# Novel Casimir wormholes in Einstein gravity

1Mohammad Reza Mehdizadeh<EMAIL_ADDRESS>1Department of Physics,
Shahid Bahonar University, P. O. Box 76175, Kerman, Iran  2Amir Hadi Ziaie
<EMAIL_ADDRESS>1Department of Physics, Shahid Bahonar University, P.
O. Box 76175, Kerman, Iran

###### Abstract

In the context of General Relativity (GR), violation of the null energy
condition (NEC) is necessary for existence of static spherically symmetric
wormhole solutions. Also, it is a well-known fact that the energy conditions
are violated by certain quantum fields, such as the Casimir effect. The
magnitude and sign of the Casimir energy depend on Dirichlet or Neumann
boundary conditions and geometrical configuration of the objects involved in a
Casimir setup. The Casimir energy may act as an ideal candidate for the matter
that supports the wormhole geometry. In the present work, we firstly find
traversable wormhole solutions supported by a general form for the Casimir
energy density assuming a constant redshift function. As well, in this
framework, assuming that the radial pressure and energy density obey a linear
equation of state, we derive for the first time Casimir traversable wormhole
solutions admitting suitable shape function. Then, we consider three geometric
configurations of the Casimir effect such as (i) two parallel plates, (ii) two
parallel cylindrical shells, and (iii) two spheres. We study wormhole
solutions for each case and their property in detail. We also check the weak
and strong energy conditions in the spacetime for the obtained wormhole
solutions. The stability of the Casimir traversable wormhole solutions are
investigated using the Tolman-Oppenheimer-Volkoff (TOV) equation. Finally, we
study trajectory of null as well as timelike particles in the wormhole
spacetime.

##  1 Introduction

Wormholes are theoretical passages between two different universes, or
sometimes between two distant parts of the same universe. The concept of
wormhole was first put forward by Einstein and Rosen in their famous Einstein-
Rosen bridge in 1935 [1, 2]. Later in 1957, Misner and Wheeler coined the term
“wormhole” in their seminal works, as an attempt to present a mechanism for
having “charge without charge” [3, 4, 5]. They found that wormholes that
connect two asymptotically flat spacetimes may render non-trivial solutions to
Einstein-Maxwell equations, where, the lines of electric field as observed in
one part of the universe could thread the throat and emerge in other part.
Traversable wormhole structures were first studied by Morris and Thorne in
1988 [6, 7]. They found exact static spherically symmetric solutions and
discussed the required conditions for physically meaningful Lorentzian
traversable wormholes. In the framework of GR, the Morris-Thorne wormholes
allow a two way communication between two regions of the spacetime through a
minimal surface called the wormhole throat. Such a two way travel requires the
fulfillment of the fundamental flare-out condition at the wormhole throat.
However, the EMT components for such a wormhole configuration always violate
the NEC [8]. The matter distribution responsible for such a situation is the
so-called “exotic matter” which has negative energy density [9, 10].
Therefore, the issue of exotic matter and the establishment of standard energy
conditions have been one of the most important challenges in wormhole physics
until today. In this respect, there have been many attempts in literature in
order to avoid or at least to minimize the usage of exotic matter in the
wormhole geometry [11]. For instance, dynamical wormhole geometries which
satisfy the energy conditions during a time period on a timelike or null
geodesic have been studied in [12, 13, 14]. Visser and Poisson have studied
construction of thin-shell wormholes which are constructed by the cut-and-
paste technique and their supporting matter is concentrated on the wormhole’s
throat [15]. Wormhole configurations with phantom or quintom-type energy as
the supporting matter have been investigated in [16, 17, 18] and wormholes
supported by nonminimal interaction between dark matter and dark energy has
been explored in [19], see also [20] for historical notes and [21] for a
comprehensive review. In GR, the thin-shell wormholes do not respect the
standard energy conditions at the throat. However, in the context of modified
theories of gravity, the presence of higher-order terms in curvature would
allow for building thin-shell wormholes supported by ordinary matter [22, 23,
24, 25]. As a matter of fact, the correction terms or additional degrees of
freedom not present in GR can provide a setting for traversable wormhole
solutions. Studies in this arena have been performed in the context of
different modified gravity theories and under various circumstances, for
example: traversable wormholes in Einstein-Gauss-Bonnet gravity [26, 27, 28,
29], higher-dimensional GR [30, 31, 32], nonsymmetric gravitational theory
[33], Lovelock [34, 35, 36] and $f(R)$ gravity theories [37, 38, 39, 40, 41,
42], modified gravities with curvature-matter coupling [43, 44, 45, 46, 47,
48], Brans-Dicke [49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62] and
other modified gravity theories [63, 64, 65, 66, 67, 68, 69, 70, 71].

It is now well known that the theoretical existence of traversable Lorentzian
wormholes in GR is accompanied by the violation of NEC and consequently the
existence of exotic types of matter. However, the quest for finding the
promising candidates of exotic matter is not a routine and easy task, and the
footprints of such type of matter have been recognized only in a small area,
such as the experimentally verified Casimir effect [72, 73, 74] and semi-
classical Hawking radiation [75]. The Casimir effect is a famous quantum field
theoretical phenomenon that appears as an attractive force between neutral
parallel conducting plates in a vacuum. The associated negative energy density
to this effect is a manifestation of the quantum fluctuations of the vacuum of
the electromagnetic field confined between the two plates [76, 77]. In view of
the exotic nature of Casimir energy, Morris and Thorne [6] and some time later
Visser [8] argued that this type of exotic energy can be considered as an
appropriate source for modeling traversable wormholes. However, wormhole
solutions in GR with Casimir energy as the supporting matter have been found
only very recently by Garattini [78], where the author studied negative energy
density due the Casimir effect and explored the consequences of quantum weak
energy conditions on the traversability of the wormhole. Subsequently, the
study of traversable wormholes in the presence of Casimir energy has been
carried out in different frameworks and under various circumstances among
which we can quote, the case of three [79] and D-dimensions [80], alternative
gravity theories [81, 82, 83], GUP corrections [84, 85, 86, 87], Casimir
source modified by a Yukawa term [88, 89, 90] and other frameworks [91, 92,
93, 94]. The Casimir effect has a strong dependence on the type of the quantum
field under investigation, shape of the objects and the boundary conditions
imposed on them [76, 77]. Our aim in the present paper is then to study
wormhole configurations with Casimir energy as the supporting matter for
different Casimir setups. The organization of the this paper is as follows: In
Sec. (2) we give a brief review on Morris-Thorne wormholes. In Sec. (3) we
proceed to find traversable wormhole solutions assuming a power-law form for
the Casimir energy density. The zero tidal force solutions are given in
Subsec. (3.1) and those of non-constant redshift function are presented in
Subsec. (3.2). Sec. (4) is devoted to the equilibrium conditions on wormhole
structure. In Sec. (5) we investigate trajectory of null and timelike
particles in the wormhole spacetime. Our conclusions are presented in Sec.
(6). Throughout the present work we set the units so that $\hbar=C=G=1$.

##  2 Morris-Thorne wormholes: a brief review

In their seminal work, Morris and Thorne introduced the following spherically
symmetric line element

$ds^{2}=-e^{2\phi(r)}dt^{2}+\left[\frac{dr^{2}}{1-b(r)/r}+r^{2}d\Omega^{2}\right],$
(1)

as a possible solution to obtain viable wormhole structure. In the above
metric, $\phi(r)$ is the redshift function as it is related to the
gravitational redshift and $b(r)$ is the wormhole shape function. The shape
function must satisfy the flare-out condition at the throat, i.e., we must
have $b^{\prime}(r_{0})<1$ and $b(r)<r$ for $r>r_{0}$ in the whole spacetime,
where $r_{0}$ is the throat radius. To obtain the components of Einstein field
equation we utilize a set of orthonormal basis vectors. These vector fields
are defined as the proper reference frame of a set of observers who remain at
rest in the coordinate system $(t,r,\theta,\phi)$, with $(r,\theta,\phi)$
fixed. If we denote the basis vectors in this coordinate system as $({\rm
e}_{t},{\rm e}_{r},{\rm e}_{\theta},{\rm
e}_{\phi})=\left(\partial/\partial_{t},\partial/\partial_{r},\partial/\partial_{\theta},\partial/\partial_{\phi}\right)$,
then the orthonormal basis vectors are given by

${\rm e}_{\hat{t}}={\rm e}^{-\phi}{\rm e}_{t},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\rm
e}_{\hat{r}}=\left(1-\frac{b}{r}\right)^{\frac{1}{2}}{\rm
e}_{r},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\rm e}_{\hat{\theta}}=\frac{{\rm
e}_{\theta}}{r},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\rm e}_{\hat{\phi}}=\frac{{\rm e}_{\phi}}{r\sin\theta},$
(2)

by the virtue of which, the components of spacetime metric (1) take on their
standard, special relativity forms as $g_{\alpha\beta}={\rm
diag}\left[-1,1,1,1\right]$. Working in this basis simplifies the mathematical
analysis and physical interpretation [6, 7]. The non-vanishing components of
Einstein tensor in this orthonormal reference frame are then found as

$\displaystyle
G_{\hat{t}\hat{t}}=\frac{b^{\prime}}{r^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
G_{\hat{r}\hat{r}}=-\frac{b}{r^{3}}+2\left[1-\frac{b}{r}\right]\frac{\phi^{\prime}}{r},$
$\displaystyle
G_{\hat{\theta}\hat{\theta}}=G_{\hat{\phi}\hat{\phi}}=\left(1-\frac{b}{r}\right)\left[\phi^{\prime\prime}+(\phi^{\prime})^{2}-\frac{rb^{\prime}-b}{2r(r-b)}\phi^{\prime}-\frac{rb^{\prime}-b}{2r^{2}(r-b)}+\frac{\phi^{\prime}}{r}\right],$
(3)

where a prime denotes differentiation with respect to $r$. Also, the nonzero
components of stress-energy tensor (SET) in the orthonormal reference frame
are given by

$T_{\hat{t}\hat{t}}=\rho(r),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
T_{\hat{r}\hat{r}}=P_{r}(r),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
T_{\hat{\theta}\hat{\theta}}=T_{\hat{\phi}\hat{\phi}}=P_{t}(r),$ (4)

where $\rho(r)$ is the energy density and $P_{r}(r)$ and $P_{t}(r)$ are the
radial and transverse pressures, respectively. Thus, the Einstein field
equation $G_{\hat{\alpha}\hat{\beta}}=8\pi T_{\hat{\alpha}\hat{\beta}}$
provides us with the following expressions

$\displaystyle 8\pi r^{2}\rho(r)$ $\displaystyle=$ $\displaystyle
1-g^{\prime}(r)r-g(r),$ (5) $\displaystyle 8\pi r^{2}P_{r}(r)$
$\displaystyle=$ $\displaystyle rg(r)\frac{f^{\prime}(r)}{f(r)}+g(r)-1,$ (6)
$\displaystyle 32\pi r^{2}P_{t}(r)$ $\displaystyle=$ $\displaystyle
2rg^{\prime}(r)-g\left[\frac{rf^{\prime}(r)}{f(r)}\right]^{2}+r^{2}g^{\prime}(r)\frac{f^{\prime}(r)}{f(r)}+\frac{2rg(r)}{f(r)}\left[f^{\prime}(r)+rf^{\prime\prime}(r)\right],$
(7)

where a prime denotes differentiation with respect to radial coordinate $r$
and we have defined the positive functions $f(r)$ and $g(r)$ as

$\displaystyle g(r)\equiv\frac{r-b}{r},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ f(r)\equiv e^{2\phi(r)}.$ (8)

The flare-out condition at the wormhole throat leads to the conditions
$g^{\prime}(r_{0})>0$ and $g(r_{0})=0$. It is well-known that existence of
traversable Lorentzian wormholes in four dimensions as solutions to the
Einstein equations requires some kind of the so-called “exotic matter”, i.e.,
matter that violates the NEC [9, 10]. This is due to the fulfillment of
flaring-out condition at the throat of the wormhole and for $r>r_{0}$.
Physically, the flare-out condition at the throat is responsible for holding
back the wormhole throat from collapsing and is crucial for its
traversability. Hence, in classical GR, a traversable wormhole configuration
requires exotic matter at or in the neighborhood of the wormhole’s throat. We
note that NEC is a part of the weak energy condition (WEC) whose physical
meaning is that the energy density is non-negative in any reference frame. In
other words, WEC requires that $T_{\mu\nu}U^{\mu}U^{\nu}\geq 0$, where
$U^{\mu}$ is a timelike vector field111The WEC is utilized in the proof of
Penrose singularity theorem [95].. For the SET (4), the WEC leads to the
following inequalities

$\rho\geq 0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \rho+P_{r}\geq
0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \rho+P_{t}\geq
0.$ (9)

Note that the last two inequalities are defined as the NEC. In addition, the
strong energy condition (SEC) is satisfied through the following inequalities
[8]

$\rho+P_{r}+2P_{t}\geq 0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \rho+P_{r}\geq
0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \rho+P_{t}\geq 0.$ (10)

Using Eqs. (5)-(7), one finds the following relationships

$\displaystyle
8\pi(\rho+P_{r})=\frac{1}{r}\left[g(r)\frac{f^{\prime}(r)}{f(r)}-g^{\prime}(r)\right],$
(11) $\displaystyle
32\pi(\rho+P_{t})=2g(r)\frac{f^{\prime\prime}(r)}{f(r)}+\frac{f^{\prime}(r)}{rf(r)}\left[2g(r)+rg^{\prime}(r)\right]-g\left[\frac{f^{\prime}(r)}{f(r)}\right]^{2}\\!\\!-\frac{2}{r^{2}}\left[rg^{\prime}(r)+2g(r)-2\right],$
(12) $\displaystyle
16\pi(\rho+P_{r}+2P_{t})=4g(r)\frac{f^{\prime}(r)}{rf(r)}+g^{\prime}(r)\frac{f^{\prime}(r)}{f(r)}+2g(r)\frac{f^{\prime\prime}(r)}{f(r)}-g(r)\left[\frac{f^{\prime}(r)}{f(r)}\right]^{2}.$
(13)

The above expressions at the throat take the following forms

$\displaystyle\rho+P_{r}\Big{|}_{r=r_{0}}$ $\displaystyle=$
$\displaystyle-\frac{g^{\prime}(r_{0})}{8\pi r_{0}},$ $\displaystyle
32\pi(\rho+P_{t})\Big{|}_{r=r_{0}}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime}(r_{0})f^{\prime}(r_{0})}{f(r_{0})}-\frac{2}{r_{0}}g^{\prime}(r_{0})+\frac{4}{r_{0}^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\rho+P_{r}+2P_{t}\Big{|}_{r=r_{0}}=\frac{g^{\prime}(r_{0})f^{\prime}(r_{0})}{16\pi
f(r_{0})},$ (14)

whence we observe that the NEC in radial direction is violated as a
consequence of flare-out condition. However, in tangential direction the
fulfillment of NEC depends on the values of metric components and their
derivatives at the throat. Also, for $f^{\prime}(r_{0})<0$ the SEC is violated
at the throat, due to the flare-out condition. Also, for the line element (1),
the Ricci $\mathcal{R}=g_{\mu\nu}\mathcal{R}^{\mu\nu}$ and the Kretschmann
$\mathcal{K}=\mathcal{R}_{\mu\nu\gamma\beta}\mathcal{R}^{\mu\nu\gamma\beta}$
scalars are given as

$\displaystyle\mathcal{R}(r)=g(r)\left[\frac{f^{\prime}(r)}{f(r)}\right]^{2}-\left[4g(r)+rg^{\prime}(r)\right]\frac{f^{\prime}(r)}{rf(r)}-2g(r)\frac{f^{\prime\prime}(r)}{f(r)}-\frac{4}{r^{2}}\left[g(r)+rg^{\prime}(r)-1\right],$
(15) $\displaystyle\mathcal{K}(r)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left[\frac{g(r)f^{\prime\prime}(r)}{f(r)}\right]^{2}-\frac{f^{\prime\prime}(r)}{f(r)}\left[\frac{g(r)f^{\prime}(r)}{f(r)}\right]^{2}+g(r)g^{\prime}(r)\frac{f^{\prime}(r)f^{\prime\prime}(r)}{f^{2}(r)}+\frac{g(r)^{2}}{4}\left[\frac{f^{\prime}(r)}{f(r)}\right]^{4}$
$\displaystyle-$
$\displaystyle\frac{1}{2}g(r)g^{\prime}(r)\left[\frac{f^{\prime}(r)}{f(r)}\right]^{3}+\left[\frac{f^{\prime}(r)g^{\prime}(r)}{2f(r)}\right]^{2}+2\left[\frac{g(r)f^{\prime}(r)}{rf(r)}\right]^{2}+\frac{4}{r^{4}}\left[\frac{1}{2}\left(rg^{\prime}(r)\right)^{2}+g(r)^{2}-2g(r)+1\right].$

The above two expressions are useful in determining the possible occurrence
(or absence) of spacetime singularities through their divergent (regular)
behaviors [96, 97].

##  3 Casimir wormhole solutions

###  3.1 Specific case: constant redshift function

The Casimir effect which was discovered more than 70 years by Dutch physicist
Hendrik Casimir, is one of the most direct manifestations of the existence of
zero-point vacuum oscillations [72]. This is a purely quantum effect which, in
its simplest form, is the attraction of a pair of neutral, parallel conducting
plates resulting from the distortion of the electromagnetic vacuum by the
boundaries. According to Casimir’s prediction, the energy per unit area of two
infinitely large, neutral parallel planes made of an ideal metal at zero
temperature, is given by [76]

$\displaystyle E(d)=-\frac{{\pi}^{2}}{720d^{3}},$ (17)

which depends on the separation distance between the planes. The Casimir
pressure can be obtained as

$\displaystyle P(d)=-\frac{\partial E}{\partial
d}=-\frac{{\pi}^{2}}{240d^{4}}.$ (18)

From the above expressions we can recognize the following form for energy
density as

$\displaystyle\rho=-\frac{{\pi}^{2}}{720d^{4}}.$ (19)

It is obvious that for the given pressure and energy density we may assume the
linear equation of state (EoS) $P=w\rho$ with EoS parameter $w=3$. The force
associated to Casimir energy Eq. (17) is attractive, i.e, under the effect of
this force, the plates tend to move toward each other. We further note that
the vacuum energy of different quantum fields (e.g., scalar, spinor,
electromagnetic and etc.) depends on the boundary conditions imposed on the
bodies that compose a Casimir set up [76, 77]. The ideal-metal planes are
idealized thin plates made of a material with an infinitely large magnitude of
the dielectric permittivity. For such arrangement, the imposed boundary
conditions that lead to Eq. (17) imply that the transverse component of the
electric field and the normal component of the magnetic field on the surface
of each plate be zero. The fulfillment of these conditions signal that an
electromagnetic field can exist only outside an ideal conductor [76]. The
problem of Casimir interaction between an ideal-metal plane and an infinitely
permeable plane was considered by Boyer [98]. An infinitely permeable plane is
characterized by an infinitely large magnetic permeability on which, the
tangential component of the magnetic induction vanishes. The Casimir energy
density for such a set up then reads

$\displaystyle\rho=\frac{7}{8}\frac{{\pi}^{2}}{720d^{4}}.$ (20)

The above result is equal to a factor of $-7/8$ times the of ideal-metal
planes. We note that the corresponding Casimir force for this case is
repulsive in the manner that the plates tend to move away from each other.
This change from attraction to repulsion occurs due to the mixed boundary
conditions [76]. Also, the Casimir energy depends on the geometry and shape of
the objects, temperature and the interplay between geometry and material
properties [76, 77, 99]. For instance, the case of a sphere in front a plane
has been discussed in [100], a plate and a cylinder has been studied in [101],
eccentric cylinders in [102, 103], a hyperboloid opposite to a plane in [104]
and, a flat and a corrugated plane in [105, 106]. The calculation of the
Casimir energy for nontrivial geometries is a complicated task. Because of
this, several approximate methods have been developed so far among which, a
powerful one for calculating the Casimir force between bodies of arbitrary
shape is the proximity force approximation (PFA) method which was suggested by
Derjaguin [107]. In this method, the Casimir energy can be computed as an
integral over infinitesimal parallel surface elements at their local distance
$L$ as measured perpendicular to a surface $\Sigma$ that can be one of the two
surfaces of the objects or an auxiliary surface placed between them. The PFA
approximation for the energy is then given by [108]

$E_{\rm PFA}=\frac{1}{A}\int_{\Sigma}E_{||}(L)dS,$ (21)

where $E_{||}(L)/A$ is the energy per unit area for two parallel plates at the
distance $L$. Using PFA method one can show that the dependence of the Casimir
force on distance for a Cylindrical shell in front of a conducting plane is
$d^{-7/2}$, which is an intermediate between the plane-spherical ($d^{-3}$)
and the parallel plate configuration ($d^{-4}$). In addition, Casimir energy
between two concentric cylinders (using PFA method) is proportional $d^{-2}$
[109] and in the case of two parallel cylinders outside each other, to
$d^{-5/2}$ [103, 110, 102]. Regarding the above considerations, we may assume
the replacement $r$ instead of $d$ and obtain the Casimir energy density in
the general form

$\displaystyle\rho=\frac{\lambda}{8\pi r^{m}},$ (22)

where $m>0$ and $\lambda=8\pi\lambda_{0}$ is a constant which depends on the
type of quantum field, shape of the objects and boundary conditions. It may
assume positive or negative values for different combinations of Dirichlet (D)
and Neumann (N) boundary conditions, e.g. in the case of two parallel
cylinders outside each other one gets $\lambda_{0}<0$ for DD and NN boundary
conditions and $\lambda_{0}>0$ For DN and ND boundary conditions [103]. From
equations (22) and (5), we obtain the following form for the shape function as

$\displaystyle b(r)=r\left[1-g(r)\right]=-\frac{\lambda
r^{3-m}+{C}_{0}\left(m-3\right)}{m-3},$ (23)

where the integration constant can be determined through the condition
$b(r_{0})=r_{0}$ as

$\displaystyle{C}_{0}=-\frac{r_{0}(m-3)+\lambda r_{0}^{3-m}}{m-3}.$ (24)

Also, the flare-out condition at the throat results in the following
inequality

$\displaystyle r_{0}-\lambda r_{0}^{3-m}>0.$ (25)

The simplest case is a model with $\phi=constant$, namely a spacetime with no
tidal forces. As is clear from Eq. (23) for $m<2$, we obtain anti-de Sitter-
like or de Sitter-like wormhole solutions for $\lambda<0$ and $\lambda>0$,
respectively. However, the spatial extension of de Sitter-like wormhole
solutions cannot be arbitrarily large. Fig. (1) shows that a decrease in the
value of parameter $\lambda$ enlarges the wormhole spatial extension.

Next, we proceed to find the expressions for radial and tangential pressure
profiles in the case of zero tidal forces. This can be achieved by using Eqs.
(6) and (7) together with the shape function (23)

$\displaystyle P_{r}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{8\pi(m-3)r^{m}}\left[1-\left(\frac{r_{0}}{r}\right)^{3-m}\right]-\frac{r_{0}}{8\pi
r^{3}},$ (26) $\displaystyle P_{t}$ $\displaystyle=$
$\displaystyle\frac{r_{0}}{16\pi
r^{3}}-\frac{\lambda}{16\pi(m-3)r^{m}}\left[m-2-\left(\frac{r_{0}}{r}\right)^{3-m}\right].$
(27)

Also, using equations (26) and (27), we get the radial ($w_{r}=P_{r}/\rho$)
and tangential ($w_{t}=P_{t}/\rho$) EoS parameters respectively, as

$\displaystyle w_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{m-3}\left[1-\left(\frac{r_{0}}{r}\right)^{3-m}\right]-\frac{r_{0}}{\lambda}r^{m-3},$
(28) $\displaystyle w_{t}$ $\displaystyle=$
$\displaystyle\frac{r_{0}r^{m-3}}{2\lambda}-\frac{1}{2(m-3)}\left[m-2-\left(\frac{r_{0}}{r}\right)^{3-m}\right].$
(29)

Furthermore, at the throat, we can see that $w_{r}^{0}=-r_{0}^{m-2}/\lambda$
and $w_{t}^{0}=-(w_{r}^{0}+1)/2$. We note that for $m<3$, in the limit
$r\rightarrow\infty$ we have $w_{r}\rightarrow-1/(m-3)$ and
$w_{t}\rightarrow-(m-2)/2(m-3)$. From Eqs. (26) and (27) we also get

$\displaystyle\rho+P_{r}$ $\displaystyle=$
$\displaystyle{\frac{\left(m-2\right)\lambda}{8\pi
r^{m}\left(m-3\right)}}-\frac{\left[\left(m-3\right)r_{0}^{m}+\lambda
r_{0}^{2}\right]r_{0}}{8\pi\left(m-3\right)r^{3}r_{0}^{m}},$ (30)
$\displaystyle\rho+P_{t}$ $\displaystyle=$
$\displaystyle{\frac{\left(m-4\right)\lambda}{16\pi{r}^{m}\left(m-3\right)}}+{\frac{\left[\left(m-3\right)r_{0}^{m}+\lambda
r_{0}^{2}\right]r_{0}}{16\pi\left(m-3\right)r^{3}r_{0}^{m}}},$ (32)
$\displaystyle\rho+P_{r}+2P_{t}=0.$

Using Eq. (LABEL:kr11) we can calculate the Kretschmann scalar for the
solution (23) as

$\displaystyle\mathcal{K}(r)$ $\displaystyle=$ $\displaystyle
6\frac{r_{0}^{2}}{{r}^{6}}+\frac{\lambda}{(m-3)r^{6}}\left[12r_{0}^{4-m}-4r_{0}m{r}^{3-m}\right]$
(33)
$\displaystyle+\frac{\lambda^{2}}{\left(m-3\right)^{2}{r}^{6}}\left[6r_{0}^{6-2m}-4mr_{0}^{3-m}{r}^{3-m}+\left(2{m}^{2}-8m+12\right){r}^{6-2m}\right],$

whereby we find that the Kretschmann scalar is finite in the whole range
$r>r_{0}$ and approaches zero asymptotically.

In what follows, we consider the energy density for the configuration of a
sphere (or a spherical lens) situated above a large disk. The closest
separation between the sphere and disk points is taken as $a\ll R$, where R is
a sphere (lens) radius. In [111], it is shown that for a configuration of a
perfectly conducting disk and lens with large separations, the energy density
is given by

$\displaystyle\rho=\frac{\lambda}{8\pi r^{3}},$ (34)

in which $\lambda=-\pi^{3}/90$. We consider the distance between the lens and
disk $a$ as the radial coordinate $r$ in wormhole spacetime. Thus, for the
special case of $m=3$ and by using equation (5) the shape function is obtained
as

$\displaystyle b(r)=\lambda\ln\left(\frac{r}{r_{0}}\right)+r_{0},$ (35)

where the integration constant has been set according to the condition
$b(r_{0})=r_{0}$. The flare-out condition at the throat implies the inequality
$\lambda<r_{0}$. It is then clear that these solutions are asymptotically
flat, i.e. $b(r)/r$ tends to zero as $r\rightarrow\infty$. For this case the
radial and tangential profiles of NEC along with the SEC are given as

$\displaystyle\rho+P_{r}$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi
r^{3}}\left[\lambda\ln\left(\frac{r_{0}}{r}\right)+\lambda-r_{0}\right],$ (36)
$\displaystyle\rho+P_{t}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
r^{3}}\left[-\lambda\ln\left(\frac{r_{0}}{r}\right)+\lambda+r_{0}\right],$
(37) $\displaystyle\rho+P_{r}+2P_{t}$ $\displaystyle=$ $\displaystyle 0.$ (38)

We then observe that the flare-out condition, i.e., $\lambda<r_{0}$, leads to
$\rho(r_{0})+P_{r}(r_{0})<0$. Hence, the radial profile of NEC is violated at
the throat. However, the tangential profile of NEC is satisfied at $r=r_{0}$.
Moreover, for this case the Kretschmann scalar reads

$\displaystyle\mathcal{K}(r)=\frac{\lambda^{2}}{r^{6}}\left[6\ln(r_{0})^{2}+6\ln(r)^{2}+4\ln\left(\frac{r_{0}}{r}\right)-12\ln(r)\ln(r_{0})+2\right]+\frac{-4\lambda
r_{0}}{r^{6}}\left[3\ln\left(\frac{r_{0}}{r}\right)+1\right]+\frac{6r_{0}^{2}}{{r}^{6}},$
(39)

whence we find that the Kretschmann scalar is finite in the whole range
$r>r_{0}$. In Fig. (2) we have shown the behavior $\rho+P_{t}$, $\rho+P_{r}$
and SEC versus radial coordinate for negative and positive values of $\lambda$
parameter in the left and right panels respectively. We see that we can choose
suitable values for the parameter $\lambda$ in order to have normal matter at
the throat and at spatial infinity. Fig. (3) shows the behavior of $w_{r}$ and
$w_{t}$ against $r$ for $\lambda<0$, where we observe that $w_{r}$ gets larger
values as we increase the value of parameter $m$, whereas $w_{t}$ takes larger
values in negative direction as $m$ increases. Fig. (4) shows a plot of EoS
parameters in radial and tangential directions for $\lambda>0$, where, we see
that the behavior of $w_{r}$ ($w_{t}$) resembles that of $w_{t}$ ($w_{r}$) for
$\lambda<0$.

Figure 1: The behavior of $1-b(r)/r$ with respect to $r$ for $r_{0}=0.1$,
$m=1.5$ and $\lambda=1,0.7,0.5$ from down to up, respectively.

Figure 2: The behavior of $\rho$ (solid curve), $\rho+p_{r}$ (dotted curve)
and $\rho+p_{t}$ (dashed curve) versus $r$. The model parameters have been set
as, $m=1.5$ and $r_{0}=0.1$, $\lambda=-1$ (left panel) and $\lambda=1$ (right
panel).

Figure 3: The behavior radial (left panel) and tangential (right panel) EoS
parameters against radial coordinate. The model parameters have been set as
$\lambda=-0.1$, $r_{0}=1$, $m=1.5$ (red curve), $m=2$ (blue curve) and $m=2.5$
(green curve).

Figure 4: The behavior radial (left panel) and tangential (right panel) EoS
parameters against radial coordinate. The model parameters have been set as
$\lambda=0.1$, $r_{0}=1$, $m=1.5$ (red curve), $m=2$ (blue curve) and $m=2.5$
(green curve).

###  3.2 Specific case: non-constant redshift function

In this section, we seek for spacetimes admitting wormhole structures with non
zero tidal force. To this aim we must adopt a strategy for specifying the
redshift function. Here, we consider a linear EoS, which provides a relation
between the EMT components, namely, $P_{r}=w\rho$. Using then Eqs. (5) and
(6), we arrive at the following differential equation

$\displaystyle
rg(r)f^{\prime}(r)+f(r)\left[rwg^{\prime}(r)+\left(g(r)-1\right)\left(1+w\right)\right]=0.$
(40)

Now, substituting for the shape function from Eq. (23) into Eq. (40), we find
the redshift function in general form as

$f(r)={\rm exp}\left[\int^{r}_{r_{0}}X(r)dr+f_{0}\right],$ (41)

where

$\displaystyle
X(r)=\frac{-C_{0}\left(m-3\right){r}^{m}+\lambda{r}^{3}\left[w(m-3)-1\right]}{r\left[\left(m-3\right)\left(r+C_{0}\right){r}^{m}+\lambda
r^{3}\right]},$ (42)

and $f_{0}$ is an integration constant. In order to check the energy
conditions, we use the field equations and the shape function Eq. (23) to get
the following expressions

$\displaystyle\rho+P_{r}$ $\displaystyle=$ $\displaystyle(1+w)\rho,$ (43)

$\displaystyle\rho+P_{t}$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{1}}{32\pi
r^{m}r_{0}^{m}}\big{[}(m-3)(r_{0}-r)-\lambda(r^{3-m}-r_{0}^{3-m})\big{]}^{-1},$
(44)

where

$\displaystyle\Sigma_{1}$ $\displaystyle=$
$\displaystyle\lambda^{2}\left[(3-(m-3)w)(w-1)r_{0}^{m}{r}^{3-m}+\left(3-\left(2m-3\right)w\right)r_{0}^{3}\right]$
(45)
$\displaystyle+\left[\left(-2m^{2}+9m-9\right)w-9+3m\right]r_{0}^{1+m}+2\left(m-3\right)\left(-2+\left(m-2\right)w\right)rr_{0}^{m}$

$\displaystyle\rho+P_{r}+2P_{t}$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{2}}{16\pi
r^{m}r_{0}^{m}}\big{[}(m-3)(r_{0}-r)-\lambda(r^{3-m}-r_{0}^{3-m})\big{]}^{-1},$
(46)

where

$\displaystyle\Sigma_{2}$ $\displaystyle=$
$\displaystyle\Sigma_{1}-2\lambda\left[{r}^{3-m}r_{0}^{m}\lambda+r_{0}^{1+m}\left(3-m\right)+r\left(m-3\right)r_{0}^{m}-\lambda
r_{0}^{3}\right]\left(w-1\right).$ (47)

In all the above equations, we have employed the solution given in Eq. (41)
and its derivative with respect to $r$. For $m>3$, the quantities $\rho+P_{t}$
and $\rho+P_{r}+2P_{t}$ in the limit of large values of radial coordinate take
the following forms

$\displaystyle\rho+P_{t}$ $\displaystyle=$ $\displaystyle-\frac{\lambda}{16\pi
r^{m}}[w(m-2)-2]+{\mathcal{O}}\left(\frac{1}{r^{m+1}}\right),$ (48)

and

$\displaystyle\rho+P_{r}+2P_{t}$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{8\pi
r^{m}}[w(m-3)-1]+{\mathcal{O}}\left(\frac{1}{r^{m+1}}\right).$ (49)

It is therefore seen that both of the above quantities tend to zero as
$r\rightarrow\infty$. Moreover, at throat we have

$\displaystyle\rho+P_{t}\Big{|}_{r=r_{0}}=\frac{m-1+3\lambda
r_{0}^{2-m}}{32\pi r0^{2}},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\rho+P_{r}+2P_{t}\Big{|}_{r=r_{0}}=\frac{m-3+\lambda r_{0}^{2-m}}{16\pi
r_{0}^{2}}.$ (50)

whence, we recognize that a suitable choice of $\lambda$ parameter and throat
radius can lead to satisfaction of $\rho+P_{r}+2P_{t}$ and NEC in tangential
direction throughout the spacetime. We now proceed to find traversable Casimir
wormholes for which $i)$ the redshift function is finite everywhere (absence
of horizon), $ii)$ any spacetime singularity is avoided at or near the
wormhole throat. This can be achieved through suitable values of EoS parameter
$w$ at the throat. We note that the second condition comes from this argument
that the presence of spacetime singularities in GR signals the break down of
the theory in the singular region [95, 112]. Also, a singular spacetime
contains incomplete paths which means that, any particle or observer traveling
through this path would experience only a finite interval of existence that in
principle cannot be continued any longer. Hence, the existence of spacetime
singularity and consequently path incompleteness could have undesirable
effects on traversability of the wormhole [6, 96]. To investigate this issue
we find Ricci scalar given in Eq. (15) using the shape function (23) and
redshift (41)

$\displaystyle\mathcal{R}(r)=\frac{\xi}{r^{m+1}r_{0}^{m}}\big{[}(m-3)(r_{0}-r)-\lambda(r^{3-m}-r_{0}^{3-m})\big{]}^{-1},$
(51)

where

$\displaystyle\xi$ $\displaystyle=$
$\displaystyle{\lambda}^{2}\left[\frac{r_{0}^{m}}{2}\left(3-\left(m-3\right){w}^{2}-\left(2-m\right)w\right){r}^{4-m}-r\left(\frac{3}{2}+\left(m-\frac{5}{2}\right)w\right)r_{0}^{3}\right]$
(52) $\displaystyle+$ $\displaystyle\lambda
r\left[-\left(\frac{3}{2}+\left(m-\frac{5}{2}\right)w\right)\left(m-3\right)r_{0}^{1+m}+r\left(1+\left(m-3\right)w\right)\left(m-3\right)r_{0}^{m}\right].$

In view of the above expression we realize that the Ricci scalar diverges in
the limit of approach to the throat. This occurs due to divergence terms
within the denominator of Eq. (51). However, it is still possible to find
suitable values of EoS parameter in order to avoid divergence in Ricci scalar.
An investigation with more scrutiny reveals that if we choose the EoS
parameter as $w=-r_{0}^{m-2}/\lambda$ then $\xi\rightarrow 0$ in the limit
$r\rightarrow r_{0}$, hence, the singularity at the wormhole throat can be
removed, using the L’Hopital’s rule. We therefore get the Ricci scalar at the
throat as

$\displaystyle\mathcal{R}(r_{0})=\frac{3\lambda r_{0}^{2-m}+3-m}{2r_{0}^{2}}.$
(53)

By the same argument, the Kretschmann scalar at the throat assumes the
following form

$\displaystyle\mathcal{K}(r_{0})$ $\displaystyle=$
$\displaystyle\frac{9\lambda^{2}r_{0}^{4-2m}+2\lambda(m-11)r_{0}^{2-m}+m^{2}-6m+33}{4r_{0}^{4}}.$
(54)

As expected, these scalar curvatures are finite at the throat signaling the
absence of spacetime singularity. It can be shown that these quantities behave
regularly in the regions with $r>r_{0}$ hence, there is no singular region in
wormhole spacetime to affect its traversability. In what follows we study some
specific wormhole solutions and their physical properties in more detail.

####  3.2.1 Case-I: Parallel plates

As we have discussed earlier, this case has been studied in various wormhole
configurations. The Casimir energy density for two parallel plates is given by
equations (19) and (20). So, the energy density $\rho$ is given by

$\displaystyle\rho=\frac{\lambda}{8\pi r^{4}},$ (55)

where $\lambda=-{8\pi^{3}}/{720}$ for parallel planes made of ideal metal at
zero temperature and, $\lambda=7\pi^{3}/720$ in the case of Casimir
interaction between an ideal metal plane and an infinitely permeable plane
characterized by an infinitely large magnetic permeability. Using then
equations (40) and (23) for $m=4$, we obtain the shape function and red-shift
function as

$\displaystyle b(r)=\frac{(r-r_{0})\lambda+r_{0}^{2}r}{r_{0}r},$ (56)

and

$\displaystyle
f(r)={C_{1}}\left(r_{0}r-\lambda\right)^{-{\frac{wr_{0}^{2}+\lambda}{r_{0}^{2}-\lambda}}}{r}^{w-1}\left(r-r_{0}\right)^{{\frac{w\lambda+r_{0}^{2}}{r_{0}^{2}-\lambda}}},$
(57)

where ${C_{1}}$ is an integration constant. Also, we see that for all the
values of the parameter $\lambda$, the quantity $b(r)/r$ tends to zero at
spatial infinity. The flare-out condition at the throat leads to the
inequality $\lambda<r_{0}^{2}$ which violates the NEC at the throat. From Eq.
(57), it is clear that $f(r_{0})=0$ and consequently the Ricci and Kretschmann
scalars diverge as we approach the throat. However, we can remove the
spacetime singularity at the throat of wormhole by choosing
$w=-r_{0}^{2}/\lambda$ in the red shift function which gives

$\displaystyle
f(r)=\left(1-\frac{\lambda}{r_{0}r}\right)^{\frac{r_{0}^{2}+\lambda}{\lambda}}.$
(58)

We therefore find out that the Ricci and Kretschmann scalars assume the
following finite values at the throat

${\mathcal{R}}(r_{0})=\frac{3\lambda-
r_{0}^{2}}{2r_{0}^{4}},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{K}(r_{0})=\frac{25r_{0}^{4}-14r_{0}^{2}\lambda+9{\lambda}^{2}}{4r_{0}^{8}}.$
(59)

In Fig. (5) the behavior of redshift function is shown where we observe that
$f(r)$ is finite everywhere and thus there is no horizon in the wormhole
spacetime. This along with the shape function (56) provide us with
asymptotically flat traversable wormhole solutions. Using equations (44) and
(46) for $m=4$ and $w=-r_{0}^{2}/\lambda$ we get

$\displaystyle\rho+P_{t}$ $\displaystyle=$
$\displaystyle{\frac{\left(4r_{0}r-r_{0}^{2}-3\lambda\right)\left(r_{0}^{2}+\lambda\right)}{32\pi\left(r_{0}r-\lambda\right)r^{4}}},$
(60) $\displaystyle\rho+P_{r}+2P_{t}$ $\displaystyle=$
$\displaystyle\frac{\left(2r_{0}r-r_{0}^{2}-\lambda\right)\left(r_{0}^{2}+\lambda\right)}{16\pi\left(r_{0}r-\lambda\right){r}^{4}}.$
(61)

In Fig. (6) the behavior of radial (left panel) and tangential (right panel)
profiles of NEC is depicted. Fig. (7) shows the behavior of SEC versus radial
coordinate. We see that the radial profile of NEC is negative everywhere
($\rho+P_{r}=\frac{(\lambda-r_{0}^{2})}{\lambda}\rho<0$), although one can
choose suitable values for the parameters $\lambda$ and $r_{0}$ so that both
$\rho+P_{t}$ and $\rho+P_{r}+2P_{t}$ take positive values throughout the
spacetime. Also, using the field equations we get the tangential EoS for $m=4$
as

$\displaystyle w_{t}=\frac{\lambda^{2}-4r_{0}^{2}\lambda-
r_{0}^{4}+4rr_{0}^{3}}{4\left(r_{0}r-\lambda\right)\lambda}.$ (62)

Fig. (8) shows the behavior of $w_{t}$, where, it is seen that the tangential
EoS parameter is a positive (negative) function of radial coordinate and
increases in positive (negative) direction for $\lambda>0\leavevmode\nobreak\
(<0)$.

Figure 5: The behavior of the metric function $f(r)$ for $r_{0}=1$, $m=4$ and
$\lambda=0.5,-0.09$ from down to up, respectively.

Figure 6: The behavior of $\rho+p_{r}$ (left panel) and $\rho+p_{t}$ (right
panel) versus $r$. The model parameters have been set as $m=4$ and $r_{0}=2$,
$\lambda=-1$ (solid curve) and $\lambda=1$ (dashed curve). Figure 7: The
behavior of $\rho+p_{r}+2p_{t}$ versus $r$. The model parameters have been set
as $m=4$ and $r_{0}=2$, $\lambda=-1$ (solid curve) and $\lambda=1$ (dashed
curve). Figure 8: The behavior of $w_{t}$ with respect to $r$ for $r_{0}=2$,
$m=4$ and $\lambda=-1,1$ from down to up, respectively.

####  3.2.2 Case-II: Two parallel eccentric cylinders

In this subsection, we consider a Casimir setup consisting of two parallel
cylinders of length $L$ and radii $a$ and $b$ respectively, so that, the
cylinder of radius $a$ lies inside that of radius $b$. Let us denote by
$\delta$ the separation between the centers of the cylinders, and $d$ the
(varying) distance between them, hence we have $\delta=b-a-d$. An exact result
for the Casimir energy for such a configuration has been presented in [103].
In the case of asymptotic behavior of Casimir interaction, i.e, in the limit
when $d\ll(b-a)$, the associated Casimir energy for Dirichlet (D) or Neumann
(N) boundary conditions on both cylinders is given by [113]

$\displaystyle
E^{cc}=-\frac{\pi^{3}\sqrt{ab}L}{1920d^{\frac{5}{2}}\sqrt{2(b-a)}}.$ (63)

For the case of DN (Dirichlet on one cylinder and Neumann on the other) or ND
boundary conditions the Casimir energy reads

$\displaystyle
E^{cc}=\frac{7\pi^{3}\sqrt{ab}L}{15360d^{\frac{5}{2}}\sqrt{2(b-a)}}.$ (64)

The Casimir energy density can then be obtained using the volume between the
two cylinders $V=\pi(b^{2}-a^{2})L$, as

$\displaystyle\rho=\frac{E^{cc}}{\pi(b^{2}-a^{2})}.$ (65)

Here, we consider the distance between the cylinders $d$ as the radial
coordinate $r$ of the wormhole. Then, energy density is found as
$\rho=\lambda/8\pi r^{\frac{5}{2}}$ where

$\displaystyle\lambda=-\frac{\pi^{3}\sqrt{ab}L}{240\sqrt{2(b-a)^{3}}(a+b)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\lambda=\frac{7\pi^{3}\sqrt{ab}L}{1920\sqrt{2(b-a)^{3}}(a+b)}.$ (66)

The first part of the above expression refers to DD or NN boundary conditions
and the second one to DN or ND boundary conditions. Using now equation (23)
for $m=5/2$, we get the shape function as

$\displaystyle b(r)=2\lambda\left(\sqrt{r}-\sqrt{r_{0}}\right)+r_{0}.$ (67)

For this solution, we find that the quantity $b(r)/r$ tends to zero at spatial
infinity, so these solutions correspond to an asymptotically flat spacetime.
The flare-out condition at the throat leads to $\lambda<\sqrt{r_{0}}$. Also,
substituting for the shape function (67) into equation (40) and setting
$w=-\sqrt{r_{0}}/\lambda$ we arrive at the following differential equation for
the redshift function

$\displaystyle
r^{\frac{3}{2}}f^{\prime}(r)+\left(\sqrt{r_{0}}-2\lambda\right)\left[f(r)+rf^{\prime}(r)\right]=0.$
(68)

The above equation admits an exact solution in the form

$f(r)=\frac{f_{0}r_{0}\left[\sqrt{r}+\sqrt{r_{0}}-2\lambda\right]^{2}}{4r\left(\sqrt{r_{0}}-\lambda\right)^{2}},$
(69)

where the integration constant has been set in such a way that the redshift
function assumes a finite value at the throat, $f(r_{0})=f_{0}$. Fig. (9)
shows the behavior of redshift function where we observe that this function is
finite for $r>r_{0}$. Using equations (44) and (46) for $m=5/2$ and
$w=-\sqrt{r_{0}}/\lambda$, we get

$\displaystyle\rho+P_{t}$ $\displaystyle=$
$\displaystyle\frac{2r_{0}-\lambda\sqrt{r_{0}}-6\lambda^{2}+\sqrt{r}\left(4\lambda+\sqrt{r_{0}}\right)}{32\pi
r^{\frac{5}{2}}\left(\sqrt{r}+\sqrt{r_{0}}-2\lambda\right)},$ (70)
$\displaystyle\rho+P_{r}+2P_{t}$ $\displaystyle=$
$\displaystyle\frac{(\sqrt{r_{0}}-2\lambda)(\lambda-\sqrt{r})}{16\pi
r^{\frac{5}{2}}\left(\sqrt{r}+\sqrt{r_{0}}-2\lambda\right)}.$ (71)

The behavior of NEC is shown in Fig. (10) where we observe its radial profile
(left panel) gets violated for both positive and negative values of $\lambda$
parameter. However, the NEC in tangential direction (right panel) is
satisfied. Also the left panel in Fig. (11) shows the behavior of SEC against
radial coordinate where we see that for $m<3$ and $\lambda<0$ this quantity is
negative at the throat and throughout the spacetime. Using the field equations
we get the EoS parameter in tangential direction as

$\displaystyle
w_{t}=\frac{\sqrt{rr_{0}}+2r_{0}-5\sqrt{r_{0}}\lambda+2\lambda^{2}}{4\lambda\left(\sqrt{r}+\sqrt{r_{0}}-2\lambda\right)}.$
(72)

The right panel in Fig. (11) presents the behavior of the above expression
where we observe that the NEC in tangential direction is fulfilled, see also
the right panel in Fig. (10). Also, from Eq. (54) the Kretschmann scalar for
$m=5/2$ is obtained as

$\displaystyle\mathcal{K}(r_{0})=\frac{97r_{0}-68\lambda\sqrt{r_{0}}+36\lambda^{2}}{16r_{0}^{5}}.$
(73)

As it is expected, the Kretschmann scalar is finite at the throat for these
traversable wormhole solutions.

Figure 9: Behavior of $f(r)$ with respect to $r$ for $r_{0}=2$ and $m=5/2$. In
the left panel we have set $\lambda=-0.1,-0.2,-0.3$ from up to down and in
right panel we have set $\lambda=0.3,0.2,0.1$ from up to down.

Figure 10: The behavior of $\rho+p_{r}$ (left panel) and $\rho+p_{t}$ (right
panel) versus $r$. The model parameters have been set as $\lambda=1$ (dashed
curve), $\lambda=-0.5$ (solid curve), $m=5/2$ and $r_{0}=2$.

Figure 11: (Left panel) The behavior of $\rho+p_{r}+2p_{t}$ versus $r$. The
model parameters have been chosen as $\lambda=1$ (dashed curve),
$\lambda=-0.5$ (solid curve), $m=5/2$ and $r_{0}=2$. (Right panel) The
behavior of EoS in tangential direction for the same values of parameters as
of the left panel.

####  3.2.3 Case-III: Two concentric spherical shells

Let us now consider two concentric spherical shells of radii $a$ and $b$ with
$\alpha=b/a>1$. The exact form for Casimir energy has been discussed in [114,
115]. However, Using PFA method, the Casimir interaction energy can be
computed for two limiting situations. In the short distance limit, i.e.,
$\alpha\rightarrow 1$, the interaction energy is given as [116]

$\displaystyle
E=-\frac{\pi^{3}a^{2}}{180(b-a)^{3}}\left[1+(\alpha-1)+O(\alpha-1)^{2}\right].$
(74)

In this case we can find the associated energy density using the fact that the
volume between two concentric spheres is $V=4\pi(b^{3}-a^{3})/3$. Hence, the
energy density is found as

$\displaystyle\rho=\frac{3E}{4\pi(b^{3}-a^{3})}.$ (75)

Considering the distance between the spheres $a$ as the radial coordinate, we
find energy density as $\rho={\lambda}/{8\pi r^{4}}$, with
$\lambda=-{\pi^{3}}/{360}$ (for both DD or NN boundary conditions). The
wormhole configuration for this type of Casimir energy is similar to the case
of two parallel plates which was studied in detail in sub Sec. 3.2.1. We
therefore proceed to study the opposite limit for which $\alpha\gg 1$. The
interaction energy is then written in the form [116]

$\displaystyle E=-\frac{2\pi^{3}}{35a\alpha^{4}},$ (76)

whence the Casimir energy density is found as

$\displaystyle\rho=-\frac{\lambda}{8\pi r^{7}}$ (77)

where $\lambda=-{12\pi^{3}a^{3}}/{35}$ and we have taken the outer radius of
the sphere i.e., $b$, as the radial coordinate in our model. Using equation
(23) for $m=7$ we obtain the shape function as

$\displaystyle
b(r)=\frac{\left(r^{4}-r_{0}^{4}\right)\lambda}{4r^{4}r_{0}^{4}}+r_{0}.$ (78)

The above solution corresponds to an asymptotically flat spacetime and the
flare-out condition at the throat leads to $\lambda<r_{0}^{5}$. Substituting
the shape function (78) into Eq. (40) along with setting
$w=-r_{0}^{5}/\lambda$ we arrive at following differential equation for the
redshift function

$\displaystyle
rf^{\prime}(r)\left(4r^{5}r_{0}^{4}-r^{4}\left(\lambda+4r_{0}^{5}\right)+\lambda
r_{0}^{4}\right)-f(r)\left(r^{4}-r_{0}^{4}\right)\left(\lambda+4r_{0}^{5}\right)=0.$
(79)

For the above differential equation, an analytic solution can not be found in
terms of elementary standard functions. We then resort to numerical
techniques. Fig. (12) shows the behavior $f(r)$ for $r_{0}=2$, $m=7$, where it
is seen that the redshift function is finite everywhere and that we have no
singularity in the wormhole spacetime. We may also examine the NEC and SEC for
these solutions. Using then equations (44) and (46) for $m=7$ with
$w=-r_{0}^{5}/\lambda$ we get

$\displaystyle\rho+P_{t}\,\,$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!\frac{r_{0}^{4}\left[r_{0}^{5}\left(7\lambda+40r^{5}\right)+\lambda\left(3\lambda+16r^{5}\right)-23\lambda
r^{4}r_{0}-44r^{4}r_{0}^{6}+4r_{0}^{10}\right]-3\lambda^{2}r^{4}}{32\pi
r^{7}\left(r_{0}^{4}\left(\lambda+4r^{5}-4r_{0}r^{4}\right)-\lambda
r^{4}\right)},$ (80) $\displaystyle\rho+P_{r}+2P_{t}\,\,$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\frac{r_{0}^{4}\left[r_{0}^{5}\left(5\lambda+32r^{5}\right)+\lambda\left(\lambda+8r^{5}\right)-13\lambda
r^{4}r_{0}-36r^{4}r_{0}^{6}+4r_{0}^{10}\right]-\lambda^{2}r^{4}}{16\pi
r^{7}\left(r_{0}^{4}\left(\lambda+4r^{5}-4r_{0}r^{4}\right)-\lambda
r^{4}\right)}.$ (81)

The left and right panels in Fig. (13) show the behavior tangential component
of NEC and SEC where we see that one can choose suitable values of $\lambda$
and $r_{0}$ parameters so that both $\rho+P_{t}$ and $\rho+P_{r}+2P_{t}$ are
satisfied at all space. Moreover, using the field equations, one can find the
EoS in tangential direction as

$\displaystyle w_{t}=\frac{\lambda
r^{4}(11w+1)+r_{0}^{4}\left[-40r^{5}w+4r_{0}r^{4}(11w+1)+\lambda(w(4w-7)-1)\right]}{4\left(4r^{5}r_{0}^{4}-\lambda
r^{4}-4r_{0}^{5}r^{4}+\lambda r_{0}^{4}\right)}.$ (82)

Fig. (14) shows the behavior of the above quantity for the same values of
model parameters as of Fig. (13). Also, from Eq. (54), one may realize that
the Kretschmann scalar is finite at the throat of wormhole.

Figure 12: The behavior of $f(r)$ with respect to $r$ for $r_{0}=2$, $m=7$ and
$\lambda=-0.1,-2,-3$ (left panel) and $\lambda=3,2,0.1$ (right panel) from up
to down.

Figure 13: The behavior of $\rho+p_{t}$ (left panel) and $\rho+2p_{t}+p_{r}$
(right panel) versus $r$. The model parameters have been set as $m=7$ and
$r_{0}=2$, $\lambda=1$ (dashed curve) and $\lambda=-4$ (solid curve). Figure
14: The behavior of tangential EoS parameter against $r$. The model parameters
have been set as of Fig. (13).

##  4 Equilibrium conditions

In the present section we examine the stability of the obtained wormhole
solutions by employing the equilibrium condition. In the context of GR, this
condition can be deduced using the famous Tolman-Oppenheimer-Volkov (TOV)
equation [117, 118, 119, 120, 121],[122, 123]. For an isotropic EMT fluid the
TOV equation is given as

$\displaystyle-{\frac{dp_{r}}{dr}}+2\frac{({\it
P_{t}}-P_{r})}{r}-\frac{\phi^{\prime}(r)}{2}(\rho+P_{r})=0.$ (83)

Given the above equation one can determine the equilibrium state of a wormhole
configuration by taking the gravitational ($F_{g}$), hydrostatic ($F_{h}$) as
well as the anisotropic ($F_{a}$) forces (arising due to anisotropy of matter)
into account. These forces are defined through the following relations

$\displaystyle
F_{g}=-\frac{\phi^{\prime}(r)}{2}(\rho+P_{r}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
F_{h}=-{\frac{dP_{r}}{dr}},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ F_{a}=\frac{2}{r}(P_{t}-P_{r}).$ (84)

In terms of the above forces the TOV equation can be rewritten as

$\displaystyle F_{g}+F_{h}+F_{a}=0.$ (85)

Now, using equations (5)-(7) and considering $P_{r}=w\rho$, we get the
corresponding relations for the forces as

$\displaystyle F_{g}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
g(r)r^{3}}\left[g^{\prime}(r)r+g(r)-1\right]\left(w+1\right)\left[wg^{\prime}(r)r+\left(g(r)-1\right)\left(w+1\right)\right],$
(86) $\displaystyle F_{h}$ $\displaystyle=$ $\displaystyle\frac{w}{8\pi
r^{3}}\left[g^{\prime\prime}(r)r^{2}-2g(r)+2\right],$ (87) $\displaystyle
F_{a}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
r^{3}}\Bigg{\\{}r^{2}g(r)w\left(w+1\right)\left[\frac{g^{\prime}(r)}{g(r)}\right]^{2}-2wr^{2}g^{\prime\prime}(r)+r\left(1+2w\right)\left(w+1\right)\left(g(r)-1\right)\frac{g^{\prime}(r)}{g(r)}$
(88) $\displaystyle+$
$\displaystyle\left[\left(w^{2}+6w+1\right)g(r)-\left(w+1\right)^{2}\right]\left(\frac{g(r)-1}{g(r)}\right)\Bigg{\\}}.$

Substituting for the shape function (23) we finally get

$\displaystyle F_{g}$ $\displaystyle=$ $\displaystyle\frac{\Sigma_{3}}{16\pi
r_{0}^{m}r^{m+1}}\left[(m-3)(r_{0}-r)-\lambda\left(r^{3-m}-r_{0}^{3-m}\right)\right]^{-1},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ F_{h}=-\frac{mr_{0}^{m-2}}{8\pi{r}^{m+1}},$
$\displaystyle F_{a}$ $\displaystyle=$ $\displaystyle\frac{\Sigma_{4}}{8\pi
r_{0}^{m}r^{m+1}}\left[(m-3)(r_{0}-r)-\lambda\left(r^{3-m}-r_{0}^{3-m}\right)\right]^{-1},$
(89)

where

$\displaystyle\Sigma_{3}$ $\displaystyle=$
$\displaystyle\left(r_{0}^{m}{r}^{3-m}-r_{0}^{3}\right){\lambda}^{2}+\left[\left(m-4\right)r_{0}^{2m-2}{r}^{3-m}-\left(m-4\right)r_{0}^{1+m}\right]\lambda$
$\displaystyle+$
$\displaystyle\left(3-m\right)r_{0}^{3m-4}{r}^{3-m}+\left(m-3\right)r_{0}^{2m-1},$
$\displaystyle\Sigma_{4}$ $\displaystyle=$ $\displaystyle
m\left[r\left(\lambda{r}^{2-m}+m-3\right)r_{0}^{2m-2}+\left(3-m\right)r_{0}^{2m-1}-r_{0}^{1+m}\lambda\right]-\frac{\Sigma_{3}}{2}.$
(90)

Figs. (15) and (16) show the graph of gravitational, hydrostatic and
anisotropic forces given in Eq. (89) for each case. It is therefore seen that
these forces cancel the effects of each other leaving thus a stable wormhole
configuration. Also, in the case for which $m=3$, we can substitute the shape
function (35) into equations (5)-(7) to find the EMT components. Then, from
Eq. (84), a simple calculation gives

$\displaystyle F_{g}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ F_{h}=\frac{1}{8\pi
r^{4}}\left[\lambda-3r_{0}+3\lambda\ln\left(\frac{r_{0}}{r}\right)\right]=-F_{a},$
(91)

whence we readily find that for this class of solutions, the gravitational
force becomes zero as a result of constant redshift function. Also, the
hydrostatic and anisotropic forces are exactly equal and opposite to each
other. Thus, the equilibrium of the three forces is achieved due to the
combined effect of them, and hence this supports the stability of the wormhole
configuration.

Figure 15: The behavior of $F_{h}$ (solid curve), $F_{a}$ (dashed curve) and
$F_{g}$ (dotted curve) against $r$. The model parameters have been set as
$m=4$ and $r_{0}=2$, $\lambda=-1$ (left panel) and $\lambda=1$ (right panel).

Figure 16: The behavior of $F_{h}$ (solid curve), $F_{a}$ (dashed curve) and
$F_{g}$ (dotted curve) versus $r$. The model parameters have been set as
$m=7$, $r_{0}=2$, $\lambda=-1$ (left panel) and $\lambda=1$ (right panel).

##  5 Particle trajectories around the wormhole

In this section we investigate geodesic equations in wormhole spacetime
described by the metric (1), using the Lagrangian formalism [124]. Due to the
spherical symmetry of the wormhole spacetime, without loss of generality we
can restrict our analysis to planar motion in the equatorial plane
$\theta=\pi/2$. The corresponding Lagrangian for metric (1) is then found as

$\mathcal{L}=g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-f(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{g(r)}+r^{2}\dot{\phi}^{2},$
(92)

where a dot denotes derivative with respect to the affine parameter $\eta$.
The above Lagrangian is constant along a geodesic, hence, we can set
$\mathcal{L}(x^{\mu},\dot{x}^{\mu})={\epsilon}$ so that time-like and null
geodesics correspond to ${\epsilon}=-1$ and ${\epsilon}=0$, respectively.
Using the Euler-Lagrange equation

$\frac{d}{d\eta}\frac{\partial{\cal
L}}{\partial\dot{x}^{\mu}}-\frac{\partial{\cal L}}{\partial x^{\mu}}=0,$ (93)

one can easily find two constants of motion given as

$\dot{t}=\frac{E}{f(r)},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
r^{2}\dot{\phi}=L,$ (94)

where $E$ and $L$ are the energy and angular momentum of the test particle,
respectively. Inserting these constants of motion into (92) we obtain

$\displaystyle\dot{r}^{2}$
$\displaystyle=g(r)\left[\frac{E^{2}}{f(r)}-\frac{L^{2}}{r^{2}}+\epsilon\right].$
(95)

It is convenient to rewrite the above equation in terms of the proper radial
distance which is defined as

$l(r)=\pm\int_{r_{0}}^{r}\frac{dr}{\sqrt{g(r)}}.$ (96)

The proper radial distance is finite for all finite values of $r$ throughout
the spacetime. We note that the extension of spacetime in terms of proper
radial distance is in such a way that, $l$ monotonically increases from
$-\infty$ in the lower universe to $l=0$ at the throat and then from zero to
$+\infty$ in the upper universe. Using the proper radial distance, Eq. (95)
takes the simple form

$\displaystyle\dot{l}^{2}=f(r)^{-1}[E^{2}-V_{{\rm eff}}(L,l)],$ (97)

where the effective potential is defined as

$\displaystyle V_{{\rm
eff}}(L,l)=f\left(r(l)\right)\left[\frac{L^{2}}{r(l)^{2}}-\epsilon\right].$
(98)

In what follows, we discuss particle trajectories in the wormhole spacetime,
using the above form for the effective potential. Indeed, geodesic equation
(97) can be viewed as a classical scattering problem with the potential
barrier $V_{{\rm eff}}(L,l)$. Moreover, using Eq. (94) we can rewrite Eq. (97)
as an ordinary differential equation for orbital motion

$\left(\frac{dl}{d\phi}\right)^{2}=\frac{{\dot{l}}^{2}}{\dot{{\phi}}^{2}}=\frac{{r(l)}^{4}}{f(r)L^{2}}\left[E^{2}-V_{{\rm
eff}}(L,l)\right].$ (99)

We note that, in traversable wormhole spacetimes, particles can travel through
the throat of the wormhole from one asymptotically flat part of the universe
to other one. Then, a geodesic can pass through the throat into the other
universe if

$\displaystyle E^{2}>V_{{\rm eff}}(L,0).$ (100)

Similarly, for those geodesics that get reflected back to the same universe by
the potential barrier, we have $E^{2}<V_{{\rm eff}}(L,0)$. In this case, there
is a turning point at $l=l_{{\rm tp}}$ which is obtained by solving the
following equation

$\displaystyle E^{2}=V_{{\rm eff}}(L,l_{{\rm tp}}).$ (101)

From Eq. (98), it is easy to verify that

$\displaystyle\frac{dV_{{\rm eff}}}{dl}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\sqrt{g(r)}\bigg{[}\left(\frac{L^{2}}{r^{2}}-\epsilon\right)f^{\prime}(r)-\frac{2L^{2}f(r)}{r^{3}}\bigg{]},$
(102) $\displaystyle\frac{d^{2}V_{{\rm eff}}}{dl^{2}}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{L^{2}}{r^{4}}f(r)\left[6g(r)-rg^{\prime}(r)\right]+\frac{f^{\prime}(r)}{2r^{4}}\left[\left(L^{2}-\epsilon
r^{2}\right)r^{2}g^{\prime}(r)-8L^{2}rg(r)\right]+\frac{f^{\prime\prime}(r)}{r^{2}}\left(L^{2}-\epsilon
r^{2}\right)g(r).$

A generic feature of this effective potential in the case $f(r)=constant$ is
that it possesses a global maximum at the throat

$\displaystyle\frac{dV_{{\rm
eff}}}{dl}\Big{|}_{l=0}=0,\quad\quad\quad\frac{d^{2}V_{{\rm
eff}}}{dl^{2}}\Big{|}_{l=0}=-\frac{L^{2}f(r_{0})g^{\prime}(r_{0})}{r_{0}^{3}}.$
(104)

From the second part of the above equation, we find that the flaring out
condition leads to $\frac{d^{2}V_{{\rm eff}}}{dl^{2}}<0$ at the throat. This
clearly provides an unstable orbit since it occurs at the maximum of the
potential for $E^{2}=V_{{\rm eff}}(L,l_{0})$. We note that these conditions
are independent of whether the geodesic is null or timelike. We now consider
the wormhole solutions presented in subsection (3.2) and restrict ourselves to
the class of wormholes with $m=4$. Substituting the shape function Eq. (56)
into Eq. (96) we find

$\displaystyle l(r)=\frac{\pm
1}{2r_{0}}\left[(r_{0}^{2}+\lambda)\ln\left(\frac{-r_{0}^{2}-\lambda+2r{r_{0}}+2\sqrt{r_{0}\left(r-r_{0}\right)\left(rr_{0}-\lambda\right)}}{r_{0}^{2}-\lambda}\right)+2\sqrt{r_{0}\left(r-r_{0}\right)\left(rr_{0}-\lambda\right)}\right].$
(105)

We can substitute the redshift function (58) into equation (98) to get the
effective potential as

$\displaystyle V_{{\rm
eff}}(L,l)=\left(1-\frac{\lambda}{r_{0}r}\right)^{\frac{r_{0}^{2}+\lambda}{\lambda}}\left[\frac{L^{2}}{r(l)^{2}}-\epsilon\right].$
(106)

Also, we calculate the derivatives of the above potential as

$\displaystyle V^{\prime}_{\rm eff}(L,l(r))$ $\displaystyle=$
$\displaystyle\frac{\left({r_{0}}-\lambda\right)^{{\frac{r_{0}^{2}}{\lambda}}}\sqrt{{\frac{\left(r-r_{0}\right)\left({r_{0}}r-\lambda\right)}{{r_{0}}r}}}}{{r}^{5}{{r_{0}}}^{{\frac{r_{0}^{2}}{\lambda}}}{r}^{{\frac{r_{0}^{2}}{\lambda}}}{r_{0}}}\left[r^{2}\epsilon\left(r_{0}^{2}+\lambda\right)+2{L}^{2}{r_{0}}r-\left(r_{0}^{2}+3\lambda\right)L^{2}\right],$
(107) $\displaystyle V^{\prime\prime}_{\rm eff}(L,l(r))$ $\displaystyle=$
$\displaystyle\frac{(rr_{0}-\lambda)^{\frac{r_{0}^{2}}{\lambda}}}{r^{\frac{7\lambda+r_{0}^{2}}{\lambda}}r_{0}^{\frac{2\lambda+r_{0}^{2}}{\lambda}}}\Big{[}L^{2}r_{0}^{2}\left(3r-2r_{0}\right)\left(4r^{2}-6r_{0}r+r_{0}^{2}\right)-2\lambda
L^{2}r_{0}\left(19r^{2}-29r_{0}r+8r_{0}^{2}\right)$ (108) $\displaystyle+$
$\displaystyle
3\lambda^{2}L^{2}\left(9r-10r_{0}\right)+r^{2}\epsilon\left(\lambda+r_{0}^{2}\right)\left(r_{0}\left(6\lambda+4r^{2}-7r_{0}r+2r_{0}^{2}\right)-5\lambda
r\right)\Big{]}.$

Next, we proceed to study null geodesics ($\epsilon=0$) for the class of
wormhole solutions with $m=4$ and nonzero redshift function. From Eq. (107) we
can find two roots that satisfy equation ${V^{\prime}_{\rm eff}}=0$, these
roots are given by $r_{1}=r_{0}$ and
$r_{2}=\frac{r_{0}}{2}+\frac{3\lambda}{2r_{0}}$. The condition $r_{2}>r_{0}$
leads to the inequality $3\lambda>{r_{0}}^{2}$, that by using the second
derivative the effective potential (108) for positive $\lambda$ we have
$V^{\prime\prime}_{\rm eff}|_{r=r_{2}}<0$, i.e., a local maximum. For this
case we have a photon sphere located outside the throat, see the left panel in
Fig. (17) where we have sketched the behavior of effective potential as a
function of proper radial distance for different values of $\lambda$
parameter. We further observe that the effective potential admits a local
minimum at the throat, i.e., $V^{\prime\prime}_{\rm eff}(L,0)>0$. In this case
the wormhole throat acts as an anti-photon sphere [125, 126]. For $\lambda<0$
or $0<\lambda<r_{0}^{2}/3$ the effective potential assumes a maximum value at
the throat, i.e. ${V^{\prime\prime}_{\rm eff}}|_{r=r_{0}}<0$, hence, the
throat acts as a photon sphere, see the right panel in Fig. (17). Fig. (18)
shows the changes in effective potential with respect to angular momentum
where we observe that the height of potential barrier increases with
increasing the angular momentum. Now, in order to discuss the photon orbits we
may eliminate $d\eta$ between the second part of Eq. (94) and Eq. (95) to find

$\displaystyle\left(\frac{dr}{d\phi}\right)^{2}=\left[1-\frac{b(r)}{r}\right]\left[\frac{r^{4}}{\mu^{2}}{\rm
e}^{-2\phi(r)}-r^{2}\right],$ (109)

where $\mu=L/E$ is the impact parameter and $\left(dr/d\phi\right)|_{r=r_{\rm
tp}}=0$ hence $\mu=r_{\rm tp}{\rm e}^{-\phi(r_{\rm tp})}$. For a photon that
comes from the polar coordinate $\lim_{r\rightarrow\infty}(r,-\pi/2-\theta/2)$
and passes through the turning point located at $(r_{tp},0)$ before reaching
the point $\lim_{r\rightarrow\infty}(r,\pi/2+\theta/2)$ one can define the
deflection angle $\theta(r_{tp})$ as [127, 128]

$\displaystyle\theta(r_{\rm tp})=-\pi+2\int_{r_{\rm
tp}}^{\infty}\frac{dr}{r\left[\left(1-\frac{b(r)}{r}\right)\left(\frac{r^{2}}{\mu^{2}}{\rm
e}^{-2\phi(r)}-1\right)\right]^{\frac{1}{2}}}.$ (110)

Figure 17: Left panel: The behavior of effective potential for null geodesics
against proper radial distance. For the model parameters we have set
$\lambda=0.7$ (blue dashed curve), $\lambda=0.6$ (red solid curve) and
$\lambda=0.5$ (black dotted curve). Right panel: The behavior of effective
potential for $\lambda=-0.7$ (blue dashed curve), $\lambda=-0.6$ (red solid
curve), $\lambda=-0.5$ (black dotted curve), $\lambda=0.2$ (green curve) and
$\lambda=0.1$ (dot-dashed curve). The throat radius and angular momentum have
been taken as $r_{0}=1$ and $L=0.5$, respectively.

It is possible that the light rays get trapped in a sphere of constant radius
and consequently may not reach asymptotically
$\lim_{r\rightarrow\infty}(r,\pi/2+\theta/2)$. In such a scenario the above
integral diverges and such a sphere is called photon sphere. As mentioned
before, the location of photon sphere can be determined through the behavior
of effective potential. The left panel in Fig. (19) shows the behavior of
deflection angle as a function of turning point for $\lambda>0$. As we observe
from the left panel of Fig. (17), the maxima of the effective potential
represent the location of photon spheres for each value of $\lambda$
parameter. These points correspond to the asymptotic values of $r_{\rm
tp}=r_{2}>r_{0}$ at which the integral (110) diverges. Hence, for each value
of the parameter $\lambda>r_{0}^{2}/3$, we have a photon sphere outside the
throat. In the right panel of Fig. (19) we have sketched the behavior of
deflection angle for $\lambda<0$ and $\lambda<r_{0}^{2}/3$. These value of the
parameter $\lambda$ determine the behavior of effective potential as shown in
the right panel of Fig. (17), where we observe that the effective potential
assumes a maximum at the throat, i.e., $l(r)|_{r_{0}=1}=0$. This maximum
indicates existence of a photon sphere at the throat where, the deflection
angle diverges in the limit $r_{\rm tp}\rightarrow r_{0}$.

In the case of timelike geodesics, we can use Eq. (107) with $\epsilon=-1$ to
get the first derivative of effective potential as

$\displaystyle V^{\prime}_{\rm
eff}(L,r(l))=\frac{\sqrt{\left(r-r_{0}\right)\left(r_{0}r-\lambda\right)}\left(r_{0}r-\lambda\right)^{{\frac{r_{0}^{2}}{\lambda}}}}{r^{5}r_{0}^{{\frac{r_{0}^{2}}{\lambda}}}{r}^{{\frac{r_{0}^{2}}{\lambda}}}{r_{0}^{\frac{3}{2}}}}\left[\left({r}^{2}+{L}^{2}\right)r_{0}^{2}-2{L}^{2}{r_{0}}r+3\lambda{L}^{2}+\lambda{r}^{2}\right],$
(111)

whereby we find the following three roots for equation $V^{\prime}_{\rm
eff}(L,r_{c})=0$ as

Figure 18: Left panel: The behavior of effective potential for null geodesics
against proper radial distance. For the model parameters we have set
$r_{0}=1$, $\lambda=-0.7$, $L=0.5$ (black dotted curve), $L=1$ (red solid
curve) and $L=2$ (blue dashed curve). Right panel: The behavior of effective
potential for $r_{0}=1$, $\lambda=0.2$, $L=0.5$ (dot-dashed curve), $L=1$
(blue long-dashed curve) and $L=2$ (blue space-dot curve).

$\displaystyle r_{\rm c}=r_{0},\leavevmode\nobreak\ \leavevmode\nobreak\
r_{c,+}=\frac{r_{0}L^{2}+L\sqrt{r_{0}^{2}L^{2}-r_{0}^{4}-4\lambda
r_{0}^{2}-3\lambda^{2}}}{r_{0}^{2}+\lambda},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
r_{c,-}=\frac{r_{0}L^{2}-L\sqrt{r_{0}^{2}L^{2}-r_{0}^{4}-4\lambda
r_{0}^{2}-3\lambda^{2}}}{r_{0}^{2}+\lambda}.$ (112)

With the help of Eq. (108), we can calculate the second derivative of
effective potential at the wormhole throat as

$\displaystyle V^{\prime\prime}_{\rm
eff}(L,0)=\frac{\left(r_{0}^{2}-\lambda\right)^{{\frac{r_{0}^{2}}{\lambda}}+1}}{2r_{0}^{{\frac{8\lambda+2r_{0}^{2}}{\lambda}}}}\left[r_{0}^{4}+\left(\lambda-{L}^{2}\right)r_{0}^{2}+3{L}^{2}\lambda\right],$
(113)

whence we find that there exists a critical value for the angular momentum at
which $V^{\prime\prime}_{\rm eff}(L,0)$ changes its sign

$\displaystyle L_{\rm
cr}=\sqrt{\frac{r_{0}^{2}(r_{0}^{2}+\lambda)}{(r_{0}^{2}-3\lambda)}}.$ (114)

For $L>L_{\rm cr}$ ($L<L_{\rm cr}$) the effective potential admits a local
maximum (local minimum) at wormhole throat. Then, unstable circular orbits can
occur due to the existence of a maximum for the effective potential and bound
orbits are possible when the effective potential gets a minimum at the throat.
In order that the roots ($r_{c,\pm}$) assume real values the angular momentum
must obey the condition $L>L_{p}$ where $L_{p}$ is given by

$\displaystyle L_{p}=\frac{\sqrt{r_{0}^{4}+4\lambda
r_{0}^{2}+3\lambda^{2}}}{r_{0}}.$ (115)

To check the stability of orbits, we may solve equation $V^{\prime}_{\rm
eff}(L,r_{c})=0$ for the square of angular momentum and substitute the result
into Eq. (108) to obtain

$\displaystyle V^{\prime\prime}_{\rm
eff}(\lambda,r_{c})=\frac{2\left(r_{0}r_{c}-\lambda\right)^{{\frac{r_{0}^{2}}{\lambda}}}\left(r_{0}^{2}+\lambda\right)\left(r_{0}-r_{c}\right)\left(r_{0}^{2}-r_{0}r_{c}+3\lambda\right)\left(r_{0}r_{c}-\lambda\right)}{r_{c}^{5}\left(r_{0}^{2}-2r_{0}r_{c}+3\lambda\right)r_{0}^{{\frac{r_{0}^{2}}{\lambda}}}r_{c}^{{\frac{r_{0}^{2}}{\lambda}}}{r_{0}}^{2}}.$
(116)

For this class of wormhole solutions, we can obtain the behavior of effective
potential as a function of the proper distance for timelike geodesics.
Generally, for circular orbits we require that $r=constant$ and so
$\dot{r}=\ddot{r}=0$. In this situation, the only possible position of the
particle will be a circle for which the conserved total energy of the particle
is equal to the extremum of the effective potential. More precisely, if the
energy corresponds to a maximum or a saddle point of the effective potential,
then the particle moves on an unstable orbit. If the energy corresponds to a
minimum of the effective potential then the trajectory of the particle will be
a stable orbit. In Fig. (20) we have plotted the behavior of effective
potential against proper radial distance for $L>L_{p}$. It is therefore seen
that for $r_{0}=1$ and $\lambda=\pm 0.1$ we have $r_{c,\pm}\in\mathbb{R}^{+}$
but only one of the roots is larger than $r_{0}$. Using then Eq. (116) one can
recognize that $V^{\prime\prime}_{\rm eff}(\lambda,r_{c})>0$ which corresponds
to a minimum of the effective potential. In Fig. (21) we choose the values of
parameters $r_{0}$ and $\lambda$ in such a way that $L<L_{p}$, therefore the
two roots $r_{c,\pm}$ are no longer real. For this case there exists only a
local minimum at wormhole throat for both $\lambda>0$ and $\lambda<0$.
Finally, in Fig. (22) we taken the values of $r_{0}$ and $\lambda$ so that
$L>L_{p}$. Then, there exist two real roots with values greater than the
throat radius. One of these roots corresponds to the maximum
($V^{\prime\prime}_{\rm eff}(\lambda,r_{c})<0$) and the other one corresponds
to the minimum ($V^{\prime\prime}_{\rm eff}(\lambda,r_{c})>0$) of the
effective potential. We also note that for $L<L_{p}$, we have only a local
maximum or a local minimum at the throat. The former occurs for $L>L_{\rm cr}$
and the latter for $L<L_{\rm cr}$.

Figure 19: Left panel: The behavior of deflection angle for $r_{0}=1$,
$L=0.5$, $\lambda=0.7$ (blue dashed curve), $\lambda=0.6$ (red solid curve)
and $\lambda=0.5$ (black dotted curve). For each value of the parameter
$\lambda$, the asymptotes of the curves are found as $r_{\rm
tp}=r_{2}=1.25,1.40,1.55$, respectively. Right panel: The behavior of
deflection angle for $r_{0}=1$, $L=0.5$, $\lambda=-0.7$ (blue dashed curve),
$\lambda=-0.6$ (red curve) and $\lambda=-0.5$ (black dotted curve),
$\lambda=0.2$ (green curve) and $\lambda=0.1$ (dot-dashed curve).

Figure 20: Behavior of Effective potential for timelike geodesics versus
proper radial distance. In the left panel we have set $\lambda=0.1$ and in the
right panel $\lambda=-0.1$. In these plots we have considered angular momentum
and throat radius as $L=1.26$ and $r_{0}=1$.

Figure 21: The behavior of effective potential for timelike geodesics. In the
left panel we have set $\lambda=1$ (blue curve) and $\lambda=0.5$ (red curve)
and in the right panel $\lambda=-1$ (blue curve) and $\lambda=-0.5$ (red
curve). In these plots the angular momentum and throat radius have been taken
as $L=0.5$ and $r_{0}=2$. Figure 22: The behavior of effective potential for
timelike geodesics. In this diagram we have set $r_{0}=1$, $\lambda=0.9$,
$L=2.9$ (blue curve) and $L=2.7$ (red curve).

##  6 Concluding Remarks

In this work, we studied static spherically symmetric wormhole spacetimes that
sustained by the Casimir energy as the source. In the first case, under zero
tidal force condition, we obtained the shape function by imposing a general
form for the Casimir energy density. In this case, we showed that the wormhole
solutions are asymptotically flat or AdS and dS so that each situation depends
on the value of model parameters, i.e $\lambda$ and power of $m$ in the
Casimir energy density. Then, we checked the WEC and SEC and showed that the
energy conditions are violated at the wormhole throat. We calculated radial
and tangential EoS parameters as the ratio of the respective pressures to the
energy density. In the second case, by using a linear EoS between the radial
pressure and energy density we derived a general form for the red shift
function. In addition, in order to be a traversable wormhole, the curvature
singularity at the wormhole throat must be absent. In this regard, we obtained
a general condition for the state parameter that meets this condition and
avoids formation of an event horizon at the throat. Hence, by choosing
specific state parameter as $w=-r_{0}^{m-2}/\lambda$, all curvature invariants
such as Ricci and Kretshmann scalars assume finite values in the range
$r_{0}\leq r$, providing then traversable wormhole solutions. Furthermore, we
study three geometric configurations of the Casimir effect in details. For the
case of Casimir effect between two parallel plates we obtained the shape and
red shift functions of the wormhole metric. Furthermore, we investigated the
WEC and SEC for positive and negative values of $\lambda$ parameter against
the radial distance from the wormhole throat. Also, for the case of parallel
cylinders and spherical shells, we obtained the shape function and numerically
solved the differential equation for the red shift function for both
$\lambda>0$ and $\lambda<0$. In addition, we checked the WEC and SEC at the
wormhole throat and showed that in general the classical energy conditions are
violated for three geometric configurations, as expected. We further
investigated the stability of wormhole solutions utilizing TOV equation and
found that our obtained wormhole solutions are stable. Also, we analyzed
trajectory of null and timelike particles for a class of wormhole solutions
with $m=4$ and nonzero redshift function. We found that for $\lambda>0$
(repulsive Casimir force) there exists a photon sphere located outside the
throat and an anti-photon sphere at the throat. For $\lambda<0$ (attractive
Casimir force) or $0<\lambda<r_{0}^{2}/3$ there exists only a photon sphere at
the wormhole throat. We also found that the deflection angle vanishes at a
critical value of turning point, namely $r_{\rm tp}^{\star}$, see Fig (19).
For $r_{\rm tp}>r_{\rm tp}^{\star}$ the deflection angle is negative and for
$r_{\rm tp}<r_{\rm tp}^{\star}$ it is positive. In the former case the
negatively deflected light rays are deflected away from the wormhole and in
the latter case the positively deflected light rays get attracted by the
wormhole lens to form stable or unstable photon orbits. It is worth mentioning
that negative deflection angle has been also reported in gravitational lensing
by naked singularities, see e.g. [129] and references therein. In the
particular case for which $r_{\rm tp}=r_{\rm tp}^{\star}$, light rays are
unaffected by the gravitating object. For timelike geodesics, we observed that
there exists a certain value for particle angular momentum, Eq. (115), with
the help of which one can determine whether the effective potential admits any
extrema outside the throat. Hence, depending on the values of $\lambda$
parameter and angular momentum, the effective potential could assume a local
minimum outside the throat allowing thus stable circular orbits for timelike
particles. Moreover, there exists a critical value for angular momentum, Eq.
(114), that decides the extrema of the effective potential at the throat. As
it is shown in Fig. (21), stable circular orbits are possible at the wormhole
throat. Finally, depending on model parameters, the effective potential could
assume a minimum as well as a maximum value outside the throat, thus, stable
(unstable) and bound orbits are possible for this situation.

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