Godel Universe from String Theory
aa r X i v : . [ h e p - t h ] D ec G¨odel Universe from String Theory
Shou-Long Li , Xing-Hui Feng , ∗ , Hao Wei and H. L¨u School of Physics,Beijing Institute of Technology, Beijing 100081, China Center for Advanced Quantum Studies, Department of Physics,Beijing Normal University, Beijing 100875, China
ABSTRACTG¨odel universe is a direct product of a line and a three-dimensional spacetime we callG α . In this paper, we show that the G¨odel metrics can arise as exact solutions in Einstein-Maxwell-Axion, Einstein-Proca-Axion, or Freedman-Schwarz gauged supergravity theories.The last allows us to embed G¨odel universe in string theory. The ten-dimensional spacetimeis a direct product of a line and the nine-dimensional one of an S × S bundle over G α ,and it can be interpreted as some decoupling limit of the rotating D1/D5/D5 intersection.For some appropriate parameter choice, the nine-dimensional metric becomes an AdS × S bundle over squashed 3-sphere. We also study the properties of the G¨odel black holes thatare constructed from the double Wick rotations of the G¨odel metrics. sllee [email protected] [email protected] ∗ [email protected] [email protected] ontents α × R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Energy condition and α value . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Metrics asymptotic to G α . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Mass and angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Squashed 3-sphere S α × T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 × S bundle over S α . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Closed time-like curves (CTCs) in solutions of General Relativity can pose challenges toHawking’s chronology protection conjecture. It is well known that Kerr metric has CTCs;however, they are hidden inside the event horizon, and hence obey the chronological cen-sorship. The G¨odel metric [1], a solution of cosmological Einstein gravity coupled to someuniform pressureless matter, describes a homogeneous rotating universe that has nakedCTCs and has no globally spatial-like Cauchy surface. However, the fine-tuning balance2etween the cosmological constant and the matter density in G¨odel universe makes it an un-realistic model to challenge the chronology protection. Nevertheless it is a good toy modelto study the effects of naked CTCs in both classical and quantum gravities.The study of CTCs has enjoyed considerable attention with the development of thestring theory and the AdS/CFT correspondence. Large number of supersymmetric or non-supersymmetric G¨odel-like solutions, including also black holes and time machines, withnaked CTCs were constructed in gauged supergravities in higher dimensions, see e.g. [2–17]. These works raise important issues whether the problems of CTCs may be resolvedby stringy or quantum considerations, or whether naked CTCs in the bulk implies thebreakdown of unitarity of the field theory.In this paper, we focus on the original four-dimensional G¨odel universe that is a directproduct of a line and a three-dimensional metric [1] ds = ℓ (cid:16) − ( dt + rdφ ) + r dφ + dr r + dz (cid:17) . (1.1)This metric is an exact solution of the Einstein equation involving some uniform pressurelessmatter R µν − Rg µν + Λ g µν = T mat µν , Λ = − ℓ , T mat µν = u µ u ν , (1.2)where u µ = ( ℓ − , , , in (1.1) is replaced by a generic constant α . To avoidpedantry, we shall refer all these metrics as G¨odel metrics, which are distinct from thoseafore-mentioned G¨odel-like metrics. It turns out that most theories associated with thesegeneral G¨odel metrics involve energy-momentum tensor of some uniform pressureless matter.Electric and magnetic fields periodic in z as a replacement for such matter were used toconstruct G¨odel metric in [21], where z is a circular coordinate. It is worth mentioning thatthe three-dimensional metric with dz removed can arise as an exact solution of Einstein-Maxwell theory with a topological term A ∧ F [13]. The embedding of the three-dimensionalmetric in heterotic string theory was first given in [11].The main purpose of this paper is to construct more Lagrangians that admit G¨odelmetrics as exact solutions. We present three Lagrangians that admit the G¨odel metrics assolutions, all involving only the fundamental matter fields. These are Einstein-Maxwell-Axion (EMA), Einstein-Proca-Axion (EPA) and Freedman-Schwarz [25] SU (2) × SU (2)gauged supergravity theories. The Freedman-Schwarz model can be obtained from theeffective actions of string theories via the Kaluza-Klein reduction on S × S . Thus the3our-dimensional G¨odel universe can be embedded in string theory.The paper is organized as follows. In section 2, we give a review of G¨odel metrics andtheir properties. We also show that for appropriate choice of parameter, such a metric canalso describe a direct product of time and a squashed 3-sphere. In section 3, we constructEMA and EPA theories that admit G¨odel metrics, and also solutions involving squashed3-sphere. We generalize the theories to higher dimensions and also give the effective three-dimensional theories by Kaluza-Klein reduction. In section 4, we consider double Wickrotations and obtain two types of black hole solutions. In section 5, we show that G¨odelmetrics are exact solutions of the Freedman-Schwarz model and hence obtain the corre-sponding ten-dimensional solutions of string theories. We conclude the paper in section6. α × R In this paper, we consider a class of metrics in the following form ds = ℓ (cid:16) − ( dt + rdφ ) + αr dφ + dr r + dz (cid:17) , (2.1)where ℓ and α are constants. The metric is a direct product of R associated with thecoordinate z and the three-dimensional metric of ( t, φ, r ), which we shall call G α . Theoriginal G¨odel metric [1] is recovered when we take α = , corresponding to G / × R . Toavoid pedantry, we shall refer the metric (2.1) with generic α also as the G¨odel metric.In addition to the constant shifting symmetry along the ( t, φ, z ) directions, the G¨odelmetric (2.1) is invariant under the constant scaling r → λr , φ → λ − φ . (2.2)Note that imposing this scaling symmetry also implies that φ describes a real line, ratherthan a circle. This scaling property indicates that the metric is homogeneous. Since themetric G α is three dimensional, its curvature is completely determined by the Ricci tensor,whose non-vanishing components are given by R ¯0¯0 = 12 αℓ , R ¯1¯1 = 1 − α αℓ = R ¯2¯2 . (2.3)Here we present the curvature in tangent space with the vielbein e ¯0 = ℓ ( dt + rdφ ) , e ¯1 = ℓ √ α rdφ , e ¯2 = ℓdrr , e ¯3 = ℓdz . (2.4)4t is clear that when α = 1, the metric is locally AdS (3-dimensional anti-de Sitter space-time), i.e. G = AdS . α value It is convenient to define the energy-momentum tensor in the vielbein basis (2.4) T ab = R ab − η ab R . (2.5)We find T ab = diag { ρ, p, p, ˜ p } , with ρ = 3 − α αℓ , p = 14 αℓ , ˜ p = 4 α − αℓ . (2.6)The original α = case gives rise to matter with uniform pressure [1]. The null-energycondition requires that ρ + p = 1 − ααℓ ≥ , ρ + ˜ p = 12 αℓ ≥ , (2.7)which is satisfied by 0 < α ≤
1. As we shall see presently this implies that G¨odel metricsin general have naked CTCs in the framework of Einstein gravity. α We now consider deformations of G α by introducing a function of two constants f = 1 + ar − br , (2.8)and deform the metric (2.1) to become ds = ℓ (cid:16) − ( dt + rdφ ) + αr f dφ + dr r f + dz (cid:17) . (2.9)It is straightforward to verify that the curvature tensors (2.3) remain unchanged. Thisimplies that the metric (2.9) is locally the same as (2.3). However, globally, the deformedmetric (2.9) is different from (2.1). An important difference is that in the deformed metric(2.9), the coordinate φ is periodic, namely∆ φ = 4 π √ α r f ′ ( r ) , with f ( r ) = 0 . (2.10)This ensures that the metric is absent from a conical singularity at r = r . After imposingthis condition, the coordinate transformation that relates (2.1) to (2.9) breaks the scalingsymmetry (2.2) and hence the two metrics are not equivalent globally.5o demonstrate this explicitly, we note that the parameter a is trivial in that it canbe eliminated by the coordinate transformation r → r − a , without altering the globalstructure. The positive parameter b can be set to 1 without loss of generality, using thescaling r → √ b r , together with appropriate scalings of the rest coordinates. Now let r = cosh ˆ r , φ = ˆ φ/ √ α , t = ˆ t − ˆ φ/ √ α , the metric (2.9) with a = 0 and b = 1 becomes ds = ℓ (cid:16) − (cid:0) d ˆ t + √ α sinh ( ˆ r ) d ˆ φ (cid:1) + sinh ˆ r d ˆ φ + d ˆ r + dz (cid:17) . (2.11)On the other hand, we find that under the coordinate transformation r = cosh ˆ r + cos ˆ φ sinh ˆ r , rφ = 1 √ α sin ˆ φ sinh ˆ r , tan (cid:0) ˆ φ + √ α ( t − ˆ t ) (cid:1) = e ˆ r tan( ˆ φ ) , (2.12)the metric (2.1) becomes also precisely (2.11). (This coordinate transformation reduces tothe one obtained in [1] for α = .) Thus it becomes clear that when ∆ ˆ φ = 2 π is fixed, thescaling symmetry of the vacuum (2.1) is broken by the coordinate transformation (2.12),and hence the two metrics are not globally equivalent. An important consequence is thatthe metric (2.11) has CTCs for r > r c withtanh( r c ) = √ α . (2.13)Here r = r c is the velocity of light surface (VLS) for which g ˆ φ ˆ φ = 0. We shall call the metric(2.9) as the deformed G¨odel metric that is asymptotic to the G¨odel metric. For generalparameters ( a, b ), there can be two VLS’s between which g φφ >
0. The global structure ofthe metric (2.9), written in somewhat different parametrization, was analysed in [13].The emerging of the CTCs in G¨odel metrics is a consequence of that α ≤
1. If oneallows α >
1, equation (2.13) has no real solution for r c , and the metrics do not have nakedCTCs. However, as we saw earlier that in the framework of Einstein gravity, α > α ′ -correction of stringtheory, solutions with α > The general G¨odel metric has two Killing vectors ξ t = 1 ℓ ∂∂t , ξ φ = 1 ℓ ∂∂φ . (2.14)6The Killing symmetry in z direction can be broken by the matter sector in some solutions.)Following the Wald formalism [29], which computes the variation of the on-shell Hamiltonian δ H associated with a Killing vector with respect to the integration constants of the solutions,we read off the associated conversed quantities by evaluating δ H at asymptotic infinity, andfind M = √ α a π ∆ φ , J = ℓ √ α b π ∆ φ , (2.15)where ∆ φ is given by (2.10). Note that if one chooses a fixed period ∆ φ = 2 π instead,rather than given by (2.10), the solution will have naked singularity at r = r , for generic α . We have also set the convention R dz = 1. For periodic z , this means period ∆ z = 1; forreal line z , this implies that the extensive quantities such as M and J are in fact uniformdensities over the line z . S α × T In the G¨odel metrics, if we let ℓ = − ˜ ℓ <
0, the metric (2.9) becomes ds = ˜ ℓ (cid:16) ( dz + rdφ ) + αhdφ + dr h − dt (cid:17) , (2.16)where we have swapped the role of ( t, z ), and set, without loss of generality, h = 1 − r . Let r = cos θ , we have ds = ˜ ℓ (cid:16) ( dz + cos θdφ ) + α sin θdφ + dθ − dt (cid:17) . (2.17)This metric describes a direct product of time with a squashed 3-sphere, which we call S α .When the squashing parameter α = 1, S α becomes the round S , written as a U (1) bundleover S . The regularity of S α requires that∆ φ = 2 π √ α , ∆ z = 4 π √ α . (2.18)The period ∆ z can be divided by a natural number n without introducing any singularityto the manifold, giving rise to S α / Z n . (Such compact G¨odel universe was also consideredin [26].) As mentioned in the introduction, theories in literature associated with G¨odel universe(2.1) or (2.9) typically involve a matter energy-momentum tensor with unknown Lagrangian7rigin. The known example in four dimensions is the Einstein-Maxwell theory with an axionand a negative cosmological constant [21] L = √− g (cid:16) R − − ( ∂χ ) − F ) , (3.1)where F = dA is the field strength. For the metric (2.1), the solutions for matter fields are A = p − α ) ℓ sin( z √ α ) ( dt + rdφ ) , χ = z √ α , ℓ = − . (3.2)In this case, the continuous shifting symmetry along the z -direction is broken to a discretesymmetry by the Maxwell potential A that is periodic in z . (The axion χ field does notbreak this symmetry since only dχ appears in the theory.) The solution is best describedas G α × S rather than G α × R . A consequence is that G α is not a solution to the three-dimensional massless sector in the Kaluza-Klein reduction of (3.1) on z . In this section, weshall construct more examples of Lagrangians that admit the G¨odel metrics of G α × R asexact solutions. In addition to (3.1), we introduce an additional topological term: L = √− g (cid:16) R − − F − ( ∂χ ) (cid:17) + ε µνρσ χ F µν F ρσ , (3.3)where F = dA is the field strength, ε µνρσ is the density of Levi-Civita tensor whose com-ponents are ± ,
0. We choose the convention ε = 1. The axion, Maxwell and Einsteinequations of motion are given by ∂ µ (cid:0) √− g∂ µ χ (cid:1) + ε µνρσ F µν F ρσ = 0 , ∂ µ (cid:0) √− gF µν − χε µνρσ F ρσ (cid:1) = 0 ,R µν − g µν R + g µν Λ − ( F µρ F ν ρ − g µν F ) − ( ∂ µ χ∂ µ χ − g µν ( ∂χ ) ) = 0 . (3.4)For the general G¨odel metric (2.9) with (2.8), we consider the following ansatz for the axionand Maxwell field A = qr dφ , χ = kz . (3.5)We find that the equations of motion are all satisfied provided that k = 1 √ α , α = 1 − q ℓ , ℓ = − . (3.6)The general solution contains three integration constants, ( a, b, q ). The reality conditionrequires that | q | < √ ℓ . It follows that we have 0 < α ≤ k ≥
1. The original α = q = ℓ . The AdS factor arises when α = 1, correspondingto turning off the Maxwell field. In section 3.3, we consider the case with α <
0, for whichthe metric describes S α × T .It is worth pointing out that in four dimensions, the axion χ is Hodge dual to a 2-formpotential B (2) with G (3) = dB (2) = ∗ dχ + A ∧ F . (3.7)The Lagrangian (3.3) is equivalent to L = √− g (cid:16) R − − F − G (3) (cid:17) . (3.8)For the G¨odel solutions we have B (2) = rdt ∧ dφ . The 3-form field strength G (3) is suggestiveof string theory, which we shall discuss in section 5. In this subsection, we replace the previous Maxwell field by a Proca field of mass µ , withthe Lagrangian L = √− g (cid:16) R − − F − µ A − ( ∂χ ) (cid:17) . (3.9)The equations of motion are (cid:3) χ = 0 , ∇ µ F µν − µ A ν = 0 ,R µν − g µν R + g µν Λ − (cid:0) F µρ F ν ρ − g µν F (cid:1) − µ (cid:0) A µ A ν − g µν A (cid:1) − (cid:0) ∂ µ χ∂ µ χ − g µν ( ∂χ ) (cid:1) = 0 . (3.10)(There should be no confusion between the Proca mass µ and the spacetime indices.) Themetric ansatz is given in (2.9). We consider the following ansatz for A and χ : A = q ( dt + rdφ ) , χ = z . (3.11)Substituting these into the equations of motion, we find µ = 1 ℓ √ α , Λ = 3 q − ℓ ℓ ( ℓ − q ) , α = 1 − q ℓ . (3.12)Note that all the constants except for ( a, b ) appearing in the solution are fixed by thecoupling constants of the theory, namely (Λ , µ ). It follows that unlike in the earlier EMAtheory, the general solution involves only two integration constants. The original α = G¨odel metric corresponds to taking q = ℓ/ √ .3 The embedding of squashed 3-sphere In the embedding of the G¨odel metric in both EMA and EPA theories discussed above, thecosmological constant Λ is negative. When it is positive, the metric G α × R becomes S α × T ,as in (2.17). For the EMA theory, we have A = q cos θ dφ , χ = t √ α , ˜ ℓ = 12Λ , α = 1 + q ℓ . (3.13)For the EPA theory, we have A = q ( dz + cos θ dφ ) , χ = t , µ = − ℓ α , Λ = 3 q + 2˜ ℓ ℓ (˜ ℓ + q ) , α = 1 + q ˜ ℓ . (3.14)Thus we see that the embedding of the S α in EPA theory requires a tachyonic vector with µ <
0, while in the EMA theory, no exotic matter is required.
In this section, we generalize the G¨odel universe to higher dimensions by considering G α × R n , namely ds = ℓ (cid:16) − ( dt + rdφ ) + αr f dφ + dr r f + dz i dz i (cid:17) , i = 1 , , . . . , n. (3.15)The α = solution can be still solved by (1.2), but with u µ = ( ℓ − , , . . . , χ in the previous subsections by a ( n − B ( n − with the field strength G ( n ) = dB ( n − . The Lagrangian (3.3) becomes L = √− g h R − − F − n ! G n − i +
18 ( n − ε µνρσα ··· α n − B α ··· α n − F µν F ρσ . (3.16)(This should not be confused with dualizing dχ to the 3-form field strength in four dimen-sions, discussed in the end of subsection 3.1.) The axion ansatz (3.5) is replaced by G ( n ) = k dz ∧ · · · dz n . (3.17)The Lagrangian (3.9) is now replaced by L = √− g (cid:16) R − − F − µ A − n ! G n ) (cid:17) . (3.18)The corresponding ansatz for G ( n ) is given by (3.17) with k = 1.10 .5 Effective three-dimensional theories The non-trivial part of the G¨odel universe is the three-dimensional metric G α . For thesolutions in subsections 3.1 and 3.2, the Killing symmetry in z direction is maintained bythe matter fields. We can thus perform dimensional reduction on the coordinate z . TheEMA theory becomes L = √− g (cid:0) R − eff − F (cid:1) + λ eff ε µνρ A µ F νρ , (3.19)In this case, the three-dimensional G α metric is now supported by A = qrdφ , α = 1 − q ℓ , Λ eff = q − ℓ ℓ (2 ℓ − q ) , λ = 22 ℓ − q . (3.20)The G α metric of this theory was constructed in [13], where the global structure of G α wasdiscussed. Under the Kaluza-Klein reduction, the EPA theory becomes L = √− g (cid:0) R − eff − F − µ A (cid:1) . (3.21)In this case, the G α metric is supported by A = q ( dt + rdφ ) , µ = 1 ℓ √ α , α = 1 − q ℓ , Λ eff = 2 q − ℓ ℓ ( ℓ − q ) . (3.22)In both theories, the free parameters of the solutions are ( a, b ) of function f (2.8), whilst q and hence α are fixed by the coupling constants of the theories. In the above dimensionalreductions, we have performed further consistent truncations to subset of fields that arerelevant to the G α metrics. Note that the solution at the beginning of this section involvesthe z -dependent A , and hence it cannot be reduced to the three-dimensional massless sector. As was discussed in [13], the metrics G α can describe black holes in three dimensions afterdouble Wick rotations t → i t , φ → i φ . (4.1)The general metric (2.9) now becomes ds = ℓ (cid:16) ( dt + rdφ ) − αr ˜ f dφ + dr r ˜ f + dz (cid:17) , (4.2)11ith the function ˜ f now given by ˜ f = 1 − ar − br . (4.3)For the EMA theory, we find that the matter fields are given by A = qr dφ , χ = z √ α , with ℓ = − , α = 1 + q ℓ ≥ . (4.4)For the EPA theory, we have A = q ( dφ + rdt ) , χ = z , with µ = 1 ℓ √ α , Λ = 3 q + 2 ℓ ℓ ( ℓ + q ) , α = 1+ q ℓ ≥ . (4.5)Thermodynamics of three-dimensional black holes was studied in [13]. Here we wouldlike to derive the first law in our context and notations. New subtlety emerges in theEMA theory, where the parameter q is an integration constant. For simplicity, we shall set ℓ = 1 for the following discussions. We also assume that the coordinate φ is periodic with∆ φ = 2 π . Thus the solution describes a rotating metric. The null-Killing vector on thehorizon r = r with f ( r ) = 0 is given by ξ = ∂∂t − Ω + ∂∂φ , Ω + = 1 r . (4.6)It is straightforward to verify that the surface gravity and hence the temperature are givenby κ = √ αr π ˜ f ′ ( r ) , T = κ π . (4.7)The mass and angular momentum can be read off from the Wald formalism, given by M = √ α a , J = √ α b . (4.8)Two situations emerge at this stage. For black holes of the EPA theory or the effectivetheories in three dimensions, the parameter q and hence α are fixed constants. In thesecases, the first law of black hole thermodynamics reads dM = T dS + Ω + dJ . (4.9)In the EMA theory; on the other hand, the parameter q is an integration constant, andhence it can be varied and should be involved in the first law. To complete the first lawinvolving the parameter q , we first note that the electric charge of the Maxwell field vanishes,namely Z ∗ F + dχ ∧ A = 0 . (4.10)12In [13], an electric charge associated with pure gauge transformation of A was introduced.We shall not consider this here.) The linear charge density of the axion field on the otherhand is non-vanishing Q χ = Z dχ = 18 √ α . (4.11)The corresponding thermodynamical potential can be read off from the 2-form potential B (2) that is Hodge dual to the axion, as in (3.7). It is given byΦ χ = r . (4.12)We find the first law reads dM = T dS + Ω + dJ + Φ χ d ( αQ χ ) . (4.13)It is puzzling that an extra factor α is needed for the completion of the first law above. In this interpretation, we switch t and φ in (4.2) and write the metric as ds = ℓ (cid:16) ( dφ + ( r − r ) dt ) − αr ˜ f dt + dr r ˜ f + dz (cid:17) . (4.14)Note that we also made a coordinate transformation so that the null Killing vector at thedegenerate surface r = r with ˜ f ( r ) = 0 is ξ = ∂ t . In other words, the metric is non-rotatingon the horizon. The temperature is given by T = √ αr ˜ f ′ ( r )4 π . (4.15)The solution has no CTCs since g φφ = ℓ and further more g tt > r > r , and hence t isglobally defined outside the horizon. Note that in this case, the entropy is a constant sincethe radius of the φ circle is constant. We can also show, using the Wald formalism thatthe mass and angular momentum both vanish. The solution can be viewed as thermalizedvacuum. In the EMA theory, q is an integration constant, which leads to non-zero electriccharge and potential, give by Q A = 116 π Z ∗ F + dχ ∧ A = q π √ α Z dφdz = q π √ α ∆ φ , Φ A = qr . (4.16)The axion charge and its thermodynamical potential are given by (4.11) and (4.12). Thisleads to the first law of black hole “thermodynamics”Φ A dQ A + Φ χ dQ χ = 0 , (4.17)provided that ∆ φ = π . 13 Embedding in string theory
In section 3 we constructed some ad hoc theories that admit G¨odel metrics as exact solutions.The Maxwell and axion fields are common occurrence in supergravities, indicating that theremay exist an exact embedding of G¨odel universe in supergravity and hence in string theory.In this section, we consider Freedman-Schwarz SU (2) × SU (2) gauged supergravity whosebosonic sector consists of the metric, a dilaton ϕ , an axion and two SU (2) Yang-Mills fields.After truncating to the U (1) subsector, the corresponding Lagrangian is L = √− g (cid:16) R − ( ∂ϕ ) − e ϕ ( ∂χ ) + 2( g + g ) e ϕ − e − ϕ ( F + F ) (cid:17) + ε µνρσ χ ( F µν F ρσ + F µν F ρσ ) , (5.1)where ( g , g ) are the gauge coupling constants of the two SU (2) Yang-Mills fields. Thetheory admits the general deformed G¨odel metric (2.9) with the matter fields given by A i = q i rdφ , χ = z √ α , ϕ = 0 , (5.2)with the parameters ℓ = 12( g + g ) , α = 1 − ( g + g )( q + q ) . (5.3) Freedman-Schwarz model can be obtained from the Kaluza-Klein reduction on S × S [27, 28]. The relevant part of the effective Lagrangian of strings in ten dimensions is L = √− g (cid:16) R − ( ∂ Φ) − e − Φ F (3) (cid:17) , (5.4)where F (3) can be either NS-NS or R-R fields. Following the reduction ansatz given in [28],we find that the ten-dimensional solution is given by Φ = 0, together with ds = 1 g + g (cid:16) − ( dt + rdφ ) + αr f dφ + dr r f + dz (cid:17) + 1 g (cid:16) ( dψ + cos θ dφ + g q rdφ ) + dθ + sin θ dφ (cid:17) + 1 g (cid:16) ( dψ + cos θ dφ + g q rdφ ) + dθ + sin θ dφ (cid:17) , (5.5) F (3) = 1 g + g dt ∧ dr ∧ dφ − sin θ g dθ ∧ dφ ∧ ( dψ + cos θ dφ + g q rdφ ) − sin θ g dθ ∧ dφ ∧ ( dψ + cos θ dφ + g q rdφ ) − q g dr ∧ dφ ∧ ( dψ + cos θ dφ ) − q g dr ∧ dφ ∧ ( dψ + cos θ dφ ) . (5.6)14ere α is again given by (5.3). The solution involves both electric string and magneticfivebrane/fivebrane charges, given byelectric : Q ∼ Z ∗ F (3) ∼ √ αg g , magnetic : Q / ∼ Z F (3) ∼ g + 1 g . (5.7)These can be either all NS-NS charges or R-R charges, and the latter corresponds to theD1/D5/D5 configuration. When q = 0 = q , the metric becomes AdS × S × S × R , whichis the decoupling limit of the string/fivebrane/fivebrane configuration [30]. The rotationsassociated with parameters ( q , q ) turn the AdS into the G α . We thus expect that thereshould be a rotating string/fivebrane/fivebrane configuration whose decoupling limit givesrise to our ten-dimensional solution (5.6). If we set either g = 0 or g = 0, but not both,the associated S is flatten to become R . The metric configuration becomes G α × S × R .The heterotic string solution of G α × S × K was first constructed in [11].To study the global structure, we first denote r as the largest root of f ( r ). Shifting thecoordinates as t → t − r φ , ψ i → ψ i − g i q i r φ , (5.8)we find that the metric is singular at r = r , where the degenerate Killing vector is purelyspatial ξ = ∂ φ . The absence of a conical singularity requires that∆ φ = 4 π √ αr f ′ ( r ) . (5.9)Thus in this system, there are three periodic coordinates φ , and ( ψ , ψ ), with ∆ ψ = 4 π =∆ ψ . The coordinate t on the other hand is not required to be periodic. The analysis ofCTCs in ten dimensions becomes more subtle. Note that we have g φφ = αr f ′ ( r ) g + g ( r − r ) ≥ , (5.10)for the region r ≥ r ; however, naked CTCs still exist. One way to see this is to considerthe general periodic Killing vector ξ = β ∂∂φ + γ ∂∂ψ + γ ∂∂ψ . (5.11)The absence of naked CTCs requires that ξ ≥ β, γ , γ ) in the r ≥ r region.This can be easily established not true. Negative modes arise for large enough r .A simpler way to see that naked CTCs exist is the follows. Let r = r ∗ such that r ∗ ( αf ( r ∗ ) − <
0, which is always achievable since 0 < α <
1. Now making a coordinateshifting ψ i → ψ i − g i q i r ∗ φ , then we have g φφ ( r ∗ ) = r ∗ ( αf ( r ∗ ) − < .3 AdS × S bundle over S α We now consider the case with negative g = − ˆ g and ˆ g − g >
0. Performing some appro-priate analytical continuation of the coordinates on the solution (5.6) and then droppingthe hat symbol, we have ds = 1 g − g (cid:16) ( dψ + cos θ dφ ) + α sin θ dφ + dθ + dz (cid:17) + 1 g (cid:16) ( dψ + ρdt + g q cos θ dφ ) + dρ ρ + 1 − ( ρ + 1) dt (cid:17) + 1 g (cid:16) ( dψ + cos θ dφ + g q cos θ dφ ) + dθ + sin θ dφ (cid:17) (5.12) F (3) = − g − g dψ ∧ d cos θ ∧ dφ + 1 g dρ ∧ dt ∧ ( dψ + ρ dt + g q cos θ dφ ) − sin θ g dθ ∧ dφ ∧ ( dψ + cos θ dφ + g q cos θ dφ ) − q g d cos θ ∧ dφ ∧ ( dψ + ρdt ) − q g d cos θ ∧ dφ ∧ ( dψ + cos θ dφ ) , (5.13)where α = 1 + ( g − g )( q + q ). The solution describes a direct production of a line ofcoordinate z and a nine-dimensional metric of an AdS × S bundle over squashed 3-sphere S α . Again the configuration involves both electric string and magnetic fivebrane/fivebranecharges electric : Q ∼ g √ α g ( g − g ) , magnetic : Q / ∼ α ( g − g ) + 1 g . (5.14)Note that ∆ ψ ∼ ∆ φ ∼ / √ α . When q = 0 = q , the metric is again AdS × S × S ,equivalent to the previous static case. For non-vanishing q i ’s, the brane configuration is notclear and it deserves further study. Interestingly, the limit of g = g leads to the well-knownAdS × S × R vacuum of string theory, and the limit g = 0 gives rise to AdS × S α × R . Four-dimensional G¨odel metrics of G α × R are perhaps the simplest solutions that exhibitnaked CTCs with no globally spatial-like Cauchy horizon. In this paper, we showed thatthe G¨odel metrics could arise as exact solutions in Lagrangian formalism. We constructedEMA and EPA theories that admit G¨odel solutions. We also showed that G¨odel universecould emerge from Freedman-Schwarz SU (2) × SU (2) gauge supergravity. This allows us togive exact embeddings of the G¨odel metrics in string theories. The ten-dimensional solution16escribes a direct product of a line and an S × S bundle over G α . Classically, we findthat naked CTCs persist in higher dimensions. (In [11], string quantization was performedon G α × S × K and it was demonstrated that CTCs can resolved by the quantum effects.)For some appropriate choice of parameters, the nine-dimensional metric can describe anAdS × S bundle over a squashed 3-sphere S α , in which case, there is no CTC. In thesuitable limit, the solution becomes the supersymmetric AdS × S × R vacuum.The scaling symmetry of the metric (2.1) resembles that of the anti-de Sitter spacetimes.This is suggestive that there may exist a boundary field theory at the r → ∞ boundaryof G¨odel universe. The exact embedding of the G¨odel metrics in string theory, as thedecoupling limit of the rotating D1/D5/D5 intersection, provides a tool of investigating theboundary field theory in the context of string theory. Acknowlegement
S.-L.L. and H.W. are supported in part by NSFC under Grants NO. 11575022 and NO.11175016. X.-H.F. and H.L. are supported in part by NSFC grants NO. 11175269, NO.11475024 and NO. 11235003.
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