Application of Mixed Generalized Quasi-Einstein Spacetimes in General Relativity

Abstract

In the present article, some geometric and physical properties of $MG{\left(QE\right)}_n$ were investigated. Moreover, general relativistic viscous fluid $MG{\left(QE\right)}_4$ spacetimes with some physical applications were studied. Finally, through a non-trivial example of $MG{\left(QE\right)}_4$ spacetime, we proved its existence.

1. Introduction

A Riemannian or a semi-Riemannian manifold $(M^n,g)$ of dimension $n(>2)$ is termed as an Einstein manifold if its $(0,2)$-type Ricci tensor $Ric(\ne 0)$ satisfies $Ric=\frac{r}{n}$, where r stands for the scalar curvature [1]. In addition to Riemannian geometry, Einstein manifolds also have a vital contribution to the general theory of relativity (GTR).

Approximately two decades ago, Chaki and Maity introduced and studied quasi-Einstein manifolds [2]. An $(M^n,g)$, $(n>2)$ is said to be a quasi-Einstein manifold ${\left(QE\right)}_n$ if its $Ric$ (≠ 0) realizes the following condition:

$$Ric(U_1,U_2)=ag(U_1,U_2)+bA\left(U_1\right)A\left(U_2\right),$$

(1)

where $a,b\in \mathbb{R}$ such that $b\ne 0$ and $A(\ne 0)$ is the 1-form such that

$$g(U_1,\rho )=A\left(U_1\right),g(\rho ,\rho )=A\left(\rho \right)=1,$$

(2)

for any vector field $U_1$, and a unit vector field $\rho$ called the generator of $(M^n,g)$. In addition, A is named the associated 1-form. Einstein manifolds form a natural subclass of the class of ${\left(QE\right)}_n$.

Under the study of exact solutions of the Einstein field equations, as well as under the consideration of quasi-umbilical hypersurfaces of semi-Euclidean spaces, ${\left(QE\right)}_n$ came into existence. For instance, the Robertson–Walker spacetimes are ${\left(QE\right)}_n$. Thus, ${\left(QE\right)}_n$ have great importance in GTR.

An $(M^n,g),(n\ge 2)$ is said to be a generalized quasi-Einstein manifold $G{\left(QE\right)}_n$ [3] if its $Ric(\ne 0)$ realizes the following condition:

$$Ric(U_1,U_2)=ag(U_1,U_2)+bA\left(U_1\right)A\left(U_2\right)+cB\left(U_1\right)B\left(U_2\right),$$

(3)

where a, b, c are non-zero scalars and A, B are two non-zero 1-forms such that

$$g(U_1,\rho )=A\left(U_1\right),g(U_1,\sigma )=B\left(U_1\right),$$

(4)

where $\rho$ and $\sigma$ are mutually orthogonal unit vector fields, i. e., $g(\rho ,\sigma )=0$. The vector fields $\rho$ and $\sigma$ are called the generators of the manifold. If $c=0$, then the manifold reduces to a quasi-Einstein manifold.

In 2007, Bhattacharya, De and Debnath [4] introduced the notion of a mixed generalized quasi-Einstein manifold. A non-flat Riemannian manifold is said to be a mixed generalized quasi-Einstein manifold and is denoted by $MG{\left(QE\right)}_n,$ if its $Ric(\ne 0)$ satisfies the following condition:

$$\begin{array}{cc}\hfill Ric(U_1,U_2)& =ag(U_1,U_2)+bA\left(U_1\right)A\left(U_2\right)+cB\left(U_1\right)B\left(U_2\right)\hfill \\ \hfill & +d[A\left(U_1\right)B\left(U_2\right)+B\left(U_1\right)A\left(U_2\right)],\hfill \end{array}$$

(5)

where a, b, c, d are non-zero scalars and A, B are two non-zero 1-forms such that

$$g(U_1,\rho )=A\left(U_1\right),g(U_1,\sigma )=B\left(U_1\right),$$

(6)

where $\rho$ and $\sigma$ are mutually orthogonal unit vector fields and are called the generators of the manifold. Recently, $MG{\left(QE\right)}_n$ have been studied by various geometers in several ways to a different extent, such as [5,6,7,8] and many others.

Putting $U_1=U_2=e_i$ in (5), where $\left\{e_i\right\}$ is an orthonormal basis of the tangent space at each point of the manifold, and taking summation over i( $1\le i\le n$), we obtain

$$r=na+b+c.$$

(7)

A Lorentzian four-dimensional manifold is said to be a mixed generalized quasi-Einstein spacetime with the generator $\rho$ as the unit timelike vector field if its $Ric(\ne 0)$ satisfies (5). Here, A and B are non-zero 1-forms such that $\sigma$ is the heat flux vector field perpendicular to the velocity vector field $\rho$. Therefore, for any vector field $U_1$, we have

$$\begin{array}{cc}\hfill & g(U_1,\rho )=A\left(U_1\right),g(U_1,\sigma )=B\left(U_1\right),\hfill \\ \hfill & g(\rho ,\rho )=A\left(\rho \right)=-1,g(\sigma ,\sigma )=B\left(\sigma \right)=1.\hfill \end{array}$$

(8)

Further, we know that if the Riemannian curvature tensor $\overline{K}$ of type $(0,4)$ has the form

$$\overline{K}(U_1,U_2,U_3,U_4)=k[g(U_2,U_3)g(U_1,U_4)-g(U_1,U_3)g(U_2,U_4)],$$

(9)

then the manifold is said to be of constant curvature k. The generalization of this manifold is the manifold of quasi-constant curvature and, in this case, the curvature tensor has the following form:

$$\begin{array}{cc}\hfill \overline{K}(U_1,U_2,U_3,U_4)& =f_1[g(U_2,U_3)g(U_1,U_4)-g(U_1,U_3)g(U_2,U_4)]\hfill \\ \hfill & +f_2[g(U_2,U_3)A\left(U_1\right)A\left(U_4\right)-g(U_2,U_4)A\left(U_1\right)A\left(U_3\right)\hfill \\ \hfill & +g(U_1,U_4)A\left(U_2\right)A\left(U_3\right)-g(U_1,U_3)A\left(U_2\right)A\left(U_4\right)],\hfill \end{array}$$

(10)

where $g(K(U_1,U_2)U_3,U_4)=\overline{K}(U_1,U_2,U_3,U_4)$, K is the curvature tensor of type $(1,3)$ and $f_1$, $f_2$ are scalars, and $\rho$ is a unit vector field defined by

$$g(U_1,\rho )=A\left(U_1\right),$$

It can be easily seen that, if the curvature tensor $\overline{K}$ is of the form (10), then the manifold is conformally flat [3]. Thus, a Riemannian or semi-Riemannian manifold is said to be of quasi-constant curvature if the curvature tensor $\overline{K}$ satisfies the relation (10); we denote such a manifold of dimension n by ${\left(QC\right)}_n$.

A non-flat Riemannian or semi-Riemannian manifold $(M^n,g)$$(n\ge 3)$ is said to be a manifold of generalized quasi-constant curvature if the curvature tensor $\overline{K}$ of type $(0,4)$ satisfies the condition [3]

$$\begin{array}{cc}\hfill \overline{K}(U_1,U_2,U_3,U_4)& =f_1[g(U_2,U_3)g(U_1,U_4)-g(U_1,U_3)g(U_2,U_4)]\hfill \\ \hfill & +f_2[g(U_1,U_4)A\left(U_2\right)A\left(U_3\right)-g(U_2,U_4)A\left(U_1\right)A\left(U_3\right)\hfill \\ \hfill & +g(U_2,U_3)A\left(U_1\right)A\left(U_4\right)-g(U_1,U_3)A\left(U_2\right)A\left(U_4\right)]\hfill \\ \hfill & +f_3[g(U_1,U_4)B\left(U_2\right)B\left(U_3\right)-g(U_2,U_4)B\left(U_1\right)B\left(U_3\right)\hfill \\ \hfill & +g(U_2,U_3)B\left(U_1\right)B\left(U_4\right)-g(U_1,U_3)B\left(U_2\right)B\left(U_4\right)],\hfill \end{array}$$

(11)

where $f_1$, $f_2$, $f_3$ are scalars and A, B are two non-zero 1-forms. $\rho$ and $\sigma$ are orthonormal unit vectors corresponding to A and B such that $g(U_1,\rho )=A\left(X\right)$, $g(U_1,\sigma )=B\left(X\right)$ and $g(\rho ,\sigma )=0$. Such a manifold is denoted by $G{\left(QC\right)}_n$.

In [9], Bhattacharya and De introduced the notion of mixed generalized quasi-constant curvature. A non-flat Riemannian or semi-Riemannian manifold $(M^n,g)$$(n\ge 3)$ is said to be a manifold of mixed generalized quasi-constant curvature if the curvature tensor $\overline{K}$ of type $(0,4)$ satisfies the condition

$$\begin{array}{cc}\hfill \overline{K}(U_1,U_2,U_3,U_4)& =f_1[g(U_2,U_3)g(U_1,U_4)-g(U_1,U_3)g(U_2,U_4)]\hfill \\ \hfill & +f_2[g(U_1,U_4)A\left(U_2\right)A\left(U_3\right)-g(U_2,U_4)A\left(U_1\right)A\left(U_3\right)\hfill \\ \hfill & +g(U_2,U_3)A\left(U_1\right)A\left(U_4\right)-g(U_1,U_3)A\left(U_2\right)A\left(U_4\right)]\hfill \\ \hfill & +f_3[g(U_1,U_4)B\left(U_2\right)B\left(U_3\right)-g(U_2,U_4)B\left(U_1\right)B\left(U_3\right)\hfill \\ \hfill & +g(U_2,U_3)B\left(U_1\right)B\left(U_4\right)-g(U_1,U_3)B\left(U_2\right)B\left(U_4\right)]\hfill \\ \hfill & +f_4[\{A\left(U_2\right)B\left(U_3\right)+B\left(U_2\right)A\left(U_3\right)\}g(U_1,U_4)\hfill \\ \hfill & -\{A\left(U_1\right)B\left(U_3\right)+B\left(U_1\right)A\left(U_3\right)\}g(U_2,U_4)\hfill \\ \hfill & +\{A\left(U_1\right)B\left(U_4\right)+B\left(U_1\right)A\left(U_4\right)\}g(U_2,U_3)\hfill \\ \hfill & -\{A\left(U_2\right)B\left(U_4\right)+B\left(U_2\right)A\left(U_4\right)\}g(U_1,U_3)],\hfill \end{array}$$

(12)

where $f_1$, $f_2$, $f_3$, $f_4$ are scalars. A, B are two non-zero 1-forms. $\rho$ and $\sigma$ are orthonormal unit vectors corresponding to A and B such that $g(U_1,\rho )=A\left(X\right)$, $g(U_1,\sigma )=B\left(X\right)$ and $g(\rho ,\sigma )=0$. Such a manifold is denoted by $MG{\left(QC\right)}_n$.

The spacetime of general relativity and cosmology is regarded as a connected four-dimensional semi-Riemannian manifold $(M^4,g)$ with Lorentzian metric g with signature $(-,+,+,+)$. The geometry of the Lorentz manifold begins with the study of a causal character of vectors of the manifold. Due to this causality, the Lorentz manifold becomes a convenient choice for the study of general relativity. Spacetimes have been studied by various authors in several ways, such as [10,11,12,13,14] and many others.

2. $\mathit{MG}{\left(\mathit{QE}\right)}_{\mathit{n}}$ Admitting the Generators $\mathbf{\rho }$ and $\mathbf{\sigma }$ as Recurrent Vector Fields

Let us consider the generators $\rho$ and $\sigma$ corresponding to the associated recurrent 1-forms A and B. Then, we have

$$\left(D_{U_1}A\right)\left(U_2\right)=\eta \left(U_1\right)A\left(U_2\right),$$

$$\left(D_{U_1}B\right)\left(U_2\right)=\phi \left(U_1\right)B\left(U_2\right),$$

where $\eta$ and $\phi$ are non-zero 1-forms.

A non-flat Riemannian or semi-Riemannian manifold $(M^n,g)$, $(n>2)$ is said to be Ricci-recurrent [15,16] if its $Ric(\ne 0)$ satisfies the following condition:

$$\left(D_{U_1}Ric\right)(U_2,U_3)=\alpha \left(U_1\right)Ric(U_2,U_3),$$

(13)

where $\alpha$ is in non-zero 1-form. Since we know that

$$\begin{array}{ccc}\hfill \left(D_{U_1}Ric\right)(U_2,U_3)& =& U_{1}Ric(U_2,U_3)-Ric(D_{U_1}{U}_2,U_3)\hfill \\ & & -Ric(U_2,D_{U_1}{U}_3),\hfill \end{array}$$

(14)

using (14) in (13), it follows that

$$\begin{array}{ccc}\hfill \alpha \left(U_1\right)Ric(U_2,U_3)& =& U_{1}Ric(U_2,U_3)-Ric(D_{U_1}{U}_2,U_3)\hfill \\ & & -Ric(U_2,D_{U_1}{U}_3).\hfill \end{array}$$

(15)

Using (5) in (15), we obtain

$$\begin{array}{cc}\hfill & \alpha \left(U_1\right)[ag(U_2,U_3)+bA\left(U_2\right)A\left(U_3\right)+cB\left(U_2\right)B\left(U_3\right)\hfill \\ \hfill & +d\{A\left(U_2\right)B\left(U_3\right)+A\left(U_3\right)B\left(U_2\right)\}]=U_1[ag(U_2,U_3)+bA\left(U_2\right)A\left(U_3\right)\hfill \\ \hfill & +cB\left(U_2\right)B\left(U_3\right)+d\{A\left(U_3\right)B\left(U_2\right)+A\left(U_2\right)B\left(U_3\right)\}]\hfill \\ \hfill & -[ag(D_{U_1}{U}_2,U_3)+bA\left(D_{U_1}{U}_2\right)A\left(U_3\right)+cB\left(D_{U_1}{U}_2\right)B\left(U_3\right)\hfill \\ \hfill & +d\{A\left(D_{U_1}{U}_2\right)B\left(U_3\right)+A\left(U_3\right)B\left(D_{U_1}{U}_2\right)\}]\hfill \\ \hfill & -[ag(U_2,D_{U_1}{U}_3)+bA\left(U_2\right)A\left(D_{U_1}{U}_3\right)+cB\left(U_2\right)B\left(D_{U_1}{U}_3\right)\hfill \\ \hfill & +d\{A\left(U_2\right)B\left(D_{U_1}{U}_3\right)+A\left(D_{U_1}{U}_3\right)B\left(U_2\right)\}].\hfill \end{array}$$

(16)

Putting $U_2=U_3=\rho$ in (16), we obtain

$$U_1(a+b)-\alpha \left(U_1\right)(a+b)=2(a+b)A\left(D_{U_1}\rho \right)+2dB\left(D_{U_1}\rho \right).$$

(17)

By using the fact that $A\left(D_{U_1}\rho \right)=0$ and (6) in (17), we have

$$U_1(a+b)-\alpha \left(U_1\right)(a+b)=2dg(D_{U_1}\rho ,\sigma ),$$

(18)

which can be written as

$$U_1(a+b)-\alpha \left(U_1\right)(a+b)=-2dA\left(D_{U_1}\sigma \right).$$

Thus, we have $A\left(D_{U_1}\sigma \right)=0$ if and only if $U_1(a+b)-\alpha \left(U_1\right)(a+b)=0$. This implies that either $D_{U_1}\sigma \perp \rho$ or $\sigma$ is a parallel vector field.

Again, putting $U_2=U_3=\sigma$ in (16), we have

$$U_1(a+b)-\alpha \left(U_1\right)(a+b)=2(a+c)B\left(D_{U_1}\sigma \right)+2dA\left(D_{U_1}\sigma \right).$$

(19)

Again, using the fact that $B\left(D_{U_1}\sigma \right)=0$ and (6) in (19), we have

$$U_1(a+b)-\alpha \left(U_1\right)(a+b)=2dg(D_v\sigma ,\rho ),$$

(20)

$$or,U_1(a+b)-\alpha \left(U_1\right)(a+b)=-2dB\left(D_v\rho \right).$$

Thus, we have $B\left(D_{U_1}\rho \right)=0$ if and only if $U_1(a+b)-\alpha \left(U_1\right)(a+b)=0$. This implies that either $D_{U_1}\rho \perp \sigma$ or $\rho$ is a parallel vector field. Hence, we can state the following theorem:

Theorem 1.

Let a mixed generalized quasi-Einstein manifold $MG{\left(QE\right)}_n$ be Ricci-recurrent; then, the following statements are equivalent:

(i) 

ρ and σ are parallel vector fields;

(ii) 

$U_1(a+b)-\alpha \left(U_1\right)(a+b)=0$ if and only if $D_{U_1}\sigma \perp \rho$;

(iii) 

$U_1(a+b)-\alpha \left(U_1\right)(a+b)=0$ if and only if $D_{U_1}\rho \perp \sigma$.

3. $\mathit{MG}{\left(\mathit{QE}\right)}_{\mathit{n}}$ Admitting the Generators $\mathbf{\rho }$ and $\mathbf{\sigma }$ as Concurrent Vector Fields

A vector field $\pi$ is said to be concurrent if it satisfies the following condition [17,18]:

$$D_{U_1}\pi =\xi U_1,$$

(21)

where $\xi$ is constant.

Let us consider the generators $\rho$ and $\sigma$ corresponding to the associated concurrent 1-forms A and B. Then, we have

$$\left(D_{U_1}A\right)\left(U_2\right)=\lambda g(U_1,U_2),$$

(22)

$$and\left(D_{U_1}B\right)\left(U_2\right)=\mu g(U_1,U_2),$$

(23)

where $\lambda$ and $\mu$ are non-zero constants.

Taking the covariant derivative of (5) with respect to $U_3$, we obtain

$$\begin{array}{cc}\hfill \left(D_{U_3}Ric\right)U_2& =b[\left(D_{U_3}A\right)\left(U_1\right)A\left(U_2\right)+A\left(U_1\right)\left(D_{U_3}A\right)\left(U_2\right)]\hfill \\ \hfill & +c[\left(D_{U_3}B\right)\left(U_1\right)B\left(U_2\right)+B\left(U_1\right)\left(D_{U_3}B\right)\left(U_2\right)]\hfill \\ \hfill & +d[\left(D_{U_3}A\right)\left(U_1\right)B\left(U_2\right)+A\left(U_1\right)\left(D_{U_3}B\right)\left(U_2\right)\hfill \\ \hfill & +\left(D_{U_3}B\right)\left(U_1\right)A\left(U_2\right)+B\left(U_1\right)\left(D_{U_3}A\right)\left(U_2\right)].\hfill \end{array}$$

(24)

Using (22) and (23) in (24), it follows that

$$\begin{array}{cc}\hfill \left(D_{U_3}Ric\right)(U_1,U_2)& =b[\lambda g(U_1,U_3)A\left(U_2\right)+\lambda g(U_2,U_3)A\left(U_1\right)]\hfill \\ \hfill & +c[\mu g(U_1,U_3)B\left(U_2\right)+\mu g(U_2,U_3)B\left(U_1\right)]\hfill \\ \hfill & +d[\lambda g(U_1,U_3)B\left(U_2\right)+\mu g(U_1,U_3)A\left(U_2\right)\hfill \\ \hfill & +\lambda g(U_2,U_3)B\left(U_1\right)+\mu g(U_2,U_3)A\left(U_1\right)].\hfill \end{array}$$

(25)

Contracting (25) over $U_1$ and $U_2$ leads to

$$\partial r\left(U_3\right)=A\left(U_3\right)[2b\lambda +2d\mu ]+B\left(U_3\right)[2c\mu +2d\lambda ].$$

(26)

From (7), it follows that

$$\partial r\left(U_1\right)=0.$$

(27)

In view of (27), (26) turns to

$$A\left(U_3\right)[2b\lambda +2d\mu ]+B\left(U_3\right)[2c\mu +2d\lambda ]=0.$$

(28)

Thus, by virtue of (28), (5) takes the form

$$Ric(U_1,U_2)=ag(U_1,U_2)+\left[b+c{\left(\frac{(b\lambda +d\mu )}{(c\mu +d\lambda )}\right)}^2-2d\frac{(b\lambda +d\mu )}{(c\mu +d\lambda )}\right]A\left(U_1\right)A\left(U_2\right)$$

(29)

which is a quasi-Einstein manifold. Thus, we can state the following theorem:

Theorem 2.

Let $MG{\left(QE\right)}_n$ be a mixed generalized quasi-Einstein manifold. If the associated vector fields of $MG{\left(QE\right)}_n$ are concurrent and the associated scalars are constants, then the manifold reduces to a quasi-Einstein manifold.

4. $\mathit{MG}{\left(\mathit{QE}\right)}_{\mathit{n}}$ Admitting Einstein’s Field Equations

The Einstein’s field equations with and without cosmological constants are given by

$$Ric(U_1,U_2)-\frac{r}{2}g(U_1,U_2)+\lambda g(U_1,U_2)=\kappa T(U_1,U_2),$$

(30)

and

$$Ric(U_1,U_2)-\frac{r}{2}g(U_1,U_2)=\kappa T(U_1,U_2),$$

(31)

respectively; $\kappa$ is a gravitational constant, $\lambda$ is a cosmological constant, and T is the energy–momentum tensor.

Using (6) in (31), it follows that

$$\begin{array}{cc}\hfill & \left(a-\frac{r}{2}\right)g(U_1,U_2)+bA\left(U_1\right)A\left(U_2\right)+cB\left(U_1\right)B\left(U_2\right)\hfill \\ \hfill & +d[A\left(U_1\right)B\left(U_2\right)+A\left(U_2\right)B\left(U_1\right)]=\kappa T(U_1,U_2).\hfill \end{array}$$

(32)

Now, taking the covariant derivative of (32) with respect to $U_3$, we arrive at

$$\begin{array}{cc}\hfill & b[\left(D_{U_3}A\right)\left(U_1\right)A\left(U_2\right)+A\left(U_1\right)\left(D_{U_3}A\right)\left(U_2\right)]\hfill \\ \hfill & +c[\left(D_{U_3}B\right)\left(U_1\right)B\left(U_2\right)+B\left(U_1\right)\left(D_{U_3}B\right)\left(U_2\right)]\hfill \\ \hfill & +d[\left(D_{U_3}A\right)\left(U_1\right)B\left(U_2\right)+A\left(U_1\right)\left(D_{U_3}B\right)\left(U_2\right)\hfill \\ \hfill & +\left(D_{U_3}B\right)\left(U_1\right)A\left(U_2\right)+B\left(U_1\right)\left(D_{U_3}A\right)\left(U_2\right)]=\kappa \left(D_{U_3}T\right)(U_1,U_2).\hfill \end{array}$$

(33)

Thus, we have a result.

Theorem 3.

Let $MG{\left(QE\right)}_n$ admit Einstein’s field equation without a cosmological constant. If the associated 1-forms A and B are covariantly constant, then the energy–momentum tensor is also covariantly constant.

5. $\mathit{MG}{\left(\mathit{QE}\right)}_{\mathbf{4}}$ Spacetime Admitting Space-Matter Tensor

In 1969, Petrov [19] introduced and studied the space–matter tensor $\overline{P}$ of type $(0,4)$ and defined by

$$\overline{P}=\overline{K}+\frac{\kappa }{2}g\wedge T-\nu G,$$

(34)

where $\overline{K}$ is the curvature tensor of type $(0,4)$, T is the energy–momentum tensor of type $(0,2)$, $\kappa$ is the gravitational constant, and $\nu$ is the energy density. Furthermore, G and $g\wedge T$ are, respectively, defined by

$$G(U_1,U_2,U_3,U_4)=g(U_2,U_3)g(U_1,U_4)-g(U_1,U_3)g(U_2,U_4),$$

(35)

and

$$\begin{array}{cc}\hfill (g\wedge T)(U_1,U_2,U_3,U_4)& =g(U_2,U_3)T(U_1,U_4)+g(U_1,U_4)T(U_2,U_3)\hfill \\ \hfill & -g(U_1,U_3)T(U_2,U_4)-g(U_2,U_4)T(U_1,U_3),\hfill \end{array}$$

(36)

for all $U_1$, $U_2$, $U_3$, $U_4$ on M.

Using (35) and (36) in (34), it follows that

$$\begin{array}{cc}\hfill \overline{P}(U_1,U_2,U_3,U_4)& =\overline{K}(U_1,U_2,U_3,U_4)+\frac{\kappa }{2}[g(U_2,U_3)T(U_1,U_4)\hfill \\ \hfill & +g(U_1,U_4)T(U_2,U_3)-g(U_1,U_3)T(U_2,U_4)\hfill \\ \hfill & -g(U_2,U_4)T(U_1,U_3)]-\nu [g(U_2,U_3)g(U_1,U_4)\hfill \\ \hfill & -g(U_1,U_3)g(U_2,U_4)].\hfill \end{array}$$

(37)

If $\overline{P}=0$, then (37) gives

$$\begin{array}{cc}\hfill \overline{K}(U_1,U_2,U_3,U_4)& =-\frac{\kappa }{2}[g(U_2,U_3)T(U_1,U_4)+g(U_1,U_4)T(U_2,U_3)\hfill \\ \hfill & -g(U_1,U_3)T(U_2,U_4)-g(U_2,U_4)T(U_1,U_3)]\hfill \\ \hfill & +\nu [g(U_2,U_3)g(U_1,U_4)-g(U_1,U_3)g(U_2,U_4)].\hfill \end{array}$$

(38)

In view of (5), from (31), it follows that

$$\begin{array}{cc}\hfill \kappa T(U_1,U_2)& =\left(a-\frac{r}{2}\right)g(U_1,U_2)+bA\left(U_1\right)A\left(U_2\right)+cB\left(U_1\right)B\left(U_2\right)\hfill \\ \hfill & +d[A\left(U_1\right)B\left(U_2\right)+A\left(U_2\right)B\left(U_1\right)].\hfill \end{array}$$

(39)

Thus, from (38) and (39), we obtain

$$\begin{array}{cc}\hfill \overline{K}(U_1,U_2,U_3,U_4)& =f_1[g(U_2,U_3)g(U_1,U_4)-g(U_1,U_3)g(U_2,U_4)]\hfill \\ \hfill & +f_2[g(U_1,U_4)A\left(U_2\right)A\left(U_3\right)-g(U_2,U_4)A\left(U_1\right)A\left(U_3\right)\hfill \\ \hfill & +g(U_2,U_3)A\left(U_1\right)A\left(U_4\right)-g(U_1,U_3)A\left(U_2\right)A\left(U_4\right)]\hfill \\ \hfill & +f_3[g(U_1,U_4)B\left(U_2\right)B\left(U_3\right)-g(U_2,U_4)B\left(U_1\right)B\left(U_3\right)\hfill \\ \hfill & +g(U_2,U_3)B\left(U_1\right)B\left(U_4\right)-g(U_1,U_3)B\left(U_2\right)B\left(U_4\right)]\hfill \\ \hfill & +f_4[g(U_1,U_4)\{A\left(U_2\right)B\left(U_3\right)+B\left(U_2\right)A\left(U_3\right)\}\hfill \\ \hfill & -g(U_2,U_4)\{A\left(U_1\right)B\left(U_3\right)+B\left(U_1\right)A\left(U_3\right)\}\hfill \\ \hfill & +g(U_2,U_3)\{A\left(U_1\right)B\left(U_4\right)+B\left(U_1\right)A\left(U_4\right)\}\hfill \\ \hfill & -g(U_1,U_3)\{A\left(U_2\right)B\left(U_4\right)+B\left(U_2\right)A\left(U_4\right)\}],\hfill \end{array}$$

(40)

where $f_1=(\nu -a+\frac{r}{2})$, $f_2=-\frac{b}{2}$, $f_3=-\frac{c}{2}$, $f_4=-\frac{d}{2}$. Thus, we can state the following theorem:

Theorem 4.

For a vanishing space–matter tensor, $MG{\left(QE\right)}_4$ spacetime satisfying Einstein’s field equation without a cosmological constant is a $MG{\left(QC\right)}_4$ spacetime.

Next, we investigate the existence of a sufficient condition under which $MG{\left(QE\right)}_4$ can be a divergence-free space–matter tensor.

From (31) and (37), we obtain

$$\begin{array}{cc}\hfill \left(div\overline{P}\right)(U_1,U_2,U_3)& =\left(divK\right)(U_1,U_2,U_3)+\frac{1}{2}[\left(D_{U_1}Ric\right)(U_2,U_3)\hfill \\ \hfill & -\left(D_{U_2}Ric\right)(U_1,U_3)]-g(U_2,U_3)[\frac{1}{4}\partial r\left(U_1\right)+\partial \nu \left(U_1\right)]\hfill \\ \hfill & +g(U_1,U_3)[\frac{1}{4}\partial r\left(U_2\right)+\partial \nu \left(U_2\right)].\hfill \end{array}$$

(41)

By using $\left(divK\right)(U_1,U_2,U_3)=\left(D_{U_1}Ric\right)(U_2,U_3)-\left(D_{U_2}Ric\right)(U_1,U_3)$ in (41), we obtain

$$\begin{array}{cc}\hfill \left(div\overline{P}\right)(U_1,U_2,U_3)& =\frac{3}{2}[\left(D_{U_1}Ric\right)(U_2,U_3)-\left(D_{U_2}Ric\right)(U_1,U_3)]\hfill \\ \hfill & -g(U_2,U_3)[\frac{1}{4}\partial r\left(U_1\right)+\partial \nu \left(U_1\right)]\hfill \\ \hfill & +g(U_1,U_3)[\frac{1}{4}\partial r\left(U_2\right)+\partial \nu \left(U_2\right)].\hfill \end{array}$$

(42)

Let $\left(div\overline{P}\right)(U_1,U_2,U_3)=0$; then, contracting (42) over $U_2$ and $U_3$, we obtain $\partial \nu \left(U_1\right)=0$, where (27) is used. Hence, we can state the following theorem:

Theorem 5.

For a divergence-free space–matter tensor, the energy density in $MG{\left(QE\right)}_4$ spacetime satisfying Einstein’s field equation without a cosmological constant is constant.

Now, by using (5) in (42), we obtain

$$\begin{array}{cc}\hfill \left(div\overline{P}\right)(U_1,U_2,U_3)& =\frac{3}{2}[\partial a\left(U_1\right)g(U_2,U_3)-\partial a\left(U_2\right)g(U_1,U_3)]\hfill \\ \hfill & +\frac{3}{2}[\partial b\left(U_1\right)A\left(U_2\right)A\left(U_3\right)-\partial b\left(U_2\right)A\left(U_1\right)A\left(U_3\right)]\hfill \\ \hfill & +\frac{3b}{2}[\left(D_{U_1}A\right)\left(U_2\right)A\left(U_3\right)+A\left(U_2\right)\left(D_{U_1}A\right)\left(U_3\right)\hfill \\ \hfill & -\left(D_{U_2}A\right)\left(U_1\right)A\left(U_3\right)-\left(D_{U_2}A\right)\left(U_3\right)A\left(U_1\right)]\hfill \\ \hfill & +\frac{3}{2}[\partial c\left(U_1\right)B\left(U_2\right)B\left(U_3\right)-\partial c\left(U_2\right)B\left(U_1\right)B\left(U_3\right)]\hfill \\ \hfill & +\frac{3c}{2}[\left(D_{U_1}B\right)\left(U_2\right)B\left(U_3\right)+B\left(U_2\right)\left(D_{U_1}B\right)\left(U_3\right)\hfill \\ \hfill & -\left(D_{U_2}B\right)\left(U_1\right)B\left(U_3\right)-\left(D_{U_2}B\right)\left(U_3\right)B\left(U_1\right)]\hfill \\ \hfill & +\frac{3}{2}[\partial d\left(U_1\right)\{A\left(U_2\right)B\left(U_3\right)+B\left(U_2\right)A\left(U_3\right)\}\hfill \\ \hfill & -\partial d\left(U_2\right)\{A\left(U_1\right)B\left(U_3\right)+B\left(U_1\right)A\left(U_3\right)\}]\hfill \\ \hfill & +\frac{3d}{2}[\left(D_{U_1}A\right)\left(U_2\right)B\left(U_3\right)+A\left(U_2\right)\left(D_{U_1}B\right)\left(U_3\right)\hfill \\ \hfill & +\left(D_{U_1}A\right)\left(U_3\right)B\left(U_2\right)+A\left(U_3\right)\left(D_{U_1}B\right)\left(U_2\right)\hfill \\ \hfill & -\left(D_{U_2}A\right)\left(U_1\right)B\left(U_3\right)-A\left(U_1\right)\left(D_{U_2}B\right)\left(U_3\right)\hfill \\ \hfill & -\left(D_{U_2}A\right)\left(U_3\right)B\left(U_1\right)-A\left(U_3\right)\left(D_{U_2}B\right)\left(U_1\right)]\hfill \\ \hfill & -g(U_2,U_3)[\frac{1}{4}\partial r\left(U_1\right)+\partial \nu \left(U_1\right)]\hfill \\ \hfill & +g(U_1,U_3)[\frac{1}{4}\partial r\left(U_2\right)+\partial \nu \left(U_2\right)].\hfill \end{array}$$

(43)

By assuming that $\nu$, a, b, c, and d are constants and the generator $\rho$ is a parallel vector field, i.e., $D_{U_1}\rho =0$, we obtain

$$\begin{array}{cc}\hfill & \partial r\left(U_1\right)=0,\partial \nu \left(U_1\right)=0,\left(D_{U_1}A\right)\left(U_2\right)=0.\hfill \end{array}$$

(44)

In view of (44), we derive

$$a+b=0,c=0,d=0.$$

(45)

Using (44) and (45), (43) reduces to

$$\left(div\overline{P}\right)(U_1,U_2,U_3)=0.$$

Thus, we can state the following theorem:

Theorem 6.

In $MG{\left(QE\right)}_4$ spacetimes admitting parallel vector field ρ satisfying Einstein’s field equation without a cosmological constant, if the energy density and associated scalars constant are constants, then the divergence of the space–matter tensor vanishes.

6. $\mathit{MG}{\left(\mathit{QE}\right)}_{\mathbf{4}}$ Spacetime Admitting General Relativistic Viscous Fluid

Ellis [20] defined the energy–momentum tensor for a perfect fluid distribution with heat conduction as

$$\begin{array}{cc}\hfill T(U_1,U_2)& =\omega g(U_1,U_2)+(\nu +\omega )A\left(U_1\right)A\left(U_2\right)+B\left(U_1\right)B\left(U_2\right)\hfill \\ \hfill & +A\left(U_1\right)B\left(U_2\right)+A\left(U_2\right)B\left(U_1\right),\hfill \end{array}$$

(46)

where $g(U_1,\rho )=A\left(U_1\right)$, $g(U_1,\sigma )=B\left(U_1\right)$, $A\left(\rho \right)=-1$, $B\left(\sigma \right)>0$, $g(\rho ,\sigma )=0$, and $\nu$, $\omega$ are called the isotropic pressure and the energy density, respectively. $\sigma$ is the heat conduction vector field perpendicular to the velocity vector field $\rho$. Assuming a mixed generalized quasi-Einstein spacetime satisfying Einstein’s field equation without a cosmological constant whose matter content is viscous fluid, then, from (31) and (46), the Ricci tensor takes the form

$$\begin{array}{cc}\hfill Ric(U_1,U_2)& =(\kappa \omega +\frac{r}{2})g(U_1,U_2)+\kappa (\nu +\omega )A\left(U_1\right)A\left(U_2\right)\hfill \\ \hfill & +\kappa B\left(U_1\right)B\left(U_2\right)+\kappa [A\left(U_1\right)B\left(U_2\right)+A\left(U_2\right)B\left(U_1\right)].\hfill \end{array}$$

(47)

By comparing (5) and (47), we obtain

$$a=\kappa \omega +\frac{r}{2},b=\kappa (\nu +\omega ),c=\kappa ,d=\kappa .$$

(48)

Taking a frame field to contract (48) over $U_1$ and $U_2$, we obtai

$$r=\kappa (\nu -3\omega ).$$

(49)

In view of (49), (47) turns to

$$\begin{array}{cc}\hfill Ric(U_1,U_2)& =\frac{\kappa (\nu -\omega )}{2}g(U_1,U_2)+\kappa (\nu +\omega )A\left(U_1\right)A\left(U_2\right)\hfill \\ \hfill & +\kappa B\left(U_1\right)B\left(U_2\right)+\kappa [A\left(U_1\right)B\left(U_2\right)+A\left(U_2\right)B\left(U_1\right)].\hfill \end{array}$$

(50)

Now, let R be the Ricci operator given by $g(R\left(U_1\right),U_2)=Ric(U_1,U_2)$ and $Ric(R\left(U_1\right),U_2)=Ri{c}^2(U_1,U_2)$. Then, we have $A\left(R\left(U_1\right)\right)=g(R\left(U_1\right),\rho )=Ric(U_1,\rho )$ and $B\left(R\left(U_1\right)\right)=g(R\left(U_1\right),\sigma )=Ric(U_1,\sigma )$. Thus, we obtain

$$\begin{array}{cc}\hfill Ric(R\left(U_1\right),U_2)& =\frac{\kappa (\nu -\omega )}{2}Ric(U_1,U_2)+\kappa (\nu +\omega )Ric(U_1,\rho )A\left(U_2\right)\hfill \\ \hfill & +\kappa Ric(U_1,\sigma )B\left(U_2\right)+\kappa [Ric(U_1,\rho )B\left(U_2\right)\hfill \\ \hfill & +A\left(U_2\right)Ric(U_1,\sigma )].\hfill \end{array}$$

(51)

Now, contracting (51) over $U_1$ and $U_2$, we obtain

$$\begin{array}{cc}\hfill Ric(U_1,U_1)={\left|\right|R\left|\right|}^2& =\frac{\kappa (\nu -\omega )r}{2}+\kappa (\nu +\omega )Ric(\rho ,\rho )\hfill \\ \hfill & +\kappa Ric(\sigma ,\sigma )+\kappa [Ric(\rho ,\sigma )+Ric(\sigma ,\rho )].\hfill \end{array}$$

(52)

For a mixed generalized quasi-Einstein spacetime, from (5), it follows that

$$Ric(U_1,\rho )=(a-b)A\left(U_1\right)-dB\left(U_1\right),Ric(U_1,\sigma )=(a+c)B\left(U_1\right)+dA\left(U_1\right).$$

(53)

In view of (48), (49), and (53), we find that

$$Ric(\rho ,\rho )=\frac{\kappa (\nu +3\omega )}{2},Ric(\sigma ,\rho )=Ric(\rho ,\sigma )=-\kappa ,Ric(\sigma ,\sigma )=\frac{\kappa (\nu -\omega +2)}{2}.$$

(54)

By making use of (54), from (52), it follows that

$${\left|\right|R\left|\right|}^2={\kappa }^2({\nu }^3{\omega }^2+\nu +\omega -3).$$

(55)

Thus, we can state the following theorem:

Theorem 7.

If $MG{\left(QE\right)}_4$ spacetime admitting viscous fluid satisfies Einstein’s field equation without a cosmological constant, then the square of the length of Ricci operator is ${\kappa }^2({\nu }^3{\omega }^2+\nu +\omega -3)$.

7. Example of $\mathit{MG}{\left(\mathit{QE}\right)}_{\mathbf{4}}$ Spacetime

In this section, we constructed a non-trivial concrete example to prove the existence of a $MG{\left(QE\right)}_4$ spacetime.

We assume a Lorentzian manifold $(M^4,g)$ endowed with the Lorentzian metric g given by

$$d{s}^2=g_{ij}d{u}^{i}d{u}^j=(1+2p)[{\left(d{u}^1\right)}^2+{\left(d{u}^2\right)}^2+{\left(d{u}^3\right)}^2-{\left(d{u}^4\right)}^2],$$

(56)

where $u^1,u^2,u^3,u^4$ are standard coordinates of $M^4$, i, j = $1,2,3,4$, and $p=e^{u^1}{k}^{-2}$, and k is a non-zero constant. Here, the signature of g is $(+,+,+,-)$, which is Lorentzian. Then, the only non-vanishing components of the Christoffel symbols and the curvature tensors are

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}1\\ 11\end{array}\right\}=\left\{\begin{array}{c}1\\ 44\end{array}\right\}=\left\{\begin{array}{c}2\\ 12\end{array}\right\}=\left\{\begin{array}{c}3\\ 13\end{array}\right\}=\left\{\begin{array}{c}4\\ 14\end{array}\right\}={\displaystyle \frac{p}{1+2p}},\left\{\begin{array}{c}1\\ 22\end{array}\right\}=\left\{\begin{array}{c}1\\ 33\end{array}\right\}={\displaystyle \frac{-p}{1+2p}}.\hfill \end{array}$$

(57)

$${\overline{K}}_{1212}={\overline{K}}_{1313}={\displaystyle \frac{-p}{1+2p}},K_{1414}={\displaystyle \frac{p}{1+2p}},$$

$${\overline{K}}_{3232}={\displaystyle \frac{-p^2}{1+2p}},{\overline{K}}_{4242}={\overline{K}}_{4343}={\displaystyle \frac{p^2}{1+2p}}$$

and the components are obtained by the symmetry properties.

The non-vanishing components of the Ricci tensors are

$$R_{11}={\displaystyle \frac{3p}{{(1+2p)}^2}},R_{22}=R_{33}={\displaystyle \frac{p}{{(1+2p)}^2}},R_{44}={\displaystyle \frac{-p}{{(1+2p)}^2}},$$

Thus, the scalar curvature r is $\frac{6q(1+q)}{{(1+2q)}^3}$.

Let us consider the associated scalars $a,b,c$, and d defined by

$$a={\displaystyle \frac{p}{{(1+2p)}^3}},b={\displaystyle \frac{1}{(1+2p)}},c={\displaystyle \frac{-1}{{(1+2p)}^3}},d={\displaystyle \frac{-p}{{(1+2p)}^2}}$$

and the 1-forms are defined by

$$A_1=B_1=\sqrt{1+2p},A_i=B_i=0\forall i=2,3,4,$$

where the generators are unit vector fields; then, from (5), we have

$$R_{11}=a{g}_{11}+b{A}_1{A}_1+c{B}_1{B}_1+d(A_1{B}_1+A_1{B}_1),$$

(58)

$$R_{22}=a{g}_{22}+b{A}_2{A}_2+c{B}_2{B}_2+d(A_2{B}_2+A_2{B}_2),$$

(59)

$$R_{33}=a{g}_{33}+b{A}_3{A}_3+c{B}_3{B}_3+d(A_3{B}_3+A_3{B}_3),$$

(60)

$$R_{44}=a{g}_{44}+b{A}_4{A}_4+c{B}_4{B}_4+d(A_4{B}_4+A_4{B}_4).$$

(61)

$$\begin{array}{ccc}\hfill Now,R.H.S.of\left(58\right)& =& a{g}_{11}+b{A}_1{A}_1+c{B}_1{B}_1+d(A_1{B}_1+A_1{B}_1)\hfill \\ \hfill & & ={\displaystyle \frac{3p}{{(1+2p)}^2}}\hfill \\ \hfill & & =R_{11}\hfill \\ \hfill & & =L.H.S.of\left(58\right).\hfill \end{array}$$

Similarly, it can easily be show that (59), (60), and (61) are also true. Hence, ($\mathrm{I}{\mathrm{R}}^4,g$) is a $MG{\left(QE\right)}_4$.