Ring Wormholes Via Duality Rotations ∗ Gary W

Physics Letters B 760 (2016) 324–328

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Physics Letters B

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Ring wormholes via duality rotations ∗ Gary W. Gibbons a,b, Mikhail S. Volkov b,c, a DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK b Laboratoire de Mathématiques et Physique Théorique, LMPT CNRS – UMR 7350, Université de Tours, Parc de Grandmont, 37200 Tours, France c Department of General Relativity and Gravitation, Institute of Physics, Kazan Federal University, Kremlevskaya street 18, 420008 Kazan, Russia a r t i c l e i n f o a b s t r a c t

Article history: We apply duality rotations and complex transformations to the Schwarzschild metric to obtain wormhole Received 27 June 2016 geometries with two asymptotically flat regions connected by a throat. In the simplest case these are the Accepted 5 July 2016 well-known wormholes supported by phantom scalar field. Further duality rotations remove the scalar Available online 9 July 2016 field to yield less well known vacuum metrics of the oblate Zipoy–Voorhees–Weyl class, which describe Editor: M. Cveticˇ ring wormholes. The ring encircles the wormhole throat and can have any radius, whereas its tension is always negative and should be less than −c4/4G. If the tension reaches the maximal value, the geometry becomes exactly flat, but the topology remains non-trivial and corresponds to two copies of Minkowski space glued together along the disk encircled by the ring. The geodesics are straight lines, and those which traverse the ring get to the other universe. The ring therefore literally produces a hole in space. Such wormholes could perhaps be created by negative energies concentrated in toroidal volumes, for example by vacuum fluctuations. © 2016 The Authors. 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.

Wormholes are bridges or tunnels between different universes approach unity as r →±∞ then the metric describes two asymp- or different parts of the same universe. They were first introduced totically flat regions connected by a throat of radius R(0). The Ein- μ = μ =− 0 by Einstein and Rosen (ER) [1], who noticed that the Schwarzschild stein equations Gν Tν imply that the energy density ρ T0 = r = black hole actually has two exterior regions connected by a bridge. and the radial pressure p Tr satisfy at r 0 The ER bridge is spacelike and cannot be traversed by classical objects, but it has been argued that it may connect quantum parti- R 1 ρ + p =−2 < 0, p =− < 0. (2) cles to produce quantum entanglement and the Einstein–Pololsky– R R2 Rosen (EPR) effect [2], hence ER=EPR [3]. Wormholes were also It follows that for a static wormhole to be a solution of the considered as geometric models of elementary particles – handles μ ν Einstein equations, the Null Energy Condition (NEC), Tμν v v = of space trapping inside an electric flux, say, which description μ ν μ Rμν v v ≥ 0for any null v , must be violated. Another demon- may indeed be valid at the Planck scale [4]. Wormholes can also stration [8] of the violation of the NEC uses the Raychaudhuri describe initial data for the Einstein equations [5] (see [6] for a re- equation [10] for a bundle of light rays described by θ, σ , ω: the cent review) whose time evolution corresponds to the black hole expansion, shear and vorticity. In the spherically symmetric case collisions of the type observed in the recent GW150914 event [7]. one has ω = σ = 0 [9], hence An interesting topic is traversable wormholes – globally static bridges accessible for ordinary classical particles or light [8] dθ μ ν 1 2 (see [9] for a review). In the simplest case such a wormhole is =−Rμν v v − θ . (3) dλ 2 described by a static, spherically symmetric line element If rays pass through a wormhole throat, there is a moment of min- 2 2 2 2 2 2 2 2 μ ν ds =−Q (r)dt + dr + R (r)(dϑ + sin ϑdϕ ), (1) imal cross-section area, θ = 0but dθ/dλ > 0, hence Rμν v v < 0 and the NEC is violated. where Q (r) and R(r) are symmetric under r →−r and R(r) at- If the spacetime is not spherically symmetric then the above tains a non-zero global minimum at r = 0. If both Q and R/r arguments do not apply, but there are more subtle geometric con- siderations showing that the wormhole throat – a compact two- surface of minimal area – can exist if only the NEC is violated * Corresponding author. E-mail addresses: [email protected] (G.W. Gibbons), [email protected] [11,12]. As a result, traversable wormholes are possible if only (M.S. Volkov). the energy density becomes negative, for example due to vacuum http://dx.doi.org/10.1016/j.physletb.2016.07.012 0370-2693/© 2016 The Authors. 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. G.W. Gibbons, M.S. Volkov / Physics Letters B 760 (2016) 324–328 325 polarization [8], or due to exotic matter types as for example phan- where k and also U , depend only on ρ, z, then the equations tom fields with a negative kinetic energy [13,14]. Otherwise, one reduce to1 can search for wormholes in the alternative theories of gravity, as 1 1 for example in the Gauss–Bonnet theory [15,16], in the brainworld U + U + U zz = 0, + + zz = 0, ρρ ρ ρ ρρ ρ ρ models [17], in theories with non-minimally coupled fields [18], or = [ + ] = [ 2 − 2 + 2 − 2 ] in massive (bi)gravity [19]. kz 2ρ Uρ U z ρ z , kρ ρ Uρ U z ( ρ z ) . The aim of this Letter is to present a method of constructing (10) traversable wormholes starting from the Schwarzschild metric and applying duality rotations and complex transformations. As a first These equations admit a scaling symmetry mapping solutions to step this gives solutions with a non-trivial scalar field, which, after solutions, a complexification, reduce to the well-known wormhole solution of → → → 2 Bronnikov and Ellis (BE) [13,14]. A further duality rotation puts the U λ U, λ , k λ k, (11) scalar field to zero again, yielding less well known vacuum metrics where λ is a constant parameter. In addition, they are invariant describing ring wormholes. They circumvent the above no-go ar- under guments because they are not spherically symmetric and secondly because they are not globally regular and exhibit a conical sin- U ↔ , k → k . (12) gularity along the ring. The ring encircles the wormhole throat and has a negative tension which should be less than −c4/4G. 2. BE wormhole Strikingly, if the tension reaches the maximal value, the geometry becomes exactly flat, but the topology remains non-trivial and cor- We start from the Schwarzschild metric (here d2 = dϑ 2 + responds to two copies of Minkowski space glued together along sin2 ϑdϕ2) the disk inside the ring. The geodesics are straight lines, and those 2m dr2 which intersect the ring get to the other universe. The ring there- ds2 =− 1 − dt2 + + r2d2. (13) fore literally creates a hole in space. r 1 − 2m/r It is well-known that the Kerr black hole contains inside the Introducing x = r − m it can be put to the form (5) with horizon a ring singularity which can also be viewed as a worm- 1 x − m hole [20]. However, its surrounding region contains closed timelike U = ln , dl2 = dx2 + (x2 − m2) d2, = 0. curves (CTCs). One should emphasize that our ring is different – it 2 x + m is static and does not create any CTCs and moreover is not shielded (14) by any horizons. Applying to this the symmetry (8) gives a new solution which is ultrastatic but has a non-trivial scalar field2 1. Model 1 x − m U = 0, dl2 = dx2 + (x2 − m2) d2,φ= ln . (15) We consider the theory with a scalar field, 2 x + m √ If we now continue the parameter m to the imaginary region, m → L =[ − 2] − R 2 (∂ ) g . (4) iμ, then the metric remains real, while Here the parameter takes two values, either =+1 corresponding 1 x − iμ =− φ → ln = iψ, (16) to the conventional scalar field, or 1 corresponding to the 2 x + iμ phantom field. We shall denote = φ for = 1 and = ψ for =−1. The phantom field ψ can formally be viewed as φ contin- where ψ is real. This gives the solution for the phantom field, ued to imaginary values. We use units in which G = c = 1, unless ds2 =−dt2 + dx2 + (x2 + μ2)(dϑ 2 + sin2 ϑdϕ2), otherwise stated. x Assume the spacetime metric to be static, ψ = arctan , (17) μ 2 2U 2 −2U 2 ds =−e dt + e dl , (5) which is precisely the BE wormhole [13,14]. Here x ∈ (−∞, +∞) and the limits x →±∞ correspond to two asymptotically flat re- where dl2 = γ dxidxk and U , γ , depend on the spatial coordi- ik ik gions, while x = 0is the wormhole throat – the 2-sphere of min- nates xk. Denoting by U ≡ ∂ U , the Lagrangian becomes k k imal radius μ. The geodesics can get through the throat from one asymptotic region to the other. (3) √ L =[R −2γ ik(U U + )] γ , (6) Passing to the coordinates i k i k (3) z = x cos ϑ, ρ = x2 + μ2 sin ϑ where R is the Ricci scalar for γik. The field equations are ↔ ± = 2 + ± 2 (3) x iμ cos ϑ ρ (z iμ) , (18) R ik= 2(U i Uk + i k), U = 0, = 0. (7)

We notice that for = 1these equations are invariant under 1 One can check that this reduction is indeed consistent, which would not be the case if the scalar field had a potential. The Weyl formulation is also consistent for an electrostatic vector field, so that it applies, for example, within the electrostatic U ↔ = φ, γik → γik . (8) sector of dilaton gravity. 2 This is a member of the Fisher–Janis–Newman–Winicour solution family If the 3-metric γik is chosen to be of the Weyl form, [21,22]. The rest of this family can be similarly obtained from the Schwarzschild metric by applying, instead of the discrete symmetry (8), continuous SO(2) rota- 2 2k 2 2 2 2 dl = e dρ + dz + ρ dϕ , (9) tions in the U , φ space. 326 G.W. Gibbons, M.S. Volkov / Physics Letters B 760 (2016) 324–328 the wormhole metric assumes the Weyl form, These metrics have been relatively well studied (see for example [25,26]) since they can be used to describe deformations of the ds2 =−dt2 + e2k(dρ2 + dz2) + ρ2dϕ2 (19) Schwarzschild metric. If we now continue the parameters to the imaginary region, with δ → iσ , m → iμ, (28) x2 + μ2 e2k = . (20) x2 + μ2 cos2 ϑ then metrics (27) remain real and reduce to the oblate ZV solutions [23,24], which are precisely the metrics (22). Much less is Since known about these solutions. Their wormhole nature has been x2 + μ2 cos2 ϑ discussed [27,28] but no systematic description is currently avail- dρ2 + dz2 = dx2 + (x2 + μ2)dϑ 2 , (21) able and moreover the solutions remain largely unknown. We shall 2 + 2 x μ therefore describe their essential properties and find new surpris- the metric (19) is indeed equivalent to the one in Eq. (17). ing features, notably the existence of a non-trivial flat space limit.

3. Ring wormhole 4. Properties of the ring wormhole

Let us apply to the solution (19) the symmetry (12), U ↔ ψ, Similarly to the BE wormhole (17), solution (22) has a throat →− and two asymptotically ﬂat regions. This can be most easily seen k k, and then act on the result with the scaling symmetry (11) 2 = = by noting that at the symmetry axis, where cos ϑ = 1, the line el- with λ σ . This gives the axially symmetric metrics with 0, 2 ement dl reduces precisely to that in (17), while U is a bounded − x function. Therefore, the solution indeed describes a wormhole in- ds2 =−e2U dt2 + e 2U dl2, U = σ arctan , (22) μ terpolating between two asymptotically ﬂat regions. One has for ⎡ ⎤ x →±∞ (assuming that μ > 0) 2+ 2 + 2 2 σ 1 2 x μ cos ϑ ⎣ 2 2 2 2⎦ dl = dx + (x + μ )dϑ 2U ±σπ 2σμ 2 + 2 e → e 1 − + ... , (29) x μ x

+ (x2 + μ2) sin2 ϑ dϕ2. hence the time coordinates in both limits differ by a factor of e2σπ while the ADM mass M =±σμ is positive when seen from one 2 →− We note that the 3-metric dl is invariant under x x while the wormhole side and negative from the other. Newtonian potential U changes sign. As we shall see below, these The wormhole is traversable, which can be easily seen by con- solutions describe ring wormholes. We thought initially these solu- sidering geodesics along the symmetry axis. A particle of mass m tions were new, but actually they were described before although and energy E follows a trajectory x(s) defined by remain very little known. We obtained them from the BE wormhole by rotating away dx 2 + m2e2U (x) = E2 , (30) the scalar field. However, they can also be obtained from the ds Schwarzschild metric without introducing any scalar fields at all. 2 2 σπ Passing to the coordinates hence x(s) ∈ (−∞, +∞) if E > m e . Since U (x) grows with x, it follows that the wormhole attracts particles in the x > 0region z = x cos ϑ, ρ = x2 − m2 sin ϑ but repels them in the x < 0region. Therefore, it acts as a “drainhole” sucking in matter from the x > 0region and spitting it out ↔ x ± m cos ϑ = ρ2 + (z ± m)2 , (23) to the x < 0region. The metric (22) degenerates at x = 0, ϑ = π/2, which cor- the Schwarzschild metric (14) assumes the Weyl form with responds to a circle of radius μ. In its vicinity one can define y = μ cos ϑ and then dl2 in (22) becomes dl2 = dx2 + (x2 − m2) d2 = e2k(dρ2 + dz2) + ρ2dϕ2, (24) σ 2+1 x2 + y2 where dl2 = (dx2 + dy2) + μ2dϕ2 (31) μ2 2 − 2 2k x m 2+ e = , (25) = μ2 r2σ 2(dr2 + r2dα2) + μ2dϕ2 x2 − m2 cos2 ϑ = 2 2 + 2 2 + 2 2 since μ (dR R dω ) μ dϕ , 2+ 2 − 2 2 where x = μr cos α, y = μr sin α while R = rσ 2/(σ 2 + 2) and 2 2 x m cos ϑ 2 2 2 2 dρ + dz = dx + (x − m )dϑ . (26) ω = (σ 2 + 2) α. The metric in the last line in (31) looks flat, how- x2 − m2 ever, since α ∈[0, 2π), the angle ω ranges from zero to ωmax = Acting on this with the scaling symmetry (11) with λ =−δ gives 2π (σ 2 + 2) > 2π . Therefore, there is a negative angle deficit vacuum metrics of the prolate Zipoy–Voorhees (ZV) class [23,24] 2 ω = 2π − ωmax =−(σ + 1) 2π (32) − x − m δ x − m δ ds2 =− + dl2, (27) and hence a conical singularity at R = 0. Since the singularity x + m x + m ⎡ ⎤ stretches in the ϕ-direction, it sweeps a ring of radius μ. Such 1−δ2 line singularities are known to be generated by singular matter x2 − m2 cos2 ϑ dl2 = ⎣ dx2 + (x2 − m2)dϑ 2⎦ sources distributed along lines – cosmic strings. Their energy per x2 − m2 unit length (tension) T is related to the angle deficit via ω = (8π G/c4) T (we restore for a moment the correct physical dimen- + (x2 − m2) sin2 ϑ dϕ2 . sions), hence the string has a negative tension G.W. Gibbons, M.S. Volkov / Physics Letters B 760 (2016) 324–328 327

(1 + σ 2) c4 T =− . (33) 4G Therefore, the wormhole solution (22) can be viewed as sourced by a ring with negative tension. The solution carries the free parameter μ, which determines the radius of the ring, and also σ determining the value of the ring tension. It is quite surprising that the self-gravitating ring can be in equilibrium and moreover admits an exact solution. If the string tension was positive then the string loop would be shrinking and the system would not be static (we are unaware of any exact solutions in this case, although one can construct exact initial data for a string loop [29]). For the solution (22) the ring has a negative tension and is similar to a strut, hence it should rather tend to expand, which tendency is counterbalanced by the gravitational attraction. This must be the reason for which such an equilibrium system exists (we do not know if the equilibrium is stable). Even more remarkable is the following. Setting σ = 0in (22) yields x2 + μ2 cos2 ϑ ds2 =−dt2 + dx2 + (x2 + μ2)dϑ 2 x2 + μ2 Fig. 1. Wormhole topology. The x, ϑ coordinates cover the whole of the manifold, + (x2 + 2) sin2 ϑ d 2. (34) μ ϕ the throat being at x = 0, ϑ ∈[0, π] and the ring is at x = 0, θ = π/2. The Weyl charts D+ and D− cover, respectively, the x > 0and x < 0 regions. Lines of constant This metric is vacuum and ultrastatic hence it must be ﬂat, so one x are the (half)-ellipses in the Weyl coordinates, the ring corresponds to the branch may think this limit is trivial. However, close to the symmetry axis, points at z = 0, ρ = μ, while the throat corresponds to the branch cuts O μ. The 2 where cos ϑ ≈ 1, the geometry reduces to that for the BE worm- upper edge of the cut in the D+ patch is identiﬁed with the lower edge of the cut hole (17).The wormhole throat is located at x = 0, but, unlike in in the D− patch and vice-versa. A winding around the ring in the x, ϑ coordinates the BE case, this is not a sphere but rather a sphere squashed to a corresponds to two windings in Weyl coordinates. disk with the geometry

μ2(cos2 ϑdϑ 2 + sin2 ϑdϕ2) = μ2(dξ 2 + ξ 2dϕ2), (35) where ξ = sin ϑ ∈[0, 1]. The disk is encircled by the ring. The coordinates x, ϑ, ϕ are global, but to study the geodesics it is convenient to pass to the Weyl coordinates (18). The metric then becomes manifestly ﬂat ds2 =−dt2 + dρ2 + dz2 + ρ2dϕ2. (36)

However, this does not mean the solution describes empty Min- kowski space. Indeed, the ring is still present and has the tension Fig. 2. Particles entering the ring are not seen coming out from the other side. c4 T =− . (37) 4G B, C in Fig. 1) continue from D+ to D− thus traversing the wormhole. Therefore, the negative tension ring genuinely creates a hole As a consequence of this the topology is non-trivial. This follows in space through which one can observe another universe as well from the fact that the Weyl coordinates ρ, z cover either only the as get there.3 This reminds one of Alice observing the room be- x < 0part or only the x > 0part of the wormhole. Indeed, one has hind the looking glass and next jumping there. An object falling through the ring can be seen from behind, while viewed from the z2 ρ2 + = 1, (38) side it is not seen coming from the other side (see Fig. 2). 2 2 2 x x + μ It is striking that the line source is enough to create the worm- hence lines of constant x in the ρ, z plane are ellipses insensi- hole. Usually the negative energy supporting the wormhole is distributed over the 3-volume, as for the BE solution, or at least over tive to the sign of x, therefore one needs two Weyl charts, D± : {ρ ≥ 0, −∞ < z < ∞}, to cover the whole of the manifold – one a 2-surface, as for the thin-shell wormholes [32]. We see however for x < 0 and one for x > 0(see Fig. 1) [30]. The coordinate trans- that it is sufficient to distribute the negative energy only along the one-dimensional ring, which is presumably easier to achieve formation (x, ϑ) → (ρ, z) degenerates at the branch point z = 0, than in other cases since a smaller amount of the NEC violation is ρ = μ corresponding to the position of the ring. Each chart there- needed. fore has a branch cut ending at the branch point, which may be It is also interesting to note that the arguments based on the chosen to be z = 0, ρ ∈[0, μ] (see Fig. 1). The upper edge of the Raychaudhuri equation do not apply since the wormhole either D+ cut is identified with the lower edge of the D− cut and vice- does not affect the geodesics at all if they miss it, or it absorbs versa. These cuts correspond to the wormhole throat, so that we see once again that the throat is a disk encircled by the ring. The geodesics in Weyl coordinates are straight lines. 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