Eur. Phys. J. C (2017) 77:289 DOI 10.1140/epjc/s10052-017-4856-z Regular Article - Theoretical Physics Gödel universe from string theory Shou-Long Li1,a, Xing-Hui Feng2,b,HaoWei1,c,H.Lü2,d 1 School of Physics, Beijing Institute of Technology, Beijing 100081, China 2 Department of Physics, Center for Advanced Quantum Studies, Beijing Normal University, Beijing 100875, China Received: 22 February 2017 / Accepted: 24 April 2017 / Published online: 6 May 2017 © The Author(s) 2017. This article is an open access publication Abstract The Gödel universe is a direct product of a line 4.1 Type I ...................... 6 and a three-dimensional spacetime we call Gα. In this paper, 4.2 Type II ...................... 6 we show that the Gödel metrics can arise as exact solu- 5 Embedding in string theory ............. 7 tions in Einstein–Maxwell–Axion, Einstein–Proca–Axion, 5.1 Freedman–Schwarz model ........... 7 or Freedman–Schwarz gauged supergravity theories. The last 5.2 Gödel universe from string theory ....... 7 3 3 option allows us to embed the Gödel universe in string the- 5.3 AdS3 × S bundle over Sα ........... 8 ory. The ten-dimensional spacetime is a direct product of 6 Conclusions ..................... 8 a line and the nine-dimensional one of an S3 × S3 bundle References ........................ 9 over Gα, and it can be interpreted as some decoupling limit of the rotating D1/D5/D5 intersection. For some appropriate 1 Introduction parameter choice, the nine-dimensional metric becomes an AdS × S3 bundle over squashed 3-sphere. We also study the 3 Closed time-like curves (CTCs) in solutions of General Rel- properties of the Gödel black holes that are constructed from ativity can pose challenges to Hawking’s chronology pro- the double Wick rotations of the Gödel metrics. tection conjecture. It is well known that the Kerr metric has CTCs; however, they are hidden inside the event hori- zon, and hence they obey the chronological censorship. The Contents Gödel metric [1], a solution of cosmological Einstein grav- ity coupled to some uniform pressureless matter, describes a 1 Introduction ..................... 1 homogeneous rotating universe that has naked CTCs and has 2 Gödel universe .................... 2 no globally spatial-like Cauchy surface. However, the fine- × R 2.1 The metrics of Gα ............. 2 tuning balance between the cosmological constant and the α 2.2 Energy condition and value .......... 2 matter density in the Gödel universe makes it an unrealistic 2.3 Metrics asymptotic to Gα ............ 3 model to challenge the chronology protection. Nevertheless 2.4 Mass and angular momentum .......... 3 it is a good toy model to study the effects of naked CTCs in 3 × T 2.5 Squashed 3-sphere Sα ........... 3 both classical and quantum gravities. 3 Gödel solutions from Lagrangian formalism .... 4 The study of CTCs has enjoyed considerable attention 3.1 Einstein–Maxwell–Axion theory with a topo- with the development of the string theory and the AdS/CFT logical term ................... 4 correspondence. A large number of supersymmetric or non- 3.2 Einstein–Proca–Axion theory .......... 4 supersymmetric Gödel-like solutions, including also black 3.3 The embedding of squashed 3-sphere ...... 5 holes and time machines, with naked CTCs were constructed 3.4 Generalizing to higher dimensions ....... 5 in gauged supergravities in higher dimensions, see e.g. [2– 3.5 Effective three-dimensional theories ...... 5 17]. This work raises important issues of whether the prob- 4 Black holes from double Wick rotations ....... 6 lems of CTCs may be resolved by stringy or quantum con- siderations, or whether naked CTCs in the bulk implt the a e-mail: [email protected] breakdown of unitarity of the field theory. b e-mail: [email protected] In this paper, we focus on the original four-dimensional c e-mail: [email protected] Gödel universe, which is a direct product of a line and a d e-mail: [email protected] three-dimensional metric [1], 123 289 Page 2 of 9 Eur. Phys. J. C (2017) 77 :289 dr 2 2 Gödel universe ds2 = 2 −(dt + rdφ)2 + 1 r 2dφ2 + + dz2 . (1.1) 2 r 2 2.1 The metrics of Gα × R This metric is an exact solution of the Einstein equation In this paper, we consider a class of metrics in the following involving some uniform pressureless matter, form: dr 2 1 mat 2 = 2 −( + φ)2 + α 2 φ2 + + 2 , Rμν − Rgμν + gμν = T , ds dt rd r d dz (2.1) 2 μν r 2 1 mat =− , Tμν = uμuν, (1.2) α 22 where and are constants. The metric is a direct product of R associated with the coordinate z and the three-dimensional metric of (t,φ,r), which we shall call Gα. The original Gödel where uμ = (−2, 0, 0, 0). The Gödel universe has aroused α = 1 metric [1] is recovered when we take 2 , corresponding continuous interest in the general-relativity community, (see, to G1/2 ×R. To avoid pedantry, we shall refer the metric (2.1) e.g., [18–24]), and the Gödel metric (1.1) was generalized with generic α also as the Gödel metric. 1 to Gödel-type metrics where the coefficient 2 in (1.1)is In addition to the constant shifting symmetry along the replaced by a generic constant α. To avoid pedantry, we shall (t,φ,z) directions, the Gödel metric (2.1) is invariant under refer to all these metrics as Gödel metrics, which are distinct the constant scaling from the afore-mentioned Gödel-like metrics. It turns out −1 that most theories associated with these general Gödel met- r → λr,φ→ λ φ. (2.2) rics involve the energy-momentum tensor of some uniform Note that imposing this scaling symmetry also implies that φ pressureless matter. Electric and magnetic fields periodic in describes a real line, rather than a circle. This scaling property z as a replacement for such matter were used to construct indicates that the metric is homogeneous. Since the metric Gα Gödel metric in [21]. It is worth mentioning that the three- is three dimensional, its curvature is completely determined dimensional metric with dz2 removed can arise as an exact by the Ricci tensor, whose non-vanishing components are solution of Einstein–Maxwell theory with a topological term given by A ∧ F [13]. The embedding of the three-dimensional metric in heterotic string theory was first given in [11]. ¯ ¯ 1 ¯ ¯ 1 − 2α ¯ ¯ R00 = , R11 = = R22. (2.3) The main purpose of this paper is to construct more 2α2 2α2 Lagrangians that admit Gödel metrics as exact solutions. We Here we represent the curvature in tangent space by the viel- present three Lagrangians that admit the Gödel metrics as bein solutions, all involving only the fundamental matter fields. These are the Einstein–Maxwell–Axion (EMA), Einstein– ¯ ¯ √ ¯ dr e0 = (dt + rdφ), e1 = α rdφ, e2 = , Proca–Axion (EPA) and Freedman–Schwarz [25] SU(2) × r ¯ SU(2) gauged supergravity theories. The Freedman–Schwarz e3 = dz. (2.4) model can be obtained from the effective actions of string the- 3 3 ories via the Kaluza–Klein reduction on S × S . Thus the It is clear that when α = 1, the metric is locally AdS3 (three- four-dimensional Gödel universe can be embedded in string dimensional anti-de Sitter spacetime), i.e. G1 = AdS3. theory. The paper is organized as follows. In Sect. 2,wegivea 2.2 Energy condition and α value review of Gödel metrics and their properties. We also show that for an appropriate choice of the parameter, such a metric Defining the energy-momentum tensor in the vielbein basis, can also describe a direct product of time and a squashed 3- ab = ab − 1 ηab , sphere. In Sect. 3, we construct EMA and EPA theories that T R 2 R (2.5) admit Gödel metrics, and also solutions involving a squashed we find T ab = diag{ρ, p, p, p˜}, with 3-sphere. We generalize the theories to higher dimensions and also give the effective three-dimensional theories by 3 − 4α 1 4α − 1 ρ = , p = , p˜ = . (2.6) Kaluza–Klein reduction. In Sect. 4, we consider double Wick 4α2 4α2 4α2 rotations and obtain two types of black hole solutions. In α = 1 Sect. 5, we show that Gödel metrics are exact solutions of Thus one sees that 2 gives rise to matter with uniform the Freedman–Schwarz model and hence obtain the corre- pressure [1]. The null-energy condition requires that sponding ten-dimensional solutions of string theories. We 1 − α 1 ρ + p = ≥ 0,ρ+˜p = ≥ 0. (2.7) conclude the paper in Sect. 6. α2 2α2 123 Eur. Phys. J. C (2017) 77 :289 Page 3 of 9 289 Thus the null-energy condition requires that 0 <α≤ 1. coordinate transformation (2.12), and hence the two metrics As we shall see presently this implies that Gödel metrics in are not globally equivalent. An important consequence is that general have naked CTCs. themetric(2.11) has CTCs for r > rc with 1 2 α coth( r ) = √ . (2.13) 2.3 Metrics asymptotic to G 2 c α We now consider deformations of Gα by introducing a func- Here r = rc is the velocity of light surface (VLS) for which tion of two constants, gφˆφˆ = 0. We shall call the metric (2.9) the deformed Gödel a b metric that is asymptotic to the Gödel metric. For general f = 1 + − , (2.8) ( , ) r r 2 parameters a b , there can be two VLS’s between which gφφ > 0. The global structure of the metric (2.9), written in and deform the metric (2.1) to become somewhat different parametrization, was analysed in [13].