# Casimir wormholes

## Abstract

Casimir energy is always indicated as a potential source to generate a traversable wormhole. It is also used to prove the existence of negative energy which can be built in the laboratory. However, in the scientific literature there is no trace of the consequences on the traversable wormhole itself. In this work, we would like to consider such a source to see if and which kind of traversable wormhole can be produced. In a further analysis, we examine also the consequences of quantum weak energy conditions on the traversability of the wormhole. We find agreement with the original Casimir traversable wormhole. Nevertheless, despite the traversability result, one finds once again that the traversability is only in principle but not in practice.

## 1 Introduction

*S*is the surface of the plates and

*a*is the separation between them. The force can be obtained with the computation of

*P*and the energy density \(\rho \) described by an equation of state (EoS) of the form \(P=\omega \rho \) with \(\omega =3\). The nature of this effect is connected with the zero point energy (ZPE) of the quantum electrodynamics vacuum distorted by the plates. It is important to observe that this effect shows a strong dependence on the geometry of the boundaries. Indeed, Boyer [7] proved the positivity of the Casimir effect for a conducting spherical shell of radius

*r*. The same positivity has been proved also in Ref. [8], by means of heat kernel and zeta regularization techniques. To the best of our knowledge, the Casimir energy represents the only artificial source of

*exotic matter*realizable in a laboratory.

^{1}Exotic matter violates the null energy condition (NEC), namely for any null vector \(k^{\mu }\), we have \(T_{\mu \nu }k^{\mu }k^{\nu }\ge 0\). Violation of the NEC is related to the existence of a bizarre but amazing object predicted by General Relativity: a

*traversable wormhole*. Traversable wormholes (TWs) are solutions of the Einstein’s field equations (EFEs) powered by classical sources [10, 11]. However, given the quantum nature of the Casimir effect, the EFE must be replaced with the semiclassical EFE, namely

*exotic matter*and a TW, we introduce the following spacetime metric:

*b*(

*r*) are arbitrary functions of the radial coordinate \(r\in \left[ r_{0},+\infty \right) \), denoted as the redshift function, and the shape function, respectively [10, 11]. 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 requirement 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 \(e^{2\phi }\rightarrow 0\), so that \(\phi (r)\) must be finite everywhere. With the help of the line element (6), we can write the EFE in an orthonormal reference frame, leading to the following set of equations:

^{2}\(p_{r}\left( r\right) \) is the radial pressure, and \(p_{t}\left( r\right) \) is the lateral pressure. 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

*quantum weak energy condition*(QWEC),

*A*is an appropriate constant introduced to describe an energy density. Note that the condition (14) has a direct connection with the

*volume integral quantifier*, which provides information about the

*total amount*of averaged null energy condition (ANEC) violating matter in the spacetime [21]. This is defined by

*dV*has been changed into \(r^{2}dr\). For example, the calculation of \(I_{V}\) for the traversable wormhole [10],

*zero radial tides,*namely \(\phi \left( r\right) =0\) and

*zero density*wormholes, namely \(b\left( r\right) =\) constant, implying \(\rho (r)=0\). The interesting feature of the QWEC (14) is that \(b\left( r\right) \) can be determined exactly, not only for the form introduced in Ref. [19], but even for a generic \(f\left( r\right) \). Indeed, with the help of Eqs. (7) and (8), we can write

*zero tidal force*(ZTF) condition. It is immediate to recognize that the previous shape function does not represent a traversable wormhole [19]. It is also possible to apply the reverse procedure, namely we fix the form of the shape function and we compute the redshift function and we obtain

## 2 The Casimir traversable wormhole

*a*to a radial coordinate

*r*. Even if the authors of Ref. [12] assume a Casimir device made by spherical plates, we have to say that the SET form they use is the same as obtained with the flat plate assumption. Therefore the curvature of the plates introduces some modification which, in this first approximation, will be neglected. We have also to observe that the replacement of

*a*with

*r*could make the stress-energy tensor (11), with \(\sigma =0\), potentially not conserved, as expected. Our strategy begins with the examination of Eq. (7) leading to the following form of the shape function:

^{3}

^{4}

^{5}The corresponding SET (37) becomesNote that the inhomogeneous function (36) is such that

^{6}[30, 31]. Even in this case, the SET is conserved, but to establish a connection with the Casimir SET, we need to write the SET of (37) in the following way

^{7}:

## 3 Properties of the Casimir wormhole

*r*. One may relate the proper distance traveled

*dl*, the radius traveled

*dr*, the coordinate time lapse

*dt*, and the proper time lapse as measured by the observer \(d\tau \), by the following relationships:

*v*, then the total time is given by

*M*is the total mass

*M*and \(M^{P}\) is the proper mass, respectively. Even in this case, the “±” depends on the wormhole side we are. In particular

*m*. Close to the throat, the radial tidal constraint (61) becomes

^{8}

*v*, then the tidal forces are null. We can use these last estimates to complete the evaluation of the crossing time which approximately is

## 4 Global monopoles from QWEC

*global monopole*[20]. Nevertheless, an interesting shape function can be obtained if one plugs the redshift function (40) into Eq. (22). Indeed, one finds

*global monopole*. Indeed, plugging the shape function (51) into the original metric (6), we can write

*throat*we find

## 5 The return of the traversable wormhole and the disappearance of the global monopole

*A*. Indeed, one finds

*A*is so huge that it cannot have a physical meaning. Moreover, we can also impose

*a*is the fixed plate distance. This further constraint is even worse than that found in Ref. [12]. Nevertheless if we constrain only the relationship (89),

*fm*, to have a throat of the order of \(10^{5}m\). One could be tempted to use the arbitrariness of \(\alpha \). However, \(\alpha >3\), otherwise the energy density becomes positive. The other case where the TW returns and the global monopole disappears is represented by the shape function (51), whose asymptotic behavior is represented by Eq. (72). With the assumption

## 6 Conclusions

*phantom energy*” in practice. I recall that the phantom energy obeys the following relationship:

*exotic matter*which, however, can contribute only at the Planck scale. Always on the side of phantom energy we proposed the idea of

*self-sustained traversable wormholes*, namely TW sustained by their own quantum fluctuations [25, 26, 27, 28, 29]. Even in this case, because the quantum fluctuation carried by the graviton behaves like the ordinary Casimir effect, we found there to be no need for a phantom contribution. In this context, in a next paper we will explore how a system behaves formed by the

*Casimir TW*, here analyzed, and the corresponding self-sustained TW version.

## Footnotes

- 1.
Actually, there exists also the possibility of taking into consideration a squeezed vacuum. See for example Ref. [9].

- 2.
However, if \(\rho \left( r\right) \) represents the mass density, then we have to replace \(\rho \left( r\right) \) with \(\rho \left( r\right) c^{2}.\)

- 3.
When \(\omega r_{1}^{2}-r_{0}^{2}<0\), we obtain a singularity. See Appendix A for details.

- 4.By imposing \({\phi \left( r_{0}\right) =0}\), the redshift function should be$$\begin{aligned} \phi \left( r\right) =\frac{1}{2}\left( {\omega -1}\right) {\ln \left( \frac{r\left( \omega +1\right) }{\left( \omega r+r_{0}\right) }\right) .} \end{aligned}$$(31)
- 5.
We have \(r_{0}=\sqrt{3}r_{1}\simeq 1.\,016\,6l_{P}\).

- 6.
\(r_{0} =\sqrt{\frac{\pi ^{3}}{90}}l_{p}=0.586\,95l_{p}.\)

- 7.
The SET is represented in an orthonormal frame.

- 8.
Note that if we put the Planckian value in \(r_{0}\) obtained in (40), then one finds \(\left| \eta ^{\hat{1}^{\prime }}\right| \le 1.\,057\,3\times 10^{-86} {\text {m}} \). This means that with a Planckian wormhole nothing can traverse it.

## Notes

### Acknowledgements

I would like to thank Francisco S.N. Lobo for useful discussions and suggestions for the traversable wormhole part and I would like to thank Enrico Calloni and Luigi Rosa for the Casimir effect part.

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