# Effects of an Electric Charge on Casimir Wormholes: Changing the Throat Size

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.

###### Abstract

In this paper we continue the investigation about the connection between
Casimir energy and the traversability of a wormhole. In addition to the
negative energy density obtained by a Casimir device, we include the effect of
an electromagnetic field generated by an electric charge. This combination
defines an electrovacuum source which has an extra parameter related to the
size of the throat. Even if the electromagnetic energy density is positive the
Null Energy Condition is still violated. The main reason is that the
electromagnetic field satisfies the property $\rho=-p_{r}$. As a consequence
the Traversable Wormhole throat can be changed as a function of the electric
charge. This means that the throat is no more Planckian and the traversability
is little less in principle but little more in practice.

##  I Introduction

Casimir wormholes are Traversable Wormholes (TW) obtained by solving the
semiclassical Einstein’s Field Equations (EFE)

$G_{\mu\nu}=\kappa\left\langle
T_{\mu\nu}\right\rangle^{\text{Ren}}\qquad\kappa=\frac{8\pi G}{c^{4}}$ (1)

with a source of the form

$\rho_{C}\left(d\right)=-\frac{\hbar c\pi^{2}}{720d^{4}},$ (2)
$P\left(d\right)=\frac{F\left(d\right)}{S}=-3\frac{\hbar c\pi^{2}}{720d^{4}},$
(3)

representing the energy density and the pressure, respectively. $d$ is the
plates separationCW . $\left\langle T_{\mu\nu}\right\rangle^{\text{Ren}}$
describes the renormalized stress-energy tensor of some matter fields which,
in this specific case, is obtained by the Zero Point Energy (ZPE) contribution
of the electromagnetic field. The two key ingredients useful to form a Casimir
wormhole are in the relationship $P\left(d\right)/\rho_{C}\left(d\right)=3$.
This particular number is the cornerstone of a Casimir wormhole. In addition,
for a Casimir wormhole, the plates separation $d$ has been promoted to be the
radial coordinate $r$ dealt with a variable. For this reason the pressure
$P\left(d\right)$ will be interpreted as a radial pressure
$p_{r}\left(r\right)$ To build a Casimir wormhole we need to introduce the
following spacetime metric

$ds^{2}=-e^{2\Phi(r)}\,dt^{2}+\frac{dr^{2}}{1-b(r)/r}+r^{2}d\Omega^{2}\,,$ (4)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ is the line element
of the unit sphere. $\Phi(r)$ and $b(r)$ are two arbitrary functions of the
radial coordinate $r\in\left[r_{0},+\infty\right)$, denoted as the redshift
function and the shape function, respectively MT ; MTY ; Visser . With the
help of the metric $\left(\ref{metric}\right)$ the EFE, written in an
orthonormal frame, are

$\frac{b^{\prime}\left(r\right)}{r^{2}}=\kappa\rho\left(r\right),$ (5)
$\frac{2}{r}\left(1-\frac{b\left(r\right)}{r}\right)\Phi^{\prime}(r)-\frac{b\left(r\right)}{r^{3}}=\kappa
p_{r}\left(r\right)$ (6)

and

$\displaystyle\Bigg{\\{}\left(1-\frac{b\left(r\right)}{r}\right)\left[\Phi^{\prime\prime}(r)+\Phi^{\prime}(r)\left(\Phi^{\prime}(r)+\frac{1}{r}\right)\right]$
$\displaystyle-\frac{b^{\prime}\left(r\right)r-b\left(r\right)}{2r^{2}}\left(\Phi^{\prime}(r)+\frac{1}{r}\right)\Bigg{\\}}=\kappa
p_{t}(r).$ (7)

$\rho\left(r\right)$ is the energy density111However, if $\rho\left(r\right)$
represents the mass density, then we have to replace $\rho\left(r\right)$ with
$\rho\left(r\right)c^{2}.$, $p_{r}\left(r\right)$ is the radial pressure, and
$p_{t}\left(r\right)$ is the lateral pressure. The line element
$\left(\ref{metric}\right)$ represents a spherically symmetric and static
wormhole and $r_{0}$ is the location of the throat. We can complete the EFE
with the expression of the conservation of the stress-energy tensor which can
be written in the same orthonormal reference frame

$p_{r}^{\prime}\left(r\right)=\frac{2}{r}\left(p_{t}\left(r\right)-p_{r}\left(r\right)\right)-\left(\rho\left(r\right)+p_{r}\left(r\right)\right)\Phi^{\prime}(r).$
(8)

If we assume that an Equation of State (EoS)
$p_{r}\left(r\right)=\omega\rho_{C}\left(r\right)$ is imposed, then we find
the following solution to the semiclassical EFE

$\displaystyle\Phi(r)$
$\displaystyle=\frac{1}{2}\left({\omega-1}\right){\ln}\left({\frac{r\omega}{\left(\omega
r+r_{0}\right)}}\right)$ (9) $\displaystyle b(r)$
$\displaystyle=\left(1-\frac{1}{\omega}\right)r_{0}+\frac{r_{0}^{2}}{\omega
r},$ (10)

where we have used the energy density $\rho_{C}\left(r\right)$ of
Eq.$\left(\ref{rhoC}\right)$ and the radial pressure $p_{r}\left(r\right)$
described by Eq.$\left(\ref{P(a)}\right)$ as a source with $d$ replaced by
$r$. A fundamental property of a wormhole is that a flaring out condition of
the throat, given by $(b-b^{\prime}r)/b^{2}>0$, must be satisfied as well as
the request that $1-b(r)/r>0$. Furthermore, at the throat $b(r_{0})=r_{0}$ and
the condition $b^{\prime}(r_{0})<1$ is imposed to have wormhole solutions. It
is also fundamental that there are no horizons present, which are identified
as the surfaces with $e^{2\Phi(r)}\rightarrow 0$, so that $\Phi(r)$ must be
finite everywhere. The procedure used to obtain a Casimir wormhole can be
extended to include also the electromagnetic field as additional source. The
key point is in the following observation: the algebraic structure of stress-
energy tensors for electromagnetic fields is determined byDymnikova ; DG

$T_{0}^{0}=T_{1}^{1}$ (11)

that it means

$\rho=-p_{r}.$ (12)

As a warm up exercise we can consider a pure spherically symmetric
electromagnetic field without the contribution of the Casimir energy, to see
if it is possible to build a TW even if the energy density is positive. In an
orthonormal frame, the SET we are going to consider is the following

$T_{\mu\nu}^{EM}=\frac{Q^{2}}{2\left(4\pi\right)^{2}\varepsilon_{0}r^{4}}diag(1,-1,1,1)$
(13)

which is conserved and traceless. From the energy density $\rho$ of the SET,
it is possible to obtain the following shape function

$b\left(r\right)=r_{0}+\frac{r_{2}^{2}}{r_{0}}-\frac{r_{2}^{2}}{r};\qquad
r_{2}^{2}=\frac{GQ^{2}}{4\pi c^{4}\varepsilon_{0}},$ (14)

which has the correct properties. Indeed

$b^{\prime}\left(r_{0}\right)=\frac{r_{2}^{2}}{r_{0}^{2}}<1\qquad\Longrightarrow\qquad
r_{2}<r_{0}$ (15)

and

$\displaystyle 1-\frac{b(r)}{r}$
$\displaystyle=\frac{\left(r-r_{0}\right)\left(rr_{0}-r_{2}^{2}\right)}{r_{0}r}>0\qquad
if\qquad r>r_{0};r>\frac{r_{2}^{2}}{r_{0}}$ (16)
$\displaystyle\qquad\frac{b-b^{\prime}r}{b^{2}}$
$\displaystyle=\frac{r_{0}r^{2}\left(r_{0}^{2}+r_{2}^{2}\right)}{\left(r_{0}^{2}r+r_{2}^{2}r-r_{2}^{2}r_{0}\right)^{2}}>0.$
(17)

Eq.$\left(\ref{pr0}\right)$ together with the shape function
$\left(\ref{b(r)E}\right)$ allow the computation of the redshift function

$\Phi(r)=\frac{1}{2}\ln\left(\frac{\left(r-r_{0}\right)\left(rr_{0}-r_{2}^{2}\right)}{r^{2}}\right)+C{,}$
(18)

where $C$ is an integration constant222Actually, because of the property
$\left(\ref{prho}\right)$, from Eq.$\left(\ref{pr0}\right)$, one can write
$\frac{2}{r}\left(1-\frac{b\left(r\right)}{r}\right)\Phi^{\prime}(r)-\frac{b\left(r\right)}{r^{3}}+\frac{b^{\prime}\left(r\right)}{r^{2}}=0$
(19) which has the following solution
$\Phi(r)=\frac{1}{2}\ln\left(1-\frac{b\left(r\right)}{r}\right)+C,$ (20) which
is a signature of a Black Hole.. As expected, we cannot adopt the strategy of
Ref.CW  because the $\ln\left(r-r_{0}\right)$ never disappears for every
choice of $r_{0}$, even if property $\left(\ref{prho}\right)$ is satisfied.
One could insist by imposing an inhomogeneous EoS of the form

$\omega\left(r\right)=-\frac{b\left(r\right)}{rb^{\prime}\left(r\right)}=\frac{r_{0}r_{2}^{2}-\left(r_{0}^{2}+r_{2}^{2}\right)r}{r_{2}^{2}r_{0}}$
(21)

allowing to fix $\Phi(r)=0$. Nevertheless, Eq.$\left(\ref{o1}\right)$ must be
compatible with

$\omega\left(r\right)=\frac{p_{r}\left(r\right)}{\rho\left(r\right)}=-1.$ (22)

This leads to

$\omega\left(r_{0}\right)=-1$ (23)

which is incompatible with the flare out condition $\left(\ref{fo}\right)$,
because one gets

$r_{0}=r_{2}.$ (24)

This means that a pure electromagnetic field cannot support a TW even if
$\rho+p_{r}=0$. It is necessary to have $\rho+p_{r}<0$. For this reason, we
are led to consider the superposition of the Casimir source with the
electromagnetic field. Such a combination potentially could produce a
different result thanks to the property $\left(\ref{prho}\right)$ which
defines an electrovacuum source. Note that such an electrovacuum source has
been investigated earlier in Ref.STP  even in the context of G.U.P.
distortions. Note also that the idea of including an electric charge or an
electromagnetic field in a TW configuration is not new. Indeed, the first
proposal is due to Kim and LeeKimLee  who considered a combination between the
Morris-Thorne wormhole and the Reissner-Nordström spacetime. Balakin et al.
discussed a nonminimal Einstein-Maxwell model BLZ . KuhfittigKuhfittig
considered a modification of the Kim and Lee charged wormhole to have
compatibility with the quantum inequality of Ford and RomanFR . The purpose of
this paper is to repeat the same procedure which has led to the original
Casimir wormhole spacetime to see if it is possible to obtain new EFE
solutions with an additional electric field. The rest of the paper is
structured as follows, in section II we continue the investigation to
determine if the Casimir energy density with an additional electric charge can
be considered as a source for a traversable wormhole, in section III we
examine the features of the Casimir wormhole with the additional electric
charge, in section IV we consider the Casimir energy and the additional
electric charge with the plates separation regarded as a parameter instead of
a variable. We summarize and conclude in section V.

##  II The Casimir Traversable Wormhole with an Additional Electric Charge

In this section we assume that our exotic matter will be represented by the
Casimir energy density $\left(\ref{rhoC}\right)$. Following Ref.CW , we
promote the constant plates separation $d$ to a radial coordinate $r$. In
addition to the Casimir source we include the contribution of an electric
field generated by a point charge. Since it is the NEC that must be violated,
the following inequality $\rho\left(r\right)+p_{r}\left(r\right)<0$ must hold.
We want to draw the reader’s attention that, thanks to the property
$\left(\ref{prho}\right)$, we can write

$\rho\left(r\right)+p_{r}\left(r\right)=\rho_{C}\left(r\right)+p_{r,C}\left(r\right)+\rho_{E}\left(r\right)+p_{r,E}\left(r\right)=-\frac{4\hbar
c\pi^{2}}{720r^{4}}<0,$ (25)

where

$\rho_{C}\left(r\right)=-\frac{\hbar c\pi^{2}}{720r^{4}};\qquad
p_{r,C}\left(r\right)=-\frac{3\hbar
c\pi^{2}}{720r^{4}};\qquad\rho_{E}\left(r\right)=\frac{Q^{2}}{2\left(4\pi\right)^{2}\varepsilon_{0}r^{4}};\qquad
p_{r,E}\left(r\right)=-\frac{Q^{2}}{2\left(4\pi\right)^{2}\varepsilon_{0}r^{4}}.$
(26)

In this context, the total energy density is represented by

$\rho\left(r\right)=\rho_{C}\left(r\right)+\rho_{E}\left(r\right)=-\frac{\hbar
c\pi^{2}}{720r^{4}}+\frac{Q^{2}}{2\left(4\pi\right)^{2}\varepsilon_{0}r^{4}}=-\frac{r_{1}^{2}}{\kappa
r^{4}}+\frac{r_{2}^{2}}{\kappa r^{4}},$ (27)

where

$\displaystyle r_{1}^{2}$ $\displaystyle=\frac{\pi^{3}l_{p}^{2}}{90},$ (28)
$\displaystyle r_{2}^{2}$ $\displaystyle=\frac{GQ^{2}}{4\pi
c^{4}\varepsilon_{0}}.$ (29)

Thus

$\begin{array}[c]{c}\rho\left(r\right)<0\\\ \rho\left(r\right)=0\\\
\rho\left(r\right)>0\end{array}\qquad
when\qquad\begin{array}[c]{c}r_{1}>r_{2}\\\ r_{1}=r_{2}\\\
r_{1}<r_{2}\end{array}.$ (30)

The shape function $b\left(r\right)$ can be obtained plugging
$\rho\left(r\right)$ of Eq.$\left(\ref{rho(r)}\right)$ into
Eq.$\left(\ref{rhoEFE}\right)$, whose solution leads to

$b\left(r\right)=r_{0}+\left(r_{2}^{2}-r_{1}^{2}\right)\int_{r_{0}}^{r}\frac{dr^{\prime}}{r^{\prime
2}}=r_{0}-\frac{r_{1}^{2}-r_{2}^{2}}{r_{0}}+\frac{r_{1}^{2}-r_{2}^{2}}{r},$
(31)

where the throat condition $b(r_{0})=r_{0}$ has been imposed. The redshift
function can be obtained by solving Eq.$\left(\ref{pr0}\right)$ with the help
of the shape function $\left(\ref{b(r)0}\right)$. One finds

$\frac{2}{r}\left(1-\frac{r_{0}}{r}+\frac{r_{1}^{2}-r_{2}^{2}}{r_{0}r}-\frac{r_{1}^{2}-r_{2}^{2}}{r^{2}}\right)\Phi^{\prime}\\!\left(r\right)-\frac{r_{0}}{r^{3}}+\frac{r_{1}^{2}-r_{2}^{2}}{r_{0}r^{3}}-\left(1-\omega\right)\frac{r_{1}^{2}-r_{2}^{2}}{r^{4}}=0,$
(32)

where we have used an EoS of the form
$p_{r}\left(r\right)=\omega\rho\left(r\right)$. The solution can be written as

$\displaystyle\Phi\left(r\right)=\left(\frac{r_{0}^{2}-\omega\left(r_{1}^{2}-r_{2}^{2}\right)}{2\left(r_{0}^{2}+r_{1}^{2}-r_{2}^{2}\right)}\right)\ln\\!\left(r-r_{0}\right)$
$\displaystyle-\left(\frac{\omega
r_{0}^{2}-r_{1}^{2}+r_{2}^{2}}{2\left(r_{0}^{2}+r_{1}^{2}-r_{2}^{2}\right)}\right)\ln\\!\left(rr_{0}+r_{1}^{2}-r_{2}^{2}\right)+\frac{\omega-1}{2}\ln\\!\left(r\right)+C.$
(33)

If $Q=0$, one recovers the familiar form of the Casimir wormhole redshift
function. It is possible to eliminate the horizon if we constrain $\omega$ to
be

$\omega=\omega_{0}=\frac{r_{0}^{2}}{r_{1}^{2}-r_{2}^{2}}$ (34)

and the redshift function $\left(\ref{Phi(r)}\right)$ becomes

$\Phi\left(r\right)=-\left(\frac{\omega-1}{2}\right)\ln\\!\left(\frac{\omega
rr_{0}+r_{0}^{2}}{\omega r}\right)+C.$ (35)

By assuming that $\Phi\left(r\right)\rightarrow 0$ for $r\rightarrow\infty$,
then we find

$\Phi\left(r\right)=\frac{\omega-1}{2}\ln\\!\left(\frac{\omega r}{\omega
r+r_{0}}\right),$ (36)

which is formally the same result of Ref.CW . The shape function
$\left(\ref{b(r)0}\right)$ can be rearranged to obtain the familiar form of
the Casimir wormholeCW

$b\\!\left(r\right)=r_{0}-\frac{r_{1}^{2}-r_{2}^{2}}{r_{0}}+\frac{r_{1}^{2}-r_{2}^{2}}{r}=r_{0}\left(1-\frac{1}{\omega}\right)+\frac{r_{0}^{2}}{\omega
r}.$ (37)

On the other hand, since the ratio in Eq.$\left(\ref{o}\right)$ is not
constrained, we can use the ratio $p_{r}\left(r\right)/\rho\left(r\right)$ to
extract information about the size of the throat. It is immediate to see that
for

$\omega=\frac{p_{r}\left(r\right)}{\rho\left(r\right)}=\frac{3r_{1}^{2}+r_{2}^{2}}{r_{1}^{2}-r_{2}^{2}};\qquad
r_{1}\neq r_{2},$ (38)

the EoS is satisfied and is independent on $r$. Plugging the value of $\omega$
in Eq.$\left(\ref{or1r2}\right)$ into Eq.$\left(\ref{o}\right)$, one finds

$r_{0}=\sqrt{3r_{1}^{2}+r_{2}^{2}}.$ (39)

Note that $r_{2}$ can be variable, while $r_{1}$ is not. It is convenient to
take $r_{1}$ as a reference scale. Thus, if we introduce a dimensionless
variable

$x=\frac{r_{2}}{r_{1}}=\sqrt{\frac{90GQ^{2}}{4\pi
c^{4}\varepsilon_{0}\pi^{3}l_{p}^{2}}}=\sqrt{\frac{90}{\pi^{3}}n^{2}\frac{e^{2}}{4\pi\varepsilon_{0}\hbar
c}}=\frac{3n}{\pi}\sqrt{\frac{10}{\pi}\alpha},$ (40)

one finds

$\omega=\frac{3+x^{2}}{1-x^{2}};\qquad r_{0}=r_{1}\sqrt{3+x^{2}};\qquad x\neq
1,$ (41)

where $Q=ne$, $e$ is the electron charge, $\alpha$ is the fine structure
constant and $n$ is the total number of the electron charges. Note that, for

$Q\rightarrow 0,\qquad r_{2}\rightarrow 0\qquad\Longrightarrow x\rightarrow
0,$ (42)

one recovers the shape function of Ref.CW  with $\omega=3$. On the other hand,
when

$Q\rightarrow\infty,\qquad r_{2}\rightarrow\infty\qquad\Longrightarrow
x\rightarrow\infty$ (43)

and $\omega\rightarrow-1$. In conclusion, we can write

$\displaystyle\omega$ $\displaystyle\in\left(3,+\infty\right);\qquad
x\in\left(0,1^{-}\right)$ (44) $\displaystyle\omega$
$\displaystyle\in\left(-\infty,-1\right);\qquad
x\in\left(1^{+},+\infty\right).$ (45)

A comment on the limit $\left(\ref{Limit}\right)$ is in order. Indeed, when
$\omega\rightarrow-1$, the shape function assumes the form

$b\left(r\right)=2r_{0}-\frac{r_{0}^{2}}{r}$ (46)

and the redshift function is

$\Phi\left(r\right)=\ln\\!\left(\frac{r-r_{0}}{r}\right)$ (47)

representing no more a TW, but a black hole. However, $\omega\rightarrow-1$ is
a limiting value which never will be reached. This means that $Q$ can be
arbitrarily large but finite. Note that in this range $\omega<-1$. This means
that the pure electromagnetic field, in this context, acts as a phantom energy
source. For completeness, we report the expression of the transverse pressure
which, in terms of $\omega$, is identical to $p_{t}\left(r\right)$ of Ref.CW .
One finds

$p_{t}\left(r\right)=\frac{r_{0}^{2}}{\kappa\omega{r}^{4}}\left[\omega+\frac{r_{0}\left(1-{\omega}^{2}\right)}{4\omega\,\left(\omega\,r+r_{0}\right)}\right]=\omega_{t}\left(r\right)\left(\frac{r_{0}^{2}}{\kappa\omega\,{r}^{4}}\right)=\omega_{t}\left(r\right)\rho(r),$
(48)

where we have introduced an inhomogeneous EoS on the transverse pressure with

$\omega_{t}\left(r\right)=-\left(\omega+\frac{r_{0}\left(1-{\omega}^{2}\right)}{4\omega\,\left(\omega\,r+r_{0}\right)}\right),$
(49)

and the final form of the SET is

$T_{\mu\nu}=\frac{r_{0}^{2}}{\kappa\omega\,{r}^{4}}\left[diag\left(-1,-\omega,\omega_{t}\left(r\right),\omega_{t}\left(r\right)\right)\right].$
(50)

The conservation of the SET is satisfied but a comparison with the SET source
shows that

$T_{\mu\nu}=T_{\mu\nu}^{Source}-\frac{1}{\kappa
r^{4}}\left[diag\left(0,0,\omega_{t}\left(r\right)-\left(r_{1}^{2}+r_{2}^{2}\right),\omega_{t}\left(r\right)-\left(r_{1}^{2}+r_{2}^{2}\right)\right)\right],$
(51)

namely a discrepancy on the transverse pressure with respect to the SET source
is present. We recall that the SET source is defined by

$T_{\mu\nu}^{Source}=T_{\mu\nu}^{Casimir}+T_{\mu\nu}^{EM}=\frac{1}{\kappa
r^{4}}\left[diag\left(-r_{1}^{2}+r_{2}^{2},-\left(3r_{1}^{2}+r_{2}^{2}\right),r_{1}^{2}+r_{2}^{2},r_{1}^{2}+r_{2}^{2}\right)\right].$
(52)

It is important to observe that there exists another interesting value for
$\omega$, namely when $\omega=1$. For this special choice, one finds that the
line element reduces to the Ellis-Bronnikov (EB) wormholeellisGL ; Bronnikov ,
namely

$ds^{2}=-\,dt^{2}+\frac{dr^{2}}{1-\frac{r_{0}^{2}}{r^{2}}}+r^{2}\,d\Omega^{2}$
(53)

whose associated SET is

$T_{\mu\nu}^{EB}=\frac{r_{1}^{2}-r_{2}^{2}}{\kappa{r}^{4}}\left[diag\left(-1,-1,1,1\right)\right].$
(54)

Nevertheless $\omega=1$ is incompatible with the relationship
$\left(\ref{or1r2}\right)$ and therefore this option will be discarded.

###  II.1 Special case $r_{1}^{2}=r_{2}^{2}$

In the special case

$r_{1}^{2}=r_{2}^{2}=r_{e}^{2},$ (55)

we find that the energy density vanishes. Therefore the wormhole shape
function is

$b\left(r\right)=r_{0}.$ (56)

On the other hand the redshift function appears to be non trivial. Indeed,
from the EFE $\left(\ref{pr0}\right)$, we find

$\frac{2}{r}\left(1-\frac{r_{0}}{r}\right)\Phi^{\prime}\\!\left(r\right)-\frac{r_{0}}{r^{3}}+\frac{4r_{e}^{2}}{r^{4}}=0$
(57)

which can be rearranged to give

$\Phi^{\prime}\\!\left(r\right)=\frac{r_{0}r-4r_{e}^{2}}{2\left(r-r_{0}\right)r^{2}}.$
(58)

The solution is

$\Phi\left(r\right)=-\frac{\ln\\!\left(r\right)}{2}+\frac{2\ln\\!\left(r\right)r_{e}^{2}}{r_{0}^{2}}-\frac{2r_{e}^{2}}{r_{0}r}+\frac{\ln\\!\left(r-r_{0}\right)}{2}\left(1-\frac{4r_{e}^{2}}{r_{0}^{2}}\right)+C.$
(59)

It is immediate to see that for

$r_{0}=2r_{e}$ (60)

the horizon disappears and

$\Phi\left(r\right)=-\frac{r_{0}}{2r},$ (61)

where we have assumed that $\Phi\left(r\right)\rightarrow 0$ for
$r\rightarrow\infty$. Therefore in this special case we still have a TW with
the following line element

$ds^{2}=-\exp\left(-\frac{r_{0}}{r}\right)\,dt^{2}+\frac{dr^{2}}{1-r_{0}/r}+r^{2}d\Omega^{2}\,,$
(62)

which is traversable in principle but not in practice because the throat is
Planckian. To complete this special case, we compute the transverse pressure
and we find

$\Bigg{\\{}\frac{r_{0}^{2}}{r^{4}}-\frac{r_{0}^{3}}{4r^{5}}\Bigg{\\}}=\kappa
p_{t}(r).$ (63)

Note that for this special case, we cannot impose an EoS of the form
$p_{r}\left(r\right)=\omega\rho\left(r\right)$ because $\rho\left(r\right)$ is
vanishing. In the next section, we are going to explore some of the features
of the TW obtained by the Casimir source and the electromagnetic field.

##  III Properties of the Casimir wormhole with an additional electromagnetic
field

In section II, we have introduced the shape function
$\left(\ref{b(r)0}\right)$ or $\left(\ref{b(r)00}\right)$ obtained by the
Casimir energy plus the electromagnetic field. Here we want to discuss some of
its properties. The first quantity we are going to analyze is the proper
radial distance, defined by

$l\left(r\right)=\pm\int_{r_{0}}^{r}\frac{dr^{\prime}}{\sqrt{1-\frac{b\left(r^{\prime}\right)}{r^{\prime}}}}.$
(64)

In this specific case, plugging Eq.$\left(\ref{b(r)00}\right)$ into
Eq.$\left(\ref{l(r)}\right)$, one gets

$\displaystyle
l\left(r\right)=\pm\int_{r_{0}}^{r}\frac{dr^{\prime}}{\sqrt{1-\frac{r_{0}}{r^{\prime}}\left(1-\frac{1}{\omega}\right)-\frac{r_{0}^{2}}{\omega
r^{\prime 2}}}}=\pm\left(\frac{\sqrt{r-r_{0}}\,\sqrt{\omega
r+r_{0}}\,}{\sqrt{\omega}}\right.$
$\displaystyle\left.+r_{0}\frac{\omega-1}{2\omega}\ln\\!\left(\frac{r_{0}+\left(2r-r_{0}\right)\omega+2\sqrt{r-r_{0}}\,\sqrt{\omega
r+r_{0}}\,\sqrt{\omega}}{\\!\left(\omega+1\right)r_{0}}\right)\right).$ (65)

We find

$l\left(r\right)\simeq\begin{array}[c]{ll}\pm\left(r+\frac{r_{0}\left(\omega-1\right)}{2\omega}\left(\ln\\!\left(\frac{4\omega
r}{r_{0}\left(\omega+1\right)}\right)\\!-1\right)+O\left(\frac{1}{r}\right)\right)&r\rightarrow\infty\\\
\pm
2\sqrt{\frac{r_{0}\omega}{\omega+1}\left(r-r_{0}\right)}+O\left(\left(r-r_{0}\right)^{\frac{3}{2}}\right)&r\rightarrow
r_{0}\end{array},$ (66)

where the“$\pm$” depends on the wormhole side we are. The proper radial
distance is an essential tool to estimate the possible time trip in going from
one station located in the lower universe, say at $l=-l_{1}$, and ending up in
the upper universe station, say at $l=l_{2}$. Following Ref.MT , we shall
locate $l_{1}$ and $l_{2}$ at a value of the radius such that $l_{1}\simeq
l_{2}\simeq 10^{4}r_{0}$ that it means $1-b\left(r\right)/r\simeq 1$. Assume
that the traveller has a radial velocity $v\left(r\right)$, as measured by a
static observer positioned at $r$. One may relate the proper distance
travelled $dl$, radius travelled $dr$, coordinate time lapse $dt$, and proper
time lapse as measured by the observer $d\tau$, by the following relationships

$v=e^{-\Phi\left(r\right)}\frac{dl}{dt}=e^{-\Phi\left(r\right)}\left(1-\frac{b\left(r\right)}{r}\right)^{-\frac{1}{2}}\frac{dr}{dt}$
(67)

and

$v\gamma=\frac{dl}{d\tau}=\mp\left(1-\frac{b\left(r\right)}{r}\right)^{-\frac{1}{2}}\frac{dr}{d\tau};\qquad\gamma=\left(1-\frac{v^{2}\left(r\right)}{c^{2}}\right)^{-\frac{1}{2}}$
(68)

respectively. If the traveler journeys with constant speed $v$, then the total
time is given by

$\Delta
t=\int_{r_{0}}^{r}\frac{e^{-\Phi\left(r^{\prime}\right)}dr^{\prime}}{v\sqrt{1-\frac{b\left(r^{\prime}\right)}{r^{\prime}}}}=\omega^{1-\frac{\omega}{2}}\int_{r_{0}}^{r}\frac{r^{{}^{\prime}\frac{3}{2}-\frac{\omega}{2}}\left(\omega
r^{\prime}+r_{0}\right)^{-1+\frac{\omega}{2}}}{v\sqrt{r^{\prime}-r_{0}}}dr^{\prime}$
(69)

while the proper total time is

$\displaystyle\Delta\tau=\int_{r_{0}}^{r}\frac{dr^{\prime}}{v\sqrt{1-\frac{b\left(r^{\prime}\right)}{r^{\prime}}}}=\frac{\sqrt{r-r_{0}}\,\sqrt{\omega
r+r_{0}}\,}{v\sqrt{\omega}}$
$\displaystyle+r_{0}\frac{\omega-1}{2\omega}\ln\\!\left(\frac{r_{0}+\left(2r-r_{0}\right)\omega+2\sqrt{r-r_{0}}\,\sqrt{\omega
r+r_{0}}\,\sqrt{\omega}}{\\!\left(\omega+1\right)r_{0}}\right).$ (70)

To evaluate $\Delta t$ we can proceed with the following approximations. Close
to the throat, one finds

$\displaystyle\Delta t$
$\displaystyle\simeq\omega^{1-\frac{\omega}{2}}\int_{r_{0}}^{r}\frac{r^{{}^{\prime}\frac{3}{2}-\frac{\omega}{2}}\left(\omega
r^{\prime}+r_{0}\right)^{-1+\frac{\omega}{2}}}{v\sqrt{r^{\prime}-r_{0}}}dr^{\prime}$
$\displaystyle\simeq\frac{2\sqrt{r_{0}}}{v}\left(\frac{\omega}{\omega+1}\right)^{1-\frac{\omega}{2}}\sqrt{r-r_{0}}$
(71)

and, in this range, $\Delta t\simeq\Delta\tau$ except for the value
$\omega=-1$, where the TW turns into a Black Hole. On the other hand, when
$r\rightarrow\infty$, one finds

$\Delta
t\simeq\int_{r_{0}}^{r}\frac{r^{{}^{\prime}\frac{1}{2}}}{v\sqrt{r^{\prime}-r_{0}}}dr^{\prime}\simeq\frac{r_{0}}{v}\left(1-\frac{1}{\omega}\right)\ln\\!\left(\frac{r}{r_{0}}\right)+\frac{r}{v}$
(72)

and even with this approximation, the leading term is the same of
$\Delta\tau$. On the same ground, we can compute the embedded surface, which
is defined by

$z\left(r\right)=\pm\int_{r_{0}}^{r}\frac{dr^{\prime}}{\sqrt{\frac{r^{\prime}}{b\left(r^{\prime}\right)}-1}}$
(73)

and, in the present case, we find

$\displaystyle
z\left(r\right)=\pm\int_{r_{0}}^{r}\frac{\sqrt{r_{0}}\,\sqrt{\left(\omega-1\right)r^{\prime}+r_{0}}}{\sqrt{\omega
r^{\prime}+r_{0}}\,\sqrt{r^{\prime}-r_{0}}}dr^{\prime}$
$\displaystyle\pm\frac{2r_{0}}{\omega^{2}}\left(F\\!\left(\frac{\sqrt{r-r_{0}}\,\sqrt{\omega}}{\sqrt{\omega
r+r_{0}}},\frac{1}{\omega}\right)+\Pi\\!\left(\frac{\sqrt{r-r_{0}}\,\sqrt{\omega}}{\sqrt{\omega
r+r_{0}}},1,\frac{1}{\omega}\right)\left(\omega^{2}-1\right)\right),$ (74)

where $F\left(\varphi,k\right)$ is the elliptic integral of the first kind and
$\Pi\left(\varphi,n,k\right)$ is the elliptic integral of the third kind.
Close to the throat, one can write

$z\left(r\right)\simeq\pm
2\sqrt{r_{0}}\sqrt{\frac{\omega}{1+\omega}}\sqrt{r-r_{0}}.$ (75)

It is interesting to note the singularity appearing when $\omega=-1$, showing
the presence of a Black Hole. To further investigate the properties of the
shape function $\left(\ref{b(r)00}\right)$, we consider the computation of the
total gravitational energy for a wormholeNZCP , defined as

$\displaystyle
E_{G}\left(r\right)=\int_{r_{0}}^{r}\left[1-\sqrt{\frac{1}{1-b\left(r^{\prime}\right)/r^{\prime}}}\right]\rho\left(r^{\prime}\right)dr^{\prime}r^{\prime
2}$ $\displaystyle+\frac{r_{0}}{2G}=\left(M-M_{\pm}^{P}\right)c^{2},$ (76)

where $M$ is the total mass $M$ and $M^{P}$ is the proper mass, respectively.
Even in this case, the “$\pm$” depends one the wormhole side we are. In
particular

$\displaystyle
M=\int_{r_{0}}^{r}\frac{4\pi}{c^{2}}\rho\left(r^{\prime}\right)r^{\prime
2}dr^{\prime}+\frac{r_{0}}{2Gc^{2}}=\frac{c^{2}}{2G}\left(r_{0}\left(1-\frac{1}{\omega}\right)+\frac{r_{0}^{2}}{\omega
r}-r_{0}\right)+\frac{r_{0}c^{2}}{2G}$
$\displaystyle=\frac{c^{2}}{2G}\left(-\frac{r_{0}}{\omega}+\frac{r_{0}^{2}}{\omega
r}\right)+\frac{r_{0}}{2Gc^{2}}\underset{r\rightarrow\infty}{\simeq}-\frac{r_{0}c^{2}}{2G\omega}+\frac{r_{0}c^{2}}{2G}=\frac{r_{0}c^{2}}{2G}\left(1-\frac{1}{\omega}\right)$
(77)

and

$\displaystyle M_{\pm}^{P}$
$\displaystyle=\pm\frac{4\pi}{c^{2}}\int_{r_{0}}^{r}\frac{\rho\left(r^{\prime}\right)r^{\prime
2}}{\sqrt{1-b\left(r^{\prime}\right)/r^{\prime}}}dr^{\prime}=\pm\frac{c^{2}}{2G}\int_{r_{0}}^{r}\frac{b^{\prime}\left(r^{\prime}\right)}{\sqrt{1-b\left(r^{\prime}\right)/r^{\prime}}}dr^{\prime}$
$\displaystyle\mp$
$\displaystyle\frac{r_{0}c^{2}}{4G\sqrt{\omega}}\left(\pi-2\arctan\\!\left(\frac{r\omega-r+2r_{0}}{2\sqrt{r\omega+r_{0}}\,\sqrt{r-r_{0}}}\right)\right)$
$\displaystyle\underset{r\rightarrow\infty}{\simeq}\mp\frac{r_{0}c^{2}}{4G\sqrt{\omega}}\left(\pi-2\arctan\\!\left(\frac{\omega-1}{2\sqrt{\omega}}\right)\right).$
(78)

Thus, at infinity one finds

$E_{G}\left(r\right)\underset{r\rightarrow\infty}{\simeq}\frac{r_{0}c^{2}}{2G}\left[\left(1-\frac{1}{\omega}\right)\mp\frac{1}{2\sqrt{\omega}}\left(\pi-2\arctan\\!\left(\frac{\omega-1}{2\sqrt{\omega}}\right)\right)\right].$
(79)

An important traversability condition is that the acceleration felt by the
traveller should not exceed Earth’s gravity $g_{\oplus}\simeq 980$ $cm/s^{2}$.
In an orthonormal basis of the traveller’s proper reference frame, we can find

$\left|\mathbf{a}\right|=\left|\sqrt{1-\frac{b\left(r\right)}{r}}e^{-\Phi\left(r\right)}\left(\gamma
e^{\Phi\left(r\right)}\right)^{\prime}\right|\leq\frac{g_{\oplus}}{c^{2}}.$
(80)

If we assume a constant speed and $\gamma\simeq 1$, then we can write

$\left|\mathbf{a}\right|=\left|\sqrt{1-\frac{r_{0}}{r}\left(1-\frac{1}{\omega}\right)-\frac{r_{0}^{2}}{\omega
r^{2}}}\frac{\left(\omega-1\right)r_{0}}{2r\left(\omega
r+r_{0}\right)}\right|\leq\frac{g_{\oplus}}{c^{2}}.$ (81)

We can see that in proximity of the throat, the traveller has a vanishing
acceleration. Always following Ref.MT , we can estimate the tidal forces by
imposing an upper bound represented by $g_{\oplus}$. The radial tidal
constraint

$\displaystyle\left|\left(1-\frac{b\left(r\right)}{r}\right)\left[\Phi^{\prime\prime}\left(r\right)+\left(\Phi^{\prime}\left(r\right)\right)^{2}-\frac{b^{\prime}\left(r\right)r-b\left(r\right)}{2r\left(r-b\left(r\right)\right)}\Phi^{\prime}\left(r\right)\right]\right|\times$
$\displaystyle c^{2}\left|\eta^{\hat{1}^{\prime}}\right|\leq g_{\oplus},$ (82)

constrains the redshift function, and the lateral tidal constraint

$\displaystyle\left|\frac{\gamma^{2}c^{2}}{2r^{2}}\left[\frac{v^{2}\left(r\right)}{c^{2}}\left(b^{\prime}\left(r\right)-\frac{b\left(r\right)}{r}\right)+2r\left(r-b\left(r\right)\right)\Phi^{\prime}\left(r\right)\right]\right|\times$
$\displaystyle\left|\eta^{\hat{2}^{\prime}}\right|\leq g_{\oplus},$ (83)

constrains the velocity with which observers traverse the wormhole.
$\eta^{\hat{1}^{\prime}}$ and $\eta^{\hat{2}^{\prime}}$ represent the size of
the traveller. In Ref.MT , they are fixed approximately equal, at the symbolic
value of $2$ $m$. Close to the throat, the radial tidal constraint
$\left(\ref{RTC}\right)$ becomes

$\displaystyle\left|\left[\frac{b\left(r\right)-b^{\prime}\left(r\right)r}{2r^{2}}\Phi^{\prime}\left(r\right)\right]\right|$
$\displaystyle=\frac{\left(\left(\omega-1\right)r+2r_{0}\right)r_{0}^{2}\left(\omega-1\right)}{4\omega\left(\omega
r+r_{0}\right)r^{4}}\underset{r\rightarrow
r_{0}}{=}\frac{1}{6r_{0}^{2}}\leq\frac{g_{\oplus}}{c^{2}\left|\eta^{\hat{1}^{\prime}}\right|}$
$\displaystyle\qquad\Longrightarrow\qquad 10^{8}m\lesssim r_{0}.$ (84)

For the lateral tidal constraint, we find

$\displaystyle\frac{v^{2}r_{0}}{2r^{4}}\left|\frac{r\left(\omega-1\right)+2r_{0}}{\omega}\right|\left|\eta^{\hat{2}^{\prime}}\right|\lesssim
g_{\oplus}\qquad\Longrightarrow\qquad v\lesssim
r_{0}\sqrt{\left|\frac{\omega}{\omega+1}\right|g_{\oplus}}$
$\displaystyle\qquad\Longrightarrow\qquad v\lesssim
3.13r_{0}\sqrt{\left|\frac{\omega}{\omega+1}\right|}\text{ }m/s.$ (85)

If the observer has a vanishing $v$, then the tidal forces are null. We can
use these last estimates to complete the evaluation of the crossing time which
approximately is

$\Delta t\simeq\frac{\Delta
l}{3.13r_{0}\sqrt{\left|\frac{\omega}{\omega+1}\right|}}\left(\frac{\omega}{\omega+1}\right)^{-\frac{\omega}{2}}\simeq
6.4\times
10^{3}\left(\left|\frac{\omega}{\omega+1}\right|\right)^{-\frac{1}{2}}\left(\frac{\omega}{\omega+1}\right)^{-\frac{\omega}{2}}s,$
(86)

which is in agreement with the estimates found in Ref.MT , even for
$\omega\rightarrow\pm\infty$. The last property we are going to discuss is the
“total amount” of ANEC violating matter in the spacetimeVKD  which is
described by

$I_{V}=\int[\rho(r)+p_{r}(r)]dV$ (87)

and for the line element $\left(\ref{metric}\right)$, one can write

$I_{V}=\frac{1}{\kappa}\int\left(r-b\left(r\right)\right)\left[\ln\left(\frac{e^{2\Phi(r)}}{1-\frac{b\left(r\right)}{r}}\right)\right]^{\prime}dr,$
(88)

where the measure $dV$ has been changed into $r^{2}dr$. For the metric
$\left(\ref{b(r)00}\right)$, one obtains

$I_{V}=-\frac{1}{\kappa}\int_{r_{0}}^{\infty}\frac{\left(\omega+1\right)r_{0}^{2}}{\omega
r^{2}}dr=-\frac{\left(\omega+1\right)r_{0}}{\omega\kappa}.$ (89)

Even in this case, $I_{V}$ is finite everywhere. The reason is that the
structure of the shape function and of the redshift function are equal to the
pure Casimir wormholeCW  Therefore we can conclude that, in proximity of the
throat the ANEC can be arbitrarily small.

###  III.1 Properties of the Casimir wormhole with an additional
electromagnetic field for the special case $r_{1}^{2}=r_{2}^{2}$

In section II.1, we have considered the special case in which the negative
Casimir energy is compensated by the positive electromagnetic field with the
assumption that $r_{1}^{2}=r_{2}^{2}$. We want to discuss some of its
properties, even if the size of this TW is Planckian. Repeating the same steps
of Section III, we find that the proper radial distance is

$\displaystyle
l\left(r\right)=\pm\int_{r_{0}}^{r}\frac{dr^{\prime}}{\sqrt{1-\frac{r_{0}}{r^{\prime}}}}=\pm\left(\sqrt{r-r_{0}}\sqrt{r}\right.$
$\displaystyle\left.+\frac{r_{0}}{2}\ln\\!\left(1+2\frac{\sqrt{r-r_{0}}\,\left(\sqrt{r}+\sqrt{r-r_{0}}\right)\,}{\\!r_{0}}\right)\right)$
(90)

and its asymptotic behavior becomes

$l\left(r\right)\simeq\begin{array}[c]{ll}\pm\left(r+\frac{r_{0}}{2}\left(\ln\\!\left(\frac{4r}{r_{0}}\right)\\!-1\right)+O\left(\frac{1}{r^{2}}\right)\right)&r\rightarrow\infty\\\
\pm
2\sqrt{r_{0}\left(r-r_{0}\right)}+O\left(\left(r-r_{0}\right)^{\frac{3}{2}}\right)&r\rightarrow
r_{0}\end{array},$ (91)

where the“$\pm$” depends on the wormhole side we are. From
Eq.$\left(\ref{vt}\right)$ and Eq.$\left(\ref{vtau}\right)$, we can compute
the total time $\Delta t$ and the proper total time $\Delta\tau$,
respectively. The total time is

$\Delta
t=\int_{r_{0}}^{r}\frac{\sqrt{r^{\prime}}\exp\left(-\frac{r_{0}}{2r^{\prime}}\right)}{v\sqrt{r^{\prime}-r_{0}}}dr^{\prime}$
(92)

and it is bounded by the following inequality chain

$\frac{1}{v\sqrt{e}}\int_{r_{0}}^{r}\frac{\sqrt{r^{\prime}}}{\sqrt{r^{\prime}-r_{0}}}dr^{\prime}\leq\Delta
t\leq\frac{1}{v}\int_{r_{0}}^{r}\frac{\sqrt{r^{\prime}}}{\sqrt{r^{\prime}-r_{0}}}dr^{\prime}.$
(93)

On the other hand the proper total time is simply

$\Delta\tau=\frac{\Delta l}{v}.$ (94)

On the same ground, we can compute the embedded surface, which is defined by

$z\left(r\right)=\pm\int_{r_{0}}^{r}\frac{dr^{\prime}}{\sqrt{\frac{r^{\prime}}{b\left(r^{\prime}\right)}-1}}=\pm
2\sqrt{r_{0}}\sqrt{r-r_{0}}$ (95)

Note that in this special case, the total gravitational energy for a
wormholeNZCP  is vanishing. As regards the acceleration felt by the traveller,
the relationship $\left(\ref{acc}\right)$ becomes

$\left|\mathbf{a}\right|=\left|\sqrt{1-\frac{r_{0}}{r}}\frac{r_{0}}{2r^{2}}\right|\leq\frac{g_{\oplus}}{c^{2}},$
(96)

where we have assumed a constant speed and $\gamma\simeq 1$. Even in this
case, in proximity of the throat, the traveller has a vanishing acceleration.
On the other hand, the radial tidal constraint $\left(\ref{RTC}\right)$ and
the lateral tidal constraint $\left(\ref{LTC}\right)$ become respectively on
the throat

$\frac{r_{0}^{2}}{4r^{4}}\underset{r\rightarrow
r_{0}}{=}\frac{1}{4r_{0}^{2}}\leq\frac{g_{\oplus}}{c^{2}\left|\eta^{\hat{1}^{\prime}}\right|}\qquad\frac{10^{8}m}{2g_{\oplus}}\leq
r_{0}\Longrightarrow\qquad 10^{8}m\lesssim r_{0}$ (97)

and

$\frac{v^{2}}{2r^{2}}\left|-\frac{r_{0}}{r}\right|\left|\eta^{\hat{2}^{\prime}}\right|\lesssim
g_{\oplus}\qquad\Longrightarrow\qquad v\lesssim
r_{0}\sqrt{2g_{\oplus}}\qquad\Longrightarrow\qquad v\lesssim 4.43r_{0}\text{
}m/s,$ (98)

where we have assumed that the size of the traveller, described by
$\eta^{\hat{1}^{\prime}}$ and $\eta^{\hat{2}^{\prime}}$, is fixed at the
symbolic value of $2$ $m$. If the observer has a vanishing $v$, then the tidal
forces are null. We can use these last estimates to complete the evaluation of
the crossing time which approximately is

$\Delta t\simeq\frac{\Delta l}{4.43r_{0}}\simeq 4.5\times 10^{3}s,$ (99)

where we have assumed that $\Delta l\simeq 2\times 10^{4}r_{0}$ like in
Section III. It is important to observe that since $r_{0}$ has a Planckian
value, then one finds $\left|\eta^{\hat{1}^{\prime}}\right|\lesssim 2.1\times
10^{-43}\operatorname{m}$. This means that with a Planckian wormhole nothing
can traverse it. The last property we are going to discuss is the “total
amount” of ANEC violating matter in the spacetimeVKD  which is described by
Eq.$\left(\ref{IV}\right)$ and for the present case, one finds

$I_{V}=\frac{1}{\kappa}\int\left(r-r_{0}\right)\left[\ln\left(\frac{e^{-r0/r}}{1-\frac{r_{0}}{r}}\right)\right]^{\prime}dr=-\frac{1}{\kappa}\int_{r_{0}}^{\infty}\frac{r_{0}^{2}}{r^{2}}dr=-\frac{r_{0}}{\kappa},$
(100)

Even in this case, $I_{V}$ is finite everywhere and this corresponds to taking
the limit $\omega\rightarrow\infty$ in Eq.$\left(\ref{IVo}\right)$.

##  IV The Casimir Traversable Wormhole with an Additional Electric Charge:
the constant Plates separation case

In this section we consider the following setting for the energy density and
radial pressure

$\rho_{C}\left(d\right)=-\frac{\hbar c\pi^{2}}{720d^{4}};\qquad
p_{r,C}\left(d\right)=-\frac{3\hbar
c\pi^{2}}{720d^{4}};\qquad\rho_{E}\left(r\right)=\frac{Q^{2}}{2\left(4\pi\right)^{2}\varepsilon_{0}r^{4}};\qquad
p_{r,E}\left(r\right)=-\frac{Q^{2}}{2\left(4\pi\right)^{2}\varepsilon_{0}r^{4}},$
(101)

namely that our exotic matter will be represented by the Casimir energy
density $\left(\ref{rhoC}\right)$ and only the electric field is variable with
a radial coordinate $r$. Of course, even in this case we find that the NEC is
violated, namely

$\rho\left(r\right)+p_{r}\left(r\right)=\rho_{C}\left(d\right)+p_{r,C}\left(d\right)+\rho_{E}\left(r\right)+p_{r,E}\left(r\right)=-\frac{4\hbar
c\pi^{2}}{720d^{4}}<0.$ (102)

The total energy density is represented by

$\rho\left(r\right)=\rho_{C}\left(d\right)+\rho_{E}\left(r\right)=-\frac{\hbar
c\pi^{2}}{720d^{4}}+\frac{Q^{2}}{2\left(4\pi\right)^{2}\varepsilon_{0}r^{4}}.$
(103)

It is interesting to observe that, differently from Eq.
$\left(\ref{rho(r)}\right)$, the energy density $\left(\ref{rho(r,d)}\right)$
vanishes when

$r=\bar{r}=d\sqrt{\frac{r_{2}}{r_{1}}}=\frac{d\sqrt{n}}{\pi}\sqrt[4]{90\pi\alpha},$
(104)

where $Q=ne$, $e$ is the electron charge, $\alpha$ is the fine structure
constant and $n$ is the total number of the electron charges. In particular,
we find that

$\rho\left(r\right)\gtreqless 0,\qquad
when\qquad\left\\{\begin{array}[c]{c}r_{0}\leq r<\bar{r}\\\ r=\bar{r}\\\
r>\bar{r}\end{array}\right..$ (105)

The shape function $b\left(r\right)$ can be obtained plugging
$\rho\left(r\right)$ $\left(\ref{rho(r)}\right)$ into
Eq.$\left(\ref{rhoEFE}\right)$, whose solution leads to

$\displaystyle b\left(r\right)$ $\displaystyle=r_{0}+\frac{GQ^{2}}{2\pi
c^{4}\varepsilon_{0}}\int_{r_{0}}^{r}\frac{dr^{\prime}}{r^{\prime
2}}-\frac{\pi^{3}}{90d^{4}}\left(\frac{\hbar
G}{c^{3}}\right)\int_{r_{0}}^{r}r^{\prime 2}dr^{\prime}$
$\displaystyle=r_{0}+\frac{r_{2}^{2}}{r_{0}}-\frac{r_{2}^{2}}{r}-\frac{r_{1}^{2}}{3d^{4}}\left(r^{3}-r_{0}^{3}\right),$
(106)

where $r_{1}$ and $r_{2}$ have the same meaning of the previous section. We
know that the shape function $\left(\ref{b(r)1}\right)$ does not represent a
TW because it is not asymptotically flat. Moreover, for large $r$,
$b\left(r\right)$ becomes negative. This means that there exists $\tilde{r}$
such that $b\left(\tilde{r}\right)=0$. However, instead of discarding
$b\left(r\right)$ of Eq. $\left(\ref{b(r)1}\right)$, we can try to establish
if there is a way to obtain a TW from Eq.$\left(\ref{b(r)1}\right)$. One
important property is the flare out condition described by

$b^{\prime}\left(r_{0}\right)<1\qquad\Longleftrightarrow\qquad\frac{r_{2}^{2}d^{4}-r_{0}^{4}r_{1}^{2}}{r_{0}^{2}d^{4}}<1,$
(107)

which is satisfied when

$r_{0}>\frac{\sqrt{2}d}{2r_{1}}\sqrt{-d^{2}+\sqrt{d^{4}+4r_{1}^{2}r_{2}^{2}}}.$
(108)

Another property that has to be satisfied is the absence of a horizon for the
redshift function. From Eq.$\left(\ref{pr0}\right)$ and
Eq.$\left(\ref{b(r)1}\right)$ one finds

$\Phi^{\prime}\\!\left(r\right)=\frac{\left(3d^{4}r_{0}^{2}+3d^{4}r_{2}^{2}+r_{0}^{4}r_{1}^{2}\right)r-6r_{2}^{2}r_{0}\,d^{4}-10r^{4}r_{1}^{2}r_{0}}{2r_{0}r_{1}^{2}r^{5}+6d^{4}r_{0}r^{3}+\left(-6d^{4}r_{0}^{2}-6d^{4}r_{2}^{2}-2r_{0}^{4}r_{1}^{2}\right)r^{2}+6d^{4}r_{0}r_{2}^{2}r}.$
(109)

Close to the throat, the r.h.s. can be approximated by

$\Phi^{\prime}\\!\left(r\right)=\frac{\left(3d^{4}r_{0}^{2}+3d^{4}r_{2}^{2}+r_{0}^{4}r_{1}^{2}\right)r_{0}-6r_{2}^{2}r_{0}d^{4}-10r_{0}^{5}r_{1}^{2}}{10r_{0}^{5}r_{1}^{2}+18d^{4}r_{0}^{3}+2\left(-6d^{4}r_{0}^{2}-6d^{4}r_{2}^{2}-2r_{0}^{4}r_{1}^{2}\right)r_{0}+6r_{2}^{2}r_{0}d^{4}}\left(r-r_{0}\right)^{-1}+\mathrm{O}\\!\left(1\right).$
(110)

It is immediate to see that a horizon will be present, unless we impose that

$\left(3d^{4}r_{0}^{2}+3d^{4}r_{2}^{2}+r_{0}^{4}r_{1}^{2}\right)r_{0}-6r_{2}^{2}r_{0}d^{4}-10r_{0}^{5}r_{1}^{2}=0.$
(111)

We have four solutions, but only two are real. They are represented by

$r_{0}=\frac{\sqrt{6}\,\sqrt{d^{2}\pm\sqrt{d^{4}-12r_{1}^{2}r_{2}^{2}}}\,d}{6r_{1}}.$
(112)

Among them, the first one is

$r_{0}=\frac{\sqrt{6}d}{6r_{1}}\sqrt{d^{2}+\sqrt{d^{4}-12r_{1}^{2}r_{2}^{2}}}\underset{d\gg
r_{1}}{\simeq}\frac{\sqrt{3}d^{2}}{3r_{1}}+O\left(\frac{1}{d^{2}}\right),$
(113)

which is independent on $r_{2}$ and therefore on the electric field. The
result has a dependence on $d$ similar to the one obtained in Refs.GABTW ; YCW
. The other interesting solution is

$r_{0}=\frac{\sqrt{6}d}{6r_{1}}\sqrt{d^{2}-\sqrt{d^{4}-12r_{1}^{2}r_{2}^{2}}}\underset{d\gg
r_{1}}{\simeq}r_{2}+O\left(\frac{1}{d^{4}}\right),$ (114)

which is independent on the plates separation but it is depending on the
electric field. However, the two solutions can be merged into one if

$d=\sqrt[4]{3}\sqrt{2r_{1}r_{2}}.$ (115)

It is straightforward to verify that

$\frac{\sqrt{6}\,\sqrt{d^{2}\pm\sqrt{d^{4}-12r_{1}^{2}r_{2}^{2}}}\,d}{6r_{1}}>\frac{\sqrt{2}d}{2r_{1}}\sqrt{-d^{2}+\sqrt{d^{4}+4r_{1}^{2}r_{2}^{2}}}$
(116)

even when the condition $\left(\ref{d}\right)$ is used. This means that the
throat location is compatible with the flare out condition. Note that
Eq.$\left(\ref{d}\right)$ establishes a relationship between the plates
separation and the charge quantity that can be used. In particular, if $d$ is
of the order of $nm$ $\left(d\simeq 10^{-9}m\right)$ and $r_{1}$ is Planckian,
one gets

$\frac{d}{r_{1}}\simeq\frac{10^{-9}m}{10^{-35}m}\simeq\frac{n}{\pi}\sqrt{\frac{180}{137\pi}}\simeq\allowbreak
0.205\,85n\qquad\Longrightarrow\qquad n\simeq 10^{26},$ (117)

where we have used the fine structure constant introduced in
$\left(\ref{fine}\right)$. Note that, for silver the average number of
conduction electrons is $5.8\cdot 10^{28}/m^{3}$. As we can see, the
relationship $\left(\ref{d}\right)$ constrains the plates separation to be a
function of the electric charge. This also implies that the throat
$\left(\ref{throat}\right)$ reduces to

$r_{0}=\sqrt{2}r_{2}.$ (118)

If we put numbers in $\left(\ref{r0d}\right)$, one finds

$r_{0}=\frac{180n^{2}}{137\pi^{3}}r_{1}\simeq\left(4.\,\allowbreak
237\,4\times 10^{-2}n^{2}\right)\left(1.6\times
10^{-37}m\right)\simeq\left(10^{26}\right)^{2}6.\,\allowbreak 779\,8\times
10^{-39}m\simeq 10^{13}m.$ (119)

Note that the equation connecting the throat with the plates separation and
the fine structure constant can be obtained also with the help of an EoS of
the form $p_{r}\left(r\right)=\omega\left(r\right)\rho\left(r\right)$ with

$\omega\left(r\right)=-\frac{b\left(r\right)}{rb^{\prime}\left(r\right)}=rd^{4}\left(\frac{r_{0}+\frac{r_{2}^{2}}{r_{0}}-\frac{r_{2}^{2}}{r}-\frac{r_{1}^{2}}{3d^{4}}\left(r^{3}-r_{0}^{3}\right)}{r_{1}^{2}r^{4}-r_{2}^{2}d^{4}}\right)$
(120)

allowing to fix $\Phi^{\prime}\\!\left(r\right)=0$. On the other hand, from
$p_{r}\left(r\right)$ and $\rho\left(r\right)$ defined in
$\left(\ref{rhoprd}\right)$, one can obtain also

$\omega\left(r\right)=\frac{p_{r}\left(r\right)}{\rho\left(r\right)}=\frac{3r_{1}^{2}r^{4}+r_{2}^{2}d^{4}}{r_{1}^{2}r^{4}-r_{2}^{2}d^{4}}.$
(121)

If we impose that $\omega\left(r\right)$ in Eq.$\left(\ref{omega(r)1}\right)$
be equal to $\omega\left(r\right)$ in Eq.$\left(\ref{omega(r)2}\right)$, one
finds that the only solution is represented by Eq.$\left(\ref{throat}\right)$.
Nevertheless, outside of the throat, the function $\omega\left(r\right)$ in
Eq.$\left(\ref{omega(r)1}\right)$ is no longer equal to the one of
Eq.$\left(\ref{omega(r)2}\right)$. Plugging the value of $r_{0}$ in
$\left(\ref{r0d}\right)$ into the shape function $\left(\ref{b(r)1}\right)$,
one finds

$b\left(r\right)=\frac{14}{9}r_{0}-\frac{r_{0}^{2}}{2r}-\frac{r^{3}}{18r_{0}^{2}}.$
(122)

In this reduced form, it is easier to see that there exists

$\tilde{r}=2.9208r_{0}\qquad where\qquad b\left(\tilde{r}\right)=0.$ (123)

For $r>\bar{r}$, $b\left(r\right)<0$. Therefore to avoid negative values, we
can cut off the region where $r>\bar{r}$. For this reason, the range of the
wormhole must be constrained to be very close to the throat. A possible
profile comes from the following setup

$\left\\{\begin{array}[c]{l}b\left(r\right)=\frac{14}{9}r_{0}-\frac{r_{0}^{2}}{2r}-\frac{r^{3}}{18r_{0}^{2}}\qquad
r_{0}\leq r\leq\tilde{r}\qquad\qquad b\left(r\right)=0\qquad r\geq\tilde{r}\\\
\Phi\\!\left(r\right)=0.\end{array}\right.$ (124)

Note that in the region $\bar{r}\leq r\leq\tilde{r}$, the constant Casimir
source becomes relevant. One can be tempted to classify such a TW as an
Absurdly Benign Traversable Wormhole (ABTW) defined byGABTW

$\left\\{\begin{array}[c]{l}b\left(r\right)=r_{0}\left(1-\mu\left(r-r_{0}\right)\right)^{2}\qquad
r_{0}\leq r\leq r_{0}+\frac{1}{\mu};\qquad\qquad b\left(r\right)=0\qquad r\geq
r_{0}+\frac{1}{\mu}\\\ \Phi\\!\left(r\right)=0.\end{array}\right.$ (125)

Even if, outside the region $r=\bar{r}$, $\rho\left(r\right)$ and
$p_{r}\left(r\right)$ tend to a constant value, one has to realize that this
behavior cannot be extended to the whole space, rather it is likely that
$\rho\left(r\right)$ and $p_{r}\left(r\right)$ vanish in proximity of the
external part of the plates, like in the model introduced by Visser who
proposed to consider the following SETVisser

$\displaystyle
T_{\sigma}^{\mu\nu}=\sigma\hat{t}^{\mu}\hat{t}^{\nu}\left[\delta\left(z\right)+\delta\left(z-a\right)\right]$
$\displaystyle+\Theta\left(z\right)\Theta\left(a-z\right)\frac{\hbar
c\pi^{2}}{720a^{4}}\left[\eta^{\mu\nu}-4\hat{z}^{\mu}\hat{z}^{\nu}\right],$
(126)

where $\hat{t}^{\mu}$ is a unit time-like vector, $\hat{z}^{\mu}$ is a normal
vector to the plates and $\sigma$ is the mass density of the plates.

##  V Conclusions

In this paper, we have extended the study begun in Ref.CW  by including an
electromagnetic source. Since the electromagnetic field satisfies the property
$\left(\ref{prho}\right)$, the NEC is still violated and it seems to be
independent on the strength of the electromagnetic field. Repeating the same
strategy adopted in Ref.CW , we have found another Casimir wormhole with a
different $\omega$ as it should be. We would like to draw the reader’s
attention that the additional electric field is a part of the source and not a
feature of the TW. The most important consequence is that the wormhole throat
becomes directly dependent on the charge in an additive way, even if under the
square root

$r_{0}=\sqrt{3r_{1}^{2}+r_{2}^{2}}.$ (127)

If this result seems to be encouraging, on the other hand we have two aspects
that must be explored. The first one is that for $r_{2}\gg r_{1}$, the energy
density becomes positive. At this stage of the analysis, we do not know if the
TW ceases to exist or not. A possible answer could come from a back reaction
investigation of the electromagnetic and gravitational fields together which
is beyond the purpose of this paper. The second aspect is that, always in the
range where $r_{2}\gg r_{1}$, $\omega\rightarrow-1$ and a horizon seems to be
appear. However, this is the result of a limiting procedure and the value
$\omega=-1$ can never be reached. To this purpose, we have to recall that, it
is the NEC that must be violated. This means that with the help of “phantom
energy”, $\rho\left(r\right)>0$phantomWH1 ; phantomWH2 ; phantomWH3 . Indeed,
the following relationship

$\displaystyle
p_{r}\left(r\right)=\omega\rho\left(r\right),\qquad\Longrightarrow\qquad
p_{r}\left(r\right)+\rho\left(r\right)<0$
$\displaystyle\qquad\Longleftrightarrow\qquad\left(1+\omega\right)\rho\left(r\right)<0,$
(128)

allows to keep $\rho\left(r\right)>0$, provided that $\omega<-1$. However,
generally speaking, it is not known how to build and manipulate such a
“phantom energy”. The electrovacuum example we have discussed in this paper
seems to be encouraging, because in this context, the electromagnetic field
seems to behave exactly like a phantom source. Note that this is not true when
the electromagnetic field is the only source. To further proceed, note that
there is an essential discontinuity when $r_{2}=r_{1}$ into the relationship
$\left(\ref{or1r2}\right)$. For this reason this case has been examined
separately in paragraph II.1. It is important to observe that even in this
case, a TW can exist at zero density with a throat of Planckian size. A
different behavior appears when one considers the mixed source case, namely
only the electromagnetic field has a variable radius, while the plates
separation has been dealt as a parameter. In this framework, one finds that it
is possible to avoid the creation of a horizon if the throat satisfies
Eq.$\left(\ref{throat}\right)$. Because we have two solutions, it is possible
to have only one solution, if we impose the constraint $\left(\ref{d}\right)$.
This constraint creates a relationship between the plates separation, the
Planck length which is not modifiable and the quantity of charge introduced
which can be changed. Unfortunately, even in this case the main limitation
comes from the plates separation which has consequences also on the throat
size. However, such a limitation is not so stringent like in Refs.GABTW ; YCW
because of the presence of the square root in Eq.$\left(\ref{d}\right)$.
Indeed, if it could be possible to push the plates separation at a distance of
the order of $pm$, one could find that the throat could be of the order of
$10^{9}m$: a gain of a factor $10^{2}$ with respect to what we have found in
Ref.GABTW . Note that the constraint $\left(\ref{d}\right)$ implies that we
have a throat radius directly proportional to the charge of the
electromagnetic field. For this reason, differently from the pure Casimir
source, the introduction of an electromagnetic field seems to go towards a
traversable wormhole which is a little less traversable in principle and a
little more traversable in practice. It is important also to observe that the
shape function $\left(\ref{b(r)1}\right)$ and subsequently the shape function
$\left(\ref{b(r)1r}\right)$ can be promoted to be a traversable wormhole shape
function if we assume that there is a smooth transition between the curved
space and flat space expressed by Eq.$\left(\ref{zero}\right)$. Alternatively,
one could use the cut and paste technique and glue the shape function
$\left(\ref{b(r)1r}\right)$ with a Schwarzschild profile. Of course the whole
analysis can be generalized to include even a magnetic field and this will be
examined in a future publication. It is interesting to observe that contrary
to the pure Casimir wormhole, the additional electromagnetic field avoids that
the TW has a throat of Planckian size. Needless to say that things could
drastically change with the inclusion of quantum corrections on the line of
Refs.RG ; RG1 ; RGFSNL ; RGFSNL1 ; RGFSNL2 .

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