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ORIGINAL RESEARCH article

Front. Phys., 03 October 2022
Sec. Statistical and Computational Physics
This article is part of the Research Topic Differential Geometric Methods in Modern Physics View all 7 articles

Spacetimes Admitting Concircular Curvature Tensor in f(R) Gravity

Updated
  • 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)

RijR2gij=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=fRRijfRfiRjRfRijR+gijfRkRkR+fR2R12fR,

where fR is an arbitrary function of the scalar curvature R and fR=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=RjklmRnn1gklgjmgkmgjl,

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=Rnn1gklgjmgkmgjl.

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

Rijf2fgij=κfTij.

In vacuum case, we have

Rijf2fgij=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

Rnf2fgij=κfTij.

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

Rnf2fLξgij=κfLξTij.

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=κfp+σuiuj+pgij+f2fgij.

The use of Eq. 2.3 implies that

R4gij=κfp+σuiuj+pgij+f2fgij.

Contracting Eq. 3.3 with ui, we get

σ=2fRf4κ.

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

p=2fRf4κ.

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=2fRf4κ and σ=2fRf4κ.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

Rij12Rgij=κfTijeff,

where

Tijeff=Tij+fRf2κgij.

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

Tijeff=peff+σeffuiuj+peffgij,

where

peff=p+fRf2κ   and     σeff=σfRf2κ.

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

peff=Rf4κ,σeff=Rf4κ.

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].

hMklmh=hRklmh112glmkRgkmlR.

It is well-known that

hRklmh=kRlmlRkm.

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

hMklmh=kRlmlRkm112glmkRgkmlR.

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

kRlmlRkm112glmkRgkmlR=0.

Contracting with glm and using jRij=12iR, 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

Rijf2fgij=κfTij.

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

hkkhRij=0,

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

hkkhTij=0.

Now, Eq. 4.7 implies

hkkhRij=κfhkkhTij.

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

hkTij=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=1TkT,

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

lλk+λkλl=1TlkT.

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

lkTij=1TlkTTij.

It follows that

lkklTij=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=12f2κp+f and b=κfp+σ.

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

R=1f3κpκσ+2f.

Since the Tij is recurrent or bi-recurrent, then hkkhTij=0 it follows that hkkhRij=0 Thus Eq. 4.13 implies that

buihkkhuj+bujhkkhui=0.

Contracting with ui, we obtain

bhkkhuj+bujuihkkhui=0.

But uihkkhui=0, thus we have

bhkkhuj=0.

Equivalently, it is

bRhkj    mum=0.

This equation implies the following cases:

Case 1. If Rhkj    mum=00, 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

abui=0.

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

σ=f2κ.

Using (4.16) in (4.15), we get

p=2fR3f6κ.

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=2fR3f3f.

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

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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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

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.