# Traversable Wormholes induced by Stress Energy Conservation: combining
Casimir Energy with a scalar field

Remo Garattini<EMAIL_ADDRESS>Università degli Studi di Bergamo,
Dipartimento di Ingegneria e Scienze Applicate, Viale Marconi 5, 24044 Dalmine
(Bergamo) Italy and I.N.F.N.- sezione di Milano, Milan, Italy.  Athanasios G.
Tzikas<EMAIL_ADDRESS>Università degli Studi di Bergamo,
Dipartimento di Ingegneria e Scienze Applicate, Viale Marconi 5, 24044 Dalmine
(Bergamo) Italy.

###### Abstract

We investigate possible manifolds characterizing traversable wormholes in the
presence of a scalar field, which is minimally coupled to gravity and has both
kinetic and potential energy. The feature of traversability requires the
violation of the null energy condition, which, in turn, signals the existence
of exotic matter with negative energy density. For this reason, we impose a
hypothetical Casimir apparatus with plates positioned at a distance either
parametrically fixed or radially varying. The main feature of all the derived
solutions is the conservation of the Stress Energy Tensor. Such a conservation
though requires the introduction of an auxiliary field, which we interpret as
a gravitational response of the Traversable Wormhole to the original source.
Interestingly, the only case that seems to avoid the necessity for such an
auxiliary field, is the one involving a scalar field with a potential, in
combination with a Casimir device with fixed plates.

##  I Introduction

What happens if a Casimir wormholeCW  is powered by a source of positive
energy density? A partial attempt to answer this question has been addressed
in CCW , where an electric field generated by a point charge has been
superposed to the original Casimir source. However, an electric field has a
Stress Energy Tensor (SET) of the following form

$T_{\mu\nu}^{\mathrm{EM}}=\frac{Q^{2}}{2\left(4\pi\right)^{2}\varepsilon_{0}r^{4}}\mathrm{diag}(1,-1,1,1)\,$
(1)

obeying the following equation of state

$\rho=-p_{\mathrm{r}}\,.$ (2)

As a consequence, the Null Energy Condition (NEC) is violated:

$\rho\left(r\right)+p_{\mathrm{r}}\left(r\right)=\rho_{\mathrm{C}}\left(r\right)+p_{\mathrm{r},\mathrm{C}}\left(r\right)+\rho_{\mathrm{E}}\left(r\right)+p_{\mathrm{r},\mathrm{E}}\left(r\right)=-\frac{4\hbar
c\pi^{2}}{720r^{4}}<0\,$ (3)

where

$\rho_{\mathrm{C}}\left(r\right)=-\frac{\hbar c\pi^{2}}{720r^{4}}\,,\qquad
p_{\mathrm{r},\mathrm{C}}\left(r\right)=-\frac{3\hbar
c\pi^{2}}{720r^{4}}\,,\qquad\rho_{\mathrm{E}}\left(r\right)=\frac{Q^{2}}{2\left(4\pi\right)^{2}\varepsilon_{0}r^{4}}\,,\qquad
p_{\mathrm{r},\mathrm{E}}\left(r\right)=-\frac{Q^{2}}{2\left(4\pi\right)^{2}\varepsilon_{0}r^{4}}\,.$
(4)

In other words, the introduction of an electric field does not change the
violation of the NEC by means of the Casimir source. Nevertheless, if one
tries to extend this analysis to a scalar field replacing the electric field,
the answer may not be trivial. It is well known that a scalar field of
opposite polarity can be a source of Traversable Wormholes (TW) even without
an additional Casimir source. Such an opposite polarity is better known as a
phantom field. The result is a very well known TW also dubbed as Ellis-
Bronniknov (EB) TW Ellis ; Bronnikov . Its line element is given by

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\mathrm{d}l^{2}+\left(l^{2}+r_{0}^{2}\right)\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\
\mathrm{d}\phi^{2}\right)$ (5)

and resembles a Minkowski spacetime written in spherically symmetric
coordinates, with the exception of the central region where a throat is
located at $l=0$ ($r=r_{0}$). Since $l$ represents the proper radial distance,
we can change variable from $l$ to a Schwarzschild-like form to find

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{1-r_{0}^{2}/r^{2}}+r^{2}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\
\mathrm{d}\phi^{2}\right)\,.$ (6)

A more general relationship is given by

$\mathrm{d}l=\pm\frac{\mathrm{d}r}{\sqrt{1-b(r)/r}}\,$ (7)

where we have introduced the shape function

$b(r)=\frac{r_{0}^{2}}{r}$ (8)

defined on the interval $\left[r_{0},\infty\right)$. Generally speaking,
$b(r)$ is correlated with the shape of the wormhole and is an arbitrary
function of $r$. Another important function is the redshift function
$\Lambda(r)$, which is correlated with the gravitational redshift and is
defined over the same interval as $b(r)$. These two functions, $b(r)$ and
$\Lambda(r)$, represent the cornerstone for defining the following general
form of a TW line element MT ; MTY ; Visser

$\mathrm{d}s^{2}=-e^{2\Lambda(r)}\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{1-b(r)/r}+r^{2}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\
\mathrm{d}\phi^{2}\right)\,.$ (9)

We recall that a fundamental property of a TW is the requirement for a
flaring-out condition of the throat, given by

$\frac{b(r)-rb^{\prime}(r)}{b^{2}(r)}>0$ (10)

along with the condition

$1-b(r)/r>0\,.$ (11)

Furthermore, at the throat the conditions

$b(r_{0})=r_{0}\qquad\mathrm{and}\qquad b^{\prime}(r_{0})<1$ (12)

are imposed, in order to have viable wormhole solutions. The prime (′) denotes
from now on differentiation with respect to the distance $r$. It is also
important to note that the absence of event horizons (surfaces with
$e^{2\Lambda(r)}\rightarrow 0$) requires $\Lambda(r)$ to be finite everywhere.
Although phantom fields represent a valid source of exotic matter with
negative energy density, one could consider the effect of an ordinary scalar
field combined with a Casimir source. A description of this kind has been
examined by S. Kim Kim , who has described the combination of a source of
exotic matter with a scalar field, whether it has a mass or not. Kim also
discussed the possibility of having a contribution from the effect of the back
reaction that could prevent the formation of a TW Kim1 . A particular relevant
result has been obtained by C. Barceló and M. Visser BV ; BV1 , who examined
the effect of a pure scalar field conformally coupled with a gravitational
field. In these papers, the authors showed that an effective coupling constant
is the key ingredient to obtain a TW with positive scalar field sources. On
the other hand, L. Butcher Butcher  proved that every form of pure scalar
field cannot be taken as a source for a TW. In this paper, we are going to
examine a specific form of negative energy density coupled with an ordinary
scalar field, considering cases with or without potential. The form of the
negative energy density will be described by the Casimir energy. We will find
that, at this level, the scalar field will not prevent the formation of a TW.
Moreover, the case with a potential will lead to an unexpected result: a
constant scalar field with a constant potential can serve as a source of a TW,
even in the absence of an additional negative energy density. This result has
a small but relevant extension when compared to the findings of Refs. BV ; BV1
. Indeed, we are going to examine the effect of a nonvanishing potential of a
positive minimally coupled scalar field on the existence of a TW. It is
important to note that such an analysis is not in contradiction with Butcher's
results because the boundary conditions are different. To further proceed, we
will consider the following SET configuration

$T_{\mu\nu}=\mathrm{Diag}\left[\rho\left(r\right),\,p_{\mathrm{r}}\left(r\right),\,p_{\mathrm{t}}\left(r\right),\,p_{\mathrm{t}}\left(r\right)\right]$
(13)

where the quantities $\rho\left(r\right)$, $p_{\mathrm{r}}\left(r\right)$ and
$p_{\mathrm{t}}(r)$ represent the total energy density, the total radial
pressure and the total tangential pressure of the anisotropic SET
respectively. The energy-momentum conservation is guaranteed by imposing the
validity of the following differential equation

$p_{\mathrm{r}}^{\prime}\left(r\right)=\frac{2}{r}\left(p_{\mathrm{t}}\left(r\right)-p_{\mathrm{r}}\left(r\right)\right)-\left(\rho\left(r\right)+p_{\mathrm{r}}\left(r\right)\right)\Lambda^{\prime}(r)\,$
(14)

which must be compatible with the Einstein Field Equations (EFE). We will
consider a SET composed by three parts: one part
$T_{\mu\nu}^{\mathrm{Scalar}}$ resulting from the scalar field, another part
resulting from the Casimir source

$T_{\mu\nu}^{\mathrm{Casimir}}=\mathrm{Diag}\left[\rho_{\mathrm{C}}\left(r\right),\,p_{\mathrm{r,C}}\left(r\right),\,p_{\mathrm{t,C}}\left(r\right),\,p_{\mathrm{t,C}}\left(r\right)\right]$
(15)

and the last part will be composed by an unknown SET, whose form is

$T_{\mu\nu}^{\mathrm{TW}}=\mathrm{Diag}\left[\rho^{\mathrm{TW}}\left(r\right),\,p_{\mathrm{r}}^{\mathrm{TW}}\left(r\right),\,p_{\mathrm{t}}^{\mathrm{TW}}\left(r\right),\,p_{\mathrm{t}}^{\mathrm{TW}}\left(r\right)\right]\,.$
(16)

This third piece will be interpreted as the TW SET necessary to have
consistency and conservation of the total SET. Contrary to the interpretation
of Ref.Kim , $T_{\mu\nu}^{\mathrm{TW}}$ will not be considered as a
contribution of an unknown matter field, rather it will be considered as a
kind of back-reaction of the TW to the matter source. Computationally, it is
convenient to introduce $T_{\mu\nu}^{\mathrm{TW}}$after having computed the
EFE with the Casimir and the scalar source. This is justified by the fact that
not every SET configuration needs the presence of $T_{\mu\nu}^{\mathrm{TW}}$.
The rest of the paper is organized as follows: In section II, we analyze the
structure of the SET for a massless scalar field without potential. Then we
derive the solutions by enforcing SET conservation for such a field,
considering two cases for the distance of the Casimir plates; one with fixed
distance and another with radially variable distance. In section III we repeat
the same procedure as in the previous section, but this time for a massive
scalar field with a potential. We summarize and conclude in section IV. Units
in which $\hbar=c=k=1$ are used throughout the paper and will be reintroduced
whenever it is necessary.

##  II Casimir wormholes for massless scalar field

The starting point of our analysis is the gravitational action governing our a
system with the form of

$S=\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{R}{2\kappa}-\frac{1}{2}(\nabla\psi)^{2}-V(\psi)+\mathcal{L}_{\mathrm{C}}\right]\,$
(17)

where $R$ is the Ricci scalar, $g$ is the determinant of the metric tensor,
$\kappa=8\pi G$, $\psi$ is the scalar field and $\mathcal{L}_{\mathrm{C}}$ is
the Lagrangian of the Casimir source. The potential of the scalar field
$V(\psi)$ is actually a functional of the distance, i.e.,

$V(\psi)=V\left[\psi\left(r\right)\right]=V(r)$ (18)

once we impose only radial dependence for the scalar field ($\psi=\psi(r)$).
In this section, we set $V(\psi)=0$ and vary $\eqref{action}$ with respect to
the scalar field and the metric to derive the differential equations of
motion. The equation for the scalar field is

$\nabla^{2}\psi=\frac{\mathrm{d}V(\psi)}{\mathrm{d}\psi}=\frac{V^{\prime}(r)}{\psi^{\prime}(r)}=0$
(19)

and in this case it can be cast into the following form

$e^{2\Lambda(r)}r^{4}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}(r)^{2}=C\
(\mathrm{const.)\quad\mathrm{with}\quad C>0\,.}$ (20)

Note that the case with $C<0$ corresponds to the introduction of a phantom
field. In an orthonormal frame, the Einstein Field Equations are

$\displaystyle\frac{b^{\prime}(r)}{r^{2}}=\kappa\rho\left(r\right)$ (21)
$\displaystyle\frac{2\Lambda^{\prime}(r)}{r}\left(1-\frac{b(r)}{r}\right)-\frac{b(r)}{r^{3}}=\kappa
p_{\mathrm{r}}\left(r\right)$ (22)
$\displaystyle\left(1-\frac{b(r)}{r}\right)\left(\Lambda^{\prime\prime}(r)+\Lambda^{\prime}(r)^{2}+\frac{\Lambda^{\prime}(r)}{r}\right)+\left(\frac{b(r)-rb^{\prime}(r)}{2r^{3}}\right)\left(r\Lambda^{\prime}(r)+1\right)=\kappa
p_{\mathrm{t}}\left(r\right)\,$ (23)

for the $tt-$, $rr-$ and $\theta\theta-$component respectively. The form of
the Casimir components is given by (4), along with the relation
$p_{\mathrm{t,C}}\left(r\right)=-\rho_{\mathrm{C}}\left(r\right)$. Before
proceeding, it is essential to emphasize that the forthcoming solutions must
also obey the conservation of the SET as represented by (14). Regarding the
Casimir apparatus, we distinguish two different cases:

a)

the fixed Casimir plates case and

b)

the variable Casimir plates case.

###  II.1 Fixed Casimir plates

For the case a), we know that the SET is formed by

$\displaystyle\rho\left(r\right)$
$\displaystyle=\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}(r)^{2}-\frac{\hbar
c\pi^{2}}{720d^{4}}=\frac{Ce^{-2\Lambda(r)}}{2r^{4}}-\frac{r_{1}^{2}}{\kappa
d^{4}}$ (24) $\displaystyle p_{\mathrm{r}}\left(r\right)$
$\displaystyle=\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}(r)^{2}-\frac{3\hbar
c\pi^{2}}{720d^{4}}=\frac{Ce^{-2\Lambda(r)}}{2r^{4}}-\frac{3r_{1}^{2}}{\kappa
d^{4}}$ (25)

and

$p_{\mathrm{t}}\left(r\right)=-\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}(r)^{2}+\frac{\hbar
c\pi^{2}}{720d^{4}}=-\frac{Ce^{-2\Lambda(r)}}{2r^{4}}+\frac{r_{1}^{2}}{\kappa
d^{4}}$ (26)

upon using (20) and after introducing the length scale

$r_{1}^{2}=\frac{\hbar
G}{c^{3}}\frac{\pi^{3}}{90}=\frac{\pi^{3}\ell_{\mathrm{P}}^{2}}{90}\,.$ (27)

The fixed separation distance of the Casimir plates is given by $d$. Inserting
Eqs. $\left(\ref{rhod},\ref{prd},\ref{ptd}\right)$ into (14), one finds

$-\frac{4\left(r\Lambda\\!^{\prime}\left(r\right)+2\right)r_{1}^{2}}{\kappa\,d^{4}r}=0$
(28)

that leads to

$\Lambda(r)=-2\ln\left(r\right)+C_{1}\,.$ (29)

The integration constant $C_{1}$ is to be determined. Plugging the redshift
function into (24), one finds

$\rho\left(r\right)=\frac{C}{2\bar{r}^{4}}-\frac{r_{1}^{2}}{\kappa
d^{4}}\gtreqless 0\qquad\mathrm{if}\qquad C\gtreqless
2\bar{r}^{4}r_{1}^{2}/\left(\kappa d^{4}\right)\,.$ (30)

Despite the simplicity of the energy density structure, it is better to
consider each case separately. We begin with the positive energy density case.

####  II.1.1 Positive Energy Density

First of all, we need to fix the integration constant. For future purposes, it
is convenient to assume $C_{1}=2\ln\left(\bar{r}\right)$ with $\bar{r}>r_{0}$,
so that

$\Lambda(r)=2\ln\left(\frac{\bar{r}}{r}\right)$ (31)

and $\Lambda(\bar{r})=0$. Therefore, when
$C>2\bar{r}^{4}r_{1}^{2}/\left(\kappa d^{4}\right)$, then

$\rho\left(r\right)=\rho_{+}=\frac{C}{2\bar{r}^{4}}-\frac{r_{1}^{2}}{\kappa
d^{4}}>0$ (32)

and the first EFE (21) leads to

$b(r)=r_{0}+\frac{\kappa\rho_{+}}{3}\left(r^{3}-r_{0}^{3}\right).$ (33)

Now $b(r)$ and $\Lambda(r)$ are exact solutions, but the redshift function has
the unpleasant feature that vanishes for large $r$ in a logarithmic way. Thus
we need a strategy to confine the whole solution in a well-defined range. To
further proceed, we insert (33) and (31) into the second EFE (22), retrieving

$-\frac{4}{r^{2}}+\frac{3r_{0}}{r^{3}}-\frac{r_{1}^{2}}{d^{4}}+\frac{r_{0}^{3}r_{1}^{2}}{d^{4}r^{3}}+\frac{\kappa
C}{2\bar{r}^{4}}-\frac{\kappa Cr_{0}^{3}}{2r^{3}\bar{r}^{4}}=\frac{\kappa
C}{2\bar{r}^{4}}-\frac{3r_{1}^{2}}{d^{4}}\,.$ (34)

We have consistency if and only if the following equality

$C=\frac{2\bar{r}^{4}\left(3r_{0}d^{4}-4\bar{r}d^{4}+r_{0}^{3}r_{1}^{2}+2r_{1}^{2}\bar{r}^{3}\right)}{\kappa
d^{4}r_{0}^{3}}\,$ (35)

is satisfied on the boundary $r=\bar{r}$. Plugging the value of $C$ into (34),
one finds

$\left(1-\frac{\bar{r}}{r}\right)\left(-\frac{4}{r^{2}}+\left(1+\frac{\bar{r}}{r}+\frac{\bar{r}^{2}}{r^{2}}\right)\frac{2r_{1}^{2}}{d^{4}}\right)=0\,.$
(36)

Finally, the third EFE (23) reduces to

$\frac{4}{r^{2}}-\frac{9r_{0}}{2r^{3}}+\frac{r_{1}^{2}}{d^{4}}-\frac{3r_{0}^{3}r_{1}^{2}}{2d^{4}r^{3}}-\frac{\kappa
C}{2\bar{r}^{4}}+\frac{3\kappa Cr_{0}^{3}}{4r^{3}\bar{r}^{4}}=-\frac{\kappa
C}{2\bar{r}^{4}}+\frac{r_{1}^{2}}{d^{4}}\,$ (37)

and, with the help of (35), we find

$\frac{4}{r^{2}}-\frac{6\bar{r}}{r^{3}}+\frac{3\bar{r}^{3}r_{1}^{2}}{d^{4}r^{3}}=0\,.$
(38)

The last equation gives the location of the boundary $r=\bar{r}$ in terms of
the plates separation and the Planck length

$\bar{r}=\sqrt{\frac{2}{3}}\frac{d^{2}}{r_{1}}\,.$ (39)

We can conclude that the EFE are satisfied and the consistency with the SET
conservation is preserved, at least on the boundary $r=\bar{r}$. Rearranging
(34) and (37) in terms of $\bar{r}$, one gets

$-\frac{4}{r^{2}}+\frac{8\bar{r}}{3r^{3}}+\frac{4}{3\bar{r}^{2}}=0\,$ (40)

for (34) and

$\frac{4}{r^{2}}-\frac{4\bar{r}}{r^{3}}=0\,$ (41)

for (37). To have full consistency, we need to include the third SET
$\left(\ref{TmnTW}\right)$ which, in this case, is composed by the following
terms

$\displaystyle p^{\mathrm{TW}}_{\mathrm{r}}\left(r\right)$
$\displaystyle=\frac{4}{\kappa r^{2}}-\frac{8\bar{r}}{3\kappa
r^{3}}-\frac{4}{3\kappa\bar{r}^{2}}\,$ (42) $\displaystyle
p^{\mathrm{TW}}_{\mathrm{t}}\left(r\right)$ $\displaystyle=-\frac{4}{\kappa
r^{2}}+\frac{4\bar{r}}{\kappa r^{3}}\,.$ (43)

Note that in $T_{\mu\nu}^{\mathrm{TW}}$ the energy density does not appear. It
is important to observe that the value of the boundary $r=\bar{r}$ is also
useful to rearrange (35) in the following way

$C=\frac{2\bar{r}^{2}\left(2r_{0}^{3}+9r_{0}\bar{r}^{2}-8\bar{r}^{3}\right)}{3\kappa
r_{0}^{3}}\,.$ (44)

Since $C>0$, we have a constraint at the range of $\bar{r}$. One finds

$C>0\qquad\mathrm{if}\qquad\bar{r}<1.278r_{0}\,.$ (45)

With the help of (44) and (39), $b(r)$ can be cast into the form

$b(r)=\frac{r^{3}}{r_{0}^{2}}-\frac{8\bar{r}r^{3}}{9r_{0}^{3}}+\frac{8\bar{r}}{9}\,.$
(46)

Substituting next (44) and (39) into $\rho_{+}$, we have to impose

$\bar{r}<\frac{9r_{0}}{8}\,.$ (47)

Note also that it is possible to estimate the throat size thanks to (39) and
(47). We obtain

$r_{0}>\sqrt{\frac{2}{3}}\frac{8d^{2}}{9r_{1}}.$ (48)

It is easy to see that with the shape function described by (46), the flaring-
out condition reads

$b^{\prime}(r_{0})=3\left(1-\frac{8\bar{r}}{9r_{0}}\right)<1\qquad\Longrightarrow\qquad
1<\frac{4\bar{r}}{3r_{0}}$ (49)

and is always satisfied.

####  II.1.2 Vanishing Energy Density

In this part of the paper, we consider the case in which
$\rho\left(r\right)=0$. This can be realized when
$C=2\bar{r}^{4}r_{1}^{2}/\left(\kappa d^{4}\right)$ and then the first EFE
(21) leads to

$b(r)=r_{0}\,.$ (50)

The other SET components can be easily deduced. Indeed, we find

$\rho\left(r\right)=p_{\mathrm{t}}\left(r\right)=0$ (51)

and

$p_{\mathrm{r}}\left(r\right)=-\frac{2r_{1}^{2}}{\kappa d^{4}}\,.$ (52)

It is also easy to check that the SET is conserved, if the redshift function
is represented by (31). Plugging (31) and $b(r)=r_{0}$ into the EFE (22) and
(23), one finds

$-\frac{4}{r^{2}}+\frac{3r_{0}}{r^{3}}=-\frac{2r_{1}^{2}}{d^{4}}\,$ (53)

for the EFE (22) and

$\frac{4}{r^{2}}-\frac{9r_{0}}{2r^{3}}=0\,$ (54)

for the EFE (23). On the boundary $r=\bar{r}$, Eq.(54) gives

$\bar{r}=\frac{9r_{0}}{8}\,$ (55)

while from (53) we find

$r_{0}=\frac{8\sqrt{6}d^{2}}{27r_{1}}\,.$ (56)

Therefore, on the boundary $r=\bar{r}$ the EFE are satisfied. Even in this
case we need to introduce $T_{\mu\nu}^{\mathrm{TW}}$ for consistency with the
following component

$\displaystyle\rho^{\mathrm{TW}}\left(r\right)$ $\displaystyle=0$ (57)
$\displaystyle p^{\mathrm{TW}}_{\mathrm{r}}\left(r\right)$
$\displaystyle=\frac{4}{\kappa r^{2}}-\frac{3r_{0}}{\kappa
r^{3}}-\frac{2r_{1}^{2}}{\kappa d^{4}}\,$ (58) $\displaystyle
p^{\mathrm{TW}}_{\mathrm{t}}\left(r\right)$ $\displaystyle=-\frac{4}{\kappa
r^{2}}+\frac{9r_{0}}{2\kappa r^{3}}\,.$ (59)

Let us mention here that it is also possible to obtain a vanishing energy
density with the help of the following redshift function

$\Lambda(r)=\frac{1}{2}\ln\left(\frac{C\kappa
d^{4}}{2r_{1}^{2}r^{4}}\right)\,$ (60)

which has the same structure of (31).

####  II.1.3 Negative Energy Density

The negative density $\rho\left(r\right)<0$ is present when
$C<2\bar{r}^{4}r_{1}^{2}/\left(\kappa d^{4}\right)$, so that

$\rho_{-}=\frac{r_{1}^{2}}{\kappa d^{4}}-\frac{C}{2\bar{r}^{4}}>0\,.$ (61)

Then the first EFE (21) will lead to

$b(r)=r_{0}-\frac{\kappa\rho_{-}}{3}\left(r^{3}-r_{0}^{3}\right).$ (62)

For this profile there exists $r=\bar{r}$ such that $b(\bar{r})=0$ and this is
located at

$\bar{r}=r_{0}\sqrt[3]{1+\frac{3}{\kappa\rho_{-}r_{0}^{2}}}\,.$ (63)

To further proceed, we substitute (62) and (31) into the second EFE (22)
retrieving

$-\frac{4}{r^{2}}+\frac{3r_{0}}{r^{3}}-\frac{r_{1}^{2}}{d^{4}}+\frac{r_{0}^{3}r_{1}^{2}}{d^{4}r^{3}}+\frac{\kappa
C}{2\bar{r}^{4}}-\frac{\kappa Cr_{0}^{3}}{2r^{3}\bar{r}^{4}}=\frac{\kappa
C}{2\bar{r}^{4}}-\frac{3r_{1}^{2}}{d^{4}}\,.$ (64)

We have consistency if and only if the above equality is satisfied on the
throat. This requirement provides a similar expression for the constant $C$
with Eq.(35). Inserting this value of $C$ into (64), one finds

$-\frac{4}{r^{2}}+\frac{4\bar{r}}{r^{3}}+\frac{2r_{1}^{2}}{d^{4}}-\frac{2\bar{r}^{3}r_{1}^{2}}{d^{4}r^{3}}=0\,.$
(65)

Finally, the third EFE (23) reduces to

$\frac{4}{r^{2}}-\frac{9r_{0}}{2r^{3}}+\frac{r_{1}^{2}}{d^{4}}-\frac{3r_{0}^{3}r_{1}^{2}}{2d^{4}r^{3}}-\frac{\kappa
C}{2\bar{r}^{4}}+\frac{3\kappa Cr_{0}^{3}}{4r^{3}\bar{r}^{4}}=-\frac{\kappa
C}{2\bar{r}^{4}}+\frac{r_{1}^{2}}{d^{4}}\,$ (66)

and, with the help of $C$, we find

$\frac{4}{r^{2}}-\frac{6\bar{r}}{r^{3}}+\frac{3\bar{r}^{3}r_{1}^{2}}{d^{4}r^{3}}=0\,.$
(67)

On the boundary $r=\bar{r}$, we obtain

$\bar{r}=\sqrt{\frac{2}{3}}\frac{d^{2}}{r_{1}}\,.$ (68)

Plugging (35) and (68) into (62), we find

$b(r)=\left(\frac{1}{r_{0}^{2}}-\frac{8\bar{r}}{9r_{0}^{3}}\right)r^{3}+\frac{8\bar{r}}{9}\,.$
(69)

The full consistency is obtained with the help of $\left(\ref{TmnTW}\right)$
where

$\displaystyle\rho^{\mathrm{TW}}\left(r\right)$ $\displaystyle=0$ (70)
$\displaystyle p^{\mathrm{TW}}_{\mathrm{r}}\left(r\right)$
$\displaystyle=-\frac{4}{\kappa r^{2}}+\frac{8\bar{r}}{3\kappa
r^{3}}+\frac{4}{3\kappa\bar{r}^{2}}\,$ (71) $\displaystyle
p^{\mathrm{TW}}_{\mathrm{t}}\left(r\right)$ $\displaystyle=\frac{4}{\kappa
r^{2}}-\frac{6\bar{r}}{\kappa r^{3}}+\frac{3\bar{r}^{3}r_{1}^{2}}{\kappa
d^{4}r^{3}}\,.$ (72)

In terms of $r_{0}$ and $\bar{r}$, we can rearrange the value of $\rho_{-}$ as

$\rho_{-}=\frac{8\bar{r}-9r_{0}}{3\kappa r_{0}^{3}}\,.$ (73)

It is easy to see that from (69), the flaring-out condition becomes

$b^{\prime}(r_{0})<1\qquad\Longleftrightarrow\qquad 1<\frac{4\bar{r}}{3r_{0}}$
(74)

and is compatible with (73). From $C>0$ and from (73), we get the validity
range for the cut-off boundary

$1.125r_{0}<\bar{r}<1.278r_{0}\,.$ (75)

###  II.2 Variable Casimir plates

For the case b), we know that the SET is formed by

$\displaystyle\rho\left(r\right)$
$\displaystyle=\frac{Ce^{-2\Lambda(r)}}{2r^{4}}-\frac{r_{1}^{2}}{\kappa
r^{4}}\,$ (76) $\displaystyle p_{\mathrm{r}}\left(r\right)$
$\displaystyle=\frac{Ce^{-2\Lambda(r)}}{2r^{4}}-\frac{3r_{1}^{2}}{\kappa
r^{4}}\,$ (77)

and

$p_{\mathrm{t}}\left(r\right)=-\frac{Ce^{-2\Lambda(r)}}{2r^{4}}+\frac{r_{1}^{2}}{\kappa
r^{4}}\,.$ (78)

Inserting Eqs.$\left(\ref{rhor},\ref{prr},\ref{ptr}\right)$ into (14), one
finds

$\frac{4\left(1-\Lambda\\!^{\prime}\left(r\right)\right)r_{1}^{2}}{\kappa
r^{4}}=0$ (79)

leading to

$\Lambda(r)=\ln\left(r\right)+C_{2}\,$ (80)

where $C_{2}$ is an integration constant to be determined. Even in the
variable case, we have two possibilities:

  1. 1.

$\Lambda(r_{0})=0$, that means

$\Lambda(r)=\ln\left(\frac{r}{r_{0}}\right)$ (81)

and

  2. 2.

$\Lambda(\bar{r})=0$, namely

$\Lambda(\bar{r})=\ln\left(\frac{r}{\bar{r}}\right)$ (82)

where $\bar{r}>r_{0}$ is an outer boundary to be determined.

The $\Lambda(r_{0})=0$ case gives no consistent solutions and so it will be
discarded. On the contrary, inserting (82) into (76), leads to

$\rho\left(r\right)=\frac{C\bar{r}^{2}}{2r^{6}}-\frac{r_{1}^{2}}{\kappa
r^{4}}\gtreqless
0\qquad\mathrm{if}\qquad\sqrt{\frac{C\kappa\bar{r}^{2}}{2r_{1}^{2}}}\frac{\bar{r}}{r_{1}}\gtreqless
r\,.$ (83)

Substituting next the redshift function (82) in (21), one finds

$b\left(r\right)=r_{0}+\frac{\bar{r}^{2}C\kappa}{6}\left(\frac{1}{r_{0}^{3}}-\frac{1}{r^{3}}\right)+r_{1}^{2}\left(\frac{1}{r}-\frac{1}{r_{0}}\right)\,.$
(84)

With the helpf of (84) and (82) and from the second EFE (22), we obtain

$\frac{2}{r^{2}}-\frac{3r_{0}}{r^{3}}-\frac{\bar{r}^{2}C\kappa}{2r^{3}r_{0}^{3}}+\frac{3r_{1}^{2}}{r^{3}r_{0}}+\frac{\bar{r}^{2}C\kappa}{2r^{6}}-\frac{3r_{1}^{2}}{r^{4}}=\frac{\kappa
C\bar{r}^{2}}{2r^{6}}-\frac{3r_{1}^{2}}{r^{4}}\,.$ (85)

Evaluating the previous equation on the boundary $r=\bar{r}$, we find the
value of $C$

$C=\frac{2r_{0}^{2}}{\kappa\bar{r}^{2}}\left(2\bar{r}r_{0}-3r_{0}^{2}+3r_{1}^{2}\right)\,.$
(86)

Since $C$ must be positive, we have an additional constraint

$\bar{r}>\frac{3\left(r_{0}^{2}-r_{1}^{2}\right)}{2r_{0}}>0\qquad\Longrightarrow\qquad
r_{0}>r_{1}\,.$ (87)

Plugging (86) into (85), we find

$\frac{2\left(r-\bar{r}\right)}{r^{3}}=0\,$ (88)

which is satisfied on the boundary as it should be. The last equation we need
to examine is the third EFE (23). Simply we get

$\frac{1}{r^{2}}=0\,$ (89)

which is meaningless. However, we have to remember that we need to consider
also $T_{\mu\nu}^{\mathrm{TW}}$ to cure such an inconsistency which, in this
case, has the following components

$\displaystyle\rho^{\mathrm{TW}}\left(r\right)$ $\displaystyle=0\,$ (90)
$\displaystyle p^{\mathrm{TW}}_{\mathrm{r}}\left(r\right)$
$\displaystyle=-\frac{2\left(r-\bar{r}\right)}{\kappa r^{3}}\,$ (91)
$\displaystyle p^{\mathrm{TW}}_{\mathrm{t}}\left(r\right)$
$\displaystyle=-\frac{1}{\kappa r^{2}}\,.$ (92)

We can rearrange the shape function with the help of (86). One then gets

$b\left(r\right)=\frac{r_{0}^{4}}{r^{3}}-\frac{2\bar{r}\,r_{0}^{3}}{3r^{3}}+\frac{2\bar{r}}{3}-\frac{r_{0}^{2}r_{1}^{2}}{r^{3}}+\frac{r_{1}^{2}}{r}\,.$
(93)

It remains to check the validity of the flaring-out condition
($b^{\prime}\left(r_{0}\right)<1$). On the one hand, from (93) we obtain

$b^{\prime}\left(r\right)=-3\frac{r_{0}^{4}}{r^{4}}+\frac{2\bar{r}\,r_{0}^{3}}{r^{4}}+3\frac{r_{0}^{2}r_{1}^{2}}{r^{4}}-\frac{r_{1}^{2}}{r^{2}}$
(94)

and so

$b^{\prime}\left(r_{0}\right)=-3+\frac{2\bar{r}}{r_{0}}+\frac{2r_{1}^{2}}{r_{0}^{2}}<1\qquad\Longrightarrow\qquad\bar{r}<2r_{0}-\frac{r_{1}^{2}}{r_{0}}\,.$
(95)

On the other hand, the inequality $\left(\ref{constr1}\right)$ must also be
satisfied and hence

$2r_{0}-\frac{r_{1}^{2}}{r_{0}}>\bar{r}>\frac{3r_{0}}{2}-\frac{3r_{1}^{2}}{2r_{0}}\,.$
(96)

Since $r_{0}>r_{1}>0$, we can assume that there exists $\alpha>1$ such that
$r_{0}=\alpha r_{1}$. Therefore, the last inequality becomes

$\frac{r_{1}}{\alpha}\left(2\alpha^{2}-1\right)>\bar{r}>\frac{3r_{1}}{2\alpha}\left(\alpha^{2}-1\right)\,.$
(97)

##  III Casimir wormholes with a non-vanishing potential for a scalar field

In this section we proceed with a non-vanishing potential for the scalar field
according to (18). Now the field equation reads

$\nabla^{2}\psi=\left(1-\frac{b(r)}{r}\right)\left[\psi^{\prime\prime}(r)+\left(\frac{2}{r}+\Lambda^{\prime}(r)\right)\psi^{\prime}(r)\right]+\left(\frac{b(r)-rb^{\prime}(r)}{2r^{2}}\right)\psi^{\prime}(r)=\frac{V^{\prime}(r)}{\psi^{\prime}(r)}\,$
(98)

where we have assumed that

$\frac{\mathrm{d}V(\psi)}{\mathrm{d}\psi}=\frac{V^{\prime}(r)}{\psi^{\prime}(r)}\qquad\mathrm{with}\qquad\psi^{\prime}(r)\neq
0\,.$ (99)

The EFE are given by (21)-(23) with the modification

$\displaystyle\rho\left(r\right)$
$\displaystyle=\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}(r)^{2}+V(r)+\rho_{\mathrm{C}}\left(r\right)\,$
(100) $\displaystyle p_{\mathrm{r}}\left(r\right)$
$\displaystyle=\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}(r)^{2}-V(r)+p_{\mathrm{r,C}}\left(r\right)\,$
(101) $\displaystyle p_{\mathrm{t}}\left(r\right)$
$\displaystyle=-\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}(r)^{2}-V(r)+p_{\mathrm{t,C}}\left(r\right)\,.$
(102)

Before going on, we have to observe that a constant scalar field with a
constant potential satisfies Eq.$\left(\ref{scalar_eom}\right)$. Moreover, in
this context, we have a behavior similar to an electric field, namely

$\rho(r)+p_{\mathrm{r}}(r)=0\,.$ (103)

Consequently, the NEC of the Casimir source, along with an additional constant
scalar field with a constant potential, is always violated. This seems to be
true even when the kinetic term can be neglected compared to the potential
term. As before, we examine two different cases; fixed and variable Casimir
plates.

###  III.1 Fixed Casimir plates

In this case, the components of the SET are

$\displaystyle\rho\left(r\right)$
$\displaystyle=\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}(r)^{2}+V(r)-\frac{r_{1}^{2}}{\kappa
d^{4}}\,$ (104) $\displaystyle p_{\mathrm{r}}\left(r\right)$
$\displaystyle=\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}(r)^{2}-V(r)-\frac{3r_{1}^{2}}{\kappa
d^{4}}\,$ (105) $\displaystyle p_{\mathrm{t}}\left(r\right)$
$\displaystyle=-\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}(r)^{2}-V(r)+\frac{r_{1}^{2}}{\kappa
d^{4}}\,.$ (106)

and its conservation (14) gives the following equation

$\left(1-\frac{b(r)}{r}\right)\left[\psi^{\prime\prime}(r)\psi^{\prime}(r)+\left(\frac{2}{r}+\Lambda^{\prime}(r)\right)\psi^{\prime}(r)^{2}\right]+\left(\frac{b(r)-rb^{\prime}(r)}{2r^{2}}\right)\psi^{\prime}(r)^{2}-V^{\prime}(r)=\frac{4r_{1}^{2}}{\kappa
d^{4}}\left(\frac{2}{r}+\Lambda^{\prime}(r)\right)\,.$ (107)

If $\psi(r)$ is a solution of the scalar field equation (98), then the left-
hand side of (107) is identically zero. As a result, the SET conservation
gives the solution for the redshift function

$\Lambda(r)=C_{3}-2\ln r\,$ (108)

where the integration constant $C_{3}$ is yet to be determined. Next, by
adding (21) and (23), we get

$\frac{b^{\prime}(r)}{r^{2}}+\left(1-\frac{b(r)}{r}\right)\left(\Lambda^{\prime\prime}(r)+\Lambda^{\prime}(r)^{2}+\frac{\Lambda^{\prime}(r)}{r}\right)+\left(\frac{b(r)-rb^{\prime}(r)}{2r^{3}}\right)\left(r\Lambda^{\prime}(r)+1\right)=0$
(109)

and by inserting (108), we solve over the shape function which is calculated
to be

$b(r)=\frac{4r}{3}-\frac{r^{3}}{3r_{0}^{2}}\,.$ (110)

The shape function does not vanish at infinity and so we need an external cut-
off boundary ($r=\bar{r}$) where $b(\bar{r})=0$. Such a boundary is simply
given by $\bar{r}=2r_{0}$, leaving us with a physically meaningful inner
region for the traversable wormhole, i.e., $r\in[r_{0},2r_{0}]$. In addition,
the flaring-out condition gives $b^{\prime}(r_{0})=1/3<1$ and is always
satisfied. Being that said, the integration constant $C_{3}$ is chosen to be
$C_{3}=2\ln\bar{r}$ so that

$\Lambda(r)=2\ln\left(\frac{\bar{r}}{r}\right)\,.$ (111)

That implies that the redshift function also vanishes at the boundary. We
proceed further by subtracting (22) from (21), getting

$\frac{b^{\prime}(r)}{r^{2}}-\frac{2\Lambda^{\prime}(r)}{r}\left(1-\frac{b(r)}{r}\right)+\frac{b(r)}{r^{3}}=2\kappa
V(r)+\frac{2r_{1}^{2}}{d^{4}}\,$ (112)

and with the help of (111) and (110), we solve over the potential

$V(r)=\frac{2}{3\kappa r^{2}}-\frac{r_{1}^{2}}{\kappa d^{4}}\,.$ (113)

To check the violation of the Weak Enrgy Condition (WEC), we calculate the
energy density (104) as

$\rho\left(r\right)=\frac{b^{\prime}(r)}{\kappa r^{2}}=\frac{4}{3\kappa
r^{2}}-\frac{1}{\kappa r_{0}^{2}}\,.$ (114)

The negative value of the energy density is guarantied if $r>2r_{0}/\sqrt{3}$.
Hence, there is a region $r\in[r_{0},2r_{0}/\sqrt{3}]$ where the WEC is
satisfied. On the contrary, the NEC is always violated because

$\rho\left(r\right)+p_{r}\left(r\right)=\frac{4}{3\kappa
r^{2}}-\frac{2}{\kappa r_{0}^{2}}<0\qquad\mathrm{for}\qquad r\geq r_{0}\,.$
(115)

Finally, we may calculate the function of the scalar field by inserting the
forms we found for $b(r)$, $\Lambda(r)$ and $V(r)$ in any of the EFE. The
result is the following differential equation

$\displaystyle\frac{r^{3}\kappa\left(\left(\psi^{\prime}(r)\right)^{2}\right)^{\prime}\left(r^{2}-r_{0}^{2}\right)+2r^{4}\left(\psi^{\prime}\\!\left(r\right)\right)^{2}\kappa+8r_{0}^{2}}{6\psi^{\prime}\\!\left(r\right)\kappa\,r^{3}r_{0}^{2}}$
$\displaystyle=\frac{r^{3}\kappa
y^{\prime}(r)\left(r^{2}-r_{0}^{2}\right)+2r^{4}y\\!\left(r\right)\kappa+8r_{0}^{2}}{6\psi^{\prime}\\!\left(r\right)\kappa\,r^{3}r_{0}^{2}}=0\,$
(116)

where $\left(\psi^{\prime}\\!\left(r\right)\right)^{2}=y\\!\left(r\right)$ and
the solution for the scalar field is calculated to be

$\displaystyle\psi(r)$
$\displaystyle=\int_{r_{0}}^{r}\sqrt{\frac{C\kappa\,x^{2}+4r_{0}^{2}}{\kappa\,x^{2}\left(x^{2}-r_{0}^{2}\right)}}dx=\frac{\pi}{2}+\arctan\\!\left(\frac{\sqrt{r^{4}+3r^{2}r_{0}^{2}-4r_{0}^{4}}}{2r_{0}^{2}}\right)+\frac{\ln\\!\left(2\sqrt{r^{2}+4r_{0}^{2}}\sqrt{r^{2}-r_{0}^{2}}+2r^{2}+3r_{0}^{2}\right)}{4}$
(117)
$\displaystyle-\arctan\\!\left(\frac{4r^{2}+6r_{0}^{2}}{3\sqrt{r^{4}+3r^{2}r_{0}^{2}-4r_{0}^{4}}}\right)-\frac{\ln\\!\left(2r^{2}+3r_{0}^{2}-2\sqrt{r^{2}+4r_{0}^{2}}\,\sqrt{r^{2}-r_{0}^{2}}\right)}{4}\,.$
(118)

Close to the throat, one finds

$\psi(r)\simeq\frac{\sqrt{10}}{\sqrt{r_{0}}}\sqrt{r-r_{0}}+\mathcal{O}\\!\left(r-r_{0}\right)\,.$
(119)

It is imperative to observe that this solution represents the only case in
which $T_{\mu\nu}^{\mathrm{TW}}$ does not come into play.

###  III.2 Variable Casimir plates

In this case, the components of the SET are

$\displaystyle\rho\left(r\right)$
$\displaystyle=\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}\left(r\right)^{2}+V(r)-\frac{r_{1}^{2}}{\kappa
r^{4}}\,$ (120) $\displaystyle p_{\mathrm{r}}\left(r\right)$
$\displaystyle=\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}(r)^{2}-V(r)-\frac{3r_{1}^{2}}{\kappa
r^{4}}\,$ (121) $\displaystyle p_{\mathrm{t}}\left(r\right)$
$\displaystyle=-\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}(r)^{2}-V(r)+\frac{r_{1}^{2}}{\kappa
r^{4}}$ (122)

and its conservation gives the following equation

$\left(1-\frac{b(r)}{r}\right)\left[\psi^{\prime\prime}(r)\psi^{\prime}(r)+\left(\frac{2}{r}+\Lambda^{\prime}(r)\right)\psi^{\prime}(r)^{2}\right]+\left(\frac{b(r)-rb^{\prime}(r)}{2r^{2}}\right)\psi^{\prime}(r)^{2}-V^{\prime}(r)=-\frac{12r_{1}^{2}}{\kappa
r^{5}}+\frac{4r_{1}^{2}}{\kappa
r^{4}}\left(\frac{2}{r}+\Lambda^{\prime}(r)\right)\,.$ (123)

Even in this case, if $\psi(r)$ is a solution of the scalar field equation
(98), then the left-hand side of (123) is identically zero. Therefore, the SET
gives the solution for the redshift function

$\Lambda(r)=\ln r+C_{4}\,$ (124)

where the integration constant can be identified as $C_{4}=-\ln\bar{r}$ with
$r=\bar{r}$ being a cut-off boundary for the wormhole. We have to solve

$\left(1-\frac{b(r)}{r}\right)\left[\frac{1}{2}\frac{d}{dr}\left(\psi^{\prime}(r)\right)^{2}+\left(\frac{2}{r}+\Lambda^{\prime}(r)\right)\psi^{\prime}(r)^{2}\right]+\left(\frac{b(r)-rb^{\prime}(r)}{2r^{2}}\right)\psi^{\prime}(r)^{2}-V^{\prime}(r)=0\,$
(125)

which can be rearranged to give

$\frac{d}{dr}\left[\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\left(\psi^{\prime}(r)\right)^{2}\right]+\left[\frac{3}{r}\left(1-\frac{b(r)}{r}\right)\psi^{\prime}(r)^{2}\right]-V^{\prime}(r)=0\,.$
(126)

This equation can be solved as a function of $V^{\prime}(r)$. Indeed, the
previous equation can be cast into the form

$\displaystyle
y(r)=\frac{1}{2}\left(1-\frac{b(r)}{r}\right)\left(\psi^{\prime}(r)\right)^{2},$
(127) $\displaystyle y^{\prime}(r)+\frac{6}{r}y(r)-V^{\prime}(r)=0\,$ (128)

with

$y(r)=\left(1-\frac{b(r)}{r}\right)\left(\psi^{\prime}(r)\right)^{2}=\frac{1}{r^{6}}\left(C_{5}+2\int\mathrm{d}r\
r^{6}V^{\prime}(r)\right)\,.$ (129)

Isolating $\psi^{\prime}(r)$, we obtain

$\psi(r)=\int\mathrm{d}r\left[\left(1-\frac{b(r)}{r}\right)^{-1}\frac{1}{r^{6}}\left(C_{5}+2\int\mathrm{d}r\
r^{6}V^{\prime}(r)\right)\right]^{\frac{1}{2}}+\psi_{0}\,.$ (130)

Unfortunately, this case does not have a general solution. Nevertheless,
specific profiles can always be examined. For instance, one can consider the
constant scalar field with a constant scalar potential. Indeed, this is a
solution of (98) and can be used to see if a TW can be supported even in
absence of a Casimir source.

###  III.3 Constant Potential without a Casimir Source

Without the Casimir source, the EFE are very simple

$\displaystyle\frac{b^{\prime}(r)}{r^{2}}=\kappa V$ (131)
$\displaystyle\frac{2\Lambda^{\prime}(r)}{r}\left(1-\frac{b(r)}{r}\right)-\frac{b(r)}{r^{3}}=-\kappa
V$ (132)
$\displaystyle\left(1-\frac{b(r)}{r}\right)\left(\Lambda^{\prime\prime}(r)+\Lambda^{\prime}(r)^{2}+\frac{\Lambda^{\prime}(r)}{r}\right)+\left(\frac{b(r)-rb^{\prime}(r)}{2r^{3}}\right)\left(r\Lambda^{\prime}(r)+1\right)=-\kappa
V\,.$ (133)

The conservation of the SET is guaranteed for any value of $V$ with a generic
redshift function which, for simplicity, we set to zero ($\Lambda(r)=0$).
Therefore, the previous system reduces to

$\displaystyle\frac{b^{\prime}(r)}{r^{2}}=\kappa V$ (134)
$\displaystyle\frac{b(r)}{r^{3}}=\kappa V$ (135)
$\displaystyle\frac{b(r)-rb^{\prime}(r)}{2r^{3}}=-\kappa V$ (136)

where we have assumed that $V>0$. Examining the right-hand side of the EFE
(134),(135),(136), one can observe that the reduced SET adopts the structure
of a cosmological constant. From Eq.(134), we find

$b(r)=r_{0}+\frac{\kappa V}{3}\left(r^{3}-r_{0}^{3}\right)\,.$ (137)

As it is, this system of equations is inconsistent. Therefore, we need to
introduce an additional unknown SET like in the previous sections. However,
even if we introduce a SET described by (16), the shape function (137) has no
bound and there is no way to introduce a physical cut-off for
$r\rightarrow\infty$. As a consequence, the case with $V>0$ will be
discarded111A different approach for a TW with a cosmological term has been
considered in Refs.DM ; LLQdO .. On the other hand, things are different when
$V<0$, because the system of EFE becomes

$\displaystyle\frac{b^{\prime}(r)}{r^{2}}$ $\displaystyle=-\kappa\bar{V}$
(138) $\displaystyle-\frac{b(r)}{r^{3}}$ $\displaystyle=\kappa\bar{V}$ (139)
$\displaystyle\frac{b(r)-rb^{\prime}(r)}{2r^{3}}$
$\displaystyle=\kappa\bar{V}\quad\mathrm{with}\quad\bar{V}=-V>0$ (140)

and the shape function is simply

$b(r)=r_{0}-\frac{\kappa\bar{V}}{3}\left(r^{3}-r_{0}^{3}\right)\,.$ (141)

This time, we have a natural boundary located at

$r_{\mathrm{bm}}=r_{0}\sqrt[3]{1+\frac{3}{\kappa\bar{V}r_{0}^{2}}}\,.$ (142)

Outside this boundary, the shape function becomes negative. Therefore, the
solution is valid into the range $\left[r_{0},r_{\mathrm{bm}}\right]$.
Moreover, the additional SET assumes the structure

$\displaystyle\rho^{\mathrm{TW}}(r)$ $\displaystyle=0$ (143) $\displaystyle
p^{\mathrm{TW}}_{\mathrm{r}}(r)$
$\displaystyle=\frac{2\bar{V}}{3}+\frac{\bar{V}r_{0}^{3}}{3r^{3}}-\frac{r_{0}}{\kappa
r^{3}}$ (144) $\displaystyle p^{\mathrm{TW}}_{\mathrm{t}}(r)$
$\displaystyle=\frac{2\bar{V}}{3}-\frac{\bar{V}r_{0}^{3}}{6r^{3}}+\frac{r_{0}}{2\kappa
r^{3}}$ (145)

and is conserved. To summarize, a proposal for a TW consistent with the
constant scalar field and, hence, with a constant potential could be the
following one:

$\left\\{\begin{array}[c]{cc}b(r)=r_{0}-\frac{\kappa\bar{V}}{3}\left(r^{3}-r_{0}^{3}\right)&r\in\left[r_{0},r_{\mathrm{bm}}\right]\\\
&\\\ \Lambda(r)=0&\forall\,r\in\left[r_{0},+\infty\right)\\\ &\\\ b(r)=0&r\geq
r_{\mathrm{bm}}\,.\end{array}\right.$ (146)

This means that the SET must be

$T_{\nu}^{\mu}=\left\\{\begin{array}[c]{cc}\mathrm{Diag}\left[-\bar{V},\bar{V}+p^{\mathrm{TW}}_{\mathrm{r}}(r),\bar{V}+p^{\mathrm{TW}}_{\mathrm{t}}(r),\,\bar{V}+p^{\mathrm{TW}}_{\mathrm{t}}(r)\right]&r\in\left[r_{0},\,r_{\mathrm{bm}}\right]\\\
&\\\ \mathrm{Diag}\left[0,0,0,0\right]&r\geq
r_{\mathrm{bm}}\,.\end{array}\right.$ (147)

Since on $r_{\mathrm{bm}}$ the SET is forced to be the Minkowski SET, then the
potential vanishes ($\bar{V}=0$) when $r\geq r_{\mathrm{bm}}$.

##  IV Conclusions

In this paper we have taken under examination the combined contribution of a
scalar field and a Casimir source. First, we considered a scalar field without
a potential, and second, we examined the case with a potential. In addition,
the distance of the Casimir plates has been considered either to be
parametrically fixed or radially variable. In every profile, a solution
appears but it must be confined in a bounded region to avoid divergences on
the redshift or the shape function. This means that our model does not satisfy
the asymptotic flatness, but it has a finite range of definition. However,
this is not a problem since it is always possible to use the junction method
and paste the inner core of the TW with a Schwarzschild metric, as
demonstrated in Ref.Sushkov  or Ref.LLQdO . To establish a consistent system
of equations, we had to consider the introduction of a third field
$T_{\mu\nu}^{\mathrm{TW}}$, which is interpreted as the reaction of the TW to
the action of the matter fields $T_{\mu\nu}^{\mathrm{Casimir}}$ and
$T_{\mu\nu}^{\mathrm{Scalar}}$. However, when considering a scalar field with
a potential term in the presence of a Casimir source with fixed plates, the
introduction of $T_{\mu\nu}^{\mathrm{TW}}$ is not necessary. At this stage of
our research, we do not know the physical reason behind this result. It is
also interesting to note that the case of the constant scalar field with a
constant potential term yields an unexpected outcome, namely the possibility
of having a TW which behaves like a Generalized Absurdly Benign Traversable
WormholeGABTW . Moreover, one finds that the constant potential acts as a
cosmological constant term, allowing the changing of the throat size as a
function of the potential well $V$. This behavior appears only when $V$ has a
negative value. Note that that the pure scalar field does not violate the NEC;
rather, Eq.$\left(\ref{prho}\right)$ is satisfied. It is also important to
observe that in the case of a scalar field with a potential and fixed Casimir
plates, the NEC is violated due to the presence of the Casimir source. In the
absence of the Casimir source, it is the combination of the constant potential
with a constant scalar field and the contribution of
$T_{\mu\nu}^{\mathrm{TW}}$ that makes the difference. To conclude, it is
noteworthy to mention that in the case described in subsection III.1, the
throat size $r_{0}$ is not fixed, while in subsection III.3 the throat depends
on the magnitude of the constant potential. On the contrary, in the case
without a potential, also termed as the massless case, the predicted throat
size is huge because it is proportional to the ratio $d^{2}/r_{1}\simeq
1.\,\allowbreak 7\times 10^{17}m$. At this stage of the investigation, we are
unaware of the nature of such behavior.

## References

  * (1) R.Garattini, Eur. Phys. J. C 79 (2019) 11, 951.
  * (2) R. Garattini, Eur. Phys. J. C 83 (2023) 5, 369.
  * (3) H. Ellis, J. Math. Phys. 14, 104 (1973).
  * (4) K.A. Bronnikov, Acta Phys. Pol. B 4, 251 (1973).
  * (5) M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988).
  * (6) M.S. Morris, K.S. Thorne and U. Yurtsever, Phys. Rev. Lett. 61, 1446 (1988).
  * (7) M. Visser, Lorentzian Wormholes: From Einstein to Hawking (American Institute of Physics, New York), 1995.
  * (8) S. W. Kim and S. P. Kim, Phys. Rev. D 58 (1998) 087703.
  * (9) S. W. Kim, Grav. Cosmol. 6 (2000) 337-340.
  * (10) C. Barcelo and M. Visser, Phys. Lett. B 466 (1999) 127-134.
  * (11) C. Barcelo and M. Visser, Class.Quant.Grav. 17 (2000) 3843-3864.
  * (12) L. M. Butcher, Phys. Rev. D 91 (2015) 124031\.
  * (13) M.S. Delgaty and R.B. Mann, Int. J. Mod. Phys. D 4 (1995) 231-246.
  * (14) J. P. S. Lemos, F. S. N. Lobo and S. Quinet de Oliveira, Phys. Rev. D 68 (2003) 064004.
  * (15) S. V. Sushkov, Phys. Rev. D 71 043520, (2005).
  * (16) R.Garattini, Eur.Phys.J.C 80 (2020) 12, 1172.