Characterization of solitons in a pseudo-quasi-conformally flat and pseudo- $W_8$ flat Lorentzian Kähler space-time manifolds

1.   Introduction

The relativistic fluid models being involved in the study of different branches of physics, like astrophysics and plasma physics. According to general relativity, a perfect fluid space-time is a four-dimensional Riemannian manifold with Lorentzian metric $g$. For the study of idealized distribution of matter, such as the interior of a star or an isotropic pressure, we use perfect fluid in cosmology. The behavior of a perfect fluid inside a spherical object is described by Einstein's field equation. A perfect fluid is said to be a radiation fluid if its mass density $\psi$ is three times that of the isotropic pressure $p$. O'Neill in 1983 studied the properties of semi-Riemannian geometry in relativity theory ^[23]. Also, Kaigorodov ^[19] studied the structure of space-time curvature. Later on, different differential geometers ^{[1,6,9,12,30]} extended their study to the curvature structure of space-time. In 2020, Panday and Chaturvedi ^[24] studied Lorentzian complex space form. In the fields of soliton theory, theory of relativity, and related topics, numerous geometers have investigated geometric and topological characteristics concerning symmetry. Their works from references ^[2,14,17,18] are a great place to start when looking for ideas and a desire to learn more about symmetry. The Ricci solitons and Einstein solitons generate self-similar solutions to partial differential equations. Ricci flow ^[16] and Einstein flow ^[5] are defined as

respectively, where $S$ is the Ricci tensor, $g$ is the Riemannian metric, and $r$ is the scalar curvature. Solitons are waves that travel across space with little energy loss and maintain their shape and speed even after colliding with other waves of the same type. The trajectory of wave transmission is characterized by nonlinear partial differential equations. The concept of $\eta$-Ricci soliton was given by Cho and Kimura ^[11]. An $\eta$-Ricci soliton equation is given by

for real constants $u$ and $v$, where $g$, $S$, and $\xi$ denote pseudo-Riemannian metric, Ricci curvature, and vector field, respectively, and $\eta$ is a 1-form. The data $(g, \xi, u, v)$ that satisfies Eq (1.2) is called $\eta$-Ricci solitons. If $v = 0$, then the data $(g, \xi, u)$ is called Ricci soliton ^[16]. It is referred to as shrinking, steady, or expanding according to whether $u$ is zero, positive, or negative, accordingly ^[10].

The Lie derivative of $g(\pi_{1}, \pi_{2})$ with respect to $\xi$ is given by

With the help of differential Eq (1.3), Eq (1.2) can be written as

for any $\pi_{1}, \pi_{2}\in \chi (M)$.

The equation of the $\eta$-Einstein soliton is introduced by Blaga ^[3] in 2018 as follows:

where $g, S, r, \xi, \eta, u$, and $v$ are stated as above. The data $(g, \xi, u, v)$ that satisfies differential Eq (1.5) is called an $\eta$-Einstein soliton in $M$; for $v = 0$, the data $(g, \xi, u)$ is called an Einstein soliton ^[5].

Using (1.3) in (1.5), we have

for any $\pi_{1}, \pi_{2}\in \chi (M)$.

Ricci soliton has been studied by Praveena and Bagewadi in ^[28,29], in which they obtained results in almost pseudo-symmetric Kähler manifolds. De et al. ^[13] in 2012 studied conformally flat almost pseudo-Ricci symmetric space-times. Ricci soliton associated with perfect fluid space-time has been discussed by Venkatesha et al. in ^[20,34]. In ^[32], Siddiqi and Siddiqi have discussed conformal Ricci soliton and geometrical structure in a perfect fluid space-time. Pratyay Debnath and Arabinada Konar studied ^[15] quasi-Einstein manifolds and quasi-Einstein space-times. In 2020, Blaga ^[4] studied solitons and geometrical structures in perfect fluid space-time. Recently, in 2023, Catuevedi et al. ^[8] studied the concept of solitons in Bochner Flat Lorentzian Kähler space-time manifold. These ideas motivated us to study the $\eta$-Ricci soliton and $\eta$-Einstein soliton in pseudo-quasi-conformally flat and pseudo-$W_8$ flat Lorentzian Kähler space-time manifolds.

2.   Basics of Lorentzian Kähler space-time manifold

A semi-Riemannian manifold (dim = $n$, even) $(M^n, g)$ endowed with a Lorentzian metric $g$ is referred to as a Lorentzian Kähler manifold if it satisfies the following conditions ^[24]:

where $F$ is a (1, 1) tensor field, $I$ is an identity matrix, $\pi_{1}$ and $\pi_{2}$ are arbitrary differentiable vector fields, and $\nabla$ is a Riemannian connection. We know that in a Lorentzian Kähler manifold, the following relation holds:

We refer to a four-dimensional Lorentzian Kähler manifold as a Lorentzian Kähler space-time manifold. This assumption is taken into consideration throughout the study.

The Einstein equation with cosmological constant for the perfect fluid space-time is as follows:

for any $\pi_{1}, \pi_{2}\in \chi (M)$, where $p$, $\psi$, $\lambda$, $K$, $S$, and $r$ are the isotropic pressure, energy density, cosmological constant, gravitational constant, Ricci tensor, and scalar curvature, respectively. $\eta$ is an associated 1-form such that $\eta(\xi) = -1$, $g$ is the metric tensor of minkowski space-time ^[22], and $\xi$ is the velocity vector of the fluid. Here, the Ricci tensor $S$ is the functional combination of $g$ and $\eta\otimes\eta$ called quasi-Einstein ^[7].

Consider an orthonormal frame field $\{E_{i}\}_{1\leq i\leq 4}$, that is, $g(E_{i}, E_{j}) = \epsilon_{ij}\delta_{ij}$, $i, j \in \{1, 2, 3, 4\}$ with $\epsilon_{11} = -1$, $\epsilon_{ii} = -1$, $i\in \{2, 3, 4\}$, $\epsilon_{ij} = 0$, $i, j \in \{1, 2, 3, 4\}$, $i \neq j$.

Let $\xi = \sum_{i = 1}^{n} \xi^{i}E_{i}$, then we can write

and

Contracting Eq (2.4), we get

The Einstein equation without cosmological constant for perfect fluid space-time is as follows:

Now, contracting Eq (2.8), we have

3.   Pseudo-quasi-conformal curvature tensor

In this section, we studied pseudo-quasi-conformal curvature tensor in a Lorentzian Kähler space-time manifold. In 2005, Shaikh and Jana ^[31] introduced and studied pseudo-quasi-conformal curvature tensor on Riemannian manifolds. Kundu and Prakash et al. ^[21,26] studied pseudo-quasi-conformal curvature tensor on $P$-sasakian manifolds. In 2021, Suh et al. ^[33], studied pseudo-quasi-conformal curvature tensor in space-times of general relativity.

The pseudo-quasi-conformal curvature tensor is defined by:

where $R$ is the curvature tensor, $S$ is the Ricci tensor, $\tilde{V}(\pi_{1}, \pi_{2}, \pi_{3}, \pi_{4}) = g(\tilde{V}(\pi_{1}, \pi_{2}), \pi_{3}, \pi_{4})$, and $R(\pi_{1}, \pi_{2}, \pi_{3}, \pi_{4}) = g(R(\pi_{1}, \pi_{2})\pi_{3}, \pi_{4})$.

If the manifold is pseudo-quasi-conformally flat, then from Eq (3.1), we can write

Now, replacing $\pi_{1}$ by $F\pi_{1}$ and $\pi_{2}$ by $F\pi_{2}$ in Eq (3.2), we get

Subtracting Eq (3.3) from Eq (3.2), we have

Now, using Eq (2.2) in Eq (3.4), we get

Taking a frame field over $\pi_{1}$ and $\pi_{4}$ in Eq (3.5) and using Eqs (2.1) and (2.3), we get

From Eqs (3.6) and (1.4), we get

Multiplying (3.7) by $\epsilon_{ii}$, taking $\pi_{2} = \pi_{3} = E_{i}$, and using Eqs (2.5) and (2.6), we get

Using Eq (2.7) in Eq (3.8), we get

Now, putting $\pi_{2} = \pi_{3} = \xi$ in (3.7), we get

Again, using Eq (2.7) in Eq (3.10), we get

From Eqs (3.9) and (3.11), we have

If $v = 0$, then we get the Ricci soliton with $u = \Big[\lambda+\frac{K}{4}(\psi-3p)\Big]\frac{\alpha_{1}}{d}$.

This will be steady if $u = 0$; therefore, from Eq (3.12), we get $p = \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, or $\frac{\alpha_{1}}{d} = 0$.

Shrinking if $u < 0$, therefore, from Eq (3.12), we get $p < \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\frac{\alpha_{1}}{d} > 0$ or $p > \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\frac{\alpha_{1}}{d} < 0$.

Expanding if $u > 0$, therefore, from Eq (3.12), we get $p > \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\frac{\alpha_{1}}{d} > 0$ or $p < \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\frac{\alpha_{1}}{d} < 0$.

Thus, we conclude:

Theorem 3.1. Ricci soliton $(g, \xi, u)$ in a pseudo-quasi-conformally flat Lorentzian Kähler space-time manifold with cosmological constant is:

(i) steady: if $p = \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$ or $\frac{\alpha_{1}}{d} = 0$,

(ii) shrinking: if $p < \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\frac{\alpha_{1}}{d} > 0$ or $p > \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\frac{\alpha_{1}}{d} < 0$,

(iii) or expanding: if $p > \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\frac{\alpha_{1}}{d} > 0$ or $p < \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\frac{\alpha_{1}}{d} < 0$.

Using Eqs (2.9) and (3.8), we get

Using Eqs (2.9) and (3.10), we get

After calculating (3.13) and (3.14), we get

and if $v = 0$, then we get the Ricci soliton with $u = \Big[\frac{K(\psi-3p)}{4}\Big]\frac{\alpha_{1}}{d}$.

This will be steady if $u = 0$, therefore, from Eq (3.15), we get $p = \frac{K\psi}{3}$ or $\frac{\alpha_{1}}{d} = 0$.

Shrinking if $u < 0$, therefore, from Eq (3.15), we get $p < \frac{K\psi}{3}$, and $\frac{\alpha_{1}}{d} > 0$ or $p > \frac{K\psi}{3}$, and $\frac{\alpha_{1}}{d} < 0$.

Expanding if $u > 0$, therefore, from Eq (3.15), we get $p < \frac{K\psi}{3}$, and $\frac{\alpha_{1}}{d} < 0$ or $p > \frac{K\psi}{3}$, and $\frac{\alpha_{1}}{d} > 0$.

Thus, we conclude:

Theorem 3.2. A Ricci soliton $(g, \xi, u)$ in a pseudo-quasi-conformally flat Lorentzian Kähler space-time manifold without cosmological constant is:

(i) steady: if $p = \frac{K\psi}{3}$ or $\frac{\alpha_{1}}{d} = 0$,

(ii) shrinking: if $p < \frac{K\psi}{3}$, and $\frac{\alpha_{1}}{d} > 0$ or $p > \frac{K\psi}{3}$, and $\frac{\alpha_{1}}{d} < 0$,

(iii) or expanding: if $p < \frac{K\psi}{3}$, and $\frac{\alpha_{1}}{d} < 0$ or $p > \frac{K\psi}{3}$, and $\frac{\alpha_{1}}{d} > 0$.

From Eqs (1.6) and (3.6), we obtain

Taking $\pi_{2} = \pi_{3} = E_{i}$, multiplying Eq (3.16) by $\epsilon_{ii}$, and using Eqs (2.5) and (2.6), we get

Using Eq (2.7) in Eq (3.17), we get

and taking $\pi_{2} = \pi_{3} = \xi$ in Eq (3.16), we have

Again, using Eq (2.7) in Eq (3.19), we get

From Eqs (3.18) and (3.20), we have

if $v = 0$, then we get the Einstein soliton with $u = \Big[\lambda+\frac{K}{4}(\psi-3p)\Big]\Big(\frac{\alpha_{1}}{d}+2\Big)$.

This will be steady if $u = 0$, therefore, from Eq (3.21), we get $p = \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$ or $\Big(\frac{\alpha_{1}}{d}+2\Big) = 0$.

Shrinking if $u < 0$, therefore, from Eq (3.21), we get $p < \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) > 0$ or $p > \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) < 0$.

Expanding if $u > 0$, therefore, from Eq (3.21), we get $p > \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) > 0$ or $p < \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) < 0$.

Thus, we conclude:

Theorem 3.3. An Einstein soliton $(g, \xi, u)$ in a pseudo-quasi-conformally flat Lorentzian Kähler space-time manifold with cosmological constant is:

(i) steady: if $p = \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$ or $\Big(\frac{\alpha_{1}}{d}+2\Big) = 0$,

(ii) shrinking: if $p < \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) > 0$ or $p > \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) < 0$,

(iii) or expanding: if $p > \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$ and $\Big(\frac{\alpha_{1}}{d}+2\Big) > 0$ or $p < \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$ and $\Big(\frac{\alpha_{1}}{d}+2\Big) < 0$.

Using Eqs (2.9) and (3.17), we get

Using Eqs (2.9) and (3.19), we get

After calculating Eqs (3.22) and (3.23), we get

if $v = 0$, then we obtain the Einstein soliton with $u = \frac{K(\psi-3p)}{4}\Big(\frac{\alpha_{1}}{d}+2\Big)$.

This will be steady if $u = 0$, therefore, from Eq (3.24), we get $p = \frac{K\psi}{3}$ or $\Big(\frac{\alpha_{1}}{d}+2\Big) = 0$.

Shrinking if $u < 0$, therefore, from Eq (3.24), we get $p < \frac{K\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) > 0$ or $p > \frac{K\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) < 0$.

Expanding if $u > 0$, therefore, from Eq (3.24), we get $p > \frac{K\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) > 0$ or $p < \frac{K\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) < 0$.

Thus, we conclude:

Theorem 3.4. An Einstein soliton $(g, \xi, u)$ in a pseudo-quasi-conformally flat Lorentzian Kähler space-time manifold without cosmological constant is:

(i) steady: if $p = \frac{K\psi}{3}$ or $\Big(\frac{\alpha_{1}}{d}+2\Big) = 0$,

(ii) shrinking: if $p < \frac{K\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) > 0$ or $p > \frac{K\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) < 0$,

(iii) or expanding: if $p > \frac{K\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) > 0$ or $p < \frac{K\psi}{3}$, and $\Big(\frac{\alpha_{1}}{d}+2\Big) < 0$.

4.   Pseudo-$W_8$ curvature tensor

In this section, we studied the pseudo-$W_8$ curvature tensor in Lorentzian Kähler space-time manifold. In 1982, Pokhariyal and Mishra ^[25] defined $W_8$ curvature tensor. Later in 2018, Pandey et al. ^[27] gave the concept of the pseudo $W_8$ curvature tensor on a Riemannian manifold. The pseudo-$W_8$ curvature tensor is defined by:

If the manifold is pseudo-$W_8$ flat, then from Eq (4.1), we can write

Now, replacing $\pi_{1}$ by $F\pi_{1}$ and $\pi_{2}$ by $F\pi_{2}$ in Eq (4.2), we get

Subtracting Eq (4.3) from Eq (4.2), we have

Using Eq (2.2) in Eq (4.4), we get

Taking a frame field over $\pi_{1}$ and $\pi_{5}$ in Eq (4.5) and using Eqs (2.1) and (2.3), we get

From Eqs (4.6) and (1.4), we get

Taking $\pi_{2} = \pi_{3} = E_{i}$, multiplying (4.7) by $\epsilon_{ii}$, and using Eqs (2.5) and (2.6), we get

Using Eq (2.7) in Eq (4.8), we get

Now, putting $\pi_{2} = \pi_{3} = \xi$ in (4.7), we have

Using Eq (2.7) in Eq (4.10), we get

From Eqs (4.9) and (4.11), we have

if $v = 0$, then we get the Ricci soliton with $u = \Big[-\lambda-\frac{K}{4}(\psi-3p)\Big]\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big)$.

This will be steady if $u = 0$, therefore, from Eq (4.12), we get $p = \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$ or $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) = 0$.

Shrinking if $u < 0$, therefore, from Eq (4.12), we get $p < \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) > 0$ or $p > \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$ or $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) < 0$.

Expanding if $u > 0$, therefore, from Eq (4.12), we get $p > \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) > 0$ or $p < \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) < 0$.

Thus, we conclude:

Theorem 4.1. A Ricci soliton $(g, \xi, u)$ in a pseudo-$W_8$ flat Lorentzian Kähler space-time manifold with cosmological constant is:

(i) steady: if $p = \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$ or $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) = 0$,

(ii) shrinking: $p < \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) > 0$ or $p > \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) < 0$,

(iii) or expanding: if $p > \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) > 0$ or $p < \frac{4}{3}\frac{\lambda}{K}+\frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) < 0$.

Using Eqs (2.9) and (4.8), we get

Using Eqs (2.9) and (4.10), we get

After calculating (4.13) and (4.14), we have

if $v = 0$, then we get the Ricci soliton with $u = -\frac{K}{4}(\psi-3p)\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big)$.

This will be steady if $u = 0$, therefore, from Eq (4.15), we get $p = \frac{\psi}{3}$ or $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) = 0$.

Shrinking if $u < 0$, therefore, from Eq (4.15), we get $p < \frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) > 0$ or $p > \frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) < 0$.

Expanding if $u > 0$, therefore, from Eq (4.15), we get $p > \frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) > 0$ or $p < \frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) < 0$.

Thus, we conclude:

Theorem 4.2. A Ricci soliton $(g, \xi, u)$ in a pseudo-$W_8$ flat Lorentzian Kähler space-time manifold with cosmological constant is:

(i) steady: if $p = \frac{\psi}{3}$ or $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) = 0$,

(ii) shrinking: $p < \frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) > 0$ or $p > \frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) < 0$,

(iii) or expanding: if $p > \frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) > 0$ or $p < \frac{\psi}{3}$, and $\Big(\frac{\alpha_{1}}{3\alpha_{2}} - 1\Big) < 0$.

From Eq (1.6) and (4.6), we get

Taking $\pi_{2} = \pi_{3} = E_{i}$, multiplying Eq (4.16) by $\epsilon_{ii}$, and using Eqs (2.5) and (2.6), we get

Using Eq (2.7) in Eq (4.17), we have

Now, putting $\pi_{2} = \pi_{3} = \xi$ in (4.16), we get

Using Eq (2.7) in Eq (4.19), we get

From Eqs (4.18) and (4.20), we have

if $v = 0$, then we get the Einstein soliton with $u = \Big[\lambda +\frac{K}{4}(\psi-3p)\Big]\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big)$.

This will be steady if $u = 0$, therefore, from Eq (4.21), we get $p = \frac{4}{3}\frac{\lambda}{K} + \frac{\psi}{3}$ or $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) = 0$.

Shrinking if $u < 0$, therefore, from Eq (4.21), we get $p < \frac{4}{3}\frac{\lambda}{K} + \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) > 0$ or $p > \frac{4}{3}\frac{\lambda}{K} + \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) < 0$.

Expanding if $u > 0$, therefore, from Eq (4.21), we get $p > \frac{4}{3}\frac{\lambda}{K} + \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) > 0$ or $p < \frac{4}{3}\frac{\lambda}{K} + \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) < 0$.

Thus, we conclude:

Theorem 4.3. An Einstein soliton $(g, \xi, u)$ in a pseudo-$W_8$ flat Lorentzian Kähler space-time manifold with cosmological constant is:

(i) steady: if $p = \frac{4}{3}\frac{\lambda}{K} + \frac{\psi}{3}$ or $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) = 0$,

(ii) shrinking: if $p < \frac{4}{3}\frac{\lambda}{K} + \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) > 0$ or $p > \frac{4}{3}\frac{\lambda}{K} + \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) < 0$,

(iii) or expanding: if $p > \frac{4}{3}\frac{\lambda}{K} + \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) > 0$ or $p < \frac{4}{3}\frac{\lambda}{K} + \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) < 0$.

Using Eqs (2.9) and (4.17), we get

Using Eqs (2.9) and (4.19), we get

After calculating Eqs (4.22) and (4.23), we get

if $v = 0$, then we get the Einstein soliton with $u = \frac{K(\psi - 3p)}{4}\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big)$.

This will be steady if $u = 0$, therefore, from Eq (4.24), we get $p = \frac{\psi}{3}$ or $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) = 0$.

Shrinking if $u < 0$, therefore, from Eq (4.24), we get $p < \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) > 0$ or $p > \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) < 0$.

Expanding if $u > 0$, therefore, from Eq (4.24), we get $p > \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) > 0$ or $p < \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) < 0$.

Thus, we conclude:

Theorem 4.4. An Einstein soliton $(g, \xi, u)$ in a pseudo-$W_8$ flat Lorentzian Kähler space-time manifold without cosmological constant is:

(i) steady: if $p = \frac{\psi}{3}$ or $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) = 0$,

(ii) shrinking: if $p < \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) > 0$ or $p > \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) < 0$,

(iii) or expanding: if $p > \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) > 0$, and or $p < \frac{\psi}{3}$, and $\Big(3 - \frac{\alpha_{1}}{3\alpha_{2}}\Big) < 0$.

Author contributions

B. B. Chaturvedi, Kunj Bihari Kaushik, Prabhawati Bhagat, Mohammad Nazrul Islam Khan: conceptualization, methodology, investigation, writing-original draft, writing-review and editing. All authors have read and agreed to the published version of the manuscript.

Use of AI tools declaration

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

Acknowledgments

The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1).

Conflict of interest

The authors declare no conflicts of interest.