Geometric Perspective of Relativistic Bulk Viscous Fluid String Spacetime

Abstract

The goal of the article is to examine the behavior of bulk viscous fluid string spacetime with a fluid density of the bulk viscous fluid string $\rho$ and a tension of the bulk viscous fluid string $\lambda$. This is known as relativistic bulk viscous fluid string spacetime. We derive some conclusions for bulk viscous fluid string with a vanishing space–matter tensor and a divergence-free matter tensor. We then focus on certain curvature properties for bulk viscous fluid string spacetime, including conformally flat, Ricci recurrent, Ricci semi-symmetric, and pseudo-Ricci-symmetric. Some physical results that align with the equation of state of Ricci semi-symmetric bulk viscous fluid string spacetime are also obtained.

1. Introduction

The General Theory of Relativity ($GTR$) is the strongest theoretical approach for examining the universe’s large-scale structure. Einstein’s version of the gravitational field equations [1,2] presents the typical approach to understanding known cosmic dynamics. With the addition of a hypothetical component of the universe known as Dark Matter ($DM$) [3], Einstein’s field equation provides the best approximation to the observable data. In addition, the cosmos has a peculiar element known as Dark Energy ($DE$) [4], which is regarded to be the main driver of the expansion of the universe and controls the matter-to-energy ratio.

The current condition of the universe is usually described by the cosmological constant in Einstein’s equation. Einstein’s gravitational field equations ($EGF{E}_s$) are also provided in their version without cosmological constants [5,6].

$${\mathit{S}}_{\mu \nu }-{\displaystyle \frac{\mathit{R}}{2}}{g}_{\mu \nu }=\kappa {\mathit{T}}_{\mu \nu },$$

(1)

where vector fields $\mu ,\nu$ in spacetime $\chi (L^n)$, the Ricci tensor, scalar curvature, Lorentzian metric, and energy–momentum tensor of spacetime are represented by $\mathit{S}$, $\mathit{R}$, g, and $\mathit{T}$, respectively. The gravitational constant, $\kappa$, is selected to be $8\pi G$, making G a universal gravitational constant. To acquire the cosmological constant and the Ricci tensor and scalar curvature required to realize Einstein’s objective of a static universe, one must use $EGF{E}_s$.

As a form of semi-Riemannian manifold with Lorentzian metric g, a spacetime can be described as a four-dimensional time-oriented Lorentzian manifold ($L^4,g$). In the Lorentzian manifold, the fundamental vector characterization served as the foundation for further research into the geometry-related characteristics of this manifold type. A Lorentzian manifold L is the ideal choice for researching cosmic models for this reason. Additionally, spacetime manifold is a Lorentzian manifold that allows for a vector field that resembles time.

The quasi-Einstein Lorentzian manifolds are referred to as perfect fluid spacetime if the Ricci tensor has the following form [5,6]:

$$\mathit{S}={\mathit{A}}_{1}g+{\mathit{A}}_2\gamma \otimes \gamma ,$$

(2)

where $\gamma$ is a 1-form metrically equal to unit time-like vector field $\zeta$, such that $\mathit{g}(\mu ,\varsigma )=-1=\gamma (\varsigma )$, and ${\mathit{A}}_1$ and ${\mathit{A}}_2$ are scalars. Furthermore, the spacetime manifold is a Lorentzian manifold that admits a time-like vector field.

The generalized quasi-Einstein manifold ${(GQE)}_n$ was first proposed in [7,8] in the manner described below.

Definition 1.

A non-flat Riemannain manifold $(L^n,g)$$(n>2)$ is said to be a generalized quasi-Einstein Lorentzian manifold $(GQE)$ if its Ricci tensor $\mathit{S}$ of type $(0,2)$ is non-zero and satisfies the expression

$$\mathit{S}=A_{1}g+A_2\gamma \otimes \gamma +A_3\mathrm{Y}\otimes \mathrm{Y}$$

(3)

wherein $A_1,A_2$, and $A_3$ are non-zero scalars of which $A_2\ne 0$, $A_3\ne 0$, and γ, Υ are 1-from, such that

$$g(\mu ,\varsigma )=\gamma (\mu )=-1,g(\mu ,\zeta )=\mathrm{Y}(\mu )=1,g(\varsigma ,\zeta )=0$$

for any vector filed $\mu \in \chi (L^n,g)$.

The 1-form γ and Υ correspond to the time-like and spacelike unit vectors ς and ζ, respectively, which are orthogonal to one another. Furthermore, ς and π are the generators of manifolds $(L^n,g)$ change to a perfect fluid spacetime if $A_3=0$.

On the other hand, fundamental causal theory has been used to analyze the evolution of viscous pressure in the context of an expanding universe that is dominated by bulk viscous matter. One can also describe the cosmos using a one-dimensional entity known as a string. There appears to be a substantial correlation between the expansion of the universe and the extension of cosmic strings, which can go to any location in the cosmos that can be found [9]. In [10], a string cloud model with bulk viscosity was also investigated by Reddy.

The energy–momentum tensor for a bulk viscous fluid string is given by [10]

$$T_{\mu \nu }=(\overline{p}+\rho ){\gamma }_{\mu }{\gamma }_{\nu }-\overline{p}{g}_{\mu \nu }-\lambda {\mathrm{Y}}_{\mu }{\mathrm{Y}}_{\nu }$$

(4)

$$\overline{p}=p-3\zeta \mathcal{H}(=\epsilon \rho ),$$

(5)

wherein $\overline{p}$ represents the total pressure, which includes the isotropic pressure p; $\lambda$ represents the string tension density; $\rho$ represents the rest energy density of the system; $\zeta (t)$ represents the bulk viscosity coefficient; $3\zeta \mathcal{H}$ is commonly referred to as the bulk viscous pressure; and $\mathcal{H}$ represents the Hubble parameter of the model.

Also, ${\gamma }_{\mu }$ is the four velocity vector, ${\mathrm{Y}}_{\mu }$ is a space-like vector, which represents the anisotropic direction of the string, and they satisfy

$$g^{\mu \nu }{\gamma }_{\mu }{\gamma }_{\nu }=-{\mathrm{Y}}^{\mu }{\mathrm{Y}}_{\mu }=1,{\gamma }^{\mu }{\mathrm{Y}}_{\mu }=0.$$

(6)

The momentum tensor is a key component of the spacetime matter content, even if spacetime matter is thought to be fluid with density, pressure, tension, and dynamical and kinematic properties, including velocity, acceleration, vorticity, shear, and expansion [11,12]. The matter composition of the cosmos is assumed to behave as a certain fluid spacetime (see [13,14,15,16]) in traditional cosmological models. These days, researchers seek to figure out the behavior of the cosmos models during the transition from a very early, high-energy state to the modern universe by fusing string theory with ideas like bulk viscosity.

Essentially, the change from a spacetime manifold to a bulk viscous fluid string spacetime entails simulating the early universe as a dynamic system in which topological flaws such as cosmic strings and fluid characteristics, such as viscosity, affect spacetime geometry. Large-scale structure development, the acceleration of the universe’s expansion, and other basic cosmic phenomena can all be investigated using this approach.

In a spacetime manifold, an event can be located using coordinates (vector fields), and the curvature is determined by the distribution of mass and energy density. On the other hand, a bulk viscous fluid string spacetime can be used to model one-dimensional spacetime defects that may have formed during phase transitions in the early universe. These strings are predicted to have enormous energy density and can act as seeds for the formation of large-scale structures like galaxies.

The shift entails using a spacetime that combines the characteristics of a bulk viscous fluid (a cosmological fluid with viscosity) and the geometry of a manifold $GTR$ to describe the evolution of the universe. Bulk viscosity can affect the expansion rate, matter distribution, and overall geometry of spacetime. This transition can be studied using a variety of cosmological models, such as those based on Bianchi type-I or Kantowski-Sachs [9,10] universes and incorporating the idea of cosmic strings.

Recently, a number of investigations based on multiple spacetimes have been published in relation to $GTR$ (see [17,18,19,20,21] for additional information). We can now further investigate bulk viscous fluid string spacetime in $GTR$ because of these studies.

The above studies inspire enough to study the relativistic bulk viscous fluid string spacetime.

2. Relativistic Bulk Viscous Fluid String Spacetime

In this section, we discuss the basic elements of spacetime equipped with the energy–momentum tensor of the bulk viscous fluid string type, or say “bulk viscous fluid string spacetime”.

Definition 2.

A four-dimensional spacetime manifold $(L^4,g)$ generated by a symmetric bulk viscous fluid string energy–momentum tensor $\mathit{T}$ is called a bulk viscous fluid string spacetime.

We now determine the $EGF{E}_s$ for a relativistic bulk viscous fluid string spacetime in accordance with (1) and (4).

$$S(\mu ,\nu )=\left\{{\displaystyle \frac{R}{2}}-\kappa \overline{p}\right\}g(\mu ,\nu )+\kappa (\overline{p}+\rho )\gamma (\mu )\gamma (\nu )-\kappa \lambda \mathrm{Y}(\mu )\mathrm{Y}(\nu ),$$

(7)

$$S(\mu ,\varsigma )=(A_1+A_2)\gamma (\mu ),$$

(8)

$$S(\mu ,\zeta )=(A_1+A_3)\mathrm{Y}(\mu ),$$

(9)

where $A_1=\frac{R}{2}-\kappa \overline{p}$, $A_2=\kappa (\overline{p}+\rho )$, and $A_3=-\kappa \lambda .$

The Lorentzian generalized quasi-Einstein manifold, which describes spacetime, is taken into consideration based on Equation (7). Relativistic bulk viscous fluid string spacetime occurs with $\frac{R}{2}-\kappa \overline{p}$, where $\kappa (\overline{p}+\rho )$ and $\kappa \lambda$ are associated scalars, and $\gamma$ and $\mathrm{Y}$ are associated 1-forms.

In this case, the total pressure $\overline{p}$, energy density $\rho$, and string tension $\lambda$ of a relativistic bulk viscous fluid string spacetime are filled with $EGF{E}_s$. We can, therefore, say the following.

Theorem 1.

A bulk viscous fluid string spacetime $(L^4,g)$ obeys the $EGF{E}_s$ without a cosmic constant and, with total pressure $\overline{p}$, energy density ρ, and string tension λ, is a generalized quasi-Einstein spacetime.

Once (2.3) has been contracted, we then find the following relations:

$$R=\kappa (3\overline{p}-\rho -\lambda ),$$

(10)

$${\displaystyle \frac{\overline{p}}{\rho }}={\displaystyle \frac{4\overline{p}}{\kappa }}+\lambda -\left\{{\displaystyle \frac{R}{\kappa }}+1\right\}.$$

(11)

Theorem 2.

If a bulk viscous fluid string spacetime $(L^4,g)$ obeys the $EGF{E}_s$ without a cosmic constant and with total pressure $\overline{p}$, energy density ρ, and string tension λ, the scalar curvature is given by (10).

Theorem 3.

If a bulk viscous fluid string spacetime $(L^4,g)$ obeys the $EGF{E}_s$ without cosmic constant and with total pressure $\overline{p}$, energy density ρ, and string tension λ, then the equation of state is furnished by (11).

3. Significance of the Equation of State Relative to Bulk Viscous Fluid String Spacetime

The equation of state $(\mathcal{E}o\mathcal{S})$ for the bulk viscous fluid string spacetime describes how the universe expands and changes, especially concerning dark energy and the accelerated expansion. The equation of state for a bulk viscous fluid string spacetime is essential to cosmology and astrophysics. Researchers may model more complicated scenarios than the straightforward perfect fluid model by adding bulk viscosity, which reflects energy dissipation within a fluid, to the equation of state. This could help them solve problems like the dynamics of the early cosmos or the source of dark energy.

In terms of $\mathcal{E}o\mathcal{S}$, the total pressure $\overline{p}$ and the energy density $\rho$ in a fluid are related by the equation of state $(\mathcal{E}o\mathcal{S})$. A term that represents the bulk viscosity in the bulk viscous fluid string spacetime is added to the effective pressure of a cosmic fluid to account for the bulk viscosity. An alternate or supplementary explanation for dark energy to explain the universe’s accelerated expansion is bulk viscosity in the bulk viscous fluid string spacetime. This acceleration can be caused by a negative effective pressure, which is produced by the bulk viscosity. Since neutron star mergers entail extremely high densities and velocities, the equation of state for bulk viscous fluids is essential in the bulk viscous fluid string spacetime. Several cosmological models, including those that include interacting dark energy and dark matter, can be constructed using the $\mathcal{E}o\mathcal{S}$ with bulk viscous fluid string spacetime [22].

Remark 1.

In light of (5) Equation of state $(\mathcal{E}o\mathcal{S})$ provides a relation between total pressure and energy density and is given by

$$\overline{p}=\epsilon \rho ,andp={\epsilon }_0\rho ,where\epsilon ={\epsilon }_0-\mathsf{\Omega },(0\le \epsilon \le 1),$$

(12)

where ${\epsilon }_0$ and Ω are constant. The values $\epsilon =-1,0,\frac{1}{3},1$ represent vacuum dominated, matter dominated, radiation dominated, and stiff fluid era [23], respectively [10].

Remark 2.

According to [24] for Takabayashi string (also known as P-string), where ε is a cosmological constant. This represents the equation of state for the Takabayashi string [24]. Only geometric strings (Nambu strings [25]) are visible when $\epsilon <0$, whereas particles predominate over strings when $\epsilon >0$.

Now, assuming that the source is of the radiation type, then $(\mathcal{E}o\mathcal{S})$ is $\epsilon =\frac{1}{3}$. Equation (11) combined with this information provides

$$\overline{p}={\displaystyle \frac{\kappa \lambda -R}{4(\kappa -1)}}and\rho ={\displaystyle \frac{\kappa \lambda -R}{12(\kappa -1)}}.$$

(13)

Corollary 1.

Let the source of a bulk viscous fluid string spacetime $(L^4,g)$ be a radiation type, then the total pressure and density are governed by (13).

In the case of vacuum dominated era $\rho =-\overline{p}=\frac{1}{4}(\kappa \lambda -R).$

Corollary 2.

If a bulk viscous fluid string spacetime $(L^4,g)$ is dominated by the vacuum era, then the total pressure and density are evaluated as

$$\rho =-\overline{p}={\displaystyle \frac{1}{4}}(\kappa \lambda -R).$$

Corollary 3.

If the source of a bulk viscous fluid string spacetime $(L^4,g)$ be a dust type, then the density is $\rho =\lambda -\frac{R}{\kappa }.$

Corollary 4.

If a bulk viscous fluid string spacetime $(L^4,g)$ is dominated by a stiff matter era, then the total pressure and density are evaluated as

$$\rho =\overline{p}={\displaystyle \frac{\kappa \lambda -R}{2(\kappa -2)}}.$$

After comparing (12) and (11), we obtain the next Corollary.

Corollary 5.

If a bulk viscous fluid string spacetime $(L^4,g)$ obeys the $EGF{E}_s$ without a cosmic constant and with total pressure $\overline{p}$, energy density ρ, and string tension λ, then the constants of $(\mathcal{E}o\mathcal{S})$ are ${\epsilon }_0=\frac{4\overline{p}}{\kappa }+\lambda$ and $\mathsf{\Omega }=\frac{R}{\kappa }+1.$

Now, in the light of the above remark and (11), we can articulate the following outcomes:

Theorem 4.

If a bulk viscous fluid string spacetime $(L^4,g)$ obeys the $EGF{E}_s$ without a cosmic constant and with total pressure $\overline{p}$, energy density ρ, then the string tension λ and evolution of the universe are given in the following table through $(\mathcal{E}o\mathcal{S})$ (11) as

$\left(\mathcal{E}o\mathcal{S}\right)(\overline{p}=\epsilon \rho )$	String tension λ	Evolution of the universe
ε = −1	$\lambda =\frac{1}{\kappa }(R-4\overline{p})$	Vacuum – dominated era
ε = 0	$\lambda =\frac{1}{\kappa }(R+\kappa -4\overline{p})$	Matter-dominated era
$\epsilon =\frac{1}{3}$	$\lambda =\frac{R}{\kappa }+4(\frac{1}{3}-\frac{\overline{p}}{\kappa })$	Radiation-dominated era
ε = 1	$\lambda =\frac{R}{\kappa }+2(1-\frac{2\overline{p}}{\kappa })$	Stiff matter era

Next, in view of Theorem 4 and Remark 2, turn up the following classification of the string.

Theorem 5.

If a bulk viscous fluid string spacetime $(L^4,g)$ obeys the $EGF{E}_s$ without a cosmic constant and with total pressure $\overline{p}$ and energy density ρ, then the string tension λ and evolution of the P-string are given in the following table by $(\mathcal{E}o\mathcal{S})$ (11) as

$\left(\mathcal{E}o\mathcal{S}\right)(\overline{p}=\epsilon \rho )$	String tension λ	Evolution of the String
ε = −1	$\lambda =\frac{1}{\kappa }(R-4\overline{p})$	Nambu strings
ε = 0	$\lambda =\frac{1}{\kappa }(R+\kappa -4\overline{p})$	Matter – dominatedstring
$\epsilon =\frac{1}{3}$	$\lambda =\frac{R}{\kappa }+4(\frac{1}{3}-\frac{\overline{p}}{\kappa })$	Gas string
ε = 1	$\lambda =\frac{R}{\kappa }+2(1-\frac{2\overline{p}}{\kappa })$	Particles predominate string

4. Bulk Viscous Fluid String Spacetime and Vanishing Space–Matter Tensor

We consider the vanishing space–matter tensor relativistic bulk viscous fluid string spacetime in this part. The primary findings of this section are first based on the information we provide.

An expansion of the concept of quasi constant curvature put out in [8] is known as generalized quasi-constant curvature.

$$\mathfrak{S}(\mu ,\nu ,\omega ,\sigma )=\alpha [g(\nu ,\omega )g(\mu ,\sigma )-g(\mu ,\omega )g(\nu ,\sigma )]$$

$$+\beta [g(\mu ,\sigma )\eta (q)\eta (\omega )-g(\nu ,\omega )\eta (\mu )\eta (\sigma )],$$

$$+g(\nu ,\omega )\eta (\mu )\eta (\sigma )-g(\nu ,\sigma )\eta (\mu )\eta (\omega )]$$

(14)

$$+\delta [g(\mu ,\sigma )\psi (\nu )\psi (\omega )-g(\nu ,\sigma )\psi (\mu )\psi (\omega )$$

$$+g(\nu ,\omega )\psi (\mu )\psi (\sigma )-g(\mu ,\omega )\psi (\nu )\psi (\sigma )],$$

where vector fields $\mu ,\nu ,\omega ,\sigma \in \chi (L^4)$, and $\eta$ and $\psi$ are nonzero 1-forms, and $\alpha$, $\beta$, and $\delta$ are scalars.

Assume that the spacelike vector field $\zeta$ and the unit timelike vector field $\varsigma$ are orthogonal. They are defined as follows: $g(\mu ,\varsigma )=\eta (\mu )$, as well as $g(\mu ,\zeta )=\psi (\mu )$.

A fourth-rank tensor $\tilde{\mathfrak{P}}$ was introduced by Petrov [26] and is defined by the following:

$$\tilde{\mathfrak{P}}=\tilde{\mathfrak{S}}+{\displaystyle \frac{\kappa }{2}}g\wedge T-\varrho \mathfrak{G},$$

(15)

where T represents the bulk viscous fluid string energy–momentum tensor of type $(0,2)$, $\kappa$ is the gravitational constant, $\varrho$ is the energy density, and $\tilde{\mathfrak{S}}$ is a $(0,4)$ type Riemannian curvature tensor. $\mathfrak{G}$ is a $(0,4)$ tensor that may be written as

$$\mathfrak{G}(\mu ,\nu ,\omega ,\sigma )=g(\nu ,\omega )g(\mu ,\sigma )-g(\mu ,\omega )g(\mu ,\sigma ),$$

(16)

where ∧ defines the Kulkarni–Nomizu product between g and T for every $\mu ,\nu ,\omega ,\sigma \in \chi (L^4)$ (for additional information, see [27,28]).

$\tilde{\mathfrak{P}}$ is also referred to as a space–matter tensor. The distribution and velocity of the matter are represented by the second portion of this tensor, while the first part depicts the curvature of the bulk viscous fluid string spacetime [27,28].

It is now possible to represent (15) as

$$\tilde{\mathfrak{P}}(\mu ,\nu ,\omega ,\sigma )=\tilde{\mathfrak{S}}(\mu ,\nu ,\omega ,\sigma )+{\displaystyle \frac{\kappa }{2}}[g(\nu ,\omega )T(\mu ,\sigma )+g(\mu ,\sigma )T(\nu ,\omega )$$

(17)

$$-g(\mu ,\omega )T(\nu ,\sigma )-g(\nu ,\sigma )T(\mu ,\omega )]$$

$$-\varrho [g(\nu ,\omega )g(\mu ,\sigma )-g(\mu ,\sigma )g(\nu ,\sigma )].$$

If $\tilde{\mathfrak{P}}(\mu ,\nu ,\omega ,\sigma )=0$, then (17) provides

$$\tilde{\mathfrak{S}}(\mu ,\nu ,\omega ,\sigma )=-{\displaystyle \frac{\kappa }{2}}[g(\nu ,\omega )T(\mu ,\sigma )+g(\mu ,\sigma )\mathcal{T}(\nu ,\omega )$$

(18)

$$-g(\mu ,\omega )\mathcal{T}(\nu ,\sigma )-g(\nu ,\sigma )\mathcal{T}(\mu ,\omega )]$$

$$+\varrho [g(\nu ,\omega )g(\mu ,\sigma )-g(\mu ,\omega )g(\nu ,\sigma )].$$

By including (4) into (18), we now obtain

$$\tilde{\mathfrak{S}}(\mu ,\nu ,\omega ,\sigma )=\alpha [g(\nu ,\omega )g(\mu ,\sigma )-g(\mu ,\omega )g(\nu ,\sigma )]$$

(19)

$$+\beta [g(\nu ,\omega )\gamma (\mu )\gamma (\sigma )+g(\mu ,\sigma )\gamma (\nu )\gamma (\omega )$$

$$-g(\mu ,\omega )\gamma (\nu )\gamma (\sigma )-g(\nu ,\sigma )\gamma (\mu )\gamma (\omega )]$$

$$+\delta [g(\mu ,\sigma )\mathrm{Y}(\nu )\mathrm{Y}(\omega )-g(\nu ,\sigma )\mathrm{Y}(\mu )\mathrm{Y}(\omega )$$

$$+g(\nu ,\omega )\mathrm{Y}(\mu )\mathrm{Y}(\sigma )-g(\mu ,\omega )\mathrm{Y}(\nu )\mathrm{Y}(\sigma )].$$

In this case, $\alpha =-\varrho$, $\beta =-\frac{\kappa (\overline{p}+\rho )}{2}$, and $\delta =-\frac{\kappa \lambda }{2}$. It is evident from (19) that the relativistic bulk viscous fluid string spacetime considered is generalized quasi-constant curvature in light of (14). Thus, the following can be expressed.

Theorem 6.

If a bulk viscous fluid string spacetime obeys the $EGF{E}_s$ without a cosmological constant, and with vanishing space–matter tensor $\tilde{\mathfrak{P}}$, then the bulk viscous fluid string spacetime is of generalized quasi-constant curvature.

5. Relativistic Bulk Viscous Fluid String Spacetime with Divergence-Free Space–Matter Tensor

In this section, we study the requirements for a space–matter tensor [28] to be divergence-free in a bulk viscous fluid string spacetime.

For a bulk viscous fluid string spacetime, if the associated scalars $A_1$, $A_2$, and $A_3$ are constant, then given (7), we obtain

$$R=4A_1+A_2,$$

(20)

which implies that R, the scalar curvature, is constant. Consequently, $dR=0$. Using (1) now, we discover from (15) that

$$(div\tilde{\mathfrak{P}})(\mu ,\nu ,\omega )=(div\tilde{\mathfrak{S}})(\mu ,\nu ,\omega )+{\displaystyle \frac{1}{2}}[({\nabla }_{\mu }S)(\mu ,\omega )-({\nabla }_{\nu }S)(\mu ,\omega )]$$

(21)

$$-g(\nu ,\omega )[d\varrho (\mu )+{\displaystyle \frac{1}{4}}dR(\mu )]+g(\mu ,\omega )[d\varrho (\nu )+{\displaystyle \frac{1}{4}}dR(\nu )].$$

We are aware that in a spacetime Lorentzian manifold (also known as a semi-Riemannian manifold),

$$(div\tilde{\mathfrak{R}})(\mu ,\nu ,\omega )=({\nabla }_{p}S)(\nu ,\omega )-({\nabla }_{q}S)(\nu ,\omega ).$$

(22)

In view of (1), we have $({\nabla }_{\mu }S)(\nu ,\omega )=\kappa ({\nabla }_{\mu }T)(\mu ,\omega ).$

Using (21) and (22), we gain

$$(div\tilde{\mathfrak{P}})(\mu ,\nu \omega )={\displaystyle \frac{3}{2}}[({\nabla }_{\mu }S)(\nu ,\omega )-({\nabla }_{\nu }S)(\nu ,\omega )]$$

(23)

$$-g(\nu ,\omega )[d\varrho (\mu )+{\displaystyle \frac{1}{4}}dR(\mu )]+g(\mu ,\omega )[d\varrho (\nu )+{\displaystyle \frac{1}{4}}dR(\nu )].$$

Given that $(div\tilde{\mathfrak{P}})(\mu ,\nu ,\omega )=0$ and that (23) is contracted over $\mu$ and $\omega$, we obtain

$$d\varrho (\nu )=0.$$

(24)

As a result, we can state the following outcome.

Theorem 7.

If a bulk viscous fluid string spacetime obeys $EGF{E}_s$ with a divergence-free space–matter tensor, the energy density ϱ is constant.

Thus, it is possible to represent (7) and (23) as

$$(div\tilde{\mathfrak{P}})(\mu ,\nu ,\omega )=-{\displaystyle \frac{3}{2}}[d(\kappa (3\overline{p}-\rho -\lambda ))(\mu )g(\nu ,\omega )-d(\kappa (3\overline{p}-\rho -\lambda ))(\nu )g(\mu ,\omega )]$$

(25)

$$+{\displaystyle \frac{3}{2}}[d(\kappa (\overline{p}+\rho ))(p)\gamma (\nu )\gamma (\omega )-d(\kappa (\overline{p}+\rho ))(\nu )\gamma (\mu )\gamma (\omega )]$$

$$-{\displaystyle \frac{3}{2}}[d(\kappa \lambda )(\mu )\mathrm{Y}(\nu )\mathrm{Y}(\omega )-d(\kappa \lambda )(\nu )\mathrm{Y}(\mu )\mathrm{Y}(\omega )]$$

$$+{\displaystyle \frac{3\kappa (\overline{p}+\rho )}{2}}[({\nabla }_{\mu }\gamma )(\nu )\gamma (\omega )+\gamma (\nu )({\nabla }_{\mu }\gamma )(\omega )$$

$$-({\nabla }_{\nu }\gamma )(\mu )\gamma (\omega )-\gamma (\mu )({\nabla }_{\nu }\gamma )\gamma (\omega )]$$

$$-{\displaystyle \frac{3\kappa \lambda }{2}}[({\nabla }_{\mu }\mathrm{Y})(\nu )\mathrm{Y}(\omega )+\mathrm{Y}(\nu )({\nabla }_{\mu }\mathrm{Y})(\omega )$$

$$+({\nabla }_{\nu }\mathrm{Y})(\mu )(\omega )+\mathrm{Y}(\mu )({\nabla }_{\nu }\mathrm{Y})\mathrm{Y}(\omega )]$$

$$-g(\nu ,\omega )[d\varrho (\mu )+{\displaystyle \frac{1}{4}}dR(\mu )]+g(\mu ,\omega )[d\varrho (\nu )+{\displaystyle \frac{1}{4}}dR(\nu )].$$

We put restrictions on the scalar values, such as the fluid density $\rho$, string tension $\lambda$, and energy density $\varrho$, which must all be constants. Furthermore, the generators $\varsigma$ and $\zeta$ of the bulk viscous fluid string spacetime manifold being parallel vector fields implies that ${\nabla }_{\mu }\varsigma =0$ and ${\nabla }_{\mu }\zeta =0$. Consequently, for every $\mu$, $d\varrho (\mu )=0$ and $dR=0$.

Moreover,

$$g({\nabla }_{\mu }\varsigma ,\nu )=0,({\nabla }_{\mu }\gamma )(q)=0,$$

(26)

$$g({\nabla }_{\mu }\zeta ,\nu )=0,({\nabla }_{\mu }\mathrm{Y})(\nu )=0.$$

(27)

It follows that $(div\tilde{\mathfrak{P}})(\mu ,\nu ,\omega )=0$ from (25)–(27). The following can, therefore, be written.

Theorem 8.

If a bulk viscous fluid string spacetime obeys $EGF{E}_s$ and the associated density of the fluid ρ, string tension λ, and energy density ϱ is constant, then the divergence of the space–matter tensor vanishes.

6. Conformally Flat Bulk Viscous Fluid String Spacetime

According to Yano [29], a bulk viscous fluid string spacetime is considered conformally flat if the Weyl conformal curvature tensor $\mathfrak{C}$ vanishes and is expressed as

$$\mathfrak{C}(\mu ,\nu )\omega =\tilde{\mathfrak{S}}(\mu ,\nu )\omega -{\displaystyle \frac{1}{2}}[S(\nu ,\omega )\mu -S(\mu ,\omega )\nu$$

(28)

$$+g(\nu ,\omega )\mathfrak{Q}\mu -g(\mu ,\omega )\mathfrak{Q}\nu ],$$

$$+{\displaystyle \frac{R}{6}}[g(\nu ,\omega )\mu -g(\mu ,\omega )\nu ].$$

The bulk viscous fluid string spacetime is conformally flat; hence, we obtain

$$\tilde{\mathfrak{S}}(\mu ,\nu ,\omega ,\sigma )={\displaystyle \frac{1}{2}}[S(\nu ,\omega )g(\mu ,\sigma )-g(\nu ,\sigma )S(\mu ,\omega )$$

(29)

$$+g(\nu ,\omega )S(\mu ,\sigma )-g(\mu ,\sigma )S(\nu ,\sigma )],$$

$$+{\displaystyle \frac{R}{6}}[g(\mu ,\omega )g(\nu ,\sigma )-g(\nu ,\omega )g(\mu ,\sigma )].$$

Utilizing (7) and (29), we provide

$$\tilde{\mathfrak{S}}(\mu ,\nu ,\omega ,\sigma )=-{\displaystyle \frac{\kappa (3\overline{p}-\rho -\lambda )}{6}}[g(\nu ,\omega )g(\mu ,\sigma )-g(\mu ,\omega )g(\nu ,\sigma )]$$

$$+{\displaystyle \frac{\kappa (\overline{p}+\rho )}{4}}[g(\mu ,\sigma )\gamma (\nu )\gamma (\sigma )-g(\nu ,\omega )\gamma (\mu )\gamma (\sigma )],$$

$$+g(\nu ,\omega )\gamma (\mu )\gamma (\sigma )-g(\nu ,\sigma )\gamma (\nu )\gamma (\omega )]$$

(30)

$$-{\displaystyle \frac{\kappa \lambda }{2}}[g(\mu ,\sigma )\mathrm{Y}(\nu )\mathrm{Y}(\omega )-g(\nu ,\sigma )\mathrm{Y}(\mu )\mathrm{Y}(\omega )$$

$$+g(\nu ,\omega )\mathrm{Y}(\omega )\mathrm{Y}(\sigma )-g(\mu ,\omega )\mathrm{Y}(\nu )\mathrm{Y}(\sigma )].$$

It demonstrates that the spacetime of bulk viscous fluids string spacetime is a generalized quasi-constant curvature. Therefore, we say the following.

Theorem 9.

A conformally flat bulk viscous fluid string spacetime obeys $EGF{E}_s$ without a cosmological constant is of generalized quasi-constant curvature.

Remark 3.

In the bulk viscous fluid spacetime, the Weyl conformal curvature tensor is numerically analyzed to investigate its behavior under certain conditions, especially when taking into account the impact of bulk viscosity in cosmological models. Understanding the nature of spacetime and the propagation of gravitational waves requires an understanding of the Weyl tensor, a purely geometrical object that is essential to the description of the gravitational field. Viscosity has an impact on the behavior of the Weyl tensor in bulk viscous fluids, which can have an impact on the universe’s expansion and evolution.

The Einstein field equations, including the effects of the Weyl tensor and bulk viscosity, are solved computationally through numerical analysis. To find approximate solutions, this frequently entails discretizing the equations and applying iterative techniques (for more numerical analysis or visual representation, see [30,31]).

In bulk viscous fluids string spacetime, let $\varsigma$ and $\zeta$ be parallel vector fields. Next,

$${\nabla }_{\mu }\varsigma =0,{\nabla }_{\mu }\zeta =0,$$

which implies that

$$\tilde{\mathfrak{S}}(\mu ,\nu )\varsigma =0and\tilde{\mathfrak{S}}(\mu ,\nu )\zeta =0.$$

Consequently, it is clear that

$$S(\mu ,\varsigma )=0andS(\mu ,\zeta )=0.$$

Afterward, by employing (8) and (9) and Definition 1, we acquire

$$A_1+A_2=0,A_2+A_3=0.$$

$A_2=A_3=-A_1$ is suggested since $\varsigma$ and $\zeta$ are parallel vector fields. The following equation is, therefore, obtained from (7):

$$S(\mu ,\nu )=-\kappa (3\overline{p}-\rho -\lambda )[g(\mu ,\nu )-\gamma (\mu )\gamma (\nu )+\mathrm{Y}(\mu )(\nu )],$$

(31)

which shows that

$$({\nabla }_{\tau }S)(\mu ,\nu )=d(-\kappa (3\overline{p}-\rho -\lambda ))[g\mu ,\nu )-\gamma (\mu )\gamma (\nu )+\mathrm{Y}(\mu )\mathrm{Y}(\nu )]$$

$$-\kappa (3\overline{p}-\rho -\lambda )[({\nabla }_{\tau }\gamma )(\mu )\gamma (\nu )+\gamma (\mu )({\nabla }_{\tau }\gamma )(\nu )]$$

$$+({\nabla }_{\tau }\mathrm{Y})(\mu )\mathrm{Y}(\nu )+\mathrm{Y}(\mu )({\nabla }_{\tau }\mathrm{Y})(q)].$$

Since $\varsigma$ and $\zeta$ are parallel vector fields. Therefore,

$$({\nabla }_{\tau }\gamma )(\mu )=0and({\nabla }_{\tau }\mathrm{Y})(\mu )=0$$

for all $\mu ,\nu ,\omega \in \chi (L).$

Thus,

$$({\nabla }_{\tau }S)(\mu ,\nu )=d-\kappa (3{\overline{p}}_{\lambda }-\rho )(z)[g(\mu ,\nu )-\gamma (\mu )\gamma (\nu )+\mathrm{Y}(\mu )\mathrm{Y}(\nu )],$$

(32)

or

$$({\nabla }_{\tau }S)(\mu ,\nu )={\displaystyle \frac{d\left\{-\kappa (3\overline{p}-\lambda -\rho )\right\}(s)}{-\kappa (3\overline{p}-\lambda -\rho )}}S(\mu ,\nu ),$$

Applying (31),

$$({\nabla }_{\tau }S)(\mu ,\nu )=\Theta (r)S(\mu ,\nu ),$$

(33)

where $\Theta (r)=\frac{d\left\{-\kappa (3\overline{p}-\lambda -\rho )\right\}(s)}{-\kappa (3\overline{p}-\lambda -\rho )}$. Hence, we have the following.

Theorem 10.

If the timelike vector field ς and spacelike vector field ζ are parallel in a bulk viscous string spacetime, then the bulk viscous fluid string spacetime is Ricci-recurrent.

7. Ricci Semi-Symmetric Bulk Viscous Fluid String Spacetime

The bulk viscous fluid string is now considered to be Ricci semi-symmetric, with the constraint that $\tilde{\mathfrak{S}}(\mu ,\nu ).S=0$ for all $\mu ,\nu \in \chi (L^4).$

Currently, we have

$$(\tilde{\mathfrak{S}}(\mu ,\nu ).S)(\omega ,\sigma )=-S(\tilde{\mathfrak{S}}(\mu ,\nu ),\omega )-\tilde{\mathfrak{S}}(\omega ,\tilde{\mathfrak{S}}(\mu ,\nu )\sigma ).$$

(34)

Then, from (7),

$$\left\{-\kappa (3\overline{p}-\lambda -\rho )\right\}g(\tilde{\mathfrak{S}}(\mu ,\nu )\omega ,\sigma )+\kappa (\overline{p}+\rho )\gamma (\tilde{\mathfrak{S}}(\mu ,\nu )\omega )\gamma (\sigma )$$

$$-\kappa \lambda \mathrm{Y}(\tilde{\mathfrak{S}}(\mu ,\nu )\omega )\mathrm{Y}(\sigma )-\left\{\kappa (3\overline{p}-\lambda -\rho )\right\}\gamma (\omega )\gamma (\tilde{\mathfrak{S}}(\mu ,\nu )\sigma )$$

$$-\kappa \lambda \mathrm{Y}(\omega )\mathrm{Y}(\tilde{\mathfrak{S}}(\mu ,\nu )\omega )=0.$$

(35)

Putting $\sigma =\varsigma$ and $\omega =\zeta$ in (35), we find

$$-\kappa (3\overline{p}-\lambda -\rho )\gamma (\tilde{\mathfrak{S}}(\mu ,\nu )\varsigma )-\kappa \lambda \mathrm{Y}(\tilde{\mathfrak{S}}(\mu ,\nu )\varsigma )=0,$$

(36)

or

$$-\kappa (3\overline{p}-\lambda -\rho )g(\tilde{\mathfrak{S}}(\mu ,\nu )\zeta ,\varsigma ))-\kappa \lambda g(\tilde{\mathfrak{S}}(\mu ,\nu )\varsigma ,\zeta )=0$$

$$-\kappa (3\overline{p}-\lambda -\rho )\tilde{\mathfrak{S}}(\mu ,\nu ,\omega ,\sigma )=0,$$

(37)

where $\widetilde{\mathfrak{S}im}(\mu ,\nu ,\omega ,\sigma )=g(\tilde{\mathfrak{S}}(\mu ,\nu )\omega ,\sigma )$ is non-vanishing since the gravitational constant $\kappa \ne 0$. Hence, we obtain

$${\displaystyle \frac{\overline{p}}{\rho }}={\displaystyle \frac{(\lambda +1)}{3}}.$$

(38)

Theorem 11.

If a bulk viscous fluid string spacetime obeys $EGF{E}_s$ is Ricci semi-symmetric, then the $\mathcal{E}o\mathcal{S}$ is given by (38).

Theorem 12.

If a Ricci semi-symmetric bulk viscous fluid string spacetime satisfies $\mathcal{E}o\mathcal{S}$ (38), then the particles of the string predominate over strings.

8. Some Physical Interpretations

Now, for the stiff matter era, radiation, and dark matter era, Equation (38) and Theorem 15 provide the following relations, respectively:

$$\overline{p}=C_1\lambda ,whereconstant{C}_1={\displaystyle \frac{1}{2}},$$

(39)

$$\lambda =0,$$

(40)

$$\rho =C_2\lambda ,whereconstantC=-{\displaystyle \frac{1}{4}}.$$

(41)

Consequently, we can articulate the following results.

Corollary 6.

If a Ricci semi-symmetric bulk viscous fluid string spacetime satisfies $\mathcal{E}o\mathcal{S}$ (38) and is dominated by the stiff matter era, then the total pressure $\overline{p}$ is proportional to the string tension λ.

Corollary 7.

If a Ricci semi-symmetric bulk viscous fluid string spacetime satisfies $\mathcal{E}o\mathcal{S}$ (38) and is dominated by the radiation era, then the string tension λ vanishes.

Corollary 8.

If a Ricci semi-symmetric bulk viscous fluid string spacetime satisfies $\mathcal{E}o\mathcal{S}$ (38) and is dominated by the dark matter era, then the total density ρ is proportional to the string tension λ.

9. Pseudo-Ricci Symmetric Bulk Viscous Fluid String Spacetime

The following definition of pseudo-Ricci-symmetric manifolds $(L^4,g)$ was introduced by Chaki in [32].

Definition 3.

If the Ricci tensor S on the semi-Riemannian manifold $(L^4,g)$ satisfies the following equation, then the non-flat semi-Riemannian manifold is considered pseudo-Ricci symmetric:

$$({\nabla }_{\mu }S)(\nu ,\omega )=2\mho (\mu )S(\nu ,\omega )+\mho (\nu )S(\mu ,\omega )+\mho (r)S(\mu ,\nu ),$$

(42)

where ℧ is a 1-form, $\mu ,\nu ,\omega$ are vector fields on $(L^4,g)$, and ∇ is the Levi-Civita connection on $(L^4,g)$. Moreover, the semi-Riemannian manifold $(L^4,g)$ reduces to Ricci-symmetric manifold if $\mho =0$ in (42) or if $\nabla S=0.$

Assuming that timelike vector fields $\varsigma$ and spacelike vector fields $\zeta$ are parallel vector fields, we will assume that the bulk viscous fluid string spacetime is used in this section.

Let $\gamma$ and $\zeta$ be parallel, and consider $(L^4,g)$ to be a bulk viscous fluid string spacetime, a generalized quasi-Einstein manifold. Next, we have

$${\nabla }_{\mu }\varsigma =0,{\nabla }_{\mu }\zeta =0\Rightarrow \tilde{\mathfrak{S}}(\mu ,\nu )\varsigma =0and\tilde{\mathfrak{S}}(\mu ,\mu )\zeta =0.$$

(43)

We now see that when we contract this assertion concerning $\nu$,

$S(\mu ,\varsigma )=0$ and $S(\mu ,\zeta )=0$. So, from (8) and (9), we find

$$S(\mu ,\gamma )=(A_1+A_2)\gamma (\mu )=0,$$

(44)

$$S(\mu ,\zeta )=(A_1+A_3)\mathrm{Y}(\mu )=0.$$

(45)

Therefore, $A_1=-A_2=-A_3$. Then, (7) and (8) turn the form

$$S(\mu ,\nu )={\displaystyle \frac{-\kappa (3\overline{p}-\lambda -\rho )}{2}}[g(\mu ,\nu )-\gamma (\mu )\gamma (\nu )+\mathrm{Y}(\mu )\mathrm{Y}(\nu )].$$

(46)

On the other side, we know that

$$({\nabla }_{\mu }S)(\nu ,\omega )={\nabla }_{\mu }S(\nu ,\omega )-S({\nabla }_{\mu }\nu ,\omega )-S(\nu ,{\nabla }_{\mu }\omega ).$$

(47)

Since bulk viscous fluid string spacetime is a generalized quasi-Einstein manifold and given (46) and (47), we can express it as

$$({\nabla }_{\mu }\mathfrak{S})(\nu ,\omega )={\displaystyle \frac{d}{d\mu }}\left\{{\displaystyle \frac{\kappa (3\overline{p}-\lambda -\rho )}{2}}\right\}[g(\nu ,\nu )-\gamma (\mu )\gamma (\omega )+\mathrm{Y}(\mu )\mathrm{Y}(\omega )],$$

(48)

where $\frac{d}{d\mu }$ signifies the derivative of $-\kappa (3\overline{p}-\lambda -\rho )$ with respect to the vector field $\mu$. Since bulk viscous fluid string spacetime $(L^4,g)$ is pseudo-Ricci-symmetric, using (43) and (48), we can write

$${\displaystyle \frac{d}{d\mu }}\left\{{\displaystyle \frac{-\kappa (3\overline{p}-\lambda -\rho )}{2}}\right\}[g(\nu ,\omega )-\gamma (\mu )\gamma (\omega )+\mathrm{Y}(\mu )\mathrm{Y}(\omega )]$$

(49)

$$=2\left\{{\displaystyle \frac{-\kappa (\lambda +\rho )}{2}}\right\}\mho (\mu )[g(\nu ,\omega )-\gamma (\nu )\gamma (\omega )+\mathrm{Y}(\nu )\mathrm{Y}(\omega )]$$

$$+\left\{{\displaystyle \frac{-\kappa (3\overline{p}-\lambda -\rho )}{2}}\right\}\mho (\nu )[g(\mu ,\omega )-\gamma (\mu )\gamma (\omega )+\mathrm{Y}(\nu )\mathrm{Y}(\omega )]$$

$$+\left\{{\displaystyle \frac{-\kappa (3\overline{p}-\lambda -\rho )}{2}}\right\}\mho (\omega )[g(\mu ,\nu )-\gamma (\mu )\gamma (\nu )+\mathrm{Y}(\mu )\mathrm{Y}(\nu )].$$

Putting $\mu =\varsigma$ and $\mu =\zeta$ in (49), we obtain

$${\displaystyle \frac{d}{d\varsigma }}\left\{{\displaystyle \frac{-\kappa (3\overline{p}-\lambda -\rho )}{2}}\right\}=\left\{-\kappa (3\overline{p}-\lambda -\rho )\right\}\mho (\mu )$$

(50)

and

$${\displaystyle \frac{d}{d\zeta }}\left\{{\displaystyle \frac{-\kappa (3\overline{p}-\lambda -\rho )}{2}}\right\}=\left\{-\kappa (3\overline{p}-\lambda -\rho )\right\}\mho (\nu ).$$

(51)

Using $\tau =\varsigma$ and $\tau =\zeta$ in (49), we obtain

$$\mho (\mu )=0and\mho (\nu )=0.$$

(52)

Therefore, in light of (49)–(52), we obtain

$${\displaystyle \frac{d}{d\varsigma }}\left\{{\displaystyle \frac{-\kappa (3\overline{p}-\lambda -\rho )}{2}}\right\}=0,{\displaystyle \frac{d}{d\zeta }}\left\{{\displaystyle \frac{-\kappa (3\overline{\mu }-\lambda -\rho )}{2}}\right\}=0,$$

which implies $-\kappa (3\overline{\mu }-\lambda -\rho )$ is constant along vector fields $\varsigma$ and $\zeta$.

Consequently, we can infer the following.

Theorem 13.

Let a bulk viscous fluid string spacetime be a generalized quasi-Einstein spacetime with parallel vector fields ς and ζ. If the bulk viscous fluid string spacetime is pseudo-Ricci-symmetric, then $-\kappa (3\overline{p}-\lambda -\rho )$ is constant along the vector field ς and vector field ζ.

Theorem 14.

If a bulk viscous fluid string spacetime is pseudo-Ricci-symmetric, then the $\mathcal{E}o\mathcal{S}$ is $\frac{\overline{p}}{\rho }=\frac{(\lambda +1)}{3}.$

Theorem 15.

If a pseudo-Ricci-symmetric bulk viscous fluid string spacetime satisfies $\mathcal{E}o\mathcal{S}$ (38), then the particles of the string are predominant over strings.

10. Conclusions

This article examines how the spacetime manifold transforms into a bulk viscous fluid string spacetime, with a bulk viscous fluid string’s fluid density and tension. We discovered that a bulk viscous fluid string spacetime is a generalized quasi-Einstein spacetime. We obtain a unique equation of state in the string spacetime of a bulk viscous fluid. This identical equation of state is used to evaluate the values of string tension, density, and total pressure.

For a bulk viscous fluid string with a divergence-free matter tensor and a vanishing space–matter tensor, we draw some conclusions. Next, we concentrate on certain curvature qualities for bulk viscous fluid string spacetime, such as pseudo-Ricci-symmetric, Ricci semi-symmetric, Ricci recurrent, and conformally flat. Additionally, several physical results are achieved that are consistent with the Ricci semi-symmetric bulk viscous fluid string spacetime equation of state.