Uniqueness Theorem for Static Phantom Wormholes in Einstein

Physics Letters B 778 (2018) 408–413

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Uniqueness theorem for static phantom wormholes in Einstein–Maxwell-dilaton theory ∗ Boian Lazov a, Petya Nedkova a,b, , Stoytcho Yazadjiev a,c,d a Department of Theoretical Physics, Sofia University, Sofia 1164, Bulgaria b Institut für Physik, Universität Oldenburg, D-26111 Oldenburg, Germany c Theoretical Astrophysics, Eberhard-Karls University of Tübingen, Tübingen 72076, Germany d Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia 1164, Bulgaria a r t i c l e i n f o a b s t r a c t

Article history: We prove a uniqueness theorem for completely regular traversable electrically charged wormhole Received 8 November 2017 solutions in the Einstein–Maxwell-dilaton gravity with a phantom scalar field and a possible phantom Received in revised form 14 January 2018 electromagnetic field. In a certain region of the parameter space, determined by the asymptotic values Accepted 22 January 2018 of the scalar field and the lapse function, the regular wormholes are completely specified by their mass, Available online xxxx scalar charge and electric charge. The argument is based on the positive energy theorem applied on an Editor: M. Cveticˇ appropriate conformally transformed Riemannian space. © 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 regular wormholes can exist in certain alternative theories of gravity without introducing any exotic matter [7]. In contrast to black holes, wormhole solutions, although avail- Wormholes arise naturally in general relativity and alternative able in many cases, are not classified systematically. A recent work theories of gravity as solutions which form bridges between differ- proved that the Ellis–Bronnikov wormhole is the unique static ent universes, or different regions of the same universe [1]. Phys- traversable wormhole in Einstein-scalar field theory, which is free ically most interesting are the traversable wormholes, which are of any singularities [8]. The goal of the present paper is to pro- free of conical or curvature singularities, and can be considered vide a uniqueness theorem for completely regular traversable elec- in context of future spacetime travel [2]. Solutions with conical trically charged wormholes in the static sector of the Einstein– singularity localized in the equatorial plane (ring wormholes) also Maxwell-dilaton theory with a phantom dilaton field and a pos- attract attention as models of elementary particles sourced by a sible phantom electromagnetic field. We consider the case of cosmic string [3]. dilaton-Maxwell coupling constant equal to unity. It is well known that in general relativity traversable wormholes cannot be supported by matter obeying the null energy con- 2. Field equations and general definitions dition [4]. Its violation requires the existence of some exotic matter with negative energy density. Such scenarios are widely investi- We consider Einstein–Maxwell-dilaton theory with a phantom gated also in cosmological context in view of dark energy models. dilaton field defined by the following action One of the most simple models proposes the introduction of phan- √ tom matter fields, which possess kinetic terms coupled repulsively 1 4 g g μ −2ϕ μν S = dx −g R + 2 ∇μϕ ∇ ϕ − ε e Fμν F . to gravity, such as phantom scalar or electromagnetic fields. Their 16π presence agrees with current cosmological observations, and en- (1) ables a variety of wormhole solutions [5], arising also as final states in dynamical collapse [6]. We should note, however, that We further allow that the Maxwell-gravity coupling constant ε can take either positive, or negative values ε =±1. Thus, we include in our analysis also phantom Maxwell fields. The action leads to the following field equations on the spacetime manifold M(4) * Corresponding author. E-mail addresses: [email protected]fia.bg (B. Lazov), g g −2ϕ β gμν βγ [email protected]fia.bg (P. Nedkova), [email protected]fia.bg (S. Yazadjiev). Rμν =−2 ∇μϕ ∇νϕ + 2εe Fμβ Fν − Fβγ F , (2) 4 https://doi.org/10.1016/j.physletb.2018.01.059 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. B. Lazov et al. / Physics Letters B 778 (2018) 408–413 409 g ∇[β Fμν] = 0, In the present paper we will also assume that the 3-dimensional man- (3) − ifold M is simply connected. We can introduce the electric field g∇ e 2ϕ F βμ = 0 β , one-form g g β 1 −2ϕ μν ∇β ∇ ϕ = εe Fμν F , E =− F , (6) 2 ιξ where Rμν is the spacetime Ricci tensor, ϕ represents the dilaton which satisfies dE = 0as a consequence of the field equations and field, and Fμν is the Maxwell field. the staticity of the Maxwell field. Then, since M(3) is simply con- We are interested in strictly static spacetimes, i.e. spacetimes nected, there exists an electric potential defined on M(3) such possessing a Killing vector field ξ , which is everywhere timelike. that E = d . The Maxwell 2-form is given by For such spacetimes there exists a smooth Riemannian manifold + − (M(3), g(3)) and a smooth lapse function N : M(3) −→ R allowing F =−N 2ξ ∧ d , (7) the following decomposition and considering (5) we obtain the asymptotic behavior of the po- 4 = R × (3) μ ν =− 2 2 + (3) i j M M , gμνdx dx N dt gij dx dx . (3) tential Q ± Staticity of the Maxwell and scalar fields is defined by means = + + O −2 = ∂ ± (r ). (8) of the Lie derivative along the timelike Killing field ξ ∂t , r Using the metric and the electromagnetic field decomposition, Lξ F = 0, Lξ ϕ = 0. (4) we can obtain the following dimensionally reduced static EMD We will further focus on the case of purely electric field, which equations means that ιξ F = 0is satisfied. We consider static, asymptotically flat wormhole solutions to − − gN = ε N 1e 2ϕ g∇ g∇i , (9) the field equations (2) with two asymptotically flat ends. Moreover, i g g g −1 g g we restrict to purely electric configurations. In analogy to [8] we Rij =−2 ∇iϕ ∇ jϕ + N ∇i ∇ j N adopt the following formal definition: −2 −2ϕ g g k g g + ε N e (gij ∇k ∇ − 2 ∇i ∇ j ), A solution to the field equations (2) is said to be a static, asymptoti- g∇ −1 −2ϕ g∇i = cally flat completely regular traversable wormhole solution if the follow- i(N e ) 0, ing conditions are satisfied: g g i −1 −2ϕ g g i ∇i(N ∇ ϕ) =−ε N e ∇i ∇ ,

1. The spacetime is strictly static. g g where ∇ and Rij are the Levi-Civita connection and the Ricci 2. The Riemannian manifold (M(3), g(3)) is complete. (3) tensor with respect to the 3-metric gij . 3. For some compact set K, M(3) − K consists of two ends End+ and Using the maximum principle for elliptic partial differential 3 ¯ ¯ End− such that each end is diffeomorphic to R( )\Bwhere Bis the equations and from the asymptotic behavior of N it follows that closed unit ball centered at the origin in R(3), and with the following the values of N on M(3) satisfy asymptotic behavior of the 3-metric, the lapse function, the scalar field, and the electromagnetic field N− ≤ N ≤ N+ (10) = = (3) −2 2M± −2 as the equality is satisfied only in the case Q ± M± 0. g = N± 1 + δij + O(r ), ij r 3. Divergence identities and functional relations between the M± − N = N± 1 − + O(r 2), potentials r q± − (3) ϕ = ϕ± − + O(r 2), We consider the conformally transformed metric γij on M r defined by Q ± =− + O −3 ∧ F (r ) dt dr, (5) = 2 r2 γij N gij. (11) (3) with respect to the standard radial coordinate rof R , where δij is the It is also convenient to introduce further the potentials standard flat metric on R(3). U = ln(N) + ϕ, = ln(N) − ϕ, (12) The conditions in the above wormhole definition are chosen in order to reflect the expected properties of the wormhole geometry allowing to simplify the reduced field equations (9) to the form in a natural way. The first condition involves the requirement that no horizons should be present, the second one ensures that the γ −2U Rij = Di UDj + DiD j U − 2ε e Di D j , (13) constant time slices are free from singularities, while the third one i = is the standard notion of asymptotically flat regions. Di D U 0, In our notations N± > 0, M±, ϕ± = 0, q± = 0, ± = 0, and i −2U i Di D = 2ε e Di D , Q ± = 0are constants. M± and q± represent the total (ADM) mass −2U i and the scalar charge of the corresponding end End±. The parame- Di(e D ) = 0, ters Q ± are connected to the conserved electric charge associated where D denotes the covariant derivative with respect to the met- with each end, however deviating from it by a multiplicative fac- i 1 ric γ . tor. We choose N− ≤ N+. ij For a given solution (γij, U , , ) the transformation (γij → γij, C1 U → U + C1, → + C2, → e ( + C3)) where C1, C2 and C3 1 The conserved electric charge is defined by eq. (21). are constants, generates new solutions. These new solutions can 410 B. Lazov et al. / Physics Letters B 778 (2018) 408–413 however be regarded as trivial. In order to get rid of them we shall We can expand the last relation (20) between the asymp- impose the following constraints on the potentials U , and : totic properties to a functional relation valid on the whole 3-dimensional manifold M(3). We consider the potential U+ =−U−, + =−−, + =− −. (14) ˜ 2U+ 2U The field equations (13) can be interpreted as describing a χ1 = e − e − 2(M+ + q+) , (24) Q˜ + 3-dimensional gravity coupled to a non-linear σ -model parameter- A = ˜ ized by the scalar fields φ (U , , ) with a target space metric where we denote = − +, which is normalized appropriately to vanish at both asymptotic ends End±. Then, we can construct a 1 − A B 2U 2 −2U G ABdφ dφ = (dUd + ddU) − 2εe d . (15) related 1-form ω =−e dχ , given explicitly by the expression 2 1 (M+ + q+) The Killing vectors for this metric are = + −2U ω 2dU 2e ˜ d , (25) ∂ 1 ∂ Q + K (1) = 2ε + e2U , (16) ∂ 2 ∂ which obeys the relation ∂ ∂ K (2) = + , 2U i =− i =− i ∂U ∂ e ωiω Diχ1 ω Di(χ1ω ), (26) ∂ (3) K (3) =−ε , as a result of the field equations. Integrating this relation over M ∂ we obtain (4) ∂ K = , 2U i i i ∂ e ωiω dμ =− Di(χ1ω )dμ =− χ1ω dSi = 0, ∞ ∞ and the corresponding Killing one-forms are given by M(3) M(3) S+ S− (27) (1) A = − K A dφ 2 dU d , (17) where we have used the asymptotic behavior of the potential χ . (2) A −2U 1 K dφ =−d + 2ε e d , (3) A Then, it is satisfied that ωi = 0on M , and consequently the (3) − A = 2U potential χ1 vanishes identically. Hence, we obtain the following K A dφ e d , (4) relation between the potentials U and K dφ A = dU. A ˜ (a) 2U+ 2U ˜ A e − e − 2M+ = 0, (28) Using the fact that K A dφ are Killing one-forms for the metric ˜ A Q + G AB and taking into account the equations for φ one can show ˜ that the following divergence identities are satisfied where we introduce the notation M+ = M+ + q+. We can further consider the potential η, defined by (a) i A = Di(K A D φ ) 0. (18) − dη = d − 2εe 2U d , (29) We integrate these equations over M(3) and consider the behavior of the potentials φ A at the two asymptotically flat ends End+ and and construct the current J i = UDi η − ηDi U . The current J i is End− given by (5). Thus, the following relations are obtained i conserved as a consequence of the field equations, i.e. Di J = 0on M(3). Integrating this equation over M(3), and using the asymptotic + =− − M+ q+ M− q−, (19) identities (19), we get − − e 2U+ Q + =−e 2U− Q −, − + ˜ − ˜ = ˜ (M+ q+) 2ε + Q + U M+η 0, (30) M+ − q+ =−(M− − q−) − 2εQ + , where the quantities U = U+ − U− = 2U+ and η = η+ − η− e2U+ = e2U− − 2(M+ + q+) , (20) Q˜ + are defined by means of the values of the potentials U and η at the two asymptotic ends End±. The identity can be extended to a where we have introduced the notations relation valid on the whole M(3). We introduce the potential ˜ −2U± Q ± = e Q ±, (21) ˜ ˜ ˜ χ2 = (M+ − q+) + 2ε + Q + U − M+η˜, (31) = + − −. (22) ˜ where we denote U = U − U+, η˜ = η − η+, which by construction ˜ = −2U± The quantities Q ± e Q ± represent the conserved electric vanishes on both asymptotic ends. We can further consider the charges associated with the two asymptotic ends. The identities 1-form = dχ2, satisfying (19)–(20) lead to Smarr-like relations connecting the conserved i i i charges for the two asymptotic ends i = Diχ2 = Di(χ2 ), (32) ˜ M+ + M− + εQ + = 0, as a result of the field equations. Integrating this equation over (3) ˜ M and using the asymptotic behavior of the potential χ2, we get q+ + q− − εQ + = 0. (23) (3) that i = 0on M , and consequently χ2 also vanishes identically. Thus, we can retain as independent parameters, which character- Hence, we obtain the relation ize the wormhole solutions, only the asymptotic charges associated ˜ ˜ ˜ with End+. (M+ − q+) + 2ε + Q + U = M+η˜ (33) B. Lazov et al. / Physics Letters B 778 (2018) 408–413 411

(3) which leads to definition, combined with the fact that the metric gij is complete (3) 2 2 ˜ 2 2U+ and the lapse function N is bounded on M . q+ − M+ + εQ +e =− U The asymptotic behavior of γ can be obtained from the ˜ 2 ij M+ (3) asymptotic behavior of gij and the lapse function N as ˜ 2 εQ + 1 1 − + e2U − e2U+ − e 2U+ . (34) = + −2 ˜ 2 γij δij O (r ). (39) 2M+ 2 2 Let us assume that the following inequality between the con- Using the relations (U ) and (U ) we can express the field served charges and the asymptotic values of the potentials is sat- equations (13) by means of a single potential U in the form isfied γ 2 2 2 ˜ 2 2U+ 2 2 ˜ 2 2U+ R =− q+ − M+ + εQ +e D UD U, (35) q+ − M+ + εQ +e ≤ 0. (40) ij ˜ 2 i j M+ Then we have a complete asymptotically flat Riemannian mani- i D D U = 0. (3) i fold (M , γij), which possesses a non-negative scalar curvature as Using the field equations we can derive an inequality restricting can be seen from (35), and zero total mass for each of its ends. ˜ From the rigidity of the positive energy theorem [9] it follows that the asymptotic values of potential U and M+ at the asymptotic (M(3), ) is isometric to (R(3), δ ). Thus, our assumption that (40) ends End± γij ij is satisfied leads to a contradiction. Therefore, we conclude that for i i wormhole solutions the mass, the electric charge and the scalar Di UD Udμ = Di(UD U)dμ charge have to satisfy the inequality M3 M3 q2 − M2 + εQ˜ 2 e2U+ > 0. (41) i i + + + = UDi UdS + UDi UdS Under this condition we can introduce a new scalar field λ, de- ∞ ∞ S+ S− fined by ˜ ˜ =4π (U+ − U−) M+ = 8π U+M+ > 0. (36) 2 2 ˜ 2 q+ − M+ + εQ +e2U+ λ = U, (42) 4. Uniqueness theorem ˜ 2 M+

We will prove a uniqueness theorem for the static completely with asymptotic values constrained in the range regular asymptotically flat wormhole solutions in the Einstein– π 0 <λ+ =−λ− ≤ , (43) Maxwell-dilaton theory given by (1), with a phantom scalar field 2 and a possible phantom electromagnetic field. The main steps of as follows from (38). the proof are based on the ideas developed in [8]. The argument In terms of the new potential λ the field equations take the is valid only for purely electric configurations, and only for a certain range of the solution parameters satisfying a constraint given form below. R(h)ij =−2DiλD jλ, (44) The following statement can be formulated: i Di D λ = 0. ˜ Theorem. The asymptotic charges M+, q+, Q + and the asymptotic In this way we reduce the problem to the one solved in [8]. Un- value U+ of the potential Ufor static, asymptotically flat completely reg- der the asymptotic condition (43) we can perform the conformal ular traversable electrically charged wormhole solutions to the Einstein– transformation Maxwell-dilaton equations (2) with a phantom scalar field and a possible 2 phantom electromagnetic fields satisfy the inequality hij = γij, (45)

2 2 ˜ 2 2U+ where the conformal factor is given by q+ − M+ + εQ +e > 0. (37) + sin4( λ λ+ ) Moreover, for fixed value of the parameter ε =±1 there can be 2 = 2 . (46) 4 only one static, asymptotically flat traversable wormhole spacetime sin (λ+) (4) (4) (M , g , ϕ, ), which is free of any singularities, with asymptotic val- Using the field equations (44) we can show that the scalar cur- ues of the potentials satisfying (14), with given mass M+, scalar charge vature of the conformally transformed metric h vanishes. The ˜ ij q+, electric charge Q +, and asymptotic value of the potential U+ satis- conformally transformed metric can be expanded as fying the inequality 2 2 − 2 + 2U+ ˜ 2 q+ M+ ε2e Q + 2 6 2 − 2 + ˜ 2 2U+ = h = δ + O (1/r ) (47) q+ M+ εQ +e π γij ij 4 ij 0 < U+ ≤ . (38) 16 sin (λ+)r4 ˜ 2 M+ 2 near End− in the standard asymptotic coordinate r →∞. We can i = i 2 It is isometric to the spherically symmetric solution constructed be- perform a coordinate transformation y x /r , and introduce a 2 = i j low. new radial coordinate R such that R δij y y . Then, the asymptotic expansion of the metric near End− is performed when R → 0, and we obtain Proof. We consider the three-dimensional Riemannian manifold (M(3), γ ) with metric γ given by (11). It is a complete asymp- 2 ij ij 2 − 2 + 2U+ ˜ 2 ∂ ∂ q+ M+ ε2e Q + totically flat manifold with two ends possessing vanishing mass, = + 2 γ = h( , ) δij O (R ). (48) M± 0. The completeness of the metric γij is ensured by its ∂ yi ∂ y j 16 sin4(λ+) 412 B. Lazov et al. / Physics Letters B 778 (2018) 408–413

Consequently, we can add a point ∞ at R = 0, and construct a 4.1. Construction of the EMD phantom wormhole spacetimes sufficiently regular manifold M˜ (3) = M(3) ∪∞ [10]. By construction the Riemannian manifold M˜ (3) = M(3) ∪∞ is geodesically com- The explicit form of the solutions, which satisfy the presented plete, scalar flat manifold with one asymptotically flat end End+. uniqueness theorem can be obtained by using the relations (U ) According to the positive energy theorem [9] its total mass M˜ (h) and (U ) given by (28) and (34), as well as the definition (42). ˜ (h) with respect to the metric hij should be non-negative, i.e. M ≥ 0 Hence, we obtain the electromagnetic field as a function of the should be satisfied. We can obtain the mass M˜ (h) from the asymp- potential λ totic behavior of hij, which is given by ˜ ˜ ˜ Q + 2M+λ π M+ =− e Cλ − cosh , (53) 1 ˜ 2 2 2U+ ˜ 2 −2 2M+ Cλ hij = 1 − 2cot(λ+) q+ − M+ + ε2e Q + δij + O (r ), r 2 2 2U+ ˜ 2 (49) Cλ = q+ − M+ + ε2e Q + .

The metric function N and the dilaton field are further ex- when r →∞. Hence we have M˜ (h) = ϕ tracted from the potentials U and using their definition (12) − 2 − 2 + 2U+ ˜ 2 2 cot(λ+) q+ M+ ε2e Q +. Taking into account the inequality (36) it follows that M˜ (h) ≤ 0, as the equality is saturated ˜ 2 2 ε Q + 2U+ λ π ˜ h π = − only for λ+ = . Therefore, we obtain that M( ) = 0 and λ+ = . N exp 2M+ ˜ e 2 π 2 M+ Cλ + = Let us note that λ 2 through the relation (42) gives an al- ˜ 2 ˜ ˜ gebraic relation which determines U+ as a function of M+, q+ ε Q + 2M+λ π M+ ˜ + e Cλ − cosh , and Q +. ˜ 2 2M+ Cλ In summary, we constructed a geodesically complete, scalar flat ˜ (3) ˜ 2 Riemannian manifold (M , hij) with one asymptotically flat end ε Q λ = + + 2U+ and vanishing total mass. Then, by the positive energy theorem in ϕ q+ ˜ e ˜ (3) R3 2M+ Cλ the rigid case [9] it follows that (M , hij) is isometric to ( , δij). (3) ˜ 2 2M˜ +λ ˜ Consequently, the metrics γij and gij are conformally flat and ε Q + π M+ − e Cλ − cosh . (54) M(3) is diffeomorphic to R3/{0}. ˜ 2 4M+ Cλ The wormhole solutions satisfying the theorem assumptions can be constructed by straightforward integration of the field equa- In the case of positive Maxwell-gravity coupling constant ε = 1 tions (44) in spherical coordinates. More precisely we have to in- the constructed solution was previously obtained by a different tegrate (44) for the metric procedure in [11]. In this work the solution is parameterized by the integration constants b1, b2, c1, c2 and Q , which are related to ˜ i j −4 λ π 2 2 2 2 2 2 our parameters M+, q+, Q + and U+ as γijdx dx = sin ( + )(dR + R dθ + R sin θdφ ), (50) 2 4 ˜ ˜ c1 = M+, c2 = 0, Q =−Q +, with the asymptotic conditions − π ≤ λ ≤ π . As a result, we obtain 2 2 ˜ 2 ˜ 2 ˜ the solution Q + 2U+ ε Q + π M+ b = q+ + e , b = cosh , (55) 1 ˜ 2 ˜ 2 2M+ 4M+ Cλ 2 2 − 2 + 2U+ ˜ 2 i j q+ M+ ε2e Q + γijdx dx = 1 + while the notation l corresponds to our expression Cλ. The proof 4R2 of the uniqueness theorem imposes a restriction on the asymptotic π 2 2 2 2 2 2 value of the potential λ, i.e. λ+ =−λ− = , which through the × dR + R dθ + R sin θdφ , 2 relation (42) leads to a restriction on the asymptotic value U+. It 2R π should satisfy the algebraic equation λ = 2arctan( ) − , (51) 2 q2 − M2 + e2U+ Q˜ 2 + + ε2 + ˜ 2 π M+ U+ = , (56) ∈ +∞ 2 2 ˜ 2 + where R (0, ). We can further introduce another radial coor- 2 q+ − M+ + εQ +e2U dinate x taking the symmetric range x ∈ (−∞, ∞) by the coordi- 2 − 2 + 2U+ ˜ 2 ˜ = − q+ M+ ε2e Q + which can be always ensured, since M+, q+, Q + can take arbi- nate transformation x R 4R , and write the metric in the form trary values. Thus, we obtain that the solution is parameterized by three independent parameters, given by the conserved charges ˜ i j M+, q+, Q + associated with the asymptotic End+. By our theo- γijdx dx rem, in the parameter range specified by (41), they determine the = dx2 + (x2 + q2 − M2 + ε e2U+ Q˜ 2 )(dθ 2 + sin2 θdφ2), + + 2 + solution uniquely, and any completely regular traversable worm- x hole should coincide with the constructed solution (53)–(54). λ = arctan( ). (52) 2 2 ˜ 2 q+ − M+ + ε e2U+ Q + 2 Acknowledgements

By construction we have obtained the unique form of the po- The support by the Bulgarian NSF Grants DFNI T02/6 and DM tential λ corresponding to a regular wormhole solution, and satis- 18/3, Soﬁa University Research Fund under Grants 80.10-30/2017 fying the prescribed asymptotic behavior. It determines the poten- and 3258/2017, COST Action MP1304, COST Action CA15117 and tial U by the relation (42), from which we can generate explicitly COST Action CA16104, and DFG Research Training Group 1620 the wormhole solutions as described in the following section. “Models of Gravity” is gratefully acknowledged. B. Lazov et al. / Physics Letters B 778 (2018) 408–413 413

References K. Nandi, A. Potapov, R. Izmailov, A. Tamang, J. Evans, Phys. Rev. D 93 (2016) 104044; X. Chew, B. Kleihaus, J. Kunz, Phys. Rev. D 94 (2016) 104031; [1] M.S. Morris, K.S. Thorne, Am. J. Phys. 56 (1988) 395–412; G. Gibbons, M. Volkov, Phys. Lett. B 760 (2016) 324; M. Visser, Lorentzian Wormholes: From Einstein to Hawking, Woodbury, New G. Gibbons, M. Volkov, J. Cosmol. Astropart. Phys. 1705 (05) (2017) 039. York, 1996. [6] A. Nakonieczna, M. Rogatko, R. Moderski, Phys. Rev. D 86 (2012) 044043; [2] E. Teo, Phys. Rev. D 58 (1998) 024014. A. Nakonieczna, M. Rogatko, Ł. Nakonieczny, J. High Energy Phys. 11 (2015) [3] G. Clement, Phys. Rev. D 57 (1998) 4885; Phys. Lett. B 449 (1999) 12; Class. 012. Quantum Gravity 1 (1984) 275; Class. Quantum Gravity 1 (1984) 283. [7] P. Kanti, B. Kleihaus, J. Kunz, Phys. Rev. Lett. 107 (2011) 271101; P. Kanti, B. Kleihaus, J. Kunz, Phys. Rev. D 85 (2012) 044007; [4] D. Hochberg, M. Visser, Phys. Rev. Lett. 81 (1998) 746. N. Montelongo Garcia, F. Lobo, Class. Quantum Gravity 28 (2011) 085018; [5] H.G. Ellis, J. Math. Phys. 14 (1973) 104; T. Harko, F. Lobo, M. Mak, S. Sushkov, Phys. Rev. D 87 (2013) 067504. K. Bronnikov, Acta Phys. Pol. B 4 (1973) 251; [8] S. Yazadjiev, Phys. Rev. D 96 (2017) 044045. H.G. Ellis, Gen. Relativ. Gravit. 10 (1979) 105–123; [9] R. Schoen, S.T. Yau, Commun. Math. Phys. 65 (1) (1979) 45; P.E. Kashargin, S.V. Sushkov, Gravit. Cosmol. 14 (2008) 80; R. Bartnik, Commun. Pure Appl. Math. 39 (5) (1986) 661. P.E. Kashargin, S.V. Sushkov, Phys. Rev. D 78 (2008) 064071; [10] G. Bunting, A. Masood-ul Alam, Gen. Relativ. Gravit. 19 (2) (1987) 147. B. Kleihaus, J. Kunz, Phys. Rev. D 90 (2014), 121503(R); [11] P. Goulart, Phantom wormholes in Einstein–Maxwell-dilaton theory, M. Zhou, A. Cardenas-Avendano, C. Bambi, B. Kleihaus, J. Kunz, Phys. Rev. D 94 arXiv:1708.00935. (2016) 024036; N. Tsukamoto, Phys. Rev. D 94 (2016) 124001;