Two-Fluid Model for Anisotropic Fluid Spacetime with Specific Stress–Energy Tensor Constraints and $\mathit{f}\left(\mathcal{R}\right)$-Gravity

Abstract

A two-fluid model can be described by an anisotropic fluid matter, and we introduced the notion of an anisotropic fluid spacetime. The algebraic and differential properties of an anisotropic fluid spacetime equipped with several forms of the stress–energy tensor is the focus of this research. We show that an anisotropic fluid spacetime with a radial pressure $p^{\Vert }$, transverse pressure $p^{\perp }$, and the energy density $\rho$ is a generalized quasi-Einstein spacetime. We prove that a dark matter era or an anisotropic fluid spacetime with vanishing vorticity is represented by an anisotropic fluid spacetime endowed with a covariant constant stress–energy tensor; on the contrary, a dark matter era or the expansion scalar vanishes is represented by an anisotropic fluid spacetime endowed with a Codazzi type of stress–energy tensor, as long as $\mathcal{A}$ stays invariant under the velocity vector field $\zeta$. Furthermore, we use the Killing velocity vector field, parallel vector fields to characterize Ricci Semi-Symmetric, $\mathcal{T}$-recurrent, Pseudo-Ricci symmetric, and $\widehat{\mathbb{R}}$-harmonic anisotropic fluid spacetime. We find that the anisotropic fluid spacetime reflect a stiff matter and a radiation era with these geometric symmetries. Finally, we provide findings for an anisotropic fluid spacetime with a divergence-free matter tensor and the vanishing space-matter tensor and explore the dynamical aspects of cosmological epoch of an anisotropic fluid spacetime coupled with $f\left(\mathcal{R}\right)$-gravity.

1. Introduction

The General Theory of Relativity ($GTR$) is the name given to Einstein’s theory of gravity. This theory states that the gravitational field originates from the stress–energy tensor $\mathcal{T}$, which also represents spacetime curvature. The best way to understand general relativity is through the mathematical development of differential geometry and relativistic fluid models. According to the fundamental concept of $GTR$, spacetime is best described as a curved Lorentzian manifold [1,2].

For $GTR$, Lorentzian manifold has significant consequences with Lorentzian metric g and signature $(-,+,+,+)$. The first step in establishing the Lorentzian manifold ${\mathcal{M}}^4$ is to study the properties of vectors on it. Therefore, the ideal choice for addressing $GTR$ [1] is the Lorentzian manifold $({\mathcal{M}}^4,g)$.

One of the main elements (matter) of the spacetime is the stress–energy tensor $\mathcal{T}$. In addition, matter is considered fluid due to its viscosity, density, pressure, and other dynamic properties (for more information, see [3]). The universe’s matter component is thought to behave as a perfect fluid in conventional cosmological models. The absence of viscosity and heat conduction characterizes a perfect fluid, which is also known as an isotropic fluid at rest. The dust matter fluid $p=0$ ($\omega =0$) [4] is the most basic illustration of the perfect fluid. In $GTR$, idealized distributions of mass–energy are often considered with perfect fluids. Moreover, in $GTR$, “stiff matter fluid” is described by the relation $p=\sigma$ ($\omega =1$) [4]. If $\sigma =3p$ ($\omega ={\displaystyle \frac{1}{3}}$) [4], then matter in spacetime originates from a radiation fluid. Furthermore, there are important uses for Equation (3) in star structure and cosmology.

Definition 1

([3,4]). A quasi-Einstein Lorentzian manifold $({\mathcal{M}}^n,g)$$(n>2)$ is designated as a perfect fluid spacetime ($PFS$) if its Ricci tensor $\mathfrak{S}$ admits the decomposition

$$\mathfrak{S}=\mathcal{A}g+\mathcal{B}\eta \otimes \eta ,$$

(1)

wherein g is the Lorentzian metric, $\mathcal{A}$ and $\mathcal{B}$ are scalars, and 1-form η is comparable to a unit time-like vector field such that $\eta \left(\zeta \right)=-1$.

Moreover, a time-like vector field is permitted by the Lorentzian manifold [5].

Definition 2

([6,7]). A non-flat Riemannain manifold $({\mathcal{M}}^n,g)$ $(n>2)$ is designated as a generalized quasi-Einstein manifold $\left(GQE\right)$ if its Ricci tensor $\mathfrak{S}$ holds the relation:

$$\mathfrak{S}=\mathcal{A}g+\mathcal{B}\eta \otimes \eta +\mathcal{C}\mathrm{{\rm Y}}\otimes \mathrm{{\rm Y}}$$

(2)

wherein $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$ are scalars; $\mathcal{A}\ne 0$, $\mathcal{C}\ne 0$ and η, Υ are 1-form such that

$$g(a,\zeta )=\eta \left(a\right),\eta \left(\zeta \right)=g(\zeta ,\zeta )=-1,$$

$$g(a,\gamma )=\mathrm{{\rm Y}}\left(a\right),\mathrm{{\rm Y}}\left(\gamma \right)=g(\gamma ,\gamma )=1,g(\zeta ,\gamma )=0$$

for any vector field $a\in \chi ({\mathcal{M}}^n,g)$. The orthogonal time-like and space-like unit vector fields ζ and γ are represented by the 1-forms η and Υ. If $\mathcal{C}=0$, $({\mathcal{M}}^n,g)$ transforms into a perfect fluid spacetime.

2. Anisotropic Fluid Spacetime

When the direction of measurement affects the characteristics of a fluid, like pressure or viscosity, it is said to be anisotropic. Because their molecules move randomly, isotropic fluids have uniform properties in all directions, in contrast to this directional dependency. Various states of matter in cosmology and high-energy physics, such as the fluid produced in heavy-ion collisions and liquid crystals, which contain molecules aligned in a specific direction, are examples of anisotropic fluids [8].

Cosmology uses anisotropic fluid models to explain the universe, especially when it comes to processes whose attributes can change with direction, such as dark energy and structure development [9].

In general relativity, an isotropic fluid spacetime represents a spacetime containing a fluid whose pressure and stress change with direction, as opposed to an isotropic fluid, which has constant pressure in every direction [10]. This can be used to phenomena like the interior structure of black holes, galactic rotation, and the universe’s accelerating expansion. It is used to solve Einstein’s equations for objects like black holes and the cosmos [10]. An anisotropic fluid can be used to represent a two-fluid model of baryonic matter, dark energy, and their interaction [9,10].

The stress–energy tensor $\mathcal{T}$ in conjunction with an anisotropic fluid has the following form [11,12]:

$$\mathcal{T}=\rho \eta \otimes \eta +p^{\Vert }\mathrm{{\rm Y}}\otimes \mathrm{{\rm Y}}+p^{\perp }[\eta \otimes \eta -\mathrm{{\rm Y}}\otimes \mathrm{{\rm Y}}],$$

(3)

where the pressure inside compact objects or the two-fluid model can be divided into two parts: radial pressure $p^{\Vert }$ and transverse pressure $p^{\perp }$ perpendicular to $\gamma$. $\rho$ represents the energy density measured by a comoving observer with the fluid and 1-form $\eta$, which corresponds to the time-like velocity vector field $\zeta$ such that $\eta \left(\zeta \right)=-1$, as well as a unit space-like vector field $\gamma$ with 1-form $\mathrm{{\rm Y}}$ such that $\mathrm{{\rm Y}}\left(\gamma \right)=1$ and is orthogonal to $\zeta$. i.e., $g(\zeta ,\gamma )=0$.

The entire state of internal force distribution at each place within a continuous medium is described by the stress tensor as a second-order mathematical tensor. By taking into consideration both normal and shear forces operating on a point in all possible surface orientations, it expands the scalar idea of pressure into all dimensions. The stress tensor connects the internal force and it describes the flux of momentum, representing the transport of momentum across surfaces.

In this instance, the 4-velocity field is described by $\eta$. Naturally, if $\rho$, $p^{\perp }$, and $p^{\Vert }$ were completely random, Equations (2) and (6) would be identical; but in practical applications, $\rho$, $p^{\perp }$, and $p^{\Vert }$ are not freely prescribed and are subject to specific physical restrictions. They are not independent, specifically, but (6) needs to be enhanced with the equations of state. $p^{\perp }=p^{\perp }\left(\rho \right)$ and $p^{\Vert }=p^{\Vert }\left(\rho \right)$ are Friedmann–Robertson–Walker spacetime solutions, which are the most significant and well-known class of (6) solutions.

It is considered that the universe is expanding faster in contemporary $GTR$ and cosmology. Furthermore, an equation of the state in the form $p^{\perp }=p^{\perp }(\rho ,t_0)$ type, where $t_0$ denotes the absolute temperature, connects $p^{\perp }$, $p^{\Vert }$, and $\rho$. But we focus on the scenario where $t_0$ remains constant. In this instance, the spacetime is referred to as an anisotropic spacetime, and the state equation with angular and radial pressures are proportional to density, being expressed as

$$p^{\perp }={\omega }_2\rho ,p^{\Vert }={\omega }_1\rho .$$

(4)

The effect of anisotropies is an interesting property of being directionally dependent, with profound effects across various scientific and engineering disciplines. Moreover, anisotropies’ effects provide a window into the early universe and the internal structure of extreme objects. Strong magnetic fields can induce pressure anisotropy in neutrons and odd stars [10]. This has an impact on their maximum mass and radius; for instance, a radial magnetic field can reduce a star’s mass and radius, whereas a transverse magnetic field can increase them. The equation of state changes from crust (nuclei) to core (uniform liquid).

Anisotropies in the cosmological gravitational wave background encode information about the non-linearity of scalar metric perturbations in the early universe. Some models incorporate radial pressure $p^{\perp }$ and tangential pressures $p^{\Vert }$, allowing for anisotropic stress [9]:

$$\Delta =p^{\perp }-p^{\Vert }.$$

(5)

Chaki and Ray [13] studied spacetimes with a covariant constant energy–momentum tensor in 1996. Following this, perfect fluid spacetime with a killing energy–momentum tensor was researched by Sharma and Ghosh [14]. A study by Mallick et al. [15] examined spacetimes using various energy–momentum tensor types. De and De [16,17] then investigated spacetimes using semi-symmetric energy–momentum tensors. In the last five years, Siddiqi and his co-authors have recently investigated numerous spacetimes with various forms of energy–momentum tensors, such as bulk viscous string fluid matter tensor [18], magneto fluid momentum tensor [19], thermodynamical energy–momentum tensor [20], string cloud momentum tensor [21], and strange quark matter tensor [22]. The properties of tensors with respect to the connection have recently been explored using a new approach, which will be an interesting topic to address in the near future (for details, see [23,24]).

A key concept in theoretical physics and cosmology, anisotropic fluid spacetime allows pressure to fluctuate in both radial and tangential directions, providing a more flexible alternative to conventional isotropic (perfect fluid) models. By permitting inhomogeneous structures at small scales, it offers realistic, non-homogeneous models for early/late universe, galaxy dynamics, and compact objects. Anisotropic fluids are essential for finding and classifying new, stable, spherically symmetric black hole solutions, including those in modified gravity theories [25].

To the best of our knowledge, the literature yields many conclusions about perfect fluid spacetimes with energy–momentum tensors, but only a small number of an anisotropic fluid spacetime. The main goals of the paper is to close this gap: we will focus on describing the geometric manifestation of an anisotropic fluid spacetime that satisfies certain covariant constant stress–energy tensors of a two-fluid model of baryonic matter or anisotropic fluid matter.

The novel classification results in this paper were produced using very strong geometric and algebraic assumptions, including parallel vector fields, vanishing space-matter tensors, Ricci recurrence, Ricci semi-symmetry, covariant consistency of stress tensor, and Codazzi-type requirements. The ensuing equations of state (radiation, stiff matter, or dark matter-like behavior) under these assumptions are generally expected and well-documented in the literature now available on quasi-Einstein and generalized quasi-Einstein spacetime. Furthermore, it has been demonstrated that $\widehat{\mathbb{R}}$-harmonic anisotropic fluid spacetime is a radiation epoch.

3. Relativistic Anisotropic Fluid Spacetime with Covariant Constant Stress–Energy Tensor

The field equation controlling perfect fluid motion is Einstein’s gravitational field Equation ($EGFE)$ without cosmic constant in $GTR$ [4]:

$$\mathfrak{S}-{\displaystyle \frac{\mathcal{R}}{2}}g=\kappa \mathcal{T},$$

(6)

where $\kappa$ is gravitational constant, $\mathcal{R}$ is scalar curvature, and $\mathfrak{S}$ is Ricci tensor of Lorentzian metric g in a spacetime manifold $(M^4,g).$

Now, using Equation (3) as well as (6), the $EGFE$ for an anisotropic spacetime is obtained as

$$\mathfrak{S}(a,b)=\left[{\displaystyle \frac{\mathcal{R}}{2}}+\kappa \right]g(a,b)+\kappa (\rho +p^{\perp })\eta \left(a\right)\eta \left(b\right)-\kappa (p^{\perp }-p^{\Vert })\mathrm{{\rm Y}}\left(a\right)\mathrm{{\rm Y}}\left(b\right)$$

(7)

for all vector fields $a,b\in \chi \left(M^4\right).$

We can express (7) as

$$\mathfrak{S}(a,b)=\mathcal{A}g(a,b)+\mathcal{B}\eta \left(a\right)\eta \left(b\right)+\mathcal{C}\mathrm{{\rm Y}}\left(a\right)\mathrm{{\rm Y}}\left(b\right).$$

(8)

Then, we infer that

$$\mathcal{A}={\displaystyle \frac{\mathcal{R}}{2}}+\kappa ,\mathcal{B}=\kappa (p^{\perp }+\rho ),\mathcal{C}=-\kappa (p^{\perp }-p^{\Vert }).$$

(9)

Also, we have

$$\mathfrak{S}(\zeta ,\zeta )=\mathcal{A}+\mathcal{B},\mathfrak{S}(\gamma ,\gamma )=\mathcal{A}+\mathcal{C}.$$

(10)

Now, in view of (8) and Definition 2, we write the first result:

Theorem 1.

Upon coupling anisotropic fluid spacetime $(M^4,g)$ with stress–energy tensor $\mathcal{T}$ obeying $EGFE$ with a radial pressure $p^{\Vert }$, transverse pressure $p^{\perp }$, and the energy density ρ, then the anisotropic fluid spacetime is a generalized quasi-Einstein spacetime.

In light of (9) and (10), we gain

$$\mathfrak{S}(\gamma ,\gamma )={\displaystyle \frac{\mathcal{R}}{2\kappa }}+\kappa -\kappa (p^{\perp }-p^{\Vert }),$$

(11)

which implies

$$(p^{\perp }-p^{\Vert })={\displaystyle \frac{\mathcal{R}}{2}}+1-{\displaystyle \frac{\mathfrak{S}(\gamma ,\gamma )}{\kappa }}.$$

(12)

Now, (5) and (12) infer the next outcome:

Theorem 2.

Upon coupling anisotropic fluid spacetime $(M^4,g)$ with stress–energy tensor $\mathcal{T}$ obeying $EGFE$ with a radial pressure $p^{\Vert }$, transverse pressure $p^{\perp }$, and the energy density ρ, then the anisotropic stress is

$$\Delta ={\displaystyle \frac{\mathcal{R}}{2}}+1-{\displaystyle \frac{\mathfrak{S}(\gamma ,\gamma )}{\kappa }}.$$

(13)

Chaki and Ray investigated spacetime manifolds with a covariant constant stress–energy tensor in [13]. Here, we examine covariant constant stress–energy tensor $\mathcal{T}$ in an anisotropic fluid spacetime.

$EGFE$ (6) provides us the following [23]:

$$\nabla \mathfrak{S}=0\iff \nabla \mathcal{T}=0.$$

(14)

According to this, the scalar curvature $\mathcal{R}=constant$. When we contract Equation (10), we obtain

$$\mathcal{R}=4\mathcal{A}-\mathcal{B}-\mathcal{C}.$$

(15)

We can state that $4\mathcal{A}-\mathcal{B}-\mathcal{C}=K$, a constant, using the previously mentioned formulae. From the above, we may deduce using Equation (9) that

$$\kappa (\rho -3p^{\Vert })=K.$$

(16)

When $K=0$, the previous equation using (4) gives

$${\omega }_1={\displaystyle \frac{p^{\Vert }}{\rho }}={\displaystyle \frac{1}{3}},$$

(17)

which implies that the anisotropic fluid spacetime with a radial pressure $p^{\Vert }$ represents radiation era. Thus, we may assert the following:

Theorem 3.

An anisotropic fluid spacetime $(M^4,g)$ coupled with covariant constant stress–energy tensor $\mathcal{T}$ with a radial pressure $p^{\Vert }$ obeys the equation of state $\rho =3p^{\Vert }+constant.$ In particular, a radiation period is indicated by the anisotropic fluid spacetime if the constant eliminates.

Given that $\nabla \mathfrak{S}=0$, we can obtain from Equation (8)

$$\begin{array}{cc}& d\mathcal{A}\left(a\right)g\left(\right(b,c)+d\mathcal{B}(a\left)\eta \right(b\left)\eta \right(c)-d\mathcal{C}(a\left)\mathrm{{\rm Y}}\right(b\left)\mathrm{{\rm Y}}\right(c)\hfill \\ +& \mathcal{B}[\left({\nabla }_a\eta \right)b\eta \left(c\right)+\eta \left(b\right)\left({\nabla }_a\eta \right)c]\hfill \\ -& \mathcal{C}\left[{\nabla }_a\mathrm{{\rm Y}}\right)b\mathrm{{\rm Y}}\left(c\right)+\mathrm{{\rm Y}}\left(b\right)\left({\nabla }_a\mathrm{{\rm Y}}\right)c]=0\hfill \end{array}$$

for all vector fields $a,b,c\in \chi (M^4,g).$

By using a frame field and contracting over b and c, we obtain

$$4d\mathcal{A}\left(a\right)-d\mathcal{B}\left(a\right)-d\mathcal{C}\left(a\right)=0.$$

(18)

Since $\eta \left(\zeta \right)=-1$ and $\mathrm{{\rm Y}}\left(\zeta \right)=0.$ $b=\zeta$, entering into (18) produces

$$d\mathcal{A}\left(a\right)\eta \left(c\right)-d\mathcal{B}\left(a\right)\eta \left(c\right)-\mathcal{B}\left({\nabla }_a\eta \right)c=0.$$

(19)

Once more, using $c=\zeta$ in the equation above, we deduce

$$-d\mathcal{A}\left(a\right)+d\mathcal{B}\left(a\right)\eta \left(c\right)=0.$$

(20)

We obtain $\mathcal{A}=constant$ by comparing (18) and (20), and hence, $\mathcal{B}=constant$ from (18) as well.

Consequently, (19) provides

$$\mathcal{B}\left({\nabla }_a\eta \right)c=0.$$

(21)

This suggests that the 1-form $\eta$ is closed or that $\mathcal{B}=0$.

Since $\eta$ is closed and $\mathcal{B}=0$, it indicates that

$$p^{\perp }+\rho =0.$$

$$\mathfrak{S}(a,b)=\mathcal{A}g(a,b)+\mathcal{C}\mathrm{{\rm Y}}\left(a\right)\mathrm{{\rm Y}}\left(b\right).$$

(22)

It concludes that the velocity vector field $\zeta$ is irrotational. Consequently, there is no vorticity in the anisotropic fluid spacetime. Therefore, we write the following:

Theorem 4.

An anisotropic fluid spacetime $(M^4,g)$ coupled with covariant constant stress–energy tensor $\mathcal{T}$ with transverse pressure $p^{\perp }$ indicates a dark matter epoch or the matter content is a anisotropic fluid spacetime with vanishing vorticity.

Theorem 5.

If an anisotropic fluid spacetime $(M^4,g)$ coupled with covariant constant stress–energy tensor $\mathcal{T}$ with transverse pressure $p^{\perp }$ indicates a dark matter epoch, then the anisotropic fluid spacetime is a perfect fluid spacetime type.

4. Relativistic Anisotropic Fluid Spacetime with Codazzi Type of Stress– Energy Tensor

A spacetime that has a stress–energy tensor $\mathcal{T}$ of the Codazzi type is a Yang Pure Space, according to the findings of [15]. In this case, we extend this condition to an anisotropic fluid spacetime and prove that it corresponds to a $RW$-spacetime with an identical requirement.

In an anisotropic fluid spacetime, suppose the stress–energy tensor $\mathcal{T}$ is of the Codazzi type [26],

$$\left({\nabla }_a\mathcal{T}\right)(b,c)=\left({\nabla }_b\mathcal{T}\right)(a,c),$$

(23)

with all vector fields $a,b,c\in \chi \left(M^4\right).$

Equation (6) is used to obtain

$$\left({\nabla }_a\mathfrak{S}\right)(b,c)={\displaystyle \frac{d\mathcal{R}\left(a\right)}{2}}g(b,c)+\kappa \left({\nabla }_a\mathcal{T}\right)(b,c).$$

(24)

Using (23), we can deduce from (24) that

$$\left({\nabla }_a\mathfrak{S}\right)(b,c)-{\displaystyle \frac{d\mathcal{R}\left(a\right)}{2}}g(b,c)=\left({\nabla }_b\mathfrak{S}\right)(a,c)-{\displaystyle \frac{d\mathcal{R}\left(b\right)}{2}}g(a,c).$$

(25)

After applying contraction over a and b, we turn up

$$d\mathcal{R}\left(c\right)=0.$$

(26)

Thus, $\mathcal{R}$ must be constant. Therefore, from (25), we obtain

$$\left({\nabla }_a\mathfrak{S}\right)(b,c)=\left({\nabla }_b\mathfrak{S}\right)(a,c).$$

(27)

This suggests that Ricci tensor $\mathfrak{S}$ belongs to the Codazzi type. Likewise, the opposite is true.

According to Guilfoyle and Nolan [27], “Yang Pure Space” is a Lorentzian manifold $(M^4,g)$ in which Yang’s equations are solved by the metric tensor of the manifold:

$$\left({\nabla }_a\mathfrak{S}\right)(b,c)-\left({\nabla }_b\mathfrak{S}\right)(a,c)=0.$$

Additionally, if and only if the spacetime is a $RW$-spacetime, the authors of the same work demonstrated that a perfect fluid spacetime with $p+\rho \ne 0$ is a Yang Pure Space. Therefore, we deduce that a $RW$-spacetime is an anisotropic fluid spacetime with a Codazzi-type stress–energy tensor $\mathcal{T}$. As a result, we may conclude the following.

Theorem 6.

An anisotropic fluid spacetime $(M^4,g)$ attached with a Codazzi type of stress–energy tensor $\mathcal{T}$ is a Yang Pure spacetime.

Theorem 7.

If an anisotropic fluid spacetime $(M^4,g)$ attached to a Codazzi type of stress–energy tensor $\mathcal{T}$, then the anisotropic fluid spacetime is a $RW$-spacetime.

Currently, applying Equation (8), we have

$$d\mathcal{A}\left(a\right)g(b,c)+d\mathcal{B}\left(a\right)\eta \left(b\right)\eta \left(c\right)+d\mathcal{C}\left(a\right)\mathrm{{\rm Y}}\left(b\right)\mathrm{{\rm Y}}\left(c\right)$$

(28)

$$+\mathcal{B}[\left({\nabla }_a\eta \right)b\eta \left(c\right)+\eta \left(b\right)\left({\nabla }_a\eta \right)c]+\mathcal{C}[\left({\nabla }_a\mathrm{{\rm Y}}\right)b\mathrm{{\rm Y}}\left(c\right)+\mathrm{{\rm Y}}\left(b\right)\left({\nabla }_a\mathrm{{\rm Y}}\right)c]$$

$$=d\mathcal{A}\left(b\right)g(a,c)+d\mathcal{B}\left(b\right)\eta \left(a\right)\eta \left(c\right)+d\mathcal{C}\left(b\right)\upsilon \left(a\right)\mathrm{{\rm Y}}\left(c\right)$$

$$+\mathcal{B}[\left({\nabla }_b\eta \right)a\eta \left(c\right)+\eta \left(a\right)\left({\nabla }_b\eta \right)c]+\mathcal{C}[\left({\nabla }_b\mathrm{{\rm Y}}\right)a\mathrm{{\rm Y}}\left(c\right)+\mathrm{{\rm Y}}\left(a\right)\left({\nabla }_b\mathrm{{\rm Y}}\right)c].$$

By contracting over b and c in a frame field, we obtain

$$4d\mathcal{A}\left(a\right)-d\mathcal{B}\left(a\right)+\mathcal{B}[\left({\nabla }_a\eta \right)\zeta +\eta \left(b\right)\left({\nabla }_a\eta \right)\zeta ]$$

(29)

$$=d\mathcal{A}\left(a\right)+d\mathcal{B}\left(\zeta \right)\eta \left(a\right)+\mathcal{B}[\left({\nabla }_{\zeta }\eta \right)a+div\zeta \eta \left(a\right)],$$

where the divergence of $\zeta$ is indicated by $div\zeta .$

When $\zeta$ is substituted for a, the following equation produces

$$3d\mathcal{A}\left(\zeta \right)+\mathcal{B}[\left({\nabla }_{\zeta }\eta \right)\zeta +\left({\nabla }_{\zeta }\eta \right)\zeta ]=\mathcal{B}[\left({\nabla }_{\zeta }\eta \right)\zeta -div\zeta ].$$

(30)

Given that $\left({\nabla }_{\zeta }\eta \right)\zeta =0$, the preceding equation provides

$$3d\mathcal{A}\left(\zeta \right)+div\zeta =0.$$

(31)

Consider the velocity vector field $\zeta$. The scalar $\mathcal{A}$ is invariant along $\zeta$. Following the previous equation, we may conclude that either $\mathcal{B}=0$ or $div\zeta =0.$

Using (9), we now state that $p^{\perp }+\rho =0,$ or that the expansion scalar vanishes [4].

Thus, we state the subsequent result:

Theorem 8.

If an anisotropic fluid spacetime $(M^4,g)$ attached with a Codazzi type of stress–energy tensor $\mathcal{T}$, and with transverse pressure $p^{\perp }$, it indicates that the dark matter epoch or the expansion scalar disappears, provided scalar $\mathcal{A}$ remains invariant under the velocity vector field $\zeta .$

5. Recurrent Anisotropic Fluid Spacetime

In this part, we assume that an anisotropic fluid spacetime $(M^4,g)$ attached with stress–energy tensor $\mathcal{T}$ is of the recurrent type or is a $\mathcal{T}$-recurrent anisotropic fluid spacetime.

Definition 3

([28]). A semi-Riemannian manifold $(M,g)$ is designated as Ricci-recurrent if its Ricci tensor $\mathfrak{S}$ obeys

$$\left({\nabla }_{\mu }\mathfrak{S}\right)(\nu ,\varsigma )=\eta \left(\mu \right)\mathfrak{S}(\nu ,\varsigma ),$$

(32)

where η is a non-zero 1-form and vector fields $\mu ,\nu ,\varsigma \in \chi \left(M^4\right).$

Assume that the stress–energy tensor in an anisotropic fluid spacetime is of the recurrent type, which means that

$$\left({\nabla }_{\mu }\mathcal{T}\right)(\nu ,\varsigma )=\eta \left(\mu \right)\mathcal{T}(\nu ,\varsigma ).$$

(33)

By applying (24), we obtain

$$\left({\nabla }_{\mu }\mathfrak{S}\right)(\mu ,\varsigma )-{\displaystyle \frac{d\mathcal{R}\left(\mu \right)}{2}}g(\nu ,\varsigma )=\eta \left(\mu \right)\mathfrak{S}(\nu ,\varsigma )-{\displaystyle \frac{\mathcal{R}}{2}}\eta \left(\mu \right)g(\nu ,\varsigma ).$$

(34)

Now, applying Equation (8), we obtain

$$d\mathcal{A}\left(\mu \right)g(\nu ,\varsigma )+d\mathcal{B}\left(\mu \right)\eta \left(\nu \right)\eta \left(\varsigma \right)+d\mathcal{C}\left(\mu \right)\mathrm{{\rm Y}}\left(\nu \right)\mathrm{{\rm Y}}\left(\varsigma \right)$$

(35)

$$+\mathcal{B}[\left({\nabla }_{\mu }\eta \right)\nu \eta \left(\varsigma \right)+\eta \left(\varsigma \right)\left({\nabla }_{\mu }\eta \right)\varsigma ]+\mathcal{C}[\left({\nabla }_{\mu }\mathrm{{\rm Y}}\right)\nu \mathrm{{\rm Y}}\left(\varsigma \right)+\mathrm{{\rm Y}}\left(\varsigma \right)\left({\nabla }_{\mu }\mathrm{{\rm Y}}\right)\varsigma ]$$

$$-{\displaystyle \frac{d\mathcal{R}}{2}}g(\nu ,\varsigma )=\eta \left(\mu \right)\mathfrak{S}(\nu ,\varsigma )-{\displaystyle \frac{\mathcal{R}\left(\mu \right)}{2}}\eta \left(\mu \right)g(\nu ,\varsigma ).$$

Taking a frame field and contracting over the vector fields $\mu$ and $\varsigma$, we obtain

$$4d\mathcal{A}\left(\mu \right)-d\mathcal{B}\left(\mu \right)+2\mathcal{B}\left({\nabla }_{\mu }\eta \right)\zeta -2d\mathcal{R}\left(\mu \right)=-(4\mathcal{A}-\mathcal{B})\eta \left(\mu \right).$$

(36)

In addition, we have, ${\mathfrak{L}}_{\zeta }p=0$ and ${\mathfrak{L}}_{\zeta }\rho =0$, [29], where $\mathfrak{L}$ is the Lie derivative operator, if $\zeta$ is Killing. From (9), we know that

$$\mathcal{A}=-{\displaystyle \frac{\kappa (p^{\Vert }-\rho )}{2}},\mathcal{B}=\kappa (p^{\perp }+\rho ).$$

Therefore, we turn up

$$d\mathcal{A}\left(\zeta \right)=d\mathcal{B}\left(\zeta \right)=0.$$

(37)

In addition, we obtain from (15)

$$\mathcal{R}=3\mathcal{A}-\mathcal{B}-\mathcal{C}.$$

Therefore, we deduce $d\mathcal{R}\left(\zeta \right)=0$ from the information mentioned above.

• Applying (9) and (37) to (36) and entering $\mu =\zeta$ results in

$$3p^{\Vert }-\rho =0.$$

(38)

Thus, we state the following:

Theorem 9.

If an anisotropic fluid spacetime $(M^4,g)$ attached with a stress–energy tensor $\mathcal{T}$ is of the recurrent type, then the anisotropic fluid spacetime with radial pressure $p^{\Vert }$ indicates radiation epoch, provided the velocity vector field is Killing.

6. Ricci Semi-Symmetric Anisotropic Fluid Spacetime

An anisotropic fluid spacetime is termed Ricci semi-symmetric, with the constraint that [30]

$$\widehat{\mathbb{R}}(\mu ,\nu )·\mathfrak{S}=0,$$

for all vector fields $\mu ,\nu \in \chi \left(M^4\right).$

Currently, we have

$$(\widehat{\mathbb{R}}(\mu ,\nu )·\mathfrak{S})(\mu ,\nu )=-\mathfrak{S}(\widehat{\mathbb{R}}(\mu ,\nu ),\varsigma )-\widehat{\mathbb{R}}(\varsigma ,\widehat{\mathbb{R}}(\mu ,\nu )\varsigma ).$$

(39)

Then, from (8),

$$\mathcal{A}g(\widehat{\mathbb{R}}(\mu ,\nu )\varsigma ,\omega )+\mathcal{B}\eta \left(\widehat{\mathbb{R}}(\mu ,\nu )\varsigma \right)\eta \left(\omega \right)$$

$$-\mathcal{C}\left(\widehat{\mathbb{R}}(\mu ,\nu )\varsigma \right)\mathrm{{\rm Y}}\left(\omega \right)-\mathcal{A}\gamma \left(\varsigma \right)\gamma \left(\widehat{\mathbb{R}}(\mu ,\nu )\varsigma \right)$$

$$-\mathcal{C}\mathrm{{\rm Y}}\left(\varsigma \right)\mathrm{{\rm Y}}\left(\widehat{\mathbb{R}}(\mu ,\nu )\varsigma \right)=0.$$

(40)

Putting $\varsigma =\zeta$ and $\omega =\gamma$ in (40), we find

$$\mathcal{A}\eta \left(\widehat{\mathbb{R}}(\mu ,\nu )\zeta \right)-\mathcal{C}\mathrm{{\rm Y}}\left(\widehat{\mathbb{R}}(\mu ,\nu )\zeta \right)=0,$$

(41)

or

$$\mathcal{A}g(\widehat{\mathbb{R}}(\mu ,\nu )\gamma ,\zeta ))-\mathcal{C}g(\widehat{\mathbb{R}}(\mu ,\nu )\gamma ,\zeta )=0$$

$$\mathcal{A}\widehat{\mathbb{R}}(\mu ,\nu ,\varsigma ,\omega )=0,$$

(42)

where $g(\widehat{\mathbb{R}}(\mu ,\nu )\varsigma ,\omega )=\widehat{\mathbb{R}}(\mu ,\nu ,\varsigma ,\omega )$ is non-zero since $\kappa$, the gravitational constant, is not zero. Thus, from (9), we turn up

$$p^{\Vert }=\rho .$$

(43)

Theorem 10.

If an anisotropic fluid spacetime with radial pressure $p^{\Vert }$ and energy density ρ is Ricci semi-symmetric, then the anisotropic fluid spacetime indicates stiff matter fluid.

7. Pseudo-Ricci Symmetric and $\widehat{\mathbb{R}}$-Harmonic Anisotropic Fluid Spacetime

In [31], Chaki described pseudo-Ricci symmetric manifolds $(M^4,g)$ as follows:

Definition 4.

The non-flat semi-Riemannian manifold is designated as pseudo-Ricci symmetric $\left(PRS\right)$ and $\mathbb{R}$-harmonic [32] if its Ricci tensor $\mathfrak{S}$ obeys the following expressions:

$$\left({\nabla }_a\mathfrak{S}\right)(b,c)=2\theta \left(a\right)\mathfrak{S}(b,c,)+\theta \left(b\right)\mathfrak{S}(a,c)+\theta \left(c\right)\mathfrak{S}(a,b),$$

(44)

$$\left({\nabla }_a\mathfrak{S}\right)(b,c)=\left({\nabla }_c\mathfrak{S}\right)(a,c),$$

(45)

wherein θ is a 1-form, vector fields $a,b,c\in \chi \left(M^4\right)$, and ∇ is the Levi–Civita connection on $(M^4,g)$. In addition, $(M^4,g)$ reduces to Ricci symmetric manifold if $\theta =0$ in (44).

This part assumes an anisotropic fluid spacetime with time-like vector fields $\zeta$ and space-like vector fields $\gamma$ that are parallel. Let $(M^4,g)$ be an anisotropic fluid spacetime designated as a $GQE$- manifold. Subsequently, we have

$${\nabla }_a\gamma =0,{\nabla }_a\zeta =0\Rightarrow \widehat{\mathbb{R}}(a,b)\zeta =0and\widehat{\mathbb{R}}(a,b)\gamma =0.$$

(46)

We now see that when we contract b,

$\mathfrak{S}(a,\zeta )=0$ and $\mathfrak{S}(a,\gamma )=0$. Thus, from (8), we deduce

$$\mathfrak{S}(a,\zeta )=(\mathcal{A}+\mathcal{B})\eta \left(a\right)=0,$$

(47)

$$\mathfrak{S}(a,\gamma )=(\mathcal{A}+\mathcal{C})\mathrm{{\rm Y}}\left(a\right)=0.$$

(48)

Therefore, $\mathcal{A}=-\mathcal{B}=-\mathcal{C}$. Then, (8) turns the form

$$\mathfrak{S}(a,b)=-\mathcal{A}\left[g\right(a,b)+\eta (a\left)\eta \right(b)+\mathrm{{\rm Y}}(a\left)\mathrm{{\rm Y}}\right(b\left)\right].$$

(49)

On the other hand, we know that

$$\left({\nabla }_a\mathfrak{S}\right)(b,c)={\nabla }_a\mathfrak{S}(b,c)-\mathfrak{S}({\nabla }_{a}b,c)-\mathfrak{S}(b,{\nabla }_{a}c).$$

(50)

In view of Theorem 1, an anisotropic fluid spacetime is a $GQE$-spacetime. Given (49) and (50), we can put it this way:

$$\left({\nabla }_a\mathbb{R}\right)(b,c)=-{\displaystyle \frac{d\mathcal{A}}{da}}[g(b,c)+\eta \left(b\right)\eta \left(c\right)+\mathrm{{\rm Y}}\left(b\right)\mathrm{{\rm Y}}\left(c\right)],$$

(51)

where ${\displaystyle \frac{d}{da}}$ signifies the derivative of $\mathcal{A}$ with respect to the vector field a. Since an anisotropic fluid spacetime is $PRS$, by adopting (46) and (51), we are able to write

$$-{\displaystyle \frac{d\mathcal{A}}{da}}[g(a,b)-\eta \left(a\right)\eta \left(b\right)+\mathrm{{\rm Y}}\left(a\right)\mathrm{{\rm Y}}\left(b\right)]$$

(52)

$$=-2\mathcal{A}\theta \left(a\right)\left[g\right(b,c)+\eta (b\left)\eta \right(c)+\mathrm{{\rm Y}}(b\left)\mathrm{{\rm Y}}\right(c\left)\right]$$

$$-\mathcal{A}\theta \left(b\right)\left[g\right(a,c)+\eta (a\left)\eta \right(c)+\mathrm{{\rm Y}}(a\left)\mathrm{{\rm Y}}\right(c\left)\right]$$

$$-\mathcal{A}\theta \left(c\right)\left[g\right(a,b)+\eta (a\left)\eta \right(b)+\mathrm{{\rm Y}}(a\left)\mathrm{{\rm Y}}\right(b\left)\right].$$

Inserting $a=\zeta$ and $b=\gamma$ in (52), we gain

$$-{\displaystyle \frac{d\mathcal{A}}{d\zeta }}=\mathcal{A}\theta \left(a\right)$$

(53)

and

$$-{\displaystyle \frac{d\mathcal{A}}{d\gamma }}=\mathcal{A}\theta \left(b\right).$$

(54)

Using $a=\zeta$ and $b=\gamma$ in (52), we obtain

$$\theta \left(a\right)=0and\theta \left(b\right)=0.$$

(55)

Therefore, in light of (52)–(55), we obtain

$${\displaystyle \frac{d\mathcal{A}}{d\zeta }}=0,{\displaystyle \frac{d\mathcal{A}}{d\gamma }}=0,$$

(56)

which implies $-\mathcal{A}$ is constant along vector fields $\zeta$ and $\gamma$.

Thus, the following can be deduced.

Theorem 11.

If an anisotropic fluid spacetime is a $GQE$-spacetime with parallel vector fields ζ and γ, and if the anisotropic fluid spacetime is $PRS$, then $-\mathcal{A}$ is constant along the vector field ζ and vector field γ.

Theorem 12.

If an anisotropic fluid spacetime with radial pressure $p^{\Vert }$ and energy density ρ is $PRS$, then the equation of state is $p^{\Vert }=\rho +constant.$ In particular, if the constant vanishes, then the $PRS$-anisotropic fluid spacetime represents a radiation epoch.

Let $(M^4,g)$ be an anisotropic fluid spacetime that is $\mathbb{R}$-harmonic. If the vector fields $\zeta$ and $\gamma$ are parallel, then (45) and (52) give us

$$\left({\nabla }_a\mathfrak{S}\right)(b,c)-\left({\nabla }_c\mathfrak{S}\right)(a,b)={\displaystyle \frac{d\mathcal{A}}{da}}[g(b,c)+\eta \left(b\right)\eta \left(c\right)+\mathrm{{\rm Y}}\left(b\right)\mathrm{{\rm Y}}\left(c\right)]$$

(57)

$$-{\displaystyle \frac{d\mathcal{A}}{dc}}[g(a,b)+\eta \left(a\right)\eta \left(b\right)+\mathrm{{\rm Y}}\left(a\right)\mathrm{{\rm Y}}\left(b\right)].$$

Putting $a=\zeta$ and $b=\gamma$ in (57), we obtain

$$-{\displaystyle \frac{d\mathcal{A}}{d\zeta }}=\mathcal{A}\theta \left(a\right)$$

(58)

and

$$-{\displaystyle \frac{d\mathcal{A}}{d\gamma }}=\mathcal{A}\theta \left(b\right).$$

(59)

Plugging $a=\zeta$ and $b=\gamma$ in (52), we find

$$\theta \left(a\right)=0and\theta \left(b\right)=0.$$

(60)

Consequentially, in view of (52)–(60), we obtain

$${\displaystyle \frac{d\mathcal{A}}{d\zeta }}=0,{\displaystyle \frac{d\mathcal{A}}{d\gamma }}=0,$$

(61)

which implies that $\mathcal{A}$ is constant along the vector fields $\zeta$ and $\gamma$. Then, from (9), we obtain

$$p^{\Vert }=\rho +constant.$$

Thus, we have established the following theorems:

Theorem 13.

Let $(M^4,g)$ be an anisotropic fluid spacetime with radial pressure $p^{\Vert }$ and energy density ρ, and under the condition that vector fields ζ and γ are parallel. If an anisotropic fluid spacetime is $\widehat{\mathbb{R}}$-harmonic, then the scalar function $\mathcal{A}$ is constant along the vector fields ζ and γ.

Theorem 14.

If an anisotropic fluid spacetime with radial pressure $p^{\Vert }$ and energy density ρ is $\mathbb{R}$-harmonic, then the equation of state is $p^{\Vert }=\rho +constant.$ In particular, if the constant vanishes, then the $\widehat{\mathbb{R}}$-harmonic anisotropic fluid spacetime represents a radiation epoch.

Assume that anisotropic fluid spacetime $(M^4,g)$ has a cyclic parallel Ricci tensor [26]. Then, we have

$$\left({\nabla }_a\mathfrak{S}\right)(b,c)+\left({\nabla }_c\mathfrak{S}\right)(a,c)+\left({\nabla }_c\mathfrak{S}\right)(a,b)=0.$$

(62)

If an anisotropic fluid spacetime has parallel vector fields $\zeta$ and $\gamma$, then (52) and (62) give us

$$0={\displaystyle \frac{d\mathcal{A}}{da}}[g(b,c)+\eta \left(b\right)\eta \left(c\right)+\mathrm{{\rm Y}}\left(b\right)\mathrm{{\rm Y}}\left(c\right)]$$

(63)

$$+{\displaystyle \frac{d\mathcal{A}}{dc}}[g(a,b)+\eta \left(a\right)\eta \left(b\right)+\mathrm{{\rm Y}}\left(a\right)\mathrm{{\rm Y}}\left(b\right)].$$

$$+{\displaystyle \frac{d\mathcal{A}}{db}}[g(a,c)+\eta \left(a\right)\eta \left(c\right)+\mathrm{{\rm Y}}\left(a\right)\mathrm{{\rm Y}}\left(c\right)].$$

Putting $a=\zeta$ in (63), we find

$${\displaystyle \frac{d\mathcal{A}}{d\zeta }}=0.$$

(64)

Putting $a=\gamma$ in (63), we turn

$${\displaystyle \frac{d\mathcal{A}}{d\gamma }}=0,$$

(65)

which implies that $\mathcal{A}$ is constant along the parallel vector fields $\zeta$ and $\gamma$. Then, from (9), we obtain

$$p^{\Vert }=\rho +constant.$$

Consequently, the following theorems have been proven:

Theorem 15.

Let $(M^4,g)$ be an anisotropic fluid spacetime with radial pressure $p^{\Vert }$ and energy density ρ and be under the condition that vector fields ζ and γ are parallel. If the anisotropic fluid spacetime has cyclic parallel Ricci tensor, then the scalar function $\mathcal{A}$ is constant along the vector fields ζ and γ.

Theorem 16.

If an anisotropic fluid spacetime with radial pressure $p^{\Vert }$ and energy density ρ has cyclic parallel Ricci tensor, then the equation of state is $p^{\Vert }=\rho +constant.$ In particular, if the constant vanishes, then the anisotropic fluid spacetime with cyclic parallel Ricci tensor represents a radiation epoch.

Remark 1.

The study of self-gravitating systems in general relativity frequently begins with the assumption of an ideal fluid (isotropic case), in which the tangential pressure $p^{\perp }$ and radial pressure $p^{\Vert }$ are equal. This is expanded upon by an anisotropic fluid with condition $p^{\perp }\ne p^{\Vert }.$ Moreover, in view of (5), an anisotropic spacetime produces distinct, physically significant conclusions, particularly in compact stellar objects like neutron stars and quark stars due to the emergence of an anisotropy factor Δ. If $\Delta >0$, an anisotropic spacetime acts as an outward, repulsive force, assisting gravity against collapse. On other hand, isotropic spacetime appears to be less compact. Isotropic configurations may become unstable sooner during collapse.

The most significant conclusion is that anisotropic models are generally more compact and stable at higher densities, magnetic fields, super-fluidity, than isotropic models, allowing for the theoretical existence of more massive stars.

8. Vanishing Space-Matter Tensor in Anisotropic Fluid Spacetime

This section focuses on the vanishing space-matter tensor in anisotropic fluid spacetime. The principal findings are based on the details that were presented in this part.

Generalized quasi-constant curvature is a generalization of the quasi-constant curvature idea introduced by [7] as follows:

$$\mathfrak{C}(\alpha ,\beta ,\delta ,\lambda )=\mathcal{E}\left[g\right(\beta ,\delta \left)g\right(\alpha ,\lambda )-g(\alpha ,\delta \left)g\right(\beta ,\lambda \left)\right]$$

$$+\mathcal{F}\left[g\right(\alpha ,\lambda \left)\eta \right(\beta \left)\eta \right(\delta )-g(\beta ,\delta \left)\eta \right(\alpha \left)\eta \right(\lambda \left)\right],$$

$$+g(\beta ,\delta )\eta \left(\alpha \right)\eta \left(\lambda \right)-g(\beta ,\lambda )\eta \left(\alpha \right)\eta \left(\delta \right)]$$

(66)

$$+\mathcal{G}\left[g\right(\alpha ,\delta \left)\psi \right(\beta \left)\psi \right(\delta )-g(\beta ,\lambda \left)\psi \right(\alpha \left)\psi \right(\delta )$$

$$+g(\beta ,\delta )\psi \left(\alpha \right)\psi \left(\lambda \right)-g(\alpha ,\delta )\psi \left(\beta \right)\psi \left(\lambda \right)],$$

where vector fields $\alpha ,\beta ,\delta ,\lambda \in \chi \left(M^4\right)$, $\psi$ and $\eta$ are 1-forms, and $\mathcal{E}$, $\mathcal{F}$, and $\mathcal{G}$ are scalars.

Let the vector fields $\gamma$ and $\zeta$ be orthogonal.

In 1969, Petrov [33] generated the fourth-rank tensor $\widehat{\mathbb{P}}$, which is defined as follows:

$$\widehat{\mathbb{P}}={\displaystyle \frac{\kappa }{2}}g\wedge \mathcal{T}-\sigma \widehat{\mathbb{G}}+\widehat{\mathbb{R}},$$

(67)

where the stress–energy tensor of anisotropic fluid spacetime is represented by $\mathcal{T}$. The gravitational constant is denoted by $\kappa$, the energy density by $\sigma$, and the Riemannian curvature tensor by $\widehat{\mathbb{R}}$. ∧ denotes the Kulkarni–Nomizu product for all vector fields $\alpha ,\beta ,\delta ,\lambda \in \chi \left(M^4\right)$ (for more details, see [34,35]).

The $(0,4)$ tensor $\widehat{\mathbb{G}}$ can be expressed as

$$\widehat{\mathbb{G}}(\alpha ,\beta ,\delta ,\lambda )=g(\beta ,\delta )g(\alpha ,\lambda )-g(\alpha ,\delta )g(\delta ,\lambda ).$$

(68)

$\widehat{\mathbb{P}}$ is referred to as a space-matter tensor. The second element of this tensor represents the distribution and velocity of matter, whereas the first part reflects the curvature of the anisotropic fluid spacetime [34,35].

We may now represent (67) as

$$\widehat{\mathbb{P}}(\alpha ,\beta ,\delta ,\lambda )=\widehat{\mathbb{R}}(\alpha ,\beta ,\delta ,\lambda )+{\displaystyle \frac{\kappa }{2}}[g(\beta ,\delta )\mathcal{T}(\alpha ,\lambda )+g(\alpha ,\lambda )\mathcal{T}(\beta ,\delta )$$

(69)

$$-g(\alpha ,\delta )\mathcal{T}(\beta ,\lambda )-g(\beta ,\lambda )\mathcal{T}(\alpha ,\delta )]$$

$$-\sigma \left[g\right(\beta ,\delta \left)g\right(\alpha ,\lambda )-g(\alpha ,\delta \left)g\right(\beta ,\lambda \left)\right].$$

If $\widehat{\mathbb{P}}(\alpha ,\beta ,\delta ,\lambda )=0$, then (69) is represented as

$$\widehat{\mathbb{R}}(\alpha ,\beta ,\delta ,\lambda )=-{\displaystyle \frac{\kappa }{2}}[g(\beta ,\delta )\mathcal{T}(\alpha ,\lambda )+g(\beta ,\lambda )\mathcal{T}(\beta ,\delta )$$

(70)

$$-g(\alpha ,\delta )\mathcal{T}(\beta ,\delta )-g(\beta ,\lambda )\mathcal{T}(\alpha ,\delta )]$$

$$+\sigma \left[g\right(\beta ,\delta \left)g\right(\alpha ,\lambda )-g(\alpha ,\delta \left)g\right(\beta ,\lambda \left)\right].$$

Combining the definition of stress–energy tensor of anisotropic fluid spacetime (3) with (70) yields

$$\widehat{\mathbb{R}}(\alpha ,\beta ,\delta ,\lambda )=\mathcal{E}[g(\beta ,\delta )g(\alpha ,\lambda )-g(\alpha ,\delta )g(\beta ,\lambda )]$$

(71)

$$+\mathcal{F}\left[g\right(\beta ,\delta \left)\eta \right(\alpha \left)\eta \right(\lambda )+g(\alpha ,\lambda \left)\eta \right(\beta \left)\eta \right(\delta )$$

$$-g(\alpha ,\delta )\eta \left(\beta \right)\eta \left(\lambda \right)-g(\alpha ,\delta )\eta \left(\beta \right)\eta \left(\delta \right)]$$

$$+\mathcal{G}\left[g\right(\alpha ,\lambda \left)\mathrm{{\rm Y}}\right(\beta \left)\mathrm{{\rm Y}}\right(\delta )-g(\beta ,\lambda \left)\mathrm{{\rm Y}}\right(\alpha \left)\mathrm{{\rm Y}}\right(\delta )$$

$$+g(\beta ,\delta )\mathrm{{\rm Y}}\left(\alpha \right)\mathrm{{\rm Y}}\left(\lambda \right)-g(\alpha ,\delta )\mathrm{{\rm Y}}\left(\beta \right)\mathrm{{\rm Y}}\left(\lambda \right)].$$

In this instance, $\mathcal{E}=\sigma$, $\mathcal{F}=-{\displaystyle \frac{\kappa (\rho +p^{\perp })}{2}}$, and $\mathcal{G}=-{\displaystyle \frac{\kappa (p^{\perp }-p^{\Vert })}{2}}$. According to (71), the anisotropic fluid spacetime under consideration has generalized quasi-constant curvature based on (66). Thus, the following can be expressed.

Theorem 17.

Let an anisotropic fluid spacetime be coupled with stress–energy tensor $\mathcal{T}$ obeying $EGFE$ with a radial pressure $p^{\Vert }$, transverse pressure $p^{\perp }$, as well as with the diminishing space-matter tensor $\widehat{\mathbb{P}}$, then the anisotropic fluid spacetime is of generalized quasi-constant curvature.

9. Anisotropic Fluid Spacetime Attached with Divergence-Free Space-Matter Tensor

The criteria needed for a space-matter tensor to be divergence-free in an anisotropic fluid spacetime are examined in this section [35].

If the scalars $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are constant in an anisotropic fluid spacetime, then given (8), we get

$$\mathcal{R}=4\mathcal{A}-\mathcal{B}-\mathcal{C}.$$

(72)

This means that the scalar curvature, $\mathcal{R}$, is constant. So, $d\mathcal{R}=0$. Using (71), we may conclude from (67) that

$$(div\widehat{\mathbb{P}})(\alpha ,\beta ,\delta )=(div\widehat{\mathbb{R}})(\alpha ,\beta ,\delta )+{\displaystyle \frac{1}{2}}[\left({\nabla }_a\mathfrak{S}\right)(\alpha ,\delta )-\left({\nabla }_b\mathfrak{S}\right)(\alpha ,\delta )]$$

(73)

$$-g(\beta ,\delta )[d\sigma \left(\alpha \right)+{\displaystyle \frac{1}{4}}d\mathcal{R}\left(\alpha \right)]+g(\alpha ,\delta )[d\sigma \left(\beta \right)+{\displaystyle \frac{1}{4}}d\mathcal{R}\left(\beta \right)].$$

We know that in a spacetime Lorentzian manifold

$$(div\widehat{\mathbb{R}})(\alpha ,\beta ,\delta )=\left({\nabla }_{\alpha }\mathfrak{S}\right)(\beta ,\delta )-\left({\nabla }_{\beta }\mathfrak{S}\right)(\alpha ,\delta ).$$

(74)

Given (71), we have $\left({\nabla }_{\alpha }\mathfrak{S}\right)(\beta ,\delta )=\kappa \left({\nabla }_{\alpha }\mathcal{T}\right)(\beta ,\delta ).$

By using (73) and (74), we obtain

$$(div\widehat{P})(\alpha ,\beta ,\delta )={\displaystyle \frac{3}{2}}[\left({\nabla }_{\alpha }\mathfrak{S}\right)(\beta ,\delta )-\left({\nabla }_{\beta }\mathfrak{S}\right)(\alpha ,\delta )]$$

(75)

$$-g(\beta ,\delta )[d\sigma \left(\alpha \right)+{\displaystyle \frac{1}{4}}d\mathcal{R}\left(\alpha \right)]+g(\alpha ,\delta )[d\sigma \left(\beta \right)+{\displaystyle \frac{1}{4}}d\mathcal{R}\left(\beta \right)].$$

Given that $(div\widehat{\mathbb{P}})(\alpha ,\beta ,\delta )=0$, and that (75) is contracted over $\alpha$ and $\delta$, we find

$$d\sigma \left(\beta \right)=0.$$

(76)

As a consequence, we can assert the following conclusion:

Theorem 18.

In an anisotropic fluid spacetime coupled with stress–energy tensor $\mathcal{T}$ obeying $EGFE$ with a radial pressure $p^{\Vert }$ and transverse pressure $p^{\perp }$, the energy density ϱ is constant when the space-matter tensor is divergence-free.

10. A Concise Summary Table for Results

The universe’s physical state is abbreviated as the equation of state parameter. As a description of actual cosmic phases, it indicates whether the cosmos is filled with hot radiation, cold matter, or vacuum energy [4].

Table 1. The alignment of the Universes with curvature symmetry of an anisotropic fluid spacetime.

Curvature Symmetry in an anisotropic fluid spacetime	Equations of state or Geometric properties $p^{\perp }={\omega }_2\rho ,p^{\Vert }={\omega }_1\rho$	Evolution of the Universe
Ricci semi-symmetric	$p^{\Vert }=\rho$	Stiff matter era
Pseudo-Ricci Symmetric	$3p^{\Vert }-\rho =0$	Radiation era
$\mathbb{R}$-harmonic and Ricci-recurrent	$p^{\Vert }=\rho +Constant$	Radiation era
Cyclic Ricci parallel	$p^{\Vert }=-\rho$	Dark matter era
Godazzi type Ricci tensor	$p^{\perp }=-\rho$	Dark matter era

shows the relationship between the classification of matter phases in the universe and curvature symmetries, particularly when relying on Ricci semi-symmetric and pseudo-Ricci symmetric anisotropic fluid spacetimes:

Using vanishing vorticity to explain the dark matter/dark energy period is frequently linked to models in which dark phenomena are expressions of non-zero torsion, rotational dynamics, or cosmic shear. For the inflation ${\omega }_1={\omega }_2=-1$, the equation of state solves the flatness problem because the energy density $\rho$ remains constant during the expansion of the universe [25].

Therefore, Theorems 8 and 15 infer that an anisotropic fluid spacetime attached with Codazzi type of stress–energy and with cyclic Ricci parallel tensor recover the dark matter era or inflation.

• Physical Interpretation:

Because imposed equations of state in cosmology represent distinct thermodynamic states of matter or energy dominating different epochs rather than merely mathematical fitting parameters, they should be interpreted as realistic physical phases of the universe. The equation of state in contemporary physics, more especially in the context of general relativity and thermodynamics, cosmic fluid typically refers to a connection between the energy density $\rho$ and the pressures $p^{\perp }$ and $p^{\Vert }$ of the stuff that fills spacetime. But when discussing the equation of state of spacetime itself, we frequently delve into the thermodynamics of spacetime [20].

1. The deformation of surfaces caused by energy density deficits is known as curvature.

2. The Einstein field equations are interpreted by the equation of state of spacetime, which is frequently linked to it, as a thermodynamical constitutive connection rather than as fundamental laws of motion [20]. It implies that microscopic, statistical interactions of quantum spacetime degrees of freedom give rise to the emergent macroscopic behavior of gravity.

3. According to the equation of state, geometry responds to pressure and energy density, such as a thermodynamic fluid [20] in a state of local thermal equilibrium in spacetime.

• Dynamical aspects of cosmological epoch in $f\left(\mathcal{R}\right)$-gravity

Modern physical cosmology is based on dynamic and empirical evidence for cosmological models across several epochs, especially with reference to the shift from matter-dominated to dark energy-dominated eras. For a dynamical system analysis, researchers use phase space and stability tests (specifically analyzing fixed points and their stability) to study the evolution of cosmological models, such as $f\left(\mathcal{R}\right)$-gravity to determine how they behave across different epochs (e.g., radiation, matter, and dark energy-dominated phases).

It has been found that $GTR$ cannot explain the acceleration of the early and late universes without accounting for dark energy. Gravitation is not properly explained by $GTR$; hence, it is feasible to tweak it to produce theories that accept inflation and mimic the Dark Energy$\left(DE\right)$. Late-time expansion is frequently described as an “attractor” in these dynamical systems, meaning the universe naturally evolves toward an accelerated state.

By applying Einstein–Hilbert Lagrangian density to a function $f\left(R\right)$, $GTR$ can be extended to the $f\left(\mathcal{R}\right)$-gravity, where $\mathcal{R}$ is the Ricci scalar. For examples, see [36,37]. Higher-order curvature resolves the problem of massive neutron stars in the equations of motion of $f\left(\mathcal{R}\right)$-gravity. Dynamical systems analysis confirms that specific modified gravity models like $f\left(\mathcal{R}\right)$-gravity can match the observed, accelerated expansion.

Buchdahl first proposed $f\left(\mathcal{R}\right)$-gravity in 1970 [38]. Using the Einstein–Hilbert action term,

$${\mathcal{E}}_{\mathcal{H}}={\displaystyle \frac{1}{16\pi }}\int [L_m+f\left(\mathcal{R}\right)]\sqrt{(-g)}{d}^{4}x,$$

(77)

where $L_m$ is the scalar space field’s matter Lagrangian. The formula for the matter’s stress–energy tensor is

$${\mathcal{T}}_{\mu \nu }={\displaystyle \frac{-2\delta \left(\sqrt{-g}\right)L_m}{\sqrt{-g}{\delta }^{\mu \nu }}}.$$

(78)

Consider that $L_m$ is exclusively dependent on $g_{\mu \nu }$ and not on its derivatives. The action variation (77) with respect to the $g_{\mu \nu }$ implies

$$f^{\prime }\left(\mathcal{R}\right){\mathfrak{S}}_{\mu \nu }-{\displaystyle \frac{1}{2}}f\left(\mathcal{R}\right)g_{\mu \nu }-({\nabla }_{\mu }{\nabla }_{\nu }-g_{\mu \nu }{\nabla }_{\mu }{\nabla }^{\mu })f^{\prime }\left(\mathcal{R}\right)=\kappa {\mathcal{T}}_{\mu \nu }.$$

(79)

wherein $f^{\prime }\left(\mathcal{R}\right)={\displaystyle \frac{\partial f\left(\mathcal{R}\right)}{\partial \mathcal{R}}}$ and, $\square \equiv {\nabla }_{\mu }{\nabla }^{\mu }$ and ${\nabla }_{\mu }$ symbolize the d’Alembertion and the covariant derivative, respectively [23]. Changing $f\left(\mathcal{R}\right)$ by $\mathcal{R}$ can weaken the relation (79). When $\mathcal{R}$ is set to constant, relation (79) becomes

$${\mathfrak{S}}_{\mu \nu }-{\displaystyle \frac{\mathcal{R}}{2}}{g}_{\mu \nu }={\displaystyle \frac{\kappa }{f^{\prime }\left(\mathcal{R}\right)}}{\mathcal{T}}_{\mu \nu }^{\left(Eff\right)},$$

(80)

where ${\mathcal{T}}_{\mu \nu }^{\left(Eff\right)}$ is an effective stress–energy tensor such that

$${\mathcal{T}}_{\mu \nu }^{\left(Eff\right)}={\mathcal{T}}_{\mu \nu }-{\displaystyle \frac{f\left(\mathcal{R}\right)-\mathcal{R}{f}^{\prime }\left(\mathcal{R}\right)}{2\kappa }}{g}_{\mu \nu }.$$

(81)

Thus, in view of (3), (81), and (80), we obtain the Ricci tensor of an anisotropic fluid spacetime that satisfies $f\left(\mathcal{R}\right)$-gravity in terms of index-free coordinates

$$\mathcal{S}(a,b)=\alpha g(a,b)+\beta \eta \left(a\right)\eta \left(b\right)+\delta \mathrm{{\rm Y}}\left(a\right)\mathrm{{\rm Y}}\left(b\right),$$

(82)

where $\alpha ={\displaystyle \frac{f\left(\mathcal{R}\right)-\mathcal{R}{f}^{\prime }\left(\mathcal{R}\right)}{2\kappa }}-{\displaystyle \frac{\mathcal{R}}{2}},$ $\beta ={\displaystyle \frac{\kappa (p^{\Vert }+\rho )}{f^{\prime }\left(\mathcal{R}\right)}}$, and $\delta ={\displaystyle \frac{\kappa (p^{\Vert }-p^{\perp })}{f^{\prime }\left(\mathcal{R}\right)}}.$ Putting $a=b=\zeta$ in (82), we obtain

$$p^{\Vert }+\rho ={\displaystyle \frac{\mathcal{R}{f}^{\prime }\left(\mathcal{R}\right)}{2\kappa }}\left\{{\displaystyle \frac{f^{\prime }\left(\mathcal{R}\right)}{\kappa }}+1\right\}-{\displaystyle \frac{f^{\prime }\left(\mathcal{R}\right)f\left(\mathcal{R}\right)}{2{\kappa }^2}}.$$

(83)

Consequently, we can state the following results:

Theorem 19.

The equation of motion for an anisotropic fluid spacetime in $f\left(\mathcal{R}\right)$-gravity is given by (80).

Theorem 20.

The equation of state of an anisotropic fluid spacetime in $f\left(\mathcal{R}\right)$-gravity is given by (83).

Also, (82) infers the following:

Theorem 21.

An anisotropic fluid spacetime in $f\left(\mathcal{R}\right)$-gravity is a generalized quasi-Einstein spacetime.

Remark 2.

The equation of state $\left(\mathcal{E}o\mathcal{S}\right)$ for the dark matter era is defined by $p^{\perp }=-\rho +f\left(t\right)$, wherein $f\left(t\right)$ is a function of the scale factor t with the cosmic time t. The eras of dark matter, stiff matter, radiation, and dust matter are defined by $\mathcal{E}o\mathcal{S}$$\rho =-p$, $\rho =p$, $p={\displaystyle \frac{\rho }{3}}$, and $p=0$ [39].

Now, in view of Theorem 20 and Remark 2, we gain the following corollaries:

Corollary 1.

If the source of an anisotropic fluid spacetime is dark matter in $f\left(\mathcal{R}\right)$-gravity, then the scalar curvature is

$$\mathcal{R}={\displaystyle \frac{f\left(\mathcal{R}\right)}{2\kappa [f^{\prime }\left(\mathcal{R}\right)+1]}}.$$

(84)

Corollary 2.

If the source of an anisotropic fluid spacetime is stiff matter in $f\left(\mathcal{R}\right)$-gravity, then the radial pressure and the density are

$$p^{\Vert }=\rho ={\displaystyle \frac{\mathcal{R}{f}^{\prime }\left(\mathcal{R}\right)}{4\kappa }}\left\{{\displaystyle \frac{f^{\prime }\left(\mathcal{R}\right)}{\kappa }}+1\right\}-{\displaystyle \frac{f^{\prime }\left(\mathcal{R}\right)f\left(\mathcal{R}\right)}{4{\kappa }^2}}.$$

(85)

Corollary 3.

If the source of an anisotropic fluid spacetime is radiation matter in $f\left(\mathcal{R}\right)$-gravity, then the radial pressure and the density are

$$p^{\Vert }={\displaystyle \frac{\mathcal{R}{f}^{\prime }\left(\mathcal{R}\right)}{8\kappa }}\left\{{\displaystyle \frac{f^{\prime }\left(\mathcal{R}\right)}{\kappa }}+1\right\}-{\displaystyle \frac{f^{\prime }\left(\mathcal{R}\right)f\left(\mathcal{R}\right)}{8{\kappa }^2}},and$$

(86)

$$\rho ={\displaystyle \frac{3\mathcal{R}{f}^{\prime }\left(\mathcal{R}\right)}{8\kappa }}\left\{{\displaystyle \frac{f^{\prime }\left(\mathcal{R}\right)}{\kappa }}+1\right\}-{\displaystyle \frac{3f^{\prime }\left(\mathcal{R}\right)f\left(\mathcal{R}\right)}{8{\kappa }^2}}.$$

(87)

11. Conclusions

This research provides a detailed geometric characterization of two-fluid models within the framework of general relativity, specifically focusing on anisotropic fluid spacetimes. By rigorously analyzing the Einstein field equations coupled with an anisotropic stress–energy tensor, we have established the following key insights:

• Geometric Equivalence: Anisotropic fluid spacetimes are mathematically isomorphic to generalized quasi-Einstein manifolds. This equivalence allows for the application of extensive results from Riemannian geometry to astrophysical fluid dynamics.
• Cosmological Epochs via Symmetry: The imposition of geometric symmetries on the stress–energy tensor ($\nabla \mathcal{T}=0$, recurrence, Codazzi type, semi-symmetry) naturally selects specific equations of state corresponding to known cosmological eras:
  Radiation Era ($\mathit{p}={\displaystyle \frac{\mathit{\rho }}{\mathbf{3}}}$): Associated with covariant constant tensors and Ricci recurrence.
  Dark Matter/Energy ($\mathit{p}=-\mathit{\rho }$): Associated with vanishing vorticity and Codazzi symmetries.
  Stiff Matter ($\mathit{p}=\mathit{\rho }$): Enforced by Ricci semi-symmetry.
• Structure of Spacetime: The conditions of a Codazzi stress–energy tensor or a vanishing space-matter tensor constrain the spacetime to be of Robertson–Walker type or generalized quasi-constant curvature, respectively. This highlights that significant deviations from isotropy or homogeneity in the universe require the breaking of these high-level geometric symmetries.

In 1996, perfect fluid spacetime with a covariant constant energy–momentum tensor were examined in [13]. After that, in [14], authors studied perfect fluid spacetime with a Killing energy–momentum tensor. Then, De and De [16] used semi-symmetric energy–momentum tensors to study spacetimes. Mallick et al. [15] used several sorts of energy–momentum tensors to explore perfect fluid spacetime. In 2004, Fernández et al. [40] explored the stress–energy–momentum tensors for natural constrained variational problems. These studies only discussed within framework of perfect fluid spacetime and the energy–momentum tensor of perfect fluid is of the form [4]

$$\mathcal{T}=pg+(p+\rho )\eta \otimes \eta ,$$

which is completely different and new from the stress–energy tensor $\mathcal{T}$ of an anisotropic fluid (3), examined in this study. Shows a clearer comparison with existing studies.

The synthesis of these results demonstrates that the classification of matter phases in the universe is deeply intertwined with the curvature symmetries of the spacetime manifold. Future research may extend these findings to modified theories of gravity [25] or higher-dimensional spacetimes, using the generalized quasi-Einstein framework as a robust foundation. The present study consists of some new classification results obtained under very strong geometric assumptions, such as covariant constancy, Codazzi-type conditions, Ricci recurrence, Ricci semi-symmetry, parallel vector fields, or vanishing space-matter tensors. Under these assumptions, the resulting equations of state (radiation, stiff matter, or dark matter-like behavior) are largely expected and well documented in the existing literature on quasi-Einstein and generalized quasi-Einstein spacetime. Moreover, the existence of $\widehat{\mathbb{R}}$-harmonic anisotropic fluid spacetime indicates that a radiation epoch has been proven. Lastly, investigated the dynamical features of the cosmic phase of an anisotropic fluid spacetime related to $f\left(\mathcal{R}\right)$-gravity and deduced the equation of motion and equation of sate for an anisotropic fluid spacetime in $f\left(\mathcal{R}\right)$-gravity.