# Homogeneous Gödel-type solutions in hybrid metric-Palatini gravity

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## Abstract

The hybrid metric-Palatini \(f({\mathscr {R}})\) gravity is a recently devised approach to modified gravity in which it is added to the metric Ricci scalar *R*, in the Einstein–Hilbert Lagrangian, a function \(f({\mathscr {R}})\) of Palatini curvature scalar \({\mathscr {R}}\), which is constructed from an independent connection. These hybrid metric-Palatini gravity theories provide an alternative way to explain the current accelerating expansion without a dark energy matter component. If gravitation is to be described by a hybrid metric-Palatini \(f({\mathscr {R}})\) gravity theory there are a number of issues that ought to be examined in its context, including the question as to whether its equations allow homogeneous Gödel-type solutions, which necessarily leads to violation of causality. Here, to look further into the potentialities and difficulties of \(f({\mathscr {R}})\) theories, we examine whether they admit Gödel-type solutions for physically well-motivated matter source. We first show that under certain conditions on the matter sources the problem of finding out space-time homogeneous (ST-homogeneous) solutions in \(f({\mathscr {R}})\) theories reduces to the problem of determining solutions of Einstein’s field equations with a cosmological constant. Employing this far-reaching result, we determine a general ST-homogeneous Gödel-type solution whose matter source is a combination of a scalar with an electromagnetic fields plus a perfect fluid. This general Gödel-type solution contains special solutions in which the essential parameter \(m^2\) can be \(m^{2} > 0\) hyperbolic family, \(m=0\) linear class, and \(m^{2} < 0\) trigonometric family, covering thus all classes of homogeneous Gödel-type spacetimes. This general solution also contains all previously known solutions as special cases. The bare existence of these Gödel-type solutions makes apparent that hybrid metric-Palatini \(f({\mathscr {R}})\) gravity does not remedy causal anomaly in the form of closed timelike curves that are permitted in general relativity.

## 1 Introduction

A number of cosmological observations coming from different sources, including the supernovae type Ia (SNe Ia) [1, 2, 3], the cosmic microwave background radiation (CMBR) [4, 5] and baryon acoustic oscillation (BAO) surveys [6, 7, 8, 9, 10], indicate that the Universe is presently expanding with an accelerating rate. The frameworks proposed to account for this observed accelerated expansion can be roughly grouped into two families. In the first, the underlying theory, general relativity (GR), is kept unchanged, and the so-called dark energy component is invoked. In this context, the simplest way to account for the accelerating expansion of the Universe is through the introduction of a cosmological constant, \(\varLambda \), into Einstein’s field equations. This is completely consistent with the available observational data, but it faces difficulties such as the order of magnitude of the cosmological constant and its microphysical origin. In the second family, modifications of Einstein’s field equations are assumed as an alternative for explaining the accelerated expansion. This latter group includes, for example, generalized theories of gravity based upon modifications of the Einstein–Hilbert action by taking nonlinear functions, *f*(*R*), of the Ricci scalar *R* or other curvature invariants (for reviews see Refs. [11, 12, 13, 14, 15, 16]).

In dealing with *f*(*R*) gravity theories two different variational approaches, which give rise to different dynamics, have often been considered in the literature [11, 12, 13, 14, 15, 16]. In the so-called metric formalism the connection is assumed to be Levi-Civita, therefore defined by the metric. In the Palatini formalism the metric and the connection are treated as independent fields, and it is assumed that the matter fields do not couple with the independent connections. Although these approaches have been invoked as possible ways to satisfactorily deal with the observed accelerated expansion of the Universe, it has been pointed out that *f*(*R*) gravity theories can face relevant difficulties, including the evolution of cosmological perturbations and local gravity constraints (see, for example, Refs. [14, 17, 18, 19, 20, 21, 22]).

These undesirable features have motivated a recent approach to modified *f*(*R*) theories of gravity, which can be employed as a possible way to handle the observed late-time cosmic acceleration, and also circumvent some difficulties that arise in the framework of *f*(*R*) theories in both formalisms. The hybrid metric-Palatini \(f({\mathscr {R}})\) gravity is a recently devised approach to such modified theories, in which it is added to the ordinary Ricci scalar *R*, in the Einstein–Hilbert Lagrangian, a function, \(f({\mathscr {R}})\), of Palatini curvature scalar \({\mathscr {R}}\), which is constructed from the independent connection \(\varGamma _{\mu \nu }^{\rho }\) [23]. These hybrid metric-Palatini gravity theories appears to suitably unify the description of the late-time cosmic acceleration with the local solar system constraints [23, 24]. Some of the astrophysical and cosmological implications of hybrid metric-Palatini gravity have been examined in a number of papers [25, 26, 27, 28, 29]. Wormwhole solutions, Einstein static universe, linear perturbations, the Cauchy problem, dynamical system analysis and a brane model have been discussed, respectively, in the references [30, 31, 32, 33, 34, 35]. Other important matters such as Noether symmetries [36] and the thermodynamic behavior [37] have also been recently considered. For an introduction to hybrid metric-Palatini \(f({\mathscr {R}})\) gravity and a detailed list of related references, we refer the reader to the recent review article [28].

In general relativity (GR) the space-times have *locally* the same causal structure of the flat space-time of special relativity since the space-times of GR are locally Minkowskian. On nonlocal scale, however, significant differences may arise since the general relativity field equations do no provide nonlocal constraints on the underlying space-times. Indeed, it has long been known that there are solutions to Einstein’s field equations that present nonlocal causal anomalies in the form of closed time-like curves (see, for example, Refs. [38, 39, 40, 41, 42, 43].

The renowned model found by Gödel [44] is the best known example of a solution to the Einstein’s equations, with a physically well-motivated source, that makes it apparent that GR permits solutions with closed timelike world lines, regardless of its local Lorentzian character that ensures locally an inherited regular chronology and therefore the local validation of the causality principle. The Gödel model is a solution of Einstein’s equations with cosmological constant \(\varLambda \) for dust of density \(\rho \), but it can also be viewed as perfect-fluid solution with equation of state \(p=\rho \,\) with no cosmological constant term. Owing to its unexpected properties, Gödel’s solution has a recognizable importance and has motivated a considerable number of investigations on rotating Gödel-type models as well as on causal anomalies in the context of general relativity (see, e.g. Refs. [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54] and other gravity theories [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79]. In two recent papers, we have also examined Gödel-type models and the violation of causality problem for *f*(*R*) gravity in both the metric and Palatini variational approaches [71, 72], extending therefore the results of Refs. [80, 81].

If gravitation is to be described by hybrid metric-Palatini \(f({\mathscr {R}})\) gravity theory there are a number of issues that ought to be reexamined in its context, including its consistence with the recent detection of gravitational wave [82, 83, 84] and the question as to whether these gravity theories allow Gödel-type solutions, which necessarily lead to closed timelike curves, or would remedy this causal pathology by ruling out this type of solutions, which are permitted in general relativity.

In this article, to proceed further with the investigations on the potentialities, difficulties and limitations of \(f({\mathscr {R}})\), we undertake this question by examining whether the \(f({\mathscr {R}})\) gravity theories admit homogeneous Gödel-type solutions for a combination of physically well-motivated matter sources. To this end, we first examine the general problem of finding out ST-homogeneous solutions in hybrid metric-Palatini \(f({\mathscr {R}})\) gravity for matter sources with constant trace *T* (scalar) of the energy-momentum tensor, and show that it reduces to to the problem of determining ST-homogeneous solutions of Einstein’s field equations with a cosmological constant determined by \(f({\mathscr {R}})\) and its first derivative \(f'({\mathscr {R}})\). Employing this far-reaching result, we determine a general ST-homogeneous Gödel-type whose matter source is a combination of a scalar with an electromagnetic fields plus a perfect fluid. This general Gödel-type solution contains special solutions in which the essential parameter \(m^2\) defines any one of the possible classified families homogeneous Gödel-type solutions, namely \(m^{2} > 0\) hyperbolic family, \(m=0\) linear class, and \(m^{2} < 0\) trigonometric family. This general homogeneous Gödel-type solution also contains previously known solutions as special cases. There emerges from one of the particular solution of the hyperbolic family that every perfect-fluid Gödel-type solution of any \(f({\mathscr {R}})\) gravity with density \(\rho \) and pressure *p* and satisfying the weak energy conditions \(\rho >0 \; \text{ and }\; \rho +p \ge 0\) is necessarily isometric to the Gödel geometry.^{1}

The bare existence of these noncausal Gödel-type solutions makes apparent that hybrid metric-Palatini \(f({\mathscr {R}})\) gravity does not remedy causal anomaly in the form of closed timelike curves that are permitted in general relativity.

The structure of the paper is as follows. In Sect. 2 we give a brief account of the hybrid metric-Palatini \(f({{\mathscr {R}}})\) gravity theories. In Sect. 3 we present the basic properties of homogenous Gödel-type geometries. This includes the conditions for space-time homogeneity, a classification of ST-homogeneous Gödel-type geometries and a study of the existence of closed time-like curves in all ST-homogeneous Gödel-type metrics. In Sect. 4 we first examine the problem of finding out ST-homogeneous solutions in \(f({\mathscr {R}})\) gravity whose trace *T* of the energy-momentum tensor of the matter source is constant, and show that in such cases the problem reduces to that of finding out solutions of Einstein’s field equations with a cosmological constant. We then show that the hybrid metric-Palatini \(f({\mathscr {R}})\) gravity theories admit ST-homogeneous Gödel-type solutions for a general combination of physically well-motivated matter contents.

## 2 Hybrid metric-Palatini gravity

*g*is the determinant of the metric tensor \(g_{\mu \nu }\,\),

*R*is the Ricci scalar associated to the Levi-Civita connection of the metric \(g_{\mu \nu }\), \({\mathscr {L}}_{m}\) is the Lagragian density for the matter fields, and the extra term \(f({{\mathscr {R}}})\) is a function of Palatini curvature scalar \({{\mathscr {R}}}\), which depends on the metric and on an independent connection \(\varGamma _{\mu \nu }^{\rho }\) through

*X*when the \(f({\mathscr {R}})\) is given as an analytic expression. Finally, we note that the variable

*X*measures the deviation from the general relativity trace equation \(R=-\kappa ^2T\).

## 3 Homogeneous Gödel-type geometries

To make this work clear and to a certain extent self-contained, in this section we present the basic properties of homogenous Gödel-type geometries, which we use in the following sections. To this end, we first discuss the conditions for space-time homogeneity (ST-homogeneity) of these space-times, and present all non-isometric ST-homogeneous Gödel-type classes. These ST-homogeneity conditions along with the set of isometrically non-equivalent geometries are important in the determination of ST-homogeneous Gödel-type solutions, in Sect. 4, to the hybrid metric-Palatini \(f({\mathscr {R}})\) field equations. Second, we discuss the existence of closed time-like curves in the metrics of these classes. The existence of these non-causal curves are crucial to examine whether hybrid metric-Palatini \(f({\mathscr {R}})\) gravities allow violation of causality of Gödel-type.

### 3.1 Homogeneity and non-equivalent metrics

*r*, and the parameters \((\varOmega ,m)\) are constants such that \(\varOmega ^{2}>0\) and \(-\infty \le m^{2}\le \infty \).

As a matter of fact, except for the case \(m^2= 4 \omega ^2\) all locally ST-homogeneous Gödel-type space-times admit a group \(G_{5}\) of isometries acting transitively on the whole space-time [86]. The special case \(m^2= 4 \omega ^2\) admits a \(G_{7}\) of isometries [86, 87].

- (i)
- Hyperbolic, in which \(m^{2} = \text{ const } > 0\) and$$\begin{aligned} H =\frac{4\,\omega }{m^{2}} \,\sinh ^2 \left( \frac{mr}{2}\right) , \quad D=\frac{1}{m} \sinh \,(mr); \end{aligned}$$(11)
- (ii)
- Linear, in which \(m=0\) and$$\begin{aligned} H = \omega r^{2}, \quad D=r, \end{aligned}$$(12)
- (iii)
- Trigonometric, where \(m^{2}= \text{ const } \equiv - \mu ^{2} < 0 \) and$$\begin{aligned} H = \frac{4\,\omega }{\mu ^{2}} \,\sin ^2 \left( \frac{\mu r}{2}\right) , \quad D=\frac{1}{\mu } \sin \,(\mu r). \end{aligned}$$(13)

^{2}In this way, to determine whether hybrid metric-Palatini \(f({\mathscr {R}})\) gravity allows Gödel-type solutions is to find out whether its field equations can be used to specify a pair of parameters \(m^2\) and \(\omega \) for a suitably chosen matter source.

### 3.2 Closed time-like curves

*G*(

*r*). Indeed, from Eq. (14) one has that the circles, hereafter called Gödel’s circles, defined by \(t, z, r = \text {const}\) become closed timelike curves whenever \(G(r) < 0\).

^{3}

*G*(

*r*) has an infinite sequence of zeros. Thus, in the section \(t, z, r = \text {const}\), there is an sequence of alternating causal [\(\,G(r) > 0\,\)] and noncausal [\(\,G(r) < 0\,\)] regions without and with noncausal circles, depending on the value of \(r=\text {const}\) (For more details see the Appendix of Ref. [75]). In this way, if \(G(r) < 0\) for a certain range of

*r*(\( r_1< r < r_2\), say) noncausal Gödel’s circles exist, whereas for

*r*in the next circular band \(r_2< r < r_3\) (say) for which \(G(r) > 0\) no such closed timelike circles exist, and so on.

To close this section, we note that in this paper by non-causal and causal solutions we mean, respectively, solutions with and without violation of causality of Gödel-type, i.e., with and without Gödel’s circles.

## 4 Solutions in hybrid metric-Palatini gravity

The aim of this section is twofold. First, we examine the problem of finding out ST-homogeneous solutions in hybrid metric-Palatini \(f({\mathscr {R}})\) gravity whose trace *T* of the energy-momentum tensor of the matter source is constant. We show that in such cases the problem of finding out solutions in the hybrid metric-Palatini \(f({\mathscr {R}})\) gravity reduces to the problem of determining ST-homogeneous solutions of Einstein’s field equations with a cosmological constant determined by \(f({\mathscr {R}})\) and its first derivative \(F = f'({\mathscr {R}})\). Second, we examine whether hybrid metric-Palatini \(f({\mathscr {R}})\) field equations admit ST-homogeneous Gödel-type solutions for a combination of a scalar field with an electromagnetic field plus a perfect fluid.

### 4.1 Field equations

*T*is also constant, which we focus in this paper, one has \(X = \text{ const }= - \kappa ^2T^{{\mathscr {R}}}\). In such cases, one can show that the field equations of the hybrid metric-Palatini gravity reduces to Einstein’s field equations with a cosmological constant.

*f*(

*R*) theories in the metric formalism, which can clearly be rewritten in the form

^{4}

### 4.2 Gödel-type solutions

In this section we discuss ST-homogeneous Gödel-type solutions in hybrid metric-Palatini \(f({\mathscr {R}})\) gravity for well-motivated matter contents whose trace of the energy-momentum tensor is constant.

*r*. Since the Lorentz frame components of the Ricci tensor depend only on \(H'/D\) and \(D''/D\), from the ST-homogeneity conditions (10) one has that for all classes of ST-homogeneous Gödel-type metrics the frame components of the Ricci tensor are constants. Thus, the Ricci scalar is also constant and given by \(R = \eta ^{CD} R_{CD}= 2 (m^2 - \omega ^2)\).

#### 4.2.1 Combined-fields general solution

In this section we take combination of scalar and electromagnetic fields with a perfect fluid as a matter source, and find a general ST-homogeneous Gödel-type solution, which contains the a perfect fluid and a scalar field particular solutions, and whose essential parameter \(m^2\) can be \(m^{2} > 0\) (hyperbolic family), \(m=0\) (linear class) or \(m^{2} < 0\) (trigonometric family) depending on the amplitude values of the matter components.

The general solution given by Eqs. (48) and (49) contains all general relativity Gödel-type known solutions [81] as particular cases. Indeed, the perfect fluid Gödel solution is recovered when \(\varepsilon = E_0 = 0\) with \(\rho ,p \ne 0 \), whereas the scalar field causal solution [81] is retrieved for \(\rho =p=E_0=0\) with \(\varepsilon \ne 0\).^{5}

Finally, from Eq. (49) one has that the combination of scalar plus electromagnetic field with a perfect fluid gives rise to ST-homogeneous Gödel-type solutions in hyperbolic class (\(m^2>0\)) when \(E_0^2 < \rho + p + 2\varepsilon ^2\), in the linear family (\(m^2=0\)) for \(E_0^2 = \rho + p + 2\varepsilon ^2\), and also in the trigonometric class (\(m^2<0\)) when \(E_0^2 > \rho + p + 2\varepsilon ^2\). Moreover, from Eq. (49) we also have that these three families of ST-homogeneous Gödel-type solutions can also be generated with a simple combination of scalar and electromagnetic fields (thus for \(\rho =p=0\)), depending on the relative values of the amplitudes \(\varepsilon ^2\) and \(E_0^2\), which give the sign of the term \(2\varepsilon ^2 - E_0^2\) in Eq. (49).

## 5 Concluding remarks

Despite the great success of the general relativity theory, a good deal of effort has recently gone into the study of the so-called modified gravity theories. In the cosmological modeling framework, this is motivated by the fact that these theories provide an alternative way to explain the current accelerating expansion of the Universe with no need to invoke the dark energy matter component. The hybrid metric-Palatini \(f({\mathscr {R}})\) gravity is a recently devised approach to such modified theories, in which it is added to the ordinary Ricci scalar *R*, in the Einstein–Hilbert Lagrangian, a function, \(f({\mathscr {R}})\), of Palatini curvature scalar \({\mathscr {R}}\), which is constructed from the independent connection \(\varGamma _{\mu \nu }^{\rho }\).

In general relativity, the bare existence of the Gödel solution to Einstein’s equations, for a physically well-motivated perfect-fluid source, makes it apparent that this theory permits solutions with violation of causality on nonlocal scale, regardless of its local Lorentzian character that ensures the local validation of the causality principle. In the context of the hybrid metric-Palatini \(f({\mathscr {R}})\) gravity theories the space-time manifolds are also assumed to be locally Lorentzian. Hence the chronology and causality structure of special relativity are inherited locally. The nonlocal question, however, is left open, and violation of causality can in principle arise.

Since homogeneous Gödel-type geometries necessarily lead to the existence of closed timelike circles (Sect. 3.2), which is an unequivocal manifestation of violation of causality, a natural way to tackle this question is by investigating whether the hybrid metric-Palatini \(f({\mathscr {R}})\) gravity theories permit Gödel-type solutions to their field equations. Furthermore, if gravity is to be governed by a \(f({\mathscr {R}})\) there are a number of issues ought to be reexamined in its context, including the question as to whether these theories admit Gödel-type solutions, or would remedy the violation of causality problem by ruling out this type of solutions, which are permitted in general relativity.

In this article, to proceed further with the investigations on the potentialities, difficulties and limitations of \(f({\mathscr {R}})\), we have examined whether \(f({\mathscr {R}})\) gravity theories admit homogeneous Gödel-type solutions for physically well-motivated matter sources. To this end, we have first examined the problem of finding out ST-homogeneous solutions in these hybrid gravity theories whose trace *T* (invariant) of the energy-momentum tensor is constant. We have shown that under this assumption the problem of finding out solutions in the hybrid metric-Palatini \(f({\mathscr {R}})\) gravity reduces to the problem of determining ST-homogeneous solutions of Einstein’s field equations with a cosmological constant. Employing this far-reaching simplifying result, we first have found a general Gödel-type solution for a source that is a combination of scalar and electromagnetic fields with a perfect fluid. In this general Gödel-type solution solution the essential parameter \(m^2\) can be \(m^{2} > 0\) (hyperbolic family), \(m=0\) (linear class) or \(m^{2} < 0\) (trigonometric family) depending on the values of the amplitudes \(\varepsilon \) (scalar field) and \(E_0\) (electromagnetic field), and the density \(\rho \) and pressure *p* of the perfect fluid. This general homogeneous Gödel-type solution 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-fluid Gödel-type solution of any \(f({\mathscr {R}})\) gravity with density \(\rho \) and pressure *p* and satisfying the weak energy conditions \(\rho >0 \; \text{ and }\; \rho +p \ge 0\) is necessarily isometric to the Gödel geometry. This extends to the context of \(f({\mathscr {R}})\) gravity a theorem established in the context of the general relativity, which states that Gödel solution is the sole perfect fluid solution of Einstein’s equations [85].

Whether or not the physical laws permit the existence of stable time machines in the form of closed timelike curves is a research field in general relativity and other gravity theories. Violation of causality on the other hand raises intriguing logical paradoxes, and is generally seen as undesirable feature in physics. The two most known remedies to these paradoxes are Novikov’s self-consistency principle [88, 89, 90], which was designed to reconcile the logical inconsistencies by demanding that the only admissible local solutions are those which are globally self-consistent,^{6} and Hawking’s chronology protection conjecture [91], which suggests that even though closed timelike curves are classically possible to be produced, quantum effects are likely to prevent such time travel. In this way, the laws of quantum physics would prevent closed timelike curves from appearing.^{7} In this regard, Hawking and Penrose have also pointed out that severe causality assumptions could risk ’ruling out something that gravity is trying to tell us’ [94], thus, discouraging further investigations. The possible existence of closed timelike curves is also particularly interesting in the quantum realm, where, for example, the quantum systems traversing these curves have been studied [95] and experimental simulation of closed timelike curves have been undertaken [96].

To conclude, we emphasize that the bare existence of the ST-homogeneous Gödel-type solutions that we have found makes apparent that the hybrid metric-Palatini \(f({\mathscr {R}})\) gravity does not remedy at a classical level the causal pathology in the form of closed timelike curves that arises in the context of general relativity. We are not aware of a quantum gravity theory following the hybrid metric-Palatini structure, though.

## Footnotes

- 1.
This extends to the context of \(f({\mathscr {R}})\) gravity a theorem which states that every perfect-fluid Gödel-type solution of Einstein’s equations is necessarily isometric to the Gödel spacetime [85].

- 2.
Gödel geometry is a solution of Eintein’s equations in general relativity, is indeed a particular case of the hyperbolic class of geometries in which \(m^2= 2 \omega ^2\).

- 3.
The only ST-homogeneous Gödel-type space-time without these noncausal circles come about when \(m^2 = 4 \omega ^2\) (see Ref. [81]). In this case, the critical radius \(r_c \rightarrow \infty \), and hence the violation of causality of Gödel type is avoided.

- 4.
Clearly, different roots

*R*give rise to different rescales of \(\kappa ^2\), and different effective cosmological constant \(\varLambda \). - 5.
We note that Gödel metric (\(m^2 = 2\omega ^2\)) can also be generated through a particular combination of perfect fluid with a scalar field, namely for non-vanishing equal amplitude of the scalar and electromagnetic fields \(\varepsilon = E_0 \ne 0\). Indeed in this case Eqs. (48) and (49) give \(m^2 = 2\omega ^2 = {\bar{\kappa }}^2 (\rho + p + \varepsilon ^2)\).

- 6.
Clearly this principle is ultimately an ad hoc global topological constraint on admissible solutions of gravity theories, thus beyond the standard scope of the local formulation of the gravity theories.

- 7.

## Notes

### Acknowledgements

M.J. Rebouças acknowledges the support of FAPERJ under a CNE E-26/202.864/2017 Grant, and thanks CNPq for the grant under which this work was carried out. J. Santos acknowledges support of Programa de Pós-Graduação em Física-CCET/UFRN.

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