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 grav- ity 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 worm- holes 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 (3) field, and Fμν is the Maxwell field. the staticity of the Maxwell field. Then, since M is simply con- (3) We are interested in strictly static spacetimes, i.e. spacetimes nected, there exists an electric potential defined on M 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 g −1 −2ϕ g g i N = ε N e ∇i ∇ , (9) the field equations (2) with two asymptotically flat ends. Moreover, 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 (3) (3) 2. The Riemannian manifold (M , g ) is complete. (3) tensor with respect to the 3-metric gij . (3) 3. For some compact set K, M − 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) 2M± as the equality is satisfied only in the case Q ± = M± = 0. (3) = −2 + + O −2 gij N± 1 δij (r ), 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 Di denotes the covariant derivative with respect to the met- with each end, however deviating from it by a multiplicative fac- 1 ric γij. tor. We choose N− ≤ N+. 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.