# A Conformally Flat Generalized Ricci Recurrent Spacetime in F(R)-Gravity

AA CONFORMALLY FLAT GENERALIZED RICCIRECURRENT SPACETIME IN F ( R ) -GRAVITY AVIK DE, TEE-HOW LOO, RAJA SOLANKI, AND P.K.SAHOO

Abstract.

In the present paper we study a conformally ﬂat generalizedRicci recurrent perfect ﬂuid spacetime with constant Ricci scalar as asolution of modiﬁed f ( R )-gravity theory. We show that a Robertson-Walker spacetime is generalized Ricci Recurrent if and only if it is Riccisymmetric. The perfect ﬂuid type matter is shown to have EoS ω = − F ( R )-gravity, like F ( R ) = R + αR m where α, m are constants and F ( R ) = R + βRlnR where β is constant. In harmony with the recentobservational studies of accelerated expansion of the universe, both casesexhibit that the null, weak, and dominant energy conditions fulﬁll theirrequirements whereas the strong energy condition is violated. Introduction

Einstein’s ﬁeld equations (EFE) R ij − R g ij = κ T ij , where κ = 8 πG , G being Newton’s gravitational constant and R = R ii theRicci scalar, imply that the energy-momentum tensor T ij is of vanishingdivergence. This requirement is accomplished if T ij is covariantly constant.Chaki and Ray showed that a general relativistic spacetime with covariant-constant energy-momentum tensor is Ricci symmetric, that is, ∇ i R jl = 0[1]. Generalizing this concept, in [2] Patterson introduced the notion of Riccirecurrent manifolds R n as an n -dimensional non-ﬂat Riemannian manifoldof dimension n > R ij of type (0 ,

2) is not identicallyzero and satisﬁes the condition ∇ i R jl = A i R jl , where A i is a non-zero 1-form. With growing interest in the Ricci recurrent manifolds, in 1995 Deet al. [3] generalized the notion to introduce a generalized Ricci recurrentmanifold ( GR ) n as a n -dimensional non-ﬂat Riemannian manifold whoseRicci tensor satisﬁes the following: ∇ i R jl = A i R jl + B i g jl , (1) A.D. and L.T.H. are supported by the grant FRGS/1/2019/STG06/UM/02/6. R.S. ac-knowledges University Grants Commission(UGC), Govt. of India, New Delhi, for award-ing Junior Research Fellowship(NFOBC)(No.F.44-1/2018(SA-III)) for ﬁnancial support. a r X i v : . [ g r- q c ] J a n AVIK DE, TEE-HOW LOO, RAJA SOLANKI, AND P.K.SAHOO where A i and B i are non-zero 1-forms. Obviously, if the one-form B i van-ishes, a ( GR ) n reduces to a R n .Unfortunately, EFE are unable to explain the late time inﬂation of theuniverse without assuming the existence of some yet undetected componentsabbreviated as dark energy. This motivated some researchers to extend itto get some higher order ﬁeld equations of gravity. One of these modiﬁedgravity theories is obtained by replacing the Ricci scalar R in the Einstein-Hilbert action with an arbitrary function F ( R ), initially proposed by HansAdolph Buchdahl in 1970 [4]. Of course the viability of such functions areconstrained by several observational data and scalar-tensor theoretical re-sults. Additionally we can always propose some phenomenological assump-tion about the form of the function F ( R ) and later verify its validity fromthe present viability criteria. There are several models or functional formsof F ( R ) proposed in the literature, for reference see ([5], [6]). The ﬁrst timeto use a quadratic form of the Ricci scalar was given by Starobinsky [7].It was shown that the issue of massive neutron stars can be solved by thehigher order curvature of F ( R ) gravity, for references see ([8], [9], [10], [11],[12]). The equations of motion of F ( R ) gravity have higher degrees andprovide considerable solutions that are diﬀerent from general relativity.General relativity models the universe as a four-dimensional smooth, con-nected, para-compact, Hausdorﬀ spacetime manifold with a Lorentzian met-ric of signature ( − , + , + , +). A Lorentzian manifold is said to be a gener-alized Ricci recurrent spacetime if the Ricci tensor satisﬁes (1). The ﬁrstauthor [13] recently investigated a ( GR ) spacetime satisfying EFE.The matter content in the EFE is more often assumed to be a perfectﬂuid continuum. Let u i denote the four velocity vector of the ﬂuid, thenthe spatial part h ij of the metric g ij can be deﬁned as h ij = g ij + u i u j sothat h ij u i = 0. h ij thus can be called the projection operator orthogonal tothe vector u i . The energy momentum tensor T ij of type (0 ,

2) is given by T ij = ph ij + ρu i u j , where ρ = T ij u i u j and p = T ij h ij are the energy densityand the isotropic pressure, respectively.We know that the energy conditions represent paths to establish the pos-itivity of the stress-energy tensor. Also, they can be used to explore theattractive nature of gravity, besides assigning the fundamental causal andthe geodesic structure of space-time [14]. The diﬀerent models of F ( R ) grav-ity give rise to the problem of constraining model parameters. By imposingthe diﬀerent energy conditions, we may have constrains of F ( R ) model pa-rameters [15]. The diﬀerent energy conditions have been used to obtainsolutions for a plenty of problems. For example the strong and weak en-ergy condition were used in the Hawking-Penrose singularity theorems andthe null energy condition is required in order to prove the second law ofblack hole thermodynamics. The energy conditions were formulated in GR CONFORMALLY FLAT GENERALIZED RICCI RECURRENT SPACETIME... 3 [16], we can derive these conditions in F ( R ) gravity by introducing neweﬀective pressure and energy density. In this paper, we examine the null,weak, strong and dominant energy conditions for F ( R )-gravity models. Inaddition, we assume that the relation between the energy density ρ and thepressure p of the matter present in the universe is given by the equation p = ωρ . Moreover, if p = ρ , then the perfect ﬂuid is termed as stiﬀ matterwhile p = ρ and p = 0 are termed as radiation and dust matter respectively.The stiﬀ matter era preceded the radiation era, and then dust matter era.The recent observational studies favors the dark matter era with p = − ρ . Indiﬀerent gravity models, there are some works on energy conditions whichlead to accelerated expansion of the universe by constraining the modelparameters by the equation of state ([17], [18]).The present paper is organized as follows: in section 2 we show thata Robertson-Walker spacetime is generalized Ricci recurrent if and only ifit is Ricci symmetric; followed by a study of conformally ﬂat generalizedRicci recurrent spacetime with constant Ricci scalar which satisﬁes F ( R )-gravity equations. In the next section we discuss several energy conditionsin such a setting, followed by some toy models of F ( R )-gravity, investigatedin conformally ﬂat ( GR ) with constant R . In section 5 we consider twomodels, one is polynomial and the other logarithmic and investigate diﬀerentenergy conditions and ﬁnd the constraints on model parameters to satisfyrequirements of energy conditions. Finally in section 6 we end up with thediscussion. 2. Robertson-Walker spacetime as an ( GR ) The current favoured model of our universe is a spatially ﬂat Robertson-Walker (RW) spacetime, a warped product R × a ( t ) M , where M is a 3-dimensional space form of vanishing curvature. The function a ( t ) is calledthe scale factor controlling the accelerated or decelerated expansion of theuniverse. In this section we show that a RW spacetime is generalized Riccirecurrent if and only if the scale factor a ( t ) is of exponential type.The line element and the Ricci scalar in a spatially ﬂat RW spacetime arerespectively given by ds = − dt + a ( t ) (cid:0) dr + r dθ + r sin θdφ (cid:1) ,R = 6 a ¨ a + ˙ a a . The Ricci tensor takes the form R jl = ( P − Q ) u j u l + P g jl = − Qu j u l + P h jl (2) AVIK DE, TEE-HOW LOO, RAJA SOLANKI, AND P.K.SAHOO where P = a ¨ a + 2 ˙ a a , Q = 3 ¨ aa (3)and u i = ( ∂ t ) i is the four-velocity of the ﬂuid with u j u j = − ∇ j u l = ˙ aa h jl . (4)It is clear from (3) that ∇ i P = − u i ˙ P , ∇ i Q = − u i ˙ Q. (5)Taking covariant derivative on (2), with the help of (4)–(5) we obtain ∇ i R jl =( ∇ i P − ∇ i Q ) u j u l + ( P − Q ) {∇ i u j u l + ∇ i u l u j } + ∇ i P g jl = ˙ Qu i u j u l + ( P − Q ) ˙ aa { h ij u l + h il u j } − ˙ P u i h jl . (6)Now let us further assume that it is an ( GR ) spacetime. By (1) and (2),we obtain ∇ i R jl = − Q ( A i + B i ) u j u l + P ( A i + B i ) h jl . (7)By comparing (6)–(7), we have { ˙ Qu i + Q ( A i + B i ) } u j u l + ( P − Q ) ˙ aa { u l h ij + u j h il } = { ˙ P u i + P ( A i + B i ) } h jl (8)Let h jl = g jl + u j u l . Then h ij h jl = h li = δ li + u i u l . Transvecting (8) with h jl , we have˙ P u i + P ( A i + B i ) = 0 . (9)Transvecting (8) with u j h il , with the help of (9), we have P − Q = 0 (10)or equivalently, a ¨ a = ˙ a . Solving this equation gives a ( t ) = Ae (cid:15)t where A > (cid:15) are constants. By using this and (3), we have ˙ P = 0 andso (9) gives A i + B i = 0 . (11)By substituting all these in (6), we have ∇ i R jl = 0, that is the spacetime isRicci symmetric. Thus we have the following: CONFORMALLY FLAT GENERALIZED RICCI RECURRENT SPACETIME... 5

Theorem 2.1.

A spatially ﬂat RW spacetime is an ( GR ) spacetime if andonly if is is Ricci symmetric and the scale factor a ( t ) is given by a ( t ) = Ae (cid:15)t where A ( > and (cid:15) are constants and A i + B i = 0 . Conformally ﬂat ( GR ) satisfying F ( R ) -gravity This section is devoted in studying conformally ﬂat generalized Ricci re-current spacetimes with a constant Ricci scalar R . The covariant derivativeof the Ricci tensor satisﬁes (1), which on contraction of j, l produces ∇ i R = A i R + 4 B i . (12)If we consider a constant Ricci scalar R , from (12) we get, B i = − R A i . (13)On the other hand, since R is constant, for a conformally ﬂat case, we obtain ∇ i R jk = ∇ k R ji . (14)Using (1) and (13) we get A i R jk − A k R ij = − R A k g ij − A i g jk ) . (15)Contracting j, k in the above equation we ﬁnally get A j R ij = R A i . (16)We consider a modiﬁed Einstein-Hilbert action term S = 1 κ (cid:90) F ( R ) √− gd x + (cid:90) L m √− gd x, where F ( R ) is an arbitrary function of the Ricci scalar R , L m is the matterLagrangian density, and we deﬁne the stress-energy tensor of matter as T ij = − √− g δ ( √− gL m ) δg ij . By varying the action S of the gravitational ﬁeld with respect to themetric tensor components g ij and using the least action principle we obtainthe ﬁeld equation F R ( R ) R ij − F ( R ) g ij + ( g ij (cid:3) − ∇ i ∇ j ) F R ( R ) = κ T ij , (17)where (cid:3) represents the d’Alembertian operator, F R = ∂F ( R ) ∂R . Einstein’s ﬁeldequations can be obtained by putting F ( R ) = R .The trace of (17) gives3 (cid:3) F R ( R ) + RF R ( R ) − F ( R ) = k T, AVIK DE, TEE-HOW LOO, RAJA SOLANKI, AND P.K.SAHOO which we can rewrite as (cid:3) F R ( R ) = ∂V eﬀ ∂F R ( R ) . (18)On the critical points of the theory, the eﬀective potential V eﬀ has a maxi-mum (or minimum), so that (cid:3) F R ( R CP ) = 0and 2 F ( R CP ) − R CP F R ( R CP ) = − κ T. Here, R CP is the curvature of the critical point. For example, in absence ofmatter, i.e., T = 0, one has the de Sitter critical point associated with aconstant scalar curvature R dS . For a constant Ricci scalar, we can expressthe above ﬁeld equations (17) as follows: R ij − R g ij = κ F R ( R ) T eﬀ ij , (19)where T eﬀ ij = T ij + F ( R ) − RF R ( R )2 κ g ij . Remembering the term κ = 8 πG , the quantity G eﬀ = GF R ( R ) can be regardedas the eﬀective gravitational coupling strength in analogy to what is done inBrans-Dicke type scalar-tensor gravity theories and further the positivity of G eﬀ (equivalent to the requirement that the graviton is not a ghost) imposesthat the eﬀective scalar degree of freedom or the scalaron term f R ( R ) > GR ) spacetime satisﬁes (19) with the velocity vectoridentical to A i (assuming a time-like unit vector), we have R ij = k ( p + ρ ) F R ( R ) A i A j + 2 κ p + F ( R )2 F R ( R ) g ij . (20)This readily gives us R ij A j = (cid:20) F ( R ) − ρk F R ( R ) (cid:21) A i . (21)But from (16) we already know that R ij A j = R A i . Hence, we conclude that R = 2 F ( R ) − ρk F R ( R ) . (22)Therefore, ρ = 2 F ( R ) − RF R ( R )4 k , (23) CONFORMALLY FLAT GENERALIZED RICCI RECURRENT SPACETIME... 7 which from the trace equation of (20) also gives us p = − RF R ( R ) − F ( R )4 k . (24)This leads to our ﬁrst result of this section: Theorem 3.1.

In a perfect ﬂuid ( GR ) spacetime with constant R satis-fying F ( R ) -gravity, if the four-velocity vector is identical with A i ; then itsisotropic pressure p and energy density ρ are given by p = − RF R ( R ) − F ( R )4 k and ρ = F ( R ) − RF R ( R )4 k . Moreover, both the pressure and density are constantin this special scenario. Remark 3.1.

The equation of state w = − in this case, consistent withthe presently well-established Λ CDM theory. This theory is supported boththeoretically and also by the plethora of observational data in recent years.

Theorem 3.2.

A vacuum ( GR ) spacetime solution with constant R in F ( R ) -gravity is only possible when F ( R ) is a constant multiple of R .Proof. For the vacuum case, T ij = 0 in (19), the trace equation gives RF R ( R ) = 2 F ( R ) which on integration gives F ( R ) = λR for integratingconstant λ . (cid:3) Energy conditions in a ( GR ) While exploring the possibility of diﬀerent matter sources in the ﬁeldequations of gravity, in both general relativity and the extended theoriesof gravity, energy conditions come in handy to constraint the (eﬀective)energy-momentum tensor and preserve the idea that not only the gravityis attractive but also the energy density is positive. For the current studyof modiﬁed F ( R )- theories of gravity, we ﬁrst deduce the eﬀective pressure p eﬀ = − RF R ( R )4 k and the eﬀective energy density ρ eﬀ = RF R ( R )4 k from (19) toinvestigate some energy conditions as follows: • Null energy condition (NEC) : It states that T eﬀ ij X i X j ≥ X i which gives us ρ eﬀ + p eﬀ ≥

0. Hence the NEC is alwayssatisﬁed in the present setting. • Weak energy condition (WEC) : It states that T eﬀ ij Y i Y j ≥ , forall time-like vectors Y i . By continuity this will also imply the NEC.Considering the timelike vector A i we obtain ρ eﬀ ≥ ρ eﬀ + p eﬀ ≥

0. Hence, WEC is satisﬁed in the present setting if RF R ( R ) ≥ F R ( R ) >

0, it implies that R ≥ • Dominant energy condition (DEC) : It states that T eﬀ ij Y i Y j ≥ Y i together with T ij Y j is not space-like, either null ortime-like. By continuity the property should also hold true for anynull vector. So we obtain, ρ eﬀ ± p eﬀ ≥

0. In the present setting, DEC

AVIK DE, TEE-HOW LOO, RAJA SOLANKI, AND P.K.SAHOO is only satisﬁed if 2 RF R ( R ) ≥ , which implies a non-negative R asin the previous case of WEC, since F R ( R ) > • Strong energy condition (SEC) : It states that T eﬀ ij Y i Y j ≥ T ii Y j Y j , for all time-like vectors Y i which implies ρ eﬀ + 3 p eﬀ ≥

0. In thepresent setting this is only satisﬁed if RF R ( R ) ≤

0. Since F R ( R ) > R ≤ R in the present study.5. Analysis of some toy models of F ( R ) -gravity in ( GR ) Various F ( R ) models have been proposed in the literature. We considerhere two diﬀerent models of F ( R )-gravity theories to analyse our result. Case:I F ( R ) = R + αR m , α and m are constant. Some of the polyno-mial models are studied in [19]. In this case, the eﬀective energy momentumreduces to, T eﬀ ij = T ij + 12 κ (1 − m ) αR m g ij (25)which represents the eﬀective pressure and energy density for a perfect ﬂuidmatter as follows, ρ eﬀ = R + mαR m κ (26)and p eﬀ = − R + mαR m κ . (27)The equation of state parameter (EoS) in this case reads as ω = − α and m , that is, the universe is dominated by cosmological constant andthis model is consistent with the presently well established Λ CDM theory.Now the previous section states the conditions on R to satisfy the diﬀerentenergy conditions. As we know that WEC is the combination of NEC andpositive density and NEC is always zero in the present setting, so we observebehaviour of SEC, DEC and density parameter.In these ﬁgures [1] we plot the density parameter, SEC and DEC withrespect to α and m taking R = 1, 1 ≤ α ≤ ≤ m ≤ CONFORMALLY FLAT GENERALIZED RICCI RECURRENT SPACETIME... 9

Figure 1.

Energy conditions for F ( R ) = R + αR m with1 ≤ α ≤

2, 0 ≤ m ≤ R = 1.We observe the behavior of all energy conditions and density parameterfor R >

S. no. Terms Results m ∈ (0 , ∞ ) , α ∈ (0 , ∞ ) ρ eff > m ∈ ( −∞ , , α ∈ ( −∞ , m = 0, α(cid:15) ( −∞ , ∞ )2 for m ∈ (0 , ∞ ) , α ∈ (0 , ∞ )WEC, DEC > < m ∈ ( −∞ , , α ∈ ( −∞ , m = 0, α(cid:15) ( −∞ , ∞ )3 NEC = 0 for m ∈ ( −∞ , ∞ ) , α ∈ ( −∞ , ∞ )Clearly WEC and DEC satisfy the condition of positivity while SEC vi-olates it, which imply the accelerated expansion of the universe. Case:II F ( R ) = R + βRln ( R ), β is constant. Some of the logarithmicmodels are studied in [20]. In this case, the eﬀective energy momentumreduces to, T eﬀ ij = T ij − βR κ g ij (28)which represents the eﬀective pressure and energy density for a perfect ﬂuidmatter as follows, ρ eﬀ = R (1 + β + βln ( R ))4 κ (29)and p eﬀ = − R (1 + β + βln ( R ))4 κ . (30)The equation of state parameter (EoS) reads as ω = − β and R >

S. no. Terms Results R ∈ (0 , e ) , β ∈ ( −∞ , − lnR ) ρ eff > R ∈ ( e , ∞ ) , β ∈ ( − lnR , ∞ )and for R = e , β ∈ ( −∞ , ∞ )2 for R ∈ (0 , e ) , β ∈ ( −∞ , − lnR )WEC, DEC > < R ∈ ( e , ∞ ) , β ∈ ( − lnR , ∞ )and for R = e , β ∈ ( −∞ , ∞ )3 NEC = 0 for any R > , β ∈ ( −∞ , ∞ ) CONFORMALLY FLAT GENERALIZED RICCI RECURRENT SPACETIME... 11

Figure 2.

Energy conditions for F ( R ) = R + βln ( R ) with − ≤ β ≤ . . ≤ R ≤ . . ≤ R ≤ .

35 and − ≤ β ≤ .

25. From the table and above ﬁgures it is clear that WEC andDEC satisﬁes the condition of positivity while SEC violates it and NEC isalways zero. The violation of SEC complies with the accelerated expansionof the universe which is compatible with recent observational studies.6.

Discussion

The modiﬁed theories of gravity have gained much attention to study thelate time acceleration of the universe. The diﬀerent energy conditions whichcan be derived from the well known Raychaudhuri equation plays an impor-tant role to deﬁne self consistencies of modiﬁed theories of gravity. In thispaper we examine the null, weak, dominant and strong energy conditionsfor the modiﬁed F ( R ) theories of gravity under conformally ﬂat generalizedRicci recurrent perfect ﬂuid spacetime with constant Ricci scalar. In twodiﬀerent F ( R ) models, F ( R ) = R + αR m where α, m are constants and F ( R ) = R + βRlnR where β is constant, we investigate diﬀerent energyconditions. The conditions derived in the section 4 are used to constrainthe model parameters in these F ( R ) models. The model parameter must satisfy R > ≤ α ≤ ≤ m ≤ R = 1 and − ≤ β ≤ .

25 , 0 . ≤ R ≤ .

35 for the two casesrespectively. On the other hand the strong energy condition violating itscondition of non-negativity and shows the negative behaviour in both thecases with the given constraints on model parameters and which favours theaccelerated expansion scenario of the universe.

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A. De, Department of Mathematical and Actuarial Sciences, UniversitiTunku Abdul Rahman, Jalan Sungai Long, 43000 Cheras, Malaysia

Email address : [email protected] T. H. Loo, Institute of Mathematical Sciences, University of Malaya,50603 Kuala Lumpur, Malaysia

Email address : [email protected] Raja Solanki, Department of Mathematics, Birla Institute of Technol-ogy and Science-Pilani, Hyderabad Campus, Hyderabad 500078, India

Email address : [email protected] P.K.Sahoo, Department of Mathematics, Birla Institute of Technologyand Science-Pilani, Hyderabad Campus, Hyderabad 500078, India

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