Physics Letters B 778 (2018) 161–166

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Physics Letters B

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Wormholes immersed in rotating matter ∗ Christian Hoffmann a,b, Theodora Ioannidou c, Sarah Kahlen a, Burkhard Kleihaus a, , Jutta Kunz a a Institut für Physik, Universität Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany b Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4525, USA c Department of Mathematics, Physics and Computational Sciences, Faculty of Engineering, Aristotle University of Thessaloniki, Thessaloniki, 54124, Greece a r t i c l e i n f o a b s t r a c t

Article history: We demonstrate that rotating matter sets the throat of an Ellis wormhole into rotation, allowing for Received 11 December 2017 wormholes which possess full reflection symmetry with respect to the two asymptotically flat Accepted 9 January 2018 regions. We analyze the properties of this new type of rotating wormholes and show that the worm- Available online 19 January 2018 hole geometry can change from a single throat to a double throat configuration. We further discuss the Editor: M. Cveticˇ ergoregions and the lightring structure of these wormholes. © 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction a complex boson field, since this allows for the possibility to im- pose rotation on the bosonic field by the choice of an appropriate In the Ellis wormhole [1–10] connects two ansatz. The rotation of the matter then implies a rotation of the asymptotically flat spacetime regions by a throat. The non-trivial spacetime, thus dragging the wormhole throat along. is provided by the presence of a phantom field, a real As we will see, such a rotating wormhole spacetime can be scalar field with a reversed sign in front of its kinetic term. Re- fundamentally different from a rotating wormhole without matter cently, the static Ellis wormhole has been generalized to - (except for the phantom field). Namely, the rotating configuration with higher dimensions [11,12]. Moreover, rotating general- can be fully symmetric with respect to a reflection of the radial izations of the Ellis wormhole in four and five dimensions have coordinate at its throat or in the case of a double throat at its been found [12–15]. equator. This does not hold for the rotating wormholes obtained Since wormholes could be of potential relevance in astro- previously [12–15], where the rotation is enforced via a boundary physics, a number of both theoretical and observational studies condition in only one of the two asymptotically flat regions. have been performed. These include astrophysical searches for In section 2 we discuss the theoretical setting to obtain worm- wormholes [16–18], studies of their properties as gravitational holes immersed in rotating bosonic matter in four spacetime di- lenses [19–21], studies of their shadows [22,23] or of the iron line mensions. Besides presenting the action, this includes a discussion profiles of thin accretion disks surrounding them [24]. of the field equations, the boundary conditions and the worm- An Ellis wormhole carries no further fields besides the phantom hole properties. In section 3 we present the results of the nu- field. Here we are interested in the effect of matter fields on the merical calculations for this new class of rotating wormholes and wormhole. To this end, we immerse the wormhole throat inside discuss their global charges, their geometric properties, and their a lump of matter. While this question has been considered before lightrings. We conclude in section 4. for nuclear matter [25–27] and bosonic matter [28–30], the matter considered so far was non-rotating. 2. Theoretical setting We here add a new twist to this quest by immersing the worm- hole throat inside rotating matter, which we take as composed of We consider General Relativity with a minimally coupled com- plex scalar field and a phantom field in four spacetime di- mensions. Besides the Einstein–Hilbert action with curvature scalar * Corresponding author. R and coupling constant κ = 8π G, the action    E-mail addresses: [email protected] (C. Hoffmann), √ [email protected] (T. Ioannidou), [email protected] (S. Kahlen), 1 4 S = R + LM −gd x (1) [email protected] (B. Kleihaus), [email protected] (J. Kunz). 2κ https://doi.org/10.1016/j.physletb.2018.01.021 0370-2693/© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 162 C. Hoffmann et al. / Physics Letters B 778 (2018) 161–166

L = = π contains the matter Lagrangian M at θ 0and θ 2 , the boundary condition for can be chosen freely at one asymptotically flat boundary, since only derivatives 1 1   L = μ − μν ∗ + ∗ − 2| |2 of the phantom field enter the PDEs, and is determined from the M ∂μ∂ g ∂μ ∂ν ∂ν ∂μ mb 2 2 asymptotic form of the solutions at the other asymptotically flat (2) boundary. where the kinetic term of the massless phantom field carries Once the solutions are obtained we can read off their physical the reverse sign as compared to the kinetic term of the complex properties. Mass and angular momentum can be obtained from the scalar field , which possesses mass mb. asymptotic behavior of the metric functions Variation of the action with respect to the metric leads to the 2M± 2 J± Einstein equations f −→ ∓ , ω −→ as η →±∞. (10) η η3 1 Gμν = Rμν − gμνR = κ Tμν (3) The particle number Q associated with the conserved current 2 of the complex scalar field is related to the angular momentum with stress–energy tensor by [31] ∂LM Tμν = gμνLM − 2 , (4) = ∂ gμν J nQ . while variation with respect to the scalar fields yields for the Of interest are also the geometrical properties of the solutions. phantom field the equation In order to determine the presence of throats and equators we con- sider the circumferential radius in the equatorial , μ ∇ ∇μ = 0(5)√ q− f Re(η) = he 2 . (11) and for the complex scalar field θ=π/2

∇μ∇ = 2 The minima of Re correspond to throats, whereas the local max- μ mb . (6) ima correspond to equators. Since Re increases without bound in To allow for the non-trivial topology of the stationary space- the asymptotic regions, any equator resides between two throats. , we choose for the line element The ergoregion of the solutions is defined as the region where the   time–time component of the metric is positive, gtt > 0. Its bound- 2 f 2 q− f b 2 2 2 2 ds =−e dt + e e (dη + hdθ ) + h sin θ(dϕ − ωdt) , ary is referred to as the ergosurface. Here gtt(η, θ) = 0. The motion of massless particles in the equatorial (7) plane is determined from where f , q, b and ω are functions of the radial coordinate η and 2 2 2 the polar angle θ, while h = η + η is an auxiliary function con- 2 L E E 0 η˙ = − V +(η) − V −(η) , (12) b−q taining the throat parameter η0. The coordinate η then takes pos- e( ) L L −∞ ∞ itive and negative values, i.e. < η < , where the two limits f −q/2 →±∞ e η correspond to two distinct asymptotically flat regions. with V ±(η) = ω ∓ √ , (13) We parametrize the complex scalar field via [31] h + (t, η,θ,ϕ) = φ(η,θ)eiωst inϕ , (8) where the derivative is with respect to some affine parameter. For ˙ dη˙ circular orbits η and dη have to vanish. Thus the extrema of the where φ(η, θ) is a real function, ωs denotes the boson frequency, potentials V ±(η) determine the location of the lightring, ηlr, and and the integer n is the rotational quantum number, which we E E the ratio , i.e. = V ±(ηlr). employ to impose rotation on the field configuration (n = 0). The L L phantom field depends only on the coordinates η and θ, 3. Symmetric rotating wormhole solutions (t, η,θ,ϕ) = ψ(η,θ). (9) We have solved the above set of PDEs numerically, sub- Substituting the above Ansätze into the Einstein equations and ject to the above boundary conditions, employing the package the matter field equations leads to a set of six coupled partial dif- FIDISOL [32], a finite difference solver based on the Newton– ferential equations (PDEs) of second order for the unknown metric Raphson method. The problem has the following input parameters: and matter functions. Inspection of this system of PDEs shows i) the boson frequency ωs is varied in the interval 0 < ωs < mb that it allows for reflection symmetric solutions with respect to (note, that the boson mass is kept fixed throughout, mb = 1.1), η →−η. ii) the throat parameter is mostly fixed to η0 = 1, though other To solve this system of PDEs, we have to impose an appropri- values have also been employed (note, that η0 = 0leads to topo- ate set of boundary conditions at the boundaries of the domain logically trivial boson stars, where η ≥ 0), iii) for the rotational of integration. For the new class of reflection symmetric rotating quantum number the lowest values are considered, n ∈{0, 1, 2}, wormholes, which are asymptotically flat and globally regular, we where n = 0represents the non-rotating case. In the following we impose the following conditions: (i) the metric and scalar field present results for 0.3 ≤ ωs ≤ 1.03, since for values of ωs outside functions f , q, b, ω, φ vanish asymptotically for η →±∞, (ii) the this interval the numerical errors are too large to be considered as derivatives ∂θ f , ∂θ q, ∂θ ω and the functions b, φ vanish along the reliable solutions (in the rotating case). rotation axis, θ = 0, (iii) the derivatives ∂θ f , ∂θ q, ∂θ b, ∂θ ω, ∂θ φ Let us start our discussion of this new type of rotating worm- = π vanish in the equatorial plane, θ 2 . hole solutions by considering their global charges. In Fig. 1 we For the phantom field the boundary conditions are more in- show the mass M and the particle number Q versus the boson fre- volved. While regularity and reflection symmetry with respect to quency ωs for two families of rotating wormhole solutions, spec- the equatorial plane also require the derivative ∂θ (η, θ) to vanish ified by the lowest non-trivial rotational quantum numbers n = 1 C. Hoffmann et al. / Physics Letters B 778 (2018) 161–166 163

Fig. 1. Global charges of symmetric wormhole solutions with throat parameter η0 = 1 immersed in non-rotating (n = 0) and rotating (n = 1, 2) matter versus the boson frequency ωs : (a) the mass M; (b) the particle number Q . These configurations possess angular momentum J = nQ . The dashed lines indicate the corresponding boson star solutions.

Fig. 2. Throat(s), equator and (s) of symmetric wormhole solutions with throat parameter η0 = 1 immersed in rotating matter versus the boson frequency ωs : (a) the rotational velocity vth of the throat(s) (solid) and the rotational velocity veq of the equator (dashed); (b) the coordinate ηergo of the ergosurface(s) in the equatorial plane. and n = 2 and the throat parameter η0 = 1. The angular momen- ditions. Instead, the ansatz for the complex scalar field with a tum J of these solutions is given by J = nQ . For comparison the non-vanishing rotational quantum number n imposes rotation on figure also contains the corresponding set of non-rotating worm- the configuration. hole solutions (obtained for the same parameters except for n = 0) The rotation of the scalar field implies rotation of the spacetime immersed in bosonic matter (see e.g. [29]). and consequently also rotation of the throat of the wormholes. This The domain of existence of these solutions is limited by a rotation of the throat is demonstrated for the above sets of rotat- maximal value for the boson frequency ωs, which is reached for ing wormholes in Fig. 2(a), where the rotational velocity vth of the ωmax = mb, where the family of solutions reaches a vacuum con- throat(s) in the equatorial plane is shown versus the boson fre- figuration with M = 0 = Q . This is completely analogous to the quency ωs. Also shown is the rotational velocity veq of the equator case of boson stars (see e.g. [33–38]). As seen in the figure, for when present. large values of the frequency, the global charges of the worm- We now turn to the ergoregions of the solutions. In Fig. 2(b) holes are similar to those of boson stars, which are also exhibited we show the coordinate ηergo of the boundary of the ergoregions with dashed black lines. However the well-known spiraling be- in the equatorial plane versus the boson frequency ωs. We note havior of boson stars is largely lost. Instead the would-be spirals that ergoregions exist only if the boson frequency ωs is smaller unwind after their first backbending with respect to the frequency than some maximal value, which depends on η0 and the rotational and then continue to lower frequencies (possibly all the way to quantum number n. For n = 1the ergoregion decreases monoton- ωs = 0, where a singular configuration should be reached [29]). ically with increasing ωs and degenerates to a circle residing at This unwinding is present in the case of rotating and non-rotating ηergo = 0, when the maximal value of ωs is approached. solutions [29,30], and it has been seen as well in other systems For n = 2the ergoregion decreases monotonically with increas- with densities [39]. ing ωs also up to a maximal frequency. But since this maximal Recall that we have imposed the same boundary conditions in frequency occurs in the range of the frequencies of the unwinding both asymptotic regions for the metric and the complex scalar spiral, the presence of several branches of solutions complicates field. Thus we have not imposed rotation via the boundary con- the picture. Following the second branch (towards smaller ωs), 164 C. Hoffmann et al. / Physics Letters B 778 (2018) 161–166

Fig. 3. Throat(s) and equator of symmetric wormhole solutions with throat parameter η0 = 1 immersed in non-rotating (n = 0) and rotating (n = 1, 2) matter versus the boson frequency ωs : (a) the coordinate ηth of the throat(s) (solid) resp. the coordinate ηeq of the equator (dashed) in the equatorial plane; (b) the circumferential radius Re(ηth) of the throat(s) (solid) resp. the circumferential radius Re(ηeq) of the equator (dashed) in the equatorial plane.

Fig. 4. Embeddings of the equatorial plane of symmetric wormhole solutions with throat parameter η0 = 1 immersed in non-rotating (n = 0) and rotating (n = 1, 2) matter: (a) embeddings for n = 0, 1, 2 and ωs = 0.3, where a pseudo-euclidean embedding is indicated by the dashed lines; (b) 3D plot for n = 0; (c) 3D plot for n = 1; (d) 3D plot for n = 2.

ηergo reaches a maximum and then decreases again. The ergoregion dividing the ergoregion into two parts, until at ωs = 0.7256 the er- then disappears on the second branch before the third branch is goregion degenerates to two rings. No ergoregion exists for smaller reached. Interestingly, for the small interval 0.7256 ≤ ωs ≤ 0.7388 frequencies on the second branch of solutions. the surface of the ergoregion consists of two disconnected parts. Let us next consider the geometry of the wormhole solutions. At ωs = 0.7388, the throat at η = 0forms an inner boundary , In Fig. 3 we present the coordinate ηth of the throat(s) resp. the C. Hoffmann et al. / Physics Letters B 778 (2018) 161–166 165

Fig. 5. Lightrings of symmetric wormhole solutions with throat parameter η0 = 1 immersed in rotating (n = 1, 2) matter versus the boson frequency ωs : (a) the coordinate + − ηlr of co-rotating massless particles in the equatorial plane; (b) the coordinate ηlr of counter-rotating massless particles in the equatorial plane; (c) the circumferential + − radius Re(ηlr ); (d) the circumferential radius Re(ηlr ). Also shown are the coordinates and the circumferential radii of the ergosurfaces (black lines). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.) coordinate ηeq of the equator in the equatorial plane as well as the emerge. For the symmetric wormholes one of the lightrings is corresponding values of the circumferential radius Re. We observe always located at η = 0, i.e., at the throat, respectively, at the that for large values of the boson frequency ωs the wormholes equator, whereas the remaining ones (if present) are located sym- possess only a single throat. At a critical value of ωs an equator metrically around η = 0. We note that for symmetric wormhole emerges, when the throat degenerates to an inflection point at the solutions immersed in rotating (n = 2) matter up to five lightrings critical value of ωs. For smaller ωs the circumferential radius Re of counter-rotating massless particles can exist. possesses a maximum at η = 0, corresponding to an equator, and When inspecting the circumferential radii for co-rotating and two minima located symmetrically at around η = 0, corresponding counter-rotating massless particles, which are also exhibited in + to two throats. Fig. 5, we realize that the circumferential radius Re(ηlr ) of the In order to get a deeper insight into the geometry of the lightrings of co-rotating massless particles and the circumferential wormholes, we exhibit in Fig. 4 the isometric embedding of the radius Re(ηth) of the single throat almost coincide for n = 1 and equatorial plane for symmetric wormhole solutions immersed in are still close for n = 2. non-rotating (n = 0) and rotating (n = 1, 2) matter with η0 = 1 We further remark that the ratio E/L of the lightring residing at and ωs = 0.3as examples of double-throat wormholes. Note the the equator of the symmetric wormholes becomes negative, when = pseudo-euclidean embedding for the case n 2. ωs is smaller than a critical value. The change of sign occurs pre- Finally we would like to address the lightrings of these space- cisely, when the lightring enters the ergoregion. + times. In Fig. 5 we therefore exhibit the coordinate ηlr of co- rotating massless particles in the equatorial plane together with 4. Conclusion − the coordinate η of counter-rotating massless particles and their lr + − Whereas previously the rotation of a wormhole spacetime had circumferential radii Re(ηlr ) and Re(ηlr ), respectively. Also shown is the location of the ergosurfaces and their respective circumfer- to be imposed by hand, by requiring a set of non-identical asymp- ential radii (solid black). totic boundary conditions for the metric in the two universes The general picture is that a single lightring exists, if ωs exceeds [13,14], we have here obtained a new type of rotating wormhole a critical value, which depends on the orientation of the orbit. solutions, which possess identical asymptotic boundary conditions When ωs is smaller than this critical value two more lightrings for the metric. Indeed, instead of imposing the rotation as a bound- 166 C. Hoffmann et al. / Physics Letters B 778 (2018) 161–166 ary condition we have obtained fully symmetric configurations by [12] V. Dzhunushaliev, V. Folomeev, B. Kleihaus, J. Kunz, E. Radu, Phys. Rev. D 88 immersing the wormholes into rotating matter. (2013) 124028. For the rotating matter we have chosen for simplicity a complex [13] B. Kleihaus, J. Kunz, Phys. Rev. D 90 (2014) 121503, https://doi.org/10.1103/ PhysRevD.90.121503, arXiv:1409.1503 [gr-qc]. massive scalar field without any self-interaction. Both asymptoti- [14] X.Y. Chew, B. Kleihaus, J. Kunz, Phys. Rev. D 94 (2016) 104031, https://doi.org/ cally flat regions – or universes – possess the same global charges, 10.1103/PhysRevD.94.104031, arXiv:1608.05253 [gr-qc]. consisting of the mass, the angular momentum and the particle [15] B. Kleihaus, J. Kunz, Fundam. Theor. Phys. 189 (2017) 35. number, where the latter two are related by the rotational quan- [16] F. Abe, Astrophys. J. 725 (2010) 787, arXiv:1009.6084 [astro-ph.CO]. tum number as in the case of boson stars. We have shown that the [17] Y. Toki, T. Kitamura, H. Asada, F. Abe, Astrophys. J. 740 (2011) 121, arXiv: 1107.5374 [astro-ph.CO]. resulting configurations can possess interesting physical properties. [18] R. Takahashi, H. Asada, Astrophys. J. 768 (2013) L16, arXiv:1303.1301 [astro- They can have a single throat or, depending on the parameters, ph.CO]. evolve an equator surrounded by a double throat. These rotat- [19] J.G. Cramer, R.L. Forward, M.S. Morris, M. Visser, G. Benford, G.A. Landis, Phys. ing wormhole solutions may also exhibit up to five lightrings for Rev. D 51 (1995) 3117, arXiv:astro-ph/9409051. counter-rotating massless particles. [20] V. Perlick, Phys. Rev. D 69 (2004) 064017, arXiv:gr-qc/0307072. The inclusion of a phantom field has allowed for the existence [21] N. Tsukamoto, T. Harada, K. Yajima, Phys. Rev. D 86 (2012) 104062, arXiv: 1207.0047 [gr-qc]. of these topologically non-trivial solutions. Clearly, by replacing [22] C. Bambi, Phys. Rev. D 87 (2013) 107501, arXiv:1304.5691 [gr-qc]. General Relativity by certain generalized theories the phan- G.[23] P. Nedkova, V.K. Tinchev, S.S. Yazadjiev, Phys. Rev. D 88 (2013) 124019, arXiv: tom field would no longer be needed [40–48]. A study of rotating 1307.7647 [gr-qc]. wormholes in such generalized theories would therefore be desir- [24] M. Zhou, A. Cardenas-Avendano, C. Bambi, B. Kleihaus, J. Kunz, Phys. Rev. D 94 able. (2016) 024036, arXiv:1603.07448 [gr-qc]. [25] V. Dzhunushaliev, V. Folomeev, B. Kleihaus, J. Kunz, Phys. Rev. D 87 (2013) Let us end with a short remark concerning the stability of the 104036. above . It is known, that the static Ellis wormhole in D [26] V. Dzhunushaliev, V. Folomeev, B. Kleihaus, J. Kunz, Phys. Rev. D 89 (2014) dimensions possesses an unstable radial mode [11,49–51]. Interest- 084018. ingly, when the wormhole throat rotates sufficiently fast, this ra- [27] A. Aringazin, V. Dzhunushaliev, V. Folomeev, B. Kleihaus, J. Kunz, J. Cosmol. Astropart. Phys. 1504 (2015) 005, https://doi.org/10.1088/1475-7516/2015/04/ dial instability disappears, as shown for (cohomogeneity-1) worm- 005, arXiv:1412.3194 [gr-qc]. holes in five dimensions [12]: As the throat is set into rotation, [28] E. Charalampidis, T. Ioannidou, B. Kleihaus, J. Kunz, Phys. Rev. D 87 (2013) a non-normalizable zero mode of the Ellis wormhole turns into a 084069, https://doi.org/10.1103/PhysRevD.87.084069, arXiv:1302.5560 [gr-qc]. second unstable radial mode. With increasing rotation velocity of [29] V. Dzhunushaliev, V. Folomeev, C. Hoffmann, B. Kleihaus, J. Kunz, Phys. the throat, the eigenvalue of the original negative mode decreases, Rev. D 90 (2014) 124038, https://doi.org/10.1103/PhysRevD.90.124038, arXiv: 1409.6978 [gr-qc]. while the eigenvalue of the second unstable mode increases, un- [30] C. Hoffmann, T. Ioannidou, S. Kahlen, B. Kleihaus, J. Kunz, Phys. Rev. D 95 til both merge and disappear. Thus rotation has a stabilizing effect (2017) 084010, https://doi.org/10.1103/PhysRevD.95.084010, arXiv:1703.03344 on the wormhole. A stability analysis along these lines for rotating [gr-qc]. 4-dimensional wormholes represents a challenging task. [31]E. F. Schunck, E.W. Mielke, in: F.W. Hehl, R.A. Puntigam, H. Ruder (Eds.), Rel- ativity and Scientific Computing: Computer Algebra, Numerics, Visualization, Springer, Berlin, Heidelberg, 1996. Acknowledgments [32] W. Schönauer, R. Weiß, J. Comput. Appl. Math. 27 (1989) 279; M. Schauder, R. Weiß, W. Schönauer, The CADSOL Program Package, Interner We would like to acknowledge support by the Deutsche Bericht Nr. 46/92, Universität Karlsruhe, 1992. Forschungsgemeinschaft Research Training Group 1620 Models of [33] P. Jetzer, Phys. Rep. 220 (1992) 163. Gravity as well as by FP7 People: Marie-Curie Actions, Interna- D.[34] T. Lee, Y. Pang, Phys. Rep. 221 (1992) 251. [35]E. F. Schunck, E.W. Mielke, Class. 20 (2003) R301. tional Research Staff Exchange Scheme (IRSES-606096), COST Ac- [36] S.L. Liebling, C. Palenzuela, Living Rev. Relativ. 15 (2012) 6, https://doi.org/ tion CA16104 GWverse. BK gratefully acknowledges support from 10.12942/lrr-2012-6. Fundamental Research in Natural Sciences by the Ministry of Edu- [37] B. Kleihaus, J. Kunz, M. List, Phys. Rev. D 72 (2005) 064002. cation and Science of the Republic of Kazakhstan. [38] B. Kleihaus, J. Kunz, M. List, I. Schaffer, Phys. Rev. D 77 (2008) 064025. [39] B. Hartmann, J. Riedel, R. Suciu, Phys. Lett. B 726 (2013) 906. [40] D. Hochberg, Phys. Lett. B 251 (1990) 349. References [41] H. Fukutaka, K. Tanaka, K. Ghoroku, Phys. Lett. B 222 (1989) 191. [42] K. Ghoroku, T. Soma, Phys. Rev. D 46 (1992) 1507. [1] H.G. Ellis, J. Math. Phys. 14 (1973) 104. [43] N. Furey, A. DeBenedictis, Class. Quantum Gravity 22 (2005) 313. [2]. K.A Bronnikov, Acta Phys. Pol. B 4 (1973) 251. [44]. K.A Bronnikov, E. Elizalde, Phys. Rev. D 81 (2010) 044032. [3] T. Kodama, Phys. Rev. D 18 (1978) 3529. [45] P. Kanti, B. Kleihaus, J. Kunz, Phys. Rev. Lett. 107 (2011) 271101. [4] H.G. Ellis, Gen. Relativ. Gravit. 10 (1979) 105. [46] P. Kanti, B. Kleihaus, J. Kunz, Phys. Rev. D 85 (2012) 044007. [5] M.S. Morris, K.S. Thorne, Am. J. Phys. 56 (1988) 395. [47]S.N. F. Lobo, M.A. Oliveira, Phys. Rev. D 80 (2009) 104012. [6] M.S. Morris, K.S. Thorne, U. Yurtsever, Phys. Rev. Lett. 61 (1988) 1446. [48] T. Harko, F.S.N. Lobo, M.K. Mak, S.V. Sushkov, Phys. Rev. D 87 (2013) 067504. [7] C. Armendariz-Picon, Phys. Rev. D 65 (2002) 104010. [49] H.-a. Shinkai, S.A. Hayward, Phys. Rev. D 66 (2002) 044005. [8] S.V. Sushkov, Phys. Rev. D 71 (2005) 043520. .[50] J.A Gonzalez, F.S. Guzman, O. Sarbach, Class. Quantum Gravity 26 (2009) S.N.[9] F. Lobo, Phys. Rev. D 71 (2005) 084011. 015011. [10]S.N. F. Lobo, Fundam. Theor. Phys. 189 (2017). [51]. J.A Gonzalez, F.S. Guzman, O. Sarbach, Class. Quantum Gravity 26 (2009) [11] T. Torii, H.a. Shinkai, Phys. Rev. D 88 (2013) 064027, https://doi.org/10.1103/ 015010. PhysRevD.88.064027, arXiv:1309.2058 [gr-qc].