A study of mixed generalized quasi-Einstein spacetimes with applications in general relativity

1.   Introduction

A Riemannian (or semi-Riemannian) manifold $(M^n, g)$, $(n \geq 3)$ is named an Einstein manifold if the Ricci tensor $Ric (\neq0)$ of type $(0, 2)$ satisfies: $Ric = \frac{r}{n}g,$ where $r$ represents the scalar curvature of $(M^n, g)$. Einstein manifolds form a natural subclass of several classes of $(M^n, g)$ determined by a curvature restriction imposed on their Ricci tensor ^[1]. Also, Einstein manifolds play a key role in Riemannian geometry, the general theory of relativity as well as in mathematical physics.

Approximately two decades ago, the idea of quasi-Einstein manifolds was proposed and studied by Chaki and Maity ^[2]. An $(M^n, g)$, $(n > 2)$ is said to be a quasi-Einstein manifold $(QE)_{n}$ if its $Ric(\neq0)$ satisfies

where $a, b (\neq 0) \in \mathbb R$ and $U$ is a non-zero 1-form such that

for all vector fields $\zeta_1$ and a unit vector field $\varrho$ called the generator of $(QE)_{n}$. Also, the 1-form $U$ is named the associated 1-form. From (1.1) it is clear that for $b = 0$, $(QE)_{n}$ reduces to an Einstein manifold. The notion of $(QE)_{n}$ came into existence during the study of exact solutions of Einstein's field equations as well as during considerations of quasi-umbilical hypersurfaces of semi-Euclidean spaces. For example, the Robertson-Walker spacetimes are $(QE)_{4}$. Also, $(QE)_{4}$ can be taken as a model of the perfect fluid spacetime in general relativity ^[3,4,5].

An $(M^n, g), (n \geq 3)$ is said to be a generalized quasi-Einstein manifold $G(QE)_n$ ^[6,7,8] if its $Ric (\neq 0)$ satisfies

where $a, b (\neq 0), c(\neq 0) \in \mathbb R$ and $U (\neq 0)$, $V(\neq 0)$ are 1-forms such that

where $\sigma$ and $\varrho$ are mutually orthogonal unit vector fields, i.e., $g(\varrho, \sigma) = 0$ and are known as generators of $G(QE)_n$.

In 2007, Bhattacharya et al. ^[9] introduced the notion of mixed generalized quasi-Einstein manifolds. An $(M^n, g), (n \geq 3)$ is said to be a mixed generalized quasi-Einstein manifold $MG(QE)_{n}$ if its $Ric (\neq 0)$ satisfies

where $a, b (\neq 0), c(\neq 0), d(\neq 0) \in \mathbb R$ and $U (\neq 0)$, $V(\neq 0)$ are 1-forms defined in (1.4).

$MG(QE)_{n}$ has wide applications in cosmology and the general theory of relativity and is studied by several authors, such as ^{[10,11,12,13]} and many others.

Putting $\zeta_1 = \zeta_2 = e_{j}$ in (1.5), where $\{e_j\}$ is an orthonormal basis of the tangent space at each point of $MG(QE)_{n}$ and taking summation over $i$ $(1 \leq j \leq n)$, we get

where $r$ is the scalar curvature of $MG(QE)_{n}$.

Let $\mathcal K$ be the Riemannian curvature tensor of an $(M^n, g)$. The $k$-nullity distribution $\mathcal N(k)$ of an $(M^n, g)$ is defined by ^[14,15]

for all $\zeta_1, \zeta_2 \in \mathcal T_{p}M^n$, where $k$ is some smooth function.

In a similar manner, the $k$-nullity distribution $\mathcal N(k)$ of a Lorentzian manifold can be defined. In a $(QE)_n$, if the generator $\varrho$ belongs to some $k$-nullity distribution $\mathcal N(k)$, then $(M^n, g)$ is called an $\mathcal N(k)$-$(QE)_n$ ^[16]. In 2007, Özgür and Triphati ^[17] proved that in an $\mathcal N(k)$-$(QE)_n$, $k$ is not arbitrary, that is equal to $\frac{a+b}{n-1}$.

A spacetime is a time oriented $(M^4, g)$ with Lorentz metric of signature $(+, +, +, -)$. A $4$-dimensional Lorentzian manifold is said to be $MG(QE)_4$ with the generator $\varrho$ if its $Ric(\neq 0)$ satisfies (1.5). Here $U(\neq 0)$ and $V(\neq 0)$ being1-forms such that $\sigma$ is the heat flux vector field perpendicular to the velocity vector field $\varrho$. Therefore, for any $\zeta_1$, we have

From (1.5) and (1.8) we have

In ^[18], a generalized $(0, 2)$ type symmetric $Z$ tensor was introduced by Mantica and Molinari and defined as follows

where $\phi$ is an arbitrary scalar function. The properties of the $Z$ tensor in several ways to a different extent have been studied in ^[19,20]. If the $Z$ tensor at each point of the spacetime vanishes, then the spacetime is said to be $Z$ flat.

Einstein's field equation (without cosmological constant) is given by

where $T$ and $\varLambda$ represent the energy-momentum tensor and the Einstein gravitational constant, respectively.

The idea of perfect fluid spacetime came into existence while discussing the structure of the universe. In general relativity the matter content of the spacetime is described by $T$. The matter content is supposed to be a fluid having pressure and density and possessing kinematical and dynamical quantities like acceleration, velocity, vorticity, shear and expansion. In a perfect fluid spacetime, the energy-momentum tensor $T$ is given through the relation

where $\psi$ and $\mu$ stand for the energy density and isotropic pressure, respectively. $\varrho$ is the unit timelike velocity vector field such that $g(\zeta_1, \varrho) = U(\zeta_1)$ for all $\zeta_1$. In case of fluid matter distribution, the energy momentum tensor is given by Ellis ^[21] as

where $g(\zeta_1, \varrho) = U(\zeta_1)$, $g(\zeta_1, \sigma) = V(\zeta_1)$, $A = U(\varrho) = -1$, $V(\sigma) > 0$, $g(\varrho, \sigma) = 0$. $\sigma$ is the heat conduction vector field perpendicular to the velocity vector field $\varrho$.

Definition 1.1. An $(M^n, g)$ is called Ricci-pseudosymmetric ^[22], if the tensors $\mathcal K\cdot Ric$ and $\mathcal Q(g, Ric)$ are linearly dependent, where

and

for all $\zeta_1$, $\zeta_2$, $\zeta_3$, $\zeta_4$ on $M^n$ and $\mathcal K$ denotes the curvature tensor of $\mathcal M^n$. Then $(\mathcal M^n, g)$ is Ricci-pseudosymmetric if and only if

holds on $G_{s}$, where $G_{s} = \{\zeta_1 \in M^n : Ric \neq \frac{r}{n}g \; at\; \zeta_1 \}$, where $L_{s}$ is a certain function on $G_{s}$.

The concircular curvature tensor $N$ of type $(1, 3)$ on an $(M^n, g)$ $(n\geq 3)$ is defined by ^[23]

where $r$ is the scalar curvature of the manifold.

In view of (1.18), it follows that

where $\overline{N}(\zeta_1, \zeta_2, \zeta_3, \zeta_4) = g(N(\zeta_1, \zeta_2)\zeta_3, \zeta_4)$, $\overline {\mathcal K}(\zeta_1, \zeta_2, \zeta_3, \zeta_4) = g({\mathcal K}(\zeta_1, \zeta_2)\zeta_3, \zeta_4)$ and $\zeta_1$, $\zeta_2$, $\zeta_3$, $\zeta_4$ $\in \chi(M^n)$. Here, $\overline{K}$ is the curvature tensor of type $(0, 4)$ and $\overline{N}(\zeta_1, \zeta_2, \zeta_3, \zeta_4)$ is the concircular curvature tensor of type (0, 4) which satisfies the following properties: $\overline{N}(\zeta_1, \zeta_2, \zeta_3, \zeta_4) = -\overline{N}(\zeta_2, \zeta_1, \zeta_3, \zeta_4)$ and $\overline{N}(\zeta_1, \zeta_2, \zeta_3, \zeta_4) = -\overline{N}(\zeta_1, \zeta_2, \zeta_4, \zeta_3)$.

From (1.19), we have

2.   Ricci pseudo-symmetric $MG(QE)_4$ spacetimes

In this section, we consider Ricci-pseudosymmetric $MG(QE)_4$ spacetime. Therefore, from (1.5) and (1.14)–(1.17) we have

Now, contracting the foregoing equation over $\zeta_1$ and $\zeta_4$, we obtain

Putting $\zeta_3 = \varrho$ in (2.2), we obtain

where $\overline{K}$ is the curvature tensor of type $(0, 4)$.

By putting $\zeta_3 = \varrho$ and $\zeta_4 = \sigma$ in (2.1) and using (1.8), we obtain

In view of (1.8), (2.3) and (2.4), we arrive at

Putting $\zeta_2 = \varrho$ in (2.5), we get

Again, putting $\zeta_2 = \sigma$ in (2.5), we get

If $d = 0$, then from (2.6) we find $L_{s} = \frac{a-b}{3}$, as $b \neq 0$. If $d \neq 0$, then $L_{s} = \frac{a-b+c}{3}$. Comparing this with (2.6), it follows that $c = d = 0$ and thus $MG(QE)_4$ spacetime reduces to a quasi Einstein spacetime. Therefore, from (2.4) we have

which means that the generator $\varrho$ belongs to the $\frac{(a-b)}{3}$-nullity distribution. Thus, the manifold turns into $\mathcal N\Big(\frac{a-b}{3}\Big)$ quasi-Einstein spacetime. Therefore, we can state the following result:

Theorem 2.1. Every Ricci-pseudosymmetric $MG(QE)_4$ spacetime is a $\mathcal N\Big(\frac{a-b}{3}\Big)$ quasi-Einstein spacetime, for some certain function $L_{s} = \frac{a-b}{3}$, where $b+c \neq 0$.

3.   $Z$-Ricci pseudo-symmetric $MG(QE)_{4}$ spacetime

An $(M^n, g)$, $(n \geq 3)$ is called Z-Ricci pseudo-symmetric if and only if the following relation

holds on the set $G_{Q} = \{\zeta_1 \in M^n : P(g, \mathcal Q)\neq 0 \quad at \quad \zeta_1 \}$, where $\mathcal Q$ is the Ricci operator defined by $Ric(\zeta_1, \zeta_2) = g(\mathcal Q\zeta_1, \zeta_2)$ and $f_{\mathcal Q}$ is a smooth function on $G_{\mathcal Q}$. The tensor $P(g, \mathcal Q)$ is defined by

for all vector fields $\zeta_1$, $\zeta_2$, $\zeta_4$.

Now, if $MG(QE)_4$ is a Z-Ricci pseudosymmetric, then from (3.1) we get

If $f_{\mathcal Q} = 0$, then $(M^n, g)$ reduces to a Z-Ricci semisymmetric manifold. Now considering

Also,

By virtue of (3.4) and (3.5), (3.3) turns to

From (1.5), it follows that

By virtue of (3.7), (3.6) becomes

In view of (1.5) and (1.10), (3.8) takes the form

which by putting $\zeta_1 = \varrho$, $\zeta_2 = \sigma$ yields

Taking the inner product of the foregoing equation with $\varrho$, we lead to

Now by putting $\zeta_4 = \varrho$ in (3.9), we obtain $d(f_{\mathcal Q}+b+c) = 0.$ Thus, we have either $d = 0$ or $f_{\mathcal Q} = -(b+c).$ For the first case $d = 0$, $MG(QE)_4$ spacetime reduces to a $G(QE)_4$ spacetime. Hence, we can state the following theorem:

Theorem 3.1. $A$ Z-Ricci pseudo-symmetric $MG(QE)_{4}$ spacetime is a $G(QE)_{4}$ spacetime.

4.   Concircularly pseudo-symmetric $MG(QE)_{4}$ spacetime

An $(M^n, g)$, $(n \geq 3)$ is said to be concircularly pseudo-symmetric if and only if the following relation

holds on the set $G_{s}$, where $G_{s} = \{\zeta_1 \in M^n : Ric \neq \frac{r}{n}g \; at\; \zeta_1 \}$ and $L_{s}$ is a certain function on $G_{s}$. In view of (1.14)–(1.16), (4.1) turns to

By using (1.5) in (4.2) it follows that

Now, contracting the foregoing equation over $\zeta_1$ and $\zeta_4$, we have

Putting $\zeta_3 = \varrho$ in (4.4), we get

Putting $\zeta_3 = \varrho$ and $\zeta_4 = \sigma$ in (4.3) and using (1.8), we can easily find

On simplification, we obtain

From (1.18) and (4.6), we obtain

In view of (1.9) and (4.6), from (4.5) it follows that

Putting $\zeta_2 = \varrho$ in (4.8) gives

Again, putting $\zeta_2 = \sigma$ in (4.8), we get

If $d = 0$, then from (4.9) we find $L_{s} = \frac{-3b-c}{12}$, as $b \neq 0$. If $d \neq 0$, then $L_{s} = \frac{-b+c}{4}$. Comparing this with (4.9), it follows that $c = d = 0$ and thus $MG(QE)_4$ spacetime reduces to a $(QE)_4$ spacetime. Therefore, from (4.7) we have

which means that the generator $\varrho$ belongs to the $\frac{(a-b)}{3}$-nullity distribution. Thus, the manifold turns into $\mathcal N\Big(\frac{a-b}{3}\Big)$ quasi-Einstein spacetime. Therefore, we have the following result:

Theorem 4.1. Every concircularly pseudo-symmetric $MG(QE)_4$ spacetime is a $\mathcal N\Big(\frac{a-b}{3}\Big)$ quasi-Einstein spacetime, for some certain function $L_{s} = \frac{a-b}{3}$, where $b+c \neq 0$.

5.   Heat flux in $MG(QE)_{4}$ spacetimes

Consider an $MG(QE)_4$ spacetime satisfying Einstein's field equation without cosmological constant whose matter content is viscous fluid. Then, by (1.11) and (1.13), the $Ric$ is of the form

From (1.5) and (5.1), we have

Putting $\zeta_2 = \varrho$ in (5.2), it follows that

for all $\zeta_1$. Removing $\zeta_1$ from the above equation we have

Taking the inner product in (5.3) by $\varrho$ yields

Using (5.5) in (5.3) we get $B = 0$ (which is inadmissible), provided $d\neq \varLambda$. Thus we have the following result:

Theorem 5.1. An $MG(QE)_{4}$ spacetime can not admit heat flux if the smooth function $d$ is not equal to the cosmological constant $\varLambda$.

6.   Example of $MG(QE)_{4}$ spacetime

In this section, we construct a non-trivial example to prove the existence of an $MG(QE)_{4}$ spacetime. We assume a Lorentzian manifold $(\mathcal M^4, g)$ endowed with the Lorentzian metric $g$ given by

where $i$, $j$ = $1, 2, 3, 4$ and $\omega$, $c$ are constants. Then the covariant and contravariant components of the metric are respectively given by

and

The only non-vanishing components of the Christoffel symbols are

The non-zero derivatives of (6.4) are

For the Riemannian curvature tensor,

The non-zero components of (Ⅰ) are:

and the non-zero components of (Ⅱ) are:

Adding components corresponding to (Ⅰ) and (Ⅱ), we have

Thus, the non-zero components of the curvature tensor, up to symmetry are

and the non-zero components of the Ricci tensor are

The scalar curvature $scal$ is given by

Let us consider the associated scalars $a$, $b$, $c$, $d$ defined by $a = -\dfrac{3}{r^2}, \; b = -\dfrac{4}{r^{2}}, \; c = \dfrac{2}{r^{2}}, \; d = \dfrac{3}{r^2}$, and the 1-forms are defined by

where generators are unit vector fields, then from (1.5), we have

Now,

By a similar argument it can be shown easily that (6.6), (6.7) and (6.8) are also true. Hence, (${\rm I\!R^4}$, g) is an $MG(QE)_{4}$.

7.   Conclusions

The most modern approaches to mathematical general relativity begin with the concept of a manifold. After Riemannian manifolds, the structure of Lorentzian manifold is the most significant subclass of pseudo-Riemannian manifolds. The theory of general relativity is mainly studied on a semi-Riemannian manifold which sometimes is not an Einstein spacetime. Thus, it was always necessary to expand the concept of Einstein manifolds to quasi-Einstein, generalized quasi-Einstein and mixed generalized quasi-Einstein manifolds. Mixed generalized quasi-Einstein manifolds play a key role in the general relativity and cosmology and has wide applications in general relativistic viscous fluid spacetime admitting heat flux and stress. In the present work, we investigate some geometric and physical properties of mixed generalized quasi-Einstein spacetimes in general relativity and cosmology satisfying certain conditions.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are really thankful to the learned reviewers for their careful reading of our manuscript and their many insightful comments and suggestions that have improved the quality of our manuscript. The authors would also like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by grant code 23UQU4330007DSR003.

Conflict of interest

The authors declare no conflicts of interest.