Abstract
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 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 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 . Researchers have shown the curial role of the concircular curvature tensor in mathematics and physics (for example, see [2–6] 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)with κ being the Newtonian constant and 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 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 in the Einstein-Hilbert action. The field equations of gravity are given aswhere is an arbitrary function of the scalar curvature R and 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 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 gravity theory [12–14]. 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 gravity. Also, spacetimes with divergence free concircular curvature tensor in 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 is defined locally aswhere , R, and gkl are the Riemann curvature tensor, the scalar curvature tensor, and the metric tensor [1].
Here, we will consider , thus it follows form Eq. 2.1 that
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 getIn 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 gravity becomeIn vacuum case, we haveContracting with gij and integrating the result, one getswhere λ is a constant.Conversely, if Eq. 2.5 holds, thenWe can thus state following theorem:
Theorem 2. A concircularly flat spacetime ingravity is vacuum if and only if.The vector filed ξ is called Killing ifwhereas ξ is called conformal Killing ifwhere is the Lie derivative with respect to the vector filed ξ and φ is a scalar function [15–17]. 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 satisfiesIt 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 [15–17].Now using Eq. 2.3 in Eq. 2.4, one getsIn 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 of Eq. 2.10 implies thatAssume that the vector field ξ is Killing on M, that is, Eq. 2.6 holds, thus we haveConversely, if Eq. 2.8 holds, then form Eq. 2.11 it follows thatWe thus motivate to state the following theorem:
Theorem 3. LetMbe a concircularly flat spacetime satisfyinggravity, then the vector fieldξis Killing if and only ifMadmits matter collineation with respect toξ.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 (The reader is referred to [18–22] 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 satisfyinggravity admits the maximum number of matter collineations.Let ξ be a conformal Killing vector field, that is, Eq. 2.7 holds. Eq. 2.11 impliesConversely, assume that Eq. 2.9 holds, then from Eq. 2.11 we obtainHence, we can state the following theorem:
Theorem 4. LetMbe a concircularly flat spacetime satisfyinggravity, thenMhas a conformal Killing vector filedξif only if the energy-momentum tensorTijhas the Lie inheritance property alongξ.The covariant derivative of both sides of Eq. 2.10 implies thatSince in a concircularly flat spacetime R is constant, then f and f′ are constant. Inserting Eq. 2.4 in Eq. 2.12, we getThus, we have:
Theorem 5. LetMbe a concirculary flat spacetime satisfyinggravity, thenMis Ricci symmetric.
3 Concircularly Flat Perfect Fluid Spacetimes in f(R) Gravity
In a perfect fluid 4 − dimensional spacetime, the energy-momentum tensor obeyswhere 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
The use of Eq. 2.3 implies that
Contracting Eq. 3.3 with ui, we get
Transvecting Eq. 3.3 with gij and using Eq. 3.4, one obtains
In consequence of the above we can state the following theorem:
Theorem 6. In a concircularly flat perfect fluid spacetime obeyingf(R) gravity, the isotropic pressurepand the energy densityσare constants andand.Combining Eq. 3.4 and Eq. 3.5, one easily getswhich 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 obeyingf(R) gravity represents dark matter era.In radiation era σ = 3p, therefor the energy-momentum tensor Tij takes the formEq. 3.6 implies that p = 0. It follows thatwhich means that the spacetime is devoid of matter.Thus we motivate to state the following corollary:
Corollary 4. LetMbe a concircularly flat spacetime obeyinggravity, then the Radiation era inMis vacuum.In pressureless fluid spacetime p = 0, the energy-momentum tensor is expressed as [25].From Eq. 3.6 it follows that σ = 0. And consequently from Eq. 3.8 we inferwhich means that the spacetime is vacuum.We thus can state the following:
Corollary 5. LetMbe a concircularly flat dust fluid spacetime obeyinggravity, thenMis vacuum.
3.1 Energy Conditions in Concircularly Flat Spacetime
In this subsection, some energy conditions in concircularly flat spacetimes obeying 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.[12–14]. 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 aswhere
This leads us to rewrite Eq. 3.1 in the following formwhere
The use of Eq. 3.4 and Eq. 3.5 entails that
Let us investigate certain energy conditions of a perfect fluid type effective matter in
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].
It is well-known that
The use of Eq. 4.2 in Eq. 4.1 implies that
Assume that the concircular curvature tensor is divergence free, that is , then
Contracting with glm and using , we obtain
Utilizing (4.5) in Eq. 4.4, we havewhich means that the Ricci tensor is of Codazzi type [29]. The converse is trivial. Thus we can state the following theorem:
Theorem 8. LetMbe a spacetime with concircular curvature tensor, thenMhas 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 gravity areUsing Eq. 4.7 in Eq. 4.6, we getHence, we have the following corollary:
Corollary 6. The energy-momentum tensor of a spacetime with divergence free concircular curvature tensor obeyinggravity is of Codazzi type.The spacetime M is called Ricci semi-symmetric [30] ifwhileas the energy-momentum tensor is called semi-symmetric ifNow, Eq. 4.7 impliesThus, we can state the following theorem:
Theorem 9. LetMbe a spacetime with divergence free concircular curvature tensor satisfyinggravity, thenMis Ricci semi-symmetric if and only if the energy-momentum tensor ofMis semi-symmetric.The energy-momentum tensor Tij is called recurrent if there exists a non-zero 1 − form λk such thatwhereas Tij is called bi-recurrent if there exists a non-zero tensor ɛhk such thatIn view of the above definition, it is clear that every recurrent tensor field is bi-recurrent.Now assume that is any symmetric recurrent tensor, that is,Contracting with gij, we obtainwhere .Applying the covariant derivative on both sides and using Eq. 4.11, we findTaking the covariant derivative of Eq. 4.10 and utilizing Eq. 4.12, we getIt follows thatSimilarly, the same result holds for a bi-recurrent symmetric tensor. In view of the above discussion, we have the following:
Lemma 1. A (bi-)recurrentsymmetric tensor is semi-symmetric.Assume that the energy-momentum tensor is recurrent or bi-recurrent, it follows form Lemma one that is semi-symmetric. Consequently, M is Ricci semi-symmetric.
Theorem 10. LetMbe a spacetime with divergence free concircular curvature tensor obeyinggravity. 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 thatwhereMaking a contraction of Eq. 4.13 with gij, we getSince the is recurrent or bi-recurrent, then it follows that Thus Eq. 4.13 implies thatContracting with ui, we obtainBut , thus we haveEquivalently, it isThis equation implies the following cases:
Case 1. If, thenb = 0 and hence we getp + σ = 0 which means that the spacetime represents inflation and the fluid behaves as a cosmological constant.
Case 2. Ifb ≠ 0, then, hence a contraction withghjimplies that. Contracting equation (4.13) withuiand using, one getsThena − b = 0. With the help of equation (4.14), we haveUsing (4.16) in (4.15), we get
Theorem 11.
Let the energy-momentum tensor of a perfect fluid spacetime with divergence free concircular curvature tensor obeyinggravity be recurrent or bi-recurrent. Then,Remark 1. According to the different states of cosmic evolution of the Universe we can conclude that the spacetime withobeyinggravity either represents inflation or
| The spacetime | Equation of state(EoS) | |
| represents quintessence era | σ + 3p = 0 | a constant multiple ofR |
| represents dust matter era | p = 0 | a constant multiple of |
| represents radiation era | σ − 3p = 0 | a constant multiple ofR2 |
| represents stiff matter era | p − σ = 0 | a constant multiple ofR3 |
| determines an EoS in quintessence era | 5σ + 3p = 0 | a constant multiple ofR4 |
Statements
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.
References
1.
YanoK. Concircular Geometry I. Concircular Transformations. Proc Imperial Acad (1940) 16:195–200. 10.3792/pia/1195579139
2.
DeUCShenawySÜnalB. Concircular Curvature on Warped Product Manifolds and Applications. Bull Malays Math Sci Soc (2020) 43:3395–409. 10.1007/s40840-019-00874-x
3.
BlairDEKimJ-STripathiMM. On the Concircular Curvature Tensor of a Contact Metric Manifold. J Korean Math Soc (2005) 42:883–92. 10.4134/jkms.2005.42.5.883
4.
MajhiPDeUC. Concircular Curvature Tensor on K-Contact Manifolds. Acta Mathematica Academiae Paedagogicae Nyregyhaziensis (2013) 29:89–99.
5.
OlszakKOlszakZ. On Pseudo-riemannian Manifolds with Recurrent Concircular Curvature Tensor. Acta Math Hung (2012) 137:64–71. 10.1007/s10474-012-0216-5
6.
VanliATUnalI. Conformal, Concircular, Quasi-Conformal and Conharmonic Flatness on normal Complex Contact Metric Manifolds. Int J Geom Methods Mod Phys (2017) 14:1750067. 10.1142/s0219887817500670
7.
O’NeillB. Semi-semi-Riemannian Geometry with Applications to Relativity. New York-London): Academic Press (1983).
8.
BuchdahlHA. Non-linear Lagrangians and Cosmological Theory. Monthly Notices R Astronomical Soc (1970) 150:1–8. 10.1093/mnras/150.1.1
9.
CapozzielloSManticaCAMolinariLG. General Properties of F(r) Gravity Vacuum Solutions. Int J Mod Phys D (2020) 29:2050089. 10.1142/s0218271820500893
10.
CapozzielloS. Curvature Quintessence. Int J Mod Phys D (2002) 11:483–91. 10.1142/s0218271802002025
11.
CapozzielloSManticaCAMolinariLG. Cosmological Perfect Fluids in Higher-Order Gravity. Gen Relativity Gravitation (2020) 52:1–9. 10.1007/s10714-020-02690-2
12.
DeALooT-HAroraSSahooP. Energy Conditions for a (Wrs)_4(wrs) 4 Spacetime in F(r)-Gravity. The Eur Phys J Plus (2021) 136:1–10. 10.1140/epjp/s13360-021-01216-2
13.
DeALooT-H. Almost Pseudo-ricci Symmetric Spacetime Solutions in F(r)-Gravity. Gen Relativity Gravitation (2021) 53:1–14. 10.1007/s10714-020-02775-y
14.
DeALooT-HSolankiRSahooPK. A Conformally Flat Generalized Ricci Recurrent Spacetime in F(r)-Gravity. Phys Scr (2021) 96:085001. 10.1088/1402-4896/abf9d2
15.
Abu-DoniaHShenawySSyiedAA. The W* − Curvature Tensor on Relativistic Space-Times. Kyungpook Math J (2020) 60:185–95. 10.5666/KMJ.2020.60.1.185
16.
MallickSDeUC. Spacetimes Admitting W2-Curvature Tensor. Int J Geom Methods Mod Phys (2014) 11:1450030. 10.1142/s0219887814500303
17.
ZenginFO. M-projectively Flat Spacetimes. Math Rep (2012) 14:363–70.
18.
DeUShenawySÜnalB. Sequential Warped Products: Curvature and Conformal Vector fields. Filomat (2019) 33:4071–83. 10.2298/fil1913071d
19.
El-SayiedHShenawySSyiedN. Conformal Vector fields on Doubly Warped Product Manifolds and Applications. Adv Math Phys (2016) 2016:6508309. 10.1155/2016/6508309
20.
El-SayiedHKShenawySSyiedN. Symmetries Off−associated Standard Static Spacetimes and Applications. J Egypt Math Soc (2017) 25:414–8. 10.1016/j.joems.2017.07.002
21.
El-SayiedHKShenawySSyiedN. On Symmetries of Generalized Robertson-walker Spacetimes and Applications. J Dynamical Syst Geometric Theories (2017) 15:51–69. 10.1080/1726037x.2017.1323418
22.
ShenawySÜnalB. 2-killing Vector fields on Warped Product Manifolds. Int J Math (2015) 26:1550065. 10.1142/s0129167x15500652
23.
BlagaAM. Solitons and Geometrical Structures in a Perfect Fluid Spacetime. Rocky Mountain J Math (2020) 50:41–53. 10.1216/rmj.2020.50.41
24.
StephaniHKramerDMacCallumMHoenselaersCHerltE. Exact Solutions of Einstein’s Field Equations. Cambridge, United Kingdom: Cambridge University Press (2009).
25.
SrivastavaSK. General Relativity and Cosmology. New Delhi, Delhi: PHI Learning Pvt. Ltd. (2008).
26.
CapozzielloSLoboFSNMimosoJP. Generalized Energy Conditions in Extended Theories of Gravity. Phys Rev D (2015) 91:124019. 10.1103/physrevd.91.124019
27.
SantosJAlcanizJReboucasMCarvalhoF. Energy Conditions in F(r) Gravity. Phys Rev D (2007) 76:083513. 10.1103/physrevd.76.083513
28.
AhsanZSiddiquiSA. Concircular Curvature Tensor and Fluid Spacetimes. Int J Theor Phys (2009) 48:3202–12. 10.1007/s10773-009-0121-z
29.
GrayA. Einstein-like Manifolds Which Are Not Einstein. Geometriae dedicata (1978) 7:259–80. 10.1007/bf00151525
30.
İnanÜ. On Metric Contact Pairs with Certain Semi-symmetry Conditions. Politeknik Dergisi (2021) 24:333–8. 10.2339/politeknik.769662
Summary
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
Volume
9 - 2021
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
Updates
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
This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics
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.