Soliton Geometry of Modified Gravity Models Engaged with Strange Quark Matter Fluid and Penrose Singularity Theorem

Abstract

The nature of the $F(R,T)$-gravity in conjunction with the quark matter fluid $\left(QMF\right)$ is examined in this research note. In the $F(R,T)$-gravity framework, we derive the equation of state for the $QMF$ in the form of: $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$ and the model of $F\left(R\right)$-gravity. We also discuss how the quark matter supports the Ricci solitons with a conformal vector field in $F(R,T)$-gravity. In this continuing work, we give estimates for the pressure and quark density in the phantom barrier period and the radiation epoch, respectively. Additionally, we use Ricci solitons to identify several black hole prospects and energy requirements for quark matter fluid spacetime ($QMF$-spacetime) connected with $F(R,T)$-gravity. Furthermore, in the $F(R,T)$-gravity model connected with $QMF$, we also discuss some applications of the Penrose singularity theorem in terms of Ricci solitons with a conformal vector field. Finally, we deduce the Schrödinger Equation using the equation of state of the $F(R,T)$-gravity model connected with $QMF$, and we uncover some constraints that imply the existence of compact quark stars of the $Ia$-supernova type in the $QMF$-spacetime with $F(R,T)$-gravity.

1. Introduction

Solitons reveal a wide variety of symmetries, extending from the inherent symmetries of their underlying equations to the appearance of broken symmetries in specific physical situations. Soliton symmetries are crucial for understanding the properties of these nonlinear waves and have been the subject of extensive physics and mathematics research.

Certain solitons, like those found in Ricci solitons in General Relativity $\left(GR\right)$ [1], can have a geometry that is rotationally symmetric.

The String cloud [2] is a group of cosmic strings, which are one-dimensional topological flaws claimed to have originated in the early universe. In a spherically symmetric spacetime with conformal motion, that motion is a particularly static ones, through a property called conformal flatness and related to the presence of conformal Killing vector fields [3]. The strange quark matter is thought to be attached to a cosmic string cloud. In order to explain the features of matter and spacetime under particular gravitational conditions, this scenario has been studied in physics research. Typically, this is done by solving Einstein’s field equations to obtain solutions that characterize such a system and its physical characteristics [4].

The concept of a quark matter fluid, which is upheld by the degenerate pressure of quark matter [5], has been proposed for a star smaller than a neutron star. A number of researchers have investigated such a quark star [6]. Few stars might actually be strange stars made completely of unusual materials, as suggested by Alcock et al. [7] and Haensel et al. [8]. In [9], scientists investigated the characteristics of quark stars, while authors examined into the quark matter associated with the string cloud in a spherically symmetric spacetime that permits conformal motion [10].

The MIT bag model [11] describes the behavior of quarks compressed to very high densities, like in the cores of neutron stars, where they form a degenerate Fermi gas. According to this hypothesis, the model’s vacuum energy density functions as a constant bag pressure (also known as the bag constant), keeping the quarks from dispersing by confining them in a limited area and balancing their internal Fermi pressure.

According to this concept, quark matter is made up of electrons, massless quarks u, heavy quarks d, and quarks s [12]. When quarks are massless and not interacting, the bag model predicts that quark pressure will exist [11]. A number of variables relative to the fundamental forces and the characteristics of the particles interacting with one another cause quarks to have different masses. These are the primary causes of the quark mass variation. Through their interactions with the Higgs field, quarks gain mass. Through the Higgs mechanism [13], quarks interact with this field to acquire mass. u-quarks (up-quarks) is the lightest of all quarks with weak interaction.

Quark energy density ${\rho }_q$ and quark pressure is predicted by the bag model to exist when quarks are massless and not interacting [11]

$${\rho }_q=3p_q,$$

(1)

where the quark pressure is represented by $p_q$.

In addition, the energy density and total pressure are provided as [2,14]

$${\rho }_Q={\rho }_q+{\mathcal{B}}_c,$$

(2)

$$p_Q=p_q-{\mathcal{B}}_c.$$

(3)

Eventuality, equation of state ($\mathcal{E}o\mathcal{S}$) is used to produce quark matter [5]

$$p_Q={\displaystyle \frac{{\rho }_Q-4{\mathcal{B}}_c}{3}},$$

(4)

where ${\mathcal{B}}_c$ is known as a bag constant, which is the variation in the bag constant, or energy density, between the perturbative and non-perturbative $QCD$ vacuums. Thus, the bag model [11] equation of state (4) provides the pressure-energy density connection for quark matter.

The $GR$ spacetime and cosmology both reflect the time-oriented (maintains a steady global direction for time) 4-dimensional connected Lorentzian manifold where four dimensions to accommodate three spatial dimensions and one time dimension, forming a continuous spacetime. After presenting Lorentzian manifold geometry, we probe the characteristics of the vectors at the manifold. For $GR$ analysis, Lorentzian manifolds are hence becoming the useful framework [15,16].

Definition 1. 

A quasi-Einstein Lorentzian manifold is referred to as a perfect fluid spacetime if the Ricci tensor has the composition [16,17]

$$Ric=A_{1}d+A_2\eta \otimes \eta ,$$

(5)

where d is Lorentzian metric, 1-form η and there are scalars $A_1$ and $A_2$.

The $GR$ may be rebuilt explicitly using the effective momentum tensor [18]. When a time-like vector field $\zeta$ such that $g(\zeta ,\zeta )=-1$ can be utilized in $GR$. Moreover, anisotropic pressure, isotropic pressure, and density, are represented the energy momentum tensor [15,16].

In particular, physical matter symmetry is a fundamental concept in the $GR$ and related theories such as spacetime-matter theory, and spacetime geometry is defined and influenced by matter-energy through its interaction and coupling with spacetime. The curvature of spacetime is determined by the distribution of matter and energy, and the ability to move of matter and energy is determined by this curvature, resulting in a dynamic and interdependent connection. More specifically, the categories of solutions (solitons) to field equations are typically made simpler by the metric of symmetry. One prominent symmetry pertaining to the geometrical flow of spacetime is the soliton.

The theory of a Ricci flow was put forward by Hamilton [1] in 1988. The Ricci flow, proposed in Riemannian Geometry, provides the main explanation for a self-similar solutions of its singularities.

The Ricci flow equation is [1]

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}d\left(t\right)=-2Ric\left(d\left(t\right)\right),d_0=d\left(0\right).\end{array}$$

(6)

Definition 2. 

If $(M,d)$ is a Riemannian manifold that admits a vector field $\mathcal{F}$, then $(M,d)$ is a Ricci soliton [1]

$$Ric+{\displaystyle \frac{1}{2}}{\mathfrak{L}}_{\mathcal{F}}d+\lambda d=0,$$

(7)

where ${\mathfrak{L}}_{\mathcal{F}}$, $Ric$, and λ represent the Lie derivative, the Ricci tensor, and a real number, respectively. According to (7), a Ricci soliton ($RS$) is either increasing, stable, or declining based on whether $\lambda >0$, $\lambda =0$, or $\lambda <0$, respectively.

An Einstein metric’s natural extension is a Ricci soliton. This indicates that a particular kind of Ricci soliton $({\mathfrak{L}}_{\mathcal{F}}d=0)$ is an Einstein manifold. The Ricci curvature of Einstein manifolds is proportional to the metric tensor, which is defined as [19]

$$Ric=\lambda d.$$

Ricci solitons are self-similar solutions in both contexts. In Ricci flow theory, the Ricci soliton is a geometric concept that refers to a fixed point or self-similar structure of the time-dependent Ricci flow itself, whereas in $GR$, the Ricci soliton refers to a specific solution to the Einstein field equations, frequently with a perfect fluid [16] or other matter sources like quark matter fluid [10].

The main distinction is that Ricci flow solitons are mathematical structures that characterize the singularities and long-term behavior of the metric evolution represented by the Ricci flow, while $GR$-based solitons are physical, spacetime solutions. The first of those is the Ricci soliton, which is a particular solution to the Einstein field equations that we examine in this paper.

The typical method for examining cosmic dynamics is known as Einstein’s version of the gravitational field equations [19,20]. The field equations of Einstein offer a good theoretical method to study the field equations that relate the distribution of mass, energy, momentum, and stress in spacetime to its curvature, or gravity. The acceleration of the early and late universe (universe expansion) cannot be explained by $GR$ without accounting for the Dark energy.

Therefore, many mathematicians and physicists created more complex theories of gravity as a result of this circumstance, such as $F\left(R\right)$-gravity. Sotiriou and Faraoni [21] derived the field equation and discussed some theoretical aspects for $F\left(R\right)$-gravity. In addition, Kobayashi and Maeda [22] explored the cosmological evolution in $F\left(R\right)$-gravity and the viability of $F\left(R\right)$-gravity. Several examples include $F\left(G\right)$-gravity [23] and $F(R,T)$-gravity [24]. Einstein’s $GR$ is derived from the hypothesis that the Einstein–Hilbert action is the action for $GR$. These theories may provide a reliable substitute for quantum gravity, as they depart from the conventional mainstream theory of gravity [25].

The $F\left(R\right)$ gravity can be acquired by extending $GR$ using the Einstein–Hilbert Lagrangian density and becomes a function $F\left(R\right)$, wherein R represents the Ricci scalar. In the $F\left(R\right)$, the heavy neutron stars are resolved by higher-order curvature. One of the astronomical theories that cannot sustain $F\left(R\right)$-gravity, for example, stable star structure [26,27,28]. Order reduction, also known as perturbative restrictions, is another technique to examine the feasibility of $F\left(R\right)$ theories in the strong gravity regime [26,27,28].

Currently, neutron stars are the focus of much scientific attention since they serve as actual sky laboratories for a variety of scientific disciplines, including nuclear physics. Modified gravity is a more appropriate theory that can accurately describe massive neutron stars, estimate their maximum mass limit, and determine how the sound speed affects this limit for a wide range of $EoS$.

The $F\left(R\right)$ gravity’s applicability is questioned, though, because of its limitations with regard to equilibrium with the solar system (for additional details, see [22,29]). Authors [30] developed a gravity model that was more thorough, which they called $F(R,T)$-theory of gravity. This was accomplished by considering that an arbitrary function of T and R. The rapid late-time expansion of the universe was well explained by this concept.

In [31], Blaga employed Ricci and Einstein solitons to illustrate the characteristics of the perfect fluid spacetime. Using a range of methods, Danish et al. (cf. [32,33]) used solitons to investigate strange quark matter fluid spacetime and perfect fluid spacetime in $GR$.

Quark matter fluid study is an interesting area these days. Under a number of assumptions, quark matter fluid is investigated in $GR$. Sotani et al. used gravitational waves to obtain the $EoS$ of $QMF$ in [5]. Using the Einstein–Gauss–Bonnet theory of string clouds in a five-dimensional spacetime, Herscovich and Richarte [6] examined the black hole and came up with interesting findings on singularities. A quark surface was revealed by Alcock and his coauthors [7], as they discussed the characteristics of odd stars. The stability of quark stars, which are represented by the bag constant, was investigated by the authors in [8]. Furthermore, the dynamical behavior of odd stars was discussed by Cheng et al. [9]. The Einstein field equation with weird quark matter connected to a string cloud via conformal motion was studied by Yavuz et al. [10]. Authors [12] have studied quark matter in symmetric spacetime. In $F(R,T)$-gravity theory, Agarwal and Pawar [34] examined a cosmological model utilizing quark matter. The properties of $F(R,T)$-gravity filled with perfect fluid admitted different solitons and gradient solitons were recently investigated in 2022 by Siddiqi et al., [2,35,36]. Motivated by previous research, we examine the $F(R,T)$-gravity model in this paper, which includes quark matter admitting a Ricci soliton.

Furthermore, the current study is designed differently in the following respects when compared to the published results mentioned above. In the current research article, we have considered a generalized gravity model called $F(R,T)$-gravity model, which involves a random connection between geometry and matter (represented by the trace of the stress-energy tensor), and an arbitrary function of T and the Ricci scalar for the Lagrangian. In Section 2, the field equations and $EoS$ for this $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity model attached to $QMF$ has been determined by us. In Section 3, we explore $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$. The gravity model is attached to $QMF$ with respect to the $RS$ metric, which differs from the previously published results. In $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity admitting $RS$ with a conformal vector field, we addressed the energy condition of quark matter fluid spacetime in Section 4. These criteria are different from the findings that have already been discussed. In Section 5, we prove the existence of black holes in the quark matter fluid spacetime in $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity admitting $RS$ with a conformal vector field. In Section 6, we derived the Schrödinger Equation in terms of the equation of state in the same framework, which has not been discussed in the previous studies. Lastly, we reveal some constraints that point to the existence of compact quark stars of the Ia supernova type in the $QMF$-spacetime with $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity.

2. Field Equation in $\mathit{F}\mathbf{(}\mathit{R}\mathbf{,}\mathit{T}\mathbf{)}\mathbf{=}{\mathit{F}}_{\mathbf{1}}\mathbf{\left(}\mathit{R}\mathbf{\right)}\mathbf{+}{\mathit{F}}_{\mathbf{2}}\mathbf{\left(}\mathit{T}\mathbf{\right)}$-Gravity Model Coupled with $\mathit{Q}\mathit{M}\mathit{F}$

The $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity coupled with $QMF$ is discussed in this part of the paper. We can obtain some cosmological models for distinct values of T and R [30] as this model depends on the physical characteristics of the $QMF$. We employ the following model as an example [30]

$$F(R,T)=F_1\left(R\right)+F_2\left(T\right),$$

(8)

where $F_1\left(R\right)$ and $F_2\left(T\right)$ represent the arbitrary functions of R and T, respectively.

Remark 1. 

In the case $F_2\left(T\right)=0$, we re-obtain the field equations of standard $F\left(R\right)$ gravity. Additionally, the effective, matter (and time) dependent coupling that provides the gravitational coupling is proportionate to the derivative of the function $F_2$ with respect to T.

Remark 2. 

The action in teleparallel gravity $F\left(T\right)$ depends only on the torsion scalar, represented by the symbol T. $F(R,T)$-gravity theory beyond standard teleparallelism by allowing for functional dependencies not only on the torsion scalar ’T’ but also on the matter stress-energy tensor. Moreover, standard teleparallel theories are based on modifications to the torsion tensor T and not the Ricci scalar, the particular $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity model differs from typical teleparallel gravity models such as $F\left(T\right)$ gravity in that it modifies the gravitational action with a function of both the Ricci scalar R and the trace of the stress-energy tensor T. The physical consequences of this $F(R,T)$ model are probably different, and it might not be a direct generalization of $F\left(T\right)$ gravity models, which are based on non-zero torsion and the Weitzenböck spacetime.

It is assumed that the Einstein–Hilbert action term is

$${\Pi }_E={\displaystyle \frac{1}{16\pi }}\int [{\mathbf{L}}_m+F(R,T)+]\sqrt{(-d)}{d}^{4}x.$$

(9)

The Lagrangian is represented as ${\mathbf{L}}_m$. The stress momentum tensor can be obtained from

$$T_{ab}=-{\displaystyle \frac{2}{\sqrt{-d}}}{\displaystyle \frac{\delta \left(\sqrt{-d}{\mathbf{L}}_m\right)}{\delta d^{ab}}}.$$

(10)

Assume that ${\mathbf{L}}_m$ depends only on $d_{ab}$. and is not influenced by its derivatives. The variation of (9) regarding the $d_{ab}$ suggests

$$\begin{array}{c}{F^{\prime }}_1\left(R\right)R{ic}_{ab}-{\displaystyle \frac{1}{2}}{F}_1\left(R\right)d_{ab}+(d_{ab}{\nabla }_c{\nabla }^c-{\nabla }_a{\nabla }_b){F^{\prime }}_1\left(R\right)\\ =8\pi {T_{ab}-{F^{\prime }}_2\left(T\right)T_{ab}-F^{\prime }}_2\left(T\right){\Xi }_{ab}+{\displaystyle \frac{1}{2}}{F}_2\left(T\right)d_{ab}.\end{array}$$

(11)

where $F_1^{\prime }\left(R\right)={\displaystyle \frac{\partial F(R,T)}{\partial R}}$ and $F_2^{\prime }\left(T\right)={\displaystyle \frac{\partial F(R,T)}{\partial T}}$.

The notation used is standard; ${\nabla }_a$ and $\square \equiv {\nabla }_c{\nabla }^c$ denote the d’Alembert operator.

Now, the covariant derivative, we have

$${\Xi }_{ab}=-2T_{ab}+d_{ab}{\mathbf{L}}_m-2d^{lk}{\displaystyle \frac{{\partial }^2{\mathbf{L}}_m}{\partial d^{ab}\partial d^{lk}}}.$$

(12)

The standard $F\left(R\right)$-gravity field equation can be obtained again if $F_2\left(T\right)=0$.

Let the $QMF$ with quark matter total pressure $p_Q$, total energy density ${\rho }_Q$ and velocity vector ${\eta }^{\alpha }$. Given our advantage in choosing ${\mathbf{L}}_m$. As a result, we fix ${\mathbf{L}}_m=-p_Q$.

For quark matter, the energy-momentum tensor is [34]

$$T_{ab}^{\left(Quark\right)}=(p_Q+{\rho }_Q){\eta }_a{\eta }_b-p_Q{d}_{ab},$$

(13)

where

$${\eta }^a{\nabla }_b{\eta }_a=0,{\eta }_a·{\eta }^a=1,$$

(14)

acts as the vector of four velocities. From (12) and (13), we obtain the variation of stress energy of $QMF$ as follows:

$${\Xi }_{ab}=-2T_{ab}-p_Q{d}_{ab}.$$

(15)

Following the incorporation of (8) and (11), we obtain

$$F_1^{\prime }\left(R\right)Ri{c}_{ab}={\displaystyle \frac{1}{2}}{F}_1\left(R\right)d_{ab}+8p{T}_{ab}-F_2^{\prime }\left(T\right)T_{ab}-F_2^{\prime }\left(T\right){\Xi }_{ab}+{\displaystyle \frac{1}{2}}{F}_2\left(T\right)d_{ab}.$$

(16)

The gravitational field equation for $QMF$ in light of (13), (14), and (15) in $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity model (16) becomes

$$\begin{array}{c}Ri{c}_{ab}={\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{{\displaystyle \frac{1}{2}}(F_1\left(R\right)+F_2\left(T\right))-8\pi p_Q-p_Q{F}_2^{\prime }\left(T\right)\right\}{d}_{ab}\\ +{\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{({\rho }_Q+p_Q)(8\pi -F_2^{\prime }\left(T\right))\right\}{\eta }_a{\eta }_b.\end{array}$$

(17)

reduces to

$$R={\displaystyle \frac{2[F_1\left(R\right)+F_2\left(T\right)]}{F_1^{\prime }\left(R\right)}}-{\displaystyle \frac{4p_Q{F}_2^{\prime }\left(T\right)}{F_1^{\prime }\left(R\right)}}-{\displaystyle \frac{8\pi (3p_Q-{\rho }_Q)}{F_1^{\prime }\left(R\right)}}-({\rho }_Q+p_Q){\displaystyle \frac{F_2^{\prime }\left(T\right)}{F_1^{\prime }\left(R\right)}}.$$

(18)

Thus, for a spacetime $(M^4,d)$ in with $QMF$ in $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity, the Ricci tensor assumes the shape

$$Ri{c}_{ab}=\alpha d_{ab}+\beta {\eta }_a{\eta }_b,$$

(19)

where

$$\alpha ={\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{{\displaystyle \frac{1}{2}}(F_1\left(R\right)+F_2\left(T\right))-8\pi p_Q-p_Q{F}_2^{\prime }\left(T\right)\right\},$$

(20)

$$\beta ={\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{({\rho }_Q+p_Q)(8\pi -F_2^{\prime }\left(T\right))\right\}.$$

(21)

We make the assumption that a and b are not both zero in full manuscript. A comparable technique was used in [35] to determine the expression of the Ricci tensor, but for consistency, we are also providing the proof. Consequently, we earn

Theorem 1. 

The $QMF$ spacetime in the $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity model has the following Ricci tensor

$$Ri{c}_{ab}={\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{{\displaystyle \frac{1}{2}}(F_1\left(R\right)+F_2\left(T\right))-8\pi p_Q-p_Q{F}_2^{\prime }\left(T\right)\right\}{d}_{ab}$$

$$+{\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{({\rho }_Q+p_Q)(8\pi -F_2^{\prime }\left(T\right))\right\}{\eta }_a{\eta }_b.$$

Corollary 1. 

In $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity, the scalar curvature of the $QMF$ spacetime appears by

$$R={\displaystyle \frac{2[F_1\left(R\right)+F_2\left(T\right)]}{F_1^{\prime }\left(R\right)}}-{\displaystyle \frac{4p_Q{F}_2^{\prime }\left(T\right)}{F_1^{\prime }\left(R\right)}}-{\displaystyle \frac{8\pi (3p_Q-{\rho }_Q)}{F_1^{\prime }\left(R\right)}}-({\rho }_Q+p_Q){\displaystyle \frac{F_2^{\prime }\left(T\right)}{F_1^{\prime }\left(R\right)}}.$$

Equation (17) can now be written in index-free notation as

$$Ric=\alpha d+\beta \eta \otimes \eta .$$

(22)

The following result is obtained by referring to Equations (3) and (17).

In particular, if $F_2\left(T\right)=0$ then $F(R,T)$ gravity retain the $F\left(R\right)$-gravity. Thus from the Theorem 1 and Corollary 1, we have

Corollary 2. 

The $QMF$ spacetime in the $F\left(R\right)$-gravity model has the following Ricci tensor

$$\begin{array}{c}\hfill Ri{c}_{ab}={\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{{\displaystyle \frac{1}{2}}(F_1\left(R\right)-8\pi p_Q)\right\}{d}_{ab}\\ \hfill +{\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{({\rho }_Q+p_Q)8\pi \right\}{\eta }_a{\eta }_b.\end{array}$$

Corollary 3. 

In the $F\left(R\right)$-gravity model, the scalar curvature of the $QMF$ spacetime appears by

$$R={\displaystyle \frac{2F_1\left(R\right)}{F_1^{\prime }\left(R\right)}}-{\displaystyle \frac{8\pi (3p_Q-{\rho }_Q)}{F_1^{\prime }\left(R\right)}}.$$

(23)

Theorem 2. 

A spacetime $(M^4,d)$ in $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity with $QMF$ is a perfect fluid spacetime, provided $F_1^{\prime }\left(R\right)\ne 0$.

Corollary 4. 

A $QMF$ spacetime $(M^4,d)$ in $F\left(R\right)$-gravity with $QMF$ is a perfect fluid spacetime.

Moreover, in the light of (1) and (2), we find the exact expression of stress-momentum tensor for $QMF$

$$T_{ab}^{\left(Quark\right)}=(p_q+{\rho }_q){\eta }_a{\eta }_b-p_q{d}_{ab}-{\mathcal{B}}_c{d}_{ab}.$$

(24)

Further terms that correspond to a magnetic field, potential anisotropic stresses associated with certain corrections to the bag model $EoS$ (3), super fluidity, etc., can be added to the energy-momentum tensor of the perfect quark fluid to provide further options. It is also possible to minimize the contribution of the imperfect fluid component due to the arbitrary addition of the additional term to the energy-momentum tensor. However, realistic quark star models do not exclude out a linearly growing charge density that produces a highly charged strange star surface. The solution to the gravitational field equations and solution (17) are the same if ${\mathcal{B}}_c=0$.

Now, employing Equation (18), we derive

$$p_Q+{\rho }_Q={\displaystyle \frac{2[F_1\left(R\right)+F_2\left(T\right)]}{F_2^{\prime }\left(T\right)}}-\left[4p_Q+{\displaystyle \frac{8\pi (3p_Q-{\rho }_Q)}{F_2^{\prime }\left(T\right)}}+{\displaystyle \frac{R{F}_1^{\prime }\left(R\right)}{F_2^{\prime }\left(T\right)}}\right],$$

(25)

provided $F_2^{\prime }\left(T\right)\ne 0.$

According to [37], the $EoS$ for dark energy is $p_Q=-{\rho }_Q+F\left(r\right)$, where $F\left(r\right)$ is a function of the scale factor “$r\left(t\right)$”, where t is the cosmic time. Srivastava also demonstrated that $\omega ={\displaystyle \frac{p_Q}{{\rho }_Q}}=-1$ produces dark era, while $\omega <-1$ and $\omega >-1$ imply a change from phantom to non-phantom [37].

Theorem 3. 

If the matter of $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity is $QMF$, then $\mathcal{E}o\mathcal{S}$ is given by (25).

Consider that the $QMF$$\mathcal{E}o\mathcal{S}$ is and that the source is of the radiation type $\omega ={\displaystyle \frac{({\rho }_Q-4{\mathcal{B}}_c)}{3}}$. Combining this finding with Equation (25) results in

$$p_Q={\displaystyle \frac{[F_1^{\prime }\left(T\right)+F_2^{\prime }\left(T\right)]}{F_2^{\prime }\left(T\right)}}-{\displaystyle \frac{R{F}_1^{\prime }\left(R\right)}{8F_2^{\prime }\left(T\right)}},$$

(26)

$${\rho }_Q={\displaystyle \frac{3[F_1^{\prime }\left(T\right)+F_2^{\prime }\left(T\right)]}{F_2^{\prime }\left(T\right)}}-{\displaystyle \frac{3R{F}_1^{\prime }\left(R\right)}{8F_2^{\prime }\left(T\right)}}$$

(27)

$$p_q={\displaystyle \frac{[F_1^{\prime }\left(T\right)+F_2^{\prime }\left(T\right)]}{F_2^{\prime }\left(T\right)}}-{\displaystyle \frac{R{F}_1^{\prime }\left(R\right)}{8F_2^{\prime }\left(T\right)}}+{\mathcal{B}}_c,$$

(28)

$${\rho }_q={\displaystyle \frac{3[F_1^{\prime }\left(T\right)+F_2^{\prime }\left(T\right)]}{F_2^{\prime }\left(T\right)}}-{\displaystyle \frac{3R{F}_1^{\prime }\left(R\right)}{8F_2^{\prime }\left(T\right)}}-{\mathcal{B}}_c.$$

(29)

where $F_2^{\prime }\left(T\right)\ne 0$. Consequently, we obtain the following corollaries:

Corollary 5. 

If the origin of the $QMF$ in the $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity is a radiation type. Then (26) and (27) determines the total pressure $p_Q$ and the total energy density ${\rho }_Q$.

Corollary 6. 

If the origin of the $QMF$ in the $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity is a radiation type. Then (28) and (29) determines the quark pressure $p_q$ and quark energy density ${\rho }_q$.

In the case of phantom barrier, ${\rho }_q=-p_q={\displaystyle \frac{[F_1\left(R\right)+F_2\left(T\right]}{16\pi +2F_2^{\prime }\left(T\right)}}-{\displaystyle \frac{R{F}_1^{\prime }\left(R\right)}{32\pi +4F_2^{\prime }\left(T\right)}}+{\mathcal{B}}_c$. Consequently, we may deduce

Corollary 7. 

If a $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity coupled with $QMF$ contains a phantom barrier type source of matter, then the quark pressure and quark energy density are determined as

$${\rho }_q=-p_q={\displaystyle \frac{[F_1\left(R\right)+F_2\left(T\right]}{16\pi +2F_2^{\prime }\left(T\right)}}-{\displaystyle \frac{R{F}_1^{\prime }\left(R\right)}{32\pi +4F_2^{\prime }\left(T\right)}}+{\mathcal{B}}_c.$$

(30)

While $\omega <-1$ and $\omega >-1$ imply a transition from phantom to non-phantom, $\omega ={\displaystyle \frac{p_Q}{{\rho }_Q}}=-1$ creates a phantom barrier, according to Corollary 3 and using (23). We acquire the following relations:

$$3p_Q-{\rho }_Q={\displaystyle \frac{1}{4\pi }}\left[F_1\left(R\right)-{\displaystyle \frac{R}{2}}{F}_1^{\prime }\left(R\right)\right]$$

(31)

$${\displaystyle \frac{p_Q}{{\rho }_Q}}={\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{4\pi }}\left[F_1\left(R\right)-{\displaystyle \frac{R}{2}}{F}_1^{\prime }\left(R\right)\right].$$

(32)

or

$$\omega ={\displaystyle \frac{F_1\left(R\right)}{4\pi }}+{\displaystyle \frac{1}{3}}-{\displaystyle \frac{R}{2}}{F}_1^{\prime }\left(R\right).$$

(33)

Theorem 4. 

If the matter of $F\left(R\right)$-gravity is $QMF$, then $\mathcal{E}o\mathcal{S}$ is given by (32).

Theorem 5. 

If the matter of $F\left(R\right)$-gravity is $QMF$, then evolution of the universe is given in the following table through $\mathcal{E}o\mathcal{S}$ (33) as

$\mathcal{E}o\mathcal{S}(\omega ={\displaystyle \frac{p_Q}{{\rho }_Q}})$	$Restrictions on F_1\left(R\right)and{F}_1^{\prime }\left(R\right)$	$Evolution of the Universe$
$\omega =0$	${\displaystyle \frac{F_1\left(R\right)}{4\pi }}+{\displaystyle \frac{1}{3}}={\displaystyle \frac{R}{2}}{F}_1^{\prime }\left(R\right)$	$Dustmatterera$
$\omega >-1$	${\displaystyle \frac{F_1\left(R\right)}{4\pi }}+{\displaystyle \frac{1}{3}}>{\displaystyle \frac{R}{2}}{F}_1^{\prime }\left(R\right)$	$Quintessenceera$
$\omega <-1$	${\displaystyle \frac{F_1\left(R\right)}{4\pi }}+{\displaystyle \frac{1}{3}}<{\displaystyle \frac{R}{2}}{F}_1^{\prime }\left(R\right)$	$Phantomera$
$\omega ={\displaystyle \frac{1}{3}}$	${\displaystyle \frac{F_1\left(R\right)}{4\pi }}={\displaystyle \frac{R}{2}}{F}_1^{\prime }\left(R\right)$	$Radiationera.$

In particular, the phantom barrier in $F\left(R\right)$-gravity, the total quark pressure and total quark density are given as,

$$p_Q=-{\rho }_Q={\displaystyle \frac{1}{16\pi }}\left[F_1\left(R\right)-{\displaystyle \frac{R}{2}}{F}_1^{\prime }\left(R\right)\right].$$

Thus, we can articulate the following corollaries:

Corollary 8. 

If $F\left(R\right)$-gravity coupled with $QMF$ contains a phantom barrier type source of matter, then the total quark pressure $p_Q$ and total quark energy density ${\rho }_Q$ are calculated as

$$p_Q=-{\rho }_Q={\displaystyle \frac{1}{16\pi }}\left[F_1\left(R\right)-{\displaystyle \frac{R}{2}}{F}_1^{\prime }\left(R\right)\right].$$

(34)

Corollary 9. 

If $F\left(R\right)$-gravity coupled with $QMF$ contains a phantom barrier type source of matter, then the quark pressure $p_q$ and quark energy density ${\rho }_q$ are calculated as

$$p_q=-{\rho }_q={\displaystyle \frac{1}{16\pi }}\left[F_1\left(R\right)-{\displaystyle \frac{R}{2}}{F}_1^{\prime }\left(R\right)\right]+{\mathcal{B}}_c.$$

(35)

3. $\mathit{F}\mathbf{(}\mathit{R}\mathbf{,}\mathit{T}\mathbf{)}\mathbf{=}{\mathit{F}}_{\mathbf{1}}\mathbf{\left(}\mathit{R}\mathbf{\right)}\mathbf{+}{\mathit{F}}_{\mathbf{2}}\mathbf{\left(}\mathit{T}\mathbf{\right)}$-Gravity Model Attached with $\mathit{QMF}$ Admitting Ricci Solitons

In this segment, we study the $RS$ in the $F(R,T)$-gravity here, linked to a $QMF$ whose timelike velocity vector field $\zeta$ is its conformal vector field (in brief $CVF$).

Authors [3] state that a $CVF$$\mathfrak{F}$ on the Lorentzian spacetime manifold $(M^4,d)$ is characterized as

$${\mathfrak{L}}_{\mathfrak{F}}d=2\mho d$$

(36)

wherein ℧ is a smooth function on $M^4$. $\mathfrak{F}$ is homothetic if ℧ is constant, and is Killing ($KVF)$ when $\mho =0$.

Once $\mathfrak{F}=\zeta$ is adopted, Equation (7) yields

$$\left({\mathfrak{L}}_{\zeta }d\right)({\epsilon }_1,{\epsilon }_2)+2Ric({\epsilon }_1,{\epsilon }_2)+2\lambda d({\epsilon }_1,{\epsilon }_2)=0,$$

(37)

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

Considering (36), we obtain

$$Ric({\epsilon }_1,{\epsilon }_2)+(\lambda +\mho )d({\epsilon }_1,{\epsilon }_2)=0.$$

(38)

By entering (22) into (38), we gain

$$(\alpha +\lambda +\mho )d({\epsilon }_1,{\epsilon }_2)=-\beta \eta \left({\epsilon }_1\right)\eta \left({\epsilon }_2\right).$$

(39)

Entering ${\epsilon }_1={\epsilon }_2=\zeta$ into (39), such that $d(\zeta ,\zeta )=\eta \left(\zeta \right)=-1$, where $\zeta$ is a time-like velocity vector field and $\eta$ is the dual of d. Now, we discover

$$\lambda ={\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left({\displaystyle \frac{1}{2}}(F_1\left(R\right)+F_2\left(T\right)+8\pi {\rho }_Q\right)-\left({\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{({\rho }_Q+2p_Q)F_2^{\prime }\left(T\right)\right\}+\mho \right).$$

(40)

Next, we utilize an acronym in this section for convenience, ${\Theta }^F$ to indicate “$F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity attached with $QMF$”.

This leads us to the following result.

Theorem 6. 

If a $QMF$-spacetime $(M^4,d)$ in ${\Theta }^F$ admits a $RS$$(d,\zeta ,\lambda )$ with a time-like velocity $CVF$ζ, then $RS$ is growing, stable or decreasing referring as

1.

${\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left({\displaystyle \frac{1}{2}}(F_1\left(R\right)+F_2\left(T\right))+8\pi {\rho }_Q\right)>\left({\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{({\rho }_Q+2p_Q)F_2^{\prime }\left(T\right)\right\}+\mho \right)$,

2.

${\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left({\displaystyle \frac{1}{2}}(F_1\left(R\right)+F_2\left(T\right))+8\pi {\rho }_Q\right)=\left({\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{({\rho }_Q+2p_Q)F_2^{\prime }\left(T\right)\right\}+\mho \right)$ and

3.

${\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left({\displaystyle \frac{1}{2}}(F_1\left(R\right)+F_2\left(T\right))+8\pi {\rho }_Q\right)<\left({\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{({\rho }_Q+2p_Q)F_2^{\prime }\left(T\right)\right\}+\mho \right)$, respectively, provided ${\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\ne 0.$

Corollary 10. 

If a $QMF$-spacetime $(M^4,d)$ in ${\Theta }^F$ admits a $RS$ and if time-like velocity vector field ζ is Killing, then $RS$ is growing, stable, or decreasing and referred to as

1.

${\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left({\displaystyle \frac{1}{2}}(F_1\left(R\right)+F_2\left(T\right))+8\pi {\rho }_Q\right)>{\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{({\rho }_Q+2p_Q)F_2^{\prime }\left(T\right)\right\}$,

2.

${\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left({\displaystyle \frac{1}{2}}(F_1\left(R\right)+F_2\left(T\right))+8\pi {\rho }_Q\right)={\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{({\rho }_Q+2p_Q)F_2^{\prime }\left(T\right)\right\}$ and

3.

${\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left({\displaystyle \frac{1}{2}}(F_1\left(R\right)+F_2\left(T\right))+8\pi {\rho }_Q\right)<{\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{({\rho }_Q+2p_Q)F_2^{\prime }\left(T\right)\right\}$, respectively, provided ${\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\ne 0.$

In particular, if $F_2\left(T\right)=0\Rightarrow F_2^{\prime }\left(T\right)=0$, then $F(R,T)$-gravity turn up to $F\left(R\right)$-gravity. Therefore, in view of (41), we obtain

$$\lambda ={\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left({\displaystyle \frac{1}{2}}{F}_1\left(R\right)+8\pi {\rho }_Q\right)-\mho .$$

(41)

Now, we gain the following results for $F\left(R\right)$-gravity as well.

Corollary 11. 

If a $QMF$-spacetime $(M^4,d)$ in the $F\left(R\right)$-gravity model attached with $QMF$ admits a $RS$ with a time-like velocity $CVF$ζ, then $RS$ is decreasing, stable, or growing and referred to as

1.

${\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left({\displaystyle \frac{1}{2}}{F}_1\left(R\right)+8\pi {\rho }_Q\right)<\mho$,

2.

${\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left({\displaystyle \frac{1}{2}}{F}_1\left(R\right)+8\pi {\rho }_Q\right)=\mho$ and

3.

${\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left({\displaystyle \frac{1}{2}}{F}_1\left(R\right)+8\pi {\rho }_Q\right)>\mho$, respectively, provided ${\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\ne 0.$

Corollary 12. 

If a spacetime $(M^4,d)$ in the $F\left(R\right)$-gravity model attached with $QMF$ admits a $RS$ and if time-like velocity vector field ζ is $KVF$, then $RS$ is stable.

4. Standard Energy Conditions in ${\mathbf{\Theta }}^{\mathit{F}}$ Admits Ricci Soliton

Using [4] as a reference, we know that in the spacetime, the Ricci tensor $Ric$ meets the criterion

$$Ric(\zeta ,\zeta )>0,$$

(42)

then the time-like convergence condition (TCC) is the name given to Equation (42), for all time-like vector fields $\zeta \in \chi \left(M^4\right)$. It provides from (17).

$$Ric(\zeta ,\zeta )=\alpha +\beta .$$

If $Ric(\zeta ,\zeta )>0$, which is the $TCC$ [38,39] is satisfied, the spacetime in concern is valid.

$${\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left({\displaystyle \frac{1}{2}}(F_1\left(R\right)+F_2\left(T\right))+8\pi {\rho }_Q\right)>\left({\displaystyle \frac{1}{F_1^{\prime }\left(R\right)}}\left\{({\rho }_Q+2p_Q)F_2^{\prime }\left(T\right)\right\}+\mho \right).$$

(43)

A spacetime [39] obeys the strong energy condition $\left(SEC\right)$. Using the data from (43) and the previously provided information, we can claim that

Theorem 7. 

If a $QMF$-spacetime $(M^4,d)$ in ${\Theta }^F$ admits a $RS$$(d,\zeta ,\lambda )$ with a time-like velocity $CVF$ζ and obeys $TCC$, then $RS$ is growing.

Corollary 13. 

If a $QMF$-spacetime $(M^4,d)$ in the $F\left(R\right)$-gravity attached with $QMF$ admits a $RS$ with a time-like velocity $CVF$ζ and obeys $TCC$, then $RS$ is growing.

Remark 3. 

As shown by Hawking and Ellis [38] in 1973,

$TCC\Rightarrow NCC$, $TCC\Rightarrow SEC$ (Null convergence condition), and

$SEC\Rightarrow NEC$ (Null energy condition) all result in $TCC\Rightarrow NCC$.

Together, Theorem 7 and Remark 1 yield the following results:

Theorem 8. 

If a $QMF$-spacetime $(M^4,d)$ in ${\Theta }^F$ admits the expanding $RS$ with a $CVF$ζ, if (43) holds, then the $QMF$-spacetime $(M^4,d)$ in the ${\Theta }^F$ holds $SEC$.

Corollary 14. 

If a $QMF$-spacetime $(M^4,d)$ in ${\Theta }^F$ admits the expanding $RS$ with a conformal vector field ζ, if (43) holds, then the $QMF$-spacetime $(M^4,d)$ in the ${\Theta }^F$ holds $NCC$.

Corollary 15. 

If a $QMF$-spacetime $(M^4,d)$ in ${\Theta }^F$ admits the $RS$ with a $CVF$ζ and obeys $SEC$, then the Ricci tensor $Ric$ of expanding $RS$ is of the second Segre type (type III) [38].

Corollary 16. 

If a $QMF$-spacetime $(M^4,d)$ in ${\Theta }^F$ admits the $RS$ with a $CVF$ζ and obeys $SEC$, then the Ricci tensor $Ric$ of expanding $RS$ is of Petrov type I [38].

5. Implementation of the Penrose’s Singularity Theorem in ${\mathbf{\Theta }}^{\mathit{F}}$ with Ricci Solitons

A spacetime singularity, a point of infinite gravity and curvature that is frequently observed in the center of a black hole, will unavoidably occur from a gravitational collapse under realistic physical conditions, according to Roger Penrose’s singularity theorem, a basic finding in $GR$.

According to the theorem [40], singularities are a strong prediction of Einstein’s theory and not just theoretical phenomena. It states that a singularity will always emerge once a region of spacetime turns into a “trapped surface,” where all light rays are drawn inward. Moreover, singularities are a key component of black hole physics since the theorem verifies that any black hole must have a singularity at its center.

Hawking’s work [38], which was based on Penrose’s, suggested that time started from a singularity by extending the theorems to encompass the singularity at the Big Bang.

Due to the Penrose theorem [40], if matter satisfies sufficient energy criteria, there will always be some kind of geodesic incompleteness within any black hole. Since light beams are constantly focused together by gravity and never driven apart, the black-hole singularity theorem requires a weak energy condition, which is true whenever the energy of matter is non-negative. This theorem is more specific and only applies when matter satisfies the strong energy requirement, which is when the energy exceeds the pressure.

Remark 4. 

According to Penrose’s singularity theorem [40], Vilenkin and Wall [41] demonstrated that a spacetime M obeys the $SCC\Rightarrow NCC$, then

(1) the connected Cauchy surface of spacetime M is non-compact, and

(2) there are some black holes and a trapped surface in spacetime M. (any sphere of sufficiently large radius is a trapped surface). It is not inside the black holes [40].

Considering Theorem 7, Remark 1, Remark 2, and Corollary 14, we can articulate the next result for this framework.

Theorem 9. 

If a $QMF$-spacetime $(M^4,d)$ in ${\Theta }^F$ admits a growing $RS$ with a $CVF$ζ and if the $QMF$-spacetime $(M^4,d)$ fulfills $NCC$, then the $QMF$-spacetime $(M^4,d)$ contains some black holes include a trapped surface, which is not inside the black holes in $QMF$ spacetime $(M^4,d)$ in ${\Theta }^F$.

Corollary 17. 

If a $QMF$-spacetime $(M^4,d)$ in ${\Theta }^F$ admits a growing $RS$ with a $KVF$ζ and if the $QMF$-spacetime $(M^4,d)$ fulfills $NCC$, then the $QMF$-spacetime $(M^4,d)$ contains some black holes include a trapped surface, which is not inside the black holes in $QMF$-spacetime $(M^4,d)$ in ${\Theta }^F.$

Corollary 18. 

If a $QMF$-spacetime $(M^4,d)$ in ${\Theta }^F$ admits a growing $RS$ with a $KVF$ζ and if the $QMF$-spacetime $(M^4,d)$ fulfills $NCC$, then the $QMF$-spacetime $(M^4,d)$ contains a non-compact connected Cauchy surface in ${\Theta }^F.$

6. Schrödinger Equation for ${\mathbf{\Theta }}^{\mathit{F}}$

We are aware that a well-known theory of quantum chromodynamics $\left(QCD\right)$ [42] describes the strong interactions that take place at the subatomic level. The energy eigenvalue of the quarks $\mathcal{E}o\mathcal{S}$ (25) was found by solving the Schrödinger equation using the quantum mechanics approach.

The most general form is the time-dependent Schrödinger equation

$$i\hslash {\displaystyle \frac{{\ddot{\Psi }}_n}{{\Psi }_n}}=\widehat{H},$$

(44)

where i is the imaginary unit, ℏ serves as the Plank constant of proportionality linking the quantum mechanical properties and energy momentum of matter, and $\widehat{H}$ is the Hamiltonian for the total energy of the system.

In 2009, Yurov and Astashenok showed in [43] that a Schrödinger equation has accurate solutions ${\Psi }_n$ in cosmology for gravity in such a way that

$${\displaystyle \frac{{\ddot{\Psi }}_n}{{\Psi }_n}}=m{\mathcal{U}}_n,$$

(45)

where the potential is ${\mathcal{U}}_n=m{n}^2{\rho }_Q-{\displaystyle \frac{3mn}{2}}({\rho }_Q+p_Q)$, n is the primary quantum number, and m is the mass of quarks. The potential ${\mathcal{U}}_n$ proportional to $m{n}^2{\rho }_Q-{\displaystyle \frac{3mn}{2}}({\rho }_Q+p_Q).$

Regarding $n=1,2,3,\cdots$. If it is possible to write down the energy of each quantum state. Thus, we have

$${\displaystyle \frac{{\ddot{\Psi }}_n}{{\Psi }_n}}=m{n}^2{\rho }_Q-{\displaystyle \frac{3mn}{2}}({\rho }_Q+p_Q).$$

(46)

After comparing (44) with (46), we get the value of Plank constant

$$\hslash ={\displaystyle \frac{\widehat{H}}{i\left\{m{n}^2{\rho }_Q-\frac{3mn}{2}({\rho }_Q+p_Q)\right\}}}.$$

(47)

Given (25) and (46), the Schrödinger equation for the potential of the ${\mathcal{U}}_n$ in ${\Theta }^F$ is now

$${\displaystyle \frac{{\ddot{\Psi }}_n}{{\Psi }_n}}=m{n}^2{\rho }_Q-mn{\displaystyle \frac{2[F_1\left(R\right)+F_2\left(T\right)]}{F_2^{\prime }\left(T\right)}}-{\displaystyle \frac{3mn}{2}}\left[4p_Q+{\displaystyle \frac{8\pi (3p_Q-{\rho }_Q)}{F_2^{\prime }\left(T\right)}}+{\displaystyle \frac{R{F}_1^{\prime }\left(R\right)}{F_2^{\prime }\left(T\right)}}\right]$$

(48)

where the potential is ${\mathcal{U}}_n=m{n}^2{\rho }_Q-mn{\displaystyle \frac{2[F_1\left(R\right)+F_2\left(T\right)]}{F_2^{\prime }\left(T\right)}}-{\displaystyle \frac{3mn}{2}}\left[4p_Q+{\displaystyle \frac{8\pi (3p_Q-{\rho }_Q)}{F_2^{\prime }\left(T\right)}}+{\displaystyle \frac{R{F}_1^{\prime }\left(R\right)}{F_2^{\prime }\left(T\right)}}\right].$ Consequently, we are presenting the following findings.

Theorem 10. 

If a $QMF$-spacetime $(M^4,g)$ in ${\Theta }^F$, then the Schrödinger equation for the potential ${\mathcal{U}}_n$ in ${\Theta }^F$ is given by Equation (48).

Corollary 19. 

If the origin of matter is of phantom barrier type in ${\Theta }^F$ then the Schrödinger equation for the potential ${\mathcal{U}}_n$ is

$${\displaystyle \frac{{\ddot{\Psi }}_n}{{\Psi }_n}}=m{n}^2{\rho }_Q-{\displaystyle \frac{3mn}{8\pi F_2^{\prime }\left(T\right)}}\left[{\displaystyle \frac{R}{2}}{F}_1^{\prime }\left(R\right)-(F_1\left(R\right)+F_2\left(T\right))\right],$$

(49)

where the potential is ${\mathcal{U}}_n=m{n}^2{\rho }_Q-{\displaystyle \frac{3mn}{8\pi F_2^{\prime }\left(T\right)}}\left[{\displaystyle \frac{R}{2}}{F}_1^{\prime }\left(R\right)-(F_1\left(R\right)+F_2\left(T\right))\right].$

7. Some Applications for Quark Stars

Quark stars can be used to test theories like Rastall gravity and to separate quark matter from hadronic matter using gravitational waves. They can also be used to explain super-luminous supernovae by their stability and possibly identify real pulsar candidates if they are low-mass bare quark stars. The enormous energy released in super-luminous supernovae may be explained by the higher stability of quark stars relative to neutron stars.

Potential uses for quark stars, which are thought to be compact objects made of quark matter, include serving as probes for the characteristics of quark matter and severe gravity conditions, which could help explain phenomena like cosmic gamma-ray bursts and pulsars. The impact of various equations of state (EoS) and gravity theories on their observable characteristics is revealed by comparing their structural qualities, such as mass-radius relations and stability, with theoretical restrictions and astronomical data. Studies, for example, evaluate the impacts of non-linear self-interactions and different MIT bag models on quark star features, demonstrating how these differences impact the existence of a clear mass gap and mass-radius relations.

The latter phases of stellar evolution produce some relativistic compact stars [44], which make great cosmic probes for examining the characteristics of matter in extraordinarily harsh environments. The most well-known processes for creating compact stars are supernova. When type Ia supernova are observed to be less bright than expected, this is referred to as a supernova with positive cosmological conditions $(\omega <-1)$ [19]. This phenomenon is explained by an accelerating cosmos that is ascribed to a positive cosmological constant ($\Lambda$), a type of dark energy $(\omega <-1)$.

In the context of supernova, a phantom barrier condition is defined as Type Ia supernova data when the dark energy $EoS$$\omega$ has changed from quintessence $(\omega <-1)$ to phantom energy $(\omega >-1)$ or vice versa, implying a dynamic dark energy.

Therefore, now in the light of Corollary 11, Equations (44) and (33), we obtain the existence condition of the Ia supernova type compact quarks stars in the $QMF$-spacetime in $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity.

Theorem 11. 

Let $QMF$-spacetime in $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity contains a phantom barrier type source of matter, then there exists Ia supernova type compact stars in the $QMF$-spacetime if

$${\displaystyle \frac{[F_1\left(R\right)+F_2\left(T\right]}{16\pi +2F_2^{\prime }\left(T\right)}}<{\displaystyle \frac{R{F}_1^{\prime }\left(R\right)}{32\pi +4F_2^{\prime }\left(T\right)}}+{\mathcal{B}}_c.$$

(50)

Corollary 20. 

Let $QMF$-spacetime in $F\left(R\right)$-gravity contains a phantom barrier type source of matter, then there exists Ia supernova type compact stars in the $QMF$-spacetime if

$${\displaystyle \frac{F_1\left(R\right)}{16\pi }}<{\displaystyle \frac{R{F}_1^{\prime }\left(R\right)}{32\pi }}+{\mathcal{B}}_c.$$

(51)

8. Conclusions

The modified gravity hypothesis can effectively represent the sudden growth of the Universe during the late cosmic eras. Our investigation of the characteristics of strange quark matter in modified gravity has shown us a wide range of phenomena, particularly its intriguing coupling to some solitonic symmetries such as Ricci solitons. We determine the formula for the Ricci scalar for the spacetime of the quark matter fluid in the particular case $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$-gravity, as well as the gravitational field equation of the $F(R,T)$-gravity coupled with $QMF$ admitting Ricci solitons with a conformal vector field. Additionally, using the $F(R,T)$-gravity framework, we obtain the equation of state for the $QMF$ and calculate the pressure and density across the universe’s radiation and dark eras.

Following that, we present specific requirements for the quark matter in conjunction with the provided $F(R,T)$-gravity, accounting for Penrose’s singularity theorem predicting the existence of singularities inside black holes, with matter forming closed trapped surfaces in terms of various energy conditions. This theorem has greatly impacted mathematical relativity, cosmology, and the development of theories of quantum gravity. In $F(R,T)$-gravity linked with $QMF$, we extract a Schrödinger equation. Finally, we turn up certain restrictions that suggest the presence of the Ia supernova type compact quarks stars in the $QMF$ spacetime in modified gravity.