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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