Spacetimes Admitting Concircular Curvaure Tensor in f(R) Gravity

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 f R 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.


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 [2][3][4][5][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 T ij is the energy-momentum tensor [7]. These equations imply that the energy-momentum tensor T ij is divergence-free. This condition is satisfied whenever ∇ l T ij 0, where ∇ l denotes the covariant differentiation. There are many modifications of the standard relativity theory. The f(R) 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 f(R) in the Einstein-Hilbert action. The field equations of f(R) gravity are given as where f(R) is an arbitrary function of the scalar curvature R and f′(R) df dR 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 f(R) 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 f(R) gravity theory [12][13][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 f(R) gravity. Also, spacetimes with divergence free concircular curvature tensor in f(R) 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.

CONCIRCULARLY FLAT SPACETIMES IN F(R) GRAVITY
The concircular curvature tensor of type (0, 4) is defined locally as where R jklm , R, and g kl are the Riemann curvature tensor, the scalar curvature tensor, and the metric tensor [1].
Here, we will consider M jklm 0, 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 g jm , we get 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 f(R) gravity become In vacuum case, we have Contracting with g ij and integrating the result, one gets where λ is a constant. Conversely, if Eq. 2.5 holds, then We can thus state following theorem: where L ξ is the Lie derivative with respect to the vector filed ξ and φ is a scalar function [15][16][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 energymomentum tensor T with respect to ξ satisfies It is clear that every Killing vector field is a matter collineation, but the converse is not generally true. The energy-momentum tensor T ij has the Lie inheritance property along the flow lines of the vector field ξ if the Lie derivative of T ij with respect to ξ satisfies [15][16][17].
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 L ξ of Eq. 2.10 implies that Frontiers in Physics | www.frontiersin.org January 2022 | Volume 9 | Article 800060 2 Assume that the vector field ξ is Killing on M, that is, Eq. 2.6 holds, thus we have Conversely, if Eq. 2.8 holds, then form Eq. 2.11 it follows that L ξ g ij 0.
We thus motivate to state the following theorem: Theorem 3. Let M be a concircularly flat spacetime satisfying f(R) 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 n(n+1) 2 (The reader is referred to [18][19][20][21][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 satisfying f(R) gravity admits the maximum number of matter collineations n(n+1)

.
Let ξ be a conformal Killing vector field, that is, Eq. 2.7 holds. Eq. 2.11 implies Conversely, assume that Eq. 2.9 holds, then from Eq. 2.11 we obtain Hence, we can state the following theorem: Theorem 4. Let M be a concircularly flat spacetime satisfying f(R) gravity, then M has a conformal Killing vector filed ξ if only if the energy-momentum tensor T ij has the Lie inheritance property along ξ .
The covariant derivative of both sides of Eq. 2.10 implies that 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 Thus, we have: Theorem 5. Let M be a concirculary flat spacetime satisfying f(R) gravity, then M is Ricci symmetric.

CONCIRCULARLY FLAT PERFECT FLUID SPACETIMES IN F(R) GRAVITY
In a perfect fluid 4 − dimensional spacetime, the energymomentum tensor T ij obeys where p is the isotropic pressure, σ is the energy density, and u i 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 Transvecting Eq. 3.3 with g ij 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 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, (3.6) 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 T ij takes the form T ij 4pu j u k + pg jk . (3.7) Eq. 3.6 implies that p 0. It follows that 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 f(R) gravity, then the Radiation era in M is vacuum.
In pressureless fluid spacetime p 0, the energy-momentum tensor is expressed as [25].
T ij σu i u j . (3.8) From Eq. 3.6 it follows that σ 0. And consequently from Eq. 3.8 we infer T ij 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 f(R) gravity, then M is vacuum.

Energy Conditions in Concircularly Flat Spacetime
In this subsection, some energy conditions in concircularly flat spacetimes obeying f(R) 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][13][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 p eff and the effective energy density σ eff to state some of these energy conditions. Eq. 2.4 may be rewritten as This leads us to rewrite Eq. 3.1 in the following form 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 f(R) gravity theory [12,26,27] 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.

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 ∇ h M h klm 0, then Contracting with g lm and using ∇ j R j i 1 2 ∇ i R, we obtain ∇ k R 0. (4.5) Utilizing (4.5) in Eq. 4.4, we have 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 f(R) gravity are Using Eq. 4.7 in Eq. 4.6, we get Hence, we have the following corollary: The spacetime Equation of state(EoS) f(R) represents quintessence era σ + 3p 0 a constant multiple of R represents dust matter era p 0 a constant multiple of R 3 2 represents radiation era σ − 3p 0 a constant multiple of R 2 represents stiff matter era p − σ 0 a constant multiple of R 3 determines an EoS in quintessence era 5σ + 3p 0 a constant multiple of R 4 Frontiers in Physics | www.frontiersin.org January 2022 | Volume 9 | Article 800060 Corollary 6. The energy-momentum tensor of a spacetime with divergence free concircular curvature tensor obeying f(R) gravity is of Codazzi type. The spacetime M is called Ricci semi-symmetric [30] if whileas the energy-momentum tensor is called semi-symmetric if Now, Eq. 4.7 implies Thus, we can state the following theorem: Theorem 9. Let M be a spacetime with divergence free concircular curvature tensor satisfying f(R) gravity, then M is Ricci semi-symmetric if and only if the energy-momentum tensor of M is semi-symmetric.
The energy-momentum tensor T ij is called recurrent if there exists a non-zero 1 − form λ k such that whereas T ij is called bi-recurrent if there exists a non-zero tensor ε hk such that In view of the above definition, it is clear that every recurrent tensor field is bi-recurrent. Now assume that T ij is any (0, 2) symmetric recurrent tensor, that is, Contracting with g ij , we obtain where T g ij T ij . Applying the covariant derivative on both sides and using Eq. 4.11, we find Taking the covariant derivative of Eq. 4.10 and utilizing Eq. 4.12, we get It follows that 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 semisymmetric.
Assume that the energy-momentum tensor T ij is recurrent or bi-recurrent, it follows form Lemma one that T ij is semisymmetric. Consequently, M is Ricci semi-symmetric. Theorem 10. Let M be a spacetime with divergence free concircular curvature tensor obeying f(R) gravity. If the energy-momentum (Ricci) tensor is recurrent or bi-recurrent, then the Ricci (energymomentum) tensor is semi-symmetric.
Let us now consider that M be a perfect fluid spacetime with divergence free concircular curvature tensor and whose energymomentum tensor is recurrent or bi-recurrent. Thus the use of Eq. 3.1 in Eq. 4.7 implies that where a 1 2f′ 2κp + f and b κ f′ p + σ . (4.14) Making a contraction of Eq. 4.13 with g ij , we get Since the T ij is recurrent or bi-recurrent, then (∇ h ∇ k − ∇ k ∇ h )T ij 0 it follows that (∇ h ∇ k − ∇ k ∇ h )R ij 0 Thus Eq. 4.13 implies that Contracting with u i , we obtain Equivalently, it is bR m hkj u m 0.
This equation implies the following cases: Case 1. If R m hkj u m 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 R m hkj u m 0, hence a contraction with g hj implies that R mk u m 0. Contracting equation (4.13) with u i and using R mk u m 0, one gets a − b ( )u i 0. Theorem 11. Let the energy-momentum tensor of a perfect fluid spacetime with divergence free concircular curvature tensor obeying f(R) 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.