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

From physics standpoint 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 angular 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.


I. INTRODUCTION
The seminal works on traversable wormholes and thinshell wormholes (TSWs), respectively by Morris and Thorne [1] and Visser [2] both employed spherical / cylindrical topologies at the throats. Besides instability [3] one major handicap in this venture was the necessity of the negative energy which meant exotic matter at large amounts [4]. We aim herein to overcome this problem by changing the spherical / circular topology to more general, angular dependent throats in the wormholes. Motivation for such a study originates from the consideration of Zipoy-Voorhees (ZV) metrics which surpasses spherical symmetry with a quadrupole moment by employing a distortion parameter [5]. 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 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 the challenge of exotic matter can be overcome. Recently we have shown [6] that the flare-out conditions [7] which were thought to be unquestionable can be reformulated. We must admit, however, that although topology change has positive effects on the energy content this doesn't 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 [8]. That is, when the throat consists of counterrotating 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 * habib.mazhari@emu.edu.tr † mustafa.halilsoy@emu.edu.tr angular dependent throat geometries have also been considered by the same token recently in 2 + 1−dimensions [9]. 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. We start with the 3 + 1−dimensional spherically flat line element in which a curved hypersurface is induced to act as our throat's geometry. Such a hypersurface, Σ (t, r, θ, ϕ) = 0, has an induced metric satisfying the Einstein equations at the junction with the proper conditions, satisfying the flare-out conditions. No doubt, such an ansatz is too general, for this reason they are restricted subsequently. Static case, for instance eliminates time dependence in Σ (t, r, θ, ϕ) = 0. We derive the general conditions for such throats and the investigation for practical solutions is a challenging problem to be considered.
Organization of the paper goes as follows. In Section II we present in brief the formalism for TSWs. Static TSWs follow in Section III where angular dependent constraint conditions are derived. (The details of computations can be found in Appendixes A and B). The paper ends with our conclusion in Section IV.

II. FORMALISM FOR TSWS
We start with a 3 + 1−dimensional Minkowski spacetime of the form and introduce a closed hypersurface defined by Σ (t, r, θ, ϕ) = r − R (t, θ, ϕ) = 0 (2) such that the original spacetime is divided into two parts which will form the inside and outside of the given hypersurface in (2). Now, we get two copies of the outside manifold and glue them at the hypersurface Σ (t, r, θ, ϕ) = 0. What is constructed by this procedure is a complete manifold and as had been introduced first by M. Visser it is called a thin-shell wormhole (TSW) whose throat turns out to be the closed hypersurface r = R (t, θ, ϕ) [10]. Let's choose x α = (t, r, θ, ϕ) for the 3+1−dimensional Minkowski spacetime and ξ i = (t, θ, ϕ) for the 2 + 1−dimensional hypersurface Σ (t, r, θ, ϕ) = 0. The induced metric tensor of the hypersurface h ij is defined by which yields Note that our notation R ,i means partial derivative with respect to x i . Next, we apply the Einstein equations on the shell, also called the Israel junction conditions [11] which are where are the extrinsic curvatures of the hypersurface in either sides. S j i is the energy momentum tensor on the shell with the components where σ is the energy density on the surface and S j i are the appropriate energy-momentum flux and momentum densities, respectively. Our explicit calculations reveal (see Appendix A) and In Appendix B we give the mixed tensor k j i in closed forms which are also used to determine explicit expressions for the energy momentum tensor.

III. STATIC TSW
Using the general formalism given above and in the Appendixes A and B we may consider some specific cases. First of all we consider the case in which the throat is static. This means R (t, θ, ϕ) = R (θ, ϕ) and consequently the line element on the shell / TSW becomes The non-zero components of the effective extrinsic curvature tensor are then given as and Finally in static configuration, one finds and We note that for the static TSW we have and Having the exact form of σ 0 , we would like to find the total energy which supports the static TSW. This can be done by using the following integral which becomes upon using the property of the Dirac delta function δ (r − R). To make this energy positive we have to consider an appropriate function for R (θ, ϕ) and calculate the total energy Ω.

A. R (θ, ϕ) function of θ only
Let's make the simpler choice by considering R (θ, ϕ) = Θ (θ) . Following this one gets where the only non-zero components of the extrinsic curvature are and Consequently, and One must note that r = Θ (θ) is the hypersurface of the throat, therefore Θ (θ) must be chosen such that the surface remains closed. For instance if we set Θ (θ) = a = const. then the throat will be a spherical shell of radius a and 8πσ 0 = − 4 a which is clearly negative and so is Ω. Picking more complicated functions periodic in θ is acceptable provided it makes the total energy positive. Here having σ 0 ≥ 0 is a sufficient condition to have Ω ≥ 0, but not necessary. Our main purpose is to show that there is possibility of having a TSW supported by ordinary matter in the sense thatσ 0 ≥ 0. This condition effectively reduces to B. R (θ, ϕ) function of ϕ only As in the previous section, here we consider R (θ, ϕ) = Φ (ϕ) which yields If we set Φ = a, then 8πσ 0 = −4 a once more as it should be. It is observed that even with R (θ, ϕ) = Φ (ϕ) the energy density σ 0 is a function of both θ and ϕ. Having the latter expression available for σ 0 we conclude that a proper function for Φ, which satisfies the condition and of course is periodic in ϕ provides the matter supporting the TSW to be positive.
C. R (θ, ϕ) as a general periodic function Finally we state the most general condition which is provided by a general periodic function for R (θ, ϕ) . As a matter of fact, in (17) we gave in closed form such a σ 0 and what is left is to provide a proper function for R (θ, ϕ) such that σ 0 ≥ 0.

D. Existence of solution
In [12] we have shown that for the TSW in 3 + 1−dimensions, σ relates to the trace of extrinsic curvature of spatial part of the Gaussian line element which amounts to Therefore for σ ≥ 0 the spatial extrinsic curvature must be negative, that is why for a positive curvature shape such as a sphere σ is negative. We note that an open surface with negative curvature can not be also an answer to our demand, because the throat of a TSW is defined to be closed. To show that such shapes i.e., negatively curved but closed, exist, we refer to, for instance, a concave dodecahedron. This is defined as a surface whose faces are concave individually like the cellular surface of a soccer ball with inside pressure less than outside. In such shapes, although the surface is closed, it consists of negatively curved individual patches in geometry with anti-de Sitter spacetime and hence makes σ ≥ 0. Similar argument is also valid in 2+1−dimensions which we have considered in [9].

IV. CONCLUSION
The throat geometry for TSWs is taken embedded in 3 + 1−dimensional spherically flat geometry. For static case we obtain the most general angular dependent constraints the functions have to satisfy to yield a positive total energy. Specific reduction procedures are given dependent on single angle, i.e. θ or ϕ, that simplify the constraint conditions. Once these constraint conditions are satisfied we shall not be destined to confront exotic matter in TSWs. We admit that finding exact candidate functions to satisfy our differential equation constraints doesn't seem to be an easy task at all. The details of our technical part are given in 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. To find the induced extrinsic curvature tensor on Σ we find the unit four-normal vector defined as and ± reefers to the inward and outward directions on the sides of the hypersurface. An explicit calculation yields and We also find The definition of the extrinsic curvature tensor is given by in which we have Γ θ rθ = Γ ϕ rϕ = 1 r , Γ r θθ = −r, Γ ϕ θϕ = cot θ, Γ r ϕϕ = −r sin 2 θ and Γ θ ϕϕ = sin θ cos θ. One finds and and k θ ϕ = h θϕ k ϕϕ + h θt k ϕt + h θθ k ϕθ .
Let's introduce the metric tensor and its inverse in which S = sin θ and h is the determinant of h i.e., h = −R 2 sin 2 θ R 2 1 − R 2 ,t + R 2 ,θ + R 2 ,ϕ . (B12) Considering the Israel junction conditions we find The other components of the energy momentum tensor can be found similarly.