On 4-Dimensional Lorentzian Manifolds with Pseudo W₂-Curvature Tensor in General Relativity

Abstract

The present paper is devoted to the study of Lorentzian spacetimes endowed with the pseudo $W_2$-curvature tensor. We investigate the geometric and symmetry properties of pseudo $W_2$-flat spacetimes and establish several results concerning their curvature structure and associated energy–momentum tensors. It is shown that a pseudo $W_2$-flat spacetime is a spacetime of constant curvature under suitable conditions on the defining parameters of the pseudo $W_2$-curvature tensor. Further, for a spacetime satisfying Einstein’s field equation with vanishing pseudo $W_2$-curvature tensor, it is proved that a vector field generates a matter collineation if and only if it is a Killing vector field. We also establish that the Lie inheritance property of the energy–momentum tensor is equivalent to the existence of a conformal Killing vector field. Moreover, it is shown that if the energy–momentum tensor is of Codazzi type, then the pseudo $W_2$-curvature tensor is divergence-free. To illustrate the obtained results, two explicit Lorentzian spacetime models are examined in detail, namely an isotropic warped spacetime and an FLRW-type exponential (de Sitter) spacetime. For each model, the metric structure, curvature tensors, Ricci operator, scalar curvature, and pseudo $W_2$-flatness condition are verified explicitly. These examples provide concrete realizations of the theoretical results and demonstrate the significance of the pseudo $W_2$-curvature tensor in the geometric study of Lorentzian manifolds and general relativity.

1. Introduction

This paper is concerned with the investigation of the pseudo $W_2$-curvature tensor and its geometric consequences in the context of general relativity. The study is carried out on a four-dimensional semi-Riemannian manifold $(M^4,g)$, where g is a Lorentzian metric of signature $(-,+,+,+)$. Such a metric structure distinguishes the temporal dimension from the spatial dimensions and serves as the standard mathematical model of spacetime in Einstein’s theory of gravitation. The Lorentzian character of the manifold introduces a crucial causal structure, wherein tangent vectors are classified as timelike, spacelike or null. This classification scheme is not merely a mathematical formalism, but rather embodies the physical reality of causal relationships in spacetime, governing both the dynamics of matter and the propagation of physical signals. The causal structure inherent to Lorentzian geometry makes these manifolds particularly well-adapted for modeling relativistic phenomena, as it directly encodes the light-cone structure that is fundamental to Einstein’s theory. Within this geometric framework, we examine the pseudo $W_2$-curvature tensor, analyzing both its mathematical properties and its physical interpretations. The Lorentzian signature plays a pivotal role in our analysis, as it determines the causal nature of geometric objects and ensures their physical relevance within the theory of general relativity.

The conservation of energy–momentum in Einstein’s field equations [1] is characterized by the divergence-free condition ${\nabla }_p{T}_j^p=0$. A stronger requirement is that the energy–momentum tensor be covariantly constant, i.e., $\nabla T=0$. Chaki and Roy [2] proved that such a condition implies Ricci symmetry $(\nabla S=0)$ in a Lorentzian spacetime, thereby establishing a connection between the invariance of matter fields and the geometric symmetry of spacetime.

Recent years have witnessed substantial progress in the study of curvature tensors and their applications in differential geometry, general relativity, and modified theories of gravity. Among these developments, the investigation of generalized curvature tensors has provided valuable insights into the geometric and physical properties of spacetime manifolds. In particular, the pioneering work of [3] established several fundamental results concerning the role of the $W_2$-curvature tensor in both classical relativistic spacetimes and $f\left(G\right)$-gravity models, thereby highlighting its significance in the study of gravitational phenomena.

Motivated by these developments, Mallick and De [4] carried out a detailed investigation of spacetime manifolds admitting the $W_2$-curvature tensor and obtained important geometric characterizations. Further contributions were made by Suh et al. [5], who studied pseudo-quasi-conformal curvature tensors and explored their implications in the context of general relativity. Their work revealed interesting connections between generalized curvature restrictions and the geometric structure of Lorentzian manifolds.

The growing interest in curvature-restricted geometric structures has also led to several notable investigations in related directions. For instance, ref. [6] examined the concircular curvature tensor in the framework of $f\left(R\right)$-gravity and derived significant results concerning its geometric and physical implications. Likewise, ref. [7] studied the $W^{*}$-curvature tensor in relativistic spacetimes and obtained important characterizations associated with Einstein’s field equations. Furthermore, ref. [8] investigated M-projectively flat spacetimes, while [9] analyzed spacetimes admitting Z-symmetric tensor fields, thereby enriching the theory of generalized curvature tensors and their applications. In addition, numerous researchers, including [10,11,12,13,14], have made significant contributions to this rapidly developing area by studying various curvature tensors, symmetry conditions, and geometric structures arising in semi-Riemannian and Lorentzian manifolds. Their collective efforts have considerably enhanced our understanding of the interplay between curvature, spacetime symmetries, and gravitational field equations.

Inspired by these noteworthy contributions and motivated by the continuing development of curvature-restricted geometry, the present work is devoted to the study of Lorentzian spacetimes admitting the pseudo $W_2$-curvature tensor within the framework of general relativity.

The concept of pseudo $W_2$-curvature tensor ${\overline{W}}_2$ in an n-dimensional Riemannian manifold $(M^n,g)$ was introduced by Prasad and Maurya [15] and is defined by

$$\begin{array}{cc}\hfill \overline{W_2}(X_u,X_v)X_z& =\psi R(X_u,X_v)X_z+\varphi [g(X_v,X_z)Q{X}_u-g(X_u,X_z)Q{X}_v]\hfill \\ & -{\displaystyle \frac{\theta }{n}}\left({\displaystyle \frac{\psi }{n-1}}+\varphi \right)[g(X_v,X_z)X_u-g(X_u,X_z)X_v],\hfill \end{array}$$

(1)

for all $X_u,X_v,X_z\in (M^n,g)$, where $\psi$ and $\varphi$ are non-zero constants, $\theta$ is the scalar curvature and Q is the Ricci operator defined by $g(Q{X}_u,X_v)=S(X_u,X_v)$. If $\psi =1$ and $\varphi =-{\displaystyle \frac{1}{n-1}}$, then (1) becomes $W_2$-curvature tensor defined by [16]. Moreover, the theory of the pseudo $W_2$-curvature tensor has been further developed by several researchers. In particular, Prasad [17] and Kumar [18] extended the study of pseudo $W_2$-curvature tensors by investigating their geometric properties under various curvature restrictions and symmetry conditions. Their works contributed significantly to a deeper understanding of the relationship between pseudo $W_2$-curvature structures, Einstein-type manifolds, and other classes of semi-Riemannian manifolds. These studies not only enriched the theory of generalized curvature tensors but also demonstrated the importance of pseudo $W_2$-geometry in the broader context of differential geometry and mathematical physics.

The purpose of the present paper is to investigate Lorentzian spacetimes satisfying geometric conditions involving the pseudo $W_2$-curvature tensor. In particular, we study pseudo $W_2$-flat spacetimes and examine their relationship with Einstein’s field equations, spacetime symmetries, and the energy–momentum tensor. We establish several characterizations of pseudo $W_2$-flat spacetimes and derive conditions under which such spacetimes possess constant curvature. Furthermore, we investigate the interplay between pseudo $W_2$-flatness and various symmetry properties, including matter collineations, Killing vector fields, conformal Killing vector fields, and Lie inheritance symmetries.

Another objective of this work is to analyze the behaviour of the pseudo $W_2$-curvature tensor under additional restrictions on the energy–momentum tensor. In particular, we show that when the energy–momentum tensor is of Codazzi type, the pseudo $W_2$-curvature tensor becomes divergence-free. These results provide new insights into the geometric structure of Lorentzian manifolds admitting special curvature conditions. To illustrate the theoretical developments, we construct and analyze explicit Lorentzian spacetime models. Two geometrically distinct examples are examined in detail: an isotropic warped spacetime and an FLRW-type exponential (de Sitter) spacetime. For each example, the metric tensor, Christoffel symbols, Riemann curvature tensor, Ricci tensor, Ricci operator, and scalar curvature are computed explicitly, and the pseudo $W_2$-flat condition is verified directly. These examples serve as concrete realizations of the theoretical results established in the paper.

The paper is organized as follows. By following the introductory framework that we established, Section 2 investigates pseudo $W_2$-flat spacetimes and derives their fundamental geometric properties, Einstein’s field equations and the associated symmetry conditions. Section 3 examines the divergence properties of the pseudo $W_2$-curvature tensor under Codazzi-type assumptions. The analysis then advances in Section 4 to consider Ricci-recurrent spacetimes satisfying the condition $div{\overline{W}}_2=0$. Section 5 demonstrates the physical relevance of our theoretical framework, and we present an explicit example of a pseudo $W_2$-flat spacetime that substantiates our principal results.

2. Spacetime with Vanishing Pseudo ${\mathit{W}}_{\mathbf{2}}$-Curvature Tensor

Let the spacetime of general relativity be $V_4$, then the $(0,4)$-type tensor of (1) gives

$$\begin{array}{cc}\hfill {}^{\prime }\overline{W_2}(X_u,X_v,X_z,X_w)& =\psi \overline{R}(X_u,X_v,X_z,X_w)\hfill \\ & +\varphi [g(X_v,X_z)S(X_u,X_w)-g(X_u,X_z)S(X_v,X_w)]\hfill \\ & -{\displaystyle \frac{\theta }{4}}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)[g(X_v,X_z)g(X_u,X_w)-g(X_u,X_z)g(X_v,X_w)],\hfill \end{array}$$

(2)

where

$${}^{\prime }\overline{W_2}(X_u,X_v,X_z,X_w)=g(\overline{W_2}(X_u,X_v)X_z,X_w),$$

$$\overline{R}(X_u,X_v,X_z,X_w)=g(R(X_u,X_v)X_z,X_w),$$

and S is the Ricci tensor.

If the spacetime is pseudo $W_2$-flat, i.e., ${}^{\prime }\overline{W_2}(X_u,X_v,X_z,X_w)=0$, then (2) reduces to

$$\begin{array}{cc}\hfill \psi \overline{R}(X_u,X_v,X_z,X_w)& =\varphi [g(X_u,X_z)S(X_v,X_w)-g(X_v,X_z)S(X_u,X_w)]\hfill \\ & +{\displaystyle \frac{\theta }{4}}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)[g(X_v,X_z)g(X_u,X_w)-g(X_u,X_z)g(X_v,X_w)].\hfill \end{array}$$

(3)

To provide a more rigorous derivation of Theorem 1, we consider the contraction of the fundamental equation of pseudo $W_2$-flat spacetimes. Multiplying (3) by the inverse metric tensor $g^{uw}$ and performing the summation over the indices u and w, the left-hand side of the equation yields the Ricci tensor:

$$g^{uw}\psi \overline{R}(X_u,X_v,X_z,X_w)=\psi S(X_v,X_z).$$

On the right-hand side, the terms involving the Ricci tensor and the metric tensor are contracted as follows:

$$g^{uw}\varphi [g(X_u,X_z)S(X_v,X_w)-g(X_v,X_z)S(X_u,X_w)]=\varphi [S(X_v,X_z)-g(X_v,X_z)\theta ].$$

Now setting $X_u=X_w=e_i$ in (3) and summing over $i,1\le i\le 4$, we have

$$\begin{array}{cc}\hfill \psi {\displaystyle \sum _{i=1}^4}\overline{R}(e_i,X_v,X_z,e_i)& =\varphi {\displaystyle \sum _{i=1}^4}\left[g(e_i,X_z)S(X_v,e_i)-g(X_v,X_z)S(e_i,e_i)\right]\hfill \\ & +{\displaystyle \frac{\theta }{4}}\left({\displaystyle \frac{\psi }{3}}+\varphi \right){\displaystyle \sum _{i=1}^4}\left[g(X_v,X_z)g(e_i,e_i)-g(e_i,X_z)g(X_v,e_i)\right],\hfill \end{array}$$

(4)

where

$${\displaystyle \sum _{i=1}^4}\overline{R}(e_i,X_v,X_z,e_i)=S(X_v,X_z),{\displaystyle \sum _{i=1}^4}g(e_i,X_z)S(X_v,e_i)=S(X_v,X_z),$$

(5)

$${\displaystyle \sum _{i=1}^4}g(e_i,X_z)S(X_v,e_i)=S(X_v,X_z),{\displaystyle \sum _{i=1}^4}S(e_i,e_i)=\theta ,$$

(6)

$${\displaystyle \sum _{i=1}^4}g(e_i,e_i)=4,\mathrm{and}{\displaystyle \sum _{i=1}^4}g(e_i,X_z)g(X_v,e_i)=g(X_v,X_z).$$

(7)

Combining (4)–(7), we obtain

$$(\psi -\varphi )S(X_v,X_z)=(\psi -\varphi ){\displaystyle \frac{\theta }{4}}g(X_v,X_z),$$

(8)

where $\psi -\varphi \ne 0$, where S and $\theta$ are the Ricci tensor and the scalar curvature of the manifold, respectively. This mathematical transition highlights that (8) is an Einstein manifold. Thus we have the following theorem:

Theorem 1.

A pseudo $W_2$-flat spacetime is an Einstein manifold, provided $\psi -\varphi \ne 0$.

Next, we prove the following theorem:

Theorem 2. 

A pseudo $W_2$-flat spacetime is a spacetime of constant curvature, provided $\psi -\varphi \ne 0$.

Proof. 

Consider the spacetime $(M^4,g)$ is pseudo $W_2$-flat, we have

$${\overline{W}}_2(X_u,X_v)X_z=0,$$

(9)

where ${\overline{W}}_2$ is of $(1,3)$-type tensor. From (3) and (9), it is obvious that

$$\begin{array}{cc}\hfill \psi R(X_u,X_v)X_z& =\varphi \left[g(X_u,X_z)Q{X}_v-g(X_v,X_z)Q{X}_u\right]\hfill \\ & +{\displaystyle \frac{\theta }{4}}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)\left[g(X_v,X_z)X_u-g(X_u,X_z)X_v\right].\hfill \end{array}$$

(10)

Now assuming $\psi \ne 0$, we obtain

$$\begin{array}{cc}\hfill R(X_u,X_v)X_z& ={\displaystyle \frac{\varphi }{\psi }}\left[g(X_u,X_z)Q{X}_v-g(X_v,X_z)Q{X}_u\right]\hfill \\ & +{\displaystyle \frac{\theta }{4\psi }}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)\left[g(X_v,X_z)X_u-g(X_u,X_z)X_v\right].\hfill \end{array}$$

(11)

Taking the inner product with $X_w$ in (11), we get

$$\begin{array}{cc}\hfill \overline{R}(X_u,X_v,X_z,X_w)& ={\displaystyle \frac{\varphi }{\psi }}\left[g(X_u,X_z)S(X_v,X_w)-g(X_v,X_z)S(X_u,X_w)\right]\hfill \\ & +{\displaystyle \frac{\theta }{4\psi }}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)\left[g(X_v,X_z)g(X_u,X_w)-g(X_u,X_z)g(X_v,X_w)\right].\hfill \end{array}$$

(12)

Again contracting (12) with respect to $X_u$ and $X_w$, we obtain

$$\begin{array}{cc}\hfill S(X_v,X_z)& ={\displaystyle \frac{\varphi }{\psi }}\left[S(X_v,X_z)-\theta g(X_v,X_z)\right]+{\displaystyle \frac{3\theta }{4\psi }}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)g(X_v,X_z).\hfill \end{array}$$

(13)

Multiplying (13) by $\psi$ and simplifying, we get

$$(\psi -\varphi )\left[S(X_v,X_z)-{\displaystyle \frac{\theta }{4}}g(X_v,X_z)\right]=0,$$

(14)

since $\psi -\varphi \ne 0$, it follows that

$$S(X_v,X_z)={\displaystyle \frac{\theta }{4}}g(X_v,X_z).$$

Therefore, the spacetime is an Einstein, and

$$Q{X}_u={\displaystyle \frac{\theta }{4}}{X}_u.$$

(15)

Substituting (15) into (11), we obtain

$$\begin{array}{cc}\hfill R(X_u,X_v)X_z& ={\displaystyle \frac{\varphi \theta }{4\psi }}\left[g(X_u,X_z)X_v-g(X_v,X_z)X_u\right]\hfill \\ \hfill & +{\displaystyle \frac{\theta }{4\psi }}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)\left[g(X_v,X_z)X_u-g(X_u,X_z)X_v\right].\hfill \end{array}$$

(16)

After cancellation of the $\varphi$-terms, (16) reduces to

$$R(X_u,X_v)X_z={\displaystyle \frac{\theta }{12}}\left[g(X_v,X_z)X_u-g(X_u,X_z)X_v\right].$$

Consequently,

$$\overline{R}(X_u,X_v,X_z,X_w)={\displaystyle \frac{\theta }{12}}\left[g(X_v,X_z)g(X_u,X_w)-g(X_u,X_z)g(X_v,X_w)\right].$$

(17)

Thus the curvature tensor has the form

$$\overline{R}(X_u,X_v,X_z,X_w)=K\left[g(X_v,X_z)g(X_u,X_w)-g(X_u,X_z)g(X_v,X_w)\right],$$

where

$$K={\displaystyle \frac{\theta }{12}}.$$

This is precisely the curvature tensor of a manifold of constant sectional curvature K. Hence (17) shows that the spacetime is of constant curvature. Therefore, a pseudo $W_2$-flat spacetime is a spacetime of constant curvature whenever $\psi -\varphi \ne 0$. □

Remark 1. 

Depending on the sign of the curvature, these spaces can be spherical (positive curvature), flat (zero curvature), or hyperbolic (negative curvature), each corresponding to different possible geometries of the universe. When applying Einstein’s field equations to such symmetric conditions, the resulting spacetime geometry is described by the Robertson–Walker (RW) metric, which models a dynamic universe whose spatial curvature may evolve over time. This metric is central to the Friedmann–Lemaitre-Robertson–Walker (FLRW) cosmological models, which describe expanding or contracting universes based on different energy contents. Moreover, when the condition of constant curvature is extended to the entire 4-dimensional spacetime, the solution leads to the de Sitter universe, an important model in cosmology that represents an exponentially expanding universe, often associated with the presence of a cosmological constant or dark energy [19,20].

Einstein’s field equation with cosmological constant can be written as

$$S(X_u,X_v)-{\displaystyle \frac{\theta }{2}}g(X_u,X_v)+\lambda g(X_u,X_v)=kT(X_u,X_v),$$

(18)

where $\lambda ,k$ and T denote the cosmological constant, the Einstein gravitational constant and the energy–momentum tensor, respectively. With the help of (8) and (18), we obtain

$$T(X_u,X_v)={\displaystyle \frac{1}{k}}\left[\lambda -{\displaystyle \frac{\theta }{4}}\right]g(X_u,X_v).$$

(19)

Taking the covariant derivative of Equation (19) with respect to $X_z$, we obtain

$$\left({\nabla }_{X_z}T\right)(X_u,X_v)={\displaystyle \frac{1}{4k}}d\theta \left(X_z\right)g(X_u,X_v),$$

(20)

where $d\theta \left(X_z\right)={\nabla }_{X_z}\theta$. Since a pseudo $W_2$-flat spacetime is Einstein. Therefore, the scalar curvature $\theta$ is constant. Hence

$$d\theta \left(X_z\right)=0,$$

(21)

for all $X_z$.

From (20) and (21), we get

$$\left({\nabla }_{X_z}T\right)(X_u,X_v)=0.$$

(22)

Thus we can state the following:

Theorem 3. 

In pseudo $W_2$-flat spacetime the energy–momentum tensor T, which fulfills Einstein’s field equation with cosmological constant λ, is observed to be covariantly constant.

In a pioneering work, Katzin et al. [21] carried out a detailed analysis of curvature collineations (CCs) in relation to the particle and field conservation laws arising in standard general relativity. If a spacetime admits a Killing vector field that also generates a curvature collineation, then the following result follows.

Theorem 4. 

If a spacetime admits a pseudo $W_2$-curvature tensor with respect to a Killing vector field ξ admitting curvature collineation, then the Lie derivative of pseudo $W_2$-curvature tensor vanishes along ξ.

Proof.  

The geometrical symmetries of a spacetime can be written as

$${\pounds }_{\xi }A-2\mathsf{\Omega }A=0,$$

(23)

where A is the physical or geometrical quantity, $\mathsf{\Omega }$ is a smooth scalar and ${\pounds }_{\xi }$ represents the Lie derivative with respect to $\xi$. For the metric inheritance symmetry, we put $A=g_{ij}$ in Equation (23). Therefore, we get

$$\left({\pounds }_{\xi }g\right)(X_u,X_v)=2\mathsf{\Omega }g(X_u,X_v).$$

(24)

Clearly, in this case if $\mathsf{\Omega }=0$, then $\xi$ becomes a Killing vector field. If $\mathsf{\Omega }$ is a smooth scalar, then $\xi$ becomes a conformal Killing vector field.

A spacetime is said to admit a symmetry called a curvature collineation, provided there exists a vector field $\xi$ such that [22,23]

$$\left({\pounds }_{\xi }R\right)(X_u,X_v)X_z=0,$$

(25)

where R is the Riemannian curvature tensor.

Motivated by the preceding discussion, we proceed to examine the inheritance of such symmetries in spacetimes that admit a pseudo $W_2$-curvature tensor.

Let a spacetime with a pseudo $W_2$-curvature tensor with respect to a Killing vector field $\xi$ be a curvature collineation. Then we have

$$\left({\pounds }_{\xi }g\right)(X_u,X_v)=0.$$

(26)

Again, since spacetime admitting a curvature collineation. Then from (25), we have

$$\left({\pounds }_{\xi }S\right)(X_u,X_v)=0,$$

(27)

where S is the Ricci tensor of the manifold. By taking the Lie derivative of (1) and using (25)–(27), we get

$$\left({\pounds }_{\xi }{\overline{W}}_2\right)(X_u,X_v)=0.$$

(28)

Thus the proof is complete. □

Theorem 5. 

Let a spacetime obeying Einstein’s field equation have vanishing pseudo $W_2$-curvature tensor. Then the spacetime admits a matter collineation with respect to a vector field ξ if and only if ξ is a Killing vector field.

Proof.  

Suppose that the pseudo $W_2$-curvature tensor vanishes, i.e., $\overline{W_2}=0.$ From (10), we obtain

$$\begin{array}{cc}\hfill \psi R(X_u,X_v)X_z& =\varphi \left[g(X_u,X_z)Q{X}_v-g(X_v,X_z)Q{X}_u\right]\hfill \\ \hfill & +{\displaystyle \frac{\theta }{4}}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)\left[g(X_v,X_z)X_u-g(X_u,X_z)X_v\right].\hfill \end{array}$$

(29)

Taking the inner product of (29) with $X_w$, we have

$$\begin{array}{cc}\hfill \overline{R}(X_u,X_v,X_z,X_w)& ={\displaystyle \frac{\varphi }{\psi }}\left[g(X_u,X_z)S(X_v,X_w)-g(X_v,X_z)S(X_u,X_w)\right]\hfill \\ \hfill & +{\displaystyle \frac{\theta }{4\psi }}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)\left[g(X_v,X_z)g(X_u,X_w)-g(X_u,X_z)g(X_v,X_w)\right].\hfill \end{array}$$

(30)

Contracting (30) with respect to $X_u$ and $X_w$, we obtain

$$S(X_v,X_z)={\displaystyle \frac{\varphi }{\psi }}\left[S(X_v,X_z)-\theta g(X_v,X_z)\right]+{\displaystyle \frac{3\theta }{4\psi }}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)g(X_v,X_z).$$

(31)

After simplification, (31) directly follows that

$$S(X_v,X_z)={\displaystyle \frac{\theta }{4}}g(X_v,X_z).$$

(32)

Thus the spacetime is an Einstein spacetime.

Now from (18) and (32), we obtain

$$kT(X_u,X_v)=\left(\lambda -{\displaystyle \frac{\theta }{4}}\right)g(X_u,X_v)$$

and

$$T(X_u,X_v)=\mu g(X_u,X_v),$$

(33)

where $\mu ={\displaystyle \frac{1}{k}}\left(\lambda -{\displaystyle \frac{\theta }{4}}\right)$. Now taking the Lie derivative of (33) along $\xi$, we get

$$\left({\pounds }_{\xi }T\right)(X_u,X_v)=\left({\pounds }_{\xi }\mu \right)g(X_u,X_v)+\mu \left({\pounds }_{\xi }g\right)(X_u,X_v).$$

(34)

Suppose that $\xi$ generates a matter collineation. Then (34) becomes

$$\left({\pounds }_{\xi }T\right)(X_u,X_v)=0.$$

Consequently,

$$\mu \left({\pounds }_{\xi }g\right)(X_u,X_v)=0.$$

Since $\mu \ne 0$, it follows that $\left({\pounds }_{\xi }g\right)(X_u,X_v)=0.$ Thus $\xi$ is a Killing vector field.

Conversely, if $\xi$ is a Killing vector field, then

$$\left({\pounds }_{\xi }g\right)(X_u,X_v)=0.$$

Using $T(X_u,X_v)=\mu g(X_u,X_v)$, we obtain

$$\left({\pounds }_{\xi }T\right)(X_u,X_v)=\mu \left({\pounds }_{\xi }g\right)(X_u,X_v)=0.$$

Hence $\xi$ generates a matter collineation. Therefore

$$\left({\pounds }_{\xi }T\right)(X_u,X_v)=0⟺\left({\pounds }_{\xi }g\right)(X_u,X_v)=0.$$

Thus the spacetime admits a matter collineation with respect to $\xi$ if and only if $\xi$ is a Killing vector field. Hence the theorem is proved. □

Theorem 6. 

Let a spacetime satisfying Einstein’s field equation be of vanishing pseudo $W_2$-curvature tensor. Then the vector field ξ is a conformal Killing vector field if and only if the energy–momentum tensor has the Lie inheritance property with respect to ξ.

Proof.  

Suppose that $\xi$ is a conformal Killing vector field. Then

$$\left({\pounds }_{\xi }g\right)(X_u,X_v)=2\mathsf{\Omega }g(X_u,X_v),$$

(35)

where $\mathsf{\Omega }$ is a smooth scalar function.

Taking the Lie derivative of (33) along $\xi$, we have

$$\left({\pounds }_{\xi }T\right)(X_u,X_v)=\left({\pounds }_{\xi }\mu \right)g(X_u,X_v)+\mu \left({\pounds }_{\xi }g\right)(X_u,X_v),$$

(36)

since $\mu$ is constant, i.e., ${\pounds }_{\xi }\mu =0$, then (36) becomes

$$\left({\pounds }_{\xi }T\right)(X_u,X_v)=\mu \left({\pounds }_{\xi }g\right)(X_u,X_v).$$

(37)

From (35) and (37), it follows that

$$\left({\pounds }_{\xi }T\right)(X_u,X_v)=2\mathsf{\Omega }\mu g(X_u,X_v).$$

(38)

Now, combining (33) and (38), we get

$$\left({\pounds }_{\xi }T\right)(X_u,X_v)=2\mathsf{\Omega }T(X_u,X_v).$$

Hence the energy–momentum tensor satisfies the Lie inheritance property with respect to $\xi$.

Conversely, suppose that the energy–momentum tensor admits the Lie inheritance property with respect to $\xi$. Then

$$\left({\pounds }_{\xi }T\right)(X_u,X_v)=2\mathsf{\Omega }T(X_u,X_v).$$

(39)

With the help of (33), Equation (39) becomes

$$\left({\pounds }_{\xi }\mu \right)g(X_u,X_v)+\mu \left({\pounds }_{\xi }g\right)(X_u,X_v)=2\mathsf{\Omega }\mu g(X_u,X_v),$$

(40)

since ${\pounds }_{\xi }\mu =0$. Therefore, (40) yields

$$\mu \left({\pounds }_{\xi }g\right)(X_u,X_v)=2\mathsf{\Omega }\mu g(X_u,X_v).$$

Assuming $\mu \ne 0$ in the above equation, we obtain

$$\left({\pounds }_{\xi }g\right)(X_u,X_v)=2\mathsf{\Omega }g(X_u,X_v).$$

Thus $\xi$ is a conformal Killing vector field. Therefore, we conclude that

$$\left({\pounds }_{\xi }g\right)(X_u,X_v)=2\mathsf{\Omega }g(X_u,X_v)⟺\left({\pounds }_{\xi }T\right)(X_u,X_v)=2\mathsf{\Omega }T(X_u,X_v).$$

Hence the vector field $\xi$ is a conformal Killing vector field if and only if the energy–momentum tensor has the Lie inheritance property with respect to $\xi$. □

Next, we analyze a spacetime filled with a perfect fluid in which the pseudo $W_2$-curvature tensor vanishes. The energy–momentum tensor $T(X_u,X_v)$ of a perfect fluid is specified by [1]

$$T(X_u,X_v)=(\beta +\gamma )\eta \left(X_u\right)\eta \left(X_v\right)+\gamma g(X_u,X_v),$$

(41)

where $\beta$ is the energy density, $\gamma$ is the isotropic pressure, $\eta$ is a non-zero 1-form such that $g(X_u,\xi )=\eta \left(X_u\right)$ for all $X_u$, and the characteristic vector field $\xi$ is unit time-like with respect to the Lorentzian metric, that is, $g(\xi ,\xi )=\eta \left(\xi \right)=-1$.

From (19) and (41), we obtain

$$\left[\lambda -{\displaystyle \frac{\theta }{4}}-\gamma k\right]g(X_u,X_v)=k(\beta +\gamma )\eta \left(X_u\right)\eta \left(X_v\right).$$

(42)

By applying a frame field and performing contraction over the vector fields $X_u$ and $X_v$, we obtain

$$\theta =4\lambda +k\beta -3\gamma k.$$

(43)

Setting $X_u=X_v=\xi$ in (42), we get

$$\theta =4(\lambda +k\beta ).$$

(44)

From (43) and (44), we deduce that

$$\beta +\gamma =0,$$

which implies that the isotropic pressure equals the negative of the energy density $(\gamma =-\beta )$. In the context of cosmological models, this specific equation of state characterizes a vacuum energy or a cosmological constant, which acts as dark energy driving the accelerated expansion of the universe. However, for an ordinary perfect fluid, $\beta +\gamma \ne 0$, leading to a contradiction. Thus the statement can be written as:

Theorem 7. 

If a perfect fluid spacetime obeying Einstein’s field equation with cosmological constant λ has a vanishing pseudo $W_2$-curvature tensor, then its matter content cannot be an ordinary perfect fluid with $\beta +\gamma \ne 0$, where β is the energy density, and γ is the isotropic pressure. Instead, it corresponds to a dark energy-like vacuum fluid state.

Now, Einstein’s field equation without cosmological constant $\lambda$ is given by

$$S(X_u,X_v)-{\displaystyle \frac{\theta }{2}}g(X_u,X_v)=kT(X_u,X_v),$$

(45)

where $k\ne 0$. Taking a frame field over $X_u$ and $X_v$ in (45), we have

$$\theta =-k\delta ,$$

(46)

where $\delta =traceT$. Using (46) in (45), we obtain

$$S(X_u,X_v)=k\left[T(X_u,X_v)-{\displaystyle \frac{\delta }{2}}g(X_u,X_v)\right].$$

(47)

Einstein’s field equation without cosmological constant $\lambda$ for a purely electromagnetic distribution takes the form [24]

$$S(X_u,X_v)=kT(X_u,X_v).$$

(48)

Therefore (47) and (48) give $\delta =0$, Equation (28) gives $\theta =0$ and Equation (17) yields $\overline{R}(X_u,X_v,X_z,X_w)=0$, which shows that the spacetime is a locally flat Lorentzian spacetime. Thus we conclude the following:

Theorem 8. 

A pseudo $W_2$-flat spacetime satisfying Einstein’s field equation is a locally flat Lorentzian spacetime.

In pseudo $W_2$-flat perfect fluid spacetime, the curvature tensor R of type (1, 3) is given by

$$R(X_u,X_v)X_z={\displaystyle \frac{\theta }{12}}\left[g(X_v,X_z)X_u-g(X_u,X_z)X_v\right],$$

(49)

where $\theta$ denotes the scalar curvature of the spacetime. Since pseudo $W_2$-flat spacetime is an Einstein space, it follows that the scalar curvature $\theta$ is constant.

Let $L^{\perp }$ be the 3-dimensional distribution in pseudo $W_2$-flat perfect fluid spacetime orthogonal to L. Then

$$R(X_u,X_v)X_z={\displaystyle \frac{\theta }{12}}\left[g(X_v,X_z)X_u-g(X_u,X_z)X_v\right]$$

(50)

for all $X_u,X_v,X_z\in L^{\perp }$ and

$$R(X_u,L)L=-{\displaystyle \frac{\theta }{12}}{X}_u,$$

(51)

for every $X_u\in L^{\perp }$.

In [25], a Lorentzian manifold is called infinitesimally spatially isotropic relative to time-like unit vector field L if its curvature tensor R satisfies

$$R(X_u,X_v)X_z=q\left[g(X_v,X_z)X_u-g(X_u,X_z)X_v\right],$$

(52)

for all $X_u,X_v,X_z\in L^{\perp }$ and $R(X_u,L)L=t{X}_u$ for all $X_u\in L^{\perp }$, where q and t are real valued functions on the manifold. Thus from (50) and (51), the following theorem arises:

Theorem 9. 

A pseudo $W_2$-flat perfect fluid spacetime satisfying the Einstein’s field equation without cosmological constant λ and having the vector field L as the velocity vector field is infinitesimally spatially isotropic relative to the unit time-like vector field L.

Let $X_u,X_v\in L^{\perp }$, $A_1$ denotes the sectional curvature of the spacetime determined by $X_u$ and $X_v$, whereas $A_2$ denotes the sectional curvature of the spacetime determined by $X_u$ and L. Then

$$A_1={\displaystyle \frac{g(R(X_u,X_v)X_v,X_u)}{g(X_u,X_u)g(X_v,X_v)-{\left\{g(X_u,X_v)\right\}}^2}}={\displaystyle \frac{\theta }{12}}$$

and

$$A_2={\displaystyle \frac{g(R(X_u,L)L,X_u)}{g(X_u,X_u)g(L,L)-{\left\{g(X_u,L)\right\}}^2}}={\displaystyle \frac{\theta }{12}}.$$

Thus we are able to state the following:

Theorem 10. 

In pseudo $W_2$-flat perfect fluid spacetime, the sectional curvature determined by two vectors $X_u,X_v\in L^{\perp }$ and the sectional curvature determined by two vectors $X_u$ and L are equal.

3. Divergence-Free Pseudo ${\mathit{W}}_{\mathbf{2}}$-Curvature Tensor and Fluid Spacetimes

Definition 1. 

A symmetric (0, 2) type tensor field T on a semi-Riemannian manifold $(M^n,g)$ is said to be a Codazzi tensor if it satisfies the Codazzi equation

$$\left({\nabla }_{X_u}T\right)(X_v,X_z)=\left({\nabla }_{X_v}T\right)(X_u,X_z)$$

(53)

for all $X_u,X_v$ and $X_z$.

The significance of the Codazzi tensor on a Riemannian manifold was studied by [26].

Now, taking the covariant derivative of (45) with $X_u$, we have

$$\left({\nabla }_{X_u}S\right)(X_v,X_z)-{\displaystyle \frac{1}{2}}g(X_v,X_z)d\theta \left(X_u\right)=k\left({\nabla }_{X_u}T\right)(X_v,X_z),$$

(54)

where $d\theta \left(X_u\right)={\nabla }_{X_u}\theta$. Interchanging $X_u$ and $X_v$ in (54), we get

$$\left({\nabla }_{X_v}S\right)(X_u,X_z)-{\displaystyle \frac{1}{2}}g(X_u,X_z)d\theta \left(X_v\right)=k\left({\nabla }_{X_v}T\right)(X_u,X_z).$$

(55)

Now, subtracting (55) from (54), we obtain

$$\begin{array}{cc}\hfill \left({\nabla }_{X_u}S\right)(X_v,X_z)-\left({\nabla }_{X_v}S\right)(X_u,X_z)& =k\left[\left({\nabla }_{X_u}T\right)(X_v,X_z)-\left({\nabla }_{X_v}T\right)(X_u,X_z)\right]\hfill \\ & +{\displaystyle \frac{1}{2}}\left[g(X_v,X_z)d\theta \left(X_u\right)-g(X_u,X_z)d\theta \left(X_v\right)\right].\hfill \end{array}$$

(56)

If T is a Codazzi tensor, then (56) yields

$$\begin{array}{c}\hfill \left({\nabla }_{X_u}S\right)(X_v,X_z)-\left({\nabla }_{X_v}S\right)(X_u,X_z)={\displaystyle \frac{1}{2}}\left[g(X_v,X_z)d\theta \left(X_u\right)-g(X_u,X_z)d\theta \left(X_v\right)\right].\end{array}$$

(57)

Now, contracting the Codazzi condition on T with respect to $X_v$ and $X_z$, we obtain

$${\nabla }^j{T}_{ij}={\nabla }_i\delta ,$$

where $\delta =traceT$. Since the energy–momentum tensor is divergence-free, i.e., ${\nabla }^j{T}_{ij}=0$, it follows that $\delta$ is constant. Taking the trace of (45), we get

$$\theta =-k\delta ,$$

which shows that $\theta$ is constant. Therefore, we obtain

$$d\theta \left(X_u\right)=d\theta \left(X_v\right)=0.$$

(58)

From (57) and (58), we arrive at

$$\left({\nabla }_{X_u}S\right)(X_v,X_z)=\left({\nabla }_{X_v}S\right)(X_u,X_z).$$

Thus the Ricci tensor S is of Codazzi type.

Conversely, suppose that S is of Codazzi type. Then from (56) and (58), we have

$$k\left[\left({\nabla }_{X_u}T\right)(X_v,X_z)-\left({\nabla }_{X_v}T\right)(X_u,X_z)\right]=0,$$

since $k\ne 0$. Therefore, the above equation becomes

$$\left({\nabla }_{X_u}T\right)(X_v,X_z)=\left({\nabla }_{X_v}T\right)(X_u,X_z),$$

which shows that T is of Codazzi type. Hence, the energy–momentum tensor T is of Codazzi type if and only if the Ricci tensor S is of Codazzi type. Thus the following theorem can be stated as:

Theorem 11. 

If a spacetime satisfying Einstein’s field equation without cosmological constant λ has an energy–momentum tensor T of Codazzi type, then the Ricci tensor S is of Codazzi type and conversely.

Next, we take the covariant derivative of (1) with respect to $X_u$, we get

$$\begin{array}{cc}\hfill \left({\nabla }_{X_u}{{}^{\prime }\overline{W}}_2\right)(X_v,X_z,X_w,X_p)& =\psi \left({\nabla }_{X_u}{}^{\prime }R\right)(X_v,X_z,X_w,X_p)\hfill \\ & +\varphi [g(X_z,X_w)\left({\nabla }_{X_u}S\right)(X_v,X_p)\hfill \\ & -g(X_v,X_w)\left({\nabla }_{X_u}S\right)(X_z,X_p)]\hfill \\ & -{\displaystyle \frac{\left({\nabla }_{X_u}\theta \right)}{4}}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)[g(X_z,X_w)g(X_v,X_p)\hfill \\ & -g(X_v,X_w)g(X_z,X_p)],\hfill \end{array}$$

(59)

for all the vector fields $X_u,X_v,X_z,X_w,X_p$ on $(M^4,g)$. Contracting (59) over $X_u=e_i$ and $X_p=e_i$, we obtain

$$\begin{array}{cc}\hfill \left(div{\overline{W}}_2\right)(X_v,X_z,X_w)& =\psi \left(divR\right)(X_v,X_z,X_w)\hfill \\ & +\varphi \left[g(X_z,X_w)\left(divS\right)\left(X_v\right)-g(X_v,X_w)\left(divS\right)\left(X_z\right)\right]\hfill \\ & -{\displaystyle \frac{1}{4}}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)\left[g(X_z,X_w)\left({\nabla }_{X_v}\theta \right)-g(X_v,X_w)\left({\nabla }_{X_z}\theta \right)\right],\hfill \end{array}$$

(60)

where $div$ represents divergence [27]. The divergence of ${\overline{W}}_2$ is given by

$$\left(div{\overline{W}}_2\right)(X_v,X_z,X_w)={\displaystyle \sum _{i=1}^4}\left({\nabla }_{e_i}{}^{\prime }\overline{W}_2\right)(X_v,X_z,X_w,e_i),$$

where ${\left\{e_i\right\}}_{i=1}^4$ is a local orthonormal frame. By the contracted second Bianchi identity, it is known that

$$\left(divR\right)(X_v,X_z,X_w)=\left({\nabla }_{X_v}S\right)(X_z,X_w)-\left({\nabla }_{X_z}S\right)(X_v,X_w).$$

(61)

Since the spacetime satisfies Einstein’s field equation. Therefore, taking covariant differentiation of (18) with respect to $X_u$, we obtain

$$\left({\nabla }_{X_u}S\right)(X_v,X_z)=k\left({\nabla }_{X_u}T\right)(X_v,X_z)+{\displaystyle \frac{1}{2}}\left({\nabla }_{X_u}\theta \right)g(X_v,X_z).$$

(62)

As T is of Codazzi type, then we have

$$\left({\nabla }_{X_u}T\right)(X_v,X_z)=\left({\nabla }_{X_v}T\right)(X_u,X_z).$$

(63)

Using (62) and (63), we obtain

$$\left({\nabla }_{X_u}S\right)(X_v,X_z)=\left({\nabla }_{X_v}S\right)(X_u,X_z),$$

(64)

which shows that the Ricci tensor S is also of Codazzi type. Hence

$$\left({\nabla }_{X_v}S\right)(X_z,X_w)=\left({\nabla }_{X_z}S\right)(X_v,X_w).$$

(65)

Therefore, from (61), (64) and (65), it is obvious that

$$\left(divR\right)(X_v,X_z,X_w)=0.$$

(66)

Substituting (66) into (60), we get

$$\begin{array}{cc}\hfill \left(div{\overline{W}}_2\right)(X_v,X_z,X_w)& =\varphi \left[g(X_z,X_w)\left(divS\right)\left(X_v\right)-g(X_v,X_w)\left(divS\right)\left(X_z\right)\right]\hfill \\ & -{\displaystyle \frac{1}{4}}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)\left[g(X_z,X_w)\left({\nabla }_{X_v}\theta \right)-g(X_v,X_w)\left({\nabla }_{X_z}\theta \right)\right].\hfill \end{array}$$

(67)

Again, by the contracted Bianchi identity, we have

$$\left(divS\right)\left(X_u\right)={\displaystyle \frac{1}{2}}\left({\nabla }_{X_u}\theta \right)={\displaystyle \frac{1}{2}}d\theta \left(X_u\right).$$

(68)

Now, from (67) and (68), we get

$$\begin{array}{c}\hfill \left(div{\overline{W}}_2\right)(X_v,X_z,X_w)=\left[{\displaystyle \frac{\varphi }{2}}-{\displaystyle \frac{1}{4}}\left({\displaystyle \frac{\psi }{3}}+\varphi \right)\right]\left[g(X_z,X_w)\left({\nabla }_{X_v}\theta \right)-g(X_v,X_w)\left({\nabla }_{X_z}\theta \right)\right].\end{array}$$

(69)

Since T is a Codazzi tensor, it follows from (18) that the scalar curvature $\theta$ is constant. Therefore, we obtain

$${\nabla }_{X_v}\theta =d\theta \left(X_v\right)=0.$$

(70)

Consequently, (69) and (70) follow that

$$\left(div{\overline{W}}_2\right)(X_v,X_z,X_w)=0,$$

which implies

$$div{\overline{W}}_2=0.$$

Hence the pseudo $W_2$-curvature tensor is divergence-free whenever the energy–momentum tensor is of Codazzi type and $\psi \ne 0$. Thus the theorem states:

Theorem 12. 

If in a spacetime the energy–momentum tensor T is of Codazzi type, then the pseudo $W_2$-curvature tensor is divergence-free, provided $\psi \ne 0$.

Also, taking the frame field after contraction and from (41), we have

$$\delta =3\gamma -\beta .$$

(71)

If $\left(div\overline{W_2}\right)(X_v,X_z,X_w)=0$, and $T(X_v,X_z)$ is of Codazzi type, then from (47), (60) and (71), we get

$$g(X_z,X_w)d\delta \left(X_v\right)-g(X_v,X_w)d\delta \left(X_z\right)=0,$$

(72)

which means that $\delta$ = constant, i.e., $3\gamma -\beta$ = constant. Hence we can state the following:

Theorem 13. 

If in a perfect fluid spacetime the divergence of pseudo $W_2$-curvature tensor vanishes and the energy–momentum tensor T is of Codazzi type, then $3\gamma -\beta$ is constant.

If $3\gamma -\beta =0$, then the isotropic pressure relates to the energy density by $\gamma ={\displaystyle \frac{\beta }{3}}$. In relativistic cosmology, this specific equation of state indicates that the spacetime is filled with pure radiation. This represents the radiation-dominated era of the early universe, where relativistic particles and photons dictated the expansion dynamics [24].

4. Divergence-Free Pseudo ${\mathit{W}}_{\mathbf{2}}$-Curvature Tensor in Ricci-Recurrent Spacetimes

Definition 2. 

A semi-Riemannian manifold $(M^n,g)(n>2)$ is called a Ricci-recurrent manifold [28] if the Ricci tensor S satisfies

$$\left({\nabla }_{X_u}S\right)(X_v,X_z)=\eta \left(X_u\right)S(X_v,X_z),$$

(73)

where η is a non-vanishing 1-form associated with the vector field ξ, as $\eta \left(X_u\right)=g(X_u,\xi )$. If the Ricci tensor fulfills the condition (73), a Lorentzian manifold is referred to as Ricci-recurrent spacetime. In this case, suppose the vector field ξ related to the 1-form η is treated as a unit time-like vector field, i.e., $g(\xi ,\xi )=\eta \left(\xi \right)=-1$.

If $div{\overline{W}}_2=0$, then from Theorem 11, we find

$$\left({\nabla }_{X_u}S\right)(X_v,X_z)=\left({\nabla }_{X_v}S\right)(X_u,X_z).$$

(74)

Applying (73) in (74), we have

$$\eta \left(X_u\right)S(X_v,X_z)=\eta \left(X_v\right)S(X_u,X_z).$$

(75)

Setting $X_u=\xi$ in (75), we get

$$S(X_v,X_z)=-\eta \left(X_v\right)S(X_z,\xi ).$$

(76)

Performing $X_v=X_z=e_i$ in (75) and summing over i, $1\le i\le 4$, we have

$${\displaystyle \sum _{i=1}^4}S(e_i,e_i)=\theta ,$$

and

$${\displaystyle \sum _{i=1}^4}g(e_i,\xi )S(X_u,e_i)=S(X_u,\xi ).$$

Therefore, with the help of the above equations and (75), we obtain

$$\eta \left(X_u\right)\theta =S(X_u,\xi ),$$

(77)

where $\theta$ is the scalar curvature of the manifold. Inserting (77) in (76), we deduce that

$$S(X_v,X_z)=-\theta \eta \left(X_v\right)\eta \left(X_z\right),$$

which implies that the spacetime is Ricci simple [29].

As a consequence of the above discussion, we obtain the following proposition:

Proposition 1. 

Let $(M^4,g)$ be a Ricci-recurrent spacetime admitting a divergence-free pseudo $W_2$-curvature tensor, i.e., $div{\overline{W}}_2=0.$ Then the Ricci tensor S is of the Ricci simple form

$$S(X_u,X_v)=-\theta \eta \left(X_u\right)\eta \left(X_v\right),$$

for some non-zero 1-form η and scalar curvature θ. If $\theta \ne 0$ and η is non-null, then $(M^4,g)$ is a quasi-Einstein spacetime of rank-one type, i.e.,

$$S(X_u,X_v)=ag(X_u,X_v)+b\omega \left(X_u\right)\omega \left(X_v\right),$$

with $a=0$, $b=-\theta$, and $\omega =\eta$.

Remark 2. 

The physical interpretation of a Ricci simple spacetime is rigorously explored in [29], where the authors proved that a Ricci simple spacetime dynamically behaves as a stiff matter fluid. A stiff fluid, characterized by the equation of state where pressure equals energy density $(\beta =\gamma )$, corresponds to the Zel’dovich cosmological model. This model is critical for describing the ultra-dense, extreme conditions of the very early universe. Thus, we conclude that a Ricci-recurrent spacetime admitting a divergence-free pseudo $W_2$-curvature tensor represents a Zel’dovich stiff matter universe. A stiff matter fluid is characterized by an equation of state where the pressure γ is exactly equal to the energy density β ($\gamma =\beta$). This implies that the speed of sound in such a medium is equal to the speed of light ($v_s=c$). Such extreme conditions are not found in the contemporary universe but are theorized to have existed during the very early stages of the Big Bang, specifically during the Planck epoch or within the core of ultra-dense relativistic stars. It is established that a Ricci-recurrent spacetime with divergence-free pseudo $W_2$-curvature tensor corresponds to a Zel’dovich stiff matter universe.

5. Examples of Pseudo ${\mathit{W}}_{\mathbf{2}}$-Flat Spacetimes and Comparative Discussion

To illustrate the theoretical results, we present two explicit Lorentzian spacetimes whose pseudo $W_2$-curvature tensor vanishes. The first example is an isotropic warped spacetime, while the second is the de Sitter spacetime. In each case, the metric, Christoffel symbols, curvature tensors, Ricci operator, scalar curvature, and pseudo $W_2$-flatness condition are verified explicitly.

Example 1. 

Isotropic Warped Spacetime:

Consider the 4-dimensional Lorentzian manifold $({\mathbb{R}}^4,g)$ with metric expressed in the coordinates $(x^1,x^2,x^3,x^4)$ as follows

$$d{s}^2=g_{ij}d{x}^{i}d{x}^j=-{\left(d{x}^1\right)}^2+e^{x^1}\left[{\left(d{x}^2\right)}^2+{\left(d{x}^3\right)}^2+{\left(d{x}^4\right)}^2\right],$$

(78)

for $i,j=1,2,3,4.$ The metric components are

$$g_{11}=-1,g_{22}=g_{33}=g_{44}=e^{x^1},g_{ij}=0\mathrm{for}i\ne j,$$

with inverse

$$g^{11}=-1,g^{22}=g^{33}=g^{44}=e^{-x^1}.$$

The non-zero Christoffel symbols of the second kind are

$${\mathsf{\Gamma }}_{22}^1={\mathsf{\Gamma }}_{33}^1={\mathsf{\Gamma }}_{44}^1=-{\displaystyle \frac{e^{x^1}}{2}},{\mathsf{\Gamma }}_{12}^2={\mathsf{\Gamma }}_{13}^3={\mathsf{\Gamma }}_{14}^4={\displaystyle \frac{1}{2}}.$$

The non-zero Riemann curvature components are

$$R_{1212}=R_{1313}=R_{1414}=-{\displaystyle \frac{e^{x^1}}{4}},R_{2323}=R_{2424}=R_{3434}={\displaystyle \frac{e^{2x^1}}{4}}.$$

Also, the non-zero Ricci components are

$$R_{11}={\displaystyle \frac{3}{4}},R_{22}=R_{33}=R_{44}=-{\displaystyle \frac{3e^{x^1}}{4}}.$$

For the metric given in (78) and with the help of the above results, the scalar curvature $\theta$ is

$$\theta =g^{11}{R}_{11}+g^{22}{R}_{22}+g^{33}{R}_{33}+g^{44}{R}_{44}=-3,$$

which can be easily seen from (17). Now, substituting $Q{X}_u=-{\displaystyle \frac{3}{4}}{X}_u$, $\theta =-3$ and $n=4$ in (2), we obtain

$${\overline{W}}_2(X_u,X_v)X_z=0.$$

Hence the isotropic warped spacetime is pseudo ${\overline{W}}_2$-flat.

Example 2. 

FLRW-type exponential spacetime (de Sitter):

Consider the de Sitter metric with the standard Cartesian spatial coordinates $(x,y,z)$ given by

$$d{s}^2=-d{t}^2+e^{2\alpha t}\left(d{x}^2+d{y}^2+d{z}^2\right),\alpha \ne 0,$$

where $\alpha$ is the Hubble constant, and t is the cosmological time coordinate. The metric components are

$$g_{00}=-1,g_{11}=g_{22}=g_{33}=e^{2\alpha t},g_{ij}=0(i\ne j),$$

$$g^{00}=-1,g^{11}=g^{22}=g^{33}=e^{-2\alpha t}.$$

The non-zero Christoffel symbols are obtained as follows

$${\mathsf{\Gamma }}_{11}^0={\mathsf{\Gamma }}_{22}^0={\mathsf{\Gamma }}_{33}^0=\alpha e^{2\alpha t},{\mathsf{\Gamma }}_{10}^1={\mathsf{\Gamma }}_{20}^2={\mathsf{\Gamma }}_{30}^3=\alpha .$$

The non-zero Riemann curvature tensor are

$$R_{0i0i}=-{\alpha }^2{e}^{2\alpha t},R_{ijkl}={\alpha }^2{e}^{4\alpha t}({\delta }_{ik}{\delta }_{jl}-{\delta }_{il}{\delta }_{jk}),i,j,k,l=1,2,3.$$

Moreover, the Ricci tensor are obtained by

$$R_{00}=-3{\alpha }^2,R_{11}=R_{22}=R_{33}=3{\alpha }^2{e}^{2\alpha t},$$

and the scalar curvature $\theta$ is

$$\theta =12{\alpha }^2.$$

Also, the Ricci operator Q is

$$Q\left({\partial }_0\right)=-3{\alpha }^2{\partial }_0,Q\left({\partial }_i\right)=3{\alpha }^2{\partial }_i,i=1,2,3.$$

Finally, from (1) and the above results, we obtain

$${\overline{W}}_2(X_u,X_v)X_z=0.$$

Hence the FLRW-type exponential spacetime (de Sitter) is pseudo ${\overline{W}}_2$-flat.

Proposition 2. 

The spacetimes described in Examples 1 and 2 are pseudo $W_2$-flat spacetimes of constant sectional curvature. Consequently, they satisfy the conclusion of Theorem 1.

Proof.  

For Example 1, direct computation shows that

$$K_{12}={\displaystyle \frac{R_{1212}}{g_{11}{g}_{22}-g_{12}^2}}={\displaystyle \frac{-e^{x^1}/4}{-e^{x^1}}}={\displaystyle \frac{1}{4}}.$$

Similarly,

$$K_{13}=K_{14}=K_{23}=K_{24}=K_{34}={\displaystyle \frac{1}{4}}.$$

Hence the sectional curvature K is constant. Moreover,

$$S=-{\displaystyle \frac{3}{4}}g,$$

so the spacetime is Einstein. Since ${\overline{W}}_2=0$, the conclusion of Theorem 1 follows.

For Example 2, the de Sitter spacetime satisfies

$$\overline{R}(X_u,X_v,X_z,X_w)={\alpha }^2\left[g(X_u,X_z)g(X_v,X_w)-g(X_u,X_w)g(X_v,X_z)\right],$$

which implies constant sectional curvature $K={\alpha }^2$. Thus de Sitter spacetime is a constant-curvature spacetime. Furthermore,

$$S=3{\alpha }^{2}g.$$

Since ${\overline{W}}_2=0$, the theorem is again verified. □

From the examples provided above, we got the results in the Table 1:

Table 1. Comparison of pseudo $W_2$-flat spacetimes.

Property	Example 1	Example 2
Metric type	Isotropic warped	FLRW exponential (de Sitter)
Non-zero Christoffel symbols	2 sets	6 sets
Riemann curvature	Non-zero, coordinate-dependent	Constant sectional curvature
Ricci tensor	$S=-{\displaystyle \frac{3}{4}}g$	$S=3{\alpha }^{2}g$
Scalar curvature	$-3$	$12{\alpha }^2$
Ricci operator	$Q=-{\displaystyle \frac{3}{4}}I$	$Q=3{\alpha }^{2}I$
Physical interpretation	Isotropic warped spacetime	de Sitter universe
Pseudo $W_2$-flat	Yes	Yes

All the above spacetimes satisfy ${\overline{W}}_2=0$, thereby confirming their pseudo $W_2$-flatness. Example 1 represents an isotropic warped spacetime with non-constant Riemann curvature components, whereas Example 2 corresponds to the de Sitter spacetime, a Lorentzian manifold of constant positive curvature satisfying $S=\lambda g$. Despite their distinct geometric structures, both examples satisfy the pseudo $W_2$-flat condition. These examples demonstrate that pseudo $W_2$-flat spacetimes arise in different geometric settings, ranging from isotropic warped geometries to FLRW-type cosmological models. Consequently, the pseudo $W_2$-flat condition provides a useful geometric framework for the study of physically relevant Lorentzian spacetimes.

6. Conclusions

The pseudo $W_2$-curvature tensor serves as a fundamental object in geometric gravity theories, extending beyond classical Riemannian geometry. By encoding richer geometric information than standard curvature tensors, it provides crucial insights into non-Einsteinian spacetimes–particularly in modified gravity, anisotropic cosmological models and scenarios involving unconventional matter. In general relativity, the curvature of spacetime is directly linked to the energy and momentum of matter through Einstein’s field equations. Additionally, investigating the role of this tensor in anisotropic Bianchi-type cosmological models could offer deeper insights into the pre-inflationary phase of the universe, where spatial isotropy was not yet established. To highlight the novelty and significance of the present work, a comparison with related studies is presented in Table 2.

Table 2. Comparison of the present work with related studies.

Aspect	${\mathit{W}}_2$-Curvature Tensor	Pseudo Quasi-Conformal Curvature Tensor	Pseudo ${\mathit{W}}_2$-Curvature Tensor (Present Work)
Geometric setting	Relativistic spacetimes and modified gravity models [3,4]	Lorentzian manifolds with pseudo quasi-conformal structures [5]	Lorentzian spacetimes satisfying Einstein’s field equations
Curvature structure	Based on the Riemann, Ricci, and scalar curvatures	Generalized conformal-type curvature tensor	Parameterized extension of the $W_2$-curvature tensor involving $\psi$ and $\varphi$
Flatness condition	$W_2$-flatness and related geometric properties	Pseudo quasi-conformal flatness	Pseudo $W_2$-flatness and its geometric consequences
Einstein field equations	Limited investigation	Not a principal focus	Direct characterization via the energy–momentum tensor
Symmetry properties	Curvature restrictions and symmetry conditions	Conformal-type geometric symmetries	Matter collineations, Killing vector fields, conformal Killing vector fields, and Lie inheritance symmetries
Divergence properties	Not emphasized	Not emphasized	Divergence-free pseudo $W_2$-curvature tensor under Codazzi-type conditions
Main characterization	Curvature properties of $W_2$-spacetimes	Pseudo quasi-conformal geometric structures	Constant-curvature characterization of pseudo $W_2$-flat spacetimes
Examples	Selected spacetime models	Mainly theoretical studies	Explicit isotropic warped and de Sitter spacetime models