Homogeneous G\" Odel-Type Solutions in Hybrid Metric-Palatini Gravity

$Homogeneous G\" Odel-Type Solutions in Hybrid Metric-Palatini Gravity$

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In the Palatini formalism the metric and the connection ple. The Gödel model is a solution of Einstein’s equa- are treated as independent fields, and it is assumed that the tions with cosmological constant Λ for dust of density ρ, matter fields do not couple with the independent connec- but it can also be viewed as perfect-fluid solution with tions. Although these approaches have been invoked as pos- equation of state p = ρ with no cosmological constant sible ways to satisfactorily deal with the observed acceler- term. Owing to its unexpected properties, Gödel’s solu- ated expansion of the Universe, it has been pointed out that tion has a recognizable importance and has motivated a f (R) gravity theories can face relevant difficulties, including considerable number of investigations on rotating Gödel- the evolution of cosmological perturbations and local grav- type models as well as on causal anomalies in the con- ity constraints (see, for example, Refs. [19,17,18,20,14,21, text of general relativity (see, e.g. Refs. [38,39,40,41,42, 22]). 43,44,45,46,47,48,49,50,51,52,53,54] and other gravity These undesirable features have motivated a recent ap- theories [55,56,57,58,59,60,61,62,63,64,65,66,67,68,69, proach to modified f (R) theories of gravity, which can be 70,71,72,73,74,75,76,77,78,79]. In two recent papers, we employed as a possible way to handle the observed late- have also examined Gödel-type models and the violation of time cosmic acceleration, and also circumvent some dif- causality problem for f (R) gravity in both the metric and ficulties that arise in the framework of f (R) theories in Palatini variational approaches [71,72], extending therefore both formalisms. The hybrid metric-Palatini f (R) gravity the results of Refs. [80] and [81]. is a recently devised approach to such modified theories, in which it is added to the ordinary Ricci scalar R, in the If gravitation is to be described by hybrid metric-Palatini Einstein-Hilbert Lagrangian, a function, f (R), of Palatini f (R) gravity theory there are a number of issues that ought curvature scalar R, which is constructed from the indepen- to be reexamined in its context, including its consistence ρ dent connection Γµν [23]. These hybrid metric-Palatini grav- with the recent detection of gravitational wave [82] and the ity theories appears to suitably unify the description of the question as to whether these gravity theories allow Gödel- late-time cosmic acceleration with the local solar system type solutions, which necessarily lead to closed timelke constraints [23,24]. Some of the astrophysical and cosmo- curves, or would remedy this causal pathology by ruling out logical implications of hybrid metric-Palatini gravity have this type of solutions, which are permitted in general rela- been examined in a number of papers [25,26,27,28,29]. tivity. Wormwhole solutions, Einstein static universe, linear perturbations, the Cauchy problem, dynamical system analy- In this article, to proceed further with the investiga- sis and a brane model have been discussed, respectively, in tions on the potentialities, difficulties and limitations of the references [30,31,32,33,34,35]. Other important mat- f (R), we undertake this question by examining whether the ters such as Noether symmetries [36] and the thermody- f (R) gravity theories admit homogeneous Gödel-type solu- namic behavior [37] have also been recently considered. For tions for a combination of physically well-motivated matter R an introduction to hybrid metric-Palatini f ( ) gravity and a sources. To this end, we first examine the general problem detailed list of related references, we refer the reader to the of finding out ST-homogeneous solutions in hybrid metric- recent review article [28]. Palatini f (R) gravity for matter sources with constant trace In general relativity (GR) the space-times have lo- T (scalar) of the energy-momentum tensor, and show that it cally the same causal structure of the flat space-time of reduces to to the problem of determining ST-homogeneous special relativity since the space-times of GR are locally solutions of Einstein’s field equations with a cosmological Minkowskian. On nonlocal scale, however, significant dif- constant determined by f (R) and its first derivative f ′(R). ferences may arise since the general relativity field equa- Employing this far-reaching result, we determine a general tions do no provide nonlocal constraints on the underlying ST-homogeneous Gödel-type whose matter source is a com- space-times. Indeed, it has long been known that there are bination of a scalar with an electromagneticfields plus a per- solutions to Einstein’s field equations that present nonlocal fect fluid. This general Gödel-type solution contains special causal anomalies in the form of closed time-like curves (see, solutions in which the essential parameter m2 defines any for example, Refs. [38,39,40,41,42,43]. one of the possible classified families homogeneous Gödel- The renowned model found by Gödel [44] is the type solutions, namely m2 > 0 hyperbolic family, m = 0 best known example of a solution to the Einstein’s equa- linear class, and m2 < 0 trigonometric family. This general tions, with a physically well-motivated source, that makes homogeneous Gödel-type solution also contains previously it apparent that GR permits solutions with closed time- known solutions as special cases. There emerges from one like world lines, regardless of its local Lorentzian char- of the particular solution of the hyperbolic family that ev- acter that ensures locally an inherited regular chronology ery perfect-fluid Gödel-type solution of any f (R) gravity and therefore the local validation of the causality princi- with density ρ and pressure p and satisfying the weak en- 3

µν ergy conditions ρ > 0 and ρ + p 0 is necessarily isometric 2/√ g δ(√ gLm)/δg is the energy-momentum ten- ≥ − − − to the Gödel geometry1. sor of the matter fields. The bare existence of these noncausal Gödel-type solu- The variation of the action (1) with respect to the inde- ρ tions makes apparent that hybrid metric-Palatini f (R) grav- pendent connection Γµν yields ity does not remedy causal anomaly in the form of closed ∇ √ gF(R)gµν = 0, (4) timelike curves that are permitted in general relativity. β − The structure of the paper is as follows. In Section 2 we wheree ∇ denotes the covariant derivative associated with R β give a brief account of the hybrid metric-Palatini f ( ) grav- Γ ρ . If onedefines a metric h = F(R)g , it can be easily µν e µν µν ity theories. In Section 3 we present the basic properties of shown that Eq. (4) determines a Levi-Civita connection of homogenous Gödel-type geometries. This includes the con- hµν , which in turn can be rewritten in terms of gµν and its ditions for space-time homogeneity, a classification of ST- ρ Levi-Civita connection µν in the form homogeneous Gödel-type geometries and a study of the existence of closed time-like curves in all ST-homogeneous ρ ρ 1 ρ ρ ρσ Γ = + δ ∂ν + δ ∂µ gµν g ∂σ lnF(R). (5) µν µν 2 µ ν − Gödel-type metrics. In Section 4 we first examine the problem of finding out ST-homogeneoussolutions in f (R) grav- Using now Eq. (5) one finds the relation between the two ity whose trace T of the energy-momentum tensor of the Ricci tensors, which is given by matter source is constant, and show that in such cases the 3 1 1 Rµν = Rµν + ∂µ F ∂ν F (∇µ ∇ν + gµν ✷)F , (6) problem reduces to that of finding out solutions of Einstein’s 2F2 − F 2 field equations with a cosmological constant. We then show where ∇µ denotes the covariant derivative associated to that the hybrid metric-Palatini f R gravity theories admit ρ ( ) . ST-homogeneous Gödel-type solutions for a general combi- µν Equation (6) in turn gives rise to the following relation nation of physically well-motivated matter contents. between the two Ricci scalars: R 3 2 3 ✷ = R + 2 (∂F) F , (7) 2 Hybrid metric-Palatini gravity 2F − F 2 αβ αβ where (∂F) = g ∂α F∂β F and ✷F = g ∇α ∇β F. The action that defines a hybrid metric-Palatini gravity is The Palatini curvature R can be obtained from the trace given by of the field Eq. (3), which yields 2 1 4 RF(R) 2 f (R)= κ T + R X . (8) S = d x√ g [R + f (R)+ Lm] , (1) 2κ2 Z − − ≡ This trace equationcan be used to express R algebraically in where κ2 = 8πG, g is the determinant of the metric tensor terms of X when the f (R) is given as an analytic expression. gµν , R is the Ricci scalar associated to the Levi-Civita con- Finally, we note that the variable X measures the deviation 2 nection of the metric gµν , Lm is the Lagragian density for from the general relativity trace equation R = κ T . the matter fields, and the extra term f (R) is a function of − Palatini curvature scalar R, which depends on the metric ρ 3 Homogeneous Godel-type¨ geometries and on an independent connection Γµν through To make this work clear and to a certain extent self- R gµν R gµν ρ ρ ρ λ ρ λ µν = ∂ρΓµν ∂νΓµρ +Γρλ Γµν Γµλ Γρν . contained, in this section we present the basic properties ≡ − − (2) of homogenous Gödel-type geometries, which we use in the following sections. To this end, we first discuss the The variation of the action (1) with respect to the metric conditions for space-time homogeneity (ST-homogeneity) gives the field equations of these space-times, and present all non-isometric ST- R homogeneous Gödel-type classes. These ST-homogeneity f ( ) 2 Gµν + F(R)Rµν gµν = κ Tµν , (3) conditions along with the set of isometrically non-equivalent − 2 geometries are important in the determination of ST- where Gµν = Rµν R/2gµν and Rµν are, respec- − homogeneous Gödel-type solutions, in Section 4, to the hy- tively, Einstein and Ricci tensor associated with the brid metric-Palatini f (R) field equations. Second, we dis- Levi-Civita connection of gµν , F(R) d f /dR, Tµν = ≡ cuss the existence of closed time-like curves in the metrics of these classes. The existence of these non-causal curves 1 This extends to the context of f (R) gravity a theorem which states that every perfect-fluid Gödel-type solution of Einstein’s equations is are crucial to examine whether hybrid metric-Palatini f (R) necessarily isometric to the Gödel spacetime [83]. gravities allow violation of causality of Gödel-type. 4

3.1 Homogeneity and non-equivalent metrics 3.2 Closed time-like curves

Gödel solution to the general relativity field equations is a We begin by noting that the presence of a single closed time- particular member of the broad family of geometries whose like curve in a space-time is an unequivocal manifestation general form in cylindrical coordinates, (r,φ,z), is given of violation of causality. However, a space-time may ad- by [81] mit non-causal closed curves other than Gödel’s circles we discuss in this Section. To examine the existence of closed 2 2 2 2 2 2 ds = [dt + H(r)dφ] D (r)dφ dr dz . (9) time-like curves in ST-homogeneous Gödel-type metrics we − − − first rewrite the line element (9) as The necessary and sufficient conditions for the Gödel- type metric (9) to be space-time homogeneous (ST- ds2 = dt2 + 2H(r)dtdφ dr2 G(r)dφ 2 dz2 , (14) − − − homogeneous) are given by [81,84] where G(r) = D2 H2. In this form it is easy to show − H D that existence of closed time-like curves, which allows for ′ = 2ω and ′′ = m2, (10) D D violation of causality in homogeneous Gödel-type space- times, depends on the sign of the metric function G(r). In- where the prime denote derivative with respect r, and the deed, from Eq. (14) one has that the circles, hereafter called parameters (Ω,m) are constants such that Ω 2 > 0 and ∞ − ≤ Gödel’s circles, defined by t,z,r = const become closed m2 ∞. ≤ timelike curves whenever G(r) < 0. As a matter of fact, except for the case m2 = 4ω2 all For the hyperbolic (m2 > 0) class of homogeneous locally ST-homogeneous Gödel-type space-times admit a Gödel-type metrics, from Eqs. (11) one has that group G5 of isometries acting transitively on the whole 2 2 space-time [84]. The special case m = 4ω admits a G7 4 mr 4ω2 mr G(r)= sinh2( ) (1 ) sinh2( )+ 1 . (15) of isometries [85,84]. m2 2 − m2 2 The irreducible set of isometrically nonequivalent ST- 2 2 homogeneous Gödel-type metrics can be obtained by inte- Therefore for 0 < m < 4ω there is a critical radius rc de- grating equations (10) and suitably eliminating nonessen- fined by G(r)= 0, which is given by tial integration constants. The final result is that ST- 1 mr 2 − homogeneous Gödel-type geometries can be grouped in the 2 c 4ω sinh = 2 1 , (16) following three classes [81]: 2 m −

i. Hyperbolic, in which m2 = const > 0 and such that for r < rc one has G(r) > 0, and for r > rc one has G(r) < 0. Thus, the circles t,r,z = const with r > rc are 4ω mr 1 closed timelike curves.3 H = sinh2( ), D = sinh(mr); (11) m2 2 m For linear class (m = 0) of homogeneous Gödel-type space-times, from Eq. (12) one easily finds ii. Linear, in which m = 0 and G(r)= r2 r4 ω2 = r2 (r ω 1) (r ω + 1) . (17) H = ωr2, D = r , (12) − − − Thus, there is a critical radius, defined by G(r)= 0, and iii. Trigonometric, where m2 = const µ2 < 0 and ≡− given by rc = 1/ω, such that for any radius r > rc one has 4ω µr 1 G(r) < 0, and then the circles defined by t,z,r = const are H = sin2( ), D = sin(µr). (13) µ2 2 µ closed timelike curves. Finally for the trigonometric class (m2 = const µ2 < ≡− Thus, clearly all ST-homogeneous Gödel-type geometries 0), from the metric functions given by Eq. (13) one finds are characterized by the two independent parameters m2 4 µ r µ r and ω — identical pairs (m2,ω2) specify isometric space- G(r)= sin2( )[ µ2 (4ω2 + µ2) sin2( )], (18) µ4 2 2 times [81,85,84].2 In this way, to determine whether hybrid − metric-Palatini f (R) gravity allows Gödel-type solutions is and therefore G(r) has an infinite sequence of zeros. Thus, in to find out whether its field equations can be used to specify the section t,z,r = const, there is an sequence of alternating a pair of parameters m2 and ω for a suitably chosen matter causal [G(r) > 0] and noncausal[G(r) < 0] regionswithout source. 3The only ST-homogeneous Gödel-type space-time without these non- 2Gödel geometry is a solution of Eintein’s equations in general relativ- causal circles come about when m2 = 4ω2 (see Ref. [81]). In this case, ity, is indeed a particular case of the hyperbolic class of geometries in the critical radius rc ∞, and hence the violation of causality of Gödel which m2 = 2ω2. type is avoided. → 5 and with noncausal circles, depending on the value of r = whose trace of the energy-momentum T is also constant, const (For more details see the Appendix of Ref. [75]). In which we focus in this paper, one has X = const = κ2T R . − this way, if G(r) < 0 for a certain range of r (r1 < r < r2, In such cases, one can show that the field equations of say) noncausal Gödel’s circles exist, whereas for r in the the hybrid metric-Palatini gravity reduces to Einstein’s field next circular band r2 < r < r3 (say) for which G(r) > 0 no equations with a cosmological constant. such closed timelike circles exist, and so on. To this end, we first examine the second term on the To close this section, we note that in this paper by non- right hand side of equation (5) which gives the departures ρ causal and causal solutions we mean, respectively, solutions of the independent connection Γµν from Levi-Civita connec- ρ with and without violation of causality of Gödel-type, i.e., tion µν . Clearly, each individual part of this second term with and without Gödel’s circles. in this equation is proportional to F 1 ∂ lnF(R)= ′ ∂ R = ∂ F(R). (23) µ F µ F µ 4 Solutions in hybrid metric-Palatini gravity On the other hand, to calculate ∂µ F(R) we note that from the trace equation (8) one has ∂µ R = ∂µ X /[RF (R) The aim of this section is twofold. First, we examine the ′ − problem of finding out ST-homogeneous solutions in hybrid F(R)], which together with equation (23) furnishes metric-Palatini f (R) gravity whose trace T of the energy- F R ∂ X R ′( ) µ momentum tensor of the matter source is constant. We show ∂µ F( )= R R R , (24) F′( ) F( ) that in such cases the problem of finding out solutions in − provided that RF′(R) F(R) = 0.From equations (23) the hybrid metric-Palatini f (R) gravity reduces to the prob- − 6 ρ and (24) one has that for X = const the connection Γ lem of determining ST-homogeneous solutions of Einstein’s µν reduces to Levi-Civita connection ρ . Furthemore, from field equations with a cosmological constant determined by µν equations (6) and (7) one can easily show that ∂ F(R)= 0 f (R) and its first derivative F = f (R). Second, we exam- µ ′ also ensures that R = R and R = R. This makes appar- ine whether hybrid metric-Palatini f (R) field equations ad- µν µν ent that ST-homogeneous spacetimes solutions whose trace mit ST-homogeneous Gödel-type solutions for a combina- of the energy-momentumtensor is constant (R+κ2T = X = tion of a scalar field with an electromagnetic field plus a const) the field equations of the hybrid metric-Palatini grav- perfect fluid. ity (3) reduce formally to field equations of f (R) theories in the metric formalism, which can clearly be rewritten in the 4.1 Field equations form 1 2 [1 + F(R)] Gµν [ f (R) RF(R)] gµν = κ Tµν , (25) We begin by noting that using equation (6) the field equa- − 2 − tions (3) of the hybrid metric-Palatini f (R) gravity can be with associated trace equation rewritten in the form RF(R) 2 f (R)= κ2T + R = const. (26) 2 eff 2 R − Gµν = κ Tµν = κ Tµν + Tµν , (19) However, for an explicitly given f (R), solving the alge- braic equation (26) one finds constant roots R’s. Thus, for where each explicit root the field equations (25) can be rewritten in 2 R 1 4 κ T = [ f (R)+ ✷F(R)] gµν F(R)Rµν the form µν 2 − 3 +∇µ ∇ν F(R) ∂µ F(R)∂ν F(R). (20) 2 − 2F(R) Gµν = κ¯ Tµν +Λ gµν , (27) In this context, an important constraint comes from the trace where of the field equations (19), which can be written in the form f (R) RF(R) κ2 Λ = − and κ¯ 2 = . (28) R 2[1 + F(R)] 1 + F(R) R + κ2T = X = κ2T , (21) − The trace equation becomes where from equation (20) one has R + κ¯ 2 T + 4Λ = 0. (29) R 2 2 R R R ✷ R 3 [∂F( )] Clearly, the factor [1 + F(R)] in equations (28) is a constant κ T = 2 f ( ) RF( )+ 3 F( ) R . (22) − − 2 F( ) that simply rescales the units of κ2 and the effective cosmo- For ST-homogeneous spacetimes, which we are con- logical constant Λ. cerned with in this paper, one has that the Ricci scalar is 4Clearly, different roots R give rise to different rescales of κ2, and dif- necessarily constant. On the other hand, for matter sources ferent effective cosmological constant Λ. 6

4.2 G¨odel-type solutions 4.2.1 Combined-ﬁelds general solution

In this section we discuss ST-homogeneous Gödel-type so- In this section we take combination of scalar and electro- lutions in hybrid metric-Palatini f (R) gravity for well- magnetic fields with a perfect fluid as a matter source, and motivated matter contents whose trace of the energy- find a general ST-homogeneous Gödel-type solution, which momentum tensor is constant. contains the a perfect fluid and a scalar field particular solu- 2 2 We begin by noting that the search for ST-homogeneous tions, and whose essential parameter m can be m > 0 (hy- 2 Gödel-type solutions to the hybrid metric-Palatini gravity perbolic family), m = 0 (linear class) or m < 0 (trigonomet- field equations is greatly simplified if instead of using co- ric family) depending on the amplitude values of the matter ordinates basis one uses a new basis given by the following components. set of linearly independent one-forms (tetrad frame) Θ A: In the Lorentzian basis (30) the energy-momentum tensor of combined matter sources takes the form θ 0 = dt + H(r)dφ , θ 1 = dr , θ 2 = D(r)dφ , θ 3 = dz, (30) (M) (S) (EM) TAB = TAB + TAB + TAB , (36)

(M) (S) (EM) relative to which the Gödel-type line element (9) takes the where TAB , TAB and TAB are, respectively, the energy local Lorentzian form momentum tensors of a perfect fluid, a scalar field, and an electromagnetic field, which we discuss in what follows. (M) 2 A B 0 2 1 2 2 2 3 2 For a perfect fluid of density ρ and pressure p, T one ds = ηAB θ θ = (θ ) (θ ) (θ ) (θ ) . (31) AB − − − has Here and in what follows capital letters are tetrad indices (or (M) TAB = (ρ + p)uAuB pηAB . (37) Lorentz frame indices) and run from 0 to 3. These Lorentz − frame indices are raised and lowered with Lorentz matrices The energy-momentum tensor of a single scalar field is AB η = ηAB = diag(+1, 1, 1, 1), respectively. given by − − − In the tetrad frame (30) the nonvanishing components of 1 CD (S) MN the Ricci tensor, RAB = η RCADB, are given by TAB = Φ A Φ B ηAB Φ M Φ N η , (38) | | − 2 | | 2 1 H′ ′ 1 H′ where vertical bar denotes components of covariant deriva- R02 = , R00 = , (32) A A α 2 D 2 D tives relative to the local basis θ = e α dx [see Eqs. (30) A 2 and (31)], i.e. Φ A = e µ ∇µ Φ . Following Ref. [81] it is 1 H′ D′′ | R11 = R22 = , (33) straightforward to show that a scalar field of the form 2 D − D Φ(z) = ε z + ε, with ε,ε = const, fulfills the scalar field AB where the prime denotes derivative with respect to r. Since equation ✷Φ = η ∇A∇B Φ = 0. Thus, the non-vanishing the Lorentz frame components of the Ricci tensor depend componentsof the energy-momenttensor for this scalar field only on H′/D and D′′/D, from the ST-homogeneity condi- are tions (10) one has that for all classes of ST-homogeneous 2 (S) (S) (S) (S) ε Gödel-type metrics the frame components of the Ricci ten- T = T = T = T = , (39) 00 − 11 − 22 33 2 sor are constants. Thus, the Ricci scalar is also constant and CD 2 2 given by R = η RCD = 2(m ω ). As for the electromagnetic part of energy momentum − In this Lorentzian basis the field equations (27) reduce tensor (36), following Ref. [81], the electromagnetic field to tensor FAB given by

F03 = F30 = E0 sin[2ω(z z0)], (40) − − ¯ 2 GAB = κ TAB +Λ ηAB , (34) F12 = F21 = E0 cos[2ω(z z0)], (41) − − satisfies the source-free Maxwell equations which, in the where from equations (32) and (33) along with the condi- tetrad frame (30), take the form tions (10) one has that the only nonvanishing Lorentz frame AB A MB C AM components of the Einstein tensor GAB for ST-homogeneous F B + γ MBF + γ MCF = 0, (42) | M Gödel-type metrics take the very simple form F[AB C] + 2FM[Cγ AB] = 0, (43) | where the brackets denote total anti-symmetrization and 2 2 2 2 2 G00 = 3ω m , G11 = G22 = ω , G33 = m ω . the Ricci rotation coefficients are defined by γA = − − BC (35) ∇ θ A θ α θ β . The non-vanishing components of the − β α B C 7

(EM) C associated energy-momentum tensor TAB = FA FBC + of ST-homogeneous Gödel-type solutions can also be gen- 1 CD erated with a simple combination of scalar and electromag- 4 ηABF FCD are given by netic fields (thus for ρ = p = 0), depending on the relative 2 2 2 (EM) (EM) (EM) (EM) E values of the amplitudes ε and E , which give the sign of T = T = T = T = 0 . (44) 0 00 11 22 − 33 2 the term 2ε2 E2 in equation (49). − 0 Thus, taking into account equations(35), (37), (39) and (44), one has that for the combined-fields matter source (36) the 5 Concluding remarks field equations (34) reduce to ε2 E2 Despite the great success of the general relativity theory, a 3ω2 m2 = κ¯ 2 ρ + + 0 +Λ , (45) good deal of efforthas recently goneinto the study of the so- − 2 2 called modified gravity theories. In the cosmological model- ε2 E2 ω2 = κ¯ 2 p + 0 Λ , (46) ing framework, this is motivated by the fact that these theo- − 2 2 − ries provide an alternative way to explain the current acceler- ε2 E2 ating expansion of the Universe with no need to invoke the m2 ω2 = κ¯ 2 p + 0 Λ . (47) − 2 − 2 − dark energy matter component. The hybrid metric-Palatini f (R) gravity is a recently devised approach to such modi- To determine the essential parameters ω2 and m2, we fied theories, in which it is added to the ordinary Ricci scalar first substitue (47) into (45) to obtain R, in the Einstein-Hilbert Lagrangian, a function, f (R), of R 2 ¯ 2 p 2 Palatini curvature scalar , which is constructed from the 2ω = κ (ρ + + ε ) (48) ρ independent connection Γµν . Second, we use (46), (47) and (48) to find In general relativity, the bare existence of the Gödel solution to Einstein’s equations, for a physically well- 2 2 2 2 m = κ¯ (ρ + p + 2ε E0 ). (49) motivated perfect-fluid source, makes it apparent that this − theory permits solutions with violation of causality on non- The trace equation (29) gives rise to the following constrain local scale, regardless of its local Lorentzian character that for the cosmological constant ensures the local validation of the causality principle. In the context of the hybrid metric-Palatini f R gravity theo- 2Λ = κ¯ 2(p ρ 2ε2 + E2) (50) ( ) − − 0 ries the space-time manifolds are also assumed to be locally Since ST-homogeneous Gödel-type geometries are char- Lorentzian. Hence the chronology and causality structure of acterized by the two essential parameters m2 and ω2, the special relativity are inherited locally. The nonlocal ques- above equations (48) and (49) make explicit how the f (R) tion, however, is left open, and violation of causality can in gravity specifies a pair of parameters (m2,ω2), and therefore principle arise. determines a general ST-homogeneous Gödel-type solution, Since homogeneous Gödel-type geometries necessar- for the combined-fields matter source (36). ily lead to the existence of closed timelike circles (Sec- The general solution given by equations (48) and (49) tion 3.2), which is an unequivocal manifestation of violation contains all general relativity Gödel-type known solu- of causality, a natural way to tackle this question is by in- tions [81] as particular cases. Indeed, the perfect fluid Gödel vestigating whether the hybrid metric-Palatini f (R) gravity solution is recovered when ε = E0 = 0 with ρ, p = 0, theories permit Gödel-type solutions to their field equations. 6 whereas the scalar field causal solution [81] is retrieved for Furthermore, if gravity is to be governed by a f (R) there 5 ρ = p = E0 = 0 with ε = 0. are a number of issues ought to be reexamined in its con- 6 Finally, from equation (49) one has that the combination text, including the question as to whether these theories ad- of scalar plus electromagnetic field with a perfect fluid gives mit Gödel-type solutions, or would remedy the violation of rise to ST-homogeneous Gödel-type solutions in hyperbolic causality problem by ruling out this type of solutions, which 2 2 2 are permitted in general relativity. class (m > 0) when E0 < ρ + p + 2ε , in the linear fam- 2 2 2 In this article, to proceed further with the investigations ily (m = 0) for E0 = ρ + p + 2ε , and also in the trigono- 2 2 2 on the potentialities, difficulties and limitations of f (R), we metric class (m < 0) when E0 > ρ + p + 2ε . Moreover, from equation (49) we also have that these three families have examined whether f (R) gravity theories admit homogeneous Gödel-type solutions for physically well-motivated 5 We note that Gödel metric (m2 = 2ω2) can also be generated through matter sources. To this end, we have first examined the prob- a particular combination of perfect fluid with a scalar field, namely lem of finding out ST-homogeneous solutions in these hy- for non-vanishing equal amplitude of the scalar and electromagnetic fields ε = E0 = 0. Indeed in this case equations (48) and (49) give brid gravity theories whose trace T (invariant) of the energy- m2 = 2ω2 = κ¯6 2(ρ + p + ε2). momentum tensor is constant. We have shown that under 8 this assumption the problem of finding out solutions in the To conclude, we emphasize that the bare existence of the hybrid metric-Palatini f (R) gravity reduces to the prob- ST-homogeneous Gödel-type solutions that we have found lem of determining ST-homogeneous solutions of Einstein’s makes apparent that the hybrid metric-Palatini f (R) grav- field equations with a cosmological constant. Employing ity does not remedy at a classical level the causal pathol- this far-reaching simplifying result, we first have found a ogy in the form of closed timelike curves that arises in the general Gödel-type solution for a source that is a combina- context of general relativity. We are not aware of a quantum tion of scalar and electromagnetic fields with a perfect fluid. gravity theory following the hybrid metric-Palatini structure, In this general Gödel-type solution solution the essential pa- though. rameter m2 can be m2 > 0 (hyperbolic family), m = 0 (lin- 2 ear class) or m < 0 (trigonometric family) depending on Acknowledgements M.J. Rebouças acknowledges the support of the values of the amplitudes ε (scalar field) and E0 (elec- FAPERJ under a CNE E-26/202.864/2017 grant, and thanks CNPq tromagnetic field), and the density ρ and pressure p of the for the grant under which this work was carried out. J. Santos acknowledges support of Programa de Pós-Graduação em F´ısica - perfect fluid. This general homogeneous Gödel-type solu- CCET/UFRN. tion also contains previously known solutions of the literature as special cases. There emerges from one of the particular solution of the hyperbolic family that every perfect- References fluid Gödel-type solution of any f (R) gravity with density ρ and pressure p and satisfying the weak energy conditions 1. A.G. Riess et al., Observational evidence from supernovae for an ρ > 0and ρ + p 0 is necessarily isometric to the Gödel ge- accelerating universe and a cosmological constant, Astron. J. 116, ≥ ometry. This extends to the context of f (R) gravity a theo- 1009 (1998) rem established in the contextof the general relativity, which 2. S. Perlmutter, et al., Discovery of a supernova explosion at half the age of the Universe, Nature 391, 51 (1998) states that Gödel solution is the sole perfect fluid solution of 3. S. Perlmutter et al., Measurements of Ω and Λ from 42 high- Einstein’s equations [83]. redshift supernovae, Astrophys. J. 517, 565 (1999) Whether or not the physical laws permit the existence of 4. R. Adam et al., Planck 2015 results. I. Overview of products and stable time machines in the form of closed timelike curves scientific results, Astron. & Astrophys. 594, A1 (2016) 5. D.N. Spergel et al., Three-Year Wilkinson Microwave Anisotropy is a research field in general relativity and other gravity the- Probe (WMAP) Observations: Implications for Cosmology, As- ories. Violation of causality on the other hand raises intrigu- trophys. J. Suppl. 170, 377S (2007) ing logical paradoxes, and is generally seen as undesirable 6. S. Cole et al., The 2dF Galaxy Redshift Survey: Power-spectrum feature in physics. The two most known remedies to these analysis of the final data set and cosmological implications, Mon. paradoxes are Novikov’s self-consistency principle [86,87, Not. Roy. Astron. Soc. 362, 505 (2005) 7. D.J. Eisenstein et al., Detection of the Baryon Acoustic Peak in the 88], which was designed to reconcile the logical inconsis- Large-Scale Correlation Function of SDSS Luminous Red Galax- tences by demanding that the only admissible local solutions ies, Astrophys. J. 633, 560 (2005) are those which are globally self-consistent,6 and Hawking’s 8. W.J. Percival et al., Baryon acoustic oscillations in the Sloan Dig- chronology protection conjecture[89], which suggests that ital Sky Survey Data Release 7 galaxy sample, Mon. Not. Roy. Astron. Soc. 401, 2148 (2010) even though closed timelike curves are classically possible 9. C. Blake et al, The Wiggle Z Dark Energy Survey: mapping the to be produced, quantum effects are likely to prevent such distance-redshift relation with baryon acoustic oscillations, Mon. time travel. In this way, the laws of quantum physics would Not. Roy. Astron. Soc. 418, 1707 (2011) prevent closed timelike curves from appearing.7 In this re- 10. L. Anderson et al, The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Baryon Acoustic Oscil- gard, Hawking and Penrose have also pointed out that se- lations in the Data Release 9 Spectroscopic Galaxy Sample, Mon. vere causality assumptions could risk ’ruling out something Not. Roy. Astron. Soc. 428, 1036 (2013) that gravity is trying to tell us’ [92], thus, discouraging fur- 11. A. De Felice and S. Tsujikawa, f(R) Theories, Living Rev. Rel. 13, ther investigations. The possible existence of closed timelike 3 (2010) curves is also particularly interesting in the quantum realm, 12. T.P. Sotiriou and V. Faraoni, f(R) theories of gravity, Rev. Mod. Phys. 82, 451 (2010) where, for example, the quantum systems traversing these 13. S. Nojiri and S.D. Odintsov, Unified cosmic history in modified curves have been studied [93] and experimental simulation gravity: From F(R) theory to Lorentz non-invariant models, Phys. of closed timelike curves have been undertaken [94]. Rep. 505, 59 (2011) 14. G.J. Olmo, Palatini approach to modified gravity: f(R) theories and 6Clearly this principle is ultimately an ad hoc global topological con- beyond, Int. J. Mod. Phys. D, 20, 413 (2011) straint on admissible solutions of gravity theories, thus beyond the 15. S. Capozziello and M. De Laurentis, Extended Theories of Grav- standard scope of the local formulation of the gravity theories. ity, Phys. Rep. 509, 167 (2011) 7For a good pedagogical overview with a fair list of references on the 16. S. Capozziello and V. Faraoni, Beyond Einstein Gravity, Funda- chronology protection conjecture and Novikov’s self-consistency prin- mental Theories of Physics 170, (Springer, Dordrecht, 2011). ciple see Visser [90], and the recent review article on closed timelike 17. W. Hu and I. Sawicki, Models of f (R) cosmic acceleration that curves and violation of causaltiy by Lobo [91] evade solar system tests, Phys. Rev. D 76, 064004 (2007) 9

18. S. Tsujikawa, Observational signatures of f (R) dark energy model 44. K. Gödel, An example of a new type of cosmological solutions of that satisfy cosmological and local gravity constraints, Phys. Rev. Einstein’s field equations of gravitation, Rev. Mod. Phys. 21, 447 D 77, 023507 (2008) (1949) 19. T. Koivisto and H. Kurki-Suonio, Cosmological perturbations in 45. M.M. Som and A.K. Raychaudhuri, Cylindrically Symmetric the Palatini formulation of modified gravity, Class. Quantum Grav. Charged Dust Distributions in Rigid Rotation in General Relativ- 23, 2355 (2006) ity, Proc. Roy. Soc. London A 304, 81 (1968) 20. T. Koivisto, The matter power spectrum in f (R) gravity, Phys. Rev. 46. A.K. Raychaudhuri and S.N. Guha Thakurta, Homogeneous D 73, 083517 (2006) space-times of the Gödel-type, Phys. Rev. D 22, 802 (1980) 21. G.J. Olmo, Violation of the equivalence principle in modified the- 47. M.J. Rebouças, J.E. Aman˚ and A.F.F. Teixeira, A note on Gödel- ories of gravity, Phys. Rev. Lett. 98, 061101 (2007) type space-times, J. Math. Phys. 27, 1370 (1986) 22. G.J. Olmo, Hydrogen atom in Palatini theories of gravity, Phys. 48. M.J. Rebouças and A.F.F. Teixeira, Features of a Relativistic Rev. D 77, 084021 (2008) Space-Time with Seven Isometries, Phys. Rev. D 34, 2985 (1986) 23. T. Harko, T.S. Koivisto, F.S.N. Lobo and G.J. Olmo, Metric- 49. F.M. Paiva, M.J. Rebouças and A.F.F. Teixeira, Time Travel in the Palatini gravity unifying local constraints and late-time cosmic ac- Homogeneous Som-Raychaudhuri Universe, Phys. Lett. A 126, celeration, Phys. Rev. D 85, 084016 (2012) 168 (1987) 50. A. Krasiński, Rotating Dust Solutions of Einsteins Equations With 24. S. Capozziello, T. Harko, F.S.N. Lobo and G.J. Olmo, Hybrid 3-Dimensinal Symmetry Groups - III - All Killing Fields Linearly modified gravity unifying local tests, galactic dynamics and late- Independent of U(Alpha) And W(Alpha), J. Math. Phys. 39, 2148 time cosmic acceleration, Int. J. Modern Phys. D 22, 1342006 (1998) (2013) 51. S. Carneiro, A Gödel-Friedman cosmology?, Phys. Rev. D 61, 25. S. Capozziello et al., Cosmology of hybrid metric-Palatini f(X)- 083506 (2000) gravity, J. Cosmol. Astropart. Phys. 04, 011 (2013) 52. Y.N. Obukhov, On physical foundations and observational effects 26. S. Capozziello et al., The virial theorem and the dark matter prob- of cosmic rotation, in ’Colloquium on Cosmic Rotation’ Eds. M. lem in hybrid metric-Palatini gravity, J. Cosmol. Astropart. Phys. Scherfner, T. Chrobok and M. Shefaat, pp 23–96 (Wissenschaft 07, 024 (2013) und Technik Verlag: Berlin, 2000) astro-ph/0008106 27. S. Capozziello et al., Galactic rotation curves in hybrid metric- 53. J.D. Barrow and C.G. Tsagas, Dynamics and stability of the Gödel Palatini gravity, Astropart. Phys. 50, 65 (2013) universe, Class. Quantum Grav. 21, 1773 (2004) 28. S. Capozziello et al., Hybrid Metric-Palatini Gravity, Universe, 1, 54. M.P. Dabrowski and J. Garecki, Energy momentum and angular 199 (2015) momentum of Gödel universes, Phys. Rev. D 70, 043511 (2004) 29. N.A. Lima and V.S. Barreto, Constraints on hybrid metric-Palatini 55. L.L. Smalley, Gödel cosmology in Riemann-Cartan spacetime gravity from background evolution, Astrophys. J. 818, 186 (2016) with spin density, Phys. Rev. D 32, 3124 (1985) 30. S. Capozziello et al., Wormholes supported by hybrid metric- 56. J. Duarte de Oliveira, A.F.F. Teixeira and J. Tiomno, Homoge- Palatini gravity, Phys. Rev. D 86, 127504 (2012) neous Cosmos of Weyssenhoff Fluid in Einstein-Cartan Space, 31. C.G. Böhmer, F.S.N. Lobo and N. Tamanini, Einstein static uni- Phys. Rev. D 34, 3661 (1986) verse in hybrid metric-Palatini gravity, Phys. Rev. D 88, 104019 57. A.J. Accioly and G.E.A. Matsas, Are there causal vacuum so- (2013) lutions with the symmetries of the Gödel universe in higher- 32. N.A. Lima, Dynamics of linear perturbations in the hybrid metric- derivative gravity?, Phys. Rev. D 38, 1083 (1988) Palatini gravity, Phys. Rev. D 89, 083527 (2014) 58. J.D. Barrow and M.P. Dabrowski, Gödel universes in string theory, 33. S. Capozziello et al., The Cauchy problem in hybrid metric- Phys. Rev. D 58, 103502 (1998) Palatini f (X)-gravity, Int. J. Geom. Methods Mod. Phys. 11, 59. J.E. Aman,˚ J.B. Fonseca-Neto, M.A.H. MacCallum and M.J. 1450042 (2014) Rebouças, Riemann-Cartan spacetimes of Gödel-type, Class. 34. S. Carloni, T. Koivisto and F.S.N. Lobo, Dynamical system anal- Quantum Grav. 15, 1089 (1998) ysis of hybrid metric-Palatini cosmologies, Phys. Rev. D 92, 60. M.J. Rebouças and A.F.F. Teixeira, Riemannian Space-times of 064035 (2015) Gödel Type in Five Dimensions, J. Math. Phys. 39, 2180 (1998) 35. Qi-Ming Fu et al., Hybrid metric-Palatini brane system, Phys. Rev. 61. M. J. Rebouças and A.F.F. Teixeira, Causal Anomalies in Kaluza- D 94, 024020 (2016) Klein Gravity Theories, Int. J. Mod. Phys. A 13, 3181 (1998) 62. P. Kanti and C.E. Vayonakis, Gödel-type universes in string- 36. A. Borowiec et al., Invariant solutions and Noether symmetries in inspired charged gravity, Phys. Rev. D 60, 103519 (1999) hybrid gravity, Phys. Rev. D 91, 023517 (2015) 63. H.L. Carrion, M.J. Rebouças and A.F.F. Teixeira, Gödel-type 37. T. Azizi and N. Borhani, Thermodynamics in hybrid metric- Spacetimes in Induced Matter Gravity Theory, J. Math. Phys. 40, Palatini gravity, Astrophys. Space Sci. 357, 146 (2015) 4011 (1999) 38. W.J. van Stockum, The gravitational field of a distribution of par- 64. E.K. Boyda, S. Ganguli, P. Horava and U. Varadarajan, Holo- ticles rotating around an axis of symmetry, Proc. R. Soc. Edin. 57, graphic protection of chronology in universes of the Gödel type, 135 (1937). Phys. Rev. D 67, 106003 (2003) 39. M.S. Morris, K.S. Thorne and U. Yurtsever, Wormholes, time ma- 65. J.D. Barrow and C.G. Tsagas, The Gödel brane, Phys. Rev. D 69, chines, and the weak energy condition, Phys. Rev. Lett. 61, 1446 064007 (2004) (1988) 66. M. Banados, G. Barnich, G. Compere and A. Gomberoff,Three- 40. M.S. Morris and K.S. Thorne, Wormholes in spacetime and their dimensional origin of Gödel spacetimes and black holes, Phys. use for interstellar travel: A tool for teaching general relativity, Rev. D 73, 044006 (2006) Am. J. Phys. 56, (1988) 395 (1988) 67. W.-H. Huang, Instability of Tachyon Supertube in Type IIA Gödel 41. F.J. Tipler, Rotating cylinders and the possibility of global causal- Spacetime, Phys. Lett. B 615, 266 (2005) ity violation, Phys. Rev. D 9, 2203 (1974) 68. D. Astefanesei, R.B. Mann and E. Radu, Nut Charged Space- 42. J.R. Gott III, Closed timelike curves produced by pairs of moving times and Closed Timelike Curves on the Boundary, JHEP 01, 049 cosmic strings: Exact solutions, Phys. Rev. Lett. 66, 1126 (1991) (2005) 43. M. Alcubierre, The warp drive: hyper-fast travel within general 69. Y. Brihaye, J. Kunz and E. Radu, From black strings to black relativity, Class. Quant. Grav. 11, L73 (1994) holes: nuttier and squashed AdS5 solutions, JHEP 08, 025 (2009) 10

70. C. Furtado et al., Gödel solution in modified gravity, Phys. Rev. D 84. M.J. Rebouças and J.E. Aman,˚ Computer-aided study of a class of 79, 124039 (2009) Riemannian space-times, J. Math. Phys. 28, 888 (1987) 71. M.J. Rebouças and J. Santos, Gödel-type universes in f(R) gravity, 85. A.F.F. Teixeira, M.J. Rebouças and J.E. Aman,˚ Isometries of ho- Phys. Rev. D 80, 063009 (2009) mogeneous Gödel-type spacetimes, Phys. Rev. D 32, 3309 (1985) 72. J. Santos, M.J. Rebouças and T.B.R.F. Oliveira, Gödel-type universes in Palatini f(R) gravity, Phys. Rev. D 81, 123017 (2010) 86. I.D. Novikov, Time machine and selfconsistent evolution in prob- 73. Z. Tao, W. Pu-Xun and Yu Hong-Wei,Gödel-type universes in f(R) lems with selfinteraction, Phys. Rev. D 45, 1989 (1992) gravity with an arbitrary coupling between matter and geometry, 87. A. Carlini, V.P. Frolov, M.B. Mensky, I.D. Novikov and H.H. Chin. Phys. Lett. 28, 120401 (2011) Soleng, Time machines: The principle of selfconsistency as a con- 74. D. Liu, P. Wu and H. Yu, Gödel-type universes in f(T) gravity, Int. sequence of the principle of minimal action, Int. J. Mod. Phys. D J. Mod. Phys. D 21, 1250074 (2012) 4, 557 (1995) 75. J.B. Fonseca-Neto, A.Yu. Petrov and M.J. Rebouças, Gödel-type 88. A. Carlini, and I.D. Novikov, Time machines and the principle of universes and chronology protection in Hoˇrava-Lifshitz gravity, self-consistency as a consequence of the principle of stationary Phys. Lett. B 725, 412 (2013) action. ii: The Cauchy problem for a self-interacting relativistic 76. P.J. Porf´ırio et al.,Chern-Simons modified gravity and closed time- particle, Int. J. Mod. Phys. D 5, 445 (1996) like curves, Phys. Rev. D 94, 044044 (2016) 89. S.W. Hawking, Chronology protection conjecture, Phys. Rev. D 77. J.A. Agudelo et al., Gödel and Gödel-type universes in Brans- 46, 603 (1992) Dicke theory, Phys. Lett. B 762, 96 (2016) 78. M. Gürses and Ç. S¸entürk, Gödel-type metrics in Einstein-Aether 90. M. Visser, The quantum physics of chronology protection in theory II: nonflat background in arbitrary dimensions, Gen. Rel. The Future of Theoretical Physics and Cosmology: Celebrating Grav. 48, 63 (2016) Stephen Hawking’s 60th Birthday, pp. 161-175, edited by G. W. 79. Otalora and M.J. Rebouças, Violation of causality in f (T) gravity, Gibbons, E.P.S. Shellard, S.J. Rankin (Cambridge U.P., Cam- Eur. Phys. J. C 77, 799 (2017) bridge 2003) 80. T. Clifton and J.D. Barrow, The existence of Gödel, Einstein, and 91. F.S. Lobo, Closed timelike curves and causality violation, Clas- de Sitter universes, Phys. Rev. D 72, 123003 (2005) sical and Quantum Gravity: Theory, Analysis and Applications, 81. M.J. Rebouças and J. Tiomno, Homogeneity of Riemannian Chapter 6, Nova Science Publisher (2008) space-times of Gödel type, Phys. Rev. D 28, 1251 (1983) 92. S. Hawking and R.Penrose, The Nature of Space and Time, 82. B.P. Abbott et al. (Virgo, LIGO Scientific), Phys. Rev. Lett. 116, Princeton University Press, Princeton (1996) 061102 (2016); B.P. Abbott et al. (Virgo, LIGO Scientific), Phys. Rev. Lett. 119, 161101 (2017); B.P. Abbott et al. (Virgo, Fermi- 93. D. Deutsch, Quantum mechanics near closed timelike lines, Phys. GBM, INTEGRAL) Astrophys. J. 848, L13 (2017). Rev. D 44, 3197 (1991) 83. F. Bampi and C. Zordan, A note on Gödel’s metric, Gen. Rel. Grav. 94. M. Ringbauer et al., Experimental simulation of closed timelike 9, 393 (1978) curves, Nat. Commun. 5:4145 (2014)