aainstitutetext: Khon Kaen Particle Physics and Cosmology Theory Group
(KKPaCT), Department of Physics, Faculty of Science, Khon Kaen University, 123
Mitraphap road, Khon Kaen, 40002, Thailandbbinstitutetext: Theoretical and
Computational Physics Group, Theoretical and Computational Science Center
(TaCS), Faculty of Science, King Mongkut’s University of Technology Thonburi,
Prachauthid Rd., Bangkok 10140, Thailandccinstitutetext: School of Science,
Walailak University, Thasala, Nakhon Si Thammarat, 80160,
Thailandddinstitutetext: College of Graduate Studies, Walailak University,
Thasala, Nakhon Si Thammarat,
80160, Thailand

# Charged traversable wormholes supported by Casimir energy with and without
GUP corrections

Daris Samart b Takol Tangphati c,d Phongpichit Channuie
darisa@kku.ac.th,takoltang<EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate new exact and analytic solutions of the
Einstein–Maxwell field equations describing Casimir wormholes with and without
the effect of the Generalized Uncertainty Principle (GUP). We consider a
specific type of the GUP relations and study three specific models of the
redshift function along with two different EoS of state given by
$p_{r}(r)=\omega_{r}(r)\rho(r)$ and $p_{t}(r)=\omega_{t}(r)p_{r}(r)$. Here we
obtain a class of asymptotically flat Casimir wormhole solutions with and
without GUP corrections under the effect of electric charge. Furthermore we
check the null, weak, strong and dominant energy conditions at the wormhole
throat of radius $r_{0}$, and show at the wormhole throat that the classical
energy conditions are violated by some arbitrary small quantity. We also
examine the wormhole geometry with semi-classical corrections by using
embedding diagrams. We use the volume integral quantifier to quantify the
amount of the exotic matter near the wormhole throat. Additionally, we
investigate exotic fluid near WH throat using the so-called exoticity
parameter and discuss the speed of sound.

##  1 Introduction

Wormholes are hypothetical objects that are used to connect different
universes or different spatial points in the same universe Visser1995 . The
first theoretical investigation of the existences of the wormholes received
significant attention after the discovery of GR shortly by Flamm Flamm:1916 .
Subsequently, a detailed study of the wormholes has been further extended by
Einstein and Rosen Einstein:1935tc  and a so-called Einstein-Rosen bridge was
used to represent wormholes at that moment. However, it was realized later
that the Einstein-Rosen bridge can not be traversable. Interestingly, the
effect of the electromagnetic interaction in Einstein field equation might
play a crucial role on creating wormholes by using geon Wheeler:1955zz ;
Misner:1957mt . However, early studies of the wormholes are just thought
experiments or theoretical playgrounds which are unable to contact to physical
realities. Until, Morris and Thorne first studied the possible existence and
demonstrated how to use the wormholes for interstellar travel Morris:1988cz .
Nevertheless, a so-called exotic matter with negative pressure is required for
supporting the throat of traversable wormholes. This leads to violations of
several (classical) energy conditions in GR. Therefore, searching for the
forms of exotic matter compatible with the energy conditions becomes the main
topics in this research field.

A number of exotic matter forms has been studying in the traversable
wormholes. On the one hand, modified theories of gravity are used to generate
effective exotic fluid in order to support the throat of traversable
wormholes. For instances, these include higher order gravity and its
motivations by string theory Giribet:2019dmg ; Ovgun:2017dik ;
KordZangeneh:2015dks ; Mehdizadeh:2012zz ; Dehghani:2011fa ; Dehghani:2009zza
; Rahaman:2006xb ; Kim:1996np ; Ghoroku:1992tz ; MenaMarugan:1991ea ;
Hochberg:1990is , alternative theories of quantum gravity, e.g., Horava-
Lifshitz Garcia-Compean:2020aaa ; Bellorin:2016nvh ; Bellorin:2015oja ;
Bellorin:2014qca ; Botta-Cantcheff:2009ffi  and massive gravity Kamma:2021wam
; Amirabi:2020kfr ; Tangphati:2019pxh ; Forghani:2018svt ; Paul:2018ppy ;
Jusufi:2017drg , $f(R)$ gravity Mishra:2021ato ; Eid:2020hvf ; Shamir:2020uzy
; Fayyaz:2020jzh ; Tangphati:2020mir ; Godani:2019kgy ; Samanta:2019tjb ;
Godani:2018blx ; Sharif:2018jdj ; Saeidi:2011zz ; Bronnikov:2010tt ;
Lobo:2009ip , scalar-tensor (Brans-Dicke) gravity and its generalizations
(both Horndeski and Galileon theories) Korolev:2020ohi ;
Papantonopoulos:2019ugr ; Franciolini:2018aad ; Mironov:2018uou ;
Mironov:2018pjk ; Evseev:2017jek ; Rubakov:2016zah ; Kolevatov:2016ppi ;
Bhattacharya:2009rt ; Eiroa:2008hv ; Lu:2003bg ; He:1999nd ; Nandi:1997en ;
Agnese:1995kd ; Xiao:1991nv , Einstein-Gauss-Bonnet thoery Sharif:2020xxj ;
Ibadov:2020btp ; Antoniou:2019awm ; Mehdizadeh:2015jra ; Kanti:2011yv ;
Kanti:2011jz ; Mazharimousavi:2010bm ; Maeda:2008nz ; Bhawal:1992sz ,
teleparallel gravity Singh:2020rai ; Mustafa:2021vqz ; Saaidi:2020rrb ;
Bahamonde:2016jqq ; Jawad:2015uea ; Salti:2013bha ; Boehmer:2012uyw ;
Aygun:2007zzb , and etc. On the other hand, exotic matter motivated from dark
energy with standard Einstein-Hilbert action also receive a lot of attentions,
e.g., vacuum energy and cosmological constant Santos:2021jjs ; Ambjorn:2021wdm
; Jusufi:2020rpw ; Garattini:2019ivd ; Heydarzade:2014ada ; Rahaman:2006xa ;
Lemos:2003jb ; Liu:1993ej ; Nishioka:1992xy ; Myers:1990mjh ; Klebanov:1988eh
, quintessence and phantom scalar fields Aounallah:2020rlf ; Zubair:2020uyb ;
Manna:2019tpn ; Dzhunushaliev:2017syc ; Kocuper:2017aap ; Dzhunushaliev:2010bv
; Cataldo:2008ku ; Lobo:2006cx ; Rahaman:2005qi ; Zaslavskii:2005fs ;
Popov:2001kk ; Popov:2000id ; Kim:1992wd , vector gauge field and its higher-
order forms Anand:2020wlk ; Barros:2018lca ; Cox:2015pga ; Rahaman:2006qm ;
Tan:1997zc ; Yoshida:1990quk ; Shen:1990cb , and Chaplygin gas fluids
Lobo:2005vc ; Jamil:2008wu ; Eiroa:2009hm ; Kuhfittig:2009mx ; Sharif:2013lna
; Sharif:2013tva ; Elizalde:2018frj . In result, both modified gravity
theories and inclusion of special forms of exotic matter could stabilize the
wormhole throat. Although most of the models are still very toy models that
are not capable of being tested in the laboratories as well as suffering from
some theoretical inconsistencies in the models themselves. However, a so-
called Casimir energy is one type of the vacuum energy which has been
confirmed by the experiment. Moreover, the Casimir energy is the most likely
candidate of matter forms that can be used to stabilize the traversable
wormholes. In addition, it is well known that the classical electric and
magnetic fields could be useful to assist the stabilization of the wormholes.
Possibility of wormholes having charge was proposed by Ref.Kim:2001ri . The
charges play the role of the additional matter to the static wormhole which is
stabilized by the exotic matter. Moreover, wormholes with charge are thought
to be a natural extension of the original Morris-Thorne wormhole
Kuhfittig:2011xh .

Then the main purpose of this work is to study the physically possible
construction of the Casimir wormholes with the electric charge underlying a
so-called Generalized Uncertainty Principle (GUP). In this work, we would like
to extend the analysis present in Ref. Jusufi:2020rpw  where the effect of GUP
have been investigated in the Casimir wormholes. In addition, Ref.Boyer:1968uf
have studied the physical implications of the Casimir effect on a charge
particle in a conducting spherical shell. Therefore, inclusion of the electric
charge in the GUP Casimir wormholes is worth studying in more details in order
to see how the electric field plays the role in wormhole constructions both
with and without the GUP by invoking several models of the red shift
functions.

This work is organized as follows: We consider the charged Casimir wormholes
with and without the GUP corrections in Sec.2. Subsequently, the embedding
diagrams of the wormholes are illustrated in Sec.3. In Sec.4, we study the
energy conditions of the charge Casimir wormholes. The amount of the exotic
matter near the wormhole throat will be calculated in Sec.5. Additionally, in
Sec.6 we investigate exotic fluid near WH throat using the so-called exoticity
parameter and discuss the speed of sound. We close this work in Sec.7 for our
conclusion of findings and discussion relevant physical consequences and
future outlook.

##  2 Casimir wormhole spacetime with electric charge

The existence of charged black holes has suggested that wormholes may also be
charged. A generic static and spherically symmetric WH can be described by the
Morris-Thorne (MT) metric Morris:1988cz . Here we begin with MT wormhole
spacetime in the Schwarzschild coordinates given by

$\displaystyle
ds^{2}=-e^{2\Phi(r)}dt^{2}+\frac{1}{1-\frac{b(r)}{r}}dr^{2}+r^{2}d\Omega^{2}\,,$
(1)

with $d\Omega^{2}\equiv d\theta^{2}+\sin^{2}\theta d\phi^{2}$. Notice that the
above metric was recently extended to a general Morris-Thorne wormhole having
an electric charge by considering a spacetime of embedding class I
Kuhfittig:2011xh . In the wormhole geometry, the redshift function $\Phi(r)$
has to be finite in order to avoid the formation of an event horizon.
Additionally, the shape function $b(r)$ determines the wormhole geometry. In
order to have a consistent WH construction, the shape function should satisfy
the following properties: ($i$) $b(r)/r<1$ for $r>r_{0}$, ($ii$) $b(r)=r_{0}$
at $r=r_{0}$, ($iii$) $b(r)/r\rightarrow 0$ as $r\rightarrow\infty$, ($iv$)
$b^{\prime}(r)r-b(r)<0$ and ($v$) $b^{\prime}(r)<1$ at $r=r_{0}$. The
Einstein–Maxwell equations for static charged Casimir wormholes can be
parametrized in terms of density $\rho(r)$, radial pressure $p_{r}$,
tangential pressure $p_{t}$ and electric charge $E(r)=Q/r^{2}$. Here the
electromagnetic stress–energy tensor is given by
$T_{\mu\nu}^{EM}=\tfrac{1}{4\pi}(F_{\mu\lambda}F^{\lambda}_{\,\,\,\,\nu}-\tfrac{1}{4}g_{\mu\nu}F_{\lambda\sigma}F^{\lambda\sigma})$
and $F_{\mu\nu}$ represents the electromagnetic strength tensor. With the help
of the line element given in Eq.(1), we obtain the following set of equations
resulting from the energy-momentum components to yield Maurya:2020l

$\displaystyle\frac{1}{r^{2}}-e^{-\lambda}\left[\frac{1}{r^{2}}-\frac{\lambda^{\prime}}{r}\right]$
$\displaystyle=$ $\displaystyle 8\pi\rho+E^{2}\,,$ (2)
$\displaystyle-\frac{1}{r^{2}}+e^{-\lambda}\left[\frac{1}{r^{2}}+\frac{\nu^{\prime}}{r}\right]$
$\displaystyle=$ $\displaystyle 8\pi p_{r}-E^{2}\,,$ (3)
$\displaystyle\frac{e^{-\lambda}}{4}\left[2\nu^{\prime\prime}+\nu^{\prime
2}-\lambda^{\prime}\nu^{\prime}+\frac{2(\nu^{\prime}-\lambda^{\prime})}{r}\right]$
$\displaystyle=$ $\displaystyle 8\pi p_{t}+E^{2}\,.$ (4)

where $\nu(r)\equiv 2\Phi(r)$ and $\exp(\lambda)\equiv(1-b(r)/r)^{-1}$. In
terms of $\Phi,\,Q$ and $b(r)$, they read

$\displaystyle\frac{b^{\prime}(r)}{r^{2}}$ $\displaystyle=$ $\displaystyle
8\pi\rho^{\text{eff}}\,,$ (5) $\displaystyle
2\Big{(}1-\frac{b(r)}{r}\Big{)}\frac{\Phi^{\prime}}{r}-\frac{b(r)}{r^{3}}$
$\displaystyle=$ $\displaystyle 8\pi p_{r}^{\text{eff}}\,,$ (6)
$\displaystyle\Big{(}1-\frac{b(r)}{r}\Big{)}\bigg{[}\Phi^{\prime\prime}+(\Phi^{\prime})^{2}-\frac{b^{\prime}(r)r-b(r)}{2r(r-b(r))}\Phi^{\prime}-\frac{b^{\prime}(r)r-b(r)}{2r^{2}(r-b(r))}+\frac{\Phi^{\prime}}{r}\bigg{]}$
$\displaystyle=$ $\displaystyle 8\pi p_{t}^{\text{eff}}\,.$ (7)

Here $p_{t}=p_{\theta}=p_{\phi}$ is assumed. For convenience, we have defined
effective parameters $\rho^{\rm eff}\equiv\rho+\tfrac{Q^{2}}{8\pi
r^{4}}\,,p_{r}^{\rm eff}\equiv p_{r}-\tfrac{Q^{2}}{8\pi r^{4}}$ and
$p_{t}^{\rm eff}\equiv p_{t}+\tfrac{Q^{2}}{8\pi r^{4}}$.

###  2.1 Charged Casimir wormholes without GUP

The Casimir effect was predicted theoretically by Hendrik B. G. Casimir
Casimir:1948dh  in 1948. The Casimir force is a time-honored example of the
mechanical effect of vacuum fluctuations. It is an interaction of a pair of
neutral, parallel conducting planes caused by the vacuum fluctuations of the
electromagnetic field, and however only in recent years reliable experimental
investigations have confirmed such a phenomenon Lamoreaux:1996wh . The Casimir
effect is a macroscopic quantum effect which causes the plates to attract each
other by negative energy. It was found that the energy per unit surface is
given

$\displaystyle{\cal E}=-\frac{\pi^{2}\hbar c}{720a^{3}}\,,$ (8)

where $a$ is a distance between plates along the $z$-axis, the direction
perpendicular to the plate. The full expression of the energy-momentum tensor
for the Casimir effect and a suitable form applying in TW can be found in
Ref.Garattini:2019ivd  and we do not repeat it here. From now on we will
consider $\hbar=1=c$, unless otherwise stated. Consequently, the finite force
per unit area acting between the plates can be determined to obtain ${\cal
F}=-d{\cal E}(a)/da=-3{\cal E}(a)/a$ producing also a pressure of the form
$p=-3\pi^{2}/720a^{3}$. Defining an equation-of-state (EoS) parameter
$\omega=p/\rho$, in the case of Casimir energy there is a natural EoS
establishing fundamental relationship by choosing $\omega=3$. Therefore, we
obtain the energy density of the form $\rho=-\pi^{2}/720a^{3}$. It has been
proven in Refs.Boyer:1968uf  that there is an existence of the positive energy
of the Casimir effect in a conducting spherical shell with the radius $r$
leading to a thermodynamic instability. Consequently, the author of
Garattini:2019ivd  has extended the study began by Morris, Thorne and
Yurtsever in Ref.Morris:1988cz  and subsequently explored by Visser Visser1995
on the nagative Casimir energy as a possible source for constructing a TW. For
more detail discussions and mathematical derivations on the Casimir energy in
TW, we refer to Ref. Visser1995 .

In this section, we use the Casimir energy density to figure out the shape
function $b(r)$. We are also interested in deriving the equation of state
allowing to connect pressure with energy density for a given wormhole
geometry. In other words, we fix the geometry parameters using different
redshift functions $\Phi$ and then examine what the EoS parameter in the
corresponding case is. Having assumed a variable separation between the
plates, we write the energy density as $\rho=-\pi^{2}/720r^{4}$.

####  2.1.1 $\Phi(r)={\rm const.}$

We begin with the simplest case in which a model is described by $\Phi={\rm
const.}$ Morris:1988cz , a spacetime with no tidal forces, namely
$\Phi^{\prime}(r)=0$. In other words, this is asymptotically flat wormhole
spacetime. From (5), we can simply solve to obtain

$\displaystyle
b(r)=r_{0}+\frac{\pi^{3}-90Q^{2}}{90}\left(\frac{1}{r}-\frac{1}{r_{0}}\right),$
(9)

where the throat condition $b(r_{0})=r_{0}$ has been imposed. Using a
condition ($i$), upper-bound values of $Q$ are constrained to be

$\displaystyle r>r_{0}\land|Q|<\frac{\sqrt{90rr_{0}+\pi^{3}}}{3\sqrt{10}}\,,$
(10)

Notice from Eq.(10) that values of $Q$ increase when $r$ increases. Important
behaviors of $b(r)$ are displayed in Fig.(1). The asymptotically flat metric
can be seen also from Fig.(1) for $b^{\prime}(r)$.

Figure 1: We show behaviors of $b(r)$ (upper left), $b^{\prime}(r)$ (upper
right) and $b^{\prime}(r)r-b(r)$ (lower one) without GUP corrections of
charged Casimir wormhole against $r$. We have used $r_{0}=1.0$, $Q=0.0$ (red),
$Q=0.25$ (blue) and $Q=0.50$ (green).

The scaling of coordinate so that $\exp(2\Phi)dt^{2}\rightarrow dt^{2}$ is
introduced since $\exp(2\Phi)={\rm const.}$. Substituting $b(r)$ into Eq.(1),
the wormhole metric becomes

$\displaystyle
ds^{2}=-dt^{2}+\frac{1}{1-\frac{r_{0}}{r}-\frac{\pi^{3}-90Q^{2}}{90r}\left(\frac{1}{r}-\frac{1}{r_{0}}\right)}dr^{2}+r^{2}d\Omega^{2}\,,$
(11)

Clearly, we can simply show that $\lim_{x\to\infty}b(r)/r\rightarrow 0$.
Defining the EoS $p^{\rm eff}_{r}(r)=\omega_{r}(r)\rho^{\rm eff}(r)$ Azreg-
Ainou:2014dwa ; Moraes:2017dbs , and assuming $\Phi^{\prime}(r)=0$, we obtain

$\displaystyle-\frac{b(r)}{r^{3}}=8\pi
p_{r}^{\text{eff}}=8\pi\omega_{r}(r)\rho^{\rm eff}\,.$ (12)

We then solve the above equation for the EoS parameter to obtain

$\displaystyle\omega_{r}(r)=\frac{r_{0}\left(-180Q^{2}+90rr_{0}+\pi^{3}\right)-\left(\pi^{3}-90Q^{2}\right)r}{\pi^{3}r_{0}}\,.$
(13)

Figure 2: We show behaviors of $\omega_{r}(r)$ of charged Casimir wormhole
without GUP corrections against $r$. We have used $r_{0}=1.0$, $Q=0.0$ (red),
and $Q=0.50$ (blue).

The behavior of $\omega_{r}(r)$ of charged Casimir wormhole against $r$ is
displayed in Fig.(2)

####  2.1.2 $\Phi(r)=\frac{r_{0}}{r}$

In this form of $\Phi(r)$ suggested by Ref.Rahaman2019 , we do mainly focus on
deriving the EoS parameters. Here for given $\Phi$ we consider $p_{r}$ and
$p_{t}$.

#### EoS $p^{\rm eff}_{r}(r)=\omega_{r}(r)\rho^{\rm eff}(r)$

In this subsection, let’s first define the EoS parameter written in the form
$p^{\rm eff}_{r}(r)=\omega_{r}(r)\rho^{\rm eff}(r)$ Azreg-Ainou:2014dwa ;
Moraes:2017dbs . From the Einstein’s field given in Eq.(6), then we solve the
resulting relation to obtain

$\displaystyle\omega_{r}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi^{3}rr_{0}}\bigg{[}r_{0}\bigg{(}3\left(\pi^{3}-120Q^{2}\right)r-2r_{0}\left(-45\left(2Q^{2}+3r^{2}\right)+90rr_{0}+\pi^{3}\right)\bigg{)}$
(14)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad-\left(\pi^{3}-90Q^{2}\right)r^{2}\bigg{]}\,.$

#### EoS $p^{\rm eff}_{t}(r)=\omega_{t}(r)p^{\rm eff}_{r}(r)$

In the second type of the EoS parameter, we consider the case in which $p^{\rm
eff}_{t}(r)=\omega_{t}(r)p^{\rm eff}_{r}(r)$ Azreg-Ainou:2014dwa ;
Moraes:2017dbs , where $\omega_{t}(r)$ is as an arbitrary function of $r$. In
this case, we combine Eq.(6) with Eq.(7) and find the following relation:

$\displaystyle
r\bigg{(}\left(r\Phi^{\prime}(r)+1\right)\left(-rb^{\prime}(r)+2r(r-b(r))\Phi^{\prime}(r)+b(r)\right)+2r^{2}(r-b(r))\Phi^{\prime\prime}(r)\bigg{)}$
$\displaystyle-2\bigg{(}\omega_{t}\big{(}2r^{2}(r-b(r))\Phi^{\prime}(r)-rb(r)+Q^{2}\big{)}+Q^{2}\bigg{)}=0\,.$
(15)

Again we are going to use the shape function (9) for the EoS parameter and
then we obtain

$\displaystyle\omega_{t}(r)=\frac{r^{2}\left(\alpha_{1}-\pi^{3}\left(r-2r_{0}\right)\right)+r_{0}r\left(\pi^{3}\left(3r-4r_{0}\right)-\alpha_{3}\right)+{\cal
B}}{2r\left(r\left(\pi^{3}\left(r-r_{0}\right)-\alpha_{1}\right)-2\alpha_{2}\left(r-r_{0}\right)r_{0}\right)}\,,$
(16)

where we have defined new parameters

$\displaystyle{\cal B}$ $\displaystyle=$ $\displaystyle
2\alpha_{2}\left(r-r_{0}\right)r_{0}^{2}\,,$ $\displaystyle\alpha_{1}$
$\displaystyle=$ $\displaystyle 90\left(Q^{2}(r-2r_{0})+rr_{0}^{2}\right)\,,$
$\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle-90Q^{2}+90rr_{0}+\pi^{3}\,,$ $\displaystyle\alpha_{3}$
$\displaystyle=$ $\displaystyle
90\left(Q^{2}(3r-4r_{0})+rr_{0}(3r_{0}-2r)\right)\,.$

Figure 3: We show the EoS parameterr of $\omega_{r}(r)$ (left panel) and
$\omega_{t}(r)$ (right panel) of charged Casimir wormhole against $r$ for a
non-constant redshift function $\Phi(r)=\tfrac{r_{0}}{r}$. We have used
$r_{0}=1.0$, $Q=0.0$ (red), and $Q=0.50$ (blue).

We display the behavior of the EoS parameterr $\omega_{t}(r)$ of charged
Casimir wormhole charaterized in Fig.(3)

####  2.1.3 $\Phi(r)=\tfrac{1}{2}\ln\left(1+\frac{\gamma^{2}}{r^{2}}\right)$

Another interesting example is the following wormhole metric given by

$\displaystyle
ds^{2}=-\bigg{(}1+\frac{\gamma^{2}}{r^{2}}\bigg{)}dt^{2}+\frac{1}{1-\frac{b(r)}{r}}dr^{2}+r^{2}d\Omega^{2}\,,$
(17)

where $\gamma$ is some positive parameter with $r\geq r_{0}$. This form of
$\Phi$ was proposed in Ref.Rahaman2019 . As we presented in the preceeding
subsection, we can also assume the EoS of the form $p^{\rm
eff}_{r}(r)=\omega_{r}(r)\,\rho^{\rm eff}(r)$ Azreg-Ainou:2014dwa ;
Moraes:2017dbs . Similarly, in this case, we find

$\displaystyle\omega_{r}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{3\pi^{3}r^{2}r_{0}^{3}\left(\gamma^{2}+r^{2}\right)}\Big{(}270r^{3}r_{0}^{2}\left(r\left(Q^{2}\left(r-2r_{0}\right)+rr_{0}^{2}\right)-\gamma^{2}\left(Q^{2}+r_{0}\left(r_{0}-2r\right)\right)\right)$
(18)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad-3\pi^{3}r^{2}\left(r-r_{0}\right)r_{0}^{2}(r-\gamma)(\gamma+r)\Big{)}\,.$

In the same manner with the preceding scenario, we in this case consider the
EoS parameter which is of the form $p^{\rm eff}_{t}(r)=\omega_{t}(r)\,p^{\rm
eff}_{r}(r)$ Azreg-Ainou:2014dwa ; Moraes:2017dbs , where $\omega_{t}(r)$ is
as an arbitrary function of $r$. In this case, combining Eq.(6) with Eq.(7),
we find the following equation:

$\displaystyle\omega_{t}(r)=\frac{rb(r)\left(\gamma^{2}r_{0}\left(\gamma^{2}+5r^{2}+2\gamma^{2}r_{0}\right)-\left(\gamma^{2}+r^{2}\right)^{2}\right)+{\cal
H}_{1}-2\gamma^{4}r^{2}r_{0}^{2}}{2\left(\gamma^{2}+r^{2}\right)\left(\left(\gamma^{2}+r^{2}\right)\left(rb(r)-Q^{2}\right)+2\gamma^{2}rr_{0}(r-b(r))\right)}\,,$
(19)

where we have defined a new parameter

$\displaystyle{\cal
H}_{1}=\left(\gamma^{2}+r^{2}\right)^{2}\left(r^{2}b^{\prime}(r)+2Q^{2}\right)-\gamma^{2}r^{2}r_{0}\left(\left(\gamma^{2}+r^{2}\right)b^{\prime}(r)+4r^{2}\right)\,.$
(20)

Figure 4: We show the EoS parameterr $\omega_{r}(r)$ and $\omega_{t}(r)$ of
charged Casimir wormhole against $r$ for a non-constant redshift function
$\exp(2\Phi(r))=1+\tfrac{\gamma^{2}}{r^{2}}$. We have used
$r_{0}=1.0,\,\gamma=1$, $Q=0.0$ (red), and $Q=0.50$ (blue).

We display the behavior of the EoS parameters $\omega_{r}(r)$ and
$\omega_{t}(r)$ of charged Casimir wormhole characterized in Fig.(4)

###  2.2 Charged Casimir wormholes with GUP

In the starting point, we will follow the work present in Ref.Jusufi:2020rpw
for elaborating in more details about the GUP corrected energy density to
construct Casimir wormholes. To be more precise, the authors of
Ref.Jusufi:2020rpw  have considered three types of the GUP relations: (1) the
Kempf, Mangano and Mann (KMM) model, (2) the Detournay, Gabriel and Spindel
(DGS) model, and (3) the so called type II model for GUP principle. In a
compact form, the authors of Ref.Jusufi:2020rpw  wrote the renormalized
energies per unit surface area of the plates for three GUP cases as

$\displaystyle{\cal
E}=-\frac{\pi^{2}\hbar}{720a^{3}}\bigg{[}1+C_{i}\bigg{(}\frac{\hbar\sqrt{\beta}}{a}\bigg{)}^{2}\bigg{]}\,,$
(21)

where we have defined $C_{i}$ as

$\displaystyle
C_{1}=\pi^{2}\bigg{(}\frac{28+3\sqrt{10}}{14}\bigg{)},\,C_{2}=4\pi^{2}\bigg{(}\frac{3+\pi^{2}}{21}\bigg{)},\,C_{3}=\frac{2\pi^{2}}{3}\,.$
(22)

Therefore, the force per unit surface area can be determined to obtain

$\displaystyle{\cal F}=-\frac{d{\cal
E}}{da}=-\frac{3\pi^{2}\hbar}{720a^{4}}\bigg{[}1+C_{i}\bigg{(}\frac{\hbar\sqrt{\beta}}{a}\bigg{)}^{2}\bigg{]}\,.$
(23)

Hence the pressure can be simply obtained as

$\displaystyle p={\cal
F}=-\frac{3\pi^{2}\hbar}{720a^{4}}\bigg{[}1+C_{i}\bigg{(}\frac{\hbar\sqrt{\beta}}{a}\bigg{)}^{2}\bigg{]}\,.$
(24)

By defining an EoS, $p=\omega\rho$, we obtain the GUP corrected energy density
in a compact form as

$\displaystyle\rho=-\frac{\pi^{2}\hbar}{720a^{4}}\bigg{[}1+D_{i}\bigg{(}\frac{\hbar\sqrt{\beta}}{a}\bigg{)}^{2}\bigg{]}\,,$
(25)

with $D_{i}=5C_{i}/3$. Notice that in the case of Casimir energy there is a
natural EoS establishing fundamental relationship by choosing $\omega=3$.
Practically, the energy density can be employed to quantify the shape function
$b(r)$. Additionally, the EoS with a specific value for $\omega$ can be used
to determine the red-shift function.

####  2.2.1 $\Phi(r)={\rm const.}$

One of the simplest case is a model with $\Phi={\rm const.}$ Morris:1988cz ,
a.k.a., a spacetime with no tidal forces, namely $\Phi^{\prime}(r)=0$. In
other words, this is asymptotically flat wormhole spacetime. From (5), we find

$\displaystyle
b(r)=r_{0}+\frac{\pi^{3}-90Q^{2}}{90}\left(\frac{1}{r}-\frac{1}{r_{0}}\right)+\frac{\pi^{3}\beta
D_{i}}{270}\left(\frac{1}{r^{3}}-\frac{1}{r^{3}_{0}}\right),$ (26)

where the throat condition $b(r_{0})=r_{0}$ has been imposed. Behavior of
$b(r)$ is displayed in Fig.(6). Using a condition ($i$), we find the
conditions

$\displaystyle
r>r_{0}\land|Q|<\frac{1}{3\sqrt{30}}\bigg{[}\sqrt{\pi^{3}\left(\frac{\beta
D_{i}}{r^{2}}+3\right)+\frac{\pi^{3}\beta D_{i}}{rr_{0}}+\frac{\pi^{3}\beta
D_{i}}{r_{0}^{2}}+270rr_{0}}\bigg{]}\,.$ (27)

Solving equation for the EoS parameter, we obtain in this case

$\displaystyle\omega(r)=\frac{-\pi^{3}\beta D_{i}r^{3}+\pi^{3}\beta
D_{i}r_{0}^{3}+270Q^{2}r^{3}r_{0}^{2}-270Q^{2}r^{2}r_{0}^{3}+{\cal
F}}{3\pi^{3}r_{0}^{3}\left(\beta D_{i}+r^{2}\right)}\,,$ (28)

where we have defined a new parameter

$\displaystyle{\cal
F}=270r^{3}r_{0}^{4}-3\pi^{3}r^{3}r_{0}^{2}+3\pi^{3}r^{2}r_{0}^{3}\,.$ (29)

The behavior of $\omega_{r}(r)$ of charged Casimir wormhole with GUP
corrections against $r$ is shown in Fig.5.

Figure 5: We show behaviors of $\omega(r)$ of charged Casimir wormhole with
GUP corrections against $r$. We have used
$r_{0}=1.0,\,D_{i}=D_{1},\,\beta=0.1$, $Q=0.0$ (red), and $Q=0.50$ (blue).

Figure 6: We show the behavior of $b(r)$ (upper-left panel), $b^{\prime}(r)$
(lower-left panel), $b(r)/r$ (upper-right panel) and $b^{\prime}(r)r-b(r)$
(lower-right panel) of charged Casimir wormhole with GUP corrections against
$r$ using $\Phi={\rm const.}$. We have used $r_{0}=1.0,\,D_{i}=D_{1}$, $Q=0.0$
(red), and $Q=0.50$ (blue).

####  2.2.2 $\Phi(r)=\frac{r_{0}}{r}$

In this subsection, we first define the EoS $p^{\rm
eff}_{r}(r)=\omega_{r}(r)\rho^{\rm eff}(r)$ Azreg-Ainou:2014dwa ;
Moraes:2017dbs . Then from the Einstein’s field equations (6), we come up with

$\displaystyle\frac{rb(r)-\frac{1}{90}\pi^{3}\omega_{r}\left(\frac{\beta
D_{i}}{r^{2}}+1\right)-Q^{2}}{2r^{2}\left(-\frac{1}{270}\pi^{3}\beta
D_{i}\left(\frac{1}{r^{3}}-\frac{1}{r_{0}^{3}}\right)-\frac{1}{90}\left(\pi^{3}-90Q^{2}\right)\left(\frac{1}{r}-\frac{1}{r_{0}}\right)+r-r_{0}\right)}+\frac{r_{0}}{r^{2}}=0\,.$
(30)

We then solve the above equation on yield

$\displaystyle\omega_{r}=\frac{r_{0}\left(2\pi^{3}\beta
D_{i}r^{3}+r_{0}\left(G_{1}r_{0}-3\left(\pi^{3}-90Q^{2}\right)r^{4}\right)\right)-\pi^{3}\beta
D_{i}r^{4}}{3\pi^{3}r_{0}^{3}\left(\beta D_{i}r+r^{3}\right)}\,,$ (31)

where we have defined a new parameter

$\displaystyle G_{1}$ $\displaystyle=$ $\displaystyle-2r_{0}\left(\pi^{3}\beta
D_{i}+3r^{2}\left(\pi^{3}-45\left(2Q^{2}+3r^{2}\right)\right)+270r_{0}r^{3}\right)$
$\displaystyle+\pi^{3}\beta D_{i}r+9\left(\pi^{3}-120Q^{2}\right)r^{3}\,.$

We can follow the basic formalism given in the preceding section and consider
the scenario in which the EoS parameter is written in the form $p^{\rm
eff}_{t}(r)=\omega_{t}(r)\,p^{\rm eff}_{r}(r)$ Azreg-Ainou:2014dwa ;
Moraes:2017dbs , where $\omega_{t}(r)$ is as an arbitrary function of $r$. In
this case, we obtain

$\displaystyle\omega_{t}=\frac{r\left(r\left(r-r_{0}\right)b^{\prime}(r)+2Q^{2}-2r_{0}\left(r+r_{0}\right)\right)+\left(-r^{2}+3r_{0}r+2r_{0}^{2}\right)b(r)}{2r\left(rb(r)+2r_{0}(r-b(r))-Q^{2}\right)}\,.$
(32)

We display the behavior of the EoS parameter $\omega_{r}(r)$ and
$\omega_{t}(r)$ of charged Casimir wormhole charaterized in Fig.(7).

Figure 7: We show the EoS parameter $\omega_{r}(r)$ and $\omega_{t}(r)$ of
charged Casimir wormhole against $r$ for a non-constant redshift function
$\Phi(r)=r_{0}/r$. We have used $r_{0}=1.0,\,\beta=0.1,\,D_{i}=D_{1}$, $Q=0.0$
(red), and $Q=0.50$ (blue).

####  2.2.3 $\Phi(r)=\frac{1}{2}\ln\left(1+\tfrac{\gamma^{2}}{r^{2}}\right)$

In the last example of the redshift function, we consider the EoS parameter
which is given by $p^{\rm eff}_{r}(r)=\omega_{r}(r)\rho^{\rm eff}(r)$ Azreg-
Ainou:2014dwa ; Moraes:2017dbs . Considering the Einstein’s field equations
(6), we obtain

$\displaystyle\omega_{r}=\frac{\pi^{3}\beta
D_{i}r^{3}\left(\gamma^{2}-r^{2}\right)+r_{0}^{2}\left(F_{1}r_{0}-3\left(\pi^{3}-90Q^{2}\right)r^{3}(r-\gamma)(\gamma+r)\right)}{3\pi^{3}r_{0}^{3}\left(\gamma^{2}+r^{2}\right)\left(\beta
D_{i}+r^{2}\right)}\,,$ (33)

where we have defined a new parameter

$\displaystyle F_{1}=\pi^{3}(r-\gamma)(\gamma+r)\left(\beta
D_{i}+3r^{2}\right)+540r^{4}\left(\gamma^{2}-Q^{2}\right)+270r_{0}r^{3}(r-\gamma)(\gamma+r)\,.$

Let us now define the EoS of the form $p^{\rm
eff}_{t}(r)=\omega_{t}(r)\,p^{\rm eff}_{r}(r)$ Azreg-Ainou:2014dwa ;
Moraes:2017dbs , where $\omega_{t}(r)$ is as an arbitrary function of $r$. In
this case, we find the following relation:

$\displaystyle\omega_{t}=-\frac{r^{4}\left(\gamma^{2}+r^{2}\right)b^{\prime}(r)+b(r)\left(-r^{5}+3\gamma^{2}r^{3}+2\gamma^{4}r\right)+{\cal
O}_{1}}{2\left(\gamma^{2}+r^{2}\right)\left(rb(r)\left(\gamma^{2}-r^{2}\right)+Q^{2}\left(\gamma^{2}+r^{2}\right)-2\gamma^{2}r^{2}\right)}\,,$
(34)

where we have defined ${\cal
O}_{1}=2Q^{2}\left(\gamma^{2}+r^{2}\right)^{2}-2\gamma^{2}r^{2}\left(\gamma^{2}+2r^{2}\right)$.
We display the behavior of the EoS parameter $\omega_{r}(r)$ and
$\omega_{t}(r)$ of charged Casimir wormhole charaterized in Fig.(8).

Figure 8: We show the EoS parameterr $\omega_{r}(r)$ and $\omega_{t}(r)$ of
charged Casimir wormhole against $r$ for a non-constant redshift function
$\Phi(r)=\tfrac{1}{2}\ln{(1+\tfrac{\gamma^{2}}{r^{2}})}$. We have used
$r_{0}=1.0,\,\beta=0.1,\,D_{i}=D_{1},\,\gamma=1$, $Q=0.0$ (red), and $Q=0.50$
(blue).

##  3 Embedding diagram

In this subsection, we analyse the embedding diagrams to represent the charged
Casimir wormhole without and with GUP corrections by considering an equatorial
slice $\theta=\pi/2$ at some fix moment in time $t={\rm constant}$. To do so,
we consider the metric which can be written as

$\displaystyle ds^{2}=\frac{dr^{2}}{1-\frac{b(r)}{r}}+r^{2}d\phi^{2}\,.$ (35)

We embed the metric (1) into three-dimensional Euclidean space to visualize
this slice and the spacetime can be written in cylindrical coordinates as

$\displaystyle ds^{2}=dz^{2}+dr^{2}+r^{2}d\phi^{2}\,.$ (36)

From the last two equations we find that

$\displaystyle\frac{dz}{dr}=\pm\sqrt{\frac{r}{r-b(r)}-1}\,,$ (37)

where $b(r)$ is given by Eq.(9) and Eq.(26). Invoking numerical techniques
allows us to illustrate the wormhole shape given in Fig.9.

Figure 9: The charged Casimir wormhole without GUP corrections (left panel)
and with GUP corrections (right panel) embedded in a three-dimensional
Euclidean space. Left panel: We have used $r_{0}=1,\,Q=0.25$ for $b(r)$ given
in Eq.(9). Right panel: We have used
$r_{0}=1,\,Q=0.25,\,\beta=0.1,\,D_{i}=D_{1}$ for $b(r)$ given in Eq.(26).

##  4 Energy conditions

We can continue our discussion on the issue of energy conditions and make some
regional plots to check the validity of all energy conditions. In this work,
we consider the three types of energy conditions to examine charged wormholes.
The first one is null energy condition (NEC) which determines the non-negative
value of energy momentum tensor contracting with null vector $k_{\mu}$ where
$k^{\mu}k_{\mu}=0$. That is to say $T_{\mu\nu}k^{\mu}k^{\nu}\geq 0$. We have

$\displaystyle\rho^{\text{eff}}+p_{r}^{\text{eff}}\geq 0\,.$

Note that NEC can be interpreted as the energy of particles traveling along a
null geodesic, such as photon and massless particles, which must be non-
negative. The energy density or pressure can be negative as long as their
summation is still equal or greater than zero. In particular we recall that
the weak energy condition (WEC) is defined by $T_{\mu\nu}t^{\mu}t^{\nu}\geq
0$. It determines the non-negative value of energy momentum tensor contracting
with timelike vector $t_{\mu}$ where $t^{\mu}t_{\mu}<0$, i.e.,

$\displaystyle\rho^{\text{eff}}+p_{r}^{\text{eff}}\geq 0\,.$

The strong energy condition (SEC) decomposed into the following conditions:

$\displaystyle\rho^{\text{eff}}+2p_{t}^{\text{eff}}\geq 0\quad{\rm
and}\quad\rho^{\text{eff}}+p_{r}^{\text{eff}}+2p_{t}^{\text{eff}}\geq 0\,.$
(38)

For the sake of completeness, we also need to test the dominant energy
condition (DEC) given by

$\displaystyle\rho^{\text{eff}}-|p_{r}^{\text{eff}}|\geq 0\quad{\rm
and}\quad\rho^{\text{eff}}-|p_{t}^{\text{eff}}|\geq 0\,.$ (39)

Notice that SEC covers NEC and avoids excessively large negative pressure.
However, the traversable wormholes in some particular models Morris:1988cz
need the exotic matter which violates the energy conditions.

###  4.1 Casimir wormholes without GUP

####  4.1.1 $\Phi(r)=\frac{r_{0}}{r}$

Given the redshift function $\Phi=r_{0}/r$ and the shape function given
Eq.(9), we can compute the energy-momentum components. We find for the radial
component

$\displaystyle p^{\rm eff}_{r}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{720\pi
r^{5}r_{0}}\bigg{[}\left(\pi^{3}-90Q^{2}\right)r^{2}+r_{0}\big{(}2r_{0}\big{(}-45\left(2Q^{2}+3r^{2}\right)$
(40)
$\displaystyle+90rr_{0}+\pi^{3}\big{)}-3\left(\pi^{3}-120Q^{2}\right)r\big{)}\bigg{]}\,,$

and for the tangential component

$\displaystyle p^{\rm
eff}_{t}(r)=-\frac{C_{3}(r-1)\left(C_{2}r_{0}+\left(\pi^{3}-90Q^{2}\right)r^{2}\right)}{C_{1}}\,,$
(41)

where

$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle 1440\pi
r^{6}\left(\pi^{3}(r-2)(r-1)+90r(2-3r)\right)r_{0}\,,$ $\displaystyle C_{2}$
$\displaystyle=$ $\displaystyle
2r_{0}\left(-45\left(2Q^{2}+3r^{2}\right)+90rr_{0}+\pi^{3}\right)-3\left(\pi^{3}-120Q^{2}\right)r\,,$
$\displaystyle C_{3}$ $\displaystyle=$
$\displaystyle\pi^{3}((r-4)r-2)-90r(3r+2)\,.$

Figure 10: We show the variation of $\rho^{\rm eff}(r),\,\rho^{\rm
eff}(r)+p^{\rm eff}_{r}(r),\,\rho^{\rm eff}(r)+2p^{\rm eff}_{t}(r),\,\rho^{\rm
eff}(r)+p^{\rm eff}_{r}(r)+2p^{\rm eff}_{t}(r),\,\rho^{\rm eff}(r)-|p^{\rm
eff}_{r}(r)|$ and $\rho^{\rm eff}(r)-|p^{\rm eff}_{t}(r)|$ as a function of
$r$ using $\Phi=r_{0}/r$ of charged Casimir wormhole. We have used
$r_{0}=1.0,\,Q=0.0$ (red), and $Q=0.50$ (blue).

####  4.1.2 $\Phi(r)=\frac{1}{2}\ln\left(1+\tfrac{\gamma^{2}}{r^{2}}\right)$

In this case, we compute the energy-momentum components to obtain the radial
component as

$\displaystyle p^{\rm
eff}_{r}(r)=\frac{D_{0}-r_{0}\left(D_{1}+\pi^{3}(r-\gamma)(\gamma+r)\right)}{720\pi
r^{4}r_{0}\left(\gamma^{2}+r^{2}\right)}\,,$ (42)

where

$\displaystyle D_{0}$ $\displaystyle=$
$\displaystyle\left(\pi^{3}-90Q^{2}\right)r(r-\gamma)(\gamma+r)\,,$
$\displaystyle D_{1}$ $\displaystyle=$ $\displaystyle
180r^{2}\left(\gamma^{2}-Q^{2}\right)+90r_{0}r(r-\gamma)(\gamma+r)\,,$

and the tangential component

$\displaystyle p^{\rm
eff}_{t}(r)=\frac{E_{2}\left(E_{3}rb(r)-2\gamma^{4}r^{2}r_{0}^{2}+E_{4}r_{0}+e_{5}\right)}{16\pi
E_{1}r^{4}\left(\gamma^{2}+r^{2}\right)^{2}}\,,$ (43)

where

$\displaystyle E_{0}$ $\displaystyle=$ $\displaystyle
180r^{2}(Q-\gamma)(\gamma+Q)+90r_{0}r\left(\gamma^{2}-r^{2}\right)+\pi^{3}\left(\gamma^{2}-r^{2}\right)\,,$
$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle\left(90\left(Q^{2}\left(r-2r_{0}\right)+rr_{0}^{2}\right)+\pi^{3}\left(r_{0}-r\right)\right)\left(\gamma^{2}+r^{2}\right)+2\gamma^{2}\left(r-r_{0}\right)r_{0}\left(-90Q^{2}+90rr_{0}+\pi^{3}\right)\,,$
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\left(\pi^{3}-90Q^{2}\right)r(r-\gamma)(\gamma+r)+E_{0}r_{0}\,,$
$\displaystyle E_{3}$ $\displaystyle=$
$\displaystyle\gamma^{2}r_{0}\left(\gamma^{2}+5r^{2}+2\gamma^{2}r_{0}\right)-\left(\gamma^{2}+r^{2}\right)^{2}\,,$
$\displaystyle E_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{90}\gamma^{2}\left(\pi^{3}\left(\gamma^{2}+r^{2}\right)-90\left(Q^{2}\left(\gamma^{2}+r^{2}\right)+4r^{4}\right)\right)\,,$
$\displaystyle E_{5}$ $\displaystyle=$
$\displaystyle-\frac{1}{90}\left(\pi^{3}-270Q^{2}\right)\left(\gamma^{2}+r^{2}\right)^{2}\,.$

Figure 11: We show the variation of $\rho^{\rm eff}(r),\,\rho^{\rm
eff}(r)+p^{\rm eff}_{r}(r),\,\rho^{\rm eff}(r)+2p^{\rm eff}_{t}(r),\,\rho^{\rm
eff}(r)+p^{\rm eff}_{r}(r)+2p^{\rm eff}_{t}(r),\,\rho^{\rm eff}(r)-|p^{\rm
eff}_{r}(r)|$ and $\rho^{\rm eff}(r)-|p^{\rm eff}_{t}(r)|$ as a function of
$r$ using $\exp(2\Phi(r))=1+\tfrac{\gamma^{2}}{r^{2}}$ of charged Casimir
wormhole. We have used $r_{0}=1.0\,,\gamma=1$, $Q=0.0$ (red), and $Q=0.50$
(blue).

We see from Fig.(10) for $\Phi=r_{0}/r$, and similarly Fig.(11) for
$\Phi(r)=\tfrac{1}{2}\ln\left(1+\tfrac{\gamma^{2}}{r^{2}}\right)$, NEC, WEC,
and SEC, are violated at the wormhole throat $r=r_{0}$. In other words, we
discover that in all plots at the wormhole throat $r=r_{0}$, we have for the
energy condition $(\rho^{\rm eff}+p^{\rm eff}_{r})|_{r_{0}=1}<0$, along with
the condition $(\rho^{\rm eff}+p^{\rm eff}_{r}+2p^{\rm
eff}_{t})|_{r_{0}=1}<0$, by arbitrary small values. Moreover, we find that
$\rho^{\rm eff}(r)-|p^{\rm eff}_{r}(r)|<0$ and $\rho^{\rm eff}(r)-|p^{\rm
eff}_{t}(r)|<0$ at the WH throat. However, as mentioned in Ref.Jusufi:2020rpw
from the quantum field theory’s point of view, quantum fluctuations are
thought to violate most energy conditions without any restrictions. Therefore,
wormholes may be stabilized by this quantum fluctuations.

###  4.2 Charged Casimir wormholes with GUP

####  4.2.1 $\Phi(r)=\frac{r_{0}}{r}$

In this case, we compute the energy-momentum components to obtain the radial
component as

$\displaystyle p^{\rm eff}_{r}(r)$ $\displaystyle=$
$\displaystyle\frac{r_{0}\left(2\pi^{3}\beta
D_{i}r^{3}+r_{0}\left(G_{1}r_{0}-3\left(\pi^{3}-90Q^{2}\right)r^{4}\right)\right)-\pi^{3}\beta
D_{i}r^{4}}{3\pi^{3}r_{0}^{3}\left(\beta D_{i}r+r^{3}\right)}$ (44)
$\displaystyle\times\bigg{[}\frac{Q^{2}}{8\pi
r^{4}}-\frac{\pi^{2}\left(\frac{\beta
D_{i}}{r^{2}}+1\right)}{720r^{4}}\bigg{]}\,,$

and the tangential component

$\displaystyle p^{\rm eff}_{t}(r)$ $\displaystyle=$
$\displaystyle\frac{r\left(r\left(r-r_{0}\right)b^{\prime}(r)+2Q^{2}-2r_{0}\left(r+r_{0}\right)\right)+\left(-r^{2}+3r_{0}r+2r_{0}^{2}\right)b(r)}{2r\left(rb(r)+2r_{0}(r-b(r))-Q^{2}\right)}p_{r}(r)\,,$
(45)

where $p_{r}(r)$ has been already given in Eq.(44) and $G_{1}$ is defined as

$\displaystyle G_{1}$ $\displaystyle=$ $\displaystyle-2r_{0}\left(\pi^{3}\beta
D_{i}+3r^{2}\left(\pi^{3}-45\left(2Q^{2}+3r^{2}\right)\right)+270r_{0}r^{3}\right)$
(46) $\displaystyle+\pi^{3}\beta
D_{i}r+9\left(\pi^{3}-120Q^{2}\right)r^{3}\,.$

Figure 12: We show the variation of $\rho(r)+p_{r}(r),\,\rho(r)+2p_{t}(r)$ and
$\rho(r)+p_{r}(r)+2p_{t}(r)$ as a function of $r$ using $\Phi(r)=r_{0}/r$ of
charged Casimir wormhole with GUP corrections. We have used
$r_{0}=1.0\,,\beta=0.01,\,D_{i}=D_{1}$, $Q=0.0$ (red), and $Q=0.50$ (blue).

####  4.2.2 $\Phi(r)=\frac{1}{2}\ln\left(1+\tfrac{\gamma^{2}}{r^{2}}\right)$

Given the redshift function $\exp(2\Phi(r))=1+\tfrac{\gamma^{2}}{r^{2}}$ and
the shape function given Eq.(26), we can compute the energy-momentum
components. We find for the radial component

$\displaystyle p^{\rm eff}_{r}(r)$ $\displaystyle=$
$\displaystyle\frac{\pi^{3}\beta
D_{i}r^{3}\left(\gamma^{2}-r^{2}\right)+r_{0}^{2}\left(F_{1}r_{0}-3\left(\pi^{3}-90Q^{2}\right)r^{3}(r-\gamma)(\gamma+r)\right)}{3\pi^{3}r_{0}^{3}\left(\gamma^{2}+r^{2}\right)\left(\beta
D_{i}+r^{2}\right)}$ (47)
$\displaystyle\times\bigg{(}-\frac{\pi^{2}\left(\frac{\beta
D_{i}}{r^{2}}+1\right)}{720r^{4}}+\frac{Q^{2}}{8\pi r^{4}}\bigg{)}\,,$

and for the tangential pressure

$\displaystyle p^{\rm eff}_{t}(t)$ $\displaystyle=$
$\displaystyle-\frac{r^{4}\left(\gamma^{2}+r^{2}\right)b^{\prime}(r)+b(r)\left(-r^{5}+3\gamma^{2}r^{3}+2\gamma^{4}r\right)+2Q^{2}\left(\gamma^{2}+r^{2}\right)^{2}-2\gamma^{2}r^{2}\left(\gamma^{2}+2r^{2}\right)}{2\left(\gamma^{2}+r^{2}\right)\left(rb(r)\left(\gamma^{2}-r^{2}\right)+Q^{2}\left(\gamma^{2}+r^{2}\right)-2\gamma^{2}r^{2}\right)}$
(48) $\displaystyle\times\frac{\pi^{3}\beta
D_{i}r^{3}\left(\gamma^{2}-r^{2}\right)+r_{0}^{2}\left(F_{1}r_{0}-3\left(\pi^{3}-90Q^{2}\right)r^{3}(r-\gamma)(\gamma+r)\right)}{3\pi^{3}r_{0}^{3}\left(\gamma^{2}+r^{2}\right)\left(\beta
D_{i}+r^{2}\right)}$
$\displaystyle\times\bigg{(}-\frac{\pi^{2}}{720r^{4}}\left(\frac{\beta
D_{i}}{r^{2}}+1\right)+\frac{Q^{2}}{8\pi r^{4}}\bigg{)}\,.$

Figure 13: We show the variation of $\rho^{\rm eff}(r)+p^{\rm
eff}_{r}(r),\,\rho^{\rm eff}(r)+2p^{\rm eff}_{t}(r),\,\rho^{\rm eff}(r)+p^{\rm
eff}_{r}(r)+2p^{\rm eff}_{t}(r),\,\rho^{\rm eff}(r)-|p^{\rm eff}_{r}(r)|$ and
$\rho^{\rm eff}(r)-|p^{\rm eff}_{t}(r)|$ as a function of $r$ using
$\exp(2\Phi(r))=1+\gamma^{2}/r^{2}$ of charged Casimir wormhole with GUP
corrections. We have used $r_{0}=1.0\,,\beta=0.01,\,D_{i}=D_{1}$, $Q=0.0$
(red), and $Q=0.50$ (blue).

In the case of adding the GUP corrections, we also see from Fig.(12) for
$\Phi=r_{0}/r$, and similarly Fig.(13) for
$\Phi(r)=\tfrac{1}{2}\ln\left(1+\tfrac{\gamma^{2}}{r^{2}}\right)$, NEC, WEC,
and SEC, are violated at the wormhole throat $r=r_{0}$. In other words, we
discover that in all plots at the wormhole throat $r=r_{0}$, we have for the
energy condition $(\rho^{\rm eff}+p^{\rm eff}_{r})|_{r_{0}=1}<0$, along with
the condition $(\rho^{\rm eff}+p^{\rm eff}_{r}+2p^{\rm
eff}_{t})|_{r_{0}=1}<0$, by arbitrary small values. Additionally, we see that
$\rho^{\rm eff}(r)-|p^{\rm eff}_{r}(r)|<0$ and $\rho^{\rm eff}(r)-|p^{\rm
eff}_{t}(r)|<0$ at the WH throat. We will perform the volume integration in
Sec.5 to estimate the amount of the total exotic matter that contains in the
wormholes.

##  5 Amount of exotic matter

In this section, we consider the “volume integral” which basically provides
information about the “total amount” of averaged null energy condition (ANEC)
violating matter in the spacetime. This quantity is related only to $\rho$ and
$p_{r}$, not to the transverse components. It is defined in terms of the
following definite integral as Jusufi:2020rpw

$\displaystyle I_{V}=2\int^{\infty}_{r_{0}}\big{(}\rho^{\rm eff}(r)+p^{\rm
eff}_{r}(r)\big{)}dV\,,$ (49)

where the volume is given by $dV=r^{2}drd\Omega$ and $d\Omega$ is solid angle.
Having introduced a cut-off $r=a$ such that the wormhole extends from $r_{0}$
to $a$ with $a\geq r_{0}$, we can simply write

$\displaystyle I_{V}=8\pi\int^{a}_{r_{0}}\big{(}\rho^{\rm eff}(r)+p^{\rm
eff}_{r}(r)\big{)}r^{2}dr\,.$ (50)

In case of $\Phi=r_{0}/r$, we obtain from the above integral

$\displaystyle I_{V}$ $\displaystyle=$
$\displaystyle\frac{1}{68040r^{4}r_{0}^{3}}\Big{(}6r^{4}\log(r)\left(5\pi^{5}\left(3\sqrt{10}+28\right)\beta-11340r_{0}^{2}\left(Q^{2}+3r_{0}^{2}\right)+126\pi^{3}r_{0}^{2}\right)$
(51)
$\displaystyle+r_{0}\Big{(}-68040r^{2}r_{0}^{2}\left(Q^{2}\left(4r-r_{0}\right)+2rr_{0}^{2}\right)+5\pi^{5}\left(3\sqrt{10}+28\right)\beta\left(12r^{3}+8r_{0}^{2}r-3r_{0}^{3}\right)$
$\displaystyle+756\pi^{3}r^{2}\left(4r-r_{0}\right)r_{0}^{2}\Big{)}\Big{)}\biggr{|}_{r}^{a}$

Moreover, we can solve for $I_{V}$ of
$\exp(2\Phi(r))=1+\tfrac{\gamma^{2}}{r^{2}}$ to obtain

$\displaystyle I_{V}$ $\displaystyle=$
$\displaystyle\frac{1}{11340r_{0}^{3}}\Big{(}\log\left(r^{2}+1\right)\left(5\pi^{5}\left(3\sqrt{10}+28\right)\beta-11340r_{0}^{2}\left(Q^{2}+r_{0}^{2}\right)+126\pi^{3}r_{0}^{2}\right)$
(52)
$\displaystyle\quad\quad\quad\quad+\log(r)\left(-5\pi^{5}\left(3\sqrt{10}+28\right)\beta+11340r_{0}^{2}\left(Q^{2}+r_{0}^{2}\right)-126\pi^{3}r_{0}^{2}\right)$
$\displaystyle\quad\quad\quad\quad+2r_{0}^{3}\left(5\pi^{5}\left(3\sqrt{10}+28\right)\beta+11340\left(Q^{2}-1\right)-126\pi^{3}\right)\tan^{-1}(r)$
$\displaystyle\quad\quad\quad\quad+\frac{10\pi^{5}\left(3\sqrt{10}+28\right)\beta
r_{0}^{3}}{3r^{3}}+\frac{10\pi^{5}\left(3\sqrt{10}+28\right)\beta
r_{0}^{3}}{r}\Big{)}\biggr{|}_{r}^{a}\,,$

Figure 14: We display the variation of $I_{V}$ against $r$ for the case of
$\Phi(r)=r_{0}/r$ (left panel) where we have used $r_{0}=1,\,Q=0.5$ and
$\exp(2\Phi(r))=1+\tfrac{\gamma^{2}}{r^{2}}$ (right panel) where we have used
$r_{0}=1,\,Q=0.5,\,\gamma=1$. For all plots, we have considered $r\geq r_{0}$
and varied values of $\beta$.

In this specific case, we have used
$D_{i}=D_{1}=\frac{5}{42}\left(3\sqrt{10}+28\right)\pi^{2},\,Q=0.5,\,\gamma=1$
and $r_{0}=1$ for two models.

We observe from Fig.14 that the quantity $I_{V}$ is negative in both cases.
This demonstrates the existence of spacetime geometries containing charged
traversable wormholes which are supported by arbitrarily small exotic matter.
Such small violations perhaps may be linked to the quantum fluctuations.

##  6 Existence of exotic matter and Speed of sound

For completeness, one could also add more information of the exoticity. To
ensure the presence of exotic fluid near WH throat, we can use the so-called
exoticity parameter MVisser1995 ; Capozziello:2011nr  when $\Sigma(r)>0$
implying that exotic matter is present at that point of spacetime. We consider

$\displaystyle\Sigma\equiv-\frac{\rho^{\text{eff}}+p_{r}^{\text{eff}}}{|\rho^{\text{eff}}|}.$
(53)

The existence of an exotic matter is required in the range that $\Sigma>0$. We
find from Eq.(53) for $\Phi=r_{0}/r$:

$\displaystyle\Sigma$ $\displaystyle=$
$\displaystyle\frac{r^{4}}{126r^{7}r_{0}^{3}\left|-90Q^{2}+\pi^{3}+\frac{5\pi^{5}\beta\left(3\sqrt{10}+28\right)}{42r^{2}}\right|}\Big{(}126r^{2}r_{0}^{2}\big{(}90\big{(}Q^{2}\big{(}r^{2}-4r_{0}r+2r_{0}^{2}\big{)}$
(54)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+rr_{0}^{2}\big{(}3r-2r_{0}\big{)}\big{)}-\pi^{3}\big{(}r^{2}-4r_{0}r+2r_{0}^{2}\big{)}\big{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-5\left(3\sqrt{10}+28\right)\pi^{5}\beta\Big{(}r^{4}-2r_{0}r^{3}-4r_{0}^{3}r+2r_{0}^{4}\Big{)}\Big{)}.$

Considering points near the WH throat, $r\rightarrow r_{0}$, we find for this
case with $\Sigma>0$

$\displaystyle\beta>\frac{3780Q^{2}-42\pi^{3}}{15\pi^{5}\sqrt{10}+140\pi^{5}},\,$
(55)

which yields $Q>\pi^{3/2}/3\sqrt{10}$. Additionally, for
$\exp(2\Phi)=1+\gamma_{2}/r^{2}$, we have

$\displaystyle\Sigma$ $\displaystyle=$
$\displaystyle\frac{1}{126r^{6}\left(r^{2}+1\right)r_{0}^{3}\left|-90Q^{2}+\pi^{3}+\frac{5\pi^{5}\beta\left(3\sqrt{10}+28\right)}{42r^{2}}\right|}\Big{(}r^{4}\big{(}126r^{3}r_{0}^{2}\Big{(}-\Big{(}\pi^{3}-90Q^{2}\Big{)}\Big{(}r^{2}-1\Big{)}$
(56)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+2\Big{(}-90Q^{2}+\pi^{3}+90\Big{)}rr_{0}+90\left(r^{2}-1\right)r_{0}^{2}\Big{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-5\left(3\sqrt{10}+28\right)\pi^{5}\beta\Big{(}r^{5}-r^{3}-2\left(2r^{2}+1\right)r_{0}^{3}\big{)}\big{)}\Big{)}.$

Similarly, at points near near WH throat, $r\rightarrow r_{0}$, we find the
same condition as that of the previous case

$\displaystyle\beta>\frac{3780Q^{2}r_{0}^{2}-42\pi^{3}r_{0}^{2}}{15\pi^{5}\sqrt{10}+140\pi^{5}}\quad{\rm
and}\quad Q>\frac{\pi^{3/2}}{3\sqrt{10}}\,.$ (57)

As mentioned in Ref.Nandi:2008ij , the total gravitational energy of a
structure composed of normal baryonic matter is negative. Equivalently, we can
use total gravitational energy to quantify the fluid given by

$\displaystyle E_{g}=\frac{1}{2}\int^{r_{1}}_{r_{0}}(1-\sqrt{g_{rr}})\rho^{\rm
eff}r^{2}dr+\frac{r_{0}}{2}\,,$ (58)

where $g_{rr}$ reads

$\displaystyle g_{rr}=\Big{(}1-\frac{b(r)}{r}\Big{)}^{-1}\,.$ (59)

In our models, we can simply show that $E_{g}>0$. As well, a speed of sound
(wormhole causality) could be investigated through the equations:

$\displaystyle c_{sr}^{2}\equiv\frac{dp^{\rm eff}_{r}}{d\rho^{\rm eff}},\quad
c_{st}^{2}\equiv\frac{dp^{\rm eff}_{t}}{d\rho^{\rm eff}}.$ (60)

Let us start by the case with $\Phi(r)=r_{0}/r$. Substituting $\rho^{\rm eff}$
and $p^{\rm eff}_{r}$ for this model into Eq.(60), we find that using
$Di=D_{1},\,r=r_{0}=1$, $c_{sr}<1$ for

$\displaystyle\beta>\frac{2520Q^{2}-28\pi^{3}+1260}{15\pi^{5}\sqrt{10}+140\pi^{5}},\,$
(61)

while $c_{st}=0$ for $r\rightarrow r_{0}=1,\,Q=0.58$ and $\beta=0.02$.
However, for the case with $\exp(2\Phi)=1+\gamma_{2}/r^{2}$, we find that
$c_{sr}=0$ for

$\displaystyle\beta=7.32\times 10^{-4}\left(90\,Q^{2}-58.994\right),\,$ (62)

while $c_{st}<1$ for $r\rightarrow r_{0}=1,\,Q=0.58$ and $\beta=0.1$. We note
here that the condition of a vanishing sound speed implies that the pressure
must be constant. For instance, for model with
$\exp(2\Phi)=1+\gamma_{2}/r^{2}$, the radial pressure at the WH throat is
constant for $\beta=7.32\times 10^{-4}\left(90\,Q^{2}-58.994\right)$.

##  7 Conclusions

In this work, we investigated new exact and analytic solutions of the
Einstein–Maxwell field equations describing Casimir wormholes with and without
the effect of the GUP corrections. In particular, we considered a specific
type of the GUP relation as an example. Along this line, we implemented three
specific models of the redshift function along with two different EoS of state
given by $p_{r}(r)=\omega_{r}(r)\rho(r)$ along with
$p_{t}(r)=\omega_{t}(r)p_{r}(r)$ and obtained the specific relation for the
EoS parameter $\omega_{r}(r)$ and $\omega_{t}(r)$, respectively.

We have obtained a new class of asymptotically flat Casimir wormhole solutions
with and without GUP corrections under the effect of electric charge.
Furthermore we have checked the null, weak, strong, and dominant energy
conditions at the wormhole throat with a radius $r_{0}$, and have demonstrated
that in general the classical energy condition are violated by some arbitrary
small quantity at the wormhole throat. Importantly, we examined the wormhole
geometry with semi-classical corrections via embedding diagrams. We also
considered the volume integral quantifier to calculate the amount of the
exotic matter near the wormhole throat. For completeness, we have also
investigated exotic fluid near WH throat using the so-called exoticity
parameter and discussed the speed of sound.

However, to gain more physical interest, the study of anisotropic charged
Casimir wormholes with and without GUP corrections may be worth investigating.
To do so, we can directly follow relevant machinery proposed for Casimir
wormholes found in Ref.Jusufi:2020rpw . Additionally, based on the present
work, the study of the gravitational lensing effect in the spacetime of the
charged GUP Casimir wormholes with and without GUP correction is another
interesting issue to be worth investigating.

### Acknowledgments

DS is financially supported by the Mid-Career Research Grant 2021 from
National Research Council of Thailand under a contract No. N41A640145. T.
Tangphati was supported by King Mongkut’s University of Technology Thonburi’s
Post-doctoral Fellowship. P. Channuie acknowledged the Mid-Career Research
Grant 2020 from National Research Council of Thailand (NRCT5-RSA63019-03).
This work is partially supported by the National Science, Research and
Innovation Fund (SRF) with grant No.R2565B030.

### Data Availability

Data sharing is not applicable to this article as no data sets were generated
or analyzed during the current study.

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