# Conformally Flat Pseudoprojective Symmetric Spacetimes infR,G Gravity

Authors: Uday Chand De; Sameh Shenawy; Abdallah Abdelhameed Syied; Nasser Bin Turki
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

Sufficient conditions on a pseudoprojective symmetric spacetimePPSn whose Ricci tensor is of Codazzi type to be either a perfect fluid or Einstein spacetime are given. Also, it is shown that a PPSn is Einstein if its Ricci tensor is cyclic parallel. Next, we illustrate that a conformally flat PPSn spacetime is of constant curvature. Finally, we investigate conformally flat PPS4 spacetimes and conformally flat PPS4 perfect fluids in fR,G theory of gravity, and amongst many results, it is proved that the isotropic pressure and the energy density of conformally flat perfect fluid PPS4 spacetimes are constants and such perfect fluid behaves like a cosmological constant. Further, in this setting, we consider some energy conditions.


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## Body

## 1. Introduction

The notion of a pseudoprojective symmetric manifold, briefly denoted byPPSn, was first introduced and studied in 1989 by Chaki and Saha [1]. Such a manifold is a nonflat pseudo-Riemannian manifold whose projective curvature tensor [2]
(1)Phijk=Rhijk−1n−1ghkRij−ghjRik,satisfies the condition
(2)∇lPhijk=2λlPhijk+λhPlijk+λiPhljk+λjPhilk+λkPhijl,where Rhijk is the Riemann curvature tensor, Rij is the Ricci tensor, λl is a nonzero 1-form, and ∇ denotes the covariant differentiation with respect to the metric g. In [1], it was proved that a PPSn manifold is of constant scalar curvature, that is,
(3)∇lR=0,and λl is an eigenvector of the Ricci tensor and the corresponding eigenvalue is R/n, that is,
(4)λlRlk=Rnλk.Also, it was shown that if aPPSn manifold admits a unit parallel vector field, then it is reduced to a pseudosymmetric manifold [3].Ann-dimensional Lorentzian manifold M is said to be a pseudoprojective symmetric spacetime if its projective curvature tensor P agrees with (2). A Lorentzian manifold M is said to be perfect fluid if its Ricci tensor satisfies
(5)Rij=αgij+βuiuj,where α and β are scalar fields and uiui=−1, that is, ui is a time-like velocity vector field [4, 5]. In differential geometry, a manifold satisfying the foregoing relation of the Ricci tensor is called a quasi-Einstein manifold without any restrictions on the velocity vector field ui[6, 7]. Throughout this paper, let λl be a unit timelike vector field.The standard theory of gravity follows from Einstein’s field equations (EFE) [8, 9].
(6)Rij−R2gij=κTijm,where R, κ, and Tijm are the scalar curvature tensor and the Newtonian gravitational constant, and Tikm is the energy-momentum tensor describing the ordinary matter. These equations correlate the geometry of a spacetime with its matter content. That is, the geometry of a spacetime determines the matter content of the spacetime conversely. Many modifications of EFE have been introduced and studied on a large scale (see references [10–12] for examples of the modified gravity theories). Amongst these modified theories, there was one known under the name fR,G gravity theory [13], which is obtained by replacing the scalar curvature R with a function fR,G of the scalar curvature R and Gauss-Bonnet scalar G.
(7)G=RhijkRhijk−4RhkRhk+R2,in the gravitational action term
(8)S=12κ∫d4x−gfR,G+Smatter,with Smatter being the action term of the standard matter fields. The fR,G field equations are given by
(9)κTikm=−12fgik+fGRikR−4RhilkRhl+2RihmnRkhmn−4RihRhk+2gikR∇2+4Rih∇h∇k+4Rkh∇h∇i−2R∇i∇k−4Rik∇2fG+4Rlihk∇l∇h−4gikRhl∇h∇lfG+Rik−∇i∇k+gik∇2fR,where Tikm results from Smatter and fG=∂GfR,G, fR=∂RfR,G[14].In this paper, we investigatePPSn spacetimes whose Ricci tensor is of Codazzi type or cyclic parallel. Next, a conformally flat PPSn spacetime is studied. After that, certain investigations of conformally flat PPS4 spacetimes in fR,G modified gravity theory are carried out. Finally, we study conformally flat PPS4 perfect fluid spacetimes in fR,G gravity.


## 2. On aPPSn Spacetime Whose Ricci Tensor Is of Codazzi Type or Cyclic Parallel

In this section, aPPSn spacetime whose Ricci tensor is of Codazzi type or cyclic parallel is considered. The Ricci tensor Rij is called of Codazzi type if [15, 16]
(10)∇lRhk=∇kRhl,whereas Rij is called cyclic parallel if
(11)∇kRhl+∇lRhk+∇hRlk=0.Transvecting equation (1) with gij, one gets
(12)Phk=1n−1nRhk−ghkR,where Phk=gijPhijk.Contracting equation (2) with gij, we obtain
(13)∇lPhk=2λlPhk+λhPlk+λjPhljk+λiPhilk+λkPhl.Using (1) and (12) in (13), we have
(14)∇lRhk=2λlnnRhk−ghkR+λhnnRlk−glkR+n−1nλjRhljk+λiRhilk−1nghkλjRlj−λhRlk−1nghkλiRil−ghlλiRik+λknnRhl−ghlR+1nghk∇lR.With the help of equations (3) and (4), one finds
(15)∇lRhk=−2n−2n2Rλlghk+1−nn2Rλkghl−1nRλhglk+λkRhl+2λlRhk+n+1nλhRlk+n−1nλjRhljk+λiRhilk.First, suppose that the Ricci tensor ofPPSn spacetime is of Codazzi type; thus, we have
(16)∇lRhk−∇kRhl=0.The use of equation (15) in (16) implies that
(17)0=−n−3n2Rλlghk+n+3n2Rλkghl−λkRhl+λlRhk+n−1nλjRhljk−λjRhkjl+2λiRhilk.It is to be noted that the Riemann curvature tensor has the following properties:(18)Rhljk+Rhklj+Rhjkl=0,Rhkjl=−Rhklj.The use of the above properties of the Riemann curvature tensor in equation (17) implies
(19)0=−n−3n2Rλlghk+n+3n2Rλkghl−λkRhl+λlRhk−3n−1nλjRhjkl.Contracting withλl and using (4), we have
(20)Rhk=n+3n2Rghk+3n2Rλkλh−3n−1nλlλjRhjkl.We thus can state the following theorem:Theorem 1.
LetM be a PPSn spacetime whose Ricci tensor is of Codazzi type; then, the Ricci tensor of M is given by (20).Suppose thatλlλjRhjkl=0, then (20) becomes
(21)Rhk=n+3n2Rghk+3n2Rλkλh,which means that aPPSn spacetime is perfect fluid.Corollary 2.
LetM be a PPSn spacetime whose Ricci tensor is of Codazzi type. Then, M is perfect fluid if λlλjRhjkl=0.The conformal curvature tensor is given by [17].
(22)Chjkl=Rhjkl−1n−2ghlRjk+gjkRhl−ghkRjl−gjlRhk+Rn−1n−2ghlgjk−ghkgjl.A contraction withλlλj implies
(23)λjλlRhjkl=λjλlChjkl+Rλhλknn−1−Rghknn−1n−2+Rhkn−2.Equations (20) and (23) are combined to give
(24)Rhk=Rnghk−3n−1n−2n2+n−3λjλlChjkl,where Chk=λjλlChjkl is the contracted Weyl tensor. Hence, we can state the following theorem:Theorem 3.
LetM be a PPSn spacetime whose Ricci tensor is of Codazzi type; then, the Ricci tensor of M is of the form (24).In particular case, ifChk=0, then equation (24) is reduced to be in the following form:
(25)Rhk=Rnghk,which means a PPSn spacetime is Einstein.Corollary 4.
LetM be a PPSn spacetime whose Ricci tensor is of Codazzi type. Then, M is Einstein if the contracted Weyl tensor vanishes.Assume thatM has cyclic parallel Ricci tensor, that is, the Ricci tensor agrees with (11). Then, using (15) in (11) infers
(26)0=−4nRλkghl−4n+1n2Rλlghk−4n+2n2λhglkR+4n+1nλlRhk+4λkRhl+4n+2nλhRlk.Contracting withλl and using equation (4), we obtain
(27)Rhk=Rnghk,which means a PPSn spacetime whose Ricci tensor obeys (4) is Einstein. Hence, we motivate to state the following theorem:Theorem 5.
LetM be a PPSn spacetime whose Ricci tensor is cyclic parallel; then M is an Einstein spacetime.


## 3. Conformally FlatPPSn Spacetimes

The divergence of the conformal curvature is expressed as [18]
(28)∇hCijkh=n−3n−2∇kRij−∇jRik−12n−1gij∇kR−gik∇jR.A spacetimeM is called conformally flat if the conformal curvature tensor vanishes, that is, Cijkl=0. It is well-known that if Cijkl=0, then ∇hCijkh=0. And consequently, the following equations hold
(29)Rhijk=1n−2ghkRij+gijRhk−ghjRik−gikRhj−Rn−1n−2ghkgij−ghjgik,(30)∇kRhl−∇lRhk=12n−1gij∇kR−gik∇jR.Since inPPSn spacetime the scalar curvature is constant, then equation (30) implies that
(31)∇kRhl=∇lRhk,which shows that the Ricci tensor is of Codazzi type [19]. We thus can conclude the following theorem:Theorem 6.
LetM be a PPSn spacetime with a divergence-free conformal curvature tensor; then, the Ricci tensor of M is of Codazzi type.In view of Theorem1, we can state the following corollary:Corollary 7.
LetM be a PPSn spacetime with a divergence-free conformal curvature tensor; then, the Ricci tensor of M is given by
(32)Rhk=n+3n2Rghk+3n2Rλkλl−3n−1nλlλjRhjkl.From equation (29), we can get
(33)λjRhljk=−Rnn−1n−2ghkλl+Rnn−1n−2λhglk+1n−2λlRhk−1n−2λhRlk,(34)λiRhilk=−Rnn−1n−2ghkλl+Rnn−1n−2λkghl+1n−2λlRhk−1n−2λkRhl.Using (33) and (34) in (15), one obtains
(35)∇lRhk=−2n2−n−1n2n−2Rλlghk−n2−3n+1n2n−2Rλkghl−n2−2n−1nn−2Rλhglk+n2−3n+1nn−2λkRhl+2n2−2n−2nn−2λlRhk+n2−2n−1nn−2λhRlk.It follows that(36)∇lRhk−∇kRhl=−n2+n−3n2n−2Rλlghk+n2+n−3n2n−2Rλkghl−n2+n−3nn−2λkRhl+n2+n−3nn−2λlRhk.In a conformally flatPPSn spacetime, the Ricci tensor is of Codazzi type; therefore,
(37)0=−n2+n−3n2n−2Rλlghk+n2+n−3n2n−2Rλkghl−n2+n−3nn−2λkRhl+n2+n−3nn−2λlRhk.Contracting withλl and using equation (4), we get
(38)Rhk=Rnghk,which illustrates that a conformally flat PPSn spacetime is Einstein.Theorem 8.
A conformally flatPPSn spacetime is Einstein.The use of (38) in (1) implies that
(39)Phijk=Rhijk−Rnn−1ghkgij−ghjgik.Then, from (39) in (29), one infers
(40)Pijkl=1n−2gilRjk+gjkRil−gikRjl−gjlRik−2Rnn−2gilgjk−gjkgjl.Hence, from (38), we get
(41)Pijkl=0.From (38) and (41) in (1), we have
(42)Rhijk=Rnn−1ghkgij−ghjgik,which means that a conformally flat PPSn spacetime is of constant curvature.In consequence of the above, we can state the following theorem:Theorem 9.
A conformally flatPPSn spacetime is projectively flat and of constant curvature.


## 4. Conformally FlatPPS4 Spacetimes in fR,G Gravity

In this section, conformally flatPPS4 spacetimes in fR,G theory of gravity are investigated. For n=4, equation (38) becomes
(43)Rhk=R4ghk.It follows that(44)Rhk=R4ghk.Multiplying equations (43) and (44), one gets
(45)RhkRhk=R24.From equation (29), it follows that
(46)Rhijk=12ghkRij+gijRhk−ghjRik−gikRhj−R6ghkgij−ghjgik.Multiplying equations (29) and (46), we obtain
(47)RhijkRhijk=2RhkRhk−13R2.With the help of equation (47), the Gauss-Bonnet topological invariant is
(48)G=−2RhkRhk+23R2.The use of equation (45) implies that
(49)G=16R2.Thus, we can state the following theorem:Theorem 10.
The Gauss-Bonnet scalar in a conformally flatPPS4 spacetime is expressed as
(50)G=16R2.In a conformally flat spacetime, equation (9) can be rewritten as
(51)Rij−R2gij=κTijm+Tijcurv=κTijeff,where Tijcurv arises from the geometry of the spacetime. The tensor Tijcurv is given as [20]
(52)κTijcurv=∇i∇j−gij∇2fR+2R∇i∇j−gij∇2fG−4Rim∇m∇j+Rjm∇m∇ifG+4Rij∇2+gijRmn∇n∇m−Rnimj∇n∇mfG−12gijRfR+GfG−f+1−fRRij−R2gij.Since in a conformally flatPPS4 spacetime the scalar curvature is constant, the previous equation reduces
(53)κTijcurv=−12gijRfR+GfG−f+1−fRRij−R2gij.Utilizing equations (38) and (49) in equation (53), we get
(54)κTijcurv=f2−R4fR−R212fG−R4gij.The use of (43) and (54) in (51) implies that
(55)κTijm=R4fR+R212fG−f2gij.The vector filedξ is called Killing if
(56)Lξgij=0,whereas ξ is called conformal Killing if
(57)Lξgij=2φgij,where Lξ is the Lie derivative with respect to the vector filed ξ and φ is a scalar function [21, 22].A spacetimeM is said to admit a matter collineation with respect to a vector field ξ if the Lie derivative of the energy-momentum tensor Tij with respect to ξ satisfies
(58)LξTij=0,while it is said that 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 [21, 22]
(59)LξTij=2φTij.Applying the Lie derivative on both sides of (55), one gets
(60)κLξTijm=R4fR+R212fG−f2Lξgij.If the vector fieldξ is Killing on a conformally flat PPS4 spacetime M, hence equation (60) implies that
(61)LξTijm=0.In the contrast, if a conformally flatPPS4 spacetime M admits matter collineation with respect to ξ, it follows from equation (60) that
(62)Lξgij=0.Hence, we can state the following theorem:Theorem 11.
LetM be a conformally flat PPS4 spacetime obeying fR,G gravity theory; then, the vector field ξ is Killing if and only if M admits matter collineation with respect to ξ.Assume that the vector fieldξ is conformal Killing; then, after using (57) in (60) utilizing (55), we acquire that
(63)LξTijm=2φTijm.Conversely, suppose that the energy-momentum tensorTij has the Lie inheritance property along the flow lines of ξ, thus making use of (59) in (60) after that using (55)), we infer that
(64)Lξgij=2φgij.Thus, we can state the following theorem:Theorem 12.
LetM be a conformally flat PPS4 spacetime obeying fR,G gravity theory; then, M has a conformal Killing vector filed ξ if only if the energy-momentum tensor Tij has the Lie inheritance property along ξ.


## 5. Conformally FlatPPS4 Perfect Fluid Spacetimes in fR,G Gravity

This section is mainly organized to study conformally flatPPS4 perfect fluid spacetimes in fR,G modified gravity theory. For a perfect fluid spacetime, the energy-momentum tensor is given as
(65)Tijm=pm+σmλiλj+pmgij,(66)Tijeff=peff+σeffλiλj+peffgij,where pm and σm are the isotropic pressure and the energy density of the ordinary matter, whereas peff and σeff are the effective isotropic pressure and the effective energy density of the effective matter.In view of (55) and (65), we have
(67)R4fR+R212fG−f2−κpmgij=κpm+σmλiλj.Contracting twice withλi and gij, one finds
(68)σm=1κf2−R4fR−R212fG,(69)3κpm−κσm=4R4fR+R212fG−f2.Utilizing (68) in (69), it arises
(70)pm=−1κf2−R4fR−R212fG.We thus motivate to state the following theorem:Theorem 13.
In a conformally flat perfect fluidPPS4 spacetime obeying fR,G gravity, the isotropic pressure pm and the energy density σm are constants. Moreover, they are given by (68) and (70).The combination of (68) and (70) gives
(71)pm+σm=0,which means that the spacetime represents inflation and fluid behaves as a cosmological constant [23].Theorem 14.
LetM be a conformally flat perfect fluid PPS4 spacetime obeying fR,G gravity; then, M represents inflation and fluid behaves as a cosmological constant.Using (54), (65), and (66) in (51), one infers
(72)peff+σeffλiλj+peffgij=pm+σmλiλj+pmgij+1κf2−R4fR−R212fG−R4gij.Making a comparison of both sides, we obtain(73)peff=pm+1κf2−R4fR−R212fG−R4,σeff=σm−1κf2−R4fR−R212fG−R4.The use of (68) and (70) implies that
(74)peff=−R4κ,(75)σeff=R4κ.In the context offR,G modified gravity, let us now deduce some energy conditions of a perfect fluid type effective matter. The energy conditions are obtained as follows [24, 25]:
(1)
Null Energy Condition (NEC). peff+σeff≥0.(2)
Weak Energy Condition (WEC). σeff≥0 and peff+σeff≥0.(3)
Dominant Energy Condition (DEC). σeff≥0 and peff±σeff≥0.(4)
Strong Energy Condition (SEC). σeff+3peff≥0 and peff+σeff≥0.In view of (74) and (22), the energy conditions are always satisfied if R≻0.



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