Effects of an electric charge on Casimir wormholes: changing the throat size

In this paper we continue the investigation of 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 ρ=-pr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho =-p_{r}$$\end{document}. As a consequence, the traversable wormhole throat can be changed as a function of the electric charge. This means that the throat is no longer Planckian and the traversability is slightly less in principle but slightly greater in practice.


I. INTRODUCTION
Casimir wormholes are Traversable Wormholes (TW) obtained by solving the semiclassical Einstein's Field Equations (EFE) with a source of the form representing the energy density and the pressure, respectively. d is the plates separation [1]. T µν 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 (d) /ρ C (d) = 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 (d) will be interpreted as a radial pressure p r (r) To build a Casimir wormhole we need to introduce the following spacetime metric where dΩ 2 = dθ 2 + sin 2 θdφ 2 is the line element of the unit sphere. Φ(r) and b(r) are two arbitrary functions of the radial coordinate r ∈ [r 0 , +∞), denoted as the redshift function and the shape function, respectively [2][3][4]. With the help of the metric (4) the EFE, written in an orthonormal frame, are b ′ (r) r 2 = κρ (r) , 2 and ρ (r) is the energy density 1 , p r (r) is the radial pressure, and p t (r) is the lateral pressure. The line element (4) 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 (r) = 2 r (p t (r) − p r (r)) − (ρ (r) + p r (r)) Φ ′ (r).
If we assume that an Equation of State (EoS) p r (r) = ωρ C (r) is imposed, then we find the following solution to the semiclassical EFE where we have used the energy density ρ C (r) of Eq.(2) and the radial pressure p r (r) described by Eq.(3) 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 ′ 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 ′ (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 2Φ(r) → 0, so that Φ(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 by [5,6] T 0 that it means ρ = −p r .
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 which is conserved and traceless. From the energy density ρ of the SET, it is possible to obtain the following shape function which has the correct properties. Indeed Eq.(6) together with the shape function (14) allow the computation of the redshift function where C is an integration constant 2 . As expected, we cannot adopt the strategy of Ref. [1] because the ln (r − r 0 ) never disappears for every choice of r 0 , even if property (12) is satisfied. One could insist by imposing an inhomogeneous EoS of the form allowing to fix Φ(r) = 0. Nevertheless, Eq.(21) must be compatible with This leads to which is incompatible with the flare out condition (15), because one gets This means that a pure electromagnetic field cannot support a TW even if ρ+p r = 0. It is necessary to have ρ+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 (12) which defines an electrovacuum source. Note that such an electrovacuum source has been investigated earlier in Ref. [7] 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 Lee [8] who considered a combination between the Morris-Thorne wormhole and the Reissner-Nordström spacetime. Balakin et al. discussed a nonminimal Einstein-Maxwell model [9]. Kuhfittig [10] considered a modification of the Kim and Lee charged wormhole to have compatibility with the quantum inequality of Ford and Roman [11]. 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.
2 Actually, because of the property (12), from Eq.(6), one can write which has the following solution which is a signature of a Black Hole.

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 (2). Following Ref. [1], 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 ρ (r) + p r (r) < 0 must hold. We want to draw the reader's attention that, thanks to the property (12), we can write where In this context, the total energy density is represented by where Thus The shape function b (r) can be obtained plugging ρ (r) of Eq.(27) into Eq.(5), whose solution leads to where the throat condition b(r 0 ) = r 0 has been imposed. The redshift function can be obtained by solving Eq. (6) with the help of the shape function (31). One finds where we have used an EoS of the form p r (r) = ωρ (r). The solution can be written as If Q = 0, one recovers the familiar form of the Casimir wormhole redshift function. It is possible to eliminate the horizon if we constrain ω to be and the redshift function (33) becomes By assuming that Φ (r) → 0 for r → ∞, then we find which is formally the same result of Ref. [1]. The shape function (31) can be rearranged to obtain the familiar form of the Casimir wormhole [1] b On the other hand, since the ratio in Eq.(34) is not constrained, we can use the ratio p r (r) /ρ (r) to extract information about the size of the throat. It is immediate to see that for the EoS is satisfied and is independent on r. Plugging the value of ω in Eq.(38) into Eq.(34), one finds 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 one finds where Q = ne, e is the electron charge, α is the fine structure constant and n is the total number of the electron charges. Note that, for one recovers the shape function of Ref. [1] with ω = 3. On the other hand, when and ω → −1. In conclusion, we can write ω ∈ (3, +∞) ; A comment on the limit (43) is in order. Indeed, when ω → −1, the shape function assumes the form and the redshift function is representing no more a TW, but a black hole. However, ω → −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 ω < −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 ω, is identical to p t (r) of Ref. [1]. One finds where we have introduced an inhomogeneous EoS on the transverse pressure with and the final form of the SET is The conservation of the SET is satisfied but a comparison with the SET source shows that namely a discrepancy on the transverse pressure with respect to the SET source is present. We recall that the SET source is defined by It is important to observe that there exists another interesting value for ω, namely when ω = 1. For this special choice, one finds that the line element reduces to the Ellis-Bronnikov (EB) wormhole [12,13], namely Nevertheless ω = 1 is incompatible with the relationship (38) and therefore this option will be discarded.
A. Special case r 2 1 = r 2 2 In the special case we find that the energy density vanishes. Therefore the wormhole shape function is On the other hand the redshift function appears to be non trivial. Indeed, from the EFE (6), we find which can be rearranged to give The solution is It is immediate to see that for the horizon disappears and where we have assumed that Φ (r) → 0 for r → ∞. Therefore in this special case we still have a TW with the following line element 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 Note that for this special case, we cannot impose an EoS of the form p r (r) = ωρ (r) because ρ (r) 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 (31) or (37) 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 In this specific case, plugging Eq.(37) into Eq.(64), one gets We find where the"±" 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. [2], we shall locate l 1 and l 2 at a value of the radius such that l 1 ≃ l 2 ≃ 10 4 r 0 that it means 1 − b (r) /r ≃ 1. Assume that the traveller has a radial velocity v (r), 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 dτ , by the following relationships and respectively. If the traveler journeys with constant speed v, then the total time is given by while the proper total time is To evaluate ∆t we can proceed with the following approximations. Close to the throat, one finds and, in this range, ∆t ≃ ∆τ except for the value ω = −1, where the TW turns into a Black Hole. On the other hand, when r → ∞, one finds and even with this approximation, the leading term is the same of ∆τ . On the same ground, we can compute the embedded surface, which is defined by and, in the present case, we find where F (ϕ, k) is the elliptic integral of the first kind and Π (ϕ, n, k) is the elliptic integral of the third kind. Close to the throat, one can write It is interesting to note the singularity appearing when ω = −1, showing the presence of a Black Hole. To further investigate the properties of the shape function (37), we consider the computation of the total gravitational energy for a wormhole [14], defined as where M is the total mass M and M P is the proper mass, respectively. Even in this case, the "±" depends one the wormhole side we are. In particular and Thus, at infinity one finds An important traversability condition is that the acceleration felt by the traveller should not exceed Earth's gravity g ⊕ ≃ 980 cm/s 2 . In an orthonormal basis of the traveller's proper reference frame, we can find If we assume a constant speed and γ ≃ 1, then we can write We can see that in proximity of the throat, the traveller has a vanishing acceleration. Always following Ref. [2], we can estimate the tidal forces by imposing an upper bound represented by g ⊕ . The radial tidal constraint constrains the redshift function, and the lateral tidal constraint constrains the velocity with which observers traverse the wormhole. η1 ′ and η2 ′ represent the size of the traveller. In Ref. [2], they are fixed approximately equal, at the symbolic value of 2 m. Close to the throat, the radial tidal constraint (82) becomes For the lateral tidal constraint, we find v 2 r 0 2r 4 r (ω − 1) + 2r 0 ω η2 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 ∆t ≃ ∆l which is in agreement with the estimates found in Ref. [2], even for ω → ±∞. The last property we are going to discuss is the "total amount" of ANEC violating matter in the spacetime [15] which is described by and for the line element (4), one can write where the measure dV has been changed into r 2 dr. For the metric (37), one obtains 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 wormhole[1] Therefore we can conclude that, in proximity of the throat the ANEC can be arbitrarily small.
A. Properties of the Casimir wormhole with an additional electromagnetic field for the special case r 2 1 = r 2 2 In section II A, we have considered the special case in which the negative Casimir energy is compensated by the positive electromagnetic field with the assumption that r 2 1 = 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 and its asymptotic behavior becomes where the"±" depends on the wormhole side we are. From Eq.(67) and Eq.(68), we can compute the total time ∆t and the proper total time ∆τ , respectively. The total time is and it is bounded by the following inequality chain On the other hand the proper total time is simply It is interesting to observe that, differently from Eq. (27), the energy density (103) vanishes when where Q = ne, e is the electron charge, α is the fine structure constant and n is the total number of the electron charges. In particular, we find that The shape function b (r) can be obtained plugging ρ (r) (27) into Eq. (5), whose solution leads to where r 1 and r 2 have the same meaning of the previous section. We know that the shape function (106) does not represent a TW because it is not asymptotically flat. Moreover, for large r, b (r) becomes negative. This means that there existsr such that b (r) = 0. However, instead of discarding b (r) of Eq. (106), we can try to establish if there is a way to obtain a TW from Eq.(106). One important property is the flare out condition described by which is satisfied when Another property that has to be satisfied is the absence of a horizon for the redshift function. From Eq.(6) and Eq.(106) one finds Φ ′ (r) = 3d 4 r 2 0 + 3d 4 r 2 2 + r 4 0 r 2 1 r − 6r 2 2 r 0 d 4 − 10r 4 r 2 1 r 0 2r 0 r 2 1 r 5 + 6d 4 r 0 r 3 + (−6d 4 r 2 0 − 6d 4 r 2 2 − 2r 4 0 r 2 1 ) r 2 + 6d 4 r 0 r 2 2 r .
Close to the throat, the r.h.s. can be approximated by Φ ′ (r) = 3d 4 r 2 0 + 3d 4 r 2 2 + r 4 0 r 2 1 r 0 − 6r 2 2 r 0 d 4 − 10r 5 0 r 2 1 10r 5 0 r 2 1 + 18d 4 r 3 0 + 2 (−6d 4 r 2 0 − 6d 4 r 2 2 − 2r 4 0 r 2 1 ) r 0 + 6r 2 2 r 0 d 4 (r − r 0 ) It is immediate to see that a horizon will be present, unless we impose that We have four solutions, but only two are real. They are represented by Among them, the first one is 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. [16,17]. The other interesting solution is Even if, outside the region r =r, ρ (r) and p r (r) tend to a constant value, one has to realize that this behavior cannot be extended to the whole space, rather it is likely that ρ (r) and p r (r) vanish in proximity of the external part of the plates, like in the model introduced by Visser who proposed to consider the following SET [4] T µν σ = σt µtν [δ (z) + δ (z − a)] wheret µ is a unit time-like vector,ẑ µ is a normal vector to the plates and σ is the mass density of the plates.

V. CONCLUSIONS
In this paper, we have extended the study begun in Ref. [1] by including an electromagnetic source. Since the electromagnetic field satisfies the property (12), 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. [1], we have found another Casimir wormhole with a different ω 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 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 ≫ 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 ≫ r 1 , ω → −1 and a horizon seems to be appear. However, this is the result of a limiting procedure and the value ω = −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", ρ (r) > 0 [18][19][20]. Indeed, the following relationship allows to keep ρ (r) > 0, provided that ω < −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 (38). For this reason this case has been examined separately in paragraph II A. 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.(112). Because we have two solutions, it is possible to have only one solution, if we impose the constraint (115). 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. [16,17] because of the presence of the square root in Eq.(115). 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. [16]. Note that the constraint (115) 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 (106) and subsequently the shape function (122) 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. (123). Alternatively, one could use the cut and paste technique and glue the shape function (122) 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