1 Introduction

Let (M, g) be a smooth n-dimensional pseudo-Riemannian manifold. A Killing field is a vector field X on a Riemannian manifold (M, g) whenever \(\mathcal {L}_{X}g = 0\), \(\mathcal {L}_{X}\) is the Lie derivative in the direction of X. Killing vector fields were considered in [13]. Recently, various generalizations of Killing vector fields have been studied. A vector field X on a Riemannian manifold (M, g) is called a conformal vector field if there is a smooth function \(\psi \) on M that named a potential function, such that \(\mathcal {L}_{X}g=2\psi g\). If the potential function \(\psi =0\), X is a Killing vector field. Conformal vector fields are completely described in [20, 37]. In the theory of general relativity, Killing vector fields help to solve higher order nonlinear equations like Einstein’s equations. Also, Killing vector fields are useful in investigating gravitational waves. On the other hand, conformal vector fields have wide applications in geometric, kinematic, and dynamic levels. From the point of view of geometry, Maxwell’s law of electromagnetic theory remains stable under geometric transformations. A kinematic study including the analysis of spacetime rotation, expansion, and shear that accepts conformal vector fields [25, 34]. A vector field X on M is called a Kerr-Schild vector field if

$$\begin{aligned} \mathcal {L}_{X}g =\alpha l\otimes l,\,\,\,\,\,\,\,\,\,\mathcal {L}_{X}l=\beta l, \end{aligned}$$

where l is a null 1-form field and \(\alpha , \beta \) are smooth functions over M. Also, the generalized Kerr-Schild vector field is defined by

$$\begin{aligned} \mathcal {L}_{X}g =\alpha g+\beta l\otimes l,\,\,\,\,\,\,\,\,\,\mathcal {L}_{X}l=\gamma l, \end{aligned}$$

where \(\alpha , \beta , \gamma \) are smooth functions over M. Coll et al. [16] studied the generalized Kerr-Schild vector field. A symmetric tensor h on M is called a square root of g if \(h_{ik}h_{j}^{k}=g_{ij}\). Kerr-Schild vector fields and generalized Kerr-Schild vector fields are useful in obtaining exact solutions of the vacuum Einstein field equations and analyzed on theoretical grounds. Several examples are given in for vacuum and Einstein-Maxwell and for perfect fluids. The general vacuum to vacuum generalized Kerr-Schild vector field was also solved (see [16] and references therein).

Garcia-Parrado and Senovilla [21] using the square root of g defined bi-conformal vector fields. A vector field X is called a bi-conformal vector field if it satisfies the following equations:

$$\begin{aligned} \mathcal {L}_{X}g =\alpha g+\beta h,\,\,\,\,\,\,\,\,\,\mathcal {L}_{X}h=\alpha h+\beta g, \end{aligned}$$

where \(\alpha , \beta \) are smooth functions that are called gauges of the symmetry and h is a symmetric square root of g. The functions \(\alpha \) and \(\beta \) play a role analogous to the factor \(\psi \) appearing in the definition of the conformal vector fields [16, 21]. The bi-conformal vector fields have interesting geometric properties, such as being related to the existence of certain holomorphic structures on complex manifolds. Also, in [21] authors used the flow of bi-conformal vector fields to proved bi-conformal vector fields can be determined as generalized conformal motions of two complementary orthonormal projectors which states the geometric interpretation of these vector fields. After then, De et al. in [19] using the metric tensor field g and the Ricci tensor field S defined Ricci bi-conformal vector fields as follows:

Definition 1.1

A vector field X on a Riemannian manifold (M, g) is called Ricci bi-conformal vector field if it satisfies the following equations

$$\begin{aligned} (\mathcal {L}_{X}g)(Y,Z)=\alpha g(Y,Z)+\beta S(Y,Z), \end{aligned}$$
(1.1)

and

$$\begin{aligned} (\mathcal {L}_{X}S)(Y,Z)=\alpha S(Y,Z)+\beta g(Y,Z), \end{aligned}$$
(1.2)

for any vector fields Y, Z and some smooth functions \(\alpha \) and \(\beta \), where S is the Ricci tensor of M with respect to metric g.

Example 1.2

Let \(g=\frac{1}{1+e^{-2x}}(dx^{2}+dy^{2})\) be a metric tensor on manifold \(M=\{(x,y)\in \mathbb {R}^{2}:\,\,\, x>0\}\). The Ricci tensor of the metric g determined by \(S=\frac{2e^{-2x}}{1+e^{-2x}}g\). For arbitrary vector field \(Y=Y^{1}(x,y)\frac{\partial }{\partial x}+Y^{2}(x,y)\frac{\partial }{\partial y}\), we have

$$\begin{aligned} \mathcal {L}_{Y}g=\left( \begin{array}{cc} 2\frac{(1+e^{-2x})\partial _{x}Y^{1}+e^{-2x}Y^{1}}{(1+e^{-2x})^{2}}&{}\frac{\partial _{x}Y^{2}+\partial _{y}Y^{1}}{1+e^{-2x}}\\ \frac{\partial _{x}Y^{2}+\partial _{y}Y^{1}}{1+e^{-2x}}&{}2\frac{(1+e^{-2x})\partial _{y}Y^{2}+e^{-2x}Y^{1}}{(1+e^{-2x})^{2}} \end{array} \right) \end{aligned}$$
(1.3)

and

$$\begin{aligned} \mathcal {L}_{Y}S=\left( \begin{array}{cc} 4e^{-2x}\frac{(1+e^{-2x})\partial _{x}Y^{1}+(e^{-2x}-1)Y^{1}}{(1+e^{-2x})^{3}} &{} \frac{2e^{-2x}(\partial _{x}Y^{2}+\partial _{y}Y^{1})}{(1+e^{-2x})^{2}} \\ \frac{2e^{-2x}(\partial _{x}Y^{2}+\partial _{y}Y^{1})}{(1+e^{-2x})^{2}} &{} 4e^{-2x}\frac{(1+e^{-2x})\partial _{y}Y^{2}+(e^{-2x}-1)Y^{1}}{(1+e^{-2x})^{3}} \end{array} \right) . \end{aligned}$$
(1.4)

Substituting g, S, \(\mathcal {L}_{Y}g\), and \(\mathcal {L}_{Y}S\) in equations (1.1) and (1.2), we deduce

$$\begin{aligned}{} & {} (1+e^{-2x})\partial _{x}Y^{1}+e^{-2x}Y^{1}=\frac{1}{2}(1+e^{-2x})\alpha +e^{-2x}\beta , \end{aligned}$$
(1.5)
$$\begin{aligned}{} & {} (1+e^{-2x})\partial _{y}Y^{2}+e^{-2x}Y^{1}=\frac{1}{2}(1+e^{-2x})\alpha +e^{-2x}\beta ,\end{aligned}$$
(1.6)
$$\begin{aligned}{} & {} \partial _{x}Y^{2}+\partial _{y}Y^{1}=0,\end{aligned}$$
(1.7)
$$\begin{aligned}{} & {} (1+e^{-2x})\partial _{x}Y^{1}+(e^{-2x}-1)Y^{1}=\frac{1}{2}(1+e^{-2x})\alpha +\frac{e^{2x}(1+e^{-2x})^{2}}{4}\beta ,\end{aligned}$$
(1.8)
$$\begin{aligned}{} & {} (1+e^{-2x})\partial _{y}Y^{2}+(e^{-2x}-1)Y^{1}=\frac{1}{2}(1+e^{-2x})\alpha +\frac{e^{2x}(1+e^{-2x})^{2}}{4}\beta . \end{aligned}$$
(1.9)

We see that \(Y^{1}=x, Y^{2}=y\) and

$$\begin{aligned} \alpha =\frac{2+4e^{-2x}}{1+e^{-2x}}-\frac{8e^{-2x}x}{(1+e^{-2x})(4e^{-2x}-e^{2x}(1+e^{-2x})^{2})}, \beta = \frac{4x}{4e^{-2x}-e^{2x}(1+e^{-2x})^{2}} \end{aligned}$$

is a solution of the above system. So the manifold M has a non-trivial Ricci bi-conformal vector field.

A vector field X is called Ricci collineation vector field [12] whenever \(\mathcal {L}_{X}S=0\). Notice, if in the definition of Ricci bi-conformal vector field we have \(\alpha =\beta =0\) then Ricci bi-conformal vector field becomes Killing vector fields and Ricci collineation vector field. From [36], in this case, Ricci bi-conformal vector field is an infinitesimal harmonic transformation, because \(\mathcal {L}_{X}g=0\) implies that \(\mathcal {L}_{X}\nabla =0\) and \(\textrm{trac}_{g}(\mathcal {L}_{X}\nabla )=0\).

On the other hand, the notion of Ricci flow on a Riemannian manifold introduced by Hamilton [24] and it is defined by

$$\begin{aligned} \frac{\partial }{\partial t}g=-2S \end{aligned}$$

where S is the Ricci tensor of a manifold. The special solutions (self-similar solusions) of the Ricci flow equation are called Ricci solitons which are generalizations of Einstein metrics. A Ricci soliton [7] is a triplet \((g, X, \lambda )\) on a pseudo-Riemannian manifold M such that

$$\begin{aligned} \mathcal {L}_{X}g+S+\lambda g=0, \end{aligned}$$
(1.10)

where \(\lambda \) is a real constant. If \(\lambda \) be a smooth function on M then it is called almost Ricci soliton. Ricci solitons are significant and practical in physics and are known as quasi-Einstein [14, 15]. The Ricci soliton is called expanding, steady, and shrinking when \(\lambda \) is positive, zero, and negative, respectively. g is called a gradient Ricci soliton whenever the vector field X is the gradient of the potential function f. For more details, see [2, 3, 5, 6, 10, 28, 30]. Moreover, Ricci solitons have been considered in physics [1, 27]. Calavaruso [9] studied the Ricci soliton equation on homogenous Gödel-type spacetimes. Notice, in definition of Ricci bi-conformal vector field if \(\beta =-1\) then Ricci bi-conformal vector field is an almost Ricci soliton. In general case, if \(\beta \ne 0\) then the first equation of the Ricci bi-conformal vector field is \(\frac{-1}{\beta }\)-almost Ricci soliton. Also, homogenous Gödel-type spacetimes studied in the different topics for more details see [4, 8, 17, 26, 29, 31, 35, 38].

Motivated by [9, 19], we study the Ricci bi-conformal vector fields on homogeneous Gödel-type spacetimes and we completely solve the Eqs. (1.1) and (1.2) for homogeneous Gödel-type spacetimes. We also prove that all Ricci bi-conformal vector fields on homogeneous Gödel-type spacetimes are Killing vector fields and Ricci collineation vector fields and we obtain which of them are gradient vector fields and almost Yamabe solitons.

The paper is organized as follows. In Section 2, we recall some necessary concepts on homogeneous Gödel-type spacetimes which will be used throughout this paper. In Section 3, we give the main results and their proofs. In fact, in this section by solving Eqs. (1.1) and (1.2) on four classes of homogeneous Gödel-type spacetimes we obtain a complete classification of Ricci bi-conformal vector fields on homogeneous Gödel-type spacetimes.

2 Preliminaries

The Gödel-type spacetimes are determined by the Lorentzian metrics

$$\begin{aligned} g=[dt+H(r)d\phi ]^{2}-dr^{2}-D^{2}(r)d\phi ^{2}-dz^{2}, \end{aligned}$$
(2.1)

where \((r,\phi ,z)\) are the normal cylindrical coordinates and t is the time variable. Gödel-type spacetimes are very popular in general relativity and extensively studied in arbitrary dimensions [22, 23]. The Gödel-type homogeneous spacetimes are described by the metrics g and extra constraint

$$\begin{aligned} D''=aD,\,\,\,\,\,H'=-2bD, \end{aligned}$$
(2.2)

for some real constants a and b (see [32, 33]). Since \(det(g)=-D^{2}(r)\), we can consider (2.1) as a metric tensor (where \(D(r)\ne 0\) ) for any functions H, D of class \(C^{\infty }\). Depending on the signs of the constants in (2.1) and functions H, D with initial condition \(H(0)=D(0)=0\), \(D'(0)=1\), Gödel-type homogeneous spacetimes are classified into the following non-isometric classes [11]:

Class I: \(a=m^{2}>0\), \(b\ne 0\). Thus, the solutions to (2.2) is described by

$$\begin{aligned} D(r)=\frac{1}{m}\sinh (mr),\qquad H(r)=\frac{-2b}{m^{2}}[\cosh (mr)-1]. \end{aligned}$$

Class II: \(a=0\), \(b\ne 0\). In this case,

$$\begin{aligned} D(r)=r,\qquad H(r)=-br^{2}. \end{aligned}$$
(2.3)

Class III: \(a=-\mu ^{2}<0\), \(b\ne 0\). Thus,

$$\begin{aligned} D(r)=\frac{1}{\mu }\sin (\mu r),\qquad H(r)=\frac{-2b}{\mu ^{2}}[1-\cos (\mu r)]. \end{aligned}$$

Class IV: \(a\ne 0\), \(b=0\). Hence, D(r) is either as in (2.3) if \(\alpha =m^{2}>0\) or as in (2.3) if \(\alpha =-\mu ^{2}<0\) and \(H(r)=0\).

From [18, 29], on Gödel-type spacetimes, the non-vanishing components of the Levi-Civita connection \(\nabla \) regarding the coordinates vector fields \(\{\partial _{t}, \partial _{r}, \partial _{\phi }, \partial _{z}\}\) are obtained from the following relations

$$\begin{aligned}{} & {} \nabla _{\partial _{t}}\partial _{r}=\frac{HH'}{2D^{2}}\partial _{t}-\frac{H'}{2D^{2}}\partial _{\phi },\\ {}{} & {} \nabla _{\partial _{t}}\partial _{\phi }=\frac{1}{2}H'\partial _{r},\\ {}{} & {} \nabla _{\partial _{r}}\partial _{\phi }=\frac{H'(D^{2}+H^{2})-2HDD'}{2D^{2}}\partial _{t}+\frac{2DD'-HH'}{2D^{2}}\partial _{\phi },\\ {}{} & {} \nabla _{\partial _{\phi }}\partial _{\phi }=(HH'-DD')\partial _{r}. \end{aligned}$$

From [9] the Ricci tensor of Gödel-type spacetimes regarding the coordinates vector fields \(\{\partial _{t}, \partial _{r}, \partial _{\phi }, \partial _{z}\}\) is represented by

$$\begin{aligned} S=\left( \begin{array}{cccc} \frac{(H')^{2}}{2D^{2}} &{} 0 &{} K &{}0 \\ 0&{} L &{} 0 &{}0\\ K &{} 0 &{} E&{}0 \\ 0 &{} 0 &{} 0 &{} 0\\ \end{array} \right) , \end{aligned}$$
(2.4)

where \(K=\frac{D^{2}H''+H(H')^{2}-DD'H'}{2D^{2}}\), \( E=\frac{H^{2}(H')^{2}-2DD'HH'+D^{2}(H')^{2}+2D^{2}HH''-2D^{3}D''}{2D^{2}}\), and \(L=\frac{(H')^{2}-2DD''}{2D^{2}} \). Let \(X=X_{1}\partial _{t}+X_{2}\partial _{r}+X_{3}\partial _{\phi }+X_{4}\partial _{z}\) be an arbitrary vector field where \(X_{i}=X_{i}(t,r,\phi ,z)\), \(i=1,2,3,4\) are differentiable maps. The Lie derivative \(\mathcal {L}_{X}g\) is given by

$$\begin{aligned} {\left\{ \begin{array}{ll} (\mathcal {L}_{X}g)(\partial _{t},\partial _{t})=2\partial _{t}X_{1}+2H\partial _{t}X_{3},\\ (\mathcal {L}_{X}g)(\partial _{t},\partial _{r})=\partial _{r}X_{1}+H\partial _{r}X_{3}-\partial _{t}X_{2},\\ (\mathcal {L}_{X}g)(\partial _{t},\partial _{\phi })=H\partial _{t}X_{1}+\partial _{\phi }X_{1}+H'X_{2}+(H^{2}-D^{2})\partial _{t}X_{3}+H\partial _{\phi }X_{3}, \\ (\mathcal {L}_{X}g)(\partial _{t},\partial _{z})=\partial _{z}X_{1}+H\partial _{z}X_{3}-\partial _{t}X_{4}, \\ (\mathcal {L}_{X}g)(\partial _{r},\partial _{r})=-2\partial _{r}X_{2},\\ (\mathcal {L}_{X}g)(\partial _{r},\partial _{\phi })=H\partial _{r}X_{1}-\partial _{\phi }X_{2}+(H^{2}-D^{2})\partial _{r}X_{3}, \\ (\mathcal {L}_{X}g)(\partial _{r},\partial _{z})=-\partial _{z}X_{2}-\partial _{r}X_{4}, \\ (\mathcal {L}_{X}g)(\partial _{\phi },\partial _{\phi })=2H\partial _{\phi }X_{1}+2HH'X_{2}-2DD'X_{2}+2(H^{2}-D^{2})\partial _{\phi }X_{3}, \\ (\mathcal {L}_{X}g)(\partial _{\phi },\partial _{z})=H\partial _{z}X_{1}+(H^{2}-D^{2})\partial _{z}X_{3}-\partial _{\phi }X_{4},\\ (\mathcal {L}_{X}g)(\partial _{z},\partial _{z})=-2\partial _{z}X_{4}. \end{array}\right. } \end{aligned}$$
(2.5)

The Lie derivative \(\mathcal {L}_{X}S\) is determined by

$$\begin{aligned} {\left\{ \begin{array}{ll} (\mathcal {L}_{X}S)(\partial _{t},\partial _{t})=X_{2}\partial _{r}(\frac{(H')^{2}}{2D^{2}})+2\frac{(H')^{2}}{2D^{2}}\partial _{t}X_{1}+2K\partial _{t}X_{3},\\ (\mathcal {L}_{X}S)(\partial _{t},\partial _{r})=L\partial _{t}X_{2}+\frac{(H')^{2}}{2D^{2}}\partial _{r}X_{1}+K\partial _{r}X_{3},\\ (\mathcal {L}_{X}S)(\partial _{t},\partial _{\phi })=X_{2}\partial _{r}K+K\partial _{t}X_{1}+E\partial _{t}X_{3}+\frac{(H')^{2}}{2D^{2}} \partial _{\phi }X_{1}+K\partial _{\phi }X_{3}, \\ (\mathcal {L}_{X}S)(\partial _{t},\partial _{z})=\frac{(H')^{2}}{2D^{2}}\partial _{z}X_{1}+K\partial _{z}X_{3}, \\ (\mathcal {L}_{X}S)(\partial _{r},\partial _{r})=X_{2}\partial _{r}L+2 L\partial _{r}X_{2},,\\ (\mathcal {L}_{X}S)(\partial _{r},\partial _{\phi })=K\partial _{r}X_{1}+E\partial _{r}X_{3}+ L\partial _{\phi }X_{2}, \\ (\mathcal {L}_{X}S)(\partial _{r},\partial _{z})= L\partial _{z}X_{2}, \\ (\mathcal {L}_{X}S)(\partial _{\phi },\partial _{\phi })=X_{2}\partial _{r}E+2K\partial _{\phi }X_{1}+2E\partial _{\phi }X_{3}, \\ (\mathcal {L}_{X}S)(\partial _{\phi },\partial _{z})=K \partial _{z}X_{1}+E\partial _{z}X_{3}, \\ (\mathcal {L}_{X}S)(\partial _{z},\partial _{z})=0. \end{array}\right. } \end{aligned}$$
(2.6)

3 The Main Results and Their Proofs

In this section, we state the our results and their proofs on homogeneous Gödel-type spacetimes. Applying (2.1), (2.4), and (2.5) into (1.1), we get

$$\begin{aligned}{} & {} 2\partial _{t}X_{1}+2H\partial _{t}X_{3}=\alpha +\beta \frac{(H')^{2}}{2D^{2}},\end{aligned}$$
(3.1)
$$\begin{aligned}{} & {} \partial _{r}X_{1}+H\partial _{r}X_{3}-\partial _{t}X_{2}=0,\end{aligned}$$
(3.2)
$$\begin{aligned}{} & {} H\partial _{t}X_{1}+\partial _{\phi }X_{1}+H'X_{2}+(H^{2}-D^{2})\partial _{t}X_{3}+H\partial _{\phi }X_{3}=\alpha H+\beta K,\end{aligned}$$
(3.3)
$$\begin{aligned}{} & {} \partial _{z}X_{1}+H\partial _{z}X_{3}-\partial _{t}X_{4}=0,\end{aligned}$$
(3.4)
$$\begin{aligned}{} & {} -2\partial _{r}X_{2}=-\alpha +\beta L,\end{aligned}$$
(3.5)
$$\begin{aligned}{} & {} H\partial _{r}X_{1}-\partial _{\phi }X_{2}+(H^{2}-D^{2})\partial _{r}X_{3}=0,\end{aligned}$$
(3.6)
$$\begin{aligned}{} & {} -\partial _{z}X_{2}-\partial _{r}X_{4}=0, \end{aligned}$$
(3.7)
$$\begin{aligned}{} & {} 2H\partial _{\phi }X_{1}+2HH'X_{2}-2DD'X_{2}+2(H^{2}-D^{2})\partial _{\phi }X_{3}=\alpha (H^{2}-D^{2})+E\beta , \end{aligned}$$
(3.8)
$$\begin{aligned}{} & {} H\partial _{z}X_{1}+(H^{2}-D^{2})\partial _{z}X_{3}-\partial _{\phi }X_{4}=0,\end{aligned}$$
(3.9)
$$\begin{aligned}{} & {} -2\partial _{z}X_{4}=-\alpha . \end{aligned}$$
(3.10)

Also, using (2.1), (2.4), and (2.6) into (1.2), we have

$$\begin{aligned}{} & {} X_{2}\partial _{r}(\frac{(H')^{2}}{2D^{2}})+2\frac{(H')^{2}}{2D^{2}}\partial _{t}X_{1}+2K\partial _{t}X_{3}=\alpha \frac{(H')^{2}}{2D^{2}}+\beta ,\end{aligned}$$
(3.11)
$$\begin{aligned}{} & {} L\partial _{t}X_{2}+\frac{(H')^{2}}{2D^{2}}\partial _{r}X_{1}+K\partial _{r}X_{3}=0,\end{aligned}$$
(3.12)
$$\begin{aligned}{} & {} X_{2}\partial _{r}K+K\partial _{t}X_{1}+E\partial _{t}X_{3}+\frac{(H')^{2}}{2D^{2}}\partial _{\phi }X_{1}+K\partial _{\phi }X_{3}=\alpha K+\beta H, \end{aligned}$$
(3.13)
$$\begin{aligned}{} & {} \frac{(H')^{2}}{2D^{2}}\partial _{z}X_{1}+K\partial _{z}X_{3}=0, \end{aligned}$$
(3.14)
$$\begin{aligned}{} & {} X_{2}\partial _{r}L+2 L\partial _{r}X_{2}=\alpha L-\beta ,,\end{aligned}$$
(3.15)
$$\begin{aligned}{} & {} K\partial _{r}X_{1}+E\partial _{r}X_{3}+ L\partial _{\phi }X_{2}=0, \end{aligned}$$
(3.16)
$$\begin{aligned}{} & {} L\partial _{z}X_{2}=0, \end{aligned}$$
(3.17)
$$\begin{aligned}{} & {} X_{2}\partial _{r}E+2K\partial _{\phi }X_{1}+2E\partial _{\phi }X_{3}=E\alpha +\beta (H^{2}-D^{2}), \end{aligned}$$
(3.18)
$$\begin{aligned}{} & {} K \partial _{z}X_{1}+E\partial _{z}X_{3}=0, \end{aligned}$$
(3.19)
$$\begin{aligned}{} & {} 0=-\beta . \end{aligned}$$
(3.20)

Before we focus on the above system solutions on homogeneous Gödel-type spacetimes for class I-IV, using (2.2) on homogeneous Gödel-type spacetimes we have

$$\begin{aligned} L=2b^{2}-a,\,\,\, \frac{(H')^{2}}{2D^{2}}=2b^{2}. \end{aligned}$$

Equation (3.20) yields \(\beta =0\).

3.1 Case I

We assume that \(a=m^{2}>0\), \(b\ne 0\). Thus \( D(r)=\frac{1}{m}\sinh (mr) \) and \( H(r)=\frac{-2b}{m^{2}}[\cosh (mr)-1] \). Equation (3.17) implies that \( 2b^{2}-a=0\) or \(\partial _{z}X_{2}=0\). Let \( 2b^{2}-a\ne 0\) then

$$\begin{aligned} \partial _{z}X_{2}=0 \end{aligned}$$
(3.21)

and

$$\begin{aligned} X_{2}=f_{1}(t,r,\phi ), \end{aligned}$$

for some smooth function \(f_{1}\). From (3.7) and (3.21), we have

$$\begin{aligned} \partial _{r}X_{4}=0. \end{aligned}$$
(3.22)

Equations (3.5) and (3.21) imply that

$$\begin{aligned} \partial _{z}\alpha =0. \end{aligned}$$
(3.23)

Differentiating the equation (3.10) by z and using (3.23), we get

$$\begin{aligned} \partial _{zz}^{2}X_{4}=0. \end{aligned}$$
(3.24)

We can rewrite (3.1), (3.2), (3.3), and (3.4) as

$$\begin{aligned}{} & {} \partial _{t}(X_{1}+HX_{3})=\frac{\alpha }{2},\\{} & {} \partial _{r}(X_{1}+HX_{3})=\partial _{t}X_{2}-2bDX_{3},\\{} & {} \partial _{\phi }(X_{1}+HX_{3})=\alpha H-H\partial _{t}X_{1}+2bDX_{2}-(H^{2}-D^{2})\partial _{1}X_{3},\\{} & {} \partial _{z}(X_{1}+HX_{3})=\partial _{t}X_{4}, \end{aligned}$$

respectively. We differentiate equation (3.25) by z and using (3.21), (3.22) and (3.25), we conclude \(bD\partial _{z}X_{3}=0\). Since \(bD\ne 0\), we deduce

$$\begin{aligned} \partial _{z}X_{3}=0. \end{aligned}$$

Equation (3.19) leads to

$$\begin{aligned} \partial _{z}X_{1}=0, \end{aligned}$$

and from (3.25), we infer

$$\begin{aligned} \partial _{t}X_{4}=0. \end{aligned}$$
(3.25)

Equation (3.9) yields

$$\begin{aligned} \partial _{\phi }X_{4}=0. \end{aligned}$$
(3.26)

From (3.10), (3.22), (3.24), (3.25), and (3.26), we get

$$\begin{aligned} X_{4}=a_{1}z+a_{2} \end{aligned}$$

and \(\alpha =2a_{1}\) for some constants \(a_{1}\) and \(a_{2}\). Using equation (3.1), we have \(\partial _{t}X_{1}+H\partial _{t}X_{3}=a_{1}\) and using equation (3.11), we obtain \(\partial _{t}X_{1}+D^{2}H\partial _{t}X_{3}=a_{1}\). Thus, we get

$$\begin{aligned} \partial _{t}X_{1}=a_{1},\qquad \qquad \partial _{t}X_{3}=0. \end{aligned}$$
(3.27)

We differentiate Eq. (3.3) by t and using (3.27),

$$\begin{aligned} \partial _{t}X_{2}=0. \end{aligned}$$
(3.28)

From equation (3.2), we have \(\partial _{r}X_{1}+H\partial _{r}X_{3}=0\) and using equation (3.11), we deduce \(\partial _{r}X_{1}+D^{2}H\partial _{r}X_{3}=0\). Hence,

$$\begin{aligned} \partial _{r}X_{1}=\partial _{r}X_{3}=0. \end{aligned}$$
(3.29)

Using (3.6) and (3.29), we obtain

$$\begin{aligned} \partial _{\phi }X_{2}=0. \end{aligned}$$
(3.30)

Therefore Eqs. (3.5), (3.21), (3.28), and (3.30) imply that

$$\begin{aligned} X_{2}=a_{1}r+a_{3} \end{aligned}$$

for some constant \(a_{3}\). From Eq. (3.3), we have

$$\begin{aligned} 2H\partial _{\phi }X_{1}+2HH'X_{2}+2H^{2}\partial _{\phi }X_{3}=2a_{1} H^{2} \end{aligned}$$

and applying Eq. (3.8), we infer

$$\begin{aligned} 2H\partial _{\phi }X_{1}-4bHDX_{2}-2DD'X_{2}+2(H^{2}-D^{2})\partial _{\phi }X_{3}=2a_{1}(H^{2}-D^{2}). \end{aligned}$$

From the difference of the above two equations, we arrive at

$$\begin{aligned} \partial _{\phi }X_{3}=a_{1}-\frac{D'}{D}X_{2}, \end{aligned}$$
(3.31)

and

$$\begin{aligned} \partial _{\phi }X_{1}=2bDX_{2}+\frac{D'H}{D}X_{2} \end{aligned}$$
(3.32)

We can write Eqs. (3.13) and (3.18) as

$$\begin{aligned}{} & {} \partial _{\phi }(X_{1}+D^{2}HX_{3})=a_{1}D^{2}H-(D^{2}H)'X_{2}, \end{aligned}$$
(3.33)
$$\begin{aligned}{} & {} \partial _{\phi }(2b^{2}D^{2}HX_{1}+EX_{3})=a_{1}E-(-4b^{3}HD+2b^{2}DD'-aDD')X_{2}. \end{aligned}$$
(3.34)

Substituting (3.31) and (3.32) into (3.33) and (3.37), we get

$$\begin{aligned}{} & {} (2bD+\frac{D'H}{D}+DD'H-2bD^{3})X_{2}=0,\end{aligned}$$
(3.35)
$$\begin{aligned}{} & {} H(\frac{D'H}{D}+2bD)(D^{2}-1)X_{2}=0. \end{aligned}$$
(3.36)

Hence \(X_{2}=0\), \(X_{1}=a_{4}\), and \(X_{3}=a_{5}\), for some constants \(a_{4}\) and \(a_{5}\). Therefore, we can write

$$\begin{aligned} X_{1}=a_{4},\,\,X_{2}=0,\,\,\,X_{3}=a_{5},\,\,\,X_{4}=a_{2},\,\, \alpha =\beta =0. \end{aligned}$$

Now in Class I, we assume that \(2b^{2}-a=0\). In this case, we have

$$\begin{aligned} L=0,\,\,\, \frac{(H')^{2}}{2D^{2}}=a,\,\,\, E=aH. \end{aligned}$$

Equations (3.1) and (3.11) yield

$$\begin{aligned}{} & {} \partial _{t}X_{1}+H\partial _{t}X_{3}=\frac{\alpha }{2},\\ {}{} & {} \partial _{t}X_{1}+D^{2}H\partial _{t}X_{3}=\frac{\alpha }{2}. \end{aligned}$$

Thus \(\partial _{t}X_{1}=\frac{\alpha }{2}\) and \(\partial _{t}X_{3}=0\). Equations (3.14) and (3.19) imply that

$$\begin{aligned} \partial _{z}X_{1}+D^{2}H\partial _{z}X_{3}=0,\,\,\,\,D^{2}\partial _{z}X_{1}+\partial _{z}X_{3}=0. \end{aligned}$$
(3.37)

Since \(D^{4}H\ne 1\), using (3.37), we infer

$$\begin{aligned} \partial _{z}X_{1}=\partial _{z}X_{3}=0. \end{aligned}$$
(3.38)

Inserting (3.38) in (3.4) and (3.9), we obtain

$$\begin{aligned} \partial _{t}X_{4}=\partial _{\phi }X_{4}=0. \end{aligned}$$
(3.39)

Equations (3.12) and (3.16) yield

$$\begin{aligned} \partial _{r}X_{1}+D^{2}H\partial _{r}X_{3}=0,\,\,\,\,D^{2}\partial _{r}X_{1}+\partial _{r}X_{3}=0. \end{aligned}$$

Then

$$\begin{aligned} \partial _{r}X_{1}=\partial _{r}X_{3}=0. \end{aligned}$$
(3.40)

Inserting (3.40) in (3.2) and (3.6), we deduce

$$\begin{aligned} \partial _{t}X_{2}=\partial _{\phi }X_{2}=0. \end{aligned}$$
(3.41)

Using \(\partial _{t}X_{1}=\frac{\alpha }{2}\), (3.38) and (3.40), we conclude

$$\begin{aligned} \partial _{z}\alpha =\partial _{r}\alpha =0. \end{aligned}$$

Equation (3.10) implies that \(\partial _{z}X_{4}=\frac{\alpha }{2}\), then applying it with together (3.39), we get

$$\begin{aligned} \partial _{t}\alpha =\partial _{\phi }\alpha =0. \end{aligned}$$

Therefore \(\alpha =a_{6}\) for some constant \(a_{6}\). Differentiating Eq. (3.3) by z, we obtain

$$\begin{aligned} \partial _{z}X_{2}=0. \end{aligned}$$
(3.42)

Using the above equation in (3.7), we find

$$\begin{aligned} \partial _{r}X_{4}=0. \end{aligned}$$
(3.43)

From Eq. (3.5), we have \(\partial _{r}X_{2}=\frac{\alpha }{2}\), thus it with (3.41) and (3.42) yield

$$\begin{aligned} X_{2}=\frac{1}{2}a_{6}r+a_{7}, \end{aligned}$$

for some constant \(a_{7}\). Also, using \(\partial _{z}X_{4}=\frac{\alpha }{2}\), (3.39) and (3.43), we get

$$\begin{aligned} X_{4}=\frac{1}{2}a_{6}z+a_{8}, \end{aligned}$$

for some constant \(a_{8}\). From Eq. (3.3), we have

$$\begin{aligned} 2H\partial _{\phi }X_{1}-4bHDX_{2}+2H^{2}\partial _{\phi }X_{3}=\alpha H^{2} \end{aligned}$$

and equation (3.8) leads to

$$\begin{aligned} 2H\partial _{\phi }X_{1}-4bHDX_{2}-2DD'X_{2}+2(H^{2}-D^{2})\partial _{\phi }X_{3}=\alpha (H^{2}-D^{2}). \end{aligned}$$

From the difference of the above two equations, we arrive at

$$\begin{aligned} \partial _{\phi }X_{3}=\frac{\alpha }{2}-\frac{D'}{D}X_{2}, \end{aligned}$$
(3.44)

and

$$\begin{aligned} \partial _{\phi }X_{1}=2bDX_{2}+\frac{D'H}{D}X_{2}. \end{aligned}$$
(3.45)

We can write Eqs. (3.13) and (3.18) as

$$\begin{aligned}{} & {} \partial _{\phi }(X_{1}+D^{2}HX_{3})=\frac{\alpha }{2}D^{2}H-(D^{2}H)'X_{2},\end{aligned}$$
(3.46)
$$\begin{aligned}{} & {} \partial _{\phi }(D^{2}X_{1}+ X_{3})=\frac{\alpha }{2}+2bDX_{2}. \end{aligned}$$
(3.47)

Substituting (3.44) and (3.45) into (3.46) and (3.47), we get

$$\begin{aligned}{} & {} (2bD+\frac{D'H}{D}+DD'H-2bD^{3})X_{2}=0,\\{} & {} (2bD^{3}-2bD+DD'H-\frac{D'}{D})X_{2}=0. \end{aligned}$$

Hence \(X_{2}=0\), \(X_{1}=a_{9}\), and \(X_{3}=a_{10}\), for some constants \(a_{9}\) and \(a_{10}\). Therefore, we have

$$\begin{aligned} X_{1}=a_{9},\,\,X_{2}=0,\,\,\,X_{3}=a_{10},\,\,\,X_{4}=a_{8},\,\, \alpha =\beta =0. \end{aligned}$$

Therefore we have the following theorem:

Theorem 3.1

All homogeneous Gödel-type spacetimes in Class I, as explained by equations (2.1) with \(H'=-2bD\) and \(D''=aD\) (\(a,b\ne 0\)), has Ricci bi-conformal vector field \(X=X_{1}\partial _{t}+X_{2}\partial _{r}+X_{3}\partial _{\phi }+X_{4}\partial _{z}\) if and only if \(\alpha =\beta =0\) and \(X_{1}=k_{1}, X_{2}=0, X_{3}=k_{2}\) and \(X_{4}=k_{3}\) for some constants \(k_{1}, k_{2}\) and \(k_{3}\).

Now, we consider the vector fields as \(X=\nabla f\) for some smooth function f which are Ricci bi-conformal on homogeneous Gödel-type spacetimes in Class I. On homogeneous Gödel-type spacetimes with metric (2.1), equation \(X_{1}\partial _{t}+X_{2}\partial _{r}+X_{3}\partial _{\phi }+X_{4}\partial _{z}=X=\nabla f=g^{ij}\partial _{j}f\partial _{i}\) implies that

$$\begin{aligned} {\left\{ \begin{array}{ll} (D^{2}-H^{2})\partial _{t}f+H^{2}\partial _{\phi }f=D^{2}X_{1},\\ -\partial _{r}f=X_{2},\\ H\partial _{t}f-\partial _{\phi }f=D^{2}X_{3},\\ -\partial _{z}f=X_{4}. \end{array}\right. } \end{aligned}$$
(3.48)

From (3.48) and Theorem 3.1, on homogeneous Gödel-type spacetimes in Class I, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} (D^{2}-H^{2})\partial _{t}f+H^{2}\partial _{\phi }f=k_{1}D^{2},\\ \partial _{r}f=0,\\ H\partial _{t}f-\partial _{\phi }f=k_{2}D^{2},\\ \partial _{z}f=-k_{3}. \end{array}\right. } \end{aligned}$$
(3.49)

Deriving the third equation of the system (3.49) with respect to r and using \(H'=-2bD\) and the second equation of (3.49), we infer

$$\begin{aligned} -b\partial _{t}f=k_{2}D'. \end{aligned}$$
(3.50)

Deriving the Eq. (3.50) with respect to r and using \(D''=aD\), we get \(ak_{2} D=0\). Since \(a\ne 0\) and \(D\ne 0\) we conclude \(k_{2}=0\). Substituting it in (3.50) and the third equation of (3.49) we obtain \(\partial _{t}f=\partial _{\phi }f=0\). Therefore \(f=-k_{3}z+k_{4}\) for some constant \(k_{4}\) and we have the following corollary:

Corollary 3.2

All homogeneous Gödel-type spacetimes in Class I, as explained by equations (2.1) \(H'=-2bD\) and \(D''=aD\) (\(a,b\ne 0\)), has Ricci bi-conformal vector field as \(X=\nabla f \) if and only if \(f=-k_{3}z+k_{4}\) for some constant \(k_{4}\).

Remark 3.3

From Theorem 3.1 we conclude that all Ricci bi-conformal vector fields on homogeneous Gödel-type spacetimes in Class I are Killing vector fields and Ricci collineation vector fields. Also, (M, g) is said to be almost Yamabe soliton if it admits a vector field X such that

$$\begin{aligned} \mathcal {L}_{X}g=(\tau -\Lambda )g, \end{aligned}$$

where \(\tau \) denotes the scalar curvature of (M, g) and \(\Lambda \) is a smooth function. Moreover, we say that the almost Yamabe soliton is an almost gradient Yamabe soliton if \(X=\nabla f\) for some potential function f. Thus, by the Theorem 3.1 we deduce that all Ricci bi-conformal vector fields on homogeneous Gödel-type spacetimes in Class I admit in almost Yamabe soliton equation with \(\Lambda =\tau \).

3.2 Class II

We assume that \(a=0\), \(b\ne 0\). Then, \( H(r)=-br^{2}\) and \( D(r)=r\). In this case, we have

$$\begin{aligned} L=2b^{2},\,\,\, \frac{(H')^{2}}{2D^{2}}=2b^{2},\,\,\, E=2b^{2}r^{2}(b^{2}r^{2}+1). \end{aligned}$$

Since \(b\ne 0\) the equation (3.17) implies that

$$\begin{aligned} \partial _{z}X_{2}=0 \end{aligned}$$
(3.51)

From (3.7) and (3.51), we obtain

$$\begin{aligned} \partial _{r}X_{4}=0. \end{aligned}$$
(3.52)

Equations (3.5) and (3.51) imply that

$$\begin{aligned} \partial _{z}\alpha =0. \end{aligned}$$
(3.53)

Differentiating the Eq. (3.10) by z and using (3.53), we get

$$\begin{aligned} \partial _{zz}^{2}X_{4}=0. \end{aligned}$$
(3.54)

We can rewrite (3.1), (3.2), (3.3), and (3.4) as

$$\begin{aligned}{} & {} \partial _{t}(X_{1}+HX_{3})=\frac{\alpha }{2},\\{} & {} \partial _{r}(X_{1}+HX_{3})=\partial _{t}X_{2}+H'X_{3},\\{} & {} \partial _{\phi }(X_{1}+HX_{3})=\alpha H-H\partial _{t}X_{1}-H'X_{2}-(H^{2}-D^{2})\partial _{1}X_{3},\\{} & {} \partial _{z}(X_{1}+HX_{3})=\partial _{t}X_{4}, \end{aligned}$$

respectively. We differentiate Eq. (3.55) by z and using (3.51), (3.52) and (3.55) we conclude \(H'\partial _{z}X_{3}=0\). Since \(H'=-2bD\ne 0\), we deduce

$$\begin{aligned} \partial _{z}X_{3}=0. \end{aligned}$$

Equation (3.19) leads to

$$\begin{aligned} \partial _{z}X_{1}=0, \end{aligned}$$

and from (3.55), we infer

$$\begin{aligned} \partial _{t}X_{4}=0. \end{aligned}$$
(3.55)

Equation (3.9) yields

$$\begin{aligned} \partial _{\phi }X_{4}=0. \end{aligned}$$
(3.56)

From (3.10), (3.52), (3.54), (3.55), and (3.56), we get

$$\begin{aligned} X_{4}=b_{1}z+b_{2} \end{aligned}$$
(3.57)

and \(\alpha =2b_{1}\) for some constants \(b_{1}\) and \(b_{2}\). From equation (3.1) we have \(\partial _{t}X_{1}+H\partial _{t}X_{3}=b_{1}\) and using equation (3.11), we conclude \(\partial _{t}X_{1}+D^{2}H\partial _{t}X_{3}=b_{1}\). Thus, we infer

$$\begin{aligned} \partial _{t}X_{1}=a_{1},\qquad \qquad \partial _{t}X_{3}=0. \end{aligned}$$
(3.58)

We differentiate Eq. (3.3) by t and using (3.58),

$$\begin{aligned} \partial _{t}X_{2}=0. \end{aligned}$$
(3.59)

From equation (3.2) we have \(\partial _{r}X_{1}+H\partial _{r}X_{3}=0\) and using equation (3.11), we get \(\partial _{r}X_{1}+D^{2}H\partial _{r}X_{3}=0\). Hence,

$$\begin{aligned} \partial _{r}X_{1}=\partial _{r}X_{3}=0. \end{aligned}$$

Using (3.6) and (3.58), we obtain

$$\begin{aligned} \partial _{\phi }X_{2}=0. \end{aligned}$$
(3.60)

Therefore Eqs. (3.5), (3.51), (3.59), and (3.60) imply that

$$\begin{aligned} X_{2}=b_{1}r+b_{3} \end{aligned}$$

for some constant \(b_{3}\). Equation (3.3) leads to

$$\begin{aligned} 2H\partial _{\phi }X_{1}+2HH'X_{2}+2H^{2}\partial _{\phi }X_{3}=2b_{1} H^{2} \end{aligned}$$

and Eq. (3.8) yields

$$\begin{aligned} 2H\partial _{\phi }X_{1}-4bHDX_{2}-2DD'X_{2}+2(H^{2}-D^{2})\partial _{\phi }X_{3}=2b_{1}(H^{2}-D^{2}). \end{aligned}$$

From the difference of the above two equations, we have

$$\begin{aligned} \partial _{\phi }X_{3}=b_{1}-\frac{D'}{D}X_{2}, \end{aligned}$$
(3.61)

and

$$\begin{aligned} \partial _{\phi }X_{1}=2bDX_{2}+\frac{D'H}{D}X_{2}. \end{aligned}$$
(3.62)

We can write Eqs. (3.13) and (3.18) as

$$\begin{aligned}{} & {} \partial _{\phi }(X_{1}+D^{2}HX_{3})=b_{1}D^{2}H-(D^{2}H)'X_{2},\end{aligned}$$
(3.63)
$$\begin{aligned}{} & {} \partial _{\phi }(2b^{2}D^{2}HX_{1}+EX_{3})=b_{1}E-(-4b^{3}HD+2b^{2}DD')X_{2}. \end{aligned}$$
(3.64)

Substituting (3.61) and (3.62) into (3.63) and (3.64), we get

$$\begin{aligned}{} & {} (2bD+\frac{D'H}{D}+DD'H-2bD^{3})X_{2}=0,\end{aligned}$$
(3.65)
$$\begin{aligned}{} & {} H(\frac{D'H}{D}+2bD)(D^{2}-1)X_{2}=0. \end{aligned}$$
(3.66)

Hence \(X_{2}=0\), \(X_{1}=b_{4}\), and \(X_{3}=b_{5}\), for some constants \(a_{4}\) and \(a_{5}\). Therefore, we have

$$\begin{aligned} X_{1}=b_{4},\,\,X_{2}=0,\,\,\,X_{3}=b_{5},\,\,\,X_{4}=b_{2},\,\, \alpha =\beta =0. \end{aligned}$$

Theorem 3.4

All homogeneous Gödel-type spacetimes in Class II, as explained by (2.1) with \(H'=-2bD\) and \(D''=0\) (\(b\ne 0\)), has Ricci bi-conformal vector field \(X=X_{1}\partial _{t}+X_{2}\partial _{r}+X_{3}\partial _{\phi }+X_{4}\partial _{z}\) if and only if \(\alpha =\beta =0\) and \(X_{1}=k_{1}, X_{2}=0, X_{3}=k_{2}\) and \(X_{4}=k_{3}\) for some constants \(k_{1}, k_{2}\) and \(k_{3}\).

Suppose that the vector fields as \(X=\nabla f\) for some smooth function f which are Ricci bi-conformal vector fields on homogeneous Gödel-type spacetimes in Class II. From (3.48) and Theorem 3.4, on homogeneous Gödel-type spacetimes in Class II, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} (r^{2}-b^{2}r^{4})\partial _{t}f+b^{2}r^{4}\partial _{\phi }f=k_{1}r^{2},\\ \partial _{r}f=0,\\ -br^{2}\partial _{t}f-\partial _{\phi }f=k_{2}r^{2},\\ \partial _{z}f=-k_{3}. \end{array}\right. } \end{aligned}$$
(3.67)

Deriving the third equation of the system (3.67) regarding r and the second equation of (3.49), we infer \(-b\partial _{t}f=k_{2}\). From the first and the third equations of (3.67) we conclude \(\partial _{t}f=\partial _{\phi }f=0\). Therefore \(f=-k_{3}z+k_{4}\) for some constant \(k_{4}\) and we have the following corollary:

Corollary 3.5

All homogeneous Gödel-type spacetimes in Class II, as explained by (2.1) with \(H'=-2bD\) and \(D''=0\) (\(b\ne 0\)), has Ricci bi-conformal vector field as \(X=\nabla f \) if and only if \(f=-k_{3}z+k_{4}\) for some constant \(k_{4}\).

Remark 3.6

From Theorem 3.4 we conclude that all Ricci bi-conformal vector fields on homogeneous Gödel-type spacetimes in Class II are Killing vector fields, Ricci collineation vector fields, and admit in almost Yamabe soliton equation with \(\Lambda =\tau \).

3.3 Class III

We consider \(a=-\mu ^{2}<0\), \(b\ne 0\). Thus, \(H(r)=\frac{2b}{\mu ^{2}}[\cos (\mu r)-1]\) and \( D(r)=\frac{1}{\mu }\sin (\mu r)\). Similar Class I we have \(2b^{2}-a\ne 0\) and

$$\begin{aligned} X_{1}=a_{4},\,\,X_{2}=0,\,\,\,X_{3}=a_{5},\,\,\,X_{4}=a_{2},\,\, \alpha =\beta =0. \end{aligned}$$

3.4 Class IV

If \(a\ne 0\) and \(b=0\). Then \(H=0\). In this case, we have

$$\begin{aligned} \frac{(H')^{2}}{2D^{2}}=0,\,\,\, K=0,\,\,\, L=-a,\,\,\,\,\,\, E=-aD^{2}. \end{aligned}$$

Then equations (3.11)-(3.20) become

$$\begin{aligned}{} & {} \partial _{t}X_{2}=\partial _{t}X_{3}=\partial _{z}X_{2}=\partial _{z}X_{3}=0,\end{aligned}$$
(3.68)
$$\begin{aligned}{} & {} \partial _{r}X_{2}=\frac{\alpha }{2},\end{aligned}$$
(3.69)
$$\begin{aligned}{} & {} D^{2}\partial _{r}X_{3}+\partial _{\phi }X_{2}=0,\end{aligned}$$
(3.70)
$$\begin{aligned}{} & {} \partial _{\phi }X_{3}=\frac{\alpha }{2}-\frac{D'}{D}X_{2}. \end{aligned}$$
(3.71)

Applying (3.68) in (3.1), (3.2), (3.3), (3.7), and (3.9), we arrive at

$$\begin{aligned} \partial _{t}X_{1}=\frac{\alpha }{2},\,\,\,\partial _{r}X_{1}=\partial _{\phi }X_{1}=\partial _{r}X_{4}=\partial _{\phi }X_{4}=0, \end{aligned}$$

respectively. From Eq. (3.10), we get

$$\begin{aligned} \partial _{z}X_{4}=\frac{\alpha }{2}. \end{aligned}$$
(3.72)

Therefore \(\alpha =2c_{1}\). From (3.4), we have

$$\begin{aligned} \partial _{z}X_{1}-\partial _{t}X_{4}=0. \end{aligned}$$

Differentiating the above equation by z and using (3.72), we obtain \(\partial _{zz}^{2}X_{1}=0\). Then

$$\begin{aligned} X_{1}=c_{1}t+c_{2}z+c_{3},\,\,\,X_{4}=c_{2}t+c_{1}z+c_{4}, \end{aligned}$$

for some constants \(c_{2}, c_{3}\) and \(c_{4}\). Since \(\partial _{t}X_{2}=\partial _{z}X_{2}=0\) and \(\partial _{r}X_{2}=c_{1}\), we can write \(X_{2}=c_{1}r+F(\phi )\) for some smooth function F. Deriving equations (3.70) and (3.71) regarding \(\phi \) and r, respectively, we obtain

$$\begin{aligned}{} & {} D^{2}\partial _{r\phi }^{2}X_{3}+\partial _{\phi \phi }^{2}X_{2}=0,\end{aligned}$$
(3.73)
$$\begin{aligned}{} & {} \partial _{r\phi }^{2}X_{3}=-(\frac{D'}{D})'X_{2}-\frac{D'}{D}c_{1}. \end{aligned}$$
(3.74)

Substituting (3.74) into (3.73), we get \(F''(\phi )-D^{2}(\frac{D'}{D})'(c_{1}r+F(\phi ))-D'Dc_{1}=0\) Thus \(F(\phi )=c_{1}=0\). This yields

$$\begin{aligned} X_{2}=0, \,\,\,X_{3}=c_{5}, X_{1}=c_{2}z+c_{3},\,\,\,X_{4}=c_{2}t+c_{4},\,\,\,\alpha =\beta =0. \end{aligned}$$

Theorem 3.7

All homogeneous Gödel-type spacetimes in Class IV, as explained by (2.1) with \(H=0\) and \(D''=aD\) (\(a\ne 0\)), has Ricci bi-conformal vector field \(X=X_{1}\partial _{t}+X_{2}\partial _{r}+X_{3}\partial _{\phi }+X_{4}\partial _{z}\) if and only if \(\alpha =\beta =0\) and \(X_{1}=k_{1}z+k_{2}, X_{2}=0, X_{3}=k_{3}\) and \(X_{4}=k_{1}t+k_{4}\) for some constants \(k_{1},\cdots , k_{5}\).

Now, we consider the vector fields as \(X=\nabla f\) for some smooth function f which are Ricci bi-conformal vector fields on homogeneous Gödel-type spacetimes in Class IV. From (3.48) and Theorem 3.7, on homogeneous Gödel-type spacetimes in Class IV, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}f=k_{1}z+k_{2},\\ \partial _{r}f=0,\\ \partial _{\phi }f=-k_{3}D^{2},\\ \partial _{z}f=-k_{1}t-k_{4}. \end{array}\right. } \end{aligned}$$
(3.75)

Deriving the third equation of the system (3.75) regarding r and the second equation of (3.75), we infer \(k_{3}=0\). Also, from the first and the fourth equations we conclude \(k_{1}=0\). Therefore \(f=-k_{4}t+k_{2}z+k_{5}\) for some constant \(k_{5}\) and we have the following corollary:

Corollary 3.8

All homogeneous Gödel-type spacetimes in Class IV, as explained by (2.1) with \(H=0\) and \(D''=aD\) (\(a\ne 0\)), has Ricci bi-conformal vector field as \(X=\nabla f \) if and only if \(f=-k_{4}t+k_{2}z+k_{5}\) for some constant \(k_{5}\).

Remark 3.9

From Theorem 3.1 we conclude that all Ricci bi-conformal vector fields on homogeneous Gödel-type spacetimes in Class IV are Killing vector fields, Ricci collineation vector fields, and admit in almost Yamabe soliton equation with \(\Lambda =\tau \).

Remark 3.10

As mentioned, in the case of Gödel-type homogeneous spacetimes the conformal factors \(\alpha \), \(\beta \) are both zero. According to Theorems 3.1, 3.4 and 3.7, it can be concluded that all the vector fields keep the leaves of the (singular) foliation \(C_R:= \{r = R\}\) invariant. These are all minimal cylinders (i.e. having zero mean curvature) and the splitting \(\langle \partial _r\rangle \oplus TC_R\) is orthogonal for both the Ricci and the metric.