General properties of f(R) gravity vacuum solutions
Salvatore Capozziello, Carlo Alberto Mantica, Luca Guido Molinari
aa r X i v : . [ g r- q c ] J u l General properties of f ( R ) gravity vacuum solutions Salvatore Capozziello,
1, 2, ∗ Carlo Alberto Mantica, † and Luca Guido Molinari ‡ Dipartimento di Fisica “E. Pancini”, Universit`a di Napoli Federico II, Napoli,and INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy, Laboratory for Theoretical Cosmology, Tomsk State University of Control Systems and Radioelectronics (TUSUR),634050 Tomsk (Russia). I.I.S. Lagrange, Via L. Modignani 65, I-20161, Milano, Italyand INFN sez. di Milano, Via Celoria 16, I-20133 Milano, Italy Dipartimento di Fisica “A. Pontremoli”, Universit`a degli Studi di Milanoand INFN sez. di Milano, Via Celoria 16, I-20133 Milano, Italy. ‡ (Dated: July 28, 2020)General properties of vacuum solutions of f ( R ) gravity are obtained by the condition that thedivergence of the Weyl tensor is zero and f ′′ = 0. Specifically, a theorem states that the gradient ofthe curvature scalar, ∇ R , is an eigenvector of the Ricci tensor and, if it is time-like, the space-timeis a Generalized Friedman-Robertson-Walker metric; in dimension four, it is Friedman-Robertson-Walker. PACS numbers: 98.80.-k, 95.35.+d, 95.36.+xKeywords: Higher-order gravity; cosmology; perfect fluids
I. INTRODUCTION
The so called f ( R ) gravity is a natural extension of Einstein’s gravity where the Hilbert-Einstein action of gravita-tional field, linear in the Ricci scalar R , is substituted with a generic function f ( R ). The issues for this generalizationmainly come from inflationary cosmology [1], late-time acceleration [2] and the possibility to unify late and earlycosmic history [3, 4].Furthermore, it is the subject of a vast research as a potential alternative to the so far undetected exotic fields thatshould account for dark matter and dark energy. This alternative is geometric: the further degrees of freedom of f ( R ) gravity may produce observable effects at different astrophysical and cosmological scales that should be, other-wise, ascribed to exotic forms of matter. There are other motivations like quantum perturbative corrections on curvedspacetimes and the natural question about the consequences of a straightforward generalization of the Hilbert-Einsteinaction to consider f ( R ) gravity as the first logical step [5, 6].Starting from a general f ( R ) gravity action, the field equations can be written as G [ f ] kl = κT kl , (1)where G [ f ] replaces the Einstein tensor: G [ f ] kl = f ′ ( R ) R kl − f ′′′ ( R ) ∇ k R ∇ l R − f ′′ ( R ) ∇ k ∇ l R (2)+ g kl [ f ′′′ ( R ) ∇ j R ∇ j R + f ′′ ( R ) ∇ R − f ( R )] . It becomes the standard expression, G kl = R kl − Rg kl , for f ( R ) = R . T kl is the stress-energy tensor for standardmatter and κ is the gravitational coupling. A main requirement is that the contracted Bianchi identity ∇ k G kl = 0continues to hold in f ( R ) gravity, that is: ∇ k G [ f ] kl = f ′′ ( R )[ R kl ∇ k R − ∇ ∇ l R + ∇ l ∇ R ] = 0 . Therefore f ( R ) theories are compatible with the physical requirement of energy-momentum conservation.In Ref.[7] we obtained a sufficient condition for G [ f ] kl to have the ‘perfect-fluid’ form G [ f ] kl = g ( R ) g kl + g ( R ) u k u l , (3) ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] for any smooth f ( R ) model, where u k is some time-like unit vector field. The requirement that each tensor in theexpression (2) of G [ f ] kl have the perfect fluid form, namely the Ricci tensor R kl , ∇ k R ∇ l R and ∇ k ∇ l R , can be statedas follows:C1) ∇ k u j = ϕ ( u j u k + g jk ) with ∇ k ϕ = − ˙ ϕu k ,C2) ∇ m C jklm = 0, where C jklm is the Weyl tensor.Condition C1 characterizes the space-time as a generalized Friedman-Robertson-Walker (GFRW) space-time. Theadditional condition 2 implies that u k ∝ ∇ k R . Then ∇ k R is eigenvector of the Ricci tensor.In n = 4 dimensions, C1 and C2 imply that the space-time is Friedman-Robertson-Walker [22] Prop. 4.1. The unicityof the vector u k realizing a GFRW space-time is discussed in [8].The perfect-fluid structure of G [ f ] kl and T kl , required in large-scale cosmology, put f ( R ) geometric extension ofGeneral Relativity on the same footing as exotic modifications of standard matter. In other words, the question iswhether cosmic phenomenology can be addressed by requiring exotic forms of matter/energy beyond the StandardModel of Particles, or gravity is not scale invariant and modifications are required at galactic scales and beyond [9, 10].Hereafter, we investigate vacuum solutions ( T kl = 0) in space-times of dimension n in order to derive some generalproperties of f ( R ) gravity.Vacuum solutions have been studied based on special forms of f ( R ) or assumptions about symmetries of the metric[11]. Perturbations of models like f ( R ) = R δ in spherical symmetry are studied in [12, 13]. The non-linear equationsfor f ′ for spherical-symmetric vacuum solutions are studied in [14], and solved for some cases. Constant curvaturesolutions, i.e. f ′ ( R ) R − f ( R ) = 0, were found with cylindrical symmetry [15, 16], plane-symmetry [17], and localrotational symmetry [18]. Spherically symmetric solutions in connection to black-holes and the rotation curve ofgalaxies are studied in [19].In this work we want to characterize the vacuum solutions of any smooth f ( R ) theory with null divergence of theWeyl tensor. The results can be summarized in the following theorem: Theorem 1.
For any smooth function f ( R ) with f ′′ = 0 , the vacuum solution with null divergence of the Weyltensor ( ∇ m C jklm = 0 ), has the properties:1) ∇ k R is an eigenvector of the Ricci tensor: R jk ∇ k R = ξ ∇ j R .2) If ∇ k R = αu k with u k u k = − , the Ricci tensor has the perfect fluid (quasi-Einstein) form: R kl = R − nξn − u k u l + R − ξn − g kl . (4)
3) The vector u k is vorticity-free, acceleration-free and shear-free, and satisfies the relation: ∇ j u k = ϕ ( u j u k + g jk ) , (5) where ∇ k ϕ = − ˙ ϕu k . The property (5) characterizes the Lorentzian space-time as a generalised Robertson-Walker space-time. It is usuallydescribed as a time-warped space-time, i.e. there is a coordinate frame where the metric tensor is: ds = − dt + a ( t ) g ∗ µν ( x ) dx µ dx ν , (6)and g ∗ is a Riemannian metric [20, 21]. For such space-times we proved the special property C jklm u m = 0 iff ∇ m C jklm = 0 [22]. In n = 4 it implies that the Weyl tensor is zero if the divergence vanishes, i.e. the space-time isRobertson-Walker.The proof of the above Theorem is given in the next sections. In Sect.2, we discuss ∇ k R which is an eigenvectorof the Ricci tensor. In Sect.3 the related form of Ricci tensor is obtained. Sect.4 is devoted to the vector u k which isvorticity and acceleration-free; its shear is proportional to the electric component of the Weyl tensor σ kl ∝ C jklm u j u m .In Sect.5 an equation for the time evolution of the shear is obtained. A vorticity and acceleration free velocity fieldrestricts the metrics to the form ds = − dt + a ( t, x ) µν dx µ dx ν . By the equations in Appendix we show, in Sect.6,that the special form of the shear implies that it is zero. The consequences are discussed in the conclusions (Sect.7).All results are then simplified because the Ricci tensor is quasi-Einstein, the Ricci tensor of the space-submanifold isEinstein, the space-time is a GFRW. In Appendix are reported all the quantities used for the derivations developedin the paper. We indicate with a dot the directional derivative u k ∇ k and with a prime the derivative with respect to R . II. ∇ k R IS AN EIGENVECTOR OF THE RICCI TENSOR
Let us rewrite the field equations G [ f ] kl = 0, with G [ f ] given in Eq.(2), as: R kl = A ∇ k R ∇ l R + B ∇ k ∇ l R − Cg kl , (7)where A = f ′′′ f ′ , B = f ′′ f ′ , C = A ∇ j R ∇ j R + B ∇ R − f f ′ . The trace is: R = − ( n − A ∇ k R ∇ k R + B ∇ R ] + n f f ′ . (8)Then ( n − C = − R + f f ′ . (9) Proof : let us evaluate ∇ j R kl = ( A ′ ∇ j R ) ∇ k R ∇ l R + A ( ∇ j ∇ k R ) ∇ l R + A ∇ k R ( ∇ j ∇ l R )+( B ′ ∇ j R ) ∇ k ∇ l R + B ∇ j ∇ k ∇ l R − C ′ g kl ∇ j R .
As said, the prime indicates the derivative in R and we antisymmetrize in two indices: ∇ j R kl − ∇ k R jl =( A − B ′ )[( ∇ k R ) ∇ j ∇ l R − ( ∇ j R ) ∇ k ∇ l R ]+ BR jklm ∇ m R − C ′ ( g kl ∇ j R − g jl ∇ k R ) . The divergence of the Weyl tensor is ∇ m C jklm = − n − n − (cid:20) ∇ j R kl − ∇ k R jl − g kl ∇ j R − g jl ∇ k R n − (cid:21) . If ∇ m C jklm = 0 then: g kl ∇ j R − g jl ∇ k R n −
1) =( A − B ′ )[( ∇ k R ) ∇ j ∇ l R − ( ∇ j R ) ∇ k ∇ l R ]+ BR jklm ∇ m R − C ′ ( g kl ∇ j R − g jl ∇ k R ) . Note that A − B ′ = B and C ′ = − n − − n − ff ′ B − B g kl ∇ j R − g jl ∇ k R n − ff ′ = B [( ∇ k R ) ∇ j ∇ l R − ( ∇ j R ) ∇ k ∇ l R ] + BR jklm ∇ m R .
We can factor out B = 0. The term B ∇ j ∇ l R is obtained from the expression (7) of the Ricci tensor. In thissubstitution, terms with A cancel out: − g kl ∇ j R − g jl ∇ k R n − ff ′ = ( ∇ k R )( R jl + Cg jl ) − ( ∇ j R )( R kl + Cg kl ) + R jklm ∇ m R .
A simplification occurs with Eq.(9):0 = ( ∇ k R )( R jl − Rn − g jl ) − ( ∇ j R )( R kl − Rn − g kl ) + R jklm ∇ m R . (10)The contraction with ∇ l R cancels out terms, leaving ( ∇ k R ) R jl ∇ l R = ( ∇ j R ) R kl ∇ l R . The equation is solved by: R jl ∇ l R = ξ ∇ j R , (11)for some eigenvalue ξ . This completes the proof of point 1 of the Theorem 1. (cid:3) III. THE RICCI TENSOR
Let us obtain the structure of the Ricci tensor with the approach used in [22]. The result will be simplified aftershowing that the shear of u k is zero.In the following, we refer to a time-like unit vector: ∇ k R = αu k , where u k u k = − u k is an eigenvector of the Riccitensor, R kj u j = ξu k . Eq.(10) is R jklm u m = − u k ( R jl − Rn − g jl ) + u j ( R kl − Rn − g kl ) . (12)Contracting with u j , gives: R jklm u j u m = − ξu k u l − R kl + Rn − g kl + u k u l ) . (13)The Weyl tensor C jklm = R jklm + g jm R kl − g km R jl + g kl R jm − g jl R km n − − R g jm g kl − g km g jl ( n − n − , is contracted with u j u m to obtain the Ricci tensor, and Eq.(13) is used. It is( n − C jklm u j u m =( n − R jklm u j u m − R kl − ξ (2 u k u l + g kl ) + R g kl + u k u l n − R − nξ ) u k u l − ( n − R kl + ( R − ξ ) g kl The resulting Ricci tensor has a quasi-Einstein term and a Weyl term: R kl = R − nξn − u k u l + R − ξn − g kl − n − n − C jklm u j u m (14)The Weyl term will be shown to be zero. IV. THE VECTOR FIELD u k Let us obtain now the properties of the vector field u k ( ∇ k R = αu k ).We rewrite the Ricci tensor (7) in terms of u k : R kl = Aα u k u l + B ( ∇ k α ) u l + Bα ∇ k u l − Cg kl . The contraction with u l gives: ξu k = − Aα u k − B ( ∇ k α ) − Cu k . Then ∇ k α is proportional to u k : ∇ k α = − ˙ αu k (15)and the Ricci tensor is: R kl = ( Aα − B ˙ α ) u k u l + Bα ( ∇ k u l ) − Cg kl . (16) Lemma IV.1.
The vector field u k is vorticity-free and acceleration-free.Proof. Eq.(15) and the identity ∇ k ∇ j R = ∇ j ∇ k R , i.e. ∇ k ( αu j ) = ∇ j ( αu k ), give ∇ j u k − ∇ k u j = 0 . (17)Contraction with u j gives zero acceleration: u j ∇ j u k = 0 because u j u j = − u k has the structure ∇ j u k = ϕ ( u j u k + g jk ) + σ jk , (18)with ϕ = ∇ i u i n − , and shear tensor σ jk (traceless, symmetric and σ jk u k = 0). The Ricci tensor becomes: R kl = ( Aα − B ˙ α + Bαϕ ) u k u l + ( Bαϕ − C ) g kl + Bασ kl . (19)Comparison with the expression (14) gives Bασ kl = − n − n − C jklm u j u m , and the relations R − nξ = ( n − Aα − B ˙ α + Bαϕ ) R − ξ = ( n − Bαϕ − C ) . The second one simplifies with Eq.(9), and it is used in the first one:( n − Bαϕ = f f ′ − ξ (20)( n − Aα − B ˙ α + ξ ) = R − f f ′ . (21) Remark IV.2.
A consequence of Eq. (12) is that u k is Riemann-compatible [23], that is: u i R jklm u m + u j R kilm u m + u k R ijlm u m = 0 . It has been proven [24] that this property also implies that u k is Weyl compatible: u i C jklm u m + u j C kilm u m + u k C ijlm u m = 0 . It follows that C jklm u m = u k E jl − u j E kl , where E kl = C jklm u j u m is the electric part of the Weyl tensor. V. THE SHEAR σ kl Lemma V.1.
Let us prove that ∇ k σ kl = u l ( σ km σ km ) , (22) ∇ k ϕ = − ˙ ϕu k . (23) Proof.
Since ∇ k C jklm = 0, ∇ k ( Bα ) = ( B ′ ∇ k R ) α − B ˙ αu k = ( B ′ α − B ˙ α ) u k and u k σ jk = 0 one evaluates: Bα ∇ k σ kl = − n − n − C jklm ∇ k ( u j u m ) = − n − n − u j C jklm σ km . Now, let us use the property (see the above Remark) u j C jklm = u l E km − u m E lk : Bα ∇ k σ kl = − n − n − u l E km σ km = u l Bασ km σ km . The second statement results from the identity R jk u j = ∇ u k − ∇ k ∇ j u j , i.e. ξu k = ∇ l ( ϕ ( u l u k + g lk + σ lk ) − ( n − ∇ k ϕ = ˙ ϕu k + ( n − ϕ u k + ∇ k ϕ + ∇ l σ kl − ( n − ∇ k ϕ =[ ˙ ϕ + ( n − ϕ + σ ij σ ij ] u k − ( n − ∇ k ϕ . The contraction with u k gives: ξ = ( n − ϕ + ˙ ϕ ) + σ kl σ kl , (24)and the previous equation simplifies to ( n − ∇ k ϕ + u k ˙ ϕ ) = 0.It is now possible to obtain an equation for the shear. Eq.(12) simplifies with the expression (14) of the Ricci tensor:[ ∇ j , ∇ k ] u l = u k (cid:20) ξn − g jl − Bασ jl (cid:21) − u j (cid:20) ξn − g kl − Bασ kl (cid:21) . The left-hand side, with the aid of the expression for ∇ j u k , is:( ˙ ϕ + ϕ )( u k g jl − u j g kl ) + ϕ ( u k σ jl − u j σ kl ) + ∇ j σ kl − ∇ k σ jl . We then obtain, with (24): ∇ j σ kl − ∇ k σ jl = ( ϕ + Bα )( u j σ kl − u k σ jl ) − σ rs σ rs n − u j g kl − u k g jl ) . (25)The contraction with u j and Eq. (22) gives:˙ σ kl + σ kl + (2 ϕ + Bα ) σ kl = σ rs σ rs n − g kl + u k u l ) . (26)This equation, considered in a useful coordinate frame, will imply that σ jk = 0. VI. THE COMOVING FRAME
Since u k is vorticity-free and acceleration-free, in the coordinates ( t, x , ..., x n − ) where u = − u µ = 0, themetric of the Lorentzian manifold has the block structure [25] eq.2.19: g ij dx i dx j = − dt + a µν ( t, x ) dx µ dx ν , where, at any time, the metric a µν is Riemannian. With the formulae in Appendix, the relation ( n − ϕ = ∇ k u k becomes: ( n − ϕ = Γ µµ = a µν ˙ a µν . (27)The relation ∇ k u j = ϕ ( u i u j + g ij ) + σ ij gives σ = 0, σ µ = 0 and σ µν = Γ µν − ϕa µν = ˙ a µν − ϕa µν , (28) Proposition VI.1.
It is possible to show that σ µν = 0 .Proof. The shear is a purely spatial tensor and σ µν a µν = 0. Let us evaluate: σ µν σ µν = ( ˙ a µν − ϕa µν ) a µτ a νσ ( ˙ a τσ − ϕa τσ )= ˙ a µν a µτ a νσ ˙ a τσ − ϕ ˙ a µν a µν + ϕ ( n − − ˙ a µν ˙ a µτ a νσ a τσ − ϕ ( n −
1) + ϕ ( n − − ˙ a µν ˙ a µν − ( n − ϕ . We used 0 = ˙ a µν a νρ + ˙ a νρ a µν .The equation for the shear tensor is0 = ˙ σ µν + σ µν + (2 ϕ + Bα ) σ µν − σ rs σ rs n − a µν . The trace is 0 = ˙ σ µν a µν + σ µν a µν + (2 ϕ + Bα ) σ µν a µν − σ rs σ rs = − σ µν ˙ a µν = − ˙ a µν ˙ a µν + ϕa µν ˙ a µν = − ˙ a µν ˙ a µν − ϕ ˙ a µν a µν = − ˙ a µν ˙ a µν − n − ϕ = 2 σ µν σ µν . Since σ µν is symmetric, the eigenvalues are real. The vanishing of the sum of squared eigenvalues means σ ij = 0. Proposition VI.2.
The property a µν ( t, x ) = a ( t ) g ∗ µν ( x ) holds.Proof. In the comoving frame u µ = 0, then Eq.(12) gives R µνρ = 0. Its expression in Appendix shows that ˙ a µν isthe Codazzi tensor: D µ ˙ a νρ = D ν ˙ a µρ .Since σ µν = 0, we obtain ˙ a µν ( t, x ) = 2 ϕa µν ( t, x ) . Then a νρ D µ ϕ = a µρ D ν ϕ ; this is true if ∂ ν ϕ = 0 i.e. ϕ only depends on time. Integration gives the warped expression a µν ( t, x ) = a ( t ) g ∗ µν ( x ), where ϕ ( t ) = ˙ aa , which is nothing else but the Hubble parameter of Friedman cosmology. VII. DISCUSSION AND CONCLUSIONS
In this paper, we discussed some properties of vacuum solutions of f ( R ) gravity. Specifically, if the shear σ jk vanishes, several formulae result simplified. In particular, the Weyl term in the Ricci tensor (14) cancels and the Riccitensor becomes quasi-Einstein as in Eq. (4). Finally, the velocity has the simplest expression for the gradient (5).Furthermore, the vanishing of the Weyl tensor divergence implies C jklm u m = 0 and E kl ≡ C jklm u j u m = 0. In thecomoving frame, E kl = 0 and a µν ( t, x ) = a ( t ) g ∗ µν ( x ) give a space-submanifold that is Einstein: r µν = r ∗ n − g ∗ µν , with r ∗ = ra constant and R = r ∗ a + 2 ξ + ϕ ( n − n − . (29)The eigenvalue is ξ = ( n − ϕ + ˙ ϕ ) = ( n − ¨ aa . The parameter α is evaluated as α = − ˙ R .The parameters α , R , ξ and ϕ can all be expressed in terms of the constant r ∗ and of derivatives of the scale parameter a . The f ( R ) gravity enters in the relations (20) and (21), that contain f and its derivatives in R up to f ′′′ :( n − f ′′ ˙ Rϕ = ξf ′ − f f ′′′ ˙ R + f ′′ ¨ R + ξf ′ = 1 n − Rf ′ − f µ = 0); the second equation is Eq.9 (with p = 0) plus n − n − timesEq.10 in [7].According to these considerations, if the divergence of the Weyl tensor is zero, the vacuum solutions of f ( R ) gravityare Generalized Friedman-Robertson-Walker space-times which, in 4-dimensions, reduce to the standard Friedman-Robertson-Walker solutions. This result holds, in particular, for f ′′ = 0 and then it generalizes the fact that f ( R )gravity in vacuum can be reduced to General Relativity plus a cosmological constant as often stated. In the presentperspective, the properties of the Weyl tensor and the geodesic structure determine the solutions.In 4-dimensions, where the FRW metric is obtained, the difference with ordinary gravity is in the equations for theevolution of the scale function: the Friedmann equations for f ( R ) = R and equations (30), (31) for f ( R ) with nulldivergence of the Weyl tensor. Therefore, the effect of f ( R ) with Weyl constraint is to dress the original Friedmannequations with new geometric effects that yield the same form of metric and manifest in a different scale function.There are cosmological spaces that elude this analysis, because of physical processes that do not allow the stringentrequirement on the Weyl tensor to hold. For example, this is the case of de Sitter solutions derived in f ( R ) gravitymotivated by the inflation and the dark energy issues. In Ref.[26], the one-loop quantization approach is developedfor a family of f ( R ) gravity models and de Sitter universes are investigated, extending a similar earlier program forEinstein gravity. The authors adopt a generalized zeta regularization, and the one-loop effective action is obtainedoff-shell. In this framework, the (de)stabilization of the de Sitter background is obtained by quantum effects. In sucha context, the requirements of Theorem 1 are violated by quantum effects. In fact, they consider small fluctuationsaround a (de Sitter) maximally symmetric space with Riemann tensor R (0) jklm = R (0)
12 ( g (0) jl g (0) km − g (0) jm g (0) kl )and metric g (0) ij . In this metric the Weyl tensor is zero. However, in a perturbed metric g ij = g (0) ij + h ij the shape ofthe Ricci tensor (eq. 2.11 in [27]) in general prevents the validity of ∇ m C jklm = 0.It is important to stress that one-loop effective actions are useful also for studying black hole nucleation rates and forproviding reliable mechanisms capable of solving the cosmological constant problem.In conclusion, the mathematical results reported here highlight the decisive effect of the Weyl condition in restrictingthe form of the vacuum solutions, and this should be of guidance in the study of realistic situations where physicalprocesses are acting in the observable universe. Acknowledgments
SC acknowledges the Istituto Nazionale di Fisica Nucleare (INFN), sezione di Napoli, iniziative specifiche
MOON-LIGHT2 and QGSKY.
APPENDIX
Given the space-time metric ds = − dt + a µν ( t, x ) dx µ dx ν let γ ρµν , r µνρσ , r µν and r be the Christoffel symbols, the Riemann tensor, the Ricci tensor and the Ricci scalar of thespace-submanifold at fixed t , and let D µ be the covariant derivative with symbols γ ρµν , and define a µν a µρ = δ µρ . Therelated space-time quantities are: • Christoffel symbols: Γ kij = Γ kji = g km ( ∂ i g jm + ∂ j g im − ∂ m g ij )Γ = 0 , Γ µ = 0 , Γ µ = 0 , Γ µν = ˙ a µν , Γ νµ = a νρ ˙ a µρ , Γ ρµν = γ ρµν • Riemann tensor: R jklm = ∂ k Γ mjl − ∂ j Γ mkl + Γ ijl Γ mik − Γ ikl Γ mij R µνρσ = r µνρσ + a σλ ( ˙ a µρ ˙ a λν − ˙ a νρ ˙ a λµ ) R µνρ = ( ∂ ν ˙ a µρ − ∂ µ ˙ a νρ ) + ( γ σµρ ˙ a νσ − γ σνρ ˙ a σµ ) = ( D ν ˙ a µρ − D µ ˙ a νρ ) R µ ν = − a νρ ¨ a µρ − ˙ a νρ ˙ a µρ R µ ν = ¨ a µν − a ρσ ˙ a νσ ˙ a µρ • Ricci tensor: R ij = R ikj k R = − a µν ¨ a µν − ˙ a µν ˙ a µν R µν = r µν − ˙ a µρ a ρσ ˙ a νσ + ¨ a µν + ˙ a µν ( a ρσ ˙ a ρσ ) • Curvature scalar R = − R + a µν R µν R = r + a µν ¨ a µν + ˙ a µν ˙ a µν + ( a µν ˙ a µν ) • Electric tensor E µν = C µν ( n − E µν = − r µν + ( n − a µν − ( n − a ρσ ˙ a ρµ ˙ a σν − ˙ a µν ( a ρσ ˙ a ρσ ) + a µν ( R + Rn − ) [1] A. A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. (1980) 99[2] S. Capozziello, Curvature quintessence, Int. J. Mod. Phys. D (2002) 483[3] S. Nojiri and S. D. Odintsov, “Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariantmodels,” Phys. Rept. (2011) 59[4] S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-timeEvolution, Phys. Rept. (2017) 1[5] S. Capozziello and V. Faraoni, ”Beyond Einstein gravity”, Fundamental theories of Physics, Vol.170 (Springer, 2011)[6] S. Capozziello and M. De Laurentis, Extended theories of gravity , Phys. Rep. n.4–5 (2011) 167–321.[7] S. Capozziello, C. A. Mantica and L. G. Molinari, Cosmological perfect fluids in f ( R ) gravity, Int. J. Geom. Meth. Mod.Phys. (2019), 1950008 (14 pp).[8] L. G. Molinari, A. Tacchini and C. A. Mantica, On the uniqueness of a shear-vorticity-acceleration-free velocity field inspace-times , Gen. Relativ. Gravit. (2019) 217 (13pp.)[9] S. Capozziello, P. Jovanovic, V. B. Jovanovic and D. Borka, Addressing the missing matter problem in galaxies through anew fundamental gravitational radius, JCAP (2017) 044[10] S. Capozziello, V. B. Jovanovic, D. Borka and P. Jovanovic, Constraining theories of gravity by fundamental plane ofelliptical galaxies, Phys. Dark Univ. (2020) 100573[11] S. Capozziello, A. Stabile and A. Troisi, Spherical symmetry in f(R)-gravity, Class. Quantum Grav. (2008) 085004[12] T. Clifton, Spherically symmetric solutions to fourth-order theories of gravity , Class. Quantum Grav. n.24 (2006) 7445[13] V. Faraoni, Cliftons spherical solution in f(R) vacuo harbours a naked singularity , Class. Quantum Grav. (2009) 195013.[14] T. R. P. Caram´es and E. R. B. de Mello, Spherically symmetric vacuum solutions of modified gravity theory in higherdimensions , Eur. Phys. J. C (2009) 113.[15] A. Azadi, D. Momeni, and M. Nouri-Zonoz, Cylindrical solutions in metric f(R) gravity , Phys. Lett. B (2008) 210–214.[16] D. Momeni and H. Gholizade,
A note on constant curvature solutions in cylindrically symmetric metric f(R) gravity , Int.J. Mod. Phys. D (2009) 1.[17] M. Sharif and M. Farasat Shamir, Plane symmetric solutions in f(R) gravity , Mod. Phys. Lett. A n.15 (2010) 1281–1288.[18] M. J. Amir and S. Sattar, Locally rotationally symmetric vacuum solutions in f(R) gravity , Int. J. Theor. Phys. (2014)773–787.[19] M. Calz`a, M. Rinaldi and L. Sebastiani, A special class of solutions in F(R)-gravity , Eur. Phys. J. C (2018) 178.[20] B.-Y. Chen, A simple characterization of generalized Robertson-Walker spacetimes , Gen. Relativ. Gravit. Generalized RobertsonWalker spacetimes - a survey , Int. J. Geom. Meth. Mod. Phys. n.3 (2017) 1730001 (27 pp.).[22] C. A. Mantica and L. G. Molinari, On the Weyl and Ricci tensors of Generalized Robertson-Walker space-times , J. Math.Phys. Jordan algebras of Riemann, Weyl and curvature compatible tensors , arXiv:1910.03929[math.dg][24] C. A. Mantica and L. G. Molinari,
Extended Derdzi´nski-Shen theorem for curvature tensors , Colloq. Math. n. 1 (2012)1-6.[25] A. A. Coley and D. McManus,
On spacetimes admitting shear-free, irrotational, geodesic timelike congruences , Class.Quantum Grav. (1994) 1261–1282.[26] G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, S. Zerbini, One-loop f ( R ) gravity in de Sitter universe , JCAP Semiclassical perdurance of De Sitter space , Nucl. Phys. B222