1. Introduction
Symmetry represents a captivating element of the Universe, acting as a foundational principle that elucidates the laws of nature, encompassing general relativity and a range of other physical theories. The energy-momentum tensor plays a crucial role in illustrating the matter content within spacetime, which is frequently envisioned as a fluid defined by its density, pressure, and various dynamic and kinematic characteristics, including velocity, acceleration, vorticity, shear, and expansion. In conventional cosmological models, the Universe’s matter is generally depicted as a perfect fluid, characterized by the absence of heat conduction and viscosity. In 1915, Albert Einstein presented his theory of ‘General Relativity of gravity’ (GR), positing that the gravitational field is intrinsically linked to the curvature of spacetime and the energy-momentum tensor. Einstein’s equations, which describe the dynamics of spacetime curvature, have profound implications across various disciplines, including particle physics, nuclear physics [1], astrophysics [2], and plasma physics.
GR is recognized as one of the most significant theories in 20th-century physics, shedding light on the essential connection between physical laws and the structure of spacetime. Over the past century, it has developed into a dynamic field of inquiry within both physics and mathematics. The theory demonstrates its greatest efficacy when its equations can comprehensively account for all phenomena observed at both extragalactic and galactic scales. To enhance our understanding of GR, we investigate the model of relativistic fluids through the framework of differential geometry, recognizing that the essence of this theory is anchored in the notion of spacetime as a curved manifold.
A spacetime can be conceptualized as a Lorentzian manifold, which serves as a solid foundation for investigating the properties of the manifold’s vectors, thereby providing a significant framework for the examination of cosmological models. This Lorentzian manifold is represented as (Mn, g). When the dimension n is three or greater, (Mn, g) is categorized as a perfect fluid spacetime (PFS) if its Ricci tensor is formulated as follows:
$\begin{eqnarray*}S=ag+b\vartheta \otimes \vartheta ,\end{eqnarray*}$
where a and b are scalar functions, and ϑ denotes a 1-form. A perfect fluid is defined by its mass density and isotropic pressure in its rest frame, and within the framework of general relativity, it is generally represented as an idealized matter distribution.In 1995, Alias et al introduced the notion of generalized Robertson–Walker (GRW) spacetimes [3, 4]. A Lorentzian manifold, represented as (Mn, g), qualifies as a GRW spacetime if it can be formulated as $M=-I{\times }_{{\nu }^{2}}{M}^{* }$, where I denotes an open interval in ${\mathbb{R}}$, M* is an (n − 1)-dimensional manifold, and ν > 0 is a scalar function. It is important to note that when n = 4 and M* has constant curvature, the resulting structure aligns with the Robertson–Walker (RW) spacetime. The characteristics of GRW spacetimes have been investigated by numerous scholars [5–7]. An n-dimensional Lorentzian manifold that contains a globally timelike vector field is classified as a spacetime.
Sanchez [8] emphasizes that GRW spacetimes are relevant in homogeneous scenarios characterized by isotropic radiation. O’Neill [9] further discusses RW spacetime as a model for imperfect fluid spacetimes. It is important to note that a four-dimensional GRW spacetime can only be categorized as a perfect fluid spacetime if it adheres to the RW spacetime framework. In geometric analysis, symmetry is crucial for depicting the arrangement of physical entities, particularly concerning the geometric structure of spacetime. The metric associated with symmetry often aids in simplifying solutions across various research domains. Ricci curvature is essential in general relativity, especially in addressing the Einstein field equations. Solitons are significant symmetry patterns linked to the geometric evolution of spacetime, including Ricci flow and Yamabe flow, which are vital for comprehending energy and entropy within the context of general relativity. Furthermore, numerous studies highlight the importance of Ricci solitons and Yamabe solitons due to their self-similar curvature properties.
The energy-momentum tensor plays a vital role in the study of spacetime, representing a fluid that possesses various properties, including density, pressure, and a spectrum of dynamic and kinematic features such as velocity, acceleration, vorticity, shear, and expansion [10]. The inclusion of viscosity terms classifies this fluid as an imperfect fluid [11]. Investigating imperfect fluid spacetimes provides valuable insights for cosmological models that extend beyond conventional perfect fluid spacetimes. A comprehensive understanding of these concepts requires an examination of the behaviors of both perfect and imperfect fluid spacetimes within established cosmological frameworks. In the context of scalar-tensor gravity, particularly with respect to the Brans–Dicke-like field, the dynamics of the imperfect fluid are articulated through an effective Einstein equation. This effective formulation of imperfect fluid dynamics is applied within Einstein’s theory, especially in canonical GRW spaces related to Friedmannian cosmology [12]. Additionally, refer to [13, 14].
Research conducted in [5] has demonstrated that a Lorentzian manifold M, where the dimension satisfies dim(M)≥3, qualifies as a GRW spacetime if and only if it possesses a timelike concircular vector field. Furthermore, in 2017, Mantica and Molinari [7] identified the necessary and sufficient conditions for a Lorentzian manifold to support a unit timelike torse-forming vector field, which additionally acts as an eigenvector of the Ricci tensor, thereby confirming its classification as a GRW spacetime.
Let ρ represent the isotropic pressure, while ${ \mathcal P }$ denotes the tensor of isotropic pressure related to the viscous fluid. The symbol σ indicates the energy density of an imperfect fluid within the framework of GRW spacetime. The energy-momentum tensor ${ \mathcal T }$ is essential for characterizing the properties of the imperfect fluid in GRW spacetime, as cited in [9, 12, 15, 16]. The formulation of this tensor is expressed by the equation
$\begin{eqnarray}{ \mathcal T }({U}_{1},{U}_{2})=\rho g({U}_{1},{U}_{2})+(\sigma +\rho )\vartheta ({U}_{1})\vartheta ({U}_{2})+{ \mathcal P }({U}_{1},{U}_{2}),\end{eqnarray}$
where ${ \mathcal P }$ represents a symmetric traceless tensor such that ${ \mathcal P }(U,\zeta )=0$ and the term ϑ(U1) = g(U1, ζ) is a 1-form linked to the fluid’s velocity vector ζ, with the condition that g(ζ, ζ) = −1.Let τ represent the cosmological constant and κ denote the gravitational constant. The behavior of fluid motion is dictated by Einstein’s gravitational equation [9], which can be expressed as follows1.1 ), we can reformulate the expression in equation (1.2 ) to obtain
$\begin{eqnarray}\begin{array}{r}S({U}_{1},{U}_{2})+\left(\tau -\frac{r}{2}\right)g({U}_{1},{U}_{2})=\kappa T({U}_{1},{U}_{2}),\end{array}\end{eqnarray}$
where S signifies the Ricci tensor and r is the scalar curvature linked to the metric g. By utilizing equation ( $\begin{eqnarray}\begin{array}{l}S({U}_{1},{U}_{2})=(-\tau +\frac{r}{2}+\kappa \rho )g({U}_{1},{U}_{2})\\ \,+\,\kappa (\sigma +\rho )\vartheta ({U}_{1})\vartheta ({U}_{2})+\kappa { \mathcal P }({U}_{1},{U}_{2}).\end{array}\end{eqnarray}$
If ${ \mathcal P }=0$, then imperfect fluid spacetime becomes a PFS. Subsequently, by contracting equation (1.3 ) and applying the condition g(ζ, ζ) = −1, we derive1.4 ) into (1.3 ), we arrive at the conclusion that
$\begin{eqnarray}r=4\tau -3\kappa \rho +\kappa \sigma .\end{eqnarray}$
By substituting ( $\begin{eqnarray}\begin{array}{rcl}S({U}_{1},{U}_{2}) & = & \frac{1}{2}(2\tau -\kappa \rho +\kappa \sigma )g({U}_{1},{U}_{2})\\ & & +\kappa (\sigma +\rho )\vartheta ({U}_{1})\vartheta ({U}_{2})+\kappa { \mathcal P }({U}_{1},{U}_{2}).\end{array}\end{eqnarray}$
Also, in dark fluid spacetime [17], we have σ = − ρ. Thus, dark fluid spacetime is a subset of imperfect fluid spacetime and if any imperfect fluid spacetime becomes dark fluid then
$\begin{eqnarray*}\begin{array}{r}S({U}_{1},{U}_{2})=\frac{1}{2}(2\tau -\kappa \rho +\kappa \sigma )g({U}_{1},{U}_{2})+\kappa { \mathcal P }({U}_{1},{U}_{2}).\end{array}\end{eqnarray*}$
A semi-Riemannian manifold (M, g) with dimension greater than two is called a pseudo quasi-Einstein manifold [18] if its Ricci tensor can be expressed as
$\begin{eqnarray*}S={a}_{1}g+{a}_{2}\vartheta \otimes \vartheta +{a}_{3}D,\end{eqnarray*}$
where a1, a2, a3 are constants, ϑ is a non-zero 1-form linked to the unit timelike vector field ζ (with ϑ(U) = g(U, ζ)), and D is a symmetric trace-free tensor satisfying D(U, ζ) = 0. The study referenced shows that the imperfect fluid GRW spacetime is a specific example of a pseudo quasi-Einstein manifold. Following the method in [19], if we set ${ \mathcal P }=q(\omega \otimes \vartheta +\vartheta \otimes \omega )$, the imperfect fluid GRW spacetime exemplifies a pseudo quasi-Einstein manifold, with q as the fluid’s shear and ω associated with the unit vector field ξ (defined by ω(U) = g(U, ξ)). The Ricci tensor can be represented as $\begin{eqnarray}\begin{array}{l}S({U}_{1},{U}_{2})=ag({U}_{1},{U}_{2})+b\vartheta ({U}_{1})\vartheta ({U}_{2})\\ \,+\,c\omega ({U}_{1})\vartheta ({U}_{2})+c\vartheta ({U}_{1})\omega ({U}_{2}),\end{array}\end{eqnarray}$
where $a=\frac{1}{2}(2\tau -\kappa \rho +\kappa \sigma )$, b = κ(σ + ρ), and c = κq.The concept of ‘soliton’ was initially put forth by Kruskal and Zabusky to clarify the properties of solitary waves. Since that time, the comprehension of solitons has advanced considerably, resulting in their utilization in a variety of fields. A thorough physical and mathematical framework has been developed for solitons, underscoring their significance in modern physics (see [20, 21]). Furthermore, the symmetry metric is frequently streamlined to categorize solutions to Einstein’s field equations, with solitons serving a vital function as a symmetry associated with the geometric evolution of spacetime. Hamilton’s research on specific geometric flows, including Ricci flow [22], has enhanced our insight into kinematics, and these geometric flows have been essential in the investigation of gravitational phenomena (see [23]).
Let (M, g) represent a pseudo-Riemannian manifold, with R denoting the Riemann curvature tensor associated with the metric g. The Riemann flow on the manifold (M, g), as introduced by Udrişte [24], is expressed by the equation
$\begin{eqnarray*}\frac{\partial }{\partial t}G(t)=-2R(g(t)),\end{eqnarray*}$
where $G=\frac{1}{2}g\odot g$, and ⊙ signifies the Kulkarni–Nomizu product. This product for two (0, 2)-tensors, ω and θ, is defined as follows: for any vector fields U1, U2, U3, U4, it holds that $\begin{eqnarray*}\begin{array}{l}(\omega \odot \theta )({U}_{1},{U}_{2},{U}_{3},{U}_{4})\\ \,=\,\omega ({U}_{1},{U}_{4})\theta ({U}_{2},{U}_{3})+\omega ({U}_{2},{U}_{3})\theta ({U}_{1},{U}_{4})\\ \,-\,\omega ({U}_{1},{U}_{3})\theta ({U}_{2},{U}_{4})-\omega ({U}_{2},{U}_{4})\theta ({U}_{1},{U}_{3}).\end{array}\end{eqnarray*}$
If there exists a smooth vector field V satisfying the equation $\begin{eqnarray}2R+\mu g\odot g+g\odot {{ \mathcal L }}_{V}g=0,\end{eqnarray}$
then the structure is referred to as a Riemann soliton (or RS) and is denoted by (Mn, g, μ, V), where μ is a constant. The classification of a Riemann soliton is determined by the value of μ, where it is referred to as expanding if μ > 0, steady if μ = 0, and shrinking if μ < 0. When the vector field V is expressed as ${\rm{grad}}\phi $ for a smooth function f, the Riemann soliton can be represented by the equation $\begin{eqnarray}2R+\mu g\odot g+2g\odot {{\rm{\nabla }}}^{2}f=0,\end{eqnarray}$
which designates it as a gradient Riemann soliton (or gradient RS). Furthermore, if μ is a smooth function, the Riemann soliton and gradient Riemann soliton are referred to as almost Riemann soliton (or ARS) and almost gradient Riemann soliton (or almost gradient RS), respectively.Numerous studies have been conducted on RSs within the context of manifolds. For example, Biswas et al [25] examined RSs on almost co-Kähler manifolds, while Venkatesha et al [26, 27] focused on RSs in the realms of contact geometry and almost Kenmotsu manifolds. Additionally, De and De [28] investigated almost Ricci solitons on para-Sasakian manifolds. Further insights can be found in the works referenced in [29–32].
Recent research has highlighted the importance of geometric solitons across various spacetimes. Azami et al have studied different forms of geometric solitons on PFSs, including Riemann solitons [33, 34], gradient Ricci–Bourguignon solitons [35], hyperbolic Ricci solitons [36], and h-almost conformal ω-Ricci–Bourguignon solitons [37]. Moreover, Alkhaldi et al [15] investigated Ricci–Yamabe solitons in the context of an imperfect fluid GRW spacetime characterized by a torse-forming vector field, while Siddiqi et al [16] analyzed the dynamics of an imperfect fluid GRW spacetime concerning k-Yamabe solitons with a similar vector field. Building on these foundational studies, our research explores almost Riemann solitons within imperfect fluid GRW spacetimes, presenting several theorems that contribute to a deeper understanding of these solitons.
If an imperfect fluid GRW spacetime satisfies an ARS (M4, g, V, μ), then
$\begin{eqnarray*}{\rm{div}}V=-\frac{r}{6}-2\mu ,\end{eqnarray*}$
where the divergence operator is represented as ${\rm{div}}$.A vector field ζ is classified as a unit timelike torse-forming vector field (UTTFF) if it satisfies the condition g(ζ, ζ) = −1 and follows the relationship expressed in the equation
$\begin{eqnarray}{{\rm{\nabla }}}_{U}\zeta =\phi (U+\vartheta (U)\zeta )\end{eqnarray}$
for any vector field U. In this context, φ represents a smooth function, while ϑ is a 1-form defined by the relation g(U, ζ) = ϑ(U).In an imperfect fluid GRW spacetime with UTTFF ζ, if it permits ARS (M4, g, γζ, μ) with a differentiable function γ, then Dγ = − (ζγ)ζ. Additionally, we get
$\begin{eqnarray*}4\tau -3\kappa \rho +\kappa \sigma =-6(\phi \gamma +2\mu +2\phi )-2\zeta \gamma .\end{eqnarray*}$
In the following theorems, we consider ${ \mathcal P }\,=q(\omega \otimes \vartheta +\vartheta \otimes \omega )$ where q represents the fluid’s shear and ω is a non-zero 1-form linked to the unit vector field ξ, defined by the relation ω(W) = g(W, ξ).
In the setting of an imperfect fluid within GRW spacetime with ${ \mathcal P }=q(\omega \otimes \vartheta +\vartheta \otimes \omega )$, which admits a gradient ARS (M4, g, ∇ ψ, μ), we have
$\begin{eqnarray}\begin{array}{r}\zeta a+b\phi +\frac{1}{3}(\xi c+c({\rm{div}}\xi ))+\zeta (3\mu +{\rm{\Delta }}\psi )\\ =\,\frac{2}{3}((a-b)\zeta \psi -\xi \psi ).\end{array}\end{eqnarray}$
Furthermore, if 3μ + Δψ is constant and that ζφ + φ2 ≠ 0, it can be concluded that Dψ = − (ζψ)ζ, q = 0, $\begin{eqnarray*}\begin{array}{r}\phi (\zeta \psi )=\frac{1}{2}(3\mu +{\rm{\Delta }}\psi )+\frac{1}{4}(2\tau -\kappa \rho +\kappa \sigma ),\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{r}-3(\zeta \phi +{\phi }^{2})-\mu -\alpha +2\phi (\zeta \psi )=\kappa (\sigma +\rho ).\end{array}\end{eqnarray*}$
Let η denote a 1-form associated with the vector field V, and the tensor field A be characterized by dη(U1, U2) = g(U1, AU2). If a GRW imperfect fluid spacetime with ${ \mathcal P }=q(\omega \otimes \vartheta +\vartheta \otimes \omega )$ and UTTFF ζ, admits an ARS (M4, g, V, μ) then
$\begin{eqnarray*}\begin{array}{rcl}({\rm{div}}A)U & = & -ag(U,V)-b\vartheta (U)\vartheta (V)-c\omega (U)\vartheta (V)\\ & & -c\vartheta (U)\omega (V)+\frac{3}{2}\left(U\left(a+3\mu +{\rm{div}}V\right)\right)\\ & & -\frac{1}{2}\vartheta (U)\zeta b-\frac{1}{2}Ub-\frac{3}{2}\phi b\vartheta (U)\\ & & -c\phi \omega (U)+\frac{1}{2}cg({{\rm{\nabla }}}_{\zeta }\xi ,U)\\ & & -\frac{1}{2}c({\rm{div}}\xi )\vartheta (U)\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}{{\rm{\nabla }}}_{U}| V{| }^{2}+2g(AU,V)-({{ \mathcal L }}_{V}g)(U,V)=0.\end{eqnarray*}$
A vector field ν defined on a Lorentzian manifold M is identified as a ν(Ric)-vector field if it satisfies the equation ∇Xν = ψQ(X) for every vector field X. Here, ψ is a constant, and Q refers to the Ricci operator.
If an imperfect fluid GRW spacetime with a ν(Ric)-vector ν an ARS (M4, g, ν, μ), then $\mu =-\frac{1+6\psi }{12}\left(4\tau -3\kappa \rho +\kappa \sigma -\kappa J\right)$. Additionally, if $\psi \ne -\frac{1}{2}$ then the imperfect fluid GRW spacetime becomes a dark perfect fluid spacetime and ${ \mathcal P }=0$.
A spacetime of dimension n is deemed ${{ \mathcal W }}_{2}$-flat if its ${{ \mathcal W }}_{2}$-curvature tensor, as outlined in [38], can be expressed in the following manner:
$\begin{eqnarray*}\begin{array}{l}{{ \mathcal W }}_{2}({U}_{1},{U}_{2},{U}_{3},{U}_{4})\\ \,=\,R({U}_{1},{U}_{2},{U}_{3},{U}_{4})\\ \,+\,\frac{1}{n-1}\left[g({U}_{1},{U}_{3})S({U}_{2},{U}_{4})\right.\\ \,-\,\left.g({U}_{2},{U}_{3})S({U}_{1},{U}_{4})\right],\end{array}\end{eqnarray*}$
with the condition that this tensor must be identically zero. In this context, U1, U2, U3, and U4 represent arbitrary vector fields.If a ${{ \mathcal W }}_{2}$-flat imperfect fluid GRW spacetime satisfies ARS (M4, g, ζ, μ), then φ = 0, $\mu =-\frac{r}{12}$, and the imperfect fluid GRW spacetime is as a dark perfect fluid spacetime.
An n-dimensional spacetime is deemed pseudo-projectively flat if its pseudo-projective curvature tensor, as outlined in [39], can be expressed through the following equation:
$\begin{eqnarray*}\bar{P}({U}_{1},{U}_{2}){U}_{3}={c}_{1}R({U}_{1},{U}_{2}){U}_{3}+{c}_{2}\times \,\left(S({U}_{2},{U}_{3}){U}_{1}-S({U}_{1},{U}_{3}){U}_{2}\right)-\frac{r}{n}\Space{0ex}{2.5ex}{0ex}(\frac{{c}_{1}}{n-1}+{c}_{2}\Space{0ex}{2.5ex}{0ex})\times \,\left[g({U}_{2},{U}_{3}){U}_{1}-g({U}_{1},{U}_{3}){U}_{2}\right],\end{eqnarray*}$
with the stipulation that this tensor must equate to zero. In this context, U1, U2, and U3 represent arbitrary vector fields, while the constants c1 and c2 are both non-zero. It is noteworthy that the pseudo-projective curvature tensor simplifies to the projective curvature tensor when the specific values of c1 = 1 and ${c}_{2}=-\frac{1}{n-1}$ are utilized.If a pseudo-projectively flat imperfect fluid GRW spacetime admits an ARS (M4, g, ζ, μ) such that c1 + 3c2 ≠ 0 then φ = 0, $\mu =-\frac{r}{12}$, and the imperfect fluid GRW spacetime becomes a dark perfect fluid spacetime.
2. Proofs of main results
In the following discussion, we will treat the vector fields X, Z, U1, U2, U3, U4, and U as general entities within the framework of spacetime, unless stated otherwise.
equation (
$\begin{eqnarray}\begin{array}{rcl}2R({U}_{1},{U}_{2},{U}_{3},{U}_{4}) & = & -2\mu \left[g({U}_{1},{U}_{4})g({U}_{2},{U}_{3})\right.\\ & & -\left.g({U}_{1},{U}_{3})g({U}_{2},{U}_{4})\right]\\ & & -\left[g({U}_{1},{U}_{4}){{ \mathcal L }}_{V}g({U}_{2},{U}_{3})\right.\\ & & +\left.g({U}_{2},{U}_{3}){{ \mathcal L }}_{V}g({U}_{1},{U}_{4})\right]\\ & & +\left[g({U}_{1},{U}_{3}){{ \mathcal L }}_{V}g({U}_{2},{U}_{4})\right.\\ & & +\left.g({U}_{2},{U}_{4}){{ \mathcal L }}_{V}g({U}_{1},{U}_{3})\right].\end{array}\end{eqnarray}$
By contracting U1 and U4 in this equation, we arrive at the relation: $\begin{eqnarray}\begin{array}{rcl}S({U}_{2},{U}_{3}) & = & -(3\mu +{\rm{div}}V)g({U}_{2},{U}_{3})\\ & & -({{ \mathcal L }}_{V}g)({U}_{2},{U}_{3}).\end{array}\end{eqnarray}$
Further contracting this equation yields the result: $\begin{eqnarray*}r+12\mu +6{\rm{div}}V=0.\end{eqnarray*}$
This completes the proof of the theorem. □When a vector field V is identified as a Killing vector field, characterized by the condition ${{ \mathcal L }}_{V}g=0$, it can be concluded that divV = 0 and r = − 12μ. Additionally, the equations (1.5 ) and (2.2 ) yield the following relationships:
$\begin{eqnarray*}\begin{array}{rcl}\mu & = & \frac{-1}{6}(2\tau -\kappa \rho +\kappa \sigma ),\\ 0 & = & -\kappa (\sigma +\rho ),\,\,{ \mathcal P }=0.\end{array}\end{eqnarray*}$
Consequently, we arrive atIn the framework of an imperfect fluid GRW spacetime that accommodates an ARS (M4, g, V, μ) defined by a Killing vector field V, it can be inferred that M signifies a dark perfect fluid. The conditions for this fluid to be steady, expanding, or shrinking are determined by the equation 2τ − κρ + κσ = 0, 0 < 2τ − κρ + κσ, and 0 > 2τ − κρ + κσ, respectively.
If V is a conformal Killing vector field, that is, ${{ \mathcal L }}_{V}g=\psi g$ for some smooth function ψ, then 2.1 ) we conclude that the following corollary.
$\begin{eqnarray*}\begin{array}{l}-2\mu \left[g({U}_{1},{U}_{4})g({U}_{2},{U}_{3})\right.\\ -\left.g({U}_{1},{U}_{3})g({U}_{2},{U}_{4})\right]\\ -\left[g({U}_{1},{U}_{4}){{ \mathcal L }}_{V}g({U}_{2},{U}_{3})\right.\\ +\left.g({U}_{2},{U}_{3}){{ \mathcal L }}_{V}g({U}_{1},{U}_{4})\right]+\left[g({U}_{1},{U}_{3}){{ \mathcal L }}_{V}g({U}_{2},{U}_{4})\right.\\ +\left.g({U}_{2},{U}_{4}){{ \mathcal L }}_{V}g({U}_{1},{U}_{3})\right]\\ =-2(\mu +\psi )\left[g({U}_{1},{U}_{4})g({U}_{2},{U}_{3})\right.\\ -\left.g({U}_{1},{U}_{3})g({U}_{2},{U}_{4})\right].\end{array}\end{eqnarray*}$
Now, on an imperfect fluid GRW spacetime with constant scalar curvature k, we have $\begin{eqnarray*}\begin{array}{r}R({U}_{1},{U}_{2},{U}_{3},{U}_{4})=k\left[g({U}_{1},{U}_{4})g({U}_{2},{U}_{3})\right.\\ -\left.g({U}_{1},{U}_{3})g({U}_{2},{U}_{4})\right].\end{array}\end{eqnarray*}$
Thus, if an imperfect fluid GRW spacetime with constant scalar curvature k and a conformal Killing vector field then from (Suppose (M, g) is an imperfect fluid GRW spacetime with constant scalar curvature k and a conformal Killing vector field V such that ${{ \mathcal L }}_{V}g=\psi g$. Then M admits ARS (M, g, V, − μ − ψ).
Let ζ denote an UTTFF within the context of an imperfect fluid GRW spacetime. As shown in [40], it can be concluded that ∇ζζ = 0, and the equation $({{\rm{\nabla }}}_{{U}_{1}}\vartheta )({U}_{2})=\phi (g({U}_{1},{U}_{2})+\vartheta ({U}_{1})\vartheta ({U}_{2}))$ is valid. Additionally, further details can be found in the following lemma for a deeper understanding.
In a four-dimensional GRW spacetime defined by an imperfect fluid, we can express the following relationships
$\begin{eqnarray}R({U}_{1},{U}_{2})\zeta =(\zeta \phi +{\phi }^{2})\left(\vartheta ({U}_{2}){U}_{1}-\vartheta ({U}_{1}){U}_{2}\right)\end{eqnarray}$
and $\begin{eqnarray}S({U}_{1},\zeta )=3(\zeta \phi +{\phi }^{2})\vartheta ({U}_{1}),\end{eqnarray}$
where R denotes the curvature tensor. Let us consider ζ as a UTTF vector field. Utilizing equation (
$\begin{eqnarray}{{\rm{\nabla }}}_{{U}_{1}}\zeta =\phi ({U}_{1}+\vartheta ({U}_{1})\zeta )\end{eqnarray}$
and $\begin{eqnarray}S({U}_{1},\zeta )=G\vartheta ({U}_{1}),\end{eqnarray}$
where G denotes a non-zero function. When the operator ${{\rm{\nabla }}}_{{U}_{2}}$ is applied to both sides of equation ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}_{{U}_{2}}{{\rm{\nabla }}}_{{U}_{1}}\zeta & = & ({U}_{2}\phi )\left({U}_{1}+\vartheta ({U}_{1})\zeta \right)\\ & & +\phi \left[{{\rm{\nabla }}}_{{U}_{2}}{U}_{1}+({{\rm{\nabla }}}_{{U}_{2}}\vartheta ({U}_{1}))\zeta \right.\\ & & +\phi \left.\left({U}_{2}+\vartheta ({U}_{2})\zeta \right)\vartheta ({U}_{1})\right].\end{array}\end{eqnarray}$
By interchanging U1 and U2 in ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}_{{U}_{1}}{{\rm{\nabla }}}_{{U}_{2}}\zeta & = & ({U}_{1}\phi )\left({U}_{2}+\vartheta ({U}_{2})\zeta \right)\\ & & +\phi \left[{{\rm{\nabla }}}_{{U}_{1}}{U}_{2}+({{\rm{\nabla }}}_{{U}_{1}}\vartheta ({U}_{2}))\zeta \right.\\ & & +\phi \left.\left({U}_{1}+\vartheta ({U}_{1})\zeta \right)\vartheta ({U}_{2})\right].\end{array}\end{eqnarray}$
By substituting U1 with [U1, U2] in ( $\begin{eqnarray}\begin{array}{r}{{\rm{\nabla }}}_{[{U}_{1},{U}_{2}]}\zeta =\phi ([{U}_{1},{U}_{2}]\\ +\vartheta ([{U}_{1},{U}_{2}])\zeta ).\end{array}\end{eqnarray}$
By combining equations ( $\begin{eqnarray}\begin{array}{rcl}R({U}_{1},{U}_{2})\zeta & = & ({U}_{1}\phi )\left({U}_{2}+\vartheta ({U}_{2})\zeta \right)\\ & & -({U}_{2}\phi )\left({U}_{1}+\vartheta ({U}_{1})\zeta \right)\\ & & +{\phi }^{2}\left(\vartheta ({U}_{2}){U}_{1}-\vartheta ({U}_{1}){U}_{2}\right).\end{array}\end{eqnarray}$
By executing the contraction concerning U2 in equation ( $\begin{eqnarray}S({U}_{1},\zeta )=-2({U}_{1}\phi )+(\zeta \phi )\vartheta ({U}_{1})-3{\phi }^{2}\vartheta ({U}_{1}).\end{eqnarray}$
By analyzing equations ( $\begin{eqnarray}G\vartheta ({U}_{1})=-2({U}_{1}\phi )+(\zeta \phi )\vartheta ({U}_{1})-3{\phi }^{2}\vartheta ({U}_{1}).\end{eqnarray}$
Replacing U1 = ζ in ( $\begin{eqnarray}G=3(\zeta \phi +{\phi }^{2}).\end{eqnarray}$
By incorporating equation ( $\begin{eqnarray}{U}_{1}\phi =-(\zeta \phi )\vartheta ({U}_{1}).\end{eqnarray}$
Finally, by utilizing equation (By utilizing the expression
$\begin{eqnarray*}\left({{ \mathcal L }}_{\zeta }g\right)({U}_{2},{U}_{3})=2\phi (g({U}_{2},{U}_{3})+\vartheta ({U}_{2})\vartheta ({U}_{3})),\end{eqnarray*}$
we determine that ${\rm{div}}\zeta =3\phi $. Employing equation ( $\begin{eqnarray*}\begin{array}{rcl}\left({{ \mathcal L }}_{\gamma \zeta }\right)g({U}_{2},{U}_{3}) & = & ({U}_{2}\gamma )\vartheta ({U}_{3})+({U}_{3}\gamma )\vartheta ({U}_{2})\\ & & +2\phi \gamma (g({U}_{2},{U}_{3})+\vartheta ({U}_{2})\vartheta ({U}_{3}))\end{array}\end{eqnarray*}$
enables us to derive $\begin{eqnarray}\begin{array}{rcl}S({U}_{2},{U}_{3}) & = & -\left[2\phi \gamma +3\phi +3\mu \right]g({U}_{2},{U}_{3})\\ & & -\left[({U}_{2}\gamma )\vartheta ({U}_{3})+({U}_{3}\gamma )\vartheta ({U}_{2})\right.\\ & & +\left.(2\phi \gamma )\vartheta ({U}_{2})\vartheta ({U}_{3})\right].\end{array}\end{eqnarray}$
Substituting ζ for U3 in equation ( $\begin{eqnarray}\begin{array}{rcl}3(\zeta \phi +{\phi }^{2})\vartheta ({U}_{2}) & = & -\left[2\phi \gamma +3\phi +3\mu \right]\vartheta ({U}_{2})\\ & & -\left[-({U}_{2}\gamma )+(\zeta \gamma -2\phi \gamma )\vartheta ({U}_{2})\right].\end{array}\end{eqnarray}$
When we set U2 = ζ in equation ( $\begin{eqnarray}3(\zeta \phi +{\phi }^{2})=-2\zeta \gamma -3\phi -3\mu .\end{eqnarray}$
By combining equations ( $\begin{eqnarray*}-(\zeta \gamma )\vartheta ({U}_{2})={U}_{2}\gamma .\end{eqnarray*}$
This allows us to deduce that Dγ = − (ζγ)ζ. When we substitute this finding into equation ( $\begin{eqnarray}\begin{array}{rcl}S({U}_{2},{U}_{3}) & = & -\left[2\phi \gamma +3\phi +3\mu \right]g({U}_{2},{U}_{3})\\ & & -\left[-2\zeta \gamma +2\phi \gamma \right]\vartheta ({U}_{2})\vartheta ({U}_{3}).\end{array}\end{eqnarray}$
By contracting equation (In the framework of an imperfect fluid GRW spacetime defined by UTTFF ζ, the presence of RS (M4, g, ζ, μ) leads to the relationship r = 3(−6φ − 4μ).
In the context of the imperfect fluid GRW spacetime, which is defined as a gradient RS (M4, g, ∇ ψ, μ), we can analyze the equation presented in (
$\begin{eqnarray*}\begin{array}{r}S({U}_{1},{U}_{2})=-(3\mu +{\rm{\Delta }}\psi )g({U}_{1},{U}_{2})\\ -2g({{\rm{\nabla }}}_{{U}_{1}}D\psi ,{U}_{2}).\end{array}\end{eqnarray*}$
From this expression, we can derive the relationship given by $\begin{eqnarray}{{\rm{\nabla }}}_{{U}_{1}}D\psi =-\frac{1}{2}\left[Q{U}_{1}+(3\mu +{\rm{\Delta }}\psi ){U}_{1}\right],\end{eqnarray}$
where Q represents the Ricci operator. By applying the covariant derivative to this equation with respect to the vector field U2, we obtain the result $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}_{{U}_{1}}{{\rm{\nabla }}}_{{U}_{2}}D\psi & = & -\frac{1}{2}\left[{{\rm{\nabla }}}_{{U}_{1}}(Q{U}_{2})+({U}_{1}(3\mu +{\rm{\Delta }}\psi )){U}_{2}\right.\\ & & +\left.(3\mu +{\rm{\Delta }}\psi ){{\rm{\nabla }}}_{{U}_{1}}{U}_{2}\right].\end{array}\end{eqnarray}$
If we interchange U2 with U1 in this equation, we arrive at $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}_{{U}_{2}}{{\rm{\nabla }}}_{{U}_{1}}D\psi & = & -\frac{1}{2}\left[{{\rm{\nabla }}}_{{U}_{2}}(Q{U}_{1})\right.\\ & & +({U}_{2}(3\mu +{\rm{\Delta }}\psi )){U}_{1}\\ & & +\left.(3\mu +{\rm{\Delta }}\psi ){{\rm{\nabla }}}_{{U}_{2}}{U}_{1}\right].\end{array}\end{eqnarray}$
Additionally, from the earlier equation, we can derive the expression $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}_{[{U}_{1},{U}_{2}]}D\psi & = & -\frac{1}{2}\left(Q[{U}_{1},{U}_{2}]\right.\\ & & +\left.(3\mu +{\rm{\Delta }}\psi )[{U}_{1},{U}_{2}]\right).\end{array}\end{eqnarray}$
Incorporating equations ( $\begin{eqnarray}\begin{array}{rcl}R({U}_{1},{U}_{2})D\psi & = & -\frac{1}{2}\left[({{\rm{\nabla }}}_{{U}_{1}}Q){U}_{2}\right.\\ & & -\left.({{\rm{\nabla }}}_{{U}_{2}}Q){U}_{1}\right]\\ & & -\frac{1}{2}\left[{U}_{1}(3\mu +{\rm{\Delta }}\psi ){U}_{2}\right.\\ & & -\left.{U}_{2}(3\mu +{\rm{\Delta }}\psi ){U}_{1}\right].\end{array}\end{eqnarray}$
From equation ( $\begin{eqnarray}\begin{array}{l}({{\rm{\nabla }}}_{V}Q)(X)=V(a)X+V(b)\vartheta (X)\zeta \\ \quad +b({{\rm{\nabla }}}_{V}\vartheta )(X)\zeta +b\vartheta (X){{\rm{\nabla }}}_{V}\zeta \\ \quad +V(c)\omega (X)\zeta +c({{\rm{\nabla }}}_{V}\omega )(X)\zeta +c\omega (X){{\rm{\nabla }}}_{V}\zeta \\ \quad +V(c)\vartheta (X)\xi +c({{\rm{\nabla }}}_{V}\vartheta )(X)\xi +c\vartheta (X){{\rm{\nabla }}}_{V}\xi .\end{array}\end{eqnarray}$
By replacing equation ( $\begin{eqnarray}\begin{array}{l}R({U}_{1},{U}_{2})D\psi =-\frac{1}{2}\left[{U}_{1}(a){U}_{2}-{U}_{2}(a){U}_{1}\right.\\ \quad +\,\left.{U}_{1}(b)\vartheta ({U}_{2})\zeta -{U}_{2}(b)\vartheta ({U}_{1})\zeta \right]\\ \quad -\,\frac{1}{2}\left[b({{\rm{\nabla }}}_{{U}_{1}}\vartheta )({U}_{2})\zeta -b({{\rm{\nabla }}}_{{U}_{2}}\vartheta )({U}_{1})\zeta \right.\\ \quad +\,\left.b\vartheta ({U}_{2}){{\rm{\nabla }}}_{{U}_{1}}\zeta -b\vartheta ({U}_{1}){{\rm{\nabla }}}_{{U}_{2}}\zeta \right]\\ \quad -\,\frac{1}{2}\left[{U}_{1}(c)\omega ({U}_{2})\zeta -{U}_{2}(c)\omega ({U}_{1})\zeta \right]\\ \quad -\,\frac{1}{2}\left[c({{\rm{\nabla }}}_{{U}_{1}}\omega )({U}_{2})\zeta -c({{\rm{\nabla }}}_{{U}_{2}}\omega )({U}_{1})\zeta \right.\\ \quad +\,\left.c\omega ({U}_{2}){{\rm{\nabla }}}_{{U}_{1}}\zeta -c\omega ({U}_{1}){{\rm{\nabla }}}_{{U}_{2}}\zeta \right]\\ \quad -\,\frac{1}{2}\left[{U}_{1}(c)\vartheta ({U}_{2})\xi -{U}_{2}(c)\vartheta ({U}_{1})\xi \right]\\ \quad -\,\frac{1}{2}\left[c({{\rm{\nabla }}}_{{U}_{1}}\vartheta )({U}_{2})\xi -c({{\rm{\nabla }}}_{{U}_{2}}\vartheta )({U}_{1})\xi \right.\\ \quad +\,\left.c\vartheta ({U}_{2}){{\rm{\nabla }}}_{{U}_{1}}\xi -c\vartheta ({U}_{1}){{\rm{\nabla }}}_{{U}_{2}}\xi \right]\\ \quad -\,\frac{1}{2}\left[{U}_{1}(3\mu +{\rm{\Delta }}\psi ){U}_{2}-{U}_{2}(3\mu +{\rm{\Delta }}\psi ){U}_{1}\right].\end{array}\end{eqnarray}$
By taking the inner product of ( $\begin{eqnarray}\begin{array}{l}S({U}_{2},D\psi )=-\frac{1}{2}\left[-3{U}_{2}(a)+(\zeta b)\vartheta ({U}_{2})+{U}_{2}(b)\right.\\ \,+\,\left.b({{\rm{\nabla }}}_{\zeta }\vartheta )({U}_{2})-b({{\rm{\nabla }}}_{{U}_{2}}\vartheta )(\zeta )+3b\phi \vartheta ({U}_{2})\right]\\ \,-\,\frac{1}{2}\left[(\zeta c)\omega ({U}_{2})+c({{\rm{\nabla }}}_{\zeta }\omega )({U}_{2})\right.\\ \,\left.-\,c({{\rm{\nabla }}}_{{U}_{2}}\omega )(\zeta )+2c\phi \omega ({U}_{2})\right]\\ \,-\,\frac{1}{2}\left[(\xi c)\vartheta ({U}_{2})+c({{\rm{\nabla }}}_{\xi }\vartheta )({U}_{2})-c({{\rm{\nabla }}}_{{U}_{2}}\vartheta )(\xi )\right.\\ \,\left.+\,c({\rm{div}}\xi )\vartheta ({U}_{2})+c\phi \omega ({U}_{2})\right]\\ \,-\,\frac{1}{2}\left[-3{U}_{2}(3\mu +{\rm{\Delta }}\psi )\right].\end{array}\end{eqnarray}$
Substituting U1 = Dψ into equation ( $\begin{eqnarray}\begin{array}{rcl}S({U}_{2},D\psi ) & = & a({U}_{2}\psi )+b\vartheta ({U}_{2})(\zeta \psi )\\ & & +c\omega ({U}_{2})(\zeta \psi )+c\vartheta ({U}_{2})(\xi \psi ).\end{array}\end{eqnarray}$
By replacing U2 with ζ in both equations ( $\begin{eqnarray}\begin{array}{r}\zeta a+b\phi +\frac{1}{3}(\xi c+c({\rm{div}}\xi ))\\ +\zeta (3\mu +{\rm{\Delta }}\psi )=\frac{2}{3}((a-b)\zeta \psi -\xi \psi ).\end{array}\end{eqnarray}$
From the equation Qζ = 3(ζφ + φ2)ζ, we can derive $\begin{eqnarray}\begin{array}{rcl}g(({{\rm{\nabla }}}_{{U}_{1}}Q){U}_{2},\zeta ) & = & g(({{\rm{\nabla }}}_{{U}_{1}}Q)\zeta ,{U}_{2})\\ & = & 3\phi (\zeta \phi +{\phi }^{2})g({U}_{1},{U}_{2})\\ & & +3{U}_{1}(\zeta \phi +{\phi }^{2})\vartheta ({U}_{2})\\ & & -\phi S({U}_{1},{U}_{2}).\end{array}\end{eqnarray}$
The result obtained by taking the inner product of ( $\begin{eqnarray}\begin{array}{rcl}g(R({U}_{1},{U}_{2})D\psi ,\zeta ) & = & -\frac{3}{2}\left[{U}_{1}(\zeta \phi +{\phi }^{2})\vartheta ({U}_{2})\right.\\ & & -\left.{U}_{2}(\zeta \phi +{\phi }^{2})\vartheta ({U}_{1})\right]\\ & & -\frac{1}{2}\left[{U}_{1}(3\mu +{\rm{\Delta }}\psi )\vartheta ({U}_{2})\right.\\ & & -\left.{U}_{2}(3\mu +{\rm{\Delta }}\psi )\vartheta ({U}_{1})\right].\end{array}\end{eqnarray}$
By replacing equation ( $\begin{eqnarray}\begin{array}{l}(\zeta \phi +{\phi }^{2})\left(({U}_{2}\psi )\vartheta ({U}_{1})-({U}_{1}\psi )\vartheta ({U}_{2})\right)\\ \quad =-\frac{3}{2}\left[{U}_{1}(\zeta \phi +{\phi }^{2})\vartheta ({U}_{2})\right.\\ \quad -\left.{U}_{2}(\zeta \phi +{\phi }^{2})\vartheta ({U}_{1})\right]\\ \quad -\frac{1}{2}\left[{U}_{1}(3\mu +{\rm{\Delta }}\psi )\vartheta ({U}_{2})\right.\\ \quad -\left.{U}_{2}(3\mu +{\rm{\Delta }}\psi )\vartheta ({U}_{1})\right].\end{array}\end{eqnarray}$
By replacing U2 with ζ in equation ( $\begin{eqnarray}\begin{array}{r}(\zeta \phi +{\phi }^{2})({U}_{1}\psi +(\zeta \psi )\vartheta ({U}_{1}))\\ \,=\frac{3}{2}\left[{U}_{1}(\zeta \phi +{\phi }^{2})\right.\\ \,+\,\,\left.\zeta (\zeta \phi +{\phi }^{2})\vartheta ({U}_{1})\right]\\ \,-\,\frac{1}{2}\left[-{U}_{1}(3\mu +{\rm{\Delta }}\psi )\right.\\ \,-\,\left.\zeta (3\mu +{\rm{\Delta }}\psi )\vartheta ({U}_{1})\right].\end{array}\end{eqnarray}$
The findings presented in [41, Proposition 2.2] demonstrate that the equation U1(ζφ + φ2) + ζ(ζφ + φ2)ϑ(U1) = 0 is valid. Assuming that 3μ + Δψ is a constant, it can be deduced that the following relationship holds: $\begin{eqnarray*}(\zeta \psi )\vartheta ({U}_{1})+{U}_{1}\psi =0.\end{eqnarray*}$
This results in the formulation Dψ = − (ζψ)ζ. By utilizing the covariant derivative with respect to U1, we derive the equation: $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}_{{U}_{1}}D\psi & = & -({U}_{1}(\zeta \psi ))\zeta -\phi (\zeta \psi )\\ & & [{U}_{1}+\vartheta ({U}_{1})\zeta ].\end{array}\end{eqnarray}$
By taking the inner product of equation ( $\begin{eqnarray}\begin{array}{rcl}{U}_{1}(\zeta \psi ) & = & -\frac{1}{2}\left(3(\zeta \phi +{\phi }^{2})\right.\\ & & +\left.3\mu +{\rm{\Delta }}\psi \right)\vartheta ({U}_{1}).\end{array}\end{eqnarray}$
When we substitute equation ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}_{{U}_{1}}D\psi & = & \frac{1}{2}\left(3(\zeta \phi +{\phi }^{2})\right.\\ & & \left.+3\mu +{\rm{\Delta }}\psi \right)\vartheta ({U}_{1})\zeta \\ & & -\phi (\zeta \psi )[{U}_{1}+\vartheta ({U}_{1})\zeta ].\end{array}\end{eqnarray}$
Incorporating ( $\begin{eqnarray}\begin{array}{l}S({U}_{1},{U}_{2})=-\left[3(\zeta \phi +{\phi }^{2})+3\mu +{\rm{\Delta }}\psi -2\phi (\zeta \psi )\right]\\ \vartheta ({U}_{1})\vartheta ({U}_{2})+(2\phi (\zeta \psi )-3\mu -{\rm{\Delta }}\psi )g({U}_{1},{U}_{2}).\end{array}\end{eqnarray}$
From equations ( $\begin{eqnarray}\begin{array}{l}\phi (\zeta \psi )=\frac{1}{2}(3\mu +{\rm{\Delta }}\psi )+\frac{1}{4}(2\tau -\kappa \rho +\kappa \sigma ),\\ \quad -3(\zeta \phi +{\phi }^{2})-3\mu -{\rm{\Delta }}\psi +2\phi (\zeta \psi )=\kappa (\sigma +\rho ).\end{array}\end{eqnarray}$
□We have
$\begin{eqnarray}({{ \mathcal L }}_{V}g)({U}_{1},{U}_{2})=g({{\rm{\nabla }}}_{{U}_{1}}V,{U}_{2})+g({{\rm{\nabla }}}_{{U}_{2}}V,{U}_{1}).\end{eqnarray}$
By utilizing equations ( $\begin{eqnarray}\begin{array}{l}g({{\rm{\nabla }}}_{{U}_{1}}V,{U}_{2})+g({{\rm{\nabla }}}_{{U}_{2}}V,{U}_{1})\\ \,+\,S({U}_{1},{U}_{2})+\left[3\mu +{\rm{div}}V\right]\\ g({U}_{1},{U}_{2})=0.\end{array}\end{eqnarray}$
When we substitute equation ( $\begin{eqnarray}\begin{array}{l}g({{\rm{\nabla }}}_{{U}_{1}}V,{U}_{2})+g({{\rm{\nabla }}}_{{U}_{2}}V,{U}_{1})\\ \,+\,b\vartheta ({U}_{1})\vartheta ({U}_{2})+c\omega ({U}_{1})\vartheta ({U}_{2})\\ \,+\,c\vartheta ({U}_{1})\omega ({U}_{2})\\ \,+\,\left(a+3\mu +{\rm{div}}V\right)g({U}_{1},{U}_{2})=0.\end{array}\end{eqnarray}$
The exterior derivative of the form η yields the expression $\begin{eqnarray}\begin{array}{l}2(d\eta )({U}_{1},{U}_{2})=g({{\rm{\nabla }}}_{{U}_{1}}V,{U}_{2})\\ \,-g({{\rm{\nabla }}}_{{U}_{2}}V,{U}_{1}),\end{array}\end{eqnarray}$
which is derived from the relation η(U1) = g(U1, V). The skew-symmetry property of dη gives rise to the equation g(U1, AU2) = − g(AU1, U2), leading to the representation of dη(U1, U2) as −g(AU1, U2). Consequently, equation ( $\begin{eqnarray}\begin{array}{l}g({{\rm{\nabla }}}_{{U}_{1}}V,{U}_{2})-g({{\rm{\nabla }}}_{{U}_{2}}V,{U}_{1})\\ \,=\,-2g(A{U}_{1},{U}_{2}).\end{array}\end{eqnarray}$
By substituting ( $\begin{eqnarray}\begin{array}{l}{{\rm{\nabla }}}_{{U}_{1}}V=-A{U}_{1}-\frac{1}{2}\left(a+3\mu +{\rm{div}}V\right)\\ {U}_{1}-\frac{1}{2}b\vartheta ({U}_{1})\zeta -\frac{1}{2}c\omega ({U}_{1})\zeta -\frac{1}{2}c\vartheta ({U}_{1})\xi .\end{array}\end{eqnarray}$
The use of the covariant derivative in the direction of U2 produces $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}_{{U}_{2}}{{\rm{\nabla }}}_{{U}_{1}}V & = & -{{\rm{\nabla }}}_{{U}_{2}}(A{U}_{1})-\frac{1}{2}\left({U}_{2}\left(a+3\mu +{\rm{div}}V\right)\right){U}_{1}\\ & & -\frac{1}{2}\left(a+3\mu +{\rm{div}}V\right){{\rm{\nabla }}}_{{U}_{2}}{U}_{1}-\frac{1}{2}\vartheta ({U}_{1}){U}_{2}b\zeta \\ & & -\frac{1}{2}b{{\rm{\nabla }}}_{{U}_{2}}(\vartheta ({U}_{1})\zeta )-\frac{1}{2}\omega ({U}_{1})({U}_{2}c)\zeta \\ & & -\frac{1}{2}c{{\rm{\nabla }}}_{{U}_{2}}(\omega ({U}_{1})\zeta )-\frac{1}{2}\vartheta ({U}_{1})({U}_{2}c)\xi \\ & & -\frac{1}{2}c{{\rm{\nabla }}}_{{U}_{2}}(\vartheta ({U}_{1})\xi ).\end{array}\end{eqnarray}$
By substituting U1 for U2 in the earlier equation, we obtain $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}_{{U}_{1}}{{\rm{\nabla }}}_{{U}_{2}}V & = & -{{\rm{\nabla }}}_{{U}_{1}}(A{U}_{2})-\frac{1}{2}\left({U}_{1}\left(a+3\mu +{\rm{div}}V\right)\right){U}_{2}\\ & & -\frac{1}{2}\left(a+3\mu +{\rm{div}}V\right){{\rm{\nabla }}}_{{U}_{1}}{U}_{2}-\frac{1}{2}\vartheta ({U}_{2}){U}_{1}b\zeta \\ & & -\frac{1}{2}b{{\rm{\nabla }}}_{{U}_{1}}(\vartheta ({U}_{2})\zeta )-\frac{1}{2}\omega ({U}_{2})({U}_{1}c)\zeta \\ & & -\frac{1}{2}c{{\rm{\nabla }}}_{{U}_{1}}(\omega ({U}_{2})\zeta )-\frac{1}{2}\vartheta ({U}_{2})({U}_{1}c)\xi \\ & & -\frac{1}{2}c{{\rm{\nabla }}}_{{U}_{1}}(\vartheta ({U}_{2})\xi ).\end{array}\end{eqnarray}$
Additionally, it can be inferred from ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}_{[{U}_{1},{U}_{2}]}V & = & -A[{U}_{1},{U}_{2}]-\frac{1}{2}\left(a+3\mu +{\rm{div}}V\right)[{U}_{1},{U}_{2}]\\ & & -\frac{1}{2}b\vartheta ([{U}_{1},{U}_{2}])\zeta -\frac{1}{2}c\omega ([{U}_{1},{U}_{2}])\zeta \\ & & -\frac{1}{2}c\vartheta ([{U}_{1},{U}_{2}])\xi .\end{array}\end{eqnarray}$
The incorporation of equations ( $\begin{eqnarray}\begin{array}{rcl}R({U}_{1},{U}_{2})V & = & ({{\rm{\nabla }}}_{{U}_{2}}A){U}_{1}-({{\rm{\nabla }}}_{{U}_{1}}A){U}_{2}\\ & & -\frac{1}{2}\left({U}_{1}\left(a+3\mu +{\rm{div}}V\right)\right){U}_{2}\\ & & +\frac{1}{2}\left({U}_{2}\left(a+3\mu +{\rm{div}}V\right)\right){U}_{1}\\ & & -\frac{1}{2}\vartheta ({U}_{2}){U}_{1}b\zeta +\frac{1}{2}\omega ({U}_{1})({U}_{2}c)\zeta \\ & & +\frac{1}{2}\vartheta ({U}_{1}){U}_{2}b\zeta +\frac{\phi }{2}b\left(\vartheta ({U}_{1}){U}_{2}\right.\\ & & -\left.\vartheta ({U}_{2}){U}_{1}\right)-\frac{1}{2}\omega ({U}_{2})({U}_{1}c)\zeta \\ & & +\frac{1}{2}c\left(\omega ({U}_{1}){{\rm{\nabla }}}_{{U}_{2}}\zeta -\omega ({U}_{2}){{\rm{\nabla }}}_{{U}_{1}}\zeta \right.\\ & & +\left.g({U}_{1},{{\rm{\nabla }}}_{{U}_{2}}\xi )\zeta -g({U}_{2},{{\rm{\nabla }}}_{{U}_{1}}\xi )\zeta \right)\\ & & +\frac{1}{2}\vartheta ({U}_{1})({U}_{2}c)\xi \\ & & -\frac{1}{2}\vartheta ({U}_{2})({U}_{1}c)\xi \\ & & +\frac{1}{2}c\left(\vartheta ({U}_{1}){{\rm{\nabla }}}_{{U}_{2}}\xi -\vartheta ({U}_{2}){{\rm{\nabla }}}_{{U}_{1}}\xi \right).\end{array}\end{eqnarray}$
Taking the inner product of equation ( $\begin{eqnarray}\begin{array}{l}g(R({U}_{1},{U}_{2})V,Z)=g(({{\rm{\nabla }}}_{{U}_{2}}A){U}_{1},Z)-g(({{\rm{\nabla }}}_{{U}_{1}}A){U}_{2},Z)\\ \,-\frac{1}{2}\left({U}_{1}\left(a+3\mu +{\rm{div}}V\right)\right)g({U}_{2},Z)\\ \,+\frac{1}{2}\left({U}_{2}\left(a+3\mu +{\rm{div}}V\right)\right)g({U}_{1},Z)-\frac{1}{2}\vartheta ({U}_{2})\vartheta (Z){U}_{1}b\\ \,+\frac{1}{2}\vartheta ({U}_{1})\vartheta (Z){U}_{2}b+\frac{\phi }{2}b\left(\vartheta ({U}_{1})g({U}_{2},Z)-\vartheta ({U}_{2})g({U}_{1},Z)\right)\\ \,-\frac{1}{2}\omega ({U}_{2})\vartheta (Z){U}_{1}c+\frac{1}{2}\omega ({U}_{1})\vartheta (Z){U}_{2}c\\ \,-\frac{1}{2}\vartheta ({U}_{2})\omega (Z){U}_{1}c+\frac{1}{2}\vartheta ({U}_{1})\omega (Z){U}_{2}c\\ \,+\frac{1}{2}c\left(\omega ({U}_{1})g({{\rm{\nabla }}}_{{U}_{2}}\zeta ,Z)-\omega ({U}_{2})g({{\rm{\nabla }}}_{{U}_{1}}\zeta ,Z)\right)\\ \,+\frac{1}{2}c\left(\vartheta ({U}_{1})g({{\rm{\nabla }}}_{{U}_{2}}\xi ,Z)-\vartheta ({U}_{2})g({{\rm{\nabla }}}_{{U}_{1}}\xi ,Z)\right)\\ \,+\frac{1}{2}c\left(\vartheta (Z)g({{\rm{\nabla }}}_{{U}_{1}}\xi ,{U}_{2})-\vartheta (Z)g({{\rm{\nabla }}}_{{U}_{2}}\xi ,{U}_{1})\right).\end{array}\end{eqnarray}$
Since dη is closed, is a closed form, it can be concluded that $\begin{eqnarray}\begin{array}{l}g({U}_{1},({{\rm{\nabla }}}_{Z}A){U}_{2})+g({U}_{2},({{\rm{\nabla }}}_{{U}_{1}}A)Z)\\ \,+\,g(Z,({{\rm{\nabla }}}_{{U}_{2}}A){U}_{1})=0.\end{array}\end{eqnarray}$
The skew self-adjoint nature of the operator A ensures that ∇VA retains this same skew self-adjoint property. Consequently, we can express equation ( $\begin{eqnarray}\begin{array}{l}g(R({U}_{1},{U}_{2})V,Z)=-g({U}_{1},({{\rm{\nabla }}}_{Z}A){U}_{2})\\ \quad -\frac{1}{2}\left({U}_{1}\left(a+3\mu +{\rm{div}}V\right)\right)g({U}_{2},Z)\\ \quad +\,\frac{1}{2}\left({U}_{2}\left(a+3\mu +{\rm{div}}V\right)\right)g({U}_{1},Z)-\,\frac{1}{2}\vartheta ({U}_{2})\vartheta (Z){U}_{1}b\\ \quad +\,\frac{1}{2}\vartheta ({U}_{1})\vartheta (Z){U}_{2}b+\,\frac{\phi }{2}b\left(\vartheta ({U}_{1})g({U}_{2},Z)-\vartheta ({U}_{2})g({U}_{1},Z)\right)\\ \quad -\,\frac{1}{2}\omega ({U}_{2})\vartheta (Z){U}_{1}c+\,\frac{1}{2}\omega ({U}_{1})\vartheta (Z){U}_{2}c-\,\frac{1}{2}\vartheta ({U}_{2})\omega (Z){U}_{1}c\\ \quad +\,\frac{1}{2}\vartheta ({U}_{1})\omega (Z){U}_{2}c+\,\frac{1}{2}c\left(\omega ({U}_{1})g({{\rm{\nabla }}}_{{U}_{2}}\zeta ,Z)-\omega ({U}_{2})g({{\rm{\nabla }}}_{{U}_{1}}\zeta ,Z)\right)\\ \quad +\,\frac{1}{2}c\left(\vartheta ({U}_{1})g({{\rm{\nabla }}}_{{U}_{2}}\xi ,Z)-\vartheta ({U}_{2})g({{\rm{\nabla }}}_{{U}_{1}}\xi ,Z)\right)\\ \quad +\,\frac{1}{2}c\left(\vartheta (Z)g({{\rm{\nabla }}}_{{U}_{1}}\xi ,{U}_{2})-\vartheta (Z)g({{\rm{\nabla }}}_{{U}_{2}}\xi ,{U}_{1})\right).\end{array}\end{eqnarray}$
By differentiating equation ( $\begin{eqnarray}\begin{array}{rcl}S({U}_{2},V) & = & -({\rm{div}}A){U}_{2}+\frac{3}{2}\left({U}_{2}\left(a+3\mu \right.\right.\\ & & \left.\left.+{\rm{div}}V\right)\right)-\frac{1}{2}\vartheta ({U}_{2})\zeta b\\ & & -\frac{1}{2}{U}_{2}b-\frac{3}{2}\phi b\vartheta ({U}_{2})\\ & & -c\phi \omega ({U}_{2})+\frac{1}{2}cg({{\rm{\nabla }}}_{\zeta }\xi ,{U}_{2})\\ & & -\frac{1}{2}c({\rm{div}}\xi )\vartheta ({U}_{2}).\end{array}\end{eqnarray}$
By replacing equation ( $\begin{eqnarray}\begin{array}{rcl}({\rm{div}}A){U}_{2} & = & -ag({U}_{2},V)-b\vartheta ({U}_{2})\vartheta (V)-c\omega ({U}_{2})\vartheta (V)\\ & & -c\vartheta ({U}_{2})\omega (V)+\frac{3}{2}\left({U}_{2}\left(a+3\mu +{\rm{div}}V\right)\right)\\ & & -\frac{1}{2}\vartheta ({U}_{2})\zeta b-\frac{1}{2}{U}_{2}b-\frac{3}{2}\phi b\vartheta ({U}_{2})\\ & & -c\phi \omega ({U}_{2})+\frac{1}{2}cg({{\rm{\nabla }}}_{\zeta }\xi ,{U}_{2})-\frac{1}{2}c({\rm{div}}\xi )\vartheta ({U}_{2}).\end{array}\end{eqnarray}$
Additionally, equation ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}_{{U}_{1}}| V{| }^{2} & = & 2g({{\rm{\nabla }}}_{{U}_{1}}U,V)\\ & & =-2g(A{W}_{1},V)-b\vartheta ({U}_{1})\vartheta (V)\\ & & -c\vartheta ({U}_{1})\omega (V)-c\vartheta ({U}_{1})\omega (V)\\ & & -\left(a+3\mu +{\rm{div}}V\right)g({U}_{1},V).\end{array}\end{eqnarray}$
equations ( $\begin{eqnarray}\begin{array}{rcl}({{ \mathcal L }}_{V}g)({U}_{1},V) & = & -b\vartheta ({U}_{1})\vartheta (V)\\ & & -c\omega ({U}_{1})\vartheta (V)-c\vartheta ({U}_{1})\omega (V)\\ & & -\left(a+3\mu +{\rm{div}}V\right)g({U}_{1},V).\end{array}\end{eqnarray}$
By substituting equation ( $\begin{eqnarray*}{{\rm{\nabla }}}_{{U}_{1}}| V{| }^{2}=-2g(A{U}_{1},V)+({{ \mathcal L }}_{U}g)({U}_{1},V).\end{eqnarray*}$
□The utilization of the Lie derivative alongside the ν(Ric)-vector results in the formulation given by the equation
$\begin{eqnarray}({{ \mathcal L }}_{\nu }g)({U}_{1},{U}_{2})=2\psi S({U}_{1},{U}_{2}).\end{eqnarray}$
Then ${\rm{div}}\nu =\psi r$. By integrating equation ( $\begin{eqnarray}\begin{array}{l}S({U}_{1},{U}_{2})+(3\mu +{\rm{div}}\nu )g({U}_{1},{U}_{2})\\ \,+\,2\psi S({U}_{1},{U}_{2})=0.\end{array}\end{eqnarray}$
Through the process of contracting this equation, we can express μ in the form $\mu =-\frac{(1+6\psi )r}{12}$. In cases where $\psi \ne -\frac{1}{2}$, it can be deduced that $\begin{eqnarray*}S({U}_{1},{U}_{2})=-\frac{3\mu +\psi r}{1+2\psi }g({U}_{1},{U}_{2}).\end{eqnarray*}$
Consequently, the imperfect fluid GRW spacetime is identified as a dark perfect fluid spacetime, leading us to infer from equation (Let us examine the scenario where the imperfect fluid GRW spacetime, represented as (M4, g), is characterized as ${{ \mathcal W }}_{2}$-flat. This stipulation leads us to establish the following equation
$\begin{eqnarray}\begin{array}{rcl}R({U}_{1},{U}_{2},{U}_{3},{U}_{4}) & = & -\frac{1}{3}\left[g({U}_{1},{U}_{3})\right.\\ & & S({U}_{2},{U}_{4})-g({U}_{2},{U}_{3})\\ & & \left.S({U}_{1},{U}_{4})\right].\end{array}\end{eqnarray}$
By contracting this equation with respect to the vectors U1 and U4, we derive the expression $\begin{eqnarray}S({U}_{2},{U}_{3})=\frac{1}{4}rg({U}_{2},{U}_{3}).\end{eqnarray}$
From equation ( $\begin{eqnarray}S({U}_{2},{U}_{3})=-(5\phi +3\mu )g({U}_{2},{U}_{3})-(2\phi )\vartheta ({U}_{2})\vartheta ({U}_{3}).\end{eqnarray}$
By substituting this result into the previous equation, we arrive at $\begin{eqnarray}(5\phi +3\mu +\frac{1}{4}r)g({U}_{2},{U}_{3})+(2\phi )\vartheta ({U}_{2})\vartheta ({U}_{3})=0.\end{eqnarray}$
The implications of this equation allow us to conclude that φ = 0 and $\mu =-\frac{r}{12}$. Additionally, the earlier equation indicates that ρ + σ = 0 and ${ \mathcal P }=0$, thereby classifying the imperfect fluid GRW spacetime as a dark perfect fluid spacetime. □We will explore the properties of the imperfect fluid GRW spacetime, represented as (M4, g), which is distinguished by its pseudo-projectively flat nature. By analyzing the inner product of the pseudo-projective curvature tensor with U4, we derive the following equation
$\begin{eqnarray}\begin{array}{rcl} & & {b}_{1}R({U}_{1},{U}_{2},{U}_{3},{U}_{4})\\ & & \,=-{c}_{2}\left(S({U}_{2},{U}_{3})g({U}_{1},{U}_{4})\right.\\ & & \left.\,-S({U}_{1},{U}_{3})g({U}_{2},{U}_{4})\right)\\ & & \,+\frac{r}{4}\Space{0ex}{2.5ex}{0ex}(\frac{{c}_{1}}{3}+{c}_{2}\Space{0ex}{2.5ex}{0ex})\left[g({U}_{2},{U}_{3})g({U}_{1},{U}_{4})\right.\\ & & \left.\,-g({U}_{1},{U}_{3})g({U}_{2},{U}_{4})\right].\end{array}\end{eqnarray}$
By contracting equation ( $\begin{eqnarray}S({U}_{2},{U}_{3})=\frac{r}{4}g({U}_{2},{U}_{3}).\end{eqnarray}$
Incorporating equation ( $\begin{eqnarray}(5\phi +3\mu +\frac{1}{4}r)g({U}_{2},{U}_{3})+(2\phi )\vartheta ({U}_{2})\vartheta ({U}_{3})=0.\end{eqnarray}$
As a result, we conclude that φ = 0 and $\mu =-\frac{r}{12}$, which leads to the conditions ρ + σ = 0 and allows us to classify the imperfect fluid GRW spacetime as a dark perfect fluid spacetime. □Declarations
The authors are thankful to the reviewers and the editor for their valuable suggestions for the improvement of the paper.