Solitonic Aspect of Relativistic Magneto-Fluid Spacetime with Some Specific Vector Fields

Abstract

The target of the current research article is to investigate the solitonic attributes of relativistic magneto-fluid spacetime (MFST) if its metrics are Ricci–Yamabe soliton (RY-soliton) and gradient Ricci–Yamabe soliton (GRY-soliton). We exhibit that a magneto-fluid spacetime filled with a magneto-fluid density $\rho$, magnetic field strength H, and magnetic permeability $\mu$ obeys the Einstein field equation without the cosmic constant being a generalized quasi-Einstein spacetime manifold $\left(GQE\right)$. In such a spacetime, we obtain an EoS with a constant scalar curvature $\mathcal{R}$ in terms of the magnetic field strength H and magnetic permeability $\mu$. Next, we achieve some cauterization of the magneto-fluid spacetime in terms of Ricci–Yamabe solitons with a time-like torse-forming vector field $\xi$ and a $\phi \left(\mathcal{R}ic\right)$ vector field. We establish the existence of a black hole in the relativistic magneto-fluid spacetime by demonstrating that it admits a shrinking Ricci–Yamabe soliton and satisfies the time-like energy convergence criteria. In addition, we examine the magneto-fluid spacetime with a gradient Ricci–Yamabe soliton and deduce some conditions for an equation of state (EoS) $\omega =-\frac{1}{5}$ with a Killing vector field. Furthermore, we demonstrate that the EoS $\omega =-\frac{1}{5}$ of the magneto-fluid spacetime under some constraints represents a star model and a static, spherically symmetric perfect fluid spacetime. Finally, we prove that a gradient Ricci–Yamabe soliton with the conditions $\mu =0$ or $H=2$; $\mu \ne 0$, $H>2$ and obeying the equation of state $\omega =-\frac{1}{5}$ is conceded in a magneto-fluid spacetime, and a naked singularity with a Cauchy horizon subsequently emerges, respectively.

1. Introduction

General relativity $\left(GR\right)$, which has disclosed the basic connection between physics and the geometry of spacetimes, is one of the most successful physics theories of the twentieth century. In addition to its crucial importance in theoretical studies, $GR$ has found success in technology when applied to our daily lives. Finding multiple solutions to Einstein’s field equations became one of the most critical challenges when it was introduced.

A time-oriented, connected, four-dimensional Lorentzian manifold was modeled using both $GR$ spacetime and cosmology, which introduces a special categorization of pseudo-Riemannian manifolds among the Lorentzian metric with a signature $(-,+,+,+)$, which is crucial in $GR$ [1]. The geometry of Lorentzian manifolds is used to investigate the behavior of vectors on the manifold. Consequently, Lorentzian manifolds are emerging as the most effective study model to explain the $GR$. For the purpose of determining a generic solution to Einstein’s field equations, if the Ricci tensor bears the form,

$$\mathcal{S}=\alpha g+\beta \eta \otimes \eta$$

then a Lorentzian manifold is said to have a perfect fluid spacetime [2], where $\alpha$ and $\beta$ are scalars and $\eta$ is a 1- form metrically equal to the unit time-like vector field. Furthermore, the term “spacetime” refers to a Lorentzian manifold that admits a time-like vector field. The energy–momentum tensor plays a crucial role as the spacetime matter content. Matter is imagined to be a fluid with density and pressure, as well as dynamical and kinematic characteristics such as acceleration, speed, shear, and expansion. In traditional cosmological models, the universe’s matter content is assumed to behave as a perfect fluid [2].

A perfect fluid has no heat flux and no stiffness. In this investigation, we claim that magnetic energy–momentum tensors occupy spacetime as a matter of the magnetism-containing content, such as the magneto-fluid density, magnetic flux, and pressure [3]. Chaki [4] used a covariant constant-energy momentum tensor to study spacetime. Likewise, several researchers have investigated spacetime in various methods, which is intimately linked to this concept (for more information, see [5,6,7,8,9,10,11,12,13,14,15,16]). Symmetries in geometry, on the other hand, are quite significant, especially when it applied from a physical standpoint. This kind of symmetry relies on the spacetime geometry and matter, and its metric generally simplifies solutions with many difficulties, such as Einstein’s field equation solutions. Physical matter symmetry is, in fact, particularly related to spacetime geometry, according to $GR$. The soliton, which is related to the geometrical flow of spacetime geometry, is one of the most significant forms of symmetry. In reality, the Ricci flow, Yamabe flow, and Einstein flow are employed in $GR$ to grasp the concepts of kinematics and thermodynamics [17]. Because curvatures maintain self-similarity, the Ricci soliton, Yamabe soliton, and Einstein soliton are focused.

Many geometers have been inspired by the ideas of Ricci flow and Yamabe flow throughout the previous two decades. In the evaluation of flow singularities, a class of solutions in which the metric changes via dilation and diffeomorphisms has a substantial impact, since they appear as viable singularity models. These solutions are also known as soliton solutions.

Danish and Akif [18], using the Ricci–Yamabe maps, derived Ricci–Yamabe solitons from a geometric flow that is a scalar combination of the Ricci and Yamabe flow discussed in [19] for the very first time. This is also known as the Ricci–Yamabe flow of the type $(\delta ,\epsilon )$. The Ricci–Yamabe flow is described as follows [19]:

$${\partial }_{t}g\left(t\right)=-2\delta \mathcal{S}\left(t\right)-\epsilon \mathcal{R}\left(t\right)g\left(t\right),g_0=g\left(0\right),t\in (a,b),$$

(1)

where the Ricci tensor and scalar curvature are symbolized by $\mathcal{S}$ and $\mathcal{R}$, respectively. Additionally, in [19], the definitions describe it as the $(\delta ,\epsilon )$-type Ricci–Yamabe flow. If $\delta =1$ and $\epsilon =0$ (Ricci soliton [20]), then it is said to be a Ricci flow [20]. If $\delta =0$ and $\epsilon =1$, then it is a Yamabe flow [20] (Yamabe soliton [20]). If $\delta =1$ and $\epsilon =-1$, then it is an Einstein flow [21] (Einstein soliton [21]).

Because of the indication of the associated scalars $\delta$ and $\epsilon$, the Ricci–Yamabe flow can also be a Riemannian, semi-Riemannian, or singular Riemannian flow. Multiple options might be advantageous in certain geometrical or physical models, such as relativistic theories. As a result, the Ricci–Yamabe soliton (RY-soliton) arises with the natural limit of the flow of the Ricci–Yamabe soliton.

In addition, there is a smooth curve $t⟼g\left(t\right)$ of semi-Riemannian metrics on a fixed manifold M. The condition consists of requiring the curve to have, at every t in its domain interval, a Ricci–Yamabe flow on the manifold blow-up limits (or rescaling limits) of the metric $g\left(t\right)$ restricted to a suitable open set when the variable $t\in [o,T)⟼T$, where T is finite.

This is a great source of inspiration for learning Ricci–Yamabe solitons. RY-soliton is essentially an advanced extension of Ricci soliton, Yamabe soliton, and Einstein soliton.

In the Ricci–Yamabe flow, a RY-soliton is called a soliton if it evolves only via diffeomorphism and scales by one parameter group. A RY-soliton on the Riemannain manifold $(M,g)$ is a datum $(g,X,\Lambda ,\delta ,\epsilon )$ obeying the equation [18,22]

$$\frac{1}{2}{\mathfrak{L}}_{X}g+\delta \mathcal{S}=(\Omega -\epsilon \frac{\mathcal{R}}{2})g,$$

(2)

where ${\mathfrak{L}}_X$ depicts the Lie derivative in the direction of soliton vector field X. In $(M,g)$, the RY-soliton is called shrinking, expanding, or steady, corresponding to $\Lambda >0$, $\Lambda <0$ or $\Lambda =0$, respectively.

In addition, RY-soliton with $X=\mathcal{D}\psi$ gives the gradient Ricci–Yamabe soliton (GRY-soliton) on semi-Riemannian manifold M, where $\mathcal{D}$ depicts the gradient operator and $-\psi$ is a smooth function on M. Therefore, Equation (2) reduces to the following form [18,22]

$$Hess\left(\psi \right)+\delta \mathcal{S}=\left(\Omega -\frac{\epsilon \mathcal{R}}{2}\right)g.$$

(3)

The Hessian is indicated by $Hess$, the gradient operator of g is $\mathcal{D}$, and the smooth function $\psi$ is termed the potential function of the $GRY$-soliton.

Moreover, the RY-soliton is said to be expanding, steady or shrinking according as $\Lambda$ is negative, zero, positive, respectively. Additionally, if $\Lambda$, $\delta$, $\epsilon$ become smooth functions then (2) is known as an almost Ricci–Yamabe soliton [23].

In terms of the Ricci soliton, Ali and Ahsan [24] explored spacetimes. Using $\eta$-Ricci and $\eta$-Einstein solitons, Blaga later showed the curvature characteristics of perfect fluid spacetimes in [25]. Ricci solitons are also employed by Venkatesha and Aruna in [26] to explore perfect fluid spacetime. Siddiqi and Siddqui [27] discussed conformal $\eta$-Ricci soliton and conformal Ricci soliton in the perfect fluid spacetime. Danish and De examine the coupling of perfect fluid spacetime with Ricci–Yamabe and $\gamma$-Ricci–Yamabe solitons in [22]. The authors also explored almost Ricci–Yamabe solitons on static spacetimes, according to [28]. In similar manner, Ali et al. [29] analyzed an imperfect fluid Generalized Robertson Walker Spacetime conceding Ricci–Yamabe Metric.

Recent research was conducted in 2023 by Siddiql et al. [17], on the Ricci soliton in thermodynamical fluid spacetime. The relativistic magneto-fluid spacetimes were already described by Siddiqi and De in their paper [30].

we are sufficiently motivated by the previous solitonic works with various spacetimes. In this paper, we explore the relativistic magneto-fluid spacetimes using Ricci–Yamabe solitons and gradient Ricci–Yamabe solitons along with different vector fields, such as the torse-forming vector field, Jacobi vector field, $\phi (\mathcal{R}ic$)-vector field, and Killing vector fields.

With the help of this geometric analysis, we obtain many physically relevant findings. Additionally, our research model is comparable to spin–orbit coupling models and the Bose–Einstein condensate model.

2. Relativistic Magneto-Fluid Spacetime (MFST)

The magnetic type matter tensor occupies spacetime, also known as relativistic magneto-fluid spacetime (MFST) [30], are discussed in this section.

We begin by mentioning the following definition for further analysis.

Definition 1 

([31]). If the Ricci tensor $\mathcal{S}$ of a semi-Riemannian manifold M $(n>3)$ does not vanish identically and satisfies the equation, it is said to be a generalized quasi-Eisntein $\left(GQE\right)$.

$$\mathcal{S}=ag+b\eta \otimes \eta +c\gamma \otimes \gamma ,$$

(4)

where $a,b,and c$ are scalars with $b\ne 0$, $c\ne 0$, and η, γ are not zero 1, thus $g(E_1,\xi )=\eta \left(E_1\right)$, $g(E_1,\zeta )=\gamma \left(E_1\right)$ for any vector filed $E_1$. The unit vectors ξ and ζ are orthogonal to each other, corresponding to the 1-form η and γ. The manifold’s generators are also ξ and ζ. M reduces to a quasi-Einstein manifold if $c=0$.

Definition 2 

([32]). A vector field ζ on a Lorentzian manifold is said to be torse-forming vector field (TFV) if for $E_1\in \chi \left(M\right)$ it obeys

$${\nabla }_{E_1}\xi =\omega E_1+\gamma \left(E_1\right)\xi ,$$

(5)

where ω is a scalar function and γ is a non-vanishing 1-form.

It is observed that a time-like $TFV$ unit is $\xi =E_1$ on an n-dimensional Lorentzian manifold M takes the following form [32]:

$${\nabla }_{E_1}{F}_1=\omega [E_1+\eta \left(E_1\right)F_1],$$

(6)

where $\eta$ is a 1-form, such that $g(E_1,F_1)=\eta \left(E_1\right)$ for all $E_1$.

Definition 3 

([33]). A vector field φ on a Lorentzian manifold M is said to be a $\phi \left(\mathcal{R}ic\right)$-vector field if it obeys

$${\nabla }_{E_1}\phi =\sigma \mathcal{R}ic{E}_1,$$

(7)

where ∇, σ, and $\mathcal{R}ic$ is the Levi-Civita connection, a constant, and Ricci operator, respectively. If $\sigma \ne 0$ then vector field φ is said to be a proper $\phi \left(\mathcal{R}ic\right)$-vector field and if $\sigma =0$ in (4) then vector filed φ is said to be covariantly constant.

Space matter is defined as a fluid conveying any spacetime substance, such as pressure, volumes, heat quantities, velocity, torque, shear, and extension [34]. In typical cosmological models, the matter tensor plays an essential part; the material substance of the cosmos is thought to operate like a $MFST$ [35].

The magnetic energy momentum tensor $\mathcal{T}$ in a $MFST$ has the following shape [2,36].

$$\mathcal{T}=pg+(p+\rho )\eta \otimes \eta +\mu \left\{\left(\eta \otimes \eta +\frac{1}{2}g\right)H-\gamma \otimes \gamma \right\}$$

(8)

where $\rho$ represents magnetic-fluid density, p signifies pressure, $\mu$ refers magnetic permeability, $\gamma$ indicates magnetic flux, and H indicates magnetic field strength, and $\eta \left(E_1\right)=g(E_1,\xi )$ and $g(F_1,\zeta )=\gamma \left(V\right)$ are two non-zero 1-forms, respectively. Additionally, $\xi$ and $\zeta$ are unit timelike vector fields with $g(\xi ,\xi )=-1$ and $F_1$ spacelike magnetic flux vector fields with $g(\zeta ,\zeta )=1$, respectively. As a result, the $MFST$ is generated by orthogonal vector fields $\xi$ and $\zeta$.

The gravitational field equation of Einstein without the cosmic constant is as follows [2]:

$$\mathcal{S}(E_1,F_1)-\frac{\mathcal{R}}{2}g(E_1,F_1)=\kappa \mathcal{T}(E_1,F_1),$$

(9)

for any $E_1,F_1\in \chi \left(M\right)$, $\kappa$ symbolizes the gravitational constant (which is set to $8\pi G$, making G a universal gravitational constant), while $\mathcal{S}$ and $\mathcal{R}$ designate the Ricci tensor and the scalar curvature of spacetime, respectively. In order to achieve both $\mathcal{S}$ and $\mathcal{R}$ for the purpose of creating a static world, Einstein’s equation, in addition to the cosmological constant, is used. It is seen in modern cosmology in light of the hypothesis of dark energy, which accelerates the expansion of the universe [5,36].

We also obtain the gravitational equation of Einstein without the cosmic constant for a $MFST$ from Equations (8) and (9).

$$\mathcal{S}(E_1,F_1)=\left\{\frac{\mathcal{R}}{2}+\kappa \left(\frac{\mu H}{2}+p\right)\right\}g(E_1,F_1)+\kappa (\mu H+\rho +p)\eta \left(E_1\right)\eta \left(F_1\right)-\kappa \mu \gamma \left(E_1\right)\gamma \left(F_1\right)$$

(10)

(10), $MFST$ under study is a $GQE$-sapcetime manifold with $\frac{\mathcal{R}}{2}+\kappa \left(\frac{\mu H}{2}+p\right),$ $\kappa (\mu H+\rho +p)$ and $-\kappa \mu$ are scalars associated with 1 forms $\eta$ and $\gamma .$

A $MFST$ concedes the magnetic flow and solves the Einstein field equation ($EF{E}_s$) without the cosmic constant in this case. As a result, we have the following:

Theorem 1. 

A $MFST$ is a $GQE$-spacetime with magneto-fluid density ρ, magnetic field strength H, and magnetic permeability μ fulfilling $EF{E}_s$ without the cosmic constant.

Now, contracting (10) we present the following:

Theorem 2. 

In a $MFST$ with magneto-fluid density ρ, magnetic field strength H, and magnetic permeability μ fulfilling $EF{E}_s$ without the cosmic constant, then the scalar curvature is

$$\mathcal{R}=-\kappa \left[\right(3H-1)\mu -\rho +5p)],$$

(11)

Now the value of $\mathcal{R}$ reflects that

$$p=\frac{\rho }{5}+\frac{(1-3H)\mu -\mathcal{R}}{5\kappa }.$$

(12)

Thus, we gain the following corollary

Corollary 1. 

If a $MFST$ with magneto-fluid density ρ, magnetic field strength H, and magnetic permeability μ fulfilling $EF{E}_s$ without the cosmic constant with constant scalar curvature $\mathcal{R}$, then $EoS$ is given by (12).

Now, we assume that the matter source is of the radiation type, in which case EoS $\omega =\frac{1}{3}$. Equation (12) and this observation combined provide

$$p=\frac{(1-3H)\mu -\mathcal{R}}{2\kappa }and\rho =\frac{3(1-3H)\mu -3\mathcal{R}}{2\kappa }$$

(13)

Corollary 2. 

Let radiation serve as the matter source of $MFST$ with magnetic field strength H, and magnetic permeability μ fulfilling $EF{E}_s$ without the cosmic constant with constant scalar curvature $\mathcal{R}$. Then, the pressure p and magneto-fluid density ρ are determined by (15).

Next, when there is a phantom barrier, $\rho =-p=\frac{(1-3H)\mu -\mathcal{R}}{6\kappa }$ Thus, we may say

Corollary 3. 

If the source of matter in $MFST$ is is phantom barrier type. Then, the pressure p and magneto-fluid density ρ quantified as $\rho =-p=\frac{(1-3H)\mu -\mathcal{R}}{6\kappa }.$

Furthermore, if $\mu =0,\mathcal{R}=0$ or $H=\frac{1}{3},\mathcal{R}=0$ is fixed in (12), we obtain $\omega =\frac{p}{\rho }=\frac{1}{5}>-1$, which displays the quintessence era. We declare:

Corollary 4. 

Let $MFST$ with magnetic field strength H, and magnetic permeability μ fulfilling $EF{E}_s$ and if $\mu =0,\mathcal{R}=0$ or $H=\frac{1}{3},\mathcal{R}=0$. Then EoS $(\omega =\frac{p}{\rho }=\frac{1}{5}>-1)$ exhibits the quintessence era.

From (10), we derive $g(\xi ,\zeta )=0$ since $\zeta$ and $\xi$ are orthogonal unit vector fields.

$$\mathcal{S}(E_1,\xi )=(a+b)\eta \left(E_1\right)$$

(14)

$$\mathcal{S}(E_1,\zeta )=(a+c)\gamma \left(E_1\right)$$

(15)

wherein

$$a=\left\{\frac{\mathcal{R}}{2}+\kappa \left(\frac{\mu H}{2}+p\right)\right\},b=\kappa (\mu H+\rho +p)c=-\kappa \mu .$$

(16)

The symmetric endomorphism of the tangent space at each point of the manifold corresponding to the Ricci tensor $\mathcal{S}$ is called Ricci operator $\mathcal{R}ic$. Then, for any $E_1,F_1$, $\mathcal{S}(E_1,V_1)=g(\mathcal{R}ic{E}_1,F_1)$. As a result, we have

$$\mathcal{R}ic{E}_1=a{E}_1+b\eta \left(E_1\right)\xi +c\gamma \left(E_1\right)\zeta .$$

(17)

3. RY-Solitons on Magneto-Fluid Spacetime

In this segment, we evaluate RY-soliton of type $(\delta ,\epsilon )$ in a $MFST$ with a timelike $TFV$$\xi$.

Putting $F_1=\zeta$, (2) becomes

$$\left({\mathfrak{L}}_{\zeta }g\right)(E_1,V_1)+2\delta \mathcal{S}(E_1,V_1+(2\Omega -\epsilon \mathcal{R})g(E_1,V_1)=0.$$

(18)

where $\mathcal{R}$ is scalar curvature. Using (6), we find

$$\mathcal{S}(E_1,V_1)=-\left(\frac{\Omega }{\alpha }-\frac{\beta \mathcal{R}}{2\alpha }+\omega \right)g(E_1,V_1)-\omega \eta \left(E_1\right)\eta \left(F_1\right)=0.$$

(19)

By using (10) in above equation and plugging $U=V=\xi$ in (19), we obtain

$$\Omega =(3\delta +\epsilon )\frac{\kappa \mu }{2}+(5\epsilon -3\delta )\frac{\kappa p}{2}+(3\delta -\epsilon )\frac{\kappa \rho }{2}-3\epsilon \frac{\kappa \mu H}{2}.$$

(20)

In addition, for $F_1=\xi$ in (19) one also find

$${\nabla }_{\xi }\xi =2(\frac{\epsilon \mathcal{R}}{2}-\frac{a+b}{2}-\Omega ).$$

(21)

Since $\Omega =\frac{\epsilon \mathcal{R}}{2}-\frac{a+b}{2}$, therefore the above equation becomes ${\nabla }_{\xi }\xi =0.$ This demonstrates that the unit timelike vector field generates geodesic integral curves.

As a result, we find following outcomes.

Theorem 3. 

If a $MFST$ with a unit timelike $TFV$ξ admits a RY-soliton $(g,\xi ,\Omega )$ of type $(\delta ,\epsilon )$, then RY-soliton is shrinking, steady and expanding, according as

1. 

$(3\delta +\epsilon )\frac{\kappa \mu }{2}+(5\epsilon -3\delta )\frac{\kappa p}{2}+(3\delta -\epsilon )\frac{\kappa \rho }{2}>3\epsilon \frac{\kappa \mu H}{2}$,

2. 

$(\delta +2\epsilon )(3\delta +\epsilon )\frac{\kappa \mu }{2}+(5\epsilon -3\delta )\frac{\kappa p}{2}+(3\delta -\epsilon )\frac{\kappa \rho }{2}=3\epsilon \frac{\kappa \mu H}{2}$,

3. 

$(3\delta +\epsilon )\frac{\kappa \mu }{2}+(5\epsilon -3\delta )\frac{\kappa p}{2}+(3\delta -\epsilon )\frac{\kappa \rho }{2}<3\epsilon \frac{\kappa \mu H}{2}$, respectively. Additionally, the integral curves generated by ξ is geodesic.

Corollary 5. 

If a $MFST$ with a unit timelike $TFV$ξ admits a Ricci soliton $(g,\xi ,\Omega )$ of type $(1,0)$, then Ricci soliton is shrinking, steady, and expanding, as $\frac{3\kappa }{2}(\mu +\rho )>\frac{3}{2}\kappa p$, $\frac{3\kappa }{2}(\mu +\rho )=\frac{3}{2}\kappa p$, and $\frac{3\kappa }{2}(\mu +\rho )<\frac{3}{2}\kappa p$, respectively.

Corollary 6. 

If a $MFST$ with a unit timelike $TFV$ξ admits a Yamabe solitons $(g,\xi ,\Omega )$ of type $(0,1)$, then Yamabe soliton is shrinking, steady, and expanding, according as

1. 

$\frac{\kappa }{2}(\mu +5p)>\frac{k}{2}(3\mu H+\rho )$,

2. 

$\frac{\kappa }{2}(\mu +5p)=\frac{k}{2}(3\mu H+\rho )$,

3. 

$\frac{\kappa }{2}(\mu +5p)<\frac{k}{2}(3\mu H+\rho )$, respectively. Additionally, the integral curves generated by ξ is geodesic.

Corollary 7. 

If a $MFST$ with a unit timelike $TFV$ξ admits an Einstein solitons $(g,\xi ,\Omega )$ of type $(1,-1)$, then Einstein soliton is expanding. Additionally, the integral curves generated by ξ is geodesic.

4. Physically Relevant Results in MFST with RY-Solitons

Let the $MFST$ without cosmic constant, i.e., $\lambda =0$. By Equation (14), we gain $\mathcal{S}(\xi ,\xi )=\kappa (\mu +\frac{\rho }{2})-\frac{k}{2}(3\mu H+5p)$. In cases where the $MFST$ satisfies the $TCC$ (timelike convergence condition), i.e., $\mathcal{S}(\zeta ,\zeta )>0$, then $\kappa (\mu +\frac{\rho }{2})-\frac{k}{2}(3\mu H+5p)>0$, or $\kappa (\mu +\frac{\rho }{2})>\frac{k}{2}(3\mu H+5p)$ the spacetime satisfies cosmic $SEC$ (strong energy condition) [3,9]. Thus, the above fact and (20) entails that.

Theorem 4. 

If a $MFST$ admits a RY-soliton $(g,\xi ,\Omega )$ of type $(\delta ,\epsilon )$ with a unit timelike $TFV$ξ and obeying timelike energy convergence condition, then RY-solitons is shrinking.

Remark 1. 

According to the Hawking and Ellis the geometric form of strong energy condition $\left(SEC\right)$ is called $TCC$$\mathcal{S}(\xi ,\xi )>o$ [3]. In addition the $TCC$ implies null convergence condition $\left(NCC\right)$ (for more details see [3]).

Therefore, using Theorem 4 and Remark 1 together, we gain the following corollary.

Corollary 8. 

If a $MFST$ admits a shrinking RY-soliton $(g,\xi ,\Omega )$ of type $(\delta ,\epsilon )$ with a unit timelike $TFV$ξ, then $MFST$ obeying null convergence condition.

In 2014, Vilenkin and Wall [37] proved that if a spacetime M obeys the null convergence condition, then

(1)

M has a non-compact connected Cauchy surface,

(2)

M contains some black holes and include a trapped surface which is out side the black holes.

Thus, in the light of the above facts, we can provide a physical relevant results by using the Corollary 8.

Theorem 5. 

If a $MFST$ admits a shrinking RY-soliton $(g,\xi ,\Omega )$ of type $(\delta ,\epsilon )$ with a unit timelike $TFV$ξ and $MFST$ obeying null convergence condition, then there exists a non-compact connected Cauchy surface in the $MFST$.

Theorem 6. 

If a $MFST$ admits a shrinking RY-soliton $(g,\xi ,\Omega )$ of type $(\delta ,\epsilon )$ with a unit timelike $TFV$ξ and $MFST$ obeying null convergence condition, then the $MFST$ contains some black holes and include a trapped surface which is outside the black holes.

From (21), we have ${\mathfrak{L}}_{\xi }\xi =0$, signifies that the integral curves generated by vector field $\xi$ is a geodesic. Let X be an affine Killing vector field on $MFST$ with a $TFV$$\xi$, then integrability condition (for more details see [38], p. 24) ${\mathfrak{L}}_X\nabla =0$, where ${\mathfrak{L}}_X$ is the Lie derivative along X. This additional requirement indicates that a Killing vector field is clearly an affine Killing vector field, but the converse is not always true.

Recall the following formula [38]

$$({\mathfrak{L}}_X\nabla )(E_1,F_1)={\nabla }_{E_1}{\nabla }_{F_1}X-{\nabla }_{{\nabla }_{E_1}{F}_1}X+R(X,E_1)F_1.$$

(22)

Adopting $E_1=F_1=\xi$, in the above Equation (22), we turn up

$$({\mathfrak{L}}_X\nabla )(\xi ,\xi )={\nabla }_{\xi }{\nabla }_{\xi }X+R(X,\xi )\xi ,$$

(23)

owing to the fact that X is an affine Killing vector field. As a result, we conclude the following:

Theorem 7. 

Let X be a affine Killing vector field on $MFST$ with a $TVF$ξ, then X is a Jacobi vector field along the geodesics of ξ.

Next, in light of (2) and (10) we have the form

$$\left({\mathfrak{L}}_{X}g\right)(E_1,F_1)=-2\left\{(\delta A+\Omega -\epsilon \mathcal{R}\right\}g\left(E_1{F}_1\right)+\delta B\eta \left(E_1\right)\eta \left(E_1\right)+\delta C\gamma \left(E_1\right)\gamma \left(F_1\right).$$

(24)

Taking Lie-differentiation of (10) along X and using (24), we obtain

$$({\mathfrak{L}}_X\mathcal{S}(U,V)=\delta B\left\{\left({\mathfrak{L}}_X\eta \right)\left(E_1\right)\eta \left(F_1\right)+\eta \left(E_1\right)\left({\mathfrak{L}}_X\eta \right)\left(F_1\right)\right\}$$

(25)

$$+\delta C\left\{\left({\mathfrak{L}}_X\gamma \right)\left(E_1\right)\gamma \left(F_1\right)+\gamma \left(E_1\right)\left({\mathfrak{L}}_X\gamma \right)\left(F_1\right)\right\}$$

$$-2\left\{(\delta a+\Omega -\epsilon \mathcal{R}g(E_1,F_1)+\delta b\eta \left(E_1\right)\eta \left(F_1\right)+\delta c\gamma \left(E_1\right)\gamma \left(F_1\right)\right\}.$$

On the other hand, differentiating (10) covariantly along $G_1$ and using the relation $\left({\nabla }_{E_1}\eta \right)\left(F_1\right)=g(E_1,F_1)+\eta \left(E_1\right)\eta \left(F_1\right)$, we obtain

$$\left({\nabla }_W\mathcal{S}\right)(E_1,F_1)=\delta a\left\{g(G_1,E_1)\eta \left(F_1\right)+g(G_1,F_1)\eta \left(E_1\right)+2\eta \left(E_1\right)\eta \left(F_1\right)\eta \left(G_1\right)\right\}$$

(26)

$$+\delta c\left\{g(G_1,E_1)\gamma \left(F_1\right)+g(G_1,F_1)\gamma \left(U\right)+2\gamma \left(E_1\right)\gamma \left(F_1\right)\gamma \left(G_1\right)\right\}.$$

(27)

Once again, adopting (2) in commutation formula [38]

$$({\mathfrak{L}}_X{\nabla }_{G_1}g-{\nabla }_{G_1}{\mathfrak{L}}_{X}g-{\nabla }_{[X,G_1]})(E_1,F_1)=-g(({\mathfrak{L}}_X\nabla )(G_1,X),F_1)$$

(28)

$$-g(({\mathfrak{L}}_X\nabla )(G_1,F_1),E_1)$$

we obtain

$$g(({\mathfrak{L}}_X\nabla )(E_1,F_1),G_1)=\left({\nabla }_{G_1}\mathcal{S}\right)(E_1,F_1)-\left({\nabla }_X\mathcal{S}\right)(F_1,G_1)-\left({\nabla }_{F_1}\mathcal{S}\right)(E_1,G_1).$$

(29)

In view of (26) and (29), we obtain

$$({\mathfrak{L}}_X\nabla )(E_1,F_1)=-2\delta b\left\{g{E}_1,F_1)\xi +\eta \left(E_1\right)\eta \left(F_1\right)\xi \right\}.$$

(30)

If $E_1$ and $F_1$ are replaced with $\xi$, then it follows from (30) that

$$({\mathfrak{L}}_X\nabla )(\xi ,\xi )=0.$$

(31)

Recalling (23) together with $E_1=F_1=\xi$ and then using (6) and (31), we find that

$${\nabla }_{\xi }{\nabla }_{\xi }X+R(X,\xi )\xi =0.$$

(32)

Thus, (32) signifies that potential vector field X is a Jacobi vector field along direction of the geodesics of $\xi$. Hence, we have the following results.

Theorem 8. 

Let a $MFST$ with $TVF$ξ admits a RY-soliton of type $(\delta ,\epsilon )$ together with the potential vector field X, then X is a Jacobi vector field along the geodesics of ξ.

5. RY-Solitons on MFST along $\mathbf{\phi }\left(\mathcal{R}\mathit{i}\mathit{c}\right)$-Vector Field

In this part, we examine the nature RY-solitons on $MFST$ with a $\phi \left(\mathcal{R}ic\right)$-vector field.

In light of (2) and (10), we find that

$${\mathfrak{L}}_{X}g(E_1,F_1)+2\delta \{ag(E_1,F_1)+b\eta \left(E_1\right)\eta \left(F_1\right)$$

$$+c\gamma \left(E_1\right)\gamma \left(F_1\right)\}+\left(2\Omega -\epsilon \mathcal{R}\right)g(E_1,F_1)=0.$$

(33)

By the definition of Lie-derivative and (4) one has

$$\left({\mathfrak{L}}_{\phi }g\right)(U,V)=2\sigma \mathcal{S}(U,V)$$

(34)

for any $E,F$.

Using (34) in (33), we obtain

$$\mathcal{S}(E_1,F_1)=-\frac{\delta }{\sigma }(a+\Omega -\frac{\epsilon \mathcal{R}}{2})g(E_1,F_1)-\frac{b}{\sigma }\eta \left(E_1\right)\eta \left(F_1\right)-\frac{c}{\sigma }\gamma \left(E_1\right)\gamma \left(F_1\right),$$

(35)

this entails the following:

Theorem 9. 

If a $MFST$ M admitting a RY-solitons $(M,g,\phi ,\Omega )$ of type $(\delta ,\epsilon )$ such that the potential vector field φ is a proper $\phi \left(\mathcal{R}ic\right)$-vector field, then $(M,g,\phi ,\Omega )$ is a $\left(GQE\right)$-spacetime.

Now, adopting $E_1=F_1=\xi$ in (35) we obtain

$$\Omega =\frac{\epsilon \mathcal{R}}{2}-\left\{a(1+\frac{\sigma }{\delta })+\frac{b}{\delta }(1+\sigma )\right\}.$$

(36)

Consequently, we articulate the theorem

Theorem 10. 

Let M be a $MFST$ admitting a RY-soliton $(M,g,\xi ,\Omega )$ of type $(\delta ,\epsilon$) with a proper $\xi \left(\mathcal{R}ic\right)$-timelike velocity vector field ξ, then RY-soliton is shrinking, steady, and expanding, according as $\frac{\epsilon \mathcal{R}}{2}>\left\{a(1+\frac{\sigma }{\delta })+\frac{b}{\delta }(1+\sigma )\right\}$, $\frac{\epsilon \mathcal{R}}{2}=\left\{a(1+\frac{\sigma }{\delta })+\frac{b}{\delta }(1+\sigma )\right\}$, and $\frac{\epsilon \mathcal{R}}{2}<\left\{a(1+\frac{\sigma }{\delta })+\frac{b}{\delta }(1+\sigma )\right\}$, respectively.

Corollary 9. 

Let M be a $MFST$ admitting a RY-soliton $(M,g,\xi ,\Omega )$ of type $(\delta ,\epsilon$) with a covariantly constant $\xi \left(\mathcal{R}ic\right)$-timelike velocity vector field ξ, then RY- soliton is expanding, steady, or shrinking according to $\frac{\epsilon \mathcal{R}}{2}>\left\{a+\frac{b}{\delta }\right\}$, $\frac{\epsilon \mathcal{R}}{2}=\left\{a+\frac{b}{\delta }\right\}$, and $\frac{\epsilon \mathcal{R}}{2}<\left\{a+\frac{b}{\delta }\right\}$, respectively.

Next, from (36) we articulate the following results.

Theorem 11. 

If a $MFST$ admitting a RY-soliton $(M,g,\xi ,\Omega )$ of type $(\delta ,\epsilon$) with a proper $\xi \left(\mathcal{R}ic\right)$-timelike velocity vector field ξ, then the scalar curvature is

$$\mathcal{R}=\frac{2\Omega }{\epsilon }+\frac{2}{\epsilon }\left\{a(1+\frac{\sigma }{\delta })+\frac{b}{\delta }(1+\sigma )\right\}$$

(37)

We gain the following conclusion as a result of Theorems 9 and 11:

Corollary 10. 

A $MFST$ admitting a RY-soliton $(M,g,\xi ,\Omega )$ of type $(\delta ,\epsilon$) with a proper $\xi \left(\mathcal{R}ic\right)$-timelike velocity vector field ξ and constant scalar curvature $\mathcal{R}$ is an imperfect fluid spacetime.

Corollary 11. 

A $MFST$ admitting a RY-soliton $(M,g,\xi ,\Omega )$ of type $(\delta ,\epsilon$) with a covariantly constant $\xi \left(\mathcal{R}ic\right)$-timelike velocity vector field ξ and constant scalar curvature $\mathcal{R}$ is a viscous fluid spacetime [39].

6. Gradient Ricci–Yamabe Soliton on Magneto-Fluid Spacetime

Let X of the be the vector field of Ricci soliton in n-dimensional magneto-fluid spacetime and $\psi$ is some smooth function on magneto-fluid spacetime M, that is, $X=D\psi$, where D stands for the gradient operator. Then, in view of (3), we can express as

$${\nabla }_{E_1}\mathcal{D}\psi +\delta \mathcal{R}ic{E}_1=\left(\Omega -\frac{\epsilon \mathcal{R}}{2}\right)E_1$$

(38)

for all $E_1\in \chi \left(M\right)$. The Equation (38) along with the relation

$$R(E_1,F_1)\mathcal{D}\psi ={\nabla }_{E_1}{\nabla }_{F_1}\mathcal{D}\psi -{\nabla }_{F_1}{\nabla }_{E_1}\mathcal{D}\psi -{\nabla }_{[E_1,F_1]}\mathcal{D}\psi$$

(39)

give

$$R(E_1,F_1)\mathcal{D}\psi =\delta \left({\nabla }_{E_1}\mathcal{R}ic\right)F_1-\left({\nabla }_{F_1}\mathcal{R}\right)E_1.$$

(40)

Now, covariant derivative for (38) along the vector field $F_1$ acquires

$${\nabla }_{F_1}{\nabla }_{E_1}\mathcal{D}\psi =-\delta (\left({\nabla }_{F_1}\mathcal{R}ic\right)\left(E_1\right)-\mathcal{R}ic\left({\nabla }_{F_1}\right)E_1-\left(\Omega -\frac{\epsilon \mathcal{R}}{2}\right){\nabla }_{F_1}{E}_1$$

(41)

Now, by covariantly differentiating (17) along vector field $E_1$, we arrive to

$$\left({\nabla }_{E_1}\mathcal{R}ic\right)\left(F_1\right)=E_1\left(a\right)F_1+E_1\left(a\right)\eta \left(F_1\right)\xi +b\left({\nabla }_{E_1}\eta \right)\left(F_1\right)\xi +b\eta \left(F_1\right){\nabla }_{E_1}\xi$$

(42)

$$+c\left({\nabla }_{E_1}\gamma \right)\left(F_1\right)\zeta +b\gamma \left(F_1\right){\nabla }_{E_1}\zeta .$$

In view of (40) and (42), we lead

$$R(E_1,F_1)\mathcal{D}\psi =\delta U\left(a\right)F_1-\delta F_1\left(a\right)E_1+\delta [E_1\left(b\right)\eta \left(F_1\right)-F_1\left(b\right)\eta \left(E_1\right)+b\left({\nabla }_{E_1}\eta \right)\left(F_1\right)$$

(43)

$$-b\left({\nabla }_{F_1}\eta \right)\left(E_1\right)]\rho +\delta b[\eta \left(F_1\right){\nabla }_{E_1}\rho -\eta \left(E_1\right){\nabla }_{F_1}\rho ]$$

$$\delta [E_1\left(c\right)\gamma \left(F_1\right)-F_1\left(c\right)\gamma \left(E_1\right)+c\left({\nabla }_{E_1}\gamma \right)\left(F_1\right)-c\left({\nabla }_{F_1}\gamma \right)\left(E_1\right)]\zeta$$

(44)

$$\delta c[\gamma \left(F_1\right){\nabla }_{E_1}\zeta -\gamma \left(E_1\right){\nabla }_{F_1}\zeta ].$$

Contracting an orthonormal frame field after taking it in (43) along the vector field $E_1$, we have

$$\mathcal{S}(F_1,\mathcal{D}\psi )=\delta (1-n)F_1\left(a\right)+\delta F_1\left(b\right)+\delta F_1\left(c\right)+\delta \xi \left(b\right)\eta \left(F_1\right)+b[\left({\nabla }_{\xi }\eta \right)\left(F_1\right)$$

$$-\left({\nabla }_{F_1}\eta \right)\left(\xi \right)+\eta \left(F_1\right)div\xi ]+\delta \left(c\right)\gamma \left(V\right)$$

$$+\delta c[\left({\nabla }_{\zeta }\gamma \right)\left(F_1\right)-\left({\nabla }_{F_1}\gamma \right)\left(\zeta \right)+\gamma \left(F_1\right)div\xi ].$$

(45)

Again, from (33) we have

$$\mathcal{S}(F_1,\mathcal{D}\psi )=\delta [a{F}_1\left(\psi \right)+b\eta \left(F_1\right)\xi \left(\psi \right)+c\gamma \left(F_1\right)\zeta \left(\psi \right)].$$

(46)

Setting $F_1=\xi$ in (45) and (46) and after comparing the value of $\mathcal{R}ic(\rho ,\mathcal{D}\psi )$, one obtain

$$\delta (a-b)\xi \left(\psi \right)=\delta (1-n)\xi \left(a\right)-\delta bdiv\xi .$$

(47)

Let the $MFST$ velocity vector field $\xi$ be Killing, i.e., ${\mathcal{L}}_{\xi }g=0$, and the scalar a be invariant with the velocity vector field $\xi$, i.e., $\xi \left(a\right)=0$. The result is $div\xi =0$. As a result of Equations (33) and (47), we obtain

$$\delta (a-b)\xi \left(\psi \right)=0,$$

(48)

which exhibits that either $a=b$ or $\xi \left(\psi \right)=0$ on a $MFST$ with the GRY-soliton. Now, we classify our observation into two cases as,

Case I. We consider that $a=b$, $\xi \left(\psi \right)\ne 0$ and $\delta \ne 0$, then from (33), we conclude that

$$\mu =\frac{5p+\rho }{2-H},H=2-\left(\frac{5p+\rho }{\mu }\right),p=\frac{\mu (2-H)}{5}-\frac{\rho }{5}$$

(49)

this gives the values of magnetic permeability, strength of magnetic field, and the magnetic fluid pressure. Additionally, $\Omega -\frac{\epsilon \mathcal{R}}{2}=b-a=0$. Thus, the GRY-soliton is expanding.

Case II. Next assume, c that $\xi \left(\psi \right)=0$ and $a\ne b$. The covariant derivative of $g(\xi ,\mathcal{D}\psi )=0$ along the vector field $E_1$ gives

$$g({\nabla }_{E_1}\xi ,\mathcal{D}\psi )=-[\left(\Omega -\frac{\epsilon \mathcal{R}}{2}\right)+(a-b)]\eta \left(E_1\right),$$

(50)

wherein (10) and (38) are used. Since the velocity vector $\rho$ is Killing in a $MFST$, that is

$$g({\nabla }_{E_1}\xi ,F_1)+g(E_1,{\nabla }_{F_1}\xi ,\xi )=0.$$

Putting $F_1=\xi$ in this equation, we turn up $g(E_1,{\nabla }_{\xi }\xi )=0$ because $g({\nabla }_{E_1}\xi ,\rho )=0$. As a result, we say that ${\nabla }_{\xi }\xi =0$. Changing $E_1$ with $\xi$ in (50) and adopting the last equation, we gain that

$$\Omega -\frac{\epsilon \mathcal{R}}{2}=b-a.$$

(51)

$$\Omega =\mu H+\frac{5p}{2}+\frac{\rho }{2}+\frac{\epsilon \mathcal{R}}{2}.$$

(52)

This reflects that the GRY-soliton in a $MFST$ is expanding.

Thus, by concluding the above facts, one can articulate the following theorem.

Theorem 12. 

Let the $MFST$ admits a GRY-soliton and its velocity vector filed ξ is Killing. Then, either

(i) 

The EoS of the $MFST$ is governed by $p=-\frac{\rho }{5}+\frac{\mu (2-H)}{5}$, the magnetic permeability $\mu =\frac{5p+\rho }{2-H}$, and strength of magnetic field is $H=2-\left(\frac{5p+\rho }{\mu }\right)$.

(ii) 

The $MFST$ admits the expanding GRY-soliton.

According to Mantica and Molinari [40], a perfect fluid spacetime with conformal Killing Ricci tensor the equation of state is

$$p=-\frac{n+1}{n-1}\rho +constant.$$

(53)

In the light of (53) and (49), we have

The EoS with $constant=\frac{\mu (2-H)}{5}=0$ breach the energy condition $\left|\frac{p}{\rho }\right|\le 1$. For $H=2$ or $\mu =0$, the EoS is

$$p=-\frac{1}{5}\rho .$$

(54)

The matter with $\frac{p}{\rho }\le -1$ is named as “photon energy”.

Hence, we turn up the following consequences:

Corollary 12. 

Let the $MFST$ admits a GRY-soliton with a Killing velocity vector filed ξ, then the EoS is governed by

$$p=-\frac{\rho }{5}+\frac{\mu (2-H)}{5},$$

if $\mu =0$ or $H=2$, then the matter of $MFST$ is named as photon energy.

Recently, in 2022, Semiz [41] presented the full solution of $EFEs$ for static spherically symmetric perfect fluid matter with EoS $\frac{p}{\rho }=\omega =-\frac{1}{5}$. In addition, Fazlpour et al. [42] also proved that model regions of stars for EoS $\omega =-\frac{1}{5}$, they describe compact spaces with naked central singularities for the same EoS $\omega =-\frac{1}{5}$.

Therefore, in light of above facts and Corollary 12 we gain the following outcomes.

Theorem 13. 

Let the $MFST$ admits a GRY-soliton with a Killing velocity vector filed ξ, then the EoS (54) is governed by

$$p=-\frac{\rho }{5}+\frac{\mu (2-H)}{5},$$

if $\mu =0$ or $H=2$, then EoS of $MFST$ represents a static spherically symmetric perfect fluid matter.

Theorem 14. 

Let the $MFST$ admits a GRY-soliton with a Killing velocity vector filed ξ and if $\mu =0$ or $H=2$, then EoS (54) of $MFST$ represents a model regions of stars and refer compact spaces with naked central singularities.

Let $H=3$, if we impose the condition $H>2$ in Theorem 13, then we obtain a EoS

$$\frac{p}{\rho }=-\frac{1}{5}\left(1+\frac{\mu }{\rho }\right).$$

(55)

Therefore, (55) entails the following explanation for EoS of $MFST$ with $\mu \ne 0$ and $H>2$.

Theorem 15. 

Let the $MFST$ admits a GRY-soliton with a Killing velocity vector filed ξ and with $\mu \ne 0$ and $H>2$, then the evolution of the universe is given in the following table through EoS (55) of the $MFST$

$Equation of state \frac{p}{\rho }=\omega$	$$\begin{gathered} Restrictions on magnetic \\ permeability \mu and \\ magneto-fluid density \rho \end{gathered}$$	$Evolution of theuniverse$
$\omega =1$	$\mu =-6\rho$	$Ultra relativistic era$
$\omega >-1$	$\mu <4\rho$	$Quintessence era$
$\omega <-1$	$\mu >4\rho$	$Phantom era$
$\omega =0$	$\mu =-\rho$	$Dust era$

In relation to the physical aspect of magnetic permeability, $\mu$ is a value that expresses how a magnetic material responds to an applied magnetic field. Magnetic permeability $\mu$ is defined as (for more details see [30])

$$\mu =B/H,$$

(56)

where B is magnetic flux density and is measure the actual magnetic field and H is the straight magnetic field. Adopting the last case of EoS $\omega =0$ for dust space with (56), we observe the following result.

Corollary 13. 

Let the $MFST$ admits a GRY-soliton with a Killing velocity vector filed ξ and with $\mu \ne 0$ and $H>3$, if $MFST$ recover the dust era with EoS (55), then $MFST$ holds the physical law of magnetism (56) with magnetic fluid density ρ and magnetic field straight $H=-1$.

Corollary 14. 

Let the $MFST$ admits a GRY-soliton with a Killing velocity vector filed ξ and with $\mu \ne 0$ and $H>3$, if $MFST$ retreat the ultra relativistic era with EoS (55), then $MFST$ holds the physical law of magnetism (56) with magnetic fluid density ρ and magnetic field straight $H=-6$.

Remark 2. 

The existence of a naked singularity is typically characterize by existence of Cauchy horizon in the spacetime [3]. In addition, a spacetime-having Cauchy horizon is a globally hyperbolic spacetime, which is a fully predictable universe (for more details see [43]).

Now, by adopting Theorem 14 and Remark 2 we gain the following outcomes.

Corollary 15. 

Let the $MFST$ admits a GRY-soliton with a Killing velocity vector filed ξ and if $\mu =0$ or $H=2$, then EoS (54) of $MFST$ represents Cauchy horizon in $MFST$ for a naked singularity.

Corollary 16. 

If a $MFST$ admits a GRY-soliton with a Killing velocity vector filed ξ and if $\mu =0$ or $H=2$, and $MFST$ holds EoS (54), then the $MFST$ is a hyperbolic spacetime for a naked singularity.

7. Physical Model

As far as a physically relevant model has solitonic solution is concerned, the theory of collapse condensates with the inter atomic attraction and spin-orbit coupling (SOC) [44] is a fundamentally important effect in physical models, chiefly, Bose–Einstein condensates (BEC) [45]. The SOC emulation proceeds by mapping the spinor wave function of electrons into a pseudo-spinor mean-field wave function in BEC, whose components represents two atomic states in the condensate. While SOC in bosonic gases is a linear effect, its interplay with the intrinsic BEC non-linearity, including several types of one dimensional $\left(1D\right)$ solitons [46]. Experimental realization of SOC in two-dimensional $\left(2D\right)$ geometry was reported too [47], which suggests, in particular, a possibility of creation of $2D$ gap soliton [48], supported by a combination of SOC and a spatially periodic field.

A fundamental problem which impedes the creation of $2D$ and $3D$ solitons in BES, non-linear optics, and other non-linear settings, is that the ubiquitous cubic self-attraction, which usually rise to solitons, simultaneously derives the critical and super critical collapse in the $2D$ and $3D$ cases, respectively [49]. Although SOC modifies the conditions of the existence of the solutions and of the blow up, it does not arrest the collapse completely [46]. The collapse destabilizes formally existing solitons, which makes stabilization of $2D$ and $3D$ solitons [45].

In the presence of SOC, the evolution of the wave function is describe by a system coupled non-linear PDE in the Schrödinger form [50]

$$i\hslash \frac{\partial \Psi }{\partial }=\left[-\frac{{\hslash }^2}{2M}\Delta +{\widehat{H}}_{so}+\frac{1}{2}(B.\widehat{\sigma })-g_2{\left|\Psi \right|}^2\right]\Psi ,$$

(57)

where M is the mass of the particle, ${\widehat{H}}_{so}$ is the SOC Hamiltonian, B is the effective magnetic field, $\widehat{\sigma }$ is the spin operator and $g_2$ is the coupling constant.

The key point in the understanding of the role of the SOC in the collapse process is the modified velocity

$$v=k+{\nabla }_k{\widehat{H}}_{so},$$

(58)

where $k=-i\frac{\partial }{\partial r}$, including the velocity and ${\nabla }_k{\widehat{H}}_{so}$ (${\nabla }_k\equiv \frac{\partial }{\partial k})$ are directly related to the particle spin.

Let the first form Rashaba spin-orbit coupling

$${\widehat{H}}_{so}\equiv {\widehat{H}}_R=\alpha (k_x{\widehat{\sigma }}_y-k_y{\widehat{\sigma }}_x),$$

(59)

with coupling constant $\alpha$ and $k=(k_x,k_y)$. The corresponding spin-dependent term in the velocity operators in Equation (58) become (for more details see [46])

$$\frac{\partial {\widehat{H}}_R}{\partial k_x}=\alpha {\widehat{\sigma }}_y,\frac{\partial {\widehat{H}}_R}{\partial k_y}=-\alpha {\widehat{\sigma }}_x.$$

(60)

In particular, in the $2D$ case, the non-linear Schrödinger equation with cubic self-attraction term give rise to degenerate families of the fundamental Townes solitons [51] with vorticity $S=0$, which means decaying solutions, hence Townes solitons, that play the role of separation between type of dynamical behavior, are compatible unstable and total norm of spinor wave function does not exceeds a critical value. Further, it is also capable of producing stable dipole and quadrupole bound states of fundamental solitons with opposite signs.

8. Conclusions

In this article, we try to analyze how the spacetime manifold becomes magnetized when certain magnetic properties are present. We achieved that a generalized quasi-Einstein spacetime ${\left(GQE\right)}_4$ is a $MFST$ with the properties of magneto-fluid density, magnetic field, strength, and magnetic permeability of magnetic fluid and obeying Einstein field equation without cosmological constant. In $MFST$, we generated an unique EoS $\frac{p}{\rho }=-\frac{1}{5}$ with constant scalar curvature. We discovered the values of magnetic pressure p and magneto-fluid density $\rho$ for the various eras of the cosmos in light of this same EoS.

A criterion for growing, stable, and shrinking Ricci–Yamabe solitons with a $\phi \left(\mathcal{R}ic\right)$-vector field in $MFST$ torse-forming vector fields is obtained. The strong energy condition and existence of a black hole are gained in $MFST$ by using the shrinking Ricci–Yamabe soliton in this sequence. Moreover, if a $MFST$ yields a Ricci–Yamabe soliton with a proper $\phi \left(\mathcal{R}ic\right)$ vector field, it is generalized quasi-Einstein spacetime.

Last but not least, the gradient Ricci–Yamabe soliton on the $MFST$ with a new type of velocity vector field, called Killing, also yields an equation of state for the $MFST$ with a gradient Ricci–Yamabe solitons. A gradient Ricci–Yamabe solution is admitted in a magneto-fluid spacetime under the parameters $\mu =0$ or $H=0$ and obeys the EoS $\frac{p}{\rho }=-\frac{1}{5}$, revealing a naked singularity with a Cauchy horizon that eventually appears.