# Investigation of Pseudo-Ricci Symmetric Spacetimes in Gray’s Subspaces

Authors: Sameh Shenawy; Uday Chand De; Nasser Bin Turki; Suliman Alsaeed; Abdallah Abdelhameed Syied
Journal: Journal of Mathematics
 (2021)

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## Abstract

In the present paper, we focused our attention to study pseudo-Ricci symmetric spacetimes in Gray’s decomposition subspaces. It is proved thatPRSn spacetimes are Ricci flat in trivial, A, and B subspaces, whereas perfect fluid in subspaces I, I⊕A, and I⊕B, and have zero scalar curvature in subspace A⊕B. Finally, it is proved that pseudo-Ricci symmetric GRW spacetimes are vacuum, and as a consequence of this result, we address several corollaries.


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## Body

## 1. Introduction

A pseudo-Ricci symmetric manifold (brieflyPRSn) is a nonflat pseudo-Riemannian manifold whose Ricci tensor satisfies(1)∇kRij=2AkRij+AiRkj+AjRik,where A is a nonzero 1-form and ∇ indicates the covariant differentiation with respect to the metric g [1].The class of pseudo-Ricci symmetric manifolds is a subclass of weakly Ricci symmetric manifolds which were first introduced and studied by Tamássy and Binh [2]. There has been much focus on the concept of PRSn manifolds; for instance, a sufficient condition on PRSn manifolds to be quasi-Einstein manifolds was introduced by De and Gazi [3]. PRSn manifolds whose scalar curvature satisfies ∇kR=0 have zero scalar curvature [1]. A concrete example of pseudo-Ricci symmetric manifolds was given in [4]. There are many generalizations of PRSn manifolds, for example, see [5, 6].An invariant orthogonal decomposition of the covariant derivative of the Ricci tensor was coined and studied by Gray in [7] (see also [8–10]). The manifolds in the trivial subspace have parallel Ricci tensor; that is, ∇kRij=0. The subspace A contains manifolds whose Ricci tensor is Killing; that is,(2)∇jRki+∇kRji+∇iRkj=0.The next subspace is denoted byℬ. The Ricci tensors of manifolds in ℬ are Codazzi; that is,(3)∇kRij=∇iRkj.The subspaceA⊕ℬ is characterized by the equation ∇R=0. Manifolds with(4)∇kRij=nn−1n+2∇kRgijlie in ℐ. In ℐ⊕A, the tensor Rij−2R/n+2gij is Killing, whereas in ℐ⊕ℬ, the tensor Rji−R/2n−1gji is a Codazzi tensor. Such manifolds are called Einstein-like manifolds [11]. Recently, there has been growing interest in this decomposition. For example, generalized Robertson–Walker spacetimes are either Einstein or perfect fluid in Gray’s orthogonal subspaces except one in which the Ricci tensor is not restricted [12].Ann-dimensional Lorentzian manifold is said to be pseudo-Ricci symmetric spacetime if the Ricci tensor satisfies equation (1). Here, we assume the associated vector Ai is a unit time-like vector (AiAi=−1).In standard theory of gravity, the relation between the matter of spacetimes and the geometry of spacetimes is given by Einstein’s field equation (EFE):(5)Rij−R2gij=kTij,where Rij, R, k, and Tij are the Ricci tensor, scalar curvature tensor, Newtonian constant, and energy-momentum tensor, respectively. EFE implies that the energy-momentum tensor Tij is divergence-free. This requirement is directly satisfied if ∇lTij=0.This paper is organized as follows: In Section2, general properties of PRSn spacetimes are considered. In Section 3, PRSn spacetimes are investigated in all Gray’s orthogonal subspaces. It is proved that PRSn spacetimes in trivial, A, and ℬ subspaces are Ricci flat, in subspaces ℐ, ℐ⊕A, and ℐ⊕ℬ are perfect fluid spacetimes, and in A⊕ℬ have a zero scalar curvature. In Section 4, we prove that pseudo-Ricci symmetric GRW spacetimes are vacuum and as a consequence, we address some corollaries.


## 2. OnPRSn Spacetimes

In this section, the main properties ofPRSn spacetimes are considered. Equation (1) implies(6)∇kRik=3AjRij+AiR.The use of∇kRik=1/2∇iR yields(7)∇iR=6AjRij+2AiR.A different contraction of equation (1) with gij gives(8)∇kR=2AkR+2AjRkj.Solving equations (7) and (8) together, one gets(9)AjRkj=0,(10)∇kR=2AkR.Lemma 1.
InPRSn spacetimes, the covariant derivative of the scalar curvature is ∇kR=2AkR. Moreover, Aj is an eigenvector of the Ricci tensor Rij with zero eigenvalue.Assume that the scalar curvature is constant. Equation (10) directly leads to R=0.Lemma 2.
In (PRS) spacetimes, the scalar curvatureR is constant if and only if R=0.Let us considerR≠0; then, the use of equation (10) in equation (1) implies that(11)∇kRij=∇kRRRij+∇iR2RRkj+∇jR2RRik.This leads us to the following lemma.Lemma 3.
InPRSn spacetimes with nonzero scalar curvature, the covariant derivative of the Ricci tensor takes the form(12)∇kRij=∇kRRRij+∇iR2RRkj+∇jR2RRik,provided R≠0.The Weyl tensor of type0,4 has the form [13](13)Cijkl=Rijkl−1n−1gilRjk+gjkRil−gikRjl−gjlRik+Rn−1n−2gilgjk−gikgjl,and its divergence is(14)∇hCijkh=n−3n−2∇kRij−∇jRik−12n−1gij∇kR−gik∇jR.In virtue of (1) and (10), we have(15)∇hCijkh=n−3n−2AkRij−AjRik−1n−1gijAkR−gikAjR.Assume that the Weyl conformal curvature tensor is divergence-free, that is,∇hCijkh=0; then,(16)AkRij−AjRik=n−2n−1n−3gijAkR−gikAjR.Contracting withAk and using equation (9), we obtain(17)Rij=n−2Rn−1n−3gij+AiAj.A multiplication withgij gives R=0, and hence,(18)Rij=0.Thus, we can conclude the following theorem:Theorem 1.
APRSn spacetime with divergence-free Weyl curvature tensor is Ricci flat.The use of this result (Rij=0) in the defining property of the conformal curvature tensor entails that(19)Cijkh=Rijkh.Hence, we have the following corollary.Corollary 1.
Semisymmetric and conformally semisymmetric pseudo-Ricci symmetric spacetimes are equivalent.The covariant derivative of equation (1) gives(20)∇r∇kRij=2∇rAkRij+∇rAiRkj+∇rAjRik.Interchanging the indicesr and k in the last equation, we have(21)∇k∇rRij=2∇kArRij+∇kAiRrj+∇kAjRir.Subtracting the last two equations, we obtain(22)∇r∇k−∇k∇rRij=2∇rAkRij+∇rAiRkj+∇rAjRik−2∇kArRij−∇kAiRrj−∇kAjRir,∇r∇k−∇k∇rRij=2Rij∇rAk+Rkj∇rAi+Rik∇rAj+2Ak∇rRij+Ai∇rRkj+Aj∇rRik−2Ar∇kRij−Ai∇kRrj−Aj∇kRir−2Rij∇kAr−Rrj∇kAi−Rir∇kAj.Making use of equation (1) and simplifying, we get(23)∇r∇k−∇k∇rRij=2Rij∇rAk−∇kAr+Rkj∇rAi+Rik∇rAj−Rrj∇kAi−Rir∇kAj+AiAkRjr+AiAjRrk+AjAkRir−AiArRkj−AjArRki.Now, assume that thePRSn is Ricci semisymmetric, that is, ∇r∇k−∇k∇rRij=0; we have(24)0=2Rij∇rAk−∇kAr+Rkj∇rAi+Rik∇rAj−Rrj∇kAi−Rir∇kAj+AiAkRjr+AiAjRrk+AjAkRir−AiArRkj−AjArRki.Contracting withAj and using equation (9), we infer(25)−AiRrk+ArRki=0.Again, contracting withAi and utilizing equation (9), we get(26)Rrk=0.Thus, we have the following theorem:Theorem 2.
Ricci semisymmetricPRSn spacetimes are Ricci flat.


## 3.PRSn Spacetimes in Gray’s Decomposition Subspaces

This section is devoted to studyPRSn spacetimes in Gray’s seven subspaces. Three main results are obtained in this section. A Lorentzian manifold M is said to be perfect fluid if its Ricci tensor satisfies(27)Rij=αgij+βuiuj,where αandβ are scalar fields and ui is a time-like vector field [14].Theorem 3.
PRSn spacetimes in trivial, A, and ℬ subspaces are Ricci flat.Proof.
The trivial subspace of Gray’s decomposition contains spacetimes whose Ricci tensors are parallel and the scalar curvatures are constant. Thus, equation (10) easily gives R=0. And hence, equation (1) becomes(28)2AkRij=−AiRkj−AjRik.
A contraction of equation (28) with gij yields(29)Rij=RAiAj=0.
And consequently,(30)Rij=0,which means that PRSn spacetimes with parallel Ricci tensor are Ricci flat.
In subspaceAPRSn spacetimes have a Killing Ricci tensor; that is,(31)∇jRki+∇kRji+∇iRkj=0.
It is well known that in this subspace, the scalar curvature is covariantly constant. Equation (10) implies R=0. Using equation (1) in equation (31), we have(32)AkRij+AjRik+AiRjk=0.
Contracting equation (32) with Ak and using equation (9), we get(33)Rij=RAjAi=0,which means that PRSn spacetimes in subspace A are Ricci flat.
Next, let us consider the subspaceℬ in which PRSn has a Codazzi type of Ricci tensor [15]. The Codazzi deviation tensor Dijk of PRSn is given by(34)Dijk=∇kRij−∇iRkj=2AkRij+AiRkj+AjRik−2AiRkj+AkRij+AjRik=AkRij−AiRkj.
A contraction withgij implies(35)gijDijk=AkR−AjRkj.
But, in this subspace, the spacetimes have Codazzi-type Ricci tensor (that is,Dijk=0); then,(36)AkRij=AiRkj.
Multiplying withgij and utilizing equation (9), we get(37)R=0.
A contraction of equation (36) by Ak gives(38)Rij=0,which means that PRSn spacetimes in Gray’s subspace ℬ are Ricci flat.Theorem 4.
PRSn spacetimes in ℐ, ℐ⊕A, and ℐ⊕ℬ subspaces are perfect fluid spacetimes.Proof.
In subspaceℐ, the Ricci tensor of pseudo-Ricci symmetric manifold M satisfies the following property:(39)∇kRij=n∇kRn−1n+2gij+n−2∇iR2n−1n+2gkj+n−2∇jR2n−1n+2gik.
Applying equation (1), we obtain(40)2AkRij+AiRkj+AjRik=n∇kRn−1n+2gij+n−2∇iR2n−1n+2gkj+n−2∇jR2n−1n+2gik.
It follows that(41)2AkRij+AiRkj+AjRik=2nAkRn−1n+2gij+n−2AiRn−1n+2gkj+n−2AjRn−1n+2gik.
Contracting withAk implies(42)Rij=nRn−1n+2gij+2−nRn−1n+2AiAj,which means that PRSn spacetimes in subspace ℐ are perfect fluid.
In subspaceℐ⊕A, the Ricci curvature tensor satisfies(43)∇kRij+∇iRkj+∇jRik=2∇kRn+2gij+2∇iRn+2gkj+2∇jRn+2gik.
Using equation (1), we infer(44)AkRij+AjRik+AiRjk=2n+2∇kRgij+∇iRgkj+∇jRgik.
Now, equation (10) implies(45)AkRij+AjRik+AiRjk=2Rn+2Akgij+Aigkj+Ajgik.
A contraction withAk yields(46)Rij=2Rn+2gij−2AjAi,which means that PRSn spacetimes in subspace ℐ⊕A are perfect fluid.
Assume thatPRSn are in Gray’s subspace ℐ⊕ℬ; that is,(47)∇kRji−∇jRki=12n−1gji∇kR−gki∇jR.
Equation (1) implies(48)2n−1AkRij−AjRki=gji∇kR−gki∇jR.
The use of equation (10) gives(49)n−1AkRij−AjRki=gjiAkR−gkiAjR.
Contracting withAk, we obtain(50)Rij=Rn−1gji+AiAj,which means that PRSn spacetimes in Gray’s subspace ℐ⊕ℬ are perfect fluid.Theorem 5.
PRSn spacetimes in A⊕ℬ subspace have zero scalar curvature.Proof.
In subspaceA⊕ℬ, the scalar curvature is covariantly constant and hence equation (10) implies(51)R=0,which means PRSn spacetimes in Gray’s subspace A⊕ℬ have zero scalar curvature.


## 4. Pseudo-Ricci Symmetric GRW Spacetimes

A generalized Robertson–Walker spacetime (for simplicity, denoted by GRW spacetimes) is the warped productM=I×fM∗ of an open connected interval I,−dt2 and a Riemannian manifold M∗, where f:I⟶ℝ+ is a positive smooth function. A Lorentzian manifold M is a generalized Robertson–Walker spacetime if and only if M possesses a unit time-like vector field ui with [16, 17](52)∇kui=φgki+ukui,(53)Rijuj=ξui,where φ and ξ are scalar functions. Vector fields satisfying equation (52) are called torse-forming.Now, assume thatM is a PRSn generalized Robertson–Walker spacetime; that is,(54)∇kRij=2AkRij+AiRkj+AjRik.A contraction withuj yields(55)uj∇kRij=2AkujRij+AiujRkj+AjujRik.Using equation (1), one gets(56)uj∇kRij=2Akξui+ξAiuk+ujAjRik.Therefore,(57)uj∇kRij=2ξAkui+ξAiuk+ujAjRik.However,(58)uj∇kRij=∇kRijuj−Rij∇kuj=ui∇kξ+ξ∇kui−Rij∇kuj=ui∇kξ+φξgki−Rik.Thus,(59)ui∇kξ+φξgki−Rik=2ξAkui+ξAiuk+ujAjRik.It is well known∇kξ=−ukuj∇jξ=−ρuk, where ρ=uj∇jξ (see [12]); thus,(60)φ+ujAjRik=ξφgki−ρuiuk−2ξAkui−ξAiuk.SinceM is PRSn, equation (9) shows that(61)AjRkj=0.Multiplying both the sides byuk, that is,(62)AjRkjuk=0.Using equation (53), one gets(63)ξAjuj=0.Now, there are two different possible cases. The first oneAjuj=0 and consequently ξ does not vanish. Then, equation (59) becomes(64)φRik=ξφgki−ρuiuk−2ξAkui−ξAiuk.A contraction byAi implies that(65)0=ξφAk−ξAiAiuk,which is a contradiction. The second case is ξ=0. Then, equation (60) leads to(66)φ+ujAjRik=0.Thus, eitherRik=0 or φ=−ujAj.Theorem 6.
A pseudo-Ricci symmetric GRW spacetime is vacuum provided the one formA is not codirectional with the torse-forming vector field u.
SupposeAi≠φui. Then, the spacetime under consideration is Ricci flat, that is, Rij=0, which implies R=0. It is known that(67)∇hCijkh=n−3n−2∇kRij−∇jRik−12n−1gij∇kR−gik∇jR,where C is the conformal curvature tensor [13].Therefore, usingRij=0 and R=0, equation (67) yields ∇hCijkh=0, that is, divC=0. In [18], Mantica et al. proved that an n-dimensional GRW spacetime satisfies divC=0 if and only if the spacetime is perfect fluid. Therefore, we conclude the following.Corollary 2.
A pseudo-Ricci symmetric GRW spacetime is a perfect fluid spacetime providedAi≠φui.SinceRij=0 and R=0, from the definition of the conformal curvature tensor, it follows that Cijkh=Rijkh. Hence, semisymmetric and conformally semisymmetric manifolds are equivalent. Eriksson and Senovilla [19] considered the semisymmetric spacetime and proved that it is of Petrov types D, N, and O. Thus, we have the following.Corollary 3.
A conformally semisymmetric pseudo-Ricci symmetric GRW spacetime is of Petrov typesD, N, and O.



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