, Phys. Rev. D 98, 044033 (2018).
  * [51] K. Jusufi and A. Övgün, Phys. Rev. D 97, 024042 (2018).
  * [52] T. Ono, A. Ishihara and H. Asada, Phys. Rev. D 98, 044047 (2018).
  * [53] N. Tsukamoto and Y. Gong, Phys. Rev. D 97, 084051 (2018).
  * [54] K. Jusufi, Phys. Rev. D 98, 044016 (2018).
  * [55] K. A. Bronnikov and K. A. Baleevskikh, Gravit. Cosmol. 25, 44 (2019).
  * [56] A. Övgün, K. Jusufi, and I. Sakalli, Phys. Rev. D 99, 024042 (2019).
  * [57] W. Javed, R. Babar and A. Övgün, Phys. Rev. D 99, 084012 (2019).
  * [58] R. Shaikh, P. Banerjee, S. Paul and T. Sarkar, JCAP 07 (2019) 028.
  * [59] R. Shaikh, P. Banerjee, S. Paul and T. Sarkar, Phys. Lett. B 789, 270 (2019).
  * [60] H. Aounallah, A. R. Soares and R. L. L. Vitoria , Eur. Phys. J. C 80, 447 (2020).
  * [61] T. Manna, F. Rahaman and T. Chowdhury, New Astronomy 102, 102040 (2023).
  * [62] F. Ahmed, EPL 142, 39002 (2023).
  * [63] M. Aryal, L. H. Ford and A. Vilenkin, Phys. Rev. D 34, 2263 (1986).
  * [64] Z.-L. Wang, arXiv: 2307.01678 [gr-qc].
  * [65] A. Vilenkin and E. P. S. Shellard, Strings and Other Topological Defects, Cambridge University Press, Cambridge (1994).
  * [66] G. de A Marques and V. B. Bezerra, Phys. Rev. D 66, 105011 (2002).
  * [67] J. R. Gott III, Astrophy. J. 288, 422 (1985).
  * [68] S. Weinberg, Gravitation and Cosmology: Principles and Applications of The General Theory of Relativity, John Wiley & Sons, New York (1971).
  * [69] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover (1972).
  * [70] M. Barriola and A. Vilenkin, Phys. Rev. Lett. 63, 341 (1989).
  * [71] F. Ahmed, Sci. Rep. 12, 8794 (2022).
  * [72] R. D. Lambaga and H. S. Ramadhan, Eur. Phys. J. C 78, 436 (2018).
  * [73] F. Ahmed, EPL 141, 54001 (2023).