# Gödel-type solutions in $\displaystyle f(R,T,R_{\mu\nu}T^{\mu\nu}$) gravity

J. S. Gonçalves<EMAIL_ADDRESS>Instituto de Física, Universidade
Federal de Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil  A. F. Santos
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal de
Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil

###### Abstract

In this paper, $\displaystyle f(R,T,R_{\mu\nu}T^{\mu\nu}$) gravity is
considered. It is a modified theory of gravity that exhibits a strong coupling
of gravitational and matter fields. Therefore, if gravity is governed by this
model a number of issues must be re-examined. In this context, the question of
causality and its violation is studied. Such analysis is carried out using the
Gödel-type solutions. It is shown that this model allows both causal and non-
causal solutions. These solutions depend directly on the content of matter
present in the universe. For the non-causal solution, a critical radius is
calculated, beyond which causality is violated. Taking different matter
contents, an infinite critical radius emerges that leads to a causal solution.
In this causal solution, a natural relationship emerges between the parameters
that determine the matter considered.

##  I Introduction

A widely accepted fact in the scientific community is the accelerated
expansion of the universe which is strongly supported by observations [1, 2,
3, 4, 5, 6, 7]. Since the General Relativity theory (GR) does not adequately
explain this phenomenon, two ways have been investigated: (i) exotic component
of the matter, called dark energy, has been considered, or (ii) alternative
models to the GR have been proposed. In this paper, the study developed
considers the second case, i.e. modified gravity theory. The $\displaystyle
f(R)$ theory is the simplest and most popular way to modify GR [8]. Several
studies have already been done on this type of theory, such as the Newtonian
limit [9], gravitational stability [10], cosmological evolution of solar-
system tests [11], inflation [12], among others. Another theory is
$\displaystyle f(R,T)$ gravity [13], which is a generalization of
$\displaystyle f(R)$ theory. In this case, the Ricci scalar $\displaystyle R$
in the Hilbert-Einstein action is changed by some function dependent on the
Ricci scalar $\displaystyle R$ and the trace of the energy-momentum tensor
$\displaystyle T$. The dependence on $\displaystyle T$ can be induced by
imperfect fluids or conformal anomalies arising from quantum effects [14, 15].
This gravitational theory has also been considered in the context of Palatini
formalism. In general, Palatini-type theories are very attractive as they
present a good analysis of the initial and current dynamics of the universe
[16, 17, 18].

In the present work, an extension of $\displaystyle f(R,T)$ gravity is
proposed. The action is described by a function $\displaystyle
f(R,T,R_{\mu\nu}T^{\mu\nu})$ where $\displaystyle R_{\mu\nu}$ and
$\displaystyle T^{\mu\nu}$ are the Ricci tensor and the energy-momentum
tensor, respectively. The stability of this theory has been investigated in
various contexts, such as, stability analysis of stellar radiating filaments
[19], stability of cylindrical stellar model [20] and stability of Einstein
Universe [21]. Furthermore, $\displaystyle f(R,T,R_{\mu\nu}T^{\mu\nu})$
gravity has already been investigated in $\displaystyle\Lambda$CDM Universe
[22], cosmic evolution in the background of non-minimal coupling [23], among
others. However, study about causality and its violation has not been
investigated in this theory. This is an important test to be performed by all
alternative gravitational theories, since GR allows for such a discussion.

To investigate the causality problem in $\displaystyle
f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity the Gödel-type metric [25] is considered.
It is a generalization of the Gödel metric proposed by Kurt Gödel, in 1949
[24]. It is the first exact solution for the GR with rotating matter. This
metric leads to the possibility of Closed Timelike Curves (CTCs). These CTCs
are not exclusive to the Gödel solution, they appear in other cosmological
models that have some hyperbolic or spherical symmetry, such as Kerr black
hole, Van-Stockum model, cosmic string, among others [26, 27]. The Gödel-type
metric provides more information about the violation of causality. It allows
the calculation of a critical radius $\displaystyle r_{c}$ that defines causal
and non-causal regions. The causality problem has already been studied in
several different models. In [28] it shows that Gödel and type-Gödel are
solutions to the $\displaystyle f(R)$ theory, as well as in [29] that both
Gödel and type-Gödel are solutions to the $\displaystyle k$-essence theory.
Causality is also discussed in Chern-Simons gravity [30, 31], $\displaystyle
f(T)$ gravity [32], $\displaystyle f(R,T)$ gravity [33], bumblebee gravity
[34], Horava-Lifshitz gravity [35], Brans-Dicke theory [36] and $\displaystyle
f(R,Q)$ gravity [37]. More recently causality is discuss in $\displaystyle
f(R,\phi,X)$ and $\displaystyle f(R,T)$ Palatini gravity [38, 39]. In this
paper, the main objective is to investigate whether the $\displaystyle
f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity allows for causality violation. The
analysis is divided into two parts, first the content of matter is just a
perfect fluid and then the content of matter is a perfect fluid plus a scalar
field.

The present paper is organized as follows. In Section II, $\displaystyle
f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity is introduced and the field equations are
derived. In Section III, the Gödel-type metric is discussed. Considering a
perfect fluid as matter content, the standard Gödel solution is obtained. In
Section IV, the problem of causality is verified for a perfect fluid plus a
scalar field as the content of matter. This content of matter leads to a
causal Gödel-type solution. In Section V, remarks and conclusions are
presented.

##  II $\displaystyle f(R,T,R_{\mu\nu}T^{\mu\nu})$ Gravity

In this section, the gravitational field equations for $\displaystyle
f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity are obtained. The action that describes
this theory is

$S=\frac{1}{2\kappa^{2}}\int
d^{4}x\sqrt{-g}\left[f(R,T,R_{\mu\nu}T^{\mu\nu})+\mathcal{L}_{m}\right],$ (1)

where $\displaystyle\kappa^{2}=8\pi G$, $\displaystyle g$ is the determinant
of the metric tensor $\displaystyle g_{\mu\nu}$, $\displaystyle R$ is the
Ricci scalar, $\displaystyle f(R,T,R_{\mu\nu}T^{\mu\nu})$ is a function that
depends on the Ricci scalar, the trace of energy-momentum and the tensor
product $\displaystyle R_{\mu\nu}T^{\mu\nu}$, respectively, and
$\displaystyle\mathcal{L}_{m}$ is the matter Lagrangian.

Varying the action (1) in relation to the metric $\displaystyle g^{\mu\nu}$
leads to

$\delta S=\frac{1}{2\kappa^{2}}\int
d^{4}x\left[\delta\sqrt{-g}f+\sqrt{-g}\left(f_{R}\delta R+f_{T}\delta
T+f_{Q}\delta R_{\mu\nu}T^{\mu\nu}\right)\right]+\int
d^{4}x\delta\left(\sqrt{-g}\mathcal{L}_{m}\right),$ (2)

with $\displaystyle f_{R}\equiv\frac{\partial f}{\partial R}$, $\displaystyle
f_{T}\equiv\frac{\partial f}{\partial T}$ and $\displaystyle
f_{Q}\equiv\frac{\partial f}{\partial Q}$, where $\displaystyle Q\equiv
R_{\mu\nu}T^{\mu\nu}$.

Considering the definition of the energy-momentum tensor, we get

$T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\partial\left(\sqrt{-g}\mathcal{L}_{m}\right)}{\partial
g^{\mu\nu}}=-2\frac{\partial\mathcal{L}_{m}}{\partial
g^{\mu\nu}}+g_{\mu\nu}\mathcal{L}_{m},$ (3)

where $\displaystyle\mathcal{L}_{m}$ is assumed to be dependent only on the
metric and not on the first derivatives. The variation in the energy-momentum
tensor trace is given as

$\displaystyle\displaystyle\delta T$
$\displaystyle\displaystyle=\delta\left(g^{\alpha\beta}T_{\alpha\beta}\right),$
(4) $\displaystyle\displaystyle=\left(\frac{\delta g^{\alpha\beta}}{\delta
g^{\mu\nu}}\right)T_{\alpha\beta}+\frac{\delta T_{\alpha\beta}}{\delta
g^{\mu\nu}}g^{\alpha\beta},$ (5)
$\displaystyle\displaystyle=\left(T_{\mu\nu}+\Theta_{\mu\nu}\right),$ (6)

with $\displaystyle\Theta_{\mu\nu}$ being a tensor defined as

$\Theta_{\mu\nu}\equiv\frac{\delta T_{\alpha\beta}}{\delta
g^{\mu\nu}}g^{\alpha\beta}.$ (7)

Using Eq. (3) this tensor can be written as

$\Theta_{\mu\nu}=-2T_{\mu\nu}+g_{\mu\nu}\mathcal{L}_{m}-2g^{\alpha\beta}\frac{\partial^{2}\mathcal{L}_{m}}{\partial
g^{\mu\nu}\partial g^{\alpha\beta}}.$ (8)

The variations $\displaystyle\delta R$ and $\displaystyle\delta T$ are known
[40]. They are given as

$f_{R}\delta R+f_{T}\delta
T=\left[R_{\mu\nu}f_{R}+\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)f_{R}+\left(T_{\mu\nu}+\Theta_{\mu\nu}\right)f_{T}\right]\delta
g^{\mu\nu}.$ (9)

The variation corresponding to the term $\displaystyle R_{\mu\nu}T^{\mu\nu}$
is composed of two parts, i.e.

$f_{Q}T^{\mu\nu}\delta
R_{\mu\nu}=\left[\frac{1}{2}\Box(f_{Q}T_{\mu\nu})+g_{\mu\nu}\nabla_{\alpha}\nabla_{\beta}(f_{Q}T^{\alpha\beta})-\nabla_{\alpha}\nabla_{\nu}(f_{Q}T^{\alpha}_{\mu})\right]$
(10)

and

$f_{Q}R_{\mu\nu}\delta
T^{\mu\nu}=f_{Q}\left[-G_{\mu\nu}\mathcal{L}_{m}-\frac{1}{2}RT_{\mu\nu}+2R^{\alpha}_{\mu}T_{\alpha\nu}-2R^{\alpha\beta}\frac{\delta^{2}\mathcal{L}_{m}}{\delta
g^{\mu\nu}\delta g^{\alpha\beta}}\right]\delta g^{\mu\nu}.$ (11)

Considering the perfect fluid as matter content, it is assumed that the
Lagrangian of the matter is $\displaystyle\mathcal{L}_{m}=-p$. This choice
implies that

$\frac{\delta^{2}\mathcal{L}_{m}}{\delta g^{\mu\nu}\delta g^{\alpha\beta}}=0.$
(12)

Then the tensor $\displaystyle\Theta_{\mu\nu}$ becomes

$\Theta_{\mu\nu}=-2T_{\mu\nu}-pg_{\mu\nu}.$ (13)

Taking these results to Eq. (2), the complete set of the field equations is
given by

$\displaystyle\displaystyle
R_{\mu\nu}f_{R}-\frac{1}{2}g_{\mu\nu}f+\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)f_{R}+f_{T}\left(T_{\mu\nu}+\Theta_{\mu\nu}\right)$
$\displaystyle\displaystyle+\frac{1}{2}\left(\Box(T_{\mu\nu}f_{Q})+g_{\mu\nu}\nabla_{\alpha}\nabla_{\beta}(T^{\alpha\beta}f_{Q})\right)-\nabla_{\alpha}(\nabla_{(\mu}T^{\alpha}_{\nu)}f_{Q})+\Xi_{\mu\nu}f_{Q}=\kappa^{2}T_{\mu\nu},$
(14)

with

$\Xi_{\mu\nu}=-G_{\mu\nu}p-\frac{1}{2}RT_{\mu\nu}+2R^{\alpha}_{(\mu}T_{\nu)\alpha}.$
(15)

In the next section, these field equations will be studied considering the
cosmological background described by the Gödel-type solutions.

##  III Gödel-type Metric and Perfect Fluid

Here the modified Einstein equations are solved considering the Gödel-type
metric and a perfect fluid as the content of matter. Before solving the field
equations obtained in the previous section, the main characteristics of this
solution are presented.

The Gödel-type solution is described by the line element

$ds^{2}=-dt^{2}-2H(r)dtd\phi+dr^{2}+\left(D^{2}(r)-H^{2}(r)\right)d\phi^{2}+dz^{2},$
(16)

where $\displaystyle H(r)$ and $\displaystyle D(r)$ are functions that
satisfies the relations

$\displaystyle\displaystyle\frac{H^{\prime}(r)}{D(r)}=2\omega\quad\mathrm{and}\quad\frac{D^{\prime\prime}(r)}{D(r)}=m^{2}.$
(17)

The prime implies the derivative with respect to $\displaystyle r$. It is
important to note that $\displaystyle\omega$ and $\displaystyle m^{2}$ are
free parameters that characterize all properties of the metric [25, 41]. This
is a generalization of the solution proposed by Gödel. The Gödel metric is an
exact solution of Einstein equations for a homogeneous rotating universe. This
solution leads to the possibility of Closed Time-like Curves (CTCs) that allow
violation of causality. As an immediate consequence, this makes time travel
theoretically possible in this space-time. For a more detailed investigation
of causality, as well as its violation, the Gödel-type solution is used. From
the parameters $\displaystyle\omega$ and $\displaystyle m^{2}$ three different
classes are defined, i.e. hyperbolic, trigonometric and linear class [25].
Here only the hyperbolic class is considered. For this class, the functions
$\displaystyle H(r)$ and $\displaystyle D(r)$ are given as

$\displaystyle\displaystyle H(r)$
$\displaystyle\displaystyle=\frac{4\omega}{m^{2}}senh^{2}\left(\frac{mr}{2}\right),$
(18) $\displaystyle\displaystyle D(r)$
$\displaystyle\displaystyle=\frac{1}{m}senh(mr).$ (19)

An important quantity that is calculated from Gödel-type solution is the
critical radius, beyond which CTCs exist. It is defined as

$r_{c}=\frac{2}{m}senh^{-1}\left(\frac{4\omega^{2}}{m^{2}}-1\right)^{-1}.$
(20)

From this, two relationships between the parameters $\displaystyle\omega^{2}$
and $\displaystyle m^{2}$ are interesting. (i) $\displaystyle
m^{2}=2\omega^{2}$ that leads to the Gödel solution with a finite critical
radius given by

$r_{c}=\frac{2}{m}senh^{-1}\left(1\right).$ (21)

As a consequence, non-causal regions are allowed. And (ii) $\displaystyle
m^{2}=4\omega^{2}$ implies an infinite critical radius. In other words, this
condition leads to a causal solution.

For simplicity, a new basis (a local Lorentz co-frame) is chosen, such that
the metric becomes [25],

$ds^{2}=\eta_{AB}\theta^{A}\theta^{B}=(\theta^{(0)})^{2}-(\theta^{(1)})^{2}-(\theta^{(2)})^{2}-(\theta^{(3)})^{2},$
(22)

where $\displaystyle\eta_{AB}$ is the Minkowski metric and
$\displaystyle\theta^{A}=e^{A}\ _{\mu}\ dx^{\mu}$ with

$\displaystyle\displaystyle\theta^{(0)}$
$\displaystyle\displaystyle=dt+H(r)d\phi,$
$\displaystyle\displaystyle\theta^{(1)}$ $\displaystyle\displaystyle=dr,$
$\displaystyle\displaystyle\theta^{(2)}$
$\displaystyle\displaystyle=D(r)d\phi,$
$\displaystyle\displaystyle\theta^{(3)}$ $\displaystyle\displaystyle=dz,$ (23)

and $\displaystyle e^{A}\ _{\mu}$ is the tetrad that satisfies the relation
$\displaystyle e^{A}\,_{\mu}e^{\mu}\,_{B}=\delta^{A}_{B}$. Here the capital
Latin letters label Lorentz indices and run from 0 to 3.

In the local Lorentz co-frame (23), the non-zero Ricci tensor components are

$\displaystyle\displaystyle R_{(0)(0)}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{2}\left(\frac{H^{\prime}}{D}\right)^{2},$
$\displaystyle\displaystyle R_{(1)(1)}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle R_{(2)(2)}=R_{(0)(0)}-\frac{D^{\prime\prime}}{D}.$
(24)

It is interesting to note that all components are constants, i.e.

$R_{(0)(0)}=2\omega^{2},\ R_{(1)(1)}=\ R_{(2)(2)}=2\omega^{2}-m^{2}.$ (25)

The non-vanishing components of the Einstein tensor in flat (local) space-time
take the form,

$\displaystyle\displaystyle G_{(0)(0)}$
$\displaystyle\displaystyle=3\omega^{2}-m^{2},$ (26)
$\displaystyle\displaystyle G_{(1)(1)}$
$\displaystyle\displaystyle=\omega^{2},$ (27) $\displaystyle\displaystyle
G_{(2)(2)}$ $\displaystyle\displaystyle=\omega^{2},$ (28)
$\displaystyle\displaystyle G_{(3)(3)}$
$\displaystyle\displaystyle=m^{2}-\omega^{2},$ (29)

where $\displaystyle G_{AB}=e^{\mu}_{A}e^{\nu}_{B}G_{\mu\nu}$ has been used.
The Ricci scalar is $\displaystyle R=2(m^{2}-w^{2})$.

Assuming that the content of matter is a perfect fluid, the energy-momentum
tensor that describes it is

$\displaystyle\displaystyle T_{AB}=(\rho+p)u_{A}u_{B}-p\eta_{AB},$ (30)

with $\displaystyle u_{A}=(1,0,0,0)$ being the four-velocity of fluid. Thus,
in flat space-time the tensor given in Eq. (13) becomes

$\displaystyle\displaystyle\Theta_{AB}=-2T_{AB}-p\eta_{AB}.$ (31)

Taking the Minkowski metric as

$\eta_{AB}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{array}\right),$ (32)

the components of the tensors (30) and (31) are

$T_{AB}=\left(\begin{array}[]{cccc}\rho&0&0&0\\\ 0&p&0&0\\\ 0&0&p&0\\\
0&0&0&p\end{array}\right);\ \
\Theta_{AB}=\left(\begin{array}[]{cccc}-2\rho-p&0&0&0\\\ 0&-p&0&0\\\
0&0&-p&0\\\ 0&0&0&-p\end{array}\right).$ (33)

The trace of $\displaystyle T_{AB}$ and $\displaystyle\Theta_{AB}$ are
$\displaystyle T=\rho-3p$ and $\displaystyle\Theta=2(p-\rho)$. Using the
components of the energy-momentum tensor and the Ricci tensor, the components
of $\displaystyle Q=R_{AB}T^{AB}$ are given as

$Q=\left(\begin{matrix}2\omega^{2}\rho&0&0&0\\\
0&\left(2\omega^{2}-m^{2}\right)p&0&0\\\
0&0&\left(2\omega^{2}-m^{2}\right)p&0\\\ 0&0&0&0\end{matrix}\right).$ (34)

It should be noted that the derivatives of the functions $\displaystyle f_{R}$
and $\displaystyle f_{Q}$ vanish since the Ricci scalar and $\displaystyle Q$
are constants. Then the field equations (14) are rewritten as

$\displaystyle\displaystyle
R_{AB}f_{R}=\kappa^{2}T_{AB}+\frac{1}{2}\eta_{AB}f-f_{T}\left(T_{AB}+\Theta_{AB}\right)-\Xi_{AB}f_{Q}.$
(35)

Taking the trace of this equation leads to

$Rf_{R}=\kappa^{2}T+2f-f_{T}\left(T+\Theta\right)-\Xi f_{Q},$ (36)

where $\displaystyle\Xi=\eta^{AB}\Xi_{AB}$. Combining these equations we have

$f_{R}G_{AB}=\kappa^{2}T_{AB}-f_{T}\left(T_{AB}+\Theta_{AB}\right)-\Xi_{AB}f_{Q}-\frac{1}{2}\eta_{AB}\left[f+\kappa^{2}T-f_{T}(T+\Theta)-\Xi
f_{Q}\right]$ (37)

with $\displaystyle G_{AB}=R_{AB}-\frac{1}{2}R\eta_{AB}$ being the Einstein
tensor. In order to solve the set of field equations, let us explicitly write
Eq. (15) in flat space-time. Then

$\Xi_{AB}=\left(\begin{matrix}-(3\omega^{2}-m^{2})p-(m^{2}-5\omega^{2})\rho&0&0&0\\\
0&\left(m^{2}-4\omega^{2}\right)p&0&0\\\
0&0&\left(m^{2}-4\omega^{2}\right)p&0\\\
0&0&0&-2\left(m^{2}-\omega^{2}\right)p\end{matrix}\right).$ (38)

As a consequence
$\displaystyle\Xi=\left(m^{2}+3\omega^{2}\right)p-\left(m^{2}-5\omega^{2}\right)\rho$.

Using these results the field equations (37) become

$\displaystyle\displaystyle 2f_{R}\left(3\omega^{2}-m^{2}\right)+f$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\kappa^{2}\left(\rho+3p\right)+f_{T}\left(\rho+p\right)+f_{Q}\left[\left(9\omega^{2}-m^{2}\right)p+\left(m^{2}-5\omega^{2}\right)\rho\right],$
(39) $\displaystyle\displaystyle 2f_{R}\omega^{2}-f$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\kappa^{2}\left(\rho-p\right)+f_{T}\left(\rho+p\right)+f_{Q}\left[\left(5\omega^{2}-3m^{2}\right)p+\left(m^{2}-5\omega^{2}\right)\rho\right],$
(40) $\displaystyle\displaystyle 2f_{R}\left(m^{2}-\omega^{2}\right)-f$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\kappa^{2}\left(\rho-p\right)+f_{T}\left(\rho+p\right)+f_{Q}\left[\left(3m^{2}-7\omega^{2}\right)p+\left(m^{2}-5\omega^{2}\right)\rho\right].$
(41)

Equations (40) and (41) lead to

$\displaystyle\displaystyle\left(2\omega^{2}-m^{2}\right)\left(2f_{R}-6pf_{Q}\right)=0.$
(42)

Assuming $\displaystyle f_{R}>0$ and $\displaystyle f_{Q}>0$, Eq. (42) gives
us

$m^{2}=2\omega^{2}.$ (43)

This condition defines the Gödel solution, and the remaining field equations
read

$\displaystyle\displaystyle m^{2}f_{R}+f$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\kappa^{2}\left(\rho+3p\right)+f_{T}\left(\rho+p\right)+f_{Q}\left(7p-3\rho\right)\frac{m^{2}}{2},$
(44) $\displaystyle\displaystyle m^{2}f_{R}-f$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\kappa^{2}\left(\rho-p\right)+f_{T}\left(\rho+p\right)-f_{Q}\left(3p+9\rho\right)\frac{m^{2}}{8}.$
(45)

Since these equations allow the Gödel solution, a critical radius, which
defines regions where causality is violated, can be calculated. Then the
critical radius in $\displaystyle f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity is

$\displaystyle\displaystyle
r_{c}=2\sinh^{-1}(1)\sqrt{\frac{2f_{R}-\frac{f_{Q}}{16}\left(19p-39\rho\right)}{2\kappa^{2}\rho+f+2f_{T}(\rho+p)}}.$
(46)

Note that the $\displaystyle r_{c}$ depends on the gravity theory and the
content of matter. It is important to emphasize that this quantity is obtained
for any $\displaystyle f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity. A similar result
is obtained for other gravity theories such as $\displaystyle f(R)$ and
$\displaystyle f(R,T)$.

It is shown that for a perfect fluid as the content of matter non-causal Gödel
curves are unavoidable. From this result, a natural question arises: is there
any condition that leads to a causal Gödel-type solution in this gravitational
theory? In the next section, such an investigation is developed.

##  IV Matter content: Perfect fluid and scalar field

Here the content of matter is a combination of perfect fluid and scalar field
[43, 42]. The main idea is to investigate the possibility of finding a causal
solution in this theory. The total energy-momentum tensor that describes this
combination is given as

$\displaystyle\displaystyle T_{AB}=T_{AB}^{M}+T_{AB}^{S},$ (47)

where $\displaystyle T_{AB}^{M}$ and $\displaystyle T_{AB}^{S}$ are the
energy-momentum tensor of the perfect fluid and scalar field, respectively.
The Lagrangian for the scalar field $\displaystyle\Phi$ is

$\mathcal{L}^{S}=\eta^{AB}\nabla_{A}\Phi\nabla_{B}\Phi,$ (48)

and the corresponding energy-momentum tensor reads

$\displaystyle\displaystyle
T_{AB}^{S}=\nabla_{A}\phi\nabla_{B}\phi-\frac{1}{2}\eta_{AB}\eta^{CD}\nabla_{C}\phi\nabla_{D}\phi.$
(49)

The energy-momentum tensor for the perfect fluid is given in Eq. (30). Then
the total energy-momentum tensor is

$\displaystyle\displaystyle
T_{AB}=\left(\rho+p\right)u_{A}u_{B}-p\eta_{AB}+\nabla_{A}\phi\nabla_{B}\phi-\frac{1}{2}\eta_{AB}\eta^{CD}\nabla_{C}\phi\nabla_{D}\phi.$
(50)

The components of the energy-momentum tensor of matter are given in Eq. (33),
while to write the components associated with the scalar field it is
considered $\displaystyle\Phi=\epsilon z+\epsilon$, with
$\displaystyle\epsilon=const$. Then

$T_{00}^{S}=-T_{11}^{S}=-T_{22}^{S}=T_{33}^{S}=\frac{\epsilon^{2}}{2}.$ (51)

The trace of $\displaystyle T_{AB}$ is given by

$T=\rho-3p+\epsilon^{2}.$ (52)

In the same way, the tensor $\displaystyle\Theta_{AB}$ must be rewritten
considering the contributions of the scalar field. Thus

$\Theta_{AB}=\Theta_{AB}^{M}+\Theta_{AB}^{S},$ (53)

where $\displaystyle\Theta_{AB}^{M}$ is already known in Eq. (33) and
$\displaystyle\Theta_{AB}^{S}$ is given as

$\Theta_{AB}^{S}=-T_{AB}^{S}+\frac{1}{2}T^{S}\eta_{AB},$ (54)

with $\displaystyle T^{S}$ being the trace of $\displaystyle T_{AB}^{S}$.

Using the total Lagrangian

$\mathcal{L}_{m}=\mathcal{L}^{PF}+\mathcal{L}^{S}=p-\epsilon^{2},$ (55)

where $\displaystyle\mathcal{L}^{PF}$ and $\displaystyle\mathcal{L}^{S}$ are
the Lagrangians of the perfect fluid and the scalar field, respectively, the
tensor $\displaystyle\Xi_{AB}$ becomes

$\Xi_{AB}=-G_{AB}p+G_{AB}\epsilon^{2}-\frac{1}{2}RT_{AB}+\eta^{CD}R_{AD}T_{BC}+\eta^{CD}R_{BD}T_{AC}.$
(56)

With these results, the field equations (37) are written as

$\displaystyle\displaystyle f_{R}G_{AB}$
$\displaystyle\displaystyle=\kappa^{2}\left[\left(\rho+p\right)u_{A}u_{B}-p\eta_{AB}+T_{AB}^{S}\right]-\Xi_{AB}f_{Q}$
(57)
$\displaystyle\displaystyle-\frac{1}{2}\left[\kappa^{2}\left(\rho-3p+\epsilon^{2}\right)+f+f_{T}\left(\rho+p-2\epsilon^{2}\right)-\Xi
f_{Q}\right]\eta_{AB}$ (58)
$\displaystyle\displaystyle+f_{T}\left[\left(\rho+p\right)u_{A}u_{B}-\frac{1}{2}\epsilon^{2}\eta_{AB}\right].$
(59)

This equation leads to the following set of field equations

$\displaystyle\displaystyle 2f_{R}\left(3\omega^{2}-m^{2}\right)+f$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\kappa^{2}\left(\rho+3p\right)+f_{T}\left(\rho+p+\epsilon^{2}\right)$
(60)
$\displaystyle\displaystyle+f_{Q}\left[\left(9\omega^{2}-m^{2}\right)p+\left(m^{2}-5\omega^{2}\right)\rho-2\omega^{2}\epsilon^{2}\right],$
$\displaystyle\displaystyle 2f_{R}\omega^{2}-f$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\kappa^{2}\left(\rho-p\right)+f_{T}\left(\rho+p-\epsilon^{2}\right)$
(61)
$\displaystyle\displaystyle+f_{Q}\left[\left(-3m^{2}+5\omega^{2}\right)p+\left(m^{2}-5\omega^{2}\right)\rho+\left(m^{2}-2\omega^{2}\right)\epsilon^{2}\right],$
$\displaystyle\displaystyle 2\left(m^{2}-\omega^{2}\right)f_{R}-f$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\kappa^{2}\left(\rho-p+2\epsilon^{2}\right)+f_{T}\left(\rho+p-\epsilon^{2}\right)$
(62)
$\displaystyle\displaystyle+f_{Q}\left[\left(3m^{2}-7\omega^{2}\right)p+\left(m^{2}-5\omega^{2}\right)\rho\right].$

For simplicity, after some manipulation, these equations are rewritten as

$\displaystyle\displaystyle\kappa^{2}\epsilon^{2}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\left(m^{2}-2\omega^{2}\right)f_{R}+f_{Q}\left[-3p\left(m^{2}-\omega^{2}\right)+\frac{\epsilon^{2}}{2}\left(m^{2}-2\omega^{2}\right)\right],$
(63) $\displaystyle\displaystyle\kappa^{2}p+\frac{1}{2}f_{T}\epsilon^{2}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{2}\left(2\omega^{2}-m^{2}\right)f_{R}+\frac{1}{2}f+\frac{f_{Q}}{4}\left[m^{2}\epsilon^{2}-\left(4\omega^{2}+2m^{2}\right)p\right],$
(64)
$\displaystyle\displaystyle\kappa^{2}\rho+f_{T}\left(\rho+p-\frac{\epsilon^{2}}{2}\right)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{2}\left(6\omega^{2}-m^{2}\right)f_{R}-\frac{1}{2}f+f_{Q}\left[2\left(5\omega^{2}-2m^{2}\right)p-\frac{\epsilon^{2}}{2}\left(m^{2}+2\omega^{2}\right)\right].$
(65)

Assuming that these equations satisfy the conditions $\displaystyle f_{R}>0$,
$\displaystyle f_{T}>0$ and $\displaystyle f_{Q}>0$ implies that the set of
equations satisfy a causal solution, i.e.

$m^{2}=4\omega^{2}.$ (66)

As a consequence of Eq. (66) the critical radius goes to infinity, i.e.
$\displaystyle r_{c}\rightarrow\infty$. Therefore, for this combination of
perfect fluid and scalar field as matter content, a causal Gödel-type solution
is allowed in $\displaystyle f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity. In
addition, due to the consistency of the equations, a relationship between the
constant $\displaystyle\epsilon$ associated with the scalar field and the
pressure $\displaystyle p$ of the perfect fluid is required.

It is important to say that by making $\displaystyle f_{Q}$ go to zero in the
field equations, the results already obtained for the $\displaystyle f(R,T)$
theory are recovered. This reinforces the principle of correspondence between
theories.

##  V Discussions and Final Remarks

In this paper, $\displaystyle f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity is
considered and the causality violation is investigated using the Gödel-type
metric. It was shown that this theory generalizes GR implying a geometry-
matter coupling. An important question that must be answered by all modified
theories is: does the new theory contain all solutions of the GR? In order to
answer this question, this work investigated whether an exact solution of GR,
such as Gödel metric or Gödel-type metric, holds in $\displaystyle
f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity.

Considering the hyperbolic class of Gödel-type solution and the perfect fluid
as matter content, the non-causal Gödel solution is obtained. From this, a
finite critical radius is calculated that leads to the possible existence of
CTCs. In this case, the critical radius is a function that depends on the
function $\displaystyle f(R,T,R_{\mu\nu}T^{\mu\nu})$ and the derivatives of
its variables, $\displaystyle f_{R}$, $\displaystyle f_{T}$ and $\displaystyle
f_{Q}$. To investigate causal solutions in this gravitational theory,
different content of matter is considered, i.e., a combination between perfect
fluid and scalar field. With these considerations, a condition emerges from
the set of field equations that implies an infinite critical radius. Thus, the
violation of causality is avoided in this theory. Furthermore, the result
presented here is a generalization of the results obtained for $\displaystyle
f(R)$ and $\displaystyle f(R,T)$ gravities. However, a different condition
arises for the causal solution in $\displaystyle f(R,T,R_{\mu\nu}T^{\mu\nu})$
gravity, namely, there is a certain connection between the contents of matter,
i.e., the perfect fluid pressure and the scalar field are related.

## Acknowledgments

This work by A. F. S. is partially supported by National Council for
Scientific and Technological Development - CNPq project No. 313400/2020-2. J.
S. Gonçalves thanks CAPES for financial support.

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