Physics Letters B 778 (2018) 161–166 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 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 spacetime 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 General Relativity 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. topology 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 space- (except for the phantom field). Namely, the rotating configuration times 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 ∗ ∗ 2 2 L = μ − μν + − | | 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 plane, 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. time, 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 geodesic 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).