Gödel-type solutions in fR T RT gravity J. S. Gonçalves 1and A. F. Santos

2025-05-06 0 0 404.97KB 13 页 10玖币
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Gödel-type solutions in f(R, T, Rµν Tµν ) gravity
J. S. Gonçalves 1, ∗and A. F. Santos 1, †
1Instituto de Física, Universidade Federal de Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil
In this paper, f(R, T, Rµν Tµν ) 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 [1–7]. 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 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 f(R, T )gravity [13], which is a generalization of f(R)theory. In this case, the Ricci scalar R
in the Hilbert-Einstein action is changed by some function dependent on the Ricci scalar Rand
the trace of the energy-momentum tensor T. The dependence on Tcan 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
∗junior@fisica.ufmt.br
†alesandroferreira@fisica.ufmt.br
arXiv:2210.13251v1 [gr-qc] 24 Oct 2022
2
[16–18].
In the present work, an extension of f(R, T )gravity is proposed. The action is described by
a function f(R, T, Rµν Tµν )where Rµν and Tµν 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, f(R, T, Rµν Tµν )gravity has already been
investigated in ΛCDM 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 f(R, T, Rµν Tµν )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 rc
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 f(R)
theory, as well as in [29] that both Gödel and type-Gödel are solutions to the k-essence theory.
Causality is also discussed in Chern-Simons gravity [30,31], f(T)gravity [32], f(R, T )gravity [33],
bumblebee gravity [34], Horava-Lifshitz gravity [35], Brans-Dicke theory [36] and f(R, Q)gravity
[37]. More recently causality is discuss in f(R, φ, X)and f(R, T )Palatini gravity [38,39]. In this
paper, the main objective is to investigate whether the f(R, T, Rµν Tµν )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, f(R, T, Rµν Tµν )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.
3
II. f(R, T, Rµν Tµν )GRAVITY
In this section, the gravitational field equations for f(R, T, Rµν Tµν )gravity are obtained. The
action that describes this theory is
S=1
2κ2Zd4x√−g[f(R, T, Rµν Tµν ) + Lm],(1)
where κ2= 8πG,gis the determinant of the metric tensor gµν ,Ris the Ricci scalar, f(R, T, Rµν Tµν )
is a function that depends on the Ricci scalar, the trace of energy-momentum and the tensor product
Rµν Tµν , respectively, and Lmis the matter Lagrangian.
Varying the action (1) in relation to the metric gµν leads to
δS =1
2κ2Zd4xδ√−gf +√−g(fRδR +fTδT +fQδRµν Tµν )+Zd4xδ √−gLm,(2)
with fR≡∂f
∂R ,fT≡∂f
∂T and fQ≡∂f
∂Q , where Q≡Rµν Tµν .
Considering the definition of the energy-momentum tensor, we get
Tµν =−2
√−g
∂(√−gLm)
∂gµν =−2∂Lm
∂gµν +gµν Lm,(3)
where Lmis 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
δT =δgαβ Tαβ ,(4)
=δgαβ
δgµν Tαβ +δTαβ
δgµν gαβ ,(5)
= (Tµν +Θµν ),(6)
with Θµν being a tensor defined as
Θµν ≡δTαβ
δgµν gαβ .(7)
Using Eq. (3) this tensor can be written as
Θµν =−2Tµν +gµν Lm−2gαβ ∂2Lm
∂gµν ∂gαβ .(8)
The variations δR and δT are known [40]. They are given as
fRδR +fTδT = [Rµν fR+ (gµν 2− ∇µ∇ν)fR+ (Tµν +Θµν )fT]δgµν .(9)
The variation corresponding to the term Rµν Tµν is composed of two parts, i.e.
fQTµν δRµν =1
22(fQTµν ) + gµν ∇α∇β(fQTαβ)− ∇α∇ν(fQTα
µ)(10)
摘要：

Gödel-typesolutionsinf(R;T;RT)gravityJ.S.Gonçalves1,andA.F.Santos1,y1InstitutodeFísica,UniversidadeFederaldeMatoGrosso,78060-900,Cuiabá,MatoGrosso,BrazilInthispaper,f(R;T;RT)gravityisconsidered.Itisamodiedtheoryofgravitythatexhibitsastrongcouplingofgravitationalandmatterelds.Therefore,ifg...

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