Thin-shell wormholes supported by total normal matter

The Zipoy-Voorhees-Weyl (ZVW) spacetime characterized by mass ($M$) and oblateness ($\delta $) is proposed in the construction of viable thin-shell wormholes (TSWs). Departure from spherical / cylindrical symmetry yields positive total energy in spite of the fact that local energy density may take negative values. We show that oblateness of the bumpy sources / black holes can be incorporated as a new degree of freedom that may play role in the resolution of the exotic matter problem in TSWs. Small velocity perturbation reveals, however, that the resulting TSW is unstable.


I. INTRODUCTION
Until popularized by Morris and Thorne [1] the idea of spacetime wormhole introduced in the 1930s by Einstein and Rosen [2] was considered non-physical and largely as a fantasy. Although all kinds of spherically / cylindrically symmetric metrics known to date were tried the wormhole concept was shadowed by the required negative total energy. While it was easy to resort to quantum field theoretical negative energy for remedy the absence of large scale quantum systems persisted as another serious handicap. For such reasons relying on classical physics and searching for support within this context seems indispensable. Even to minimize the negative (exotic) energy the idea of thin-shell wormhole (TSW) was developed (See [3] and those cited therein). By construction in all these studies the otherwise non-traversable wormhole throat that connects two different universes has circular topology.
In this study we add oblateness as a new degree of freedom represented by the parameter δ (−∞ < δ < ∞) and show that for certain range of δ total energy becomes positive to avoid exotic sources. This happens in the Zipoy-Voorhees-Weyl (ZVW) spacetime [4] with a quadrupole moment Q = 1 3 M 3 δ 1 − δ 2 , where M =mass of the bumpy object (or the black hole). Naturally for δ = 1 one recovers the spherical Schwarzschild geometry. It should be supplemented that integrability and chaotic behaviours of the ZVW spacetime still is not well understood [5]. Asymptotically flat, rotating ZVW metric was discovered by Tomimatsu and Sato (TS) [6], which similar to its static predecessor remains from physics stand point yet unclear. Once the problem of gravitational wave detection is overcome we expect that non-Kerr (i.e., δ = 1) multipoles of the entire TS family can be detected. We note also that non-asymptotically flat extension of ZVW metric is also available whose physics is yet to be understood [7]. Herein we concentrate on the static * habib.mazhari@emu.edu.tr † mustafa.halilsoy@emu.edu.tr ZVW metric in general relativity and construct thin-shell wormhole (TSW) in this spacetime. We should add that in the context of the modified theories of gravity, previously, there were some attempts to introduce thin-shell wormhole supported by positive / normal matter [8]. From this token it was realized that the normal matter is possible only for the exotic branch solution of the Einstein-Gauss-Bonnet field equation [8].
The paper is organized as follows. Construction of TSW in ZVW spacetime is carried out in Section II. Integration of the total energy is achieved in Section III. Stability analysis follows in Section IV and Conclusion in Section V completes the paper.

II. ZVW THIN-SHELL WORMHOLE (TSW)
The two parameter ZVW spacetime in the prolate spheroidal coordinates is described by the line element where k = M δ and the range of coordinates are 1 < x, 0 ≤ y 2 ≤ 1, ϕ ∈ [0, 2π] and −∞ < t < ∞. We note that −∞ < δ < ∞ such that δ = 0 corresponds to a flat spacetime and with δ = 1 one finds the Schwarzschild black hole solution with the horizon located at x = 1. For the case δ = 1 the hypersurface x = 1 is a true curvature singularity (naked singularity) [9]. As we shall see, the asymptotic behaviour of the ZV spacetime for x → ∞ and δ > 1 is of our interest. It can be seen from (1) that, in the limit x → ∞, it becomes which after a redefinition of kx = r and y = cos θ, the line element becomes which is flat. Construction of thin-shell wormhole (TSW) follows the standard procedure of cutting and pasting [8]. We consider two copies of ZVW spacetime and we remove from each in which a = constant outside the singularities / horizons. We should comment here that the minimality conditions of Hochberg and Visser [10] which is also known as the generalized flare-out conditions in static wormholes do not apply in the present case of TSW [11]. At the throat two spacetimes are identified to make a complete manifold. We introduce next the induced coordinates ξ i = (τ, y, φ) on the wormhole's throat with its proper time τ . The two coordinates are related by so that induced metric on the throat Σ reads The Israel junction conditions [12] on Σ take the form (c = 8πG = 1) in which . stands for a jump across the hypersurface. K j i is the extrinsic curvature defined by with the normal unit vector Note that K = T race K j i and S j i =diag(−σ, P y , P φ ) is the energy momentum tensor on the thin-shell. The parametric equation of the hypersurface Σ is given by The normal unit vectors to M ± is found to be with ∆ = 1 B +ȧ 2 andȧ = da dτ . The resulting extrinsic curvature components are K y(±) in which a subscript a stands for ∂ ∂a . The surface energy momentum tensor has components defined as

III. POSITIVE MATTER SOURCES
The energy-momentum components at equilibrium condition, i.e. a = a 0 =constant withȧ =ä = 0 yields

FIG. 2:
A 3D plot of the positive amount of energy Ω 2 versus a and δ with constant mass parameter i.e. M = 1. This figure shows clearly that for δ > 2 there exist ac in which with 1 < a < ac the total energy is positive and therefore the resultant thin shell wormhole is supported by ordinary / normal matter. It should be added that for a given δ > 2, there exists 1 < a < ac which leads to a physically accepted TSW.  Fig. 2, here it is more clear that when the value of a → 1, the threshold δ which admits a positive total energy approaches to δ = 2. This means that δ = 2 is a critical value to have a thin shell wormhole supported by ordinary matter. For larger a the threshold δ gets larger values.
The explicit form of the energy momentum components are then found to be .
Note that since 1 < a 0 there are no singularities in the foregoing expressions. From the conditions a 0 > 1 and y < 1 one observes that P y0 and P φ0 are both positive while σ 0 may be positive, negative or zero. Fig. 1 displays σ 0 in terms of a 0 and y. We observe that for δ > 1 there exists regions that σ 0 becomes positive. This is seen clearly in Fig. 1. Also in the interval on which σ 0 ≥ 0 the weak and strong energy conditions are satisfied. We note that, in Fig. 1 the energy density σ 0 is shown in terms of y and a 0 but only y is variable and a 0 is fixed for a specific TSW. This means that at the throat x = a 0 and only y and φ are variable. As one sees from (18), there is angular symmetry and as a result, not only σ 0 but also P y0 and P φ0 are only functions of y. In addition, P y0 is a constant function of y. Therefore once we set the radius of the throat i.e., a 0 , our energy momentum tensor's components are left with the only variable y. Hence, depending on y, the energy density of the TSW i.e. σ 0 is locally positive or negative. This is what we see in Fig. 1. Of course, the situation is completely different for P y0 and P φ0 which are positive for entire domain of y.
In addition to the energy conditions we are mainly interested in the total energy supporting the TSW given by which simplifies to In Eq. (19) δ (x − a 0 ) is the Dirac delta function. In Fig. 2 we plot Ω versus a and δ with fixed value of mass M = 1. Fig. 3 reveals more details. These plots overall show that TSW supported by normal matter is possible provided δ > 2. That explains also why given ordinary matter alone in Schwarzschild spacetime with δ = 1, there was no such traversable wormhole. In this regard let us add that even an arbitrarily small energy condition violation is considered respectable [13].

IV. STABILITY ANALYSIS
In this section we apply the small velocity perturbation on the shell under the condition that the EoS of the TSW after the perturbation is same as its EoS at its static equilibrium. This is possible if the perturbation process occurs slow enough in which all the intermediate states can also be considered as static equilibrium points. Therefore the EoS after the perturbation reads and These in explicit form amount to which upon integration yieldṡ In Figs. 4 and 5 we plotȧ in terms of a and y for the initial values a 0 = 1.2 andȧ 0 = ±0.1, respectively, and δ = 2.5. As one can see in both cases the velocity does not get zero, which means that the throat does not go back to its initial position.

V. CONCLUSION
We considered the possibility of having physical wormhole solution in Einstein's general relativity which is sup- ported by normal matter and the energy conditions are satisfied. Our emphasis is on Einstein's general relativity instead the modified theories such as the Lovelock theory. In such theories there were attempts to find such physical wormholes [8] with non-physical solution and it was shown that a TSW supported by positive energy was possible. In this context we have shown that TSWs with oblate sources can be employed to admit an overall physical (i.e. non-exotic) matter even in Einstein's general relativity. As we have depicted in figures, for δ > 2 not only the total energy of the wormhole is positive but also the WECs and SECs are satisfied for limited y-interval, which increases for larger δ. Certain range of deviation from spherical symmetry can be chosen from the total energy integral to render this possible. Locally it can easily be checked from Eq. (16) for y = 0, for instance, that we have a negative energy density, however, this is compensated within the total energy expression. It is also expected that once the metric and the surface energy-momentum become time dependent energy conservation on the thin shell will not be valid any more. Another important aspect concerning TSWs which has not been discussed here is their stability against perturbations. It is shown that small velocity perturbations in the x−direction leads to an unstable wormhole throat. Finally, from curiosity we wish to ask: does the deformation parameter δ saves wormholes other than TSWs in Einstein's theory?. It remains to be seen.