# \(3+1\)-dimensional thin shell wormhole with deformed throat can be supported by normal matter

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## Abstract

From the physics standpoint the exotic matter problem is a major difficulty in thin shell wormholes (TSWs) with spherical/cylindrical throat topologies. We aim to circumvent this handicap by considering angle dependent throats in \(3+1\) dimensions. By considering the throat of the TSW to be deformed spherical, i.e., a function of \(\theta \) and \(\varphi \), we present general conditions which are to be satisfied by the shape of the throat in order to have the wormhole supported by matter with positive density in the static reference frame. We provide particular solutions/examples to the constraint conditions.

## Keywords

Constraint Condition Extrinsic Curvature Null Energy Condition Exotic Matter Traversable Wormhole## 1 Introduction

The seminal works on traversable wormholes and thin shell wormholes (TSWs), respectively, by Morris and Thorne [1] and Visser [2] both employed spherical/cylindrical [3, 4] geometry at the throats. Besides instability of TSWs [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and the wormholes supported by ghost scalar field [21, 22] one major problem in this venture is the violation of the null energy condition (NEC). Precisely, TSWs allow presence of exotic matter at the throat with \(\sigma <0\) in which \(\sigma \) is the energy density on the hypersurface of the throat. Several attempts have been made to introduce the TSWs supported by normal matter of the kind with \( \sigma >0\) in the framework of the Gauss–Bonnet theory of gravity [23, 24, 25, 26, 27]. We aim in this study to seek for \(\sigma >0\) against violation of NEC by changing the spherical/circular geometry to more general, angle dependent throats in the wormholes. That is, since NEC is violated a different choice of frame may account a negative energy density. Motivation for such a study originates from the consideration of Zipoy–Voorhees metrics which surpasses spherical symmetry with a quadrupole moment by employing a distortion parameter [28, 29, 30]. In brief, this amounts to compress a sphere into an ellipsoidal form through a distortion mechanism. This minor change contributes to the total energy and makes it positive in the static reference frame under certain conditions. In local angular intervals we confront still with negative energies in part but the integral of the total energy happens to be positive. We recall that any rotating system with spherical symmetry becomes axial in which by employing a similar refinement of the throat we may construct wormholes with a positive total energy. It is our belief that by this method of suitable choice of geometry at the throat and in a special frame we can measure a positive energy. Recently we have shown [31] that the flare-out conditions [32], which were thought to be unquestionable, can be reformulated. We must admit, however, that although geometry change has positive effects on the energy content this does not guarantee that the resulting wormhole becomes stable. For the particular case of counter-rotational effects in \(2+1\)-dimensional TSW we have shown that stability conditions are slightly improved [33]. That is, when the throat consists of counter-rotating rings in \(2+1\) dimensions the stability of the resulting TSW becomes stronger. This result has not been confirmed in \(3+1\)-dimensional TSWs yet. Arbitrary angle dependent throat geometries have also been considered by the same token recently in \(2+1\) dimensions [34]. Therein a large class of wormholes with non-circular throat shapes are pointed out in which positive energy supports the wormhole. In the same reference we explain also the distinctions (if any) by employing the ordinary time instead of the proper time. Extension of this result to the more realistic \(3+1\) dimensions makes the aim of the present study. Numerical computation of our chosen ansatzes yields a positive total energy, as promised from the outset.

We start with the \(3+1\)-dimensional flat, spherically symmetric line element in which a curved hypersurface is induced to act as our throat’s geometry. Such a hypersurface, \(\Sigma \left( t,r,\theta ,\varphi \right) =0\), has an induced metric satisfying the Einstein equations at the junction with the proper conditions, obeying the flare-out conditions. No doubt, such an ansatz is too general; for this reason they are restricted subsequently. The static case, for instance, eliminates the time dependence in \(\Sigma \left( t,r,\theta ,\varphi \right) =0\). We derive the general conditions for such throats and present particular ansatzes depending on \(\theta \) and \(\varphi \) angles alone that satisfy our constraint conditions.

The organization of the paper goes as follows. In Sect. 2 we present in brief the formalism for TSWs. Static TSWs follow in Sect. 3 where angle dependent constraint conditions are derived. (The details of computations can be found in Appendices A and B.) The paper ends with our conclusion in Sect. 4.

## 2 Formalism for TSWs

## 3 Static TSWs

### 3.1 \(\mathcal {R}\left( \theta ,\varphi \right) \) function of \(\theta \) only

*a*and \(8\pi \sigma _{0}=-\frac{4}{a}\), which is clearly negative and so is \(\Omega .\) Picking more complicated functions periodic in \(\theta \) is acceptable provided it makes the total energy positive. Here, having \(\sigma _{0}\ge 0\) is a sufficient condition to have \(\Omega \ge 0,\) but not necessary. Our main purpose as we stated in the Sect. 1 is to show that there is possibility of having a TSW supported by ordinary matter in the sense that \(\tilde{\sigma }_{0}\ge 0.\) This condition effectively reduces to

### 3.2 \(\mathcal {R}\left( \theta ,\varphi \right) \) function of \(\varphi \) only

### 3.3 \(\mathcal {R}\left( \theta ,\varphi \right) \) as a general periodic function

### 3.4 Existence of solution

## 4 Conclusion

The throat geometry for TSWs is taken embedded in \(3+1\)-dimensional flat geometry in spherical coordinates. For static case we obtain the most general angle dependent constraints the functions have to satisfy in order to yield a positive total energy. We must admit that the positivity condition refers to a static frame in which the energy density becomes positive although the NEC remains violated. Specific reduction procedures are given dependent on both \(\varphi ,\) and \(\theta \) and \(\varphi \), which simplify the constraint conditions. Once these constraint conditions are satisfied we shall not be destined to confront exotic matter in TSWs. At least in particular, static frames two particular examples are given which yield positive total energy \(\Omega \), from Eq. (23). We admit also that finding analytically general integrals for functions to satisfy our differential equation constraints does not seem an easy task at all. The details of our technical part are given in the Appendix. The argument/method can naturally be extended to cover more general wormholes, not only the TSWs. One issue that remains open, in all this endeavor which we have not discussed, is the stability of such constructions. A final warning to the traveler who intends to cross the throat: The thin edges may give harmful tidal effects from geometrical point of view, so keep away from those edges if you dream to enjoy a journey at all.

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