## ORIGINAL RESEARCH article

Front. Phys., 03 October 2022
Sec. Statistical and Computational Physics
Volume 9 - 2021 | https://doi.org/10.3389/fphy.2021.800060

# Spacetimes Admitting Concircular Curvature Tensor in f(R) Gravity

• 1Department of Pure Mathematics, University of Calcutta, West Bengal, India
• 2Basic Science Department, Modern Academy for Engineering and Technology, Maadi, Egypt
• 3Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
• 4Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia
• 5Applied Science College, Department of Mathematical Sciences, Umm Al-Qura University, Makkah, Saudi Arabia

The main object of this paper is to investigate spacetimes admitting concircular curvature tensor in f(R) gravity theory. At first, concircularly flat and concircularly flat perfect fluid spacetimes in $fR$ gravity are studied. In this case, the forms of the isotropic pressure p and the energy density σ are obtained. Next, some energy conditions are considered. Finally, perfect fluid spacetimes with divergence free concircular curvature tensor in f(R) gravity are studied; amongst many results, it is proved that if the energy-momentum tensor of such spacetimes is recurrent or bi-recurrent, then the Ricci tensor is semi-symmetric and hence these spacetimes either represent inflation or their isotropic pressure and energy density are constants.

## 1 Introduction

A concircular transformation was first coined by Yano in 1940 [1]. Such a transformation preserves geodesic circles. The geometry that deals with a concircular transformation is called concircular geometry. Under concircular transformation the concircular curvature tensor $M$ remains invariant. Every spacetime M has vanishing concircular curvature tensor is called concirculary flat. A concircularly flat spacetime is of constant curvature. As a result, the deviation of a spacetime from constant curvature is measured by the concircular curvature tensor $M$. Researchers have shown the curial role of the concircular curvature tensor in mathematics and physics (for example, see [26] and references therein).

In Einstein’s theory of gravity, the relation between the matter of spacetimes and the geometry of the spacetimes is given by Einstein’s field equations (EFE)

$Rij−R2gij=κTij,$

with κ being the Newtonian constant and $Tij$ is the energy-momentum tensor [7]. These equations imply that the energy-momentum tensor Tij is divergence-free. This condition is satisfied whenever ∇lTij = 0, where ∇l denotes the covariant differentiation. There are many modifications of the standard relativity theory. The $fR$ gravity theory is the most popular of such modification of the standard theory of gravity. This important modification was first introduced in 1970 [8]. This modified theory can be obtained by replacing the scalar curvature R with a generic function $fR$ in the Einstein-Hilbert action. The field equations of $fR$ gravity are given as

$κTij=f′RRij−f′′Rf∇iR∇jR−f′′R∇i∇jR+gijf′′′R∇kR∇kR+f′′R∇2R−12fR,$

where $fR$ is an arbitrary function of the scalar curvature R and $f′R=dfdR$ which must be positive to ensure attractive gravity [9]. The f(R) gravity represents a higher order and well-studied theory of gravity. For example, an earlier investigation of quintessence and cosmic acceleration in $fR$ gravity theory as a higher order gravity theory are considered in [10]. Also, Capoziello et al. proved that, in a generalized Robertson-Walker spacetime with divergence free conformal curvature tensor, the higher order gravity tensor has the form of perfect fluid [11].

In a series of recent studies, weakly Ricci symmetric spacetimes (WRS)4, almost pseudo-Ricci symmetric spacetimes(APRS)4, and conformally flat generalized Ricci recurrent spacetimes are investigated in $fR$ gravity theory [1214]. Motivated by these studies and many others, the main aim of this paper is to study concircularly flat and concircularly flat perfect fluid spacetimes in $fR$ gravity. Also, spacetimes with divergence free concircular curvature tensor in $fR$ gravity are considered.

This article is organized as follows. In Section 2, concircularly flat spacetimes in f(R) gravity are considered. In Section 3, we study concircularly flat perfect fluid spacetimes in f(R) gravity as well as we consider some energy conditions. Finally, spacetimes with divergence free concircular curvature tensor in f(R) gravity are investigated.

## 2 Concircularly Flat Spacetimes in f(R) Gravity

The concircular curvature tensor of type $0,4$ is defined locally as

$Mjklm=Rjklm−Rnn−1gklgjm−gkmgjl,$

where $Rjklm$, R, and gkl are the Riemann curvature tensor, the scalar curvature tensor, and the metric tensor [1].

Here, we will consider $Mjklm=0$, thus it follows form Eq. 2.1 that

$Rjklm=Rnn−1gklgjm−gkmgjl.$

This equation leads us to state the following theorem:

Theorem 1. A concircularly flat spacetime is of constant curvature.

Corollary 1. A concircularly flat spacetime is of constant scalar curvature.Contracting Eq. 2.2 with gjm, we get

$Rkl=Rnglk.$

In view of Eq. 2.3 we can state the following corollary:

Corollary 2. A concircularly flat spacetime is Einstein.In view of corollary 1, the field Eq. 1.1 in $fR$ gravity become

$Rij−f2f′gij=κf′Tij.$

In vacuum case, we have

$Rij−f2f′gij=0.$

Contracting with gij and integrating the result, one gets

$f=λRn2,$

where λ is a constant.Conversely, if Eq. 2.5 holds, then

$Tij=0.$

We can thus state following theorem:

Theorem 2. A concircularly flat spacetime in $fR$ gravity is vacuum if and only if $f=λRn2$.The vector filed ξ is called Killing if

$Lξgij=0,$

whereas ξ is called conformal Killing if

$Lξgij=2φgij,$

where $Lξ$ is the Lie derivative with respect to the vector filed ξ and φ is a scalar function [1517]. The symmetry of a spacetime is measured by the number of independent Killing vector fields the spacetime admits. A spacetime of maximum symmetry has a constant curvature.A spacetime M is said to admit a matter collineation with respect to a vector field ξ if the Lie derivative of the energy-momentum tensor T with respect to ξ satisfies

$LξTij=0.$

It is clear that every Killing vector field is a matter collineation, but the converse is not generally true. The energy-momentum tensor Tij has the Lie inheritance property along the flow lines of the vector field ξ if the Lie derivative of Tij with respect to ξ satisfies [1517].

$LξTij=2φTij.$

Now using Eq. 2.3 in Eq. 2.4, one gets

$Rn−f2f′gij=κf′Tij.$

In a concircularly flat spacetime the scalar curvature R is constant, and hence f and f′ are also constants. Now, we consider a non-vacuum concircularly flat spacetime M. Therefore the Lie derivative $Lξ$ of Eq. 2.10 implies that

$Rn−f2f′Lξgij=κf′LξTij.$

Assume that the vector field ξ is Killing on M, that is, Eq. 2.6 holds, thus we have

$LξTij=0.$

Conversely, if Eq. 2.8 holds, then form Eq. 2.11 it follows that

$Lξgij=0.$

We thus motivate to state the following theorem:

Theorem 3. Let M be a concircularly flat spacetime satisfying $fR$ gravity, then the vector field ξ is Killing if and only if M admits matter collineation with respect to ξ.The isometry of spacetimes prescriped by Killing vector fields represents a very important type of spacetime symmetry. Spacetimes of constant curvature are known to have maximum such symmetry, that is, they admit the maximum number of linearly independent Killing vector fields. The maximum numer of linearly independent Killing vector fields in an n − dimensional spacetime is $nn+12$ (The reader is referred to [1822] and references therein for a more discussion on this topic). This fact with the above theorem leads to the following corollary.

Corollary 3. A non-vacuum concircularly flat spacetime satisfying $fR$ gravity admits the maximum number of matter collineations $nn+12$.Let ξ be a conformal Killing vector field, that is, Eq. 2.7 holds. Eq. 2.11 implies

$LξTij=2φTij.$

Conversely, assume that Eq. 2.9 holds, then from Eq. 2.11 we obtain

$Lξgij=2φgij.$

Hence, we can state the following theorem:

Theorem 4. Let M be a concircularly flat spacetime satisfying $fR$ gravity, then M has a conformal Killing vector filed ξ if only if the energy-momentum tensor Tij has the Lie inheritance property along ξ .The covariant derivative of both sides of Eq. 2.10 implies that

$∇kTij=0.$

Since in a concircularly flat spacetime R is constant, then f and f′ are constant. Inserting Eq. 2.4 in Eq. 2.12, we get

$∇kRij=0.$

Thus, we have:

Theorem 5. Let M be a concirculary flat spacetime satisfying $fR$ gravity, then M is Ricci symmetric.

## 3 Concircularly Flat Perfect Fluid Spacetimes in f(R) Gravity

In a perfect fluid 4 − dimensional spacetime, the energy-momentum tensor $Tij$ obeys

$Tij=p+σuiuj+pgij,$

where p is the isotropic pressure, σ is the energy density, and ui is a unit timelike vector field [7, 23].

Making use of Eq. 3.1 in Eq. 2.4, we get

$Rij=κf′p+σuiuj+pgij+f2f′gij.$

The use of Eq. 2.3 implies that

$R4gij=κf′p+σuiuj+pgij+f2f′gij.$

Contracting Eq. 3.3 with ui, we get

$σ=2f−Rf′4κ.$

Transvecting Eq. 3.3 with gij and using Eq. 3.4, one obtains

$p=−2f−Rf′4κ.$

In consequence of the above we can state the following theorem:

Theorem 6. In a concircularly flat perfect fluid spacetime obeying f(R) gravity, the isotropic pressure p and the energy density σ are constants and $p=−2f−Rf′4κ$ and $σ=2f−Rf′4κ$.Combining Eq. 3.4 and Eq. 3.5, one easily gets

$p+σ=0,$

which means that the spacetime represents dark matter era or alternatively the perfect fluid behaves as a cosmological constant [ [24]]. Thus we can state the following theorem:

Theorem 7. A concircularly flat perfect fluid spacetime obeying f(R) gravity represents dark matter era.In radiation era σ = 3p, therefor the energy-momentum tensor Tij takes the form

$Tij=4pujuk+pgjk.$

Eq. 3.6 implies that p = 0. It follows that

$Tij=0.$

which means that the spacetime is devoid of matter.Thus we motivate to state the following corollary:

Corollary 4. Let M be a concircularly flat spacetime obeying $fR$ gravity, then the Radiation era in M is vacuum.In pressureless fluid spacetime p = 0, the energy-momentum tensor is expressed as [25].

$Tij=σuiuj.$

From Eq. 3.6 it follows that σ = 0. And consequently from Eq. 3.8 we infer

$Tij=0,$

which means that the spacetime is vacuum.We thus can state the following:

Corollary 5. Let M be a concircularly flat dust fluid spacetime obeying $fR$ gravity, then M is vacuum.

### 3.1 Energy Conditions in Concircularly Flat Spacetime

In this subsection, some energy conditions in concircularly flat spacetimes obeying $fR$ gravity are considered. Indeed, energy conditions serve as a filtration system of the energy-momentum tensor in standard theory of gravity and the modified theories of gravity.[1214]. In [26], the authors studied weak energy condition (WEC), dominant energy conditions (DEC), null energy conditions (NEC), and strong energy conditions in two extended theories of gravity. As a starting point, we need to determine the effective isotropic pressure peff and the effective energy density σeff to state some of these energy conditions.

Eq. 2.4 may be rewritten as

$Rij−12Rgij=κf′Tijeff,$

where

$Tijeff=Tij+f−Rf′2κgij.$

This leads us to rewrite Eq. 3.1 in the following form

$Tijeff=peff+σeffuiuj+peffgij,$

where

$peff=p+f−Rf′2κ and σeff=σ−f−Rf′2κ.$

The use of Eq. 3.4 and Eq. 3.5 entails that

$peff=−Rf′4κ,σeff=Rf′4κ.$

Let us investigate certain energy conditions of a perfect fluid type effective matter in $fR$ gravity theory [12, 26, 27]:

1) Null energy condition (NEC): it says that peff + σeff ≥ 0.

2) Weak energy condition (WEC): it states that σeff ≥ 0 and peff + σeff ≥ 0.

3) Dominant energy condition (DEC): it states that σeff ≥ 0 and peff ± σeff ≥ 0.

4) Strong energy condition (SEC): it states that σeff + 3peff ≥ 0 and peff + σeff ≥ 0.

In this context, all mentioned energy conditions are consistently satisfied if Rf’ ≥ 0. As mentioned earlier, f′ must be positive to ensure attractive gravity. Therefore, the previous energy conditions are always satisfied if R ≥ 0.

## 4 Spacetimes With Divergence Free Concircular Curvature Tensor in f(R) Gravity

The divergence of the concircular curvature tensor, for n = 4, is given by [28].

$∇hM klmh=∇hR klmh−112glm∇kR−gkm∇lR.$

It is well-known that

$∇hR klmh=∇kRlm−∇lRkm.$

The use of Eq. 4.2 in Eq. 4.1 implies that

$∇hM klmh=∇kRlm−∇lRkm−112glm∇kR−gkm∇lR.$

Assume that the concircular curvature tensor is divergence free, that is $∇hMklmh=0$, then

$∇kRlm−∇lRkm−112glm∇kR−gkm∇lR=0.$

Contracting with glm and using $∇jRij=12∇iR$, we obtain

$∇kR=0.$

Utilizing (4.5) in Eq. 4.4, we have

$∇kRlm=∇lRkm,$

which means that the Ricci tensor is of Codazzi type [29]. The converse is trivial. Thus we can state the following theorem:

Theorem 8. Let M be a spacetime with concircular curvature tensor, then M has Codazzi type of Ricci tensor if and only if the concircular curvature tensor is divergence free.In view of Eq. 4.5, the field Eq. 1.1 in $fR$ gravity are

$Rij−f2f′gij=κf′Tij.$

Using Eq. 4.7 in Eq. 4.6, we get

$∇kTlm=∇lTkm.$

Hence, we have the following corollary:

Corollary 6. The energy-momentum tensor of a spacetime with divergence free concircular curvature tensor obeying $fR$ gravity is of Codazzi type.The spacetime M is called Ricci semi-symmetric [30] if

$∇h∇k−∇k∇hRij=0,$

whileas the energy-momentum tensor is called semi-symmetric if

$∇h∇k−∇k∇hTij=0.$

Now, Eq. 4.7 implies

$∇h∇k−∇k∇hRij=κf′∇h∇k−∇k∇hTij.$

Thus, we can state the following theorem:

Theorem 9. Let M be a spacetime with divergence free concircular curvature tensor satisfying $fR$ gravity, then M is Ricci semi-symmetric if and only if the energy-momentum tensor of M is semi-symmetric.The energy-momentum tensor Tij is called recurrent if there exists a non-zero 1 − form λk such that

$∇kTij=λkTij,$

whereas Tij is called bi-recurrent if there exists a non-zero tensor ɛhk such that

$∇h∇kTij=εhkTij.$

In view of the above definition, it is clear that every recurrent tensor field is bi-recurrent.Now assume that $Tij$ is any $0,2$ symmetric recurrent tensor, that is,

$∇kTij=λkTij.$

Contracting with gij, we obtain

$λk=1T∇kT,$

where $T=gijTij$.Applying the covariant derivative on both sides and using Eq. 4.11, we find

$∇lλk+λkλl=1T∇l∇kT.$

Taking the covariant derivative of Eq. 4.10 and utilizing Eq. 4.12, we get

$∇l∇kTij=1T∇l∇kTTij.$

It follows that

$∇l∇k−∇k∇lTij=0.$

Similarly, the same result holds for a bi-recurrent $0,2$ symmetric tensor. In view of the above discussion, we have the following:

Lemma 1. A (bi-)recurrent $0,2$ symmetric tensor is semi-symmetric.Assume that the energy-momentum tensor $Tij$ is recurrent or bi-recurrent, it follows form Lemma one that $Tij$ is semi-symmetric. Consequently, M is Ricci semi-symmetric.

Theorem 10. Let M be a spacetime with divergence free concircular curvature tensor obeying $fR$ gravity. If the energy-momentum (Ricci) tensor is recurrent or bi-recurrent, then the Ricci (energy-momentum) tensor is semi-symmetric.Let us now consider that M be a perfect fluid spacetime with divergence free concircular curvature tensor and whose energy-momentum tensor is recurrent or bi-recurrent. Thus the use of Eq. 3.1 in Eq. 4.7 implies that

$Rij=agij+buiuj,$

where

$a=12f′2κp+f and b=κf′p+σ.$

Making a contraction of Eq. 4.13 with gij, we get

$R=1f′3κp−κσ+2f.$

Since the $Tij$ is recurrent or bi-recurrent, then $∇h∇k−∇k∇hTij=0$ it follows that $∇h∇k−∇k∇hRij=0$ Thus Eq. 4.13 implies that

$bui∇h∇k−∇k∇huj+buj∇h∇k−∇k∇hui=0.$

Contracting with ui, we obtain

$−b∇h∇k−∇k∇huj+bujui∇h∇k−∇k∇hui=0.$

But $ui∇h∇k−∇k∇hui=0$, thus we have

$b∇h∇k−∇k∇huj=0.$

Equivalently, it is

$bRhkj mum=0.$

This equation implies the following cases:

Case 1. If $Rhkj mum=0≠0$, then b = 0 and hence we get p + σ = 0 which means that the spacetime represents inflation and the fluid behaves as a cosmological constant.

Case 2. If b ≠ 0, then $Rhkj mum=0$, hence a contraction with ghj implies that $Rmkum=0$. Contracting equation (4.13) with ui and using $Rmkum=0$, one gets

$a−bui=0.$

Then ab = 0. With the help of equation (4.14), we have

$σ=f2κ.$

Using (4.16) in (4.15), we get

$p=2f′R−3f6κ.$

Theorem 11. Let the energy-momentum tensor of a perfect fluid spacetime with divergence free concircular curvature tensor obeying $fR$ gravity be recurrent or bi-recurrent. Then,

1) The spacetime represents inflation, or

2) The isotropic pressure p and the energy density σ are constants. Moreover, they are given by Eq. 4.16 and Eq. 4.17.

In virtue of Eq. 4.16 and Eq. 4.17 we have

$pσ=2f′R−3f3f.$

In view of the previous theorem we can state the following:

Remark 1. According to the different states of cosmic evolution of the Universe we can conclude that the spacetime with $∇hMklmh=0$ obeying $fR$ gravity either represents inflation or

## Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary material, further inquiries can be directed to the corresponding authors.

## Author Contributions

Conceptualization and methodology, SS, UD, AS, NT, HA-D, and SA; formal analysis, SS, UD and AS; writing original draft preparation, SS, AS and NT; writing-review and editing, SS, UD, HA-D, and SA; supervision, SS and UD; project administration, NT and AS; and funding acquisition, NT and SA. All authors have read and agreed to the published version of the manuscript.

## Funding

This project was supported by the Researchers Supporting Project number (RSP2022R413), King Saud University, Riyadh, Saudi Arabia.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s Note

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Keywords: perfect fluid, energy-momentum tensor, concircular curvature tensor, f (R) gravity theory, energy conditions in modified gravity

Citation: De UC, Shenawy S, Abu-Donia HM, Turki NB, Alsaeed S and Syied AA (2022) Spacetimes Admitting Concircular Curvature Tensor in f(R) Gravity. Front. Phys. 9:800060. doi: 10.3389/fphy.2021.800060

Received: 22 October 2021; Accepted: 09 December 2021;
Published: 12 January 2022.

Edited by:

Jae Won Lee, Gyeungsang National University, South Korea

Reviewed by:

Peibiao Zhao, Nanjing University of Science and Technology, China
Salvatore Capozziello, University of Naples Federico II, Italy
İnan Ünal, Munzur University, Turkey
Oğuzhan Bahadır, Kahramanmaras Sütçü Imam University, Turkey

Copyright © 2022 De, Shenawy, Abu-Donia, Turki, Alsaeed and Syied. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Nasser Bin Turki, nassert@ksu.edu.sa; Abdallah Abdelhameed Syied, a.a_syied@yahoo.com