F($\mathfrak{R},\mathfrak{T}$)-Gravity with Anisotropic Fluid Admitting Hyperbolic Ricci Solitons with Torse-Forming Vector Field

Abstract

This study is dedicated to a separable $F(\mathfrak{R},\mathfrak{T})$-gravity related to the anisotropic matter to extract the equation of state for $F(\mathfrak{R},\mathfrak{T})$-gravity. In this research, we offer insight into calculating the density and pressure in the phantom barrier, stiff fluid, and matter-dominated eras, respectively. As demonstrated, a spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity full of anisotropic matter is a generalized quasi-Einstein spacetime. In addition, we gain the equation of state of Codazzi type, Ricci semi-symmetric and Ricci-pseudo symmetric anisotropic fluid spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity. We prove an anisotropic spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity endowed with Codazzi-type Ricci tensor is a Yang Pure spacetime and Robertson–Walker spacetime. Furthermore, we try to give out the energy constraints of Penrose’s singularity theorem for black holes in an anisotropic fluid spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity. Lastly, we study hyperbolic Ricci solitons on anisotropic fluid spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity endowed with a torse-forming vector field, and for steady hyperbolic Ricci soliton, we deduced the equation of state of anisotropic fluid spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity.

1. Introduction

A well-known instance of symmetry is the soliton, which is connected to the geometric motion of spacetime. One of the key principles of mathematical physics and GTR is the study of solitons and their symmetry in spacetime. These confined, uniform solutions are essential for explaining the universe’s large-scale structure as well as elementary particles. Moreover, solitons are frequently examined as geometric entities associated with the evolution of spacetime metrics in general relativity [1].

The concept of hyperbolic geometric flow was first presented in 2010 by Dai et al. [2]

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{{ð}^2}{ðt^2}}\mathfrak{g}\left(t\right)=-\mathcal{S}\left(t\right)\mathfrak{g}\left(t\right),{\mathfrak{g}}_0=\mathfrak{g}\left(0\right),{\displaystyle \frac{ð}{ðt}}\mathfrak{g}\left(t\right)={\mathfrak{h}}_{ij},$$

(1)

wherein ${\mathfrak{h}}_{ij}$ is a symmetric (0, 2)-tensor field.

The theories of hyperbolic Ricci solitons (HRS) and gradient hyperbolic Ricci solitons (GHRS) have been put forward by Faraji et al. [3]:

$${\displaystyle \frac{1}{2}}{\mathfrak{L}}_{\gamma }{\mathfrak{L}}_{\gamma }\mathfrak{g}+⋋{\mathfrak{L}}_{\gamma }\mathfrak{g}+\mathcal{S}=\Theta \mathfrak{g},$$

(2)

wherein the Ricci curvature of M is $\mathcal{S}$. In (2), ⋋ and $\Theta$ denote classifications of solitons and the core type rate, respectively. Additionally, $\Theta$ indicates the rate of change of solitons and has geometrical importance. Based on the constant $\Theta$, the rate of change of the $HRS$ can be rising, steady, or decreasing regardless of whether $\Theta <0$, $\Theta >0$, or $\Theta =0$.

The $HRS$ turn into $GHRS$ [3] if there is a potential function F on Riemannian manifold M such that $\gamma =\nabla f$. Thus, (2) can be express as

$${\mathfrak{L}}_{\nabla F}\left(\mathbb{H}essf\right)+2⋋\mathbb{H}essf+\mathcal{S}=\Theta \mathfrak{g},$$

(3)

Here, the hessian of the smooth function f is indicated by $\mathbb{H}essf$.

Adara [4] used $\eta$-Ricci and $\eta$-Einstein solitons to illustrate the properties of the spacetime. In [5], Venkatesha and Kumara also discussed Ricci solitons on perfect fluid spacetime. Several researchers have also used hyperbolic Ricci solitons to study different spacetimes (for additional information, see [6,7,8]). The concept of hyperbolic Ricci solitons has lately been studied in various ways by Blaga and Özgür (see [9,10,11] for further details).

The standard method for interpreting known cosmic dynamics is presented by Einstein’s gravitational field equations [12,13]. Einstein’s field equation yields the greatest approximation to the observed data when a hypothetical component of the cosmos called Dark Matter $\left(DM\right)$ [14] is included [15]. Likewise, the universe contains an unconventional component called Dark Energy $\left(DE\right)$, which is thought to be the primary cause of the universe’s expansion and regulates the matter-to-energy ratio in $GTR$.

By applying the Einstein–Hilbert Lagrangian density to a function $F\left(\mathfrak{R}\right)$, $GTR$ can be extended to the $F\left(\mathfrak{R}\right)$-gravity, where $\mathfrak{R}$ is the Ricci scalar [16,17,18]. Nevertheless, the $F\left(\mathfrak{R}\right)$-gravity has a number of limitations on the solar system’s stability and cannot accommodate other models (see [19,20,21] for more information). The $F(\mathfrak{R},\mathfrak{T})$-gravity theory was introduced in [22] as a more comprehensive gravity by taking into account that the Lagrangian is a function of $\mathfrak{T}$ and $\mathfrak{R}$. Here, the energy–momentum tensor $\mathfrak{T}$ is denoted by its trace. This concept was used to effectively explain the universe’s late-time fast expansion (for more details, see [23,24].

A spacetime can be conceived as a 4-dimensional time-driven Lorentzian manifold (${\mathcal{M}}^4,\mathfrak{g})$ as a type of semi-Riemannian manifold with Lorentzian metric g. It is believed that scalar fields are essential to physics and cosmology in $F(\mathfrak{R},\mathfrak{T})$-gravity theory. In [25], the authors reconstructed the exponential models of $F(\mathfrak{R},\mathfrak{T})$-gravity. $F(\mathfrak{R},\mathfrak{T})$-gravity was studied by Chaubey [6], who presented some of their findings. Some characteristics of the cosmic perfect fluid in $\mathfrak{f}\left(\mathfrak{R}\right)$ gravity were examined by Capozziello et al. in [26,27].

Because $F(\mathfrak{R},\mathfrak{T})$-gravity can handle a variety of cosmological and astrophysical problems, the geometric approach in $F(\mathfrak{R},\mathfrak{T})$-gravity of gravity has drawn a lot of interest from cosmologists and astrophysicists with various approaches and models (for more details, see [28,29,30,31]). Danish et al. have also examined $F(\mathfrak{R},\mathfrak{T})$-gravity attached with perfect fluid in [32,33,34]. Siddiqi et al. examined $F(\mathfrak{R},{\mathfrak{T}}^{\mathfrak{2}})$ gravity associated with hyperbolic Ricci solitons in [35,36].

Therefore, the aforementioned research provides sufficient motivation to explore $AFST$ in $F(\mathfrak{R},\mathfrak{T})$ in terms of $HRS.$

A central object of study within this geometric framework is the stress–energy tensor, which encapsulates the flux and density of energy and momentum. A perfect fluid is physically characterized by the absence of shear stresses, viscosity, and heat conduction and is geometrically defined by an isotropic pressure $\mathfrak{p}$ and an energy density $\varrho$ measured in the rest frame of the fluid. The stress energy tensor $\mathbb{T}$ for such a fluid takes the following form [15]:

$$\mathfrak{T}=\mathfrak{p}\mathfrak{g}+(\varrho +\mathfrak{p})\eta \otimes \eta ,$$

(4)

where $\eta$ is the four-velocity vector field of the fluid.

A perfect fluid is described by the lack of viscosity and heat conduction, representing an isotropic fluid or a star-shaped fluid in its rest frame. The simplest realization of a perfect fluid is the “dust matter fluid,” characterized by vanishing pressure $\mathfrak{p}=0$ ($\omega =0)$ [15]. In GTR, perfect fluids are extensively utilized to model idealized matter distributions. Another critical equation of state is the “stiff matter fluid,” defined by the relation $\mathfrak{p}=\varrho$ ($\omega =1)$ [37]. Furthermore, if the energy density satisfies $\varrho =3\mathfrak{p}$ ($\omega ={\displaystyle \frac{1}{3}})$ [15], the matter content corresponds to a radiation fluid. These equations of state are pivotal in the study of stellar structure and early universe cosmology [38].

Definition 1 

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

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

(5)

wherein $\mathfrak{g}$ 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 $\eta \left(\zeta \right)=-1$.

Definition 2 

([39,40]). A non-flat Riemannain manifold $({\mathcal{M}}^n,\mathfrak{g})$$(n>2)$ is designated as a generalized quasi-Einstein manifold $\left(GQE\right)$ if its Ricci tensor $\mathcal{S}$ holds the relation:

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

(6)

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

$$\mathfrak{g}(\gamma ,\zeta )=\eta \left(\gamma \right),\eta \left(\zeta \right)=\mathfrak{g}(\zeta ,\zeta )=-1,$$

(7)

$$\mathfrak{g}(\gamma ,\gamma )=\sigma \left(\gamma \right),\sigma \left(\gamma \right)=g(\gamma ,\gamma )=1,\mathfrak{g}(\zeta ,\gamma )=0$$

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

Definition 3 

([41]). A vector field ζ on a semi-Riemannian manifold $(M^n,\mathfrak{g}$) is termed a torse-forming type vector field (TFVF) if it fulfills the subsequent relation:

$${\widehat{\nabla }}_{{\gamma }_1}\zeta =\phi {\gamma }_1+\eta \left({\gamma }_1\right)\zeta ,$$

(8)

for every vector field ${\gamma }_1$ on ($(M^n,\mathfrak{g}$) where $\widehat{\nabla }$ is the Levi–Civita connection, φ is a smooth function on M, and η is a one-form. The function F is referred to as the conformal scalar [42].

2. Anisotropic Fluid Spacetime

Anisotropic fluids are characterized by physical properties, such as pressure or viscosity, that exhibit directional dependence [43]. This stands in contrast to isotropic fluids, where properties are uniform in all directions due to random molecular motion. In the context of cosmology and high-energy physics, anisotropic fluids [44] are relevant for modeling phenomena such as the fluid dynamics in heavy-ion collisions and liquid crystals with aligned molecular structures with isotropic fluid, which has constant pressure in every direction [45]. In cosmology, anisotropic fluid models are employed to describe the universe, particularly for processes where attributes vary with direction, such as dark energy evolution and large-scale structure formation. In GTR, an $AFST$ represents a manifold containing a fluid where pressure and stress are directionally dependent. This framework is applicable to the interior structure of black holes, galactic rotation curves, and the accelerating expansion of the universe. Notably, a two-fluid model comprising baryonic matter, dark energy, and their interaction can be effectively described by an anisotropic fluid [44,45]. One important example is the Bowers–Liang interior solution, which models compact stars with anisotropic pressure by extending the Schwarzchild solution.

The stress–energy tensor $\mathfrak{T}$ associated with an anisotropic fluid in the two-fluid model is given by [46,47]:

$$\mathfrak{T}=\varrho \eta \otimes \eta +{\mathfrak{p}}^{\Vert }\sigma \otimes \sigma +{\mathfrak{p}}^{\perp }[\eta \otimes \eta -\sigma \otimes \sigma ],$$

(9)

where the pressure within compact objects is decomposed into a radial pressure ${\mathfrak{p}}^{\Vert }$ and a transverse pressure ${\mathfrak{p}}^{\perp }$ perpendicular to the radial direction. Here, $\varrho$ represents the density computed by a co-moving observer $\eta$ is the 1-form corresponding to the time-like fluid velocity vector $\zeta \left(\eta \right(\zeta )=-1)$, and $\sigma$ is the 1-form corresponding to a unit space-like vector field $\gamma \left(\sigma \right(\gamma )=1)$ orthogonal to $\zeta \left(\mathfrak{g}\right(\zeta ,\gamma )=0).$

Furthermore, it is assumed that the radial and transverse pressures are proportional to the density via the equations of state:

$${\mathfrak{p}}^{\perp }={\omega }_2\varrho ,{\mathfrak{p}}^{\Vert }={\omega }_1\varrho ,$$

(10)

where ${\omega }_1$ and ${\omega }_2$ are equation of state parameters.

3. $\mathit{A}\mathit{F}\mathit{S}\mathit{T}$ in $\mathit{F}(\mathfrak{R},\mathfrak{T})$-Gravity Theory

The $AFST$ in the $F(\mathfrak{R},\mathfrak{T})$-gravity depends on the physical characteristics of the matter field; hence, there are several conjecture models that can be created for distinct values of $\mathfrak{R}$ and $\mathfrak{T}$ [22]. As an illustration, we pick

$$F(\mathfrak{R},\mathfrak{T})=\mathfrak{R}+2F\left(\mathfrak{T}\right),$$

(11)

where $F\left(\mathfrak{T}\right)$ denotes a function of $\mathfrak{T}$ only. Note that the element $2F\left(\mathfrak{T}\right)$ appears in the gravitational action between space matter and Ricci scalar $\mathfrak{R}.$

Using the Einstein–Hilbert action term as our assumption

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

(12)

wherein $L_m$ is the matter Lagrangian density and $F(\mathfrak{R},\mathfrak{T})$ is a function of Ricci scalar $\mathfrak{R}$ and trace $\mathfrak{T}={\mathfrak{g}}^{\mu \nu }{\mathfrak{T}}_{\mu \nu }$ of the energy–momentum tensor ${\mathfrak{T}}_{\mu \nu }$.

The formula for the stress–energy tensor of matter as

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

(13)

Let the metric tensor components ${\mathfrak{g}}_{\mu \nu }$ be the only factors affecting the Lagrangian density $L_m$ of matter, rather than its derivatives. We get

$${\mathfrak{T}}_{\mu \nu }={\mathfrak{g}}_{\mu \nu }-L_m-2{\displaystyle \frac{\partial L_m}{\partial {\mathfrak{g}}^{\mu \nu }}}.$$

(14)

The action variation (12) with respect to the $g_{\mu \nu }$ implies

$$\delta \sigma {\mathcal{E}}_{\mathcal{H}}={\displaystyle \frac{1}{16\pi }}\int [F_{\mathfrak{R}}\delta \mathfrak{R}+F_{\mathfrak{T}}{\displaystyle \frac{\delta \mathfrak{T}}{\delta {\mathfrak{g}}^{\mu \nu }}}-{\displaystyle \frac{1}{2}}{\mathfrak{g}}_{\mu \nu }F(\mathfrak{R},\mathfrak{T})\delta {\mathfrak{g}}^{\mu \nu }+16\pi {\displaystyle \frac{1\delta \left(\sqrt{-\mathfrak{g}}{L}_m\right)}{\sqrt{-\mathfrak{g}}\delta {\mathfrak{g}}^{\mu \nu }}}]\sqrt{(-\mathfrak{g})}{d}^{4}x,$$

(15)

Here, $F_{\mathfrak{R}}={\displaystyle \frac{\partial F(\mathfrak{R},\mathfrak{T})}{\partial \mathfrak{R}}}$ and $F_{\mathfrak{T}}={\displaystyle \frac{\partial F(\mathfrak{R},\mathfrak{T})}{\partial \mathfrak{T}}}.$

Regarding the Ricci scalar variation, we get

$$\delta \mathfrak{R}=\delta \left({\mathfrak{g}}^{\mu \nu }{\mathcal{S}}_{\mu \nu }\right)={\mathcal{S}}_{\mu \nu }\delta {\mathfrak{g}}^{\mu \nu }+{\mathfrak{g}}^{\mu \nu }({\nabla }_a\delta {\Gamma }_{\mu \nu }^a-{\nabla }_{\nu }\delta {\Gamma }_{\mu a}^a),$$

(16)

where the covariant derivative with regard to the symmetric connection $\Gamma$ connected to the metric $\mathfrak{g}$ is represented by ${\nabla }_a$. When the Christoffel symbols are varied,

$$\delta {\Gamma }_{\mu \nu }^a={\displaystyle \frac{1}{2}}{\mathfrak{g}}^{am}({\nabla }_{\mu }\delta {\mathfrak{g}}_{\nu m}+{\nabla }_{\nu }\delta {\mathfrak{g}}_{m\mu -{\nabla }_m\delta {\mathfrak{g}}_{\mu \nu }}),$$

(17)

and (16) entail that

$$\delta \mathfrak{R}={\mathcal{S}}_{\mu \nu }\delta {\mathfrak{g}}^{\mu \nu }+{\mathfrak{g}}_{\mu \nu }\square \delta {\mathfrak{g}}^{\mu \nu }-{\nabla }_{\mu }{\nabla }_{\nu }\delta {\mathfrak{g}}^{\mu \nu }.$$

(18)

Consequently, we acquire the variation in the gravitational field’s action.

$$\delta {\mathcal{E}}_{\mathcal{H}}={\displaystyle \frac{1}{16\pi }}\int [F_{\mathfrak{R}}{\mathcal{S}}_{\mu \nu }\delta {\mathfrak{g}}^{\mu \nu }+F_{\mathfrak{R}}{\mathfrak{g}}_{\mu \nu }\square \delta {\mathfrak{g}}^{\mu \nu }-F_{\mathfrak{R}}{\nabla }_{\mu }{\nabla }_{\nu }\delta {\mathfrak{g}}^{\mu \nu }$$

(19)

$$+F_{\mathfrak{T}}{\displaystyle \frac{\delta \left({\mathfrak{g}}^{mn}{\mathfrak{T}}_{mn}\right)}{\delta {\mathfrak{g}}^{\mu \nu }}}\delta {\mathfrak{g}}^{\mu \nu }-{\displaystyle \frac{1}{2}}{\mathfrak{g}}_{\mu \nu }F(\mathfrak{R},\mathfrak{T})\delta {\mathfrak{g}}^{\mu \nu }+16\pi {\displaystyle \frac{\delta \left(\sqrt{-\mathfrak{g}}{L}_m\right)}{\sqrt{-\mathfrak{g}}\delta {\mathfrak{g}}^{\mu \nu }}}]\sqrt{(-\mathfrak{g})}{d}^{4}x.$$

The variation of $\mathfrak{T}$ with respect to the metric tensor is defined as

$${\displaystyle \frac{\delta \left({\mathfrak{g}}^{mn}{\mathfrak{T}}_{mn}\right)}{\delta {\mathfrak{g}}^{\mu \nu }}}={\Psi }_{\mu \nu }+{\mathfrak{T}}_{\mu \nu },$$

(20)

where

$${\Psi }_{\mu \nu }={\mathfrak{g}}^{\mu \nu }{\displaystyle \frac{\delta {\mathfrak{T}}_{\mu \nu }}{\delta {\mathfrak{g}}^{\mu \nu }}}.$$

(21)

The field equations of the $\mathfrak{f}(\mathfrak{R},\mathfrak{T})$ gravity model are obtained by partially integrating the second and third terms in (19).

$$F_{\mathfrak{R}}(\mathfrak{R},\mathfrak{T}){\mathcal{S}}_{\mu \nu }-{\displaystyle \frac{1}{2}}F(\mathfrak{R},\mathfrak{T}){\mathfrak{g}}_{\mu \nu }+({\mathfrak{g}}_{\mu \nu }{\nabla }_{\mu }{\nabla }^{\mu }-{\nabla }_{\mu }{\nabla }_{\nu })F_{\mathfrak{R}}(\mathfrak{R},\mathfrak{T})$$

(22)

$$=8\pi {\mathfrak{T}}_{\mu \nu }-F_{\mathfrak{T}}(\mathfrak{R},\mathfrak{T}){\mathfrak{T}}_{\mu \nu }-F_{\mathfrak{T}}(\mathfrak{R},\mathfrak{T}){\Psi }_{\mu \nu },$$

wherein, $\square \equiv {\nabla }_{\mu }{\nabla }^{\mu }$ and ${\nabla }_{\mu }$ indicate the d’Alembertion and the covariant derivative, respectively. In addition, we have

$${\Psi }_{\mu \nu }=-2{\mathfrak{T}}_{\mu \nu }+{\mathfrak{g}}_{\mu \nu }{L}_m-2{\mathfrak{g}}^{lk}{\displaystyle \frac{{\partial }^2{L}_m}{\partial {\mathfrak{g}}^{\mu \nu }\partial {\mathfrak{g}}^{lk}}}.$$

(23)

Equations (12) and (13) with $F(\mathfrak{R},\mathfrak{T})=F\left(\mathfrak{R}\right)$ recover the field equations for $F\left(\mathfrak{R}\right)$-gravity.

Let the anisotropic fluid matter with total isotropic pressure be ${\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert }$, rest energy density $\varrho$, four-velocity vector ${\mu }^{\mu }$, and space-like vector ${\sigma }^{\mu }$. We have the advantage to choose $L_m$. Consequently, let $L_m=-\mathfrak{p}={\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert }$. Since the term “matter Lagrangian” has no universally accepted definition, we take into consideration

$${\mathfrak{T}}_{\mu \nu }=\varrho {\eta }_{\mu }{\eta }_{\nu }+{\mathfrak{p}}^{\Vert }{\sigma }_{\mu }{\sigma }_{\nu }+{\mathfrak{p}}^{\perp }[{\eta }_{\mu }{\eta }_{\nu }-{\sigma }_{\mu }{\sigma }_{\nu }]$$

(24)

where

$${\eta }^{\mu }{\nabla }_{\nu }{\eta }_{\mu }=0,{\eta }_{\mu }·{\sigma }^{\mu }=-{\sigma }^{\mu }·{\sigma }_{\nu }=1,{\eta }^{\mu }·{\sigma }_{\mu }=0.$$

(25)

From (23) and (24), we obtain

$${\Psi }_{\mu \nu }=({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })g_{\mu \nu }-2{\mathfrak{T}}_{\mu \nu }.$$

(26)

Generally, the physical characteristics of the matter field also affect the field equations through the tensor ${\Psi }_{\mu \nu }$. Therefore, the choice of F in (11) defines the situation of $F(\mathfrak{R},\mathfrak{T})$-gravity.

Therefore, after adopting (11) and (22), we get

$${\mathcal{S}}_{\mu \nu }={\displaystyle \frac{\mathfrak{R}}{2}}{g}_{\mu \nu }-2F^{\prime }\left(\mathfrak{T}\right){\mathfrak{T}}_{\mu \nu }-2F^{\prime }\left(\mathfrak{T}\right){\Psi }_{\mu \nu }+F\left(\mathfrak{T}\right)g_{\mu \nu }+8\pi {\mathfrak{T}}_{\mu \nu }.$$

(27)

In light of (24)–(26), Equation (27) is transformed into

$${\mathcal{S}}_{\mu \nu }=\left\{{\displaystyle \frac{1}{2}}\mathfrak{R}-2({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })F^{\prime }\left(\mathfrak{T}\right)+F\left(\mathfrak{T}\right)\right\}{g}_{\mu \nu }+\left\{({\mathfrak{p}}^{\perp }+\varrho )+2(F^{\prime }\left(\mathfrak{T}\right)+4\pi )\right\}{\eta }_{\mu }{\eta }_{\nu }$$

(28)

$$+\left\{({\mathfrak{p}}^{\Vert }-\varrho )+2(4\pi +F^{\prime }\left(\mathfrak{T}\right))\right\}{\sigma }_{\mu }{\sigma }_{\nu },$$

$$\mathfrak{R}=({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })[8F^{\prime }\left(\mathfrak{T}\right)-1]-4[F^{\prime }\left(\mathfrak{T}\right)+F^{(}\mathfrak{T})+4\pi ].$$

(29)

Thus, for the $AFST$ in $\mathfrak{f}(\mathfrak{R},\mathfrak{T})$-gravity, the Ricci tensor is

$${\mathcal{S}}_{\mu \nu }=A_1{g}_{\mu \nu }+A_2{\gamma }_{\mu }{\gamma }_{\nu }+A_3{\sigma }_{\mu }{\sigma }_{\nu },$$

(30)

wherein

$$A_1=\left\{{\displaystyle \frac{1}{2}}\mathfrak{R}-2({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })F^{\prime }\left(\mathfrak{T}\right)+F\left(\mathfrak{T}\right)\right\}{A}_2=\left\{({\mathfrak{p}}^{\perp }+\varrho )+2(F^{\prime }\left(\mathfrak{T}\right)+4\pi )\right\}$$

(31)

and

$$\begin{array}{c}\hfill A_3=\left\{({\mathfrak{p}}^{\Vert }-\varrho )+2(4\pi +F^{\prime }\left(\mathfrak{T}\right))\right\}.\end{array}$$

We assume that throughout the content $A_1,A_2$ and $A_3$ are not simultaneously zero. As a result, we can draw the following conclusions:

Theorem 1. 

The Ricci curvature tensor for the $F(\mathfrak{R},\mathfrak{T})$-gravity model with anisotropic fluid matter is of the following form:

$${\mathcal{S}}_{\mu \nu }=\left\{{\displaystyle \frac{1}{2}}\mathfrak{R}-2({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })F^{\prime }\left(\mathfrak{T}\right)+F\left(\mathfrak{T}\right)\right\}{g}_{\mu \nu }+\left\{({\mathfrak{p}}^{\perp }+\varrho )+2(F^{\prime }\left(\mathfrak{T}\right)+4\pi )\right\}{\eta }_{\mu }{\eta }_{\nu }$$

$$+\left\{({\mathfrak{p}}^{\Vert }-\varrho )+2(4\pi +F^{\prime }\left(\mathfrak{T}\right))\right\}{\sigma }_{\mu }{\sigma }_{\nu }.$$

Corollary 1. 

The scalar curvature tensor of the $F(\mathfrak{R},\mathfrak{T})$-gravity model with anisotropic fluid is

$$\mathfrak{R}=({\mathfrak{p}}^{\Vert }+\left({\mathfrak{p}}^{\perp }\right)[8F^{\prime }\left(\mathfrak{T}\right)-1]-4[F^{\prime }\left(\mathfrak{T}\right)+F\left(\mathfrak{T}\right)+4\pi ].$$

Next, Theorem 4 and Definition (2) entail the following result.

Theorem 2. 

A spacetime $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity with anisotropic fluid is a $GQE$-spacetime.

We can also state the following outcome.

Theorem 3. 

An $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity attached with anisotropic fluid matter is a $GQE$-spacetime.

Example 1. 

Let us assume a semi-Riemannian metric $\mathfrak{g}$ on ${\mathbb{L}}^4$ by

$$d{\mathfrak{s}}^2={\mathfrak{g}}_{\mu \nu }d{u}^{\mu }d{u}^{\nu }=u^2[d{\left(u^1\right)}^2+{\left(d{u}^2\right)}^2+{\left(d{u}^3\right)}^2]-{\left(d{u}^4\right)}^2,$$

(32)

where $\mu ,\nu =1,2,3,4$. The only non-vanishing components of the Christoffel symbols are the curvature tensors and the derivatives of the curvature tensor components.

$${\Gamma }_{12}^1={\Gamma }_{22}^2={\Gamma }_{23}^3={\displaystyle \frac{1}{2u^2}},{\Gamma }_{11}^2={\Gamma }_{33}^2=-{\displaystyle \frac{1}{2u^2}},$$

$${\mathcal{R}}_{1221}={\mathcal{R}}_{2332}=-{\displaystyle \frac{1}{2u^2}},{\mathcal{R}}_{1331}={\displaystyle \frac{1}{4u^2}},{\mathcal{R}}_{1232}=0.$$

The components of the Ricci tensor ${\mathcal{S}}_{\mu \nu }$ that do not vanish are

$${\mathcal{S}}_{11}=-{\displaystyle \frac{1}{4{\left(u^2\right)}^2}},{\mathcal{S}}_{22}=-{\displaystyle \frac{1}{{\left(u^2\right)}^2}},{\mathcal{S}}_{33}=-{\displaystyle \frac{1}{4{\left(u^2\right)}^2}}.$$

The scalar curvature $\mathfrak{R}$ of $({\mathbb{L}}^4,\mathfrak{g})$ is $-{\displaystyle \frac{3}{2{\left(u^2\right)}^3}}\ne 0.$

We will now demonstrate that $({\mathbb{L}}^4,\mathfrak{g})$ is a $\left(GQE\right)$-manifold [48]. Now, let us examine the related scalars as follows:

$$A_1=-{\displaystyle \frac{1}{{\left(u^2\right)}^3}},A_2=-{\displaystyle \frac{5}{2{\left(u^2\right)}^3}},A_3=-{\displaystyle \frac{2}{{\left(u^2\right)}^3}}.$$

(33)

Once more, let us select the corresponding 1-forms at any point $u\in {\mathbb{L}}^4$ as follows:

$${\eta }_{\mu }\left(u\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{{\sqrt{2}}^{{\sqrt{u}}^2}}},\hfill & \mathit{for}\mu =1,3\hfill \\ & \\ 0,\hfill & \mathit{otherwise},\hfill \end{array}\right.$$

(34)

$${\sigma }_{\mu }\left(u\right)=\left\{\begin{array}{cc}{\sqrt{u}}^2,\hfill & \mathit{for}\mu =1\hfill \\ & \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(35)

To verify Relation (30), it is required to look at the following equations.

$${\mathcal{S}}_{11}=A_1{\mathfrak{g}}_{11}+A_2{\eta }_1{\eta }_1+A_3{\sigma }_1{\sigma }_1,$$

(36)

$${\mathcal{S}}_{22}=A_1{\mathfrak{g}}_{22}+A_2{\eta }_2{\eta }_2+A_3{\sigma }_2{\sigma }_2,$$

(37)

$${\mathcal{S}}_{33}=A_1{\mathfrak{g}}_{33}+A_2{\eta }_3{\eta }_3+A_3{\sigma }_3{\sigma }_3,$$

(38)

Because of (33)–(36), we obtain

$${\mathcal{S}}_{11}={\displaystyle \frac{1}{{\left(u^2\right)}^3}}{u}^2+[-{\displaystyle \frac{5}{2{\left(u^2\right)}^3}}]{\displaystyle \frac{1}{2}}\left(u^2\right)=-{\displaystyle \frac{1}{4{\left(u^2\right)}^2}}={\mathcal{S}}_{11}=L.H.S\mathit{of}\left(36\right).$$

It is possible to demonstrate that (37) and (38) are likewise true using a similar argument. We will now demonstrate that the corresponding vectors ${\eta }_{\mu }$ and ${\sigma }_{\mu }$ are unit.

Here, ${\mathfrak{g}}^{\mu \nu }{\eta }_{\mu }{\eta }_{\nu }=-1,{\mathfrak{g}}^{\mu \nu }{\sigma }_{\mu }{\sigma }_{\nu }=1,{\mathfrak{g}}^{\mu \nu }{\eta }_{\mu }{\sigma }_{\nu }=0.$ Therefore, ${\eta }_{\mu }$ and ${\sigma }_{\mu }$ are orthogonal. Thus, $({\mathbb{L}}^4,\mathfrak{g})$ is a ${\left(GQE\right)}_4$-spacetime.

Example 2. 

Let us take the metric

$$d{\mathfrak{s}}^2=-e^{V\left(\tau \right)}d{t}^2+{\displaystyle \frac{1}{f}}d{\tau }^2-{\tau }^2[d{\alpha }^2+Si{n}^2\alpha d{\phi }^2],$$

(39)

where only the radial coordinate τ determines the metric potentials V and f.

Consider the simplest linear functional

$$F(\mathfrak{R},\mathfrak{T})=\mathfrak{R}+2\chi \mathfrak{T},$$

(40)

where χ is a coupling constant. Using (22) and (40), we get

$${\mathcal{S}}_{\mu \nu }=8\pi {\mathfrak{T}}_{\mu \nu }+\chi \mathfrak{T}{\mathfrak{g}}_{\mu \nu }+2\chi ({\mathfrak{T}}_{\mu \nu }+P{\mathfrak{g}}_{\mu \nu })=8\pi {\tilde{\mathfrak{T}}}_{\mu \nu },$$

(41)

in view of (60), the modified energy tensor ${\tilde{\mathfrak{T}}}_{\mu \nu }$ is defined as

$${\tilde{\mathfrak{T}}}_{\mu \nu }={\mathfrak{T}}_{\mu \nu }\left(1+{\displaystyle \frac{\chi }{4\pi }}\right)+{\displaystyle \frac{\chi }{8\pi }}(\mathfrak{T}+2P){\mathfrak{g}}_{\mu \nu },$$

(42)

where $P={\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert }.$ Applying the covariant derivative of ${\mathfrak{T}}_{\mu \nu }$ with (40), we find

$${\nabla }^{\mu }{\mathfrak{T}}_{\mu \nu }=-{\displaystyle \frac{1}{2(4\pi +\chi )}}\chi [{\mathfrak{g}}_{\mu \nu }{\nabla }^{\mu }+2{\nabla }^{\mu }({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert }{\mathfrak{g}}_{\mu \nu })]$$

(43)

Now, (42) and (43) entail

$${\nabla }^{\mu }{\tilde{\mathfrak{T}}}_{\mu \nu }=0.$$

(44)

Now, using line element (39) with (41) and (42), the field equation for an anisotropic fluid spacetime [49] is

$$\left[{\displaystyle \frac{1}{{\tau }^2}}-{\displaystyle \frac{f}{{\tau }^2}}-{\displaystyle \frac{f^{\prime }}{\tau }}\right]=8\pi \tilde{\varrho },$$

(45)

$$\left[f({\displaystyle \frac{1}{{\tau }^2}}+{\displaystyle \frac{V^{\prime }}{\tau }})-{\displaystyle \frac{1}{{\tau }^2}}\right]=8\pi \widetilde{{\mathfrak{p}}^{\Vert }}$$

(46)

$$\left[{\displaystyle \frac{f}{4}}(2V^{″}+V^{\prime 2}+2{\displaystyle \frac{V^{\prime }}{\tau }})+{\displaystyle \frac{f^{\prime }}{4}}(f^{\prime }+{\displaystyle \frac{2}{\tau }})\right]=8\pi \widetilde{{\mathfrak{p}}^{\perp }},$$

(47)

where primes denote differentiation with respect to the radial coordinates $\tau .$ Equations (45)–(47) and (60) will help us to find $A_1,A_2$ and $A_3.$ Now, using (42), we can write $F(\mathfrak{R},\mathfrak{T})$-gravity for an anisotropic fluid spacetime

$${\mathfrak{p}}^{\perp }={\displaystyle \frac{8\pi \widetilde{{\mathfrak{p}}^{\perp }}}{(8\pi +2\chi )}}+{\displaystyle \frac{8\pi (3\tilde{\varrho }-\widetilde{{\mathfrak{p}}^{\Vert }}-2\widetilde{{\mathfrak{p}}^{\perp }})\chi }{3(8\pi +2\chi )(8\pi +4\chi )}}$$

(48)

$$\varrho ={\displaystyle \frac{8\pi \tilde{\varrho }}{(8\pi +4\chi )}}+{\displaystyle \frac{8\pi (3\tilde{\varrho }+\widetilde{{\mathfrak{p}}^{\Vert }}+2\widetilde{{\mathfrak{p}}^{\perp }})\chi }{3(8\pi +2\chi )(8\pi +4\chi )}}$$

(49)

$${\mathfrak{p}}^{\Vert }={\displaystyle \frac{8\pi \widetilde{{\mathfrak{p}}^{\Vert }}}{(8\pi +2\chi )}}+{\displaystyle \frac{8\pi (3\tilde{\varrho }-\widetilde{{\mathfrak{p}}^{\Vert }}-2\widetilde{{\mathfrak{p}}^{\perp }})\chi }{3(8\pi +2\chi )(8\pi +4\chi )}}$$

(50)

Consequently, in the light of (10) and (48)–(50) we can obtain the equation of state with ${\omega }_1$ and ${\omega }_2$ for $F(\mathfrak{R},\mathfrak{T})$-gravity model with anisotropic fluid matter as follows:

$${\displaystyle \frac{{\mathfrak{p}}^{\Vert }}{\varrho }}={\displaystyle \frac{\frac{8\pi \widetilde{{\mathfrak{p}}^{\Vert }}}{(8\pi +2\chi )}+\frac{8\pi (3\tilde{\varrho }-\widetilde{{\mathfrak{p}}^{\Vert }}-2\widetilde{{\mathfrak{p}}^{\perp }})\chi }{3(8\pi +2\chi )(8\pi +4\chi )}}{\frac{8\pi \tilde{\varrho }}{(8\pi +4\chi )}+\frac{8\pi (3\tilde{\varrho }+\widetilde{{\mathfrak{p}}^{\Vert }}+2\widetilde{{\mathfrak{p}}^{\perp }})\chi }{3(8\pi +2\chi )(8\pi +4\chi )}}}={\omega }_1$$

(51)

$${\displaystyle \frac{{\mathfrak{p}}^{\perp }}{\varrho }}={\displaystyle \frac{\frac{8\pi \widetilde{{\mathfrak{p}}^{\perp }}}{(8\pi +2\chi )}+\frac{8\pi (3\tilde{\varrho }-\widetilde{{\mathfrak{p}}^{\Vert }}-2\widetilde{{\mathfrak{p}}^{\perp }})\chi }{3(8\pi +2\chi )(8\pi +4\chi )}}{\frac{8\pi \tilde{\varrho }}{(8\pi +4\chi )}+\frac{8\pi (3\tilde{\varrho }+\widetilde{{\mathfrak{p}}^{\Vert }}+2\widetilde{{\mathfrak{p}}^{\perp }})\chi }{3(8\pi +2\chi )(8\pi +4\chi )}}}={\omega }_2.$$

(52)

4. Anisotropic Fluid in $\mathit{F}(\mathfrak{R},\mathfrak{T})$-Gravity with Codazzi-Type Ricci Tensor

In an anisotropic fluid spacetime (in shor $AFST$) in $F(\mathfrak{R},\mathfrak{T})$-gravity, suppose the Ricci tensor $\mathcal{S}$ is of the Codazzi type [50], meaning that

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

(53)

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

By using Equations (30) and (53), we currently have

$$d{A}_1\left(\mu \right)g(\nu ,\varsigma )+d{A}_2\left(\mu \right)\gamma \left(\nu \right)\gamma \left(\varsigma \right)+d{A}_3\left(\mu \right)\sigma \left(\nu \right)\sigma \left(\varsigma \right)$$

(54)

$$+A_2[\left({\nabla }_{\mu }\gamma \right)\nu \gamma \left(\varsigma \right)+\gamma \left(\nu \right)\left({\nabla }_{\mu }\gamma \right)\varsigma ]+A-3[\left({\nabla }_{\mu }\sigma \right)\nu \sigma \left(\varsigma \right)+\sigma \left(\nu \right)\left({\nabla }_{\mu }\sigma \right)\varsigma ]$$

$$=d{A}_1\left(\nu \right)g(\mu ,\varsigma )+d{A}_2\left(\nu \right)\gamma \left(\mu \right)\gamma \left(\varsigma \right)+d{A}_3\left(\nu \right)\sigma \left(\mu \right)\sigma \left(\varsigma \right)$$

$$+A_2[\left({\nabla }_{\nu }\gamma \right)\mu \gamma \left(\varsigma \right)+\gamma \left(\mu \right)\left({\nabla }_{\nu }\gamma \right)\varsigma ]+A_3[\left({\nabla }_{\nu }\sigma \right)\mu \sigma \left(\varsigma \right)+\sigma \left(\mu \right)\left({\nabla }_{\nu }\sigma \right)\varsigma ].$$

By contracting over $\nu$ and $\varsigma$ in a frame field, we obtain

$$4d{A}_1\left(\mu \right)-d{A}_2\left(\mu \right)+A_2[\left({\nabla }_{\mu }\gamma \right)\zeta +\gamma \left(\nu \right)\left({\nabla }_{\mu }\gamma \right)\zeta ]$$

(55)

$$=d{A}_1\left(\mu \right)+d{A}_2\left(\zeta \right)\gamma \left(\mu \right)+A_2[\left({\nabla }_{\zeta }\gamma \right)\mu +div\zeta \gamma \left(\mu \right)],$$

Here, the divergence of $\zeta$ is indicated by $div\left(\zeta \right).$

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

$$3d{A}_1\left(\zeta \right)+A_2[\left({\nabla }_{\zeta }\gamma \right)\zeta +\left({\nabla }_{\zeta }\gamma \right)\zeta ]=A_2[\left({\nabla }_{\zeta }\gamma \right)\zeta -div\zeta ].$$

(56)

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

$$3d{A}_1\left(\zeta \right)+div\zeta =0.$$

(57)

Consider the velocity vector field. $\zeta$ does not affect the scalar $A_1$. Following the previous equation, we may conclude that either $A_2=0$ or $div\zeta =0.$

Using (31), we get

$${\mathfrak{p}}^{\perp }+\varrho =-\left[2(F^{\prime }\left(\mathfrak{T}\right)+4\pi )\right].$$

(58)

Remark 1. 

To determine the compatibility of a theory with the Null Energy Condition (NEC) and Weak Energy Condition (WEC) based on the signs and magnitudes of $F^{\prime }$, for NEC compatibility, ${{\mathfrak{p}}^{\perp }}_{eff}+{\varrho }_{eff}\ge 0$ for $F^{\prime }\left(\mathfrak{T}\right)>0$; for WEC compatibility, ${\varrho }_{eff}\ge 0$ and ${{\mathfrak{p}}^{\perp }}_{eff}+{\varrho }_{eff}\ge 0$ for $F^{\prime }\left(\mathfrak{T}\right)>0.$ Thus, this geometric corrections (58) are not compatible with NEC and WEC.

Remark 2. 

The equation of state $\left(\mathcal{E}o\mathcal{S}\right)$ for dark matter era is defined by ${\mathfrak{p}}^{\perp }=-\varrho +F\left(\beta \right)$, wherein $F\left(\beta \right)$ is a function of the scale factor β with cosmic time t. The eras of dark matter, stiff matter, radiation, and dust matter are defined by [51] $\mathcal{E}o\mathcal{S}$$\varrho =-\mathfrak{p}$, $\varrho =\mathfrak{p}$, $\mathfrak{p}={\displaystyle \frac{\varrho }{3}}$, and $\mathfrak{p}=0$.

Given (58), $\mathcal{E}o\mathcal{S}$ gives an expression for the relationship between transverse pressure and energy density.

Now, we can state that

Theorem 4. 

A spacetime $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity filled with an anisotropic fluid matter endowed with Codazzi-type Ricci tensor represents dark matter era, provided $A_1$ is invariant under the four-velocity vector field $\zeta .$

Guilfoyle and Nolan [52] claim that “Yang Pure Space” is a Lorentzian manifold $(M^4,g)$ where the manifold’s metric tensor fulfills Yang’s equations:

$$\left({\nabla }_{\mu }\mathcal{S}\right)(\nu ,\varsigma )-\left({\nabla }_{\nu }\mathcal{S}\right)(\mu ,\varsigma )=0.$$

Additionally, the authors of the same study showed that a perfect fluid spacetime with $\mathfrak{p}+\varrho \ne 0$ is a Yang Pure Space if and only if the spacetime is a $RW$-spacetime. Consequently, we conclude that an anisotropic fluid spacetime with a Codazzi-type Ricci tensor is a $RW$-spacetime.

Thus, we can draw the following conclusions.

Theorem 5. 

Let a spacetime $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity be filled with an anisotropic fluid matter with transverse pressure ${\mathfrak{p}}^{\perp }$, rest energy density ρ, and endowed with Codazzi type Ricci tensor. Then, the $AFST$ in $F(\mathfrak{R},\mathfrak{T})$-gravity is a Yang Pure spacetime.

Theorem 6. 

Let a spacetime $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity be filled with an anisotropic fluid matter with transverse pressure ${\mathfrak{p}}^{\perp }$, rest energy density ϱ, and endowed with Codazzi-type Ricci tensor. Then, the $AFST$ is a $RW$-spacetime.

Theorem 7. 

If an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity is endowed with Codazzi-type Ricci tensor, then the $\mathcal{E}o\mathcal{S}$ of an $AFST$ in $F(\mathfrak{R},\mathfrak{T})$-gravity is given by (58).

Theorem 8. 

If an $AFST$ $(M^4,g)$ in $\mathfrak{f}(\mathfrak{R},\mathfrak{T})$-gravity is endowed with Codazzi-type Ricci tensor, then the universe’s evolution are provided as

$\mathit{Anisotropic}\mathit{Fluid}\mathit{in}F(R,\mathfrak{T})\text{-}\mathit{Gravity}\mathit{Represents}$	$\mathit{Equation}\mathit{of}\mathit{State}\left(\mathcal{E}o\mathcal{S}\right)$	$\mathit{Transve}\mathit{Pressure}{\mathfrak{p}}^{\perp }\mathit{and}\mathit{Density}\mathsf{\varrho }\mathit{Are}$
Vacuum dominated era	${\mathfrak{p}}^{\perp }=-\varrho$	${\mathfrak{p}}^{\perp }=0\mathit{and}\varrho =0$
Stiff fluid era	${\mathfrak{p}}^{\perp }=\varrho$	${\mathfrak{p}}^{\perp }=-[F^{\prime }\left(\mathfrak{T}\right)+4\pi ]\mathit{and}{\mathfrak{\varrho }}^{\perp }=-[F^{\prime }\left(\mathfrak{T}\right)+4\pi ]$
Radiation era	${\mathfrak{p}}^{\perp }={\displaystyle \frac{\varrho }{3}}$	${\mathfrak{p}}^{\perp }=-{\displaystyle \frac{1}{2}}[F^{\prime }\left(\mathfrak{T}\right)+4\pi ]\mathit{and}{\mathfrak{\varrho }}^{\perp }=-{\displaystyle \frac{3}{2}}[F^{\prime }\left(\mathfrak{T}\right)+4\pi ]$
matter-dominated era	${\mathfrak{p}}^{\perp }=0$	${\mathfrak{p}}^{\perp }=0\mathit{and}\varrho =-[F^{\prime }\left(\mathfrak{T}\right)+4\pi ]$

5. Ricci Semi-Symmetric $\mathit{A}\mathit{F}\mathit{S}\mathit{T}$ in $\mathit{F}(\mathfrak{R},\mathfrak{T})$-Gravity

The $AFST$$(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity is assumed to be Ricci semi-symmetric in this section.

Definition 4 

([53]). The Riemannian manifold is designated as Ricci semi-symmetric if its curvature tensor satisfies

$$\mathcal{R}im·\mathcal{S}=0,$$

(59)

where $\mathcal{R}im$ indicates the Riemannian curvature and $\mathcal{S}$ is Ricci curvature tensor.

The difference between the volume of an extremely small area in curved space and flat space is described by the Ricci tensor. Understanding of gravitational theories, especially $F\left(\mathfrak{R}\right)$-gravity and $F(\mathfrak{R},\mathfrak{T})$-gravity, which can describe complex astrophysical objects like massive neutron stars, is aided by the study of Ricci semi-symmetric manifolds. In particular, at higher dimensions or under modified gravity theories such as $F(\mathfrak{R},\mathfrak{T})$-gravity, the definition of Ricci semi-symmetry helps mathematicians and physicists in their research of spacetime geometry by defining a class of manifolds that are not entirely symmetric but have a significant geometric structure.

Now,

$$\left(\mathcal{R}im\right(\mu ,\nu )·\mathcal{S})(\alpha ,\beta )=\mathcal{S}\left(\mathcal{R}im\right(\mu ,\nu )\alpha ,\beta )-\mathcal{S}(\alpha ,\mathcal{R}im(\mu ,\nu \left)\beta \right)$$

(60)

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

Next, Equations (7) and (30) can be written in index-free notation as follows:

$$\mathcal{S}\left(\mu ,\nu \right)=A_{1}g(\mu ,\nu )+A_2\eta \left(\mu \right)\eta \left(\nu \right)+A_3\sigma \left(\mu \right)\sigma \left(\nu \right).$$

(61)

In addition, we have $g(\mu ,\zeta )=\gamma \left(\mu \right),g(\mu ,\gamma )=\sigma \left(\mu \right)$ and $\gamma \left(\zeta \right)=-1$, $\sigma \left(\gamma \right)=1$, where $\zeta$ and $\gamma$ are orthogonal vector fields such that $g(\zeta ,\gamma )=0$.

Now, from (60), we turn up

$$A_{1}g(\mathcal{R}im(\mu ,\nu )\alpha ,\beta )+A_2\eta \left(\mathcal{R}im(\mu ,\nu )\alpha \right)\eta \left(\beta \right)+A_3\sigma \left(\mathcal{R}im(\mu ,\nu )\alpha \right)\sigma \left(\beta \right)$$

(62)

$$+A_{1}g(\alpha ,\mathcal{R}im(\mu ,\nu )\beta )+A_2\eta \left(\alpha \right)\eta \left(\mathcal{R}im(\mu ,\nu )\beta \right)+A_3\sigma \left(\alpha \right)\sigma \left(\mathcal{R}im(\mu ,nu)\beta \right)=0.$$

inserting $\beta =\zeta$ and $\alpha =\gamma$ in (62), we gain

$$A_2\eta \left(\mathcal{R}im(\mu ,\nu )\gamma \right)+A_3\sigma \left(\mathcal{R}im(\mu ,\nu )\zeta \right)=0$$

or

$$(A_3-A_2)\overline{\mathcal{R}}im(\mu ,\nu ,\zeta ,\gamma )=0,$$

(63)

where $\overline{\mathcal{R}}im(\mu ,\nu ,\zeta ,\gamma )=g(\mathcal{R}im(\mu ,\nu )\zeta ,\gamma ).$

Consequently, Equation (63) argues either $A_3-A_2=0$ or $\overline{\mathcal{R}}im(\mu ,\nu ,\zeta ,\gamma )=0.$ Therefore, we have the following situations.

Case (i): If $A_3=A_2$, then ${\mathfrak{p}}^{\perp }+2\varrho ={\mathfrak{p}}^{\Vert }$.

Case (ii): If $\overline{\mathcal{R}}im(\mu ,\nu ,\zeta ,\gamma )=0$ for all $\mu ,\nu ,\zeta ,\gamma \in \chi \left(M^4\right)$. Again, setting $\mu =\nu =\zeta$, and after contraction, we turn up $\mathcal{S}(\zeta ,\zeta )=0,$ for all $\zeta \in \chi \left(M^4\right)$. Thus, (30) entails that $(A_1-A_2)\gamma \left(\zeta \right)=0$, so $A_1=A_2$. Hence, $A_1=A_2$ signifies

$${\mathfrak{p}}^{\perp }+\varrho ={\displaystyle \frac{1}{2}}\mathfrak{R}+F\left(\mathfrak{T}\right)-2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi ].$$

(64)

Thus, we can articulate:

Theorem 9. 

Let an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity be a Ricci semi-symmetric spacetime. Then, the $\mathcal{E}o\mathcal{S}$ is

$${\mathfrak{p}}^{\perp }+\varrho ={\displaystyle \frac{1}{2}}\mathfrak{R}+F\left(\mathfrak{T}\right)-2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi ].$$

The rotational curves of galaxies can be used to deduce the dark matter equation of state, which is a negative pressure equation of state $\mathfrak{p}=-\varrho$ [44] that characterizes the inflationary era, also known as late dark energy, which causes the Universe to expand more quickly.

Therefore, for ${\mathfrak{p}}^{\perp }=-\varrho$, Equation (64) infers

$${\mathfrak{p}}^{\perp }=-\varrho ={\displaystyle \frac{\mathfrak{R}+F\left(\mathfrak{T}\right)}{4F^{\prime }\left(\mathfrak{T}\right)}}+{\displaystyle \frac{(2\pi +1)}{F^{\prime }\left(\mathfrak{T}\right)}}-{\mathfrak{p}}^{\Vert }.$$

(65)

Thus, we can state the following:

Theorem 10. 

If the source of a Ricci semi-symmetric $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity is late dark energy or inflationary era, then the transverse pressure and rest energy density are given (65).

Corollary 2. 

If the source of a Ricci semi-symmetric $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity is radiation type, then the transverse pressure and rest energy density are given below:

$${\mathfrak{p}}^{\perp }={\displaystyle \frac{1}{8}}\mathfrak{R}+{\displaystyle \frac{F\left(\mathfrak{T}\right)}{4}}-{\displaystyle \frac{1}{2}}[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi ],$$

$$\varrho ={\displaystyle \frac{3}{2}}\left\{{\displaystyle \frac{1}{4}}\mathfrak{R}+{\displaystyle \frac{3F\left(\mathfrak{T}\right)}{4}}-{\displaystyle \frac{1}{2}}[({\mathfrak{p}}^{\perp }++{\mathfrak{p}}^{\Vert })F^{\prime }\left(\mathfrak{T}\right)-1)+4\pi ]\right\}.$$

Remark 3. 

Since any Ricci semi-symmetric manifold is Ricci-pseudo symmetric, the reverse is not true in this situation (see [53]).

Consequently, we obtain the following result in light of the aforementioned comment and Theorem 9.

Corollary 3. 

Let an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity be a Ricci-pseudo symmetric anisotropic fluid spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity. Then, $\mathcal{E}o\mathcal{S}$ is

$${\mathfrak{p}}^{\perp }+\varrho ={\displaystyle \frac{1}{2}}\mathfrak{R}+F\left(\mathfrak{T}\right)-2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi ].$$

6. Energy Constraint and Penrose Theorem in $\mathit{F}(\mathfrak{R},\mathfrak{T})$-Gravity

Any black hole will always have some kind of geodesic incompleteness if matter meets adequate energy requirements, according to the Penrose theorem [54].

The time-like convergence condition (TCC) is represented by Equation (66) if we can determine whether the criterion is satisfied by the Ricci tensor $\mathcal{S}$ in the spacetime

$$\mathcal{S}(\mu ,\mu )>0,$$

(66)

for all time-like vector fields $\mu \in \chi \left(M^4\right)$.

In light of Theorem 2, we consider the spacetime $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity model with $BVSF$ matter as a $GQE$-spacetime with unit time-like velocity vector field $\zeta$. Then, we gain $g(\zeta ,\zeta )=-1$. Now, adopting $\mu =\nu =\zeta$ in (28), we turn up

$$\mathcal{S}(\zeta ,\zeta )={\displaystyle \frac{1}{2}}\mathfrak{R}+F\left(\mathfrak{T}\right)-2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi +({\mathfrak{p}}^{\perp }+\varrho )].$$

(67)

We can now assert the subsequent theorem:

Theorem 11. 

Let an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity obey the $TCC$ if

$${\displaystyle \frac{1}{2}}\mathfrak{R}+F\left(\mathfrak{T}\right)>2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi +({\mathfrak{p}}^{\perp }+\varrho )].$$

$TCC$ is the geometric representation of the strong energy condition $\left(SEC\right)$ [38,55]. Furthermore, the null convergence condition $\left(NCC\right)$ is implied by the $TCC$, implying the null energy condition $\left(NEC\right)$ [38]. Considering (67) and Theorem 11, we now arrive at the following:

Corollary 4. 

Let an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity obey the $NCC$ if

$${\displaystyle \frac{1}{2}}\mathfrak{R}+F\left(\mathfrak{T}\right)>2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi +({\mathfrak{p}}^{\perp }+\varrho )],$$

provided the four-velocity vector field ζ.

Again, (67) and Corollary 4 entail the following:

Theorem 12. 

An $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity obeys the $NEC$ $({\mathfrak{p}}^{\perp }+\varrho )>0$ if this gives

$${\displaystyle \frac{1}{2}}\mathfrak{R}+F\left(\mathfrak{T}\right)+2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi +({\mathfrak{p}}^{\perp }+\varrho )]>0.$$

Let spacetime M fulfill the $SCC\Rightarrow NCC$ according to Penrose’s singularity theorem for geodesic incompleteness [54]. Vilenkin and Wall ([56]) proved that spacetime M obeys the $SCC\Rightarrow NCC$. This implies that:

(i)

M has a connected Cauchy surface that is not compact, and

(ii)

M has some black holes as well as a trapped surface that lies outside of them.

Thus, in light of Corollary 4 and using the above facts (1) and (2) together, we gain the following two results.

Theorem 13. 

Let $AFST$ $(M^4,g)$ in the $F(\mathfrak{R},\mathfrak{T})$-gravity model obey the $NCC$ if

$${\displaystyle \frac{1}{2}}\mathfrak{R}+F\left(\mathfrak{T}\right)>2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi +({\mathfrak{p}}^{\perp }+\varrho )],$$

(68)

then, an $AFST$ $(M^4,g)$ admits a non-compact connected Cauchy surface.

Theorem 14. 

If an $AFST$ $(M^4,g)$ in the $F(\mathfrak{R},\mathfrak{T})$-gravity model obeys the $NCC$ (68), then an $AFST$ $(M^4,g)$ admits black holes as well as a trapped surface that lies outside of them.

Additionally, the Ricci tensor $\mathcal{S}$ and the energy momentum tensor $\mathfrak{T}$ are examples of symmetric second-order tensors defined on the 4-dimensional Lorentzian spacetime manifold $(M^4,g)$ that are classified algebraically as the Segre classification (for more information, see [38,57]). Restrictions are imposed by the energy condition on energy momentum tensors or equivalently on Ricci tensor of $GTR$ for matter fields like those represented by Segre classes. The most common energy condition is $SEC$ [38]. The $SEC$ is the assertion that for any time-like vector field $\mathcal{S}(\mu ,\nu )>0$, $\mathcal{S}$ belongs to the Segre class [(11),(11)] [38,57].

Considering the above facts and using Theorem 11, we gain the following outputs.

Theorem 15. 

If an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity model obeys the $SEC$ with timelike vector field ζ with condition

$${\displaystyle \frac{1}{2}}\mathfrak{R}+F\left(\mathfrak{T}\right)>2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi +({\mathfrak{p}}^{\perp }+\varrho )]$$

, then Ricci tensor $\mathcal{S}$ belongs to the Segre class [(11), (11)].

Corollary 5. 

An $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity obeys the $SEC$ if

$${\displaystyle \frac{1}{2}}\mathfrak{R}+F\left(\mathfrak{T}\right)>2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi +({\mathfrak{p}}^{\perp }+\varrho )],$$

then $GQE$-spacetime $(M^4,g)$ is of type Ludwig–Scanlan types [58] $A_1$ and $A_2$.

For timelike vector field $\zeta$, through Equation (30), we have

$$\mathcal{S}(\mu ,\zeta )=(A_1-A_2)g(\mu ,\zeta )forall\mu .$$

(69)

$${\displaystyle \frac{{\mathfrak{p}}^{\perp }}{\varrho }}={\displaystyle \frac{1}{2}}\mathfrak{R}+F\left(\mathfrak{T}\right)-2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi +1]$$

(70)

Thus, we have

Theorem 16. 

In an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity, the generator ζ is an eigenvector of the Ricci tensor corresponding to the eigenvalue $A_1-A_2$. Then, $\mathcal{E}o\mathcal{S}$ is furnished by (70).

In addition, if we fix $\mathfrak{R}=0$ in (70), then we gain $\mathcal{E}o\mathcal{S}$,

$${\displaystyle \frac{{\mathfrak{p}}^{\perp }}{\varrho }}=F\left(\mathfrak{T}\right)-2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi +1].$$

This leads us to the next corollary.

Corollary 6. 

For an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity, if generator ζ is an eigenvector of the Ricci tensor and $\mathfrak{R}=0$, then the universe existence is seen in the table below through $\mathcal{E}o\mathcal{S}$ of an $AFST$ in $F(\mathfrak{R},\mathfrak{T})$-gravity as:

$\mathit{Equation}\mathit{of}\mathit{State}{\displaystyle \frac{{\mathfrak{p}}^{\perp }}{\varrho }}={\omega }_{\mathbf{2}}$	$\hspace{1em}\hspace{1em}\mathit{Restrictions}\mathit{on}\mathfrak{f}\left(\mathfrak{T}\right)\mathit{and}{\mathit{F}}^{\prime }\left(\mathfrak{T}\right)$	Evolution of the Universe
${\omega }_2=1$	$f\left(\mathfrak{T}\right)=2\left[F^{\prime }\right(\mathfrak{T})(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi ]+3]$	Ultra relativistic era
${\omega }_2>-1$	$F^{\prime }\left(\mathfrak{T}\right)<-2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi ]-2]$	Quintessence era
${\omega }_2<-1$	$F^{\prime }\left(\mathfrak{T}\right)>2\left[F^{\prime }\right(\mathfrak{T})(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi ]+2]$	Phantom era
${\omega }_2=0$	$F^{\prime }\left(\mathfrak{T}\right)=2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi ]+1]$	Dust era

A necessary condition for the late accelerated expansion of the Universe is that the parameter of inflation or dark energy equation of state ${\displaystyle \frac{{\mathfrak{p}}^{\perp }}{\varrho }}={\omega }_2$ must be negative. In this case, if $\mathcal{S}(\zeta ,\zeta )\le 0$, then the violation of $SEC$ is necessary. Therefore, (67) holds only for

$${\displaystyle \frac{1}{2}}\mathfrak{R}+F\left(\mathfrak{T}\right)\le 2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)+4\pi +({\mathfrak{p}}^{\perp }+\varrho )].$$

(71)

$${\displaystyle \frac{1}{2}}\mathfrak{R}+F\left(\mathfrak{T}\right)-2[F^{\prime }\left(\mathfrak{T}\right)(({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })-1)-4\pi \le ({\omega }_2+1)].$$

(72)

The right-hand side of (72) vanishes in the case of inflation or dark energy, ${\omega }_2=-1$, and we get $\varrho \le {\displaystyle \frac{{\mathfrak{p}}^{\Vert }}{F^{\prime }\left(\mathfrak{T}\right)}}-{\displaystyle \frac{2}{F^{\prime }\left(\mathfrak{T}\right)}}\{(2\pi +1)+{\displaystyle \frac{1}{4}}\mathfrak{R}+F\left(\mathfrak{T}\right)\}.$

As a result, we can express the following result.

Theorem 17. 

If the source of matter is dark energy in the $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity, then the SEC is violated, and the energy density holds the condition

$$\varrho \le {\displaystyle \frac{{\mathfrak{p}}^{\Vert }}{F^{\prime }\left(\mathfrak{T}\right)}}-{\displaystyle \frac{2}{F^{\prime }\left(\mathfrak{T}\right)}}\{(2\pi +1)+{\displaystyle \frac{1}{4}}\mathfrak{R}+F\left(\mathfrak{T}\right)\}.$$

(73)

7. HRS on an $\mathit{A}\mathit{F}\mathit{S}\mathit{T}$ in $\mathit{F}(\mathfrak{R},\mathfrak{T})$-Gravity

This section examines the $HRS$ structure in an $AFST$ in $F(\mathfrak{R},\mathfrak{T})$-gravity with a time-like torse-forming velocity vector field.

By applying the property of the Lie derivative, we have

$$\left({\mathfrak{L}}_{\zeta }\mathfrak{g}\right)({\Xi }_1,{\Xi }_2)=\mathfrak{g}({\nabla }_{{\Xi }_1}\zeta ,{\Xi }_2)+\mathfrak{g}({\Xi }_1,{\nabla }_{\zeta }{\Xi }_2),$$

(74)

for all vector fields ${\Xi }_1,{\Xi }_2\in \chi \left(M^4\right).$

Utilizing (8) in (74), we get

$$\left({\mathfrak{L}}_{\zeta }\mathfrak{g}\right)({\Xi }_1,{\Xi }_2)=2\phi [\mathfrak{g}({\Xi }_1,{\Xi }_2)+\eta \left({\Xi }_1\right)\eta \left({\Xi }_2\right)].$$

(75)

By computing the Lie derivative along $\zeta$ of (75), we get

$$\left({\mathfrak{L}}_{\zeta }\left({\mathfrak{L}}_{\zeta }\mathfrak{g}\right)\right)({\Xi }_1,{\Xi }_2)={\mathfrak{L}}_{\zeta }\left(\left({\mathfrak{L}}_{\zeta }\mathfrak{g}\right)({\Xi }_1,{\Xi }_2)\right)-\left({\mathfrak{L}}_{\zeta }\mathfrak{g}\right)({\mathfrak{L}}_{\zeta }{\Xi }_1,{\Xi }_2)$$

(76)

$$-\left({\mathfrak{L}}_{\zeta }\mathfrak{g}\right)({\Xi }_1,{\mathfrak{L}}_{\zeta }{\Xi }_2).$$

$${\mathfrak{L}}_{\zeta }\left(\left({\mathfrak{L}}_{\zeta }\mathfrak{g}\right)({\Xi }_1,{\Xi }_2)\right)=4F^2[\mathfrak{g}({\Xi }_1,{\Xi }_2)+\eta \left({\Xi }_1\right)\eta \left({\Xi }_2\right)]+2F(\mathfrak{g}({\mathfrak{L}}_{\zeta }{\Xi }_1,{\Xi }_2)$$

(77)

$$-{\mathfrak{L}}_{\zeta }\left(\eta \left({\Xi }_1\right)\eta \left({\Xi }_2\right)\right)+\mathfrak{g}({\Xi }_1,{\mathfrak{L}}_{\zeta }{\Xi }_2)).$$

Furthermore,

$$\left({\mathfrak{L}}_{\zeta }\mathfrak{g}\right)({\mathfrak{L}}_{\zeta }{\Xi }_1,{\Xi }_2)=2F[\mathfrak{g}({\mathfrak{L}}_{\zeta }{\Xi }_1,{\Xi }_2)+\eta \left({\mathfrak{L}}_{\zeta }{\Xi }_1\right)\eta \left({\Xi }_2\right)].$$

(78)

Likewise,

$$\left({\mathfrak{L}}_{\zeta }\mathfrak{g}\right)({\Xi }_1,{\mathfrak{L}}_{\zeta }{\Xi }_2)=2F[\mathfrak{g}({\Xi }_1,{\mathfrak{L}}_{\zeta }{\Xi }_2)+\eta \left({\mathfrak{L}}_{\zeta }{\Xi }_2\right)\eta \left({\Xi }_1\right)].$$

(79)

Again, we find

$$\eta \left({\mathfrak{L}}_{\zeta }{\Xi }_1\right)=\mathfrak{g}({\nabla }_{\zeta }{\Xi }_1,\zeta )-\mathfrak{g}({\nabla }_{{\Xi }_1}\zeta ,\zeta ).$$

(80)

Therefore,

$$\eta \left({\mathfrak{L}}_{\zeta }{\Xi }_1\right)\eta \left({\Xi }_2\right)+\eta \left({\mathfrak{L}}_{\zeta }{\Xi }_2\right)\eta \left({\Xi }_1\right)={\mathfrak{L}}_{\zeta }(\eta \left({\Xi }_1\right)\eta \left({\Xi }_2\right).$$

(81)

Finally, if we apply (74), (77), (78), (80) and (81) to (76), we obtain

$${\mathfrak{L}}_{\zeta }\left(\left({\mathfrak{L}}_{\zeta }\mathfrak{g}\right)({\Xi }_1,{\Xi }_2)\right)=4{\phi }^2[\mathfrak{g}({\Xi }_1,{\Xi }_2)+\eta \left({\Xi }_1\right)\eta \left({\Xi }_2\right)].$$

(82)

The $HRS$ Equation (2) is now used to derive

$$2{\phi }^2[g({\Xi }_1,{\Xi }_2)+\eta \left({\Xi }_1\right)\eta \left({\Xi }_2\right)]+2⋋F[\mathfrak{g}({\Xi }_1,{\Xi }_2)+\eta \left({\Xi }_1\right)\eta \left({\Xi }_2\right)]$$

(83)

$$+\mathcal{S}({\Xi }_1,{\Xi }_2)=\Theta \mathfrak{g}({\Xi }_1,{\Xi }_2),$$

which implies that

$$\mathcal{S}({\Xi }_1,{\Xi }_2)=(\Theta -2{\phi }^2+2⋋\phi )\mathfrak{g}({\Xi }_1,{\Xi }_2)+2\phi (\phi +⋋)\eta \left({\Xi }_1\right)\eta \left({\Xi }_2\right).$$

(84)

Now, (28) and (84) infer

$$\mathcal{S}({\Xi }_1,{\Xi }_2)=(\Theta -2{\phi }^2+2⋋\phi -A_1)\mathfrak{g}({\Xi }_1,{\Xi }_2)+2\phi (\phi +⋋-A_2)\eta \left({\Xi }_1\right)\eta \left({\Xi }_2\right)-A_3\sigma \left({\Xi }_1\right)\sigma \left({\Xi }_2\right).$$

(85)

After plugging ${\Xi }_1={\Xi }_2=\zeta$, we turn up

$$⋋=\phi +{\displaystyle \frac{A_1}{4\phi }}-\left\{{\displaystyle \frac{\Theta }{4\phi }}+{\displaystyle \frac{A_2}{2}}\right\}.$$

(86)

Now, in the light of (5), (84) and (86) entail the following results:

Theorem 18. 

If an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity admits $HRS$ with a time-like torse-forming vector field, then $AFST$ is a perfect fluid spacetime.

Theorem 19. 

If an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity admits $HRS$ with a torse-forming vector field, then $HRS$ is increasing, stable, or decreasing, referring to the following:

1. 

$\phi +{\displaystyle \frac{A_1}{4\phi }}>\left\{{\displaystyle \frac{\Theta }{4\phi }}+{\displaystyle \frac{A_2}{2}}\right\}$,

2. 

$\phi +{\displaystyle \frac{A_1}{4\phi }}=\left\{{\displaystyle \frac{\Theta }{4\phi }}+{\displaystyle \frac{A_2}{2}}\right\}$, and

3. 

$\phi +{\displaystyle \frac{A_1}{4\phi }}<\left\{{\displaystyle \frac{\Theta }{4\phi }}+{\displaystyle \frac{A_2}{2}}\right\}$, respectively.

Next, adopting a stable $HRS$ case and using (31) and (86) together, we gain

$${\mathfrak{p}}^{\perp }+\varrho ={\displaystyle \frac{1}{2\phi }}\left\{[\mathfrak{R}+4\phi +\mathfrak{f}\left(\mathfrak{T}\right)]+2[F^{\prime }\mathfrak{T}+4\pi ]-[\Theta +2({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })F^{\prime }\mathfrak{T}]\right\}.$$

(87)

Consequently, we turn up the next theorem for stable $HRS$.

Theorem 20. 

If an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity admits stable $HRS$ with a torse-forming vector field, then $\mathcal{E}o\mathcal{S}$ is

$${\mathfrak{p}}^{\perp }+\varrho ={\displaystyle \frac{1}{2\phi }}\left\{[\mathfrak{R}+4\phi +\mathfrak{f}\left(\mathfrak{T}\right)]+2[F^{\prime }\left(\mathfrak{T}\right)+4\pi ]-[\Theta +2({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })F^{\prime }\left(\mathfrak{T}\right)]\right\}.$$

Theorem 21. 

If an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity admits stable $HRS$ with a torse-forming vector field satisfying the $NEC$, ${\mathfrak{p}}^{\perp }+\varrho \ge 0$ if

$${\displaystyle \frac{1}{2\phi }}\left\{[\mathfrak{R}+4\phi +\mathfrak{f}\left(\mathfrak{T}\right)]+2[F^{\prime }\left(\mathfrak{T}\right)+4\pi ]\ge [\Theta +2({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })F^{\prime }\left(\mathfrak{T}\right)]\right\}.$$

(88)

Corollary 7. 

If an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity admits stable $HRS$ with a torse-forming vector field satisfying the $WEC$, ${\mathfrak{p}}^{\perp }+\varrho \ge 0$ if

$${\displaystyle \frac{1}{2\phi }}\left\{[\mathfrak{R}+4\phi +\mathfrak{f}\left(\mathfrak{T}\right)]+2[F^{\prime }\left(\mathfrak{T}\right)+4\pi ]\ge [\Theta +2({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })F^{\prime }\left(\mathfrak{T}\right)]\right\}.$$

Now, in the light of Remark 2 and Theorem 20, we infer the following Corollaries:

Corollary 8. 

If the source of an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity is radiation type and admits stable $HRS$ with a torse-forming vector field, then the transverse pressures and density are

$${\mathfrak{p}}^{\perp }={\displaystyle \frac{1}{[1+2F^{\prime }\left(\mathfrak{T}\right)]8\phi }}\left\{[\mathfrak{R}+4\phi +\mathfrak{f}\left(\mathfrak{T}\right)]+2[F^{\prime }\mathfrak{T}+4\pi ]-[\Theta +2{\mathfrak{p}}^{\Vert })F^{\prime }\left(\mathfrak{T}\right)]\right\},$$

$$\varrho ={\displaystyle \frac{3}{[1+2F^{\prime }\left(\mathfrak{T}\right)]8\phi }}\left\{[\mathfrak{R}+4\phi +\mathfrak{f}\left(\mathfrak{T}\right)]+2[F^{\prime }\mathfrak{T}+4\pi ]-[\Theta +2{\mathfrak{p}}^{\Vert }{F}^{\prime }\left(\mathfrak{T}\right)]\right\}.$$

(89)

Corollary 9. 

If stiff matter is the source of an $AFST$ $(M^4,g)$ in $F(\mathfrak{R},\mathfrak{T})$-gravity and admits stable $HRS$ with a torse-forming vector field, then the transverse pressures and density are

$${\mathfrak{p}}^{\perp }=\varrho ={\displaystyle \frac{1}{[1+2F^{\prime }\left(\mathfrak{T}\right)]4\phi }}\left\{[\mathfrak{R}+4\phi +\mathfrak{f}\left(\mathfrak{T}\right)]+2[F^{\prime }\left(\mathfrak{T}\right)+4\pi ]-[\Theta +2({\mathfrak{p}}^{\perp }+{\mathfrak{p}}^{\Vert })F^{\prime }\left(\mathfrak{T}\right)]\right\}.$$

(90)

The anisotropic field equations reduce directly to the standard, isotropic gravitational field equations in the isotropic limit, where radial pressure and tangential pressure equalize such that ${\mathfrak{p}}^{\Vert }={\mathfrak{p}}^{\perp }=\mathfrak{p}$. Then, the anisotropic stress tensor vanishes. These equations are consistent with perfect fluid, FLRW-type scenarios. The results are consistent, and the field equations accept perfect fluid solutions where the energy–momentum tensor is of the form

$${\mathfrak{T}}_{\mu \nu }=(\varrho +\mathfrak{p}){\eta }_{\mu }{\eta }_{\nu }+\mathfrak{p}{\mathfrak{g}}_{\mu \nu }.$$

Moreover, the usual energy conditions (WEC, NEC, SEC) that are necessary for the models’ physical viability are preserved by the isotropic limit.

8. Conclusions

The modified $F(\mathfrak{R},\mathfrak{T})$-gravity hypothesis provides an effective representation of the Universe’s abrupt expansion during the late cosmic eras. Numerous ranges of physical and geometric phenomena have been revealed that extend beyond standard General Relativity (GTR) to the $F(\mathfrak{R},\mathfrak{T})$-gravity model. Cosmological problems including dark matter, late-time acceleration, and the early inflationary epoch are mostly addressed by this $F(\mathfrak{R},\mathfrak{T})$-gravity model. In the present study of an $AFST$ in modified gravity, most notable is its fascinating relationship to certain geometric symmetries like Ricci semi-symmetry and Ricci pseudo-symmetry. We derive the gravitational field equation of the $F(\mathfrak{R},\mathfrak{T})$-gravity model coupled with the formula for the Ricci scalar for the $AFST$ in the specific $F(\mathfrak{R},\mathfrak{T})$-gravity model. An anisotropic spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity with a Codazzi-type Ricci tensor is a Robertson–Walker spacetime and a Yang Pure spacetime. For the anisotropic fluids, exact Equations of State can be derived using solitonic solutions. Furthermore, we derive the equation of state for an $AFST$ and compute the total density and transverse pressure throughout the radiation and phantom barrier eras of the universe using the $F(\mathfrak{R},\mathfrak{T})$-gravity framework with hyperbolic Ricci soliton as well. In this gravitational framework, solitons result in variants of traditional physics equations. As self-similar solutions to geometric flows (such as the hyperbolic Ricci flow), hyperbolic Ricci solitons uncover hidden symmetries in the universe’s geometry that Einstein’s original field equations were unable to describe.

Next, we propose certain conditions for the an $AFST$ using the given $F(\mathfrak{R},\mathfrak{T})$-gravity model, taking into consideration different energy sources and black holes in terms of hyperbolic Ricci solitons. The SEC and NCC, which are essential for assessing the physical feasibility of the universe’s expansion models, are tested using the measurements of hyperbolic Ricci solitons. In addition, by applying Penrose’s Singularity Theorem to the $F(\mathfrak{R},\mathfrak{T})$-gravity model, the existence of black holes and trapped surfaces that remain stable even under modified gravitational conditions is demonstrated.

Although geometric and algebraic solutions exist, determining the stability and physical importance of these gravitational solitons is a difficult task. It is still difficult to represent late-time acceleration while maintaining physical energy conditions (such as the Null Energy Condition) with the compatibility of the metric.

Future Scope

1.

The cold dark matter may be resolved by various solitons in this $F(\mathfrak{R},\mathfrak{T})$-gravity model, according to research, giving dark matter halos a geometric origin.

2.

Examining if compact, stable, and unusual stellar structures like quark stars exist within this paradigm.

3.

Investigating the $F(\mathfrak{R},\mathfrak{T})$-gravity model with solitons as a possible effective theory that approximates the effects of quantum gravity and wave dynamics.