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

In the present paper, we focused our attention to study pseudo-Ricci symmetric spacetimes in Gray’s decomposition subspaces. It is proved that ( PRS ) n 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.


Introduction
A pseudo-Ricci symmetric manifold (briefly (PRS) n ) is a nonflat pseudo-Riemannian manifold whose Ricci tensor satisfies where A is a nonzero 1-form and ∇ indicates the covariant differentiation with respect to the metric g [1]. e 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]. ere has been much focus on the concept of (PRS) n manifolds; for instance, a sufficient condition on (PRS) n manifolds to be quasi-Einstein manifolds was introduced by De and Gazi [3]. (PRS) n manifolds whose scalar curvature satisfies ∇ k R � 0 have zero scalar curvature [1]. A concrete example of pseudo-Ricci symmetric manifolds was given in [4]. ere are many generalizations of (PRS) n 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][9][10]). e manifolds in the trivial subspace have parallel Ricci tensor; that is, ∇ k R ij � 0. e subspace A contains manifolds whose Ricci tensor is Killing; that is, (2) e next subspace is denoted by B. e Ricci tensors of manifolds in B are Codazzi; that is, e subspace A⊕B is characterized by the equation ∇ kRg ij (4) lie in I. In I⊕A, the tensor R ij − (2R/(n + 2))g ij is Killing, whereas in I⊕B, the tensor R ji − (R/2(n − 1))g ji 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]. An n-dimensional Lorentzian manifold is said to be pseudo-Ricci symmetric spacetime if the Ricci tensor satisfies equation (1). Here, we assume the associated vector A i is a unit time-like vector (A i A i � − 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): where R ij , R, k, and T ij are the Ricci tensor, scalar curvature tensor, Newtonian constant, and energy-momentum tensor, respectively. EFE implies that the energy-momentum tensor is paper is organized as follows: In Section 2, general properties of (PRS) n spacetimes are considered. In Section 3, (PRS) n spacetimes are investigated in all Gray's orthogonal subspaces. It is proved that (PRS) n 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.

On (PRS) n Spacetimes
In this section, the main properties of (PRS) n spacetimes are considered. Equation (1) implies A different contraction of equation (1) with g ij gives Solving equations (7) and (8) together, one gets Assume that the scalar curvature is constant. Equation (10) directly leads to R � 0.

Lemma 2. In (PRS) spacetimes, the scalar curvature R is constant if and only if R � 0.
Let us consider R ≠ 0; then, the use of equation (10) in equation (1) implies that is leads us to the following lemma.
Lemma 3. In (PRS) n spacetimes with nonzero scalar curvature, the covariant derivative of the Ricci tensor takes the form provided R ≠ 0.
e Weyl tensor of type (0, 4) has the form [13] and its divergence is In virtue of (1) and (10), we have Assume that the Weyl conformal curvature tensor is divergence-free, that is, Contracting with A k and using equation (9), we obtain A multiplication with g ij gives R � 0, and hence, us, we can conclude the following theorem: e use of this result (R ij � 0) in the defining property of the conformal curvature tensor entails that Hence, we have the following corollary.  (1) gives Interchanging the indices r and k in the last equation, we have Subtracting the last two equations, we obtain Making use of equation (1) and simplifying, we get Now, assume that the (PRS) n is Ricci semisymmetric, that is, Contracting with A j and using equation (9), we infer Again, contracting with A i and utilizing equation (9), we get us, we have the following theorem:

ERROR!!PRS) n Spacetimes in Gray's Decomposition Subspaces
is section is devoted to study (PRS) n spacetimes in Gray's seven subspaces.
ree main results are obtained in this section. A Lorentzian manifold M is said to be perfect fluid if its Ricci tensor satisfies where α and β are scalar fields and u i is a time-like vector field [14]. Proof. e trivial subspace of Gray's decomposition contains spacetimes whose Ricci tensors are parallel and the scalar curvatures are constant.
us, equation (10) easily gives R � 0. And hence, equation (1) becomes A contraction of equation (28) with g ij yields And consequently, which means that (PRS) n spacetimes with parallel Ricci tensor are Ricci flat.
In subspace A (PRS) n spacetimes have a Killing Ricci tensor; that is, 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
Contracting equation (32) with A k and using equation (9), we get which means that (PRS) n spacetimes in subspace A are Ricci flat.
Next, let us consider the subspace B in which (PRS) n has a Codazzi type of Ricci tensor [15]. e Codazzi deviation tensor D ijk of (PRS) n is given by

Journal of Mathematics
Multiplying with g ij and utilizing equation (9), we get

A contraction of equation (36) by A k gives
which means that (PRS) n spacetimes in Gray's subspace B are Ricci flat. Proof. In subspace I, the Ricci tensor of pseudo-Ricci symmetric manifold M satisfies the following property: (39) Applying equation (1), we obtain It follows that Contracting with A k implies which means that (PRS) n spacetimes in subspace I are perfect fluid. In subspace I⊕A, the Ricci curvature tensor satisfies Using equation (1), we infer Now, equation (10) implies A k g ij + A i g kj + A j g ik .

(45)
A contraction with A k yields which means that (PRS) n spacetimes in subspace I⊕A are perfect fluid. Assume that (PRS) n are in Gray's subspace I⊕B; that is, Equation (1) implies e use of equation (10) gives Contracting with A k , we obtain which means that (PRS) n spacetimes in Gray's subspace I⊕B are perfect fluid. Proof. In subspace A⊕B, the scalar curvature is covariantly constant and hence equation (10) implies which means (PRS) n spacetimes in Gray's subspace A⊕B have zero scalar curvature.

Pseudo-Ricci Symmetric GRW Spacetimes
where φ and ξ are scalar functions. Vector fields satisfying equation (52) are called torse-forming. Now, assume that M is a (PRS) n generalized Robertson-Walker spacetime; that is, (54) A contraction with u j yields Using equation (1), one gets erefore, However, us, (59) [12]); thus, Since M is (PRS) n , equation (9) shows that Multiplying both the sides by u k , that is, Using equation (53), one gets Now, there are two different possible cases. e first one A j u j � 0 and consequently ξ does not vanish. en, equation (59) becomes A contraction by A i implies that which is a contradiction. e second case is ξ � 0. en, equation (60) leads to us, either R ik � 0 or φ � − u j A j .

Theorem 6.
A pseudo-Ricci symmetric GRW spacetime is vacuum provided the one form A is not codirectional with the torse-forming vector field u.
Suppose A i ≠ φu i . en, the spacetime under consideration is Ricci flat, that is, R ij � 0, which implies R � 0. It is known that where C is the conformal curvature tensor [13].
erefore, using R ij � 0 and R � 0, equation (67) yields ∇ h C h ijk � 0, that is, div C � 0. In [18], Mantica et al. proved that an n-dimensional GRW spacetime satisfies div C � 0 if and only if the spacetime is perfect fluid. erefore, we conclude the following.

Corollary 2. A pseudo-Ricci symmetric GRW spacetime is a perfect fluid spacetime provided
Since R ij � 0 and R � 0, from the definition of the conformal curvature tensor, it follows that C h ijk � R h ijk . 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. us, we have the following.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.