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]. dr 2 The emerging of the CTCs in Gödel metrics is a conse- ds2 = 2 −(dt + rdφ)2 + αr 2 f dφ2 + + dz2 . r 2 f quence of that α ≤ 1. If one allows α>1, the metrics do (2.9) not have naked CTCs. However, as we saw earlier that in the framework of Einstein gravity, α>1 violates the null- It is straightforward to verify that the curvature tensors (2.3) energy condition. In higher-derivative gravity due to the α remain unchanged. This implies that the metric (2.9)is correction of string theory, solutions with α>1 were con- locally the same as (2.3). However, globally, the deformed structed in [2]. However, the theory, when treated on its own, metric (2.9) is different from (2.1). An important difference involves inevitable ghost modes. is that in the deformed metric (2.9), the coordinate φ is peri- odic, namely 2.4 Mass and angular momentum 4π φ = √ , with f (r ) = 0. (2.10) α 2 ( ) 0 r0 f r0 The general Gödel metric has two Killing vectors, This ensures that the metric is absent from a conical singular- ∂ ∂ ξ = 1 ,ξ= 1 . = t φ (2.14) ity at r r0. After imposing this condition, the coordinate 2 ∂t  ∂φ transformation that relates (2.1)to(2.9) breaks the scaling symmetry (2.2) and hence the two metrics are not equivalent Following the Wald formalism [29], which computes the δH globally. variation of the on-shell Hamiltonian associated with a To demonstrate this explicitly, we note that the parameter Killing vector with respect to the integration constants of the a is trivial in that it can be eliminated by the coordinate trans- solutions, we read off the associated conversed quantities by δH formation r → r − 1 a and t → t + 1 aφ, without altering evaluating at asymptotic infinity to find 2 2 √ √ the global structure. The positive parameter b can be√ set to α a  α b 1 without loss of generality, using the scaling r → br, M = φ, J = φ, (2.15) 16π 16π together with appropriate scalings of the rest coordinates. √ √ φ Now let r = cosh rˆ, φ = φ/ˆ α, t = tˆ − φ/ˆ α, the metric where is given by (2.10). Note that if one chooses a φ = π (2.9) with a = 0 and b = 1 becomes fixed period 2 instead, rather than the one given by    (2.10), the solution will have a naked singularity at r = r0, 2  2 = 2 − ˆ + √2 2( 1 ˆ) φˆ for generic α. We have also set the convention dz = 1. For ds dt α sinh 2 r d  periodic z, this means the period is z = 1; for real line z, + sinh2 rˆ dφˆ2 + drˆ2 + dz2 . (2.11) this implies that extensive quantities such as M and J are in fact uniform densities over the line z. On the other hand, we find that, under the coordinate trans- formation 3 2.5 Squashed 3-sphere Sα × T 1 r = cosh rˆ + cos φˆ sinh rˆ, rφ = √ sin φˆ sinh rˆ, α In the Gödel metrics, if we let 2 =−˜2 < 0, the metric   √ 1 (2.9) becomes tan 1 φˆ + 1 α(t − tˆ) = tanh( 1 φ),ˆ (2.12) 2 2 ˆ 2   r 2 2 ˜2 2 2 dr 2 the metric (2.1) becomes also precisely (2.11). (This coor- ds =  (dz + rdφ) + αhdφ + − dt , (2.16) h dinate transformation reduces to the one obtained in [1]for α = 1 φˆ = π ( , ) 2 .) Thus it becomes clear that when 2 is fixed, where we have swapped the role of t z and have set, without the scaling symmetry of the vacuum (2.1)isbrokenbythe loss of generality, h = 1 − r 2. Letting r = cos θ,wehave 123 289 Page 4 of 9 Eur. Phys. J. C (2017) 77 :289   μνρσ ds2 = ˜2 (dz + cos θdφ)2 + α sin2 θdφ2 + dθ 2 − dt2 . where F = dA is the field strength, ε is the density of Levi-Civita tensor whose components are ±1, 0. We choose (2.17) the convention ε0123 = 1. The axion, Maxwell and Einstein This metric describes a direct product of time with a squashed equations of motion are given by 3 √ 3-sphere, which we call S . When the quashing parameter μ 1 μνρσ α ∂μ −g∂ χ + ε Fμν Fρσ = 0, α = 1, S3 becomes S3, written as a U(1) bundle over S2. √ 8 α μν 1 μνρσ 3 ∂μ −gF − χε Fρσ = 0, The regularity of Sα requires that 2   1 1 ρ 1 2 2π 4π Rμν − gμν R + gμν − Fμρ Fν − gμν F φ = √ , z = √ . (2.18) 2 2 4 α α   − 1 ∂ χ∂ χ − 1 (∂χ)2 = . 2 μ μ 2 gμν 0 (3.4) The period z can be divided by a natural number n without introducing any singularity to the manifold, giving rise to For the general Gödel metric (2.9) with (2.8), we consider 3 Sα/Zn. (Such a compact Gödel universe was also considered the following ansatz for the axion and Maxwell field: in [26].) A = qr dφ, χ = kz. (3.5)

We find that the equations of motion are all satisfied provided 3 Gödel solutions from Lagrangian formalism that As mentioned in the introduction, theories in the literature 1 q2 1 k = √ ,α= 1 − ,2 =− . (3.6) associated with the Gödel universe (2.1)or(2.9) typically α 22 2 involve a matter energy-momentum tensor with unknown The general solution contains three integration constants,√ Lagrangian origin. A well-known example in four dimen- (a, b, q). The reality condition requires that |q| < 2 . sions is the Einstein–Maxwell theory with an axion and a It follows that we have 0 <α≤ 1 and k ≥ 1. We have the negative cosmological constant [21], original α = 1 Gödel metric corresponding to q = .The   2 √ AdS factor arises when α = 1, corresponding to turning off L = −g R − 2 − 1 (∂χ)2 − 1 F2 , (3.1) 3 2 4 the Maxwell field. In Sect. 3.3, we consider the case with α< 3 × T where F = dA is the field strength. For the metric (2.1), the 0, for which the metric describes Sα . solutions for the matter fields are It is worth pointing out that in four dimensions, the axion χ B( )  z z is the Hodge dual to a 2-form potential 2 with A = 2(1 − α) sin(√ )(dt + rdφ), χ = √ , α α = =∗ χ + 1 ∧ . G(3) dB(2) d 2 A F (3.7) 1 2 =− . The Lagrangian (3.3) is equivalent to  (3.2) 2 √   L = − −  − 1 2 − 1 2 . In this case, the continuous shifting symmetry along the z- g R 2 4 F 12 G(3) (3.8) direction is broken to a discrete symmetry by the Maxwell For the Gödel solution we have B( ) = rdt∧ dφ. The 3-form potential A, which is periodic in z. (The axion χ field does 2 field strength G( ) is suggestive of string theory, which we dχ 3 not break this symmetry since only appears in the theory.) shall discuss in Sect. 5. The solution is best described as Gα × S1 rather than Gα × R. A consequence is that Gα is not a solution to the three- 3.2 Einstein–Proca–Axion theory dimensional massless sector in the Kaluza–Klein reduction of (3.1)onz. In this section, we shall construct more examples In this subsection, we replace the previous Maxwell field by of Lagrangians that admit Gödel metrics Gα × R as exact a Proca field of mass μ, with the Lagrangian solutions. √   L = − −  − 1 2 − 1 μ2 2 − 1 (∂χ)2 . 2 g R 2 4 F 2 A 2 (3.9) 3.1 Einstein–Maxwell–Axion theory with a topological The equations of motion are term μν 2 ν χ = 0, ∇μ F − μ A = 0,   In addition to (3.1), we introduce the additional topological 1 1 ρ 1 2 Rμν − gμν R + gμν − Fμρ Fν − gμν F term 2 2 4     √ − 1 μ2 − 1 2 L = −g R − 2 − 1 F2 − 1 (∂χ)2 2 Aμ Aν 2 gμν A 4 2   μνρσ 1 1 2 + 1 ε χ , − ∂μχ∂μχ − gμν(∂χ) = 0. (3.10) 8 Fμν Fρσ (3.3) 2 2 123 Eur. Phys. J. C (2017) 77 :289 Page 5 of 9 289

(There should be no confusion between the Proca mass μ and G(n) = dB(n−1). The Lagrangian (3.3) becomes the spacetime indices.) The metric ansatz is given by (2.9). √ L = − −  − 1 2 − 1 2 We consider the following ansatz for A and χ: g R 2 4 F 2 n! G(n−1) 1 μνρσα ···α − + ε 1 n 1 Bα ···α Fμν Fρσ . A = q(dt + rdφ), χ = z. (3.11) 8 (n−1)! 1 n−1 (3.16) (This should not be confused with dualizing dχ to the 3- Substituting these into the equations of motion, we find form field strength in four dimensions, discussed in the end 1 3q2 − 22 q2 of Sect. 3.1.) The axion ansatz (3.5) is replaced by μ = √ ,= ,α= 1 − . (3.12)  α 42(2 − q2) 2 G(n) = k dz1 ∧···dzn. (3.17) Note that all the constants except for (a, b) appearing in the The Lagrangian (3.9) is now replaced by solution are fixed by the coupling constants of the theory, √   namely (, μ). It follows that unlike in the earlier EMA L = − −  − 1 2 − 1 μ2 2 − 1 2 . 2 g R 2 4 F 2 A 2 n! G(n) theory, the general solution involves only two integration α = 1 (3.18) constants. The original√ 2 Gödel metric corresponds to taking q = / 2. The corresponding ansatz for G(n) is given by (3.17) with k = 1. 3.3 The embedding of squashed 3-sphere 3.5 Effective three-dimensional theories In the embedding of the Gödel metric in both EMA and EPA theories discussed above, the cosmological constant  is neg- The non-trivial part of the Gödel universe is the three- 3 ative. When it is positive, the metric Gα ×R becomes Sα ×T, dimensional metric Gα. We can perform dimensional reduc- as in (2.17). For the EMA theory, we have tion on the coordinate z associated with the trivial R direction. The EMA theory becomes t 1   A = q cos θ dφ, χ = √ , ˜2 = , √ α  L = − −  − 1 2 + 1 λ εμνρ . 2 g R 2 eff 4 F 4 eff Aμ Fνρ q2 (3.19) α = 1 + . (3.13) 2˜2 In this case, the three-dimensional Gα metric is supported by For EPA theory, we have q2 q2 − 2 A = qrdφ, α = 1 − ,eff = , = ( + θ φ), χ = ,μ2 =− 1 , 22 22(22 − q2) A q dz cos d t ˜ 2α 2 ˜ λ2 = . (3.20) 3q2 + 22 q2 eff 22 − q2  = ,α = 1 + . (3.14) ˜2(˜2 + 2) ˜2 4 q The Gα metric of this theory was constructed in [13], where 3 the global structure of Gα was discussed. Under the Kaluza– Thus we see that the embedding of the Sα in EPA theory requires a tachyonic vector with μ2 < 0, while in EMA Klein reduction, the EPA theory becomes √   theory, no exotic matter is required. L = − −  − 1 2 − 1 μ2 2 . g R 2 eff 4 F 2 A (3.21) 3.4 Generalizing to higher dimensions In this case, the Gα metric is supported by 1 q2 A = q(dt + rdφ), μ = √ ,α= 1 − , In this section, we generalize the Gödel universe to higher  α 2 dimensions by considering Gα × Rn, namely 2q2 − 2   = . (3.22) dr 2 eff 2(2 − 2) s2 = 2 − ( t + r φ)2 + αr 2 f φ2 + 4 q d d d d 2  r f In both theories, the free parameters of the solutions are (a, b) +dzi dzi , i = 1, 2,...,n. (3.15) of the function f (2.8), while q and hence α arefixedbythe coupling constants of the theories. In the above dimensional The α = 1 solution can be still solved by (1.2), but with reductions, we have performed further consistent truncations μ − 2 u = ( 2, 0,...,0). to the subset of fields that are relevant to the Gα metrics. Note In order for the metrics to be solutions of some Lagrangians, that the solution at the beginning of this section involves a we can replace the axion χ in the previous subsections z-dependent A, and hence it cannot be reduced to the three- by a (n − 1)-form potential B(n−1) with the field strength dimensional massless sector. 123 289 Page 6 of 9 Eur. Phys. J. C (2017) 77 :289

4 Black holes from double Wick rotations parameter q and hence α are fixed constants. In these cases, the first law of black hole thermodynamics reads 4.1 Type I dM = T dS + +dJ. (4.9)

As was discussed in [13], the metrics Gα can describe black In EMA theory, on the other hand, the parameter q is an holes in three dimensions after double Wick rotations, integration constant, and hence it can be varied and should t → it,φ→ iφ. (4.1) be involved in the first law. To complete the first law involving the parameter q, we first note that the electric charge of the The general metric (2.9) now becomes Maxwell field vanishes, namely  2  2 2 2 2 ˜ 2 dr 2 ds =  (dt + rdφ) − αr f dφ + + dz , (4.2) ∗F + dχ ∧ A = 0. (4.10) r 2 f˜ with the function f˜ now given by (In [13], an electric charge associated with pure gauge trans- a b formation of A was introduced. We shall not consider this f˜ = 1 − − . (4.3) here.) The linear charge density of the axion field on the r r 2 other hand is non-vanishing For EMA theory, we find that the matter fields are given by 1 1 z 1 Qχ = dχ = √ . (4.11) A = qr dφ, χ = √ , with 2 =− , 8 8 α α 2 2 The corresponding thermodynamical potential can be read α = + q ≥ . 1 1 (4.4) off from the 2-form potential B( ), that is, the Hodge dual to 22 2 the axion, as in (3.7). It is given by For EPA theory, we have  = . 1 χ r0 (4.12) A = q(dφ + rdt), χ = z, with μ = √ ,  α We find that the first law reads 3q2 + 22  = , = +  +  (α ). 42(2 + q2) dM T dS +dJ χ d Qχ (4.13) q2 It is puzzling that an extra factor α is needed for the comple- α = 1 + ≥ 1. (4.5) 2 tion of the first law above. The thermodynamics of three-dimensional black holes was studied in [13]. Here we would like to derive the first 4.2 Type II law in our context and notations. A new subtlety emerges in EMA theory, where the parameter q is an integration con- In this interpretation, we switch t and φ in (4.2) and write stant. For simplicity, we shall set  = 1 for the following themetricas φ discussions. We also assume that the coordinate is peri-  2  2 2 2 2 ˜ 2 dr 2 odic with φ = 2π. Thus the solution describes a rotating ds =  (dφ + (r − r0)dt) − αr f dt + + dz . r 2 f˜ metric. The null-Killing vector on the horizon r = r0 with (4.14) f (r0) = 0 is given by ∂ ∂ ξ = −  ,= 1 . Note that we also made a coordinate transformation so that + + (4.6) = ∂t ∂φ r0 the null-Killing vector at the degenerate surface r r0 with f˜(r ) = 0isξ = ∂ . In other words, the metric is non- It is straightforward to verify that the surface gravity and 0 t rotating on the horizon. The temperature is given by hence the temperature are given by √ √ α 2 ˜( ) αr  κ r0 f r0 κ = 0 f˜ (r ), T = . (4.7) T = . (4.15) 4π 0 2π 4π The mass and angular momentum can be read off from the The solution has no CTCs since gφφ = 2 and furthermore Wald formalism, given by gtt > 0forr > r0, and hence t is globally defined outside the √ √ horizon. Note that in this case, the entropy is a constant since M = 1 α a, J = 1 α b. (4.8) 8 8 the radius of the φ circle is constant. We can also show, using Two situations emerge at this stage. For black holes of the the Wald formalism, that the mass and angular momentum EPA theory or the effective theories in three dimensions, the both vanish. The solution can be viewed as a thermalized 123 Eur. Phys. J. C (2017) 77 :289 Page 7 of 9 289

vacuum. In EMA theory, q is an integration constant, which where F(3) can be either NS–NS or R–R fields. Following leads to non-zero electric charge and potential, given by the reduction ansatz given in [28], we find that the ten-  = q dimensional solution is given by 0, together with = 1 ∗ + χ ∧ = √ φ Q A F d A d dz  16π 16π α 1 q ds2 = − (dt + rdφ)2 = √ φ,  = . 10 2 + 2 A qr0 (4.16) g1 g2 16π α  dr 2 +αr 2 f dφ2 + + dz2 The axion charge and its thermodynamical potential are given r 2 f by (4.11) and (4.12). This leads to the first law of black hole  1 2 “thermodynamics”, + (dψ1 + cos θ1 dφ1 + g1q1rdφ) g2 1   +  = , AdQ A χ dQχ 0 (4.17) + θ 2 + 2 θ φ2 d 1 sin 1 d 1  provided that φ = π. 1 + (dψ2 + cos θ2 dφ2 g2 2  + φ)2 + θ 2 + 2 θ φ2 , 5 Embedding in string theory g2q2rd d 2 sin 2 d 2 (5.5) 1 F( ) = dt ∧ dr ∧ dφ 5.1 Freedman–Schwarz model 3 2 + 2 g1 g2 sin θ1 In Sect. 3 we constructed some ad hoc theories that admit − dθ1 ∧ dφ1 ∧ (dψ1 + cos θ1 dφ1 + g1q1rdφ) g2 Gödel metrics as exact solutions. The Maxwell and axion 1 θ fields are common occurrences in supergravities, indicating −sin 2 θ ∧ φ ∧ ( ψ + θ φ + φ) 2 d 2 d 2 d 2 cos 2 d 2 g2q2rd that there may exist an exact embedding of the Gödel universe g2 in supergravity and hence in string theory. In this section, we q1 − dr ∧ dφ ∧ (dψ1 + cos θ1 dφ1) consider Freedman–Schwarz SU(2) × SU(2) gauged super- g1 gravity whose bosonic sector consists of the metric, a dilaton q2 − dr ∧ dφ ∧ (dψ2 + cos θ2 dφ2). (5.6) ϕ, an axion and two SU(2) Yang–Mills fields. After trun- g2 cating to the U(1)2 subsector, the corresponding Lagrangian α is Here is again given by (5.3). The solution involves both √  electric string and magnetic fivebrane/fivebrane charges, L = − − 1 (∂ϕ)2 − 1 2ϕ(∂χ)2 g R 2 2 e given by  + ( 2 + 2) ϕ − 1 −ϕ( 2 + 2) 2 g1 g2 e 4 e F1 F2 1 electric : Q1 ∼ ∗F (3) ∼ √ , μνρσ α 3 3 + 1 ε χ( + ), g1 g2 8 F1 μν F1 ρσ F2 μν F2 ρσ (5.1) : ∼ ∼ 1 + 1 . ( , ) magnetic Q5/5 F(3) (5.7) where g1 g2 are the gauge coupling constants of the two g2 g2 SU(2) Yang–Mills fields. The theory admits the general 1 2 deformed Gödel metric (2.9) with the matter fields given by These can be either all NS–NS charges or R–R charges, and z the latter corresponds to the D1/D5/D5 configuration. When Ai = qi rdφ, χ = √ ,ϕ= 0, (5.2) 3 3 α q1 = 0 = q2, the metric becomes AdS3 ×S ×S ×R, which is the decoupling limit of the string/fivebrane/fivebrane con- with the parameters figuration [30]. The rotations associated with parameters 1 (q1, q2) turn the AdS3 into the Gα. We thus expect that there 2 = ,α= 1 − (g2 + g2)(q2 + q2). (5.3) ( 2 + 2) 1 2 1 2 should be a rotating string/fivebrane/fivebrane configuration 2 g1 g2 whose decoupling limit gives rise to our ten-dimensional = = 5.2 Gödel universe from string theory solution (5.6). If we set either g1 0org2 0, but not both, the associated S3 is flattened to become R3. The metric 3 4 The Freedman–Schwarz model can be obtained from the configuration becomes Gα × S × R . The heterotic string × 3 × Kaluza–Klein reduction on S3 × S3 [27,28]. The relevant solution of Gα S K3 was first constructed in [11]. part of the effective Lagrangian of strings in ten dimensions To study the global structure, we first denote r0 as the is largest root of f (r). Shifting the coordinates, √   L = − − 1 (∂)2 − 1 − 2 , → − φ, ψ → ψ − φ, 10 g R 2 12 e F(3) (5.4) t t r0 i i gi qi r0 (5.8) 123 289 Page 8 of 9 Eur. Phys. J. C (2017) 77 :289

sin θ we find that the metric is singular at r = r0, where the degen- − 2 θ ∧ 2 d 2 erate Killing vector is purely spatial ξ = ∂φ. The absence of g2 a conical singularity requires that dφ2 ∧ (dψ2 + cos θ2 dφ2 + g2q2 cos θ dφ) q1 4π − d cos θ ∧ dφ ∧ (dψ1 + ρdt) φ = √ . (5.9) g1 αr 2 f (r ) 0 0 q2 − d cos θ ∧ dφ ∧ (dψ2 + cos θ2 dφ2), (5.13) Thus, in this system, there are three periodic coordinates φ, g2 and (ψ ,ψ ), with ψ = 4π = ψ . The coordinate t on 1 2 1 2 where α = 1 + (g2 − g2)(q2 + q2). The solution describes the other hand is not required to be periodic. The analysis of 1 2 1 2 a direct production of a line of coordinate z and a nine- CTCs in ten dimensions becomes more subtle. Note that we 3 dimensional metric of an AdS3×S bundle over the squashed have 3 3-sphere Sα. Again the configuration involves both electric α 2 ( ) r0 f r0 string and magnetic fivebrane/fivebrane charges gφφ = (r − r ) ≥ 0, (5.10) 2 + 2 0 g1 g2 g1 electric : Q1 ∼ √ , α g2(g2 − g2)2 for the region r ≥ r0; however, naked CTCs still exist. One 2 1 2 way to see this is to consider the general periodic Killing 1 1 magnetic : Q / ∼ + . (5.14) 5 5 α( 2 − 2) 2 vector, g1 g2 g2 √ ∂ ∂ ∂ ψ ∼ φ ∼ / α = = ξ = β + γ + γ . (5.11) Note that 1 . When q1 0 q2,the ∂φ 1 ∂ψ 2 ∂ψ 3 3 1 2 metric is again AdS3 × S × S , equivalent to the previous static case. For non-vanishing q ’s, the brane configuration The absence of naked CTCs requires that ξ 2 ≥ 0 for all real i is not clear and it deserves further study. Interestingly, the (β, γ ,γ ) in the r ≥ r region. This can be easily established 1 2 0 limit of g = g leads to the well-known AdS × S3 × R4 not true. Negative modes arise for large enough r. 1 2 3 vacuum of string theory, and the limit g2 = 0 gives rise to A simpler way to see that naked CTCs exist is as follows. × 3 × R4 2 ∗ AdS3 Sα . Let r = r∗ such that r∗ (α f (r ) − 1)<0, which is always achievable since 0 <α<1. Now making a coordinate shift 2 ∗ ψi → ψi − gi qi r∗φ, then we have gφφ = r∗ (α f (r ) − 1)< 0. 6 Conclusions

Four-dimensional Gödel metrics of Gα × R are perhaps the 5.3 AdS × S3 bundle over S3 3 α simplest solutions that exhibit naked CTCs with no globally spatial-like Cauchy horizon. In this paper, we showed that the We now consider the case with negative g2 =−ˆg2 and gˆ2 − 1 1 1 Gödel metrics could arise as exact solutions in Lagrangian g2 > 0. Performing some appropriate analytical continuation 2 formalism. We constructed EMA and EPAtheories that admit of the coordinates on the solution (5.6) and then dropping the Gödel solutions. We also showed that Gödel universe could hat symbol, we have emerge from Freedman–Schwarz SU(2) × SU(2) gauge  1 supergravity. This allows us to give exact embeddings of ds2 = (dψ + cos θ dφ)2 10 2 − 2 the Gödel metrics in string theories. (It seems very closely g1 g2  related to a T-dual version of the M-theory Gödel solution +α 2 θ φ2 + θ 2 + 2 sin d d dz [31,32].) The ten-dimensional solution describes a direct  1 product of a line and an S3 × S3 bundle over Gα. Classi- + (dψ + ρdt + g q cos θ dφ)2 2 1 1 1 cally, we find that naked CTCs persist in higher dimensions. g1  (In [11], string quantization was performed on Gα × S3 × K dρ2 3 + − (ρ2 + 1) dt2 and it was demonstrated that CTCs can resolved by the quan- ρ2 + 1  tum effects.) For some appropriate choice of parameters, the 1 2 × 3 + (dψ2 + cos θ2 dφ2 + g2q2 cos θ dφ) nine-dimensional metric can describe an AdS3 S bundle g2 3 2  over a squashed 3-sphere Sα, in which case there is no CTC. +dθ 2 + sin2 θ dφ2 (5.12) In the suitable limit, the solution becomes the supersymmet- 2 2 2 3 4 ric AdS3 × S × R vacuum. 1 F( ) =− dψ ∧ d cos θ ∧ dφ The scaling symmetry of the metric (2.1) resembles that 3 2 − 2 g1 g2 of the anti-de Sitter spacetimes. This is suggestive of there 1 possibly existing a boundary field theory at the r →∞ + dρ ∧ dt ∧ (dψ + ρ dt + g q cos θ dφ) 2 1 1 1 boundary of the Gödel universe. The exact embedding of the g1 123 Eur. Phys. J. C (2017) 77 :289 Page 9 of 9 289

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