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Ring Wormholes Via Duality Rotations ∗ Gary W Physics Letters B 760 (2016) 324–328 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 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- 2 = i k where dl γikdx dx and U , γik, depend on the spatial coordi- = k gions, while x 0is the wormhole throat – the 2-sphere of min- nates x . Denoting by Uk ≡ ∂kU , the Lagrangian becomes 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.
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