On LRS Space-Times Admitting Conformal Motions

Abstract

In this paper, we study the conformal symmetry for locally rotationally symmetric Bianchi type I space-time. New exact conformal solutions of Einstein’s field equations for this space-time were obtained. The space-time geometry of these solutions is found to be non-vacuum, conformally flat, and shear-free. We show that in order for LRS Bianchi type I space-time to admit a conformal vector field it must reduce to the FRW space-time. Some physical and kinematic properties of the obtained conformal solutions are also discussed.

1. Introduction

Bianchi-type models are spatially homogeneous cosmological models that admit a group of motions $G_3$ [1]. Alternatively, these space-times often considered as a generalization of the Friedman–Robertson–Walker (FRW) model. Due to their ability to describe anisotropic universes, these space-times have been extensively studied, particularly in [2]. In general relativity (GR) and modified theories, some exact solutions to these space-times have been obtained, see for example [3,4,5,6,7]. Among the various Bianchi-type space-times, a locally rotationally symmetric (LRS) Bianchi type I space-time represents one of the simplest anisotropic cosmological models.

GR is considered an important pillar of modern physics and cosmology, it has wide applications in different branches. The basic equations of this theory are Einstein’s field equations (EFEs), $G_{ab}=-\kappa T_{ab}$, which describe the relationship between matter and energy (represented by the energy–momentum tensor $T_{ab}$) and the geometry of space-time (represented by the Einstein tensor $G_{ab}$). EFEs are non-linear differential equations of quadratic degree depending on some variables. Due to the high degree of non-linearity, obtaining exact solutions to these equations without imposing certain constraints, either on the geometry of the space-time or on the energy-momentum tensor, is difficult. These constraints assume some symmetric properties on the space-time such as spherical symmetric [8], cylindrical symmetric [9], and axial symmetric [10]. In addition to these symmetric properties, some symmetries like isometry, homothetic, conformal, Ricci collineations, etc., are assumed to obtain exact solutions of EFEs. These symmetries reduce the variables derived from the basic equation, and the partial differential equations are then reduced to ordinary differential equations with fewer variables, which, in some special cases, can be solved more easily. All the known solutions in [1] have been derived by assuming symmetry restrictions on the geometric and physical properties of space-time.

The most important symmetries in GR are those of the metric tensor $g_{ab}$, namely Killing, homothetic, and conformal symmetries. Killing symmetry is a special case of homothetic symmetry, which, in turn, generalizes to conformal symmetry. The essence of conformal symmetry lies in the fact that, unlike Killing and homothetic symmetries, it does not preserve the Einstein tensor. In this sense, conformal symmetry is often considered non-natural or accidental. Conformal symmetry preserves the angle between two vectors and scales the metric tensor by a conformal factor.

In GR, conformal symmetry is defined in terms of the conformal Killing vector field $\mathbf{\zeta }={\zeta }^a(t, x, y, z){\partial }_a$. Let $(M, g)$ be a space-time, where M is a smooth four-dimensional manifold, and g is a Lorentzian metric with signature $(+---)$. A vector field $\mathbf{\zeta }$ on space-time $(M, g)$ is called a conformal Killing vector field if it satisfies the following equation [11]:

$${\mathcal{L}}_{\zeta }{g}_{ab}={\zeta }_{a;b}+{\zeta }_{b;a}=2\psi (t, x, y, z)g_{ab},$$

(1)

where ${\mathcal{L}}_{\zeta }$ represents the Lie derivative operator relative to $\mathbf{\zeta }$, the semi-colon denotes a covariant derivative with respect to metric connection, and $\psi$ is a smooth real-valued function on M, called the conformal factor. If the conformal factor is a non-zero constant, $\psi =constant\ne 0$, the vector field $\mathbf{\zeta }$ becomes a homothetic vector field. In the case where $\psi =0$, the vector field $\mathbf{\zeta }$ is referred to as a Killing vector field. Equation (1) is known as the conformal Killing equation, and in component form, it takes the following expression:

$$g_{ab,c}{\zeta }^c+g_{ac}{\zeta }_{,b}^c+g_{cb}{\zeta }_{,a}^c=2\psi (t,x,y,z)g_{ab}.$$

(2)

The different types of space-time symmetries have received great attention in the literature on GR. In references [12,13], several viable solutions to the EFEs were obtained by assuming that the space-time admits conformal and homothetic vector fields, respectively. Some articles focused on the possibility of the existence of these vector fields [14,15,16], and others classified space-times according to them [17,18,19,20]. The solution of a spherically symmetric perfect fluid space-time admitting a conformal Killing vector field has been studied by assuming this vector is orthogonal to the four-velocity vector $u^a$ [21]. Spherically symmetric perfect fluid space-times admitting a conformal Killing vector field were discussed by Gad [22]. The general form of this vector field for such space-times was obtained by Moopanar and Maharaj [23]. Herrera et al. [24] studied non-static spherically symmetric fluids admitting a conformal symmetry, and they found several families of exact analytical solutions. The effects of the conformal symmetry in addition to GR can also be seen in astrophysics and cosmology. A new model of a gravastar has been proposed within the framework of the Mazur–Mottola model, which offers a possible substitute for the black hole, assuming that the gravastar admits conformal motions [25]. Under the assumption of spherical symmetry and static wormhole space-time admits a conformal symmetry, Böhmer et al. obtained a new exact solution of traversable wormholes in [26], and then they presented analyses of these solutions such that they obtained a direct relation between the conformal factor and an observable quantity [27].

Motivated by the above discussion, we study the conformal symmetry of the LRS Bianchi type I space-time in order to solve EFEs without making assumptions, either on variables or on physical properties. We will only assume that a space-time under study admits a conformal motion. The paper is organized as follows: in the next section, we give a brief overview of the space-time under study. In Section 3, we present the system of equations governing the conformal symmetry in LRS Bianchi type I space-time, which consists of coupled first-order partial differential equations. We explicitly solve this system to obtain the conformal Killing vector field $\zeta$ and the conformal factor $\psi$, as well as the relations between the coefficients of the metric. Section 4 specifies the exact conformal solutions for the EFEs when the material is represented by a perfect fluid. The kinematic quantities of the obtained conformal solutions are defined in Section 5. Finally, we conclude with remarks and discussion.

2. Version of Model

We consider the line element of LRS Bianchi type I space-time in Cartesian coordinates in the following form [1]:

$$d{s}^2=d{t}^2-A^{2}d{x}^2-B^2\left(d{y}^2+d{z}^2\right),$$

(3)

with the convention $x^0=t$ (cosmic time), $x^1=x$, $x^2=y$, and $x^3=z$, and the scale factors $A\left(t\right)$ and $B\left(t\right)$ are functions of t only.

The above line element (3) admits at least four linearly independent KV, which can be expressed as [28]

$${\eta }_{\left(1\right)}={\partial }_x, {\eta }_{\left(2\right)}={\partial }_y, {\eta }_{\left(3\right)}={\partial }_z, {\eta }_{\left(4\right)}=y{\partial }_z-z{\partial }_y.$$

As indicated in [29,30], the physical and kinematic quantities of the space-time under study are determined as follows:

• The contravariant and covariant components of the four-velocity vector field can be defined by $u^a=u_a=(1, 0, 0, 0)$, and they are verified as $g_{ab}{u}^a{u}^b=1$.
• The volume and the average scale factor are given by

  $$V={\tau }^3=\left(A{B}^2\right).$$
• The acceleration vector field ${\dot{u}}_a$ is

  $${\dot{u}}_a=u_{a;b}{u}^b=0.$$
• The rotation tensor ${\omega }_{ab}$ is defined as

  $${\omega }_{ab}=u_{[a;b]}+\dot{u}{[}_a{u}_b]=0.$$
• The expansion rate $\mathrm{\Theta }$ and the Hubble parameter H are given by

  $$\mathrm{\Theta }=u_{;a}^a={\displaystyle \frac{\dot{A}}{A}}+2{\displaystyle \frac{\dot{B}}{B}},$$

  and

  $$H={\displaystyle \frac{\mathrm{\Theta }}{3}}={\displaystyle \frac{1}{3}}\left({\displaystyle \frac{\dot{A}}{A}}+2{\displaystyle \frac{\dot{B}}{B}}\right),$$

  where $H_1={\displaystyle \frac{\dot{A}}{A}}\mathrm{and}{H}_2=H_3={\displaystyle \frac{\dot{B}}{B}}$ are the directional Hubble parameters, which measure the rate of expansion in the directions of x, y, and z, respectively.
• The average anisotropy parameter $\delta$, and the deceleration parameter q are

  $$\begin{array}{ccc}\hfill \delta & =& {\displaystyle \frac{1}{3}}\sum _{i=1}^3\left({\displaystyle \frac{H_i^2}{H^2}}-{\displaystyle \frac{2H_i}{H}}+1\right),\hfill \\ & =& {\displaystyle \frac{3}{{\mathrm{\Theta }}^2}}\left({\displaystyle \frac{{\dot{A}}^2}{A^2}}+2{\displaystyle \frac{{\dot{B}}^2}{B^2}}\right)-1,\hfill \end{array}$$

  and

  $$\begin{array}{cc}\hfill q& =3{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{1}{\mathrm{\Theta }}}\right)-1,\hfill \\ \hfill & =-1-3{\displaystyle \frac{B^2\left(\ddot{A}A-{\dot{A}}^2\right)+2A^2\left(\ddot{B}B-{\dot{B}}^2\right)}{{\left(\dot{A}B+2\dot{B}A\right)}^2}}.\hfill \end{array}$$
• The shear scalar $\sigma$ can also be defined as follows:

  $${\sigma }^2={\displaystyle \frac{1}{2}}{\sigma }_{ab}{\sigma }^{ab},$$

  where

  $${\sigma }_{ab}=u_{(a;b)}+\dot{u}{(}_a{u}_b)-{\displaystyle \frac{1}{3}}\mathrm{\Theta }(g_{ab}+u_a{u}_b).$$
  The non-zero components of the shear tensor are given by

  $${\sigma }_{11}=-{\displaystyle \frac{2A^2}{3}}\left[{\displaystyle \frac{\dot{A}}{A}}-{\displaystyle \frac{\dot{B}}{B}}\right].$$

  $${\sigma }_{22}={\sigma }_{33}=-{\displaystyle \frac{B^2}{3}}\left[{\displaystyle \frac{\dot{B}}{B}}-{\displaystyle \frac{\dot{A}}{A}}\right].$$

Here, the dot denotes partial differentiation with respect to time: $\dot{}={\displaystyle \frac{\partial }{\partial t}}$.

3. Conformal Killing Equations and Their Solution

The study of the conformal Killing vector field, $\mathbf{\zeta }={\left({\zeta }^a(t, x, y, z){\partial }_a\right)}_{a=0}^3$, on the LRS Bianchi type I given in (3) is based on the examination of the ten equations obtained from the conformal Equation (1). For the space-time under study (3), we obtain the following set of conformal equations:

$$\begin{array}{ccc}\hfill {\zeta }_{,0}^0& =& \psi ,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill {\zeta }_{,1}^0-A^2{\zeta }_{,0}^1& =& 0,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill {\zeta }_{,2}^0-B^2{\zeta }_{,0}^2& =& 0,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill {\zeta }_3^0-B^2{\zeta }_0^3& =& 0,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\zeta }_{,1}^1+{\displaystyle \frac{\dot{A}}{A}}{\zeta }^0& =& \psi ,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill B^2{\zeta }_{,1}^2+A^2{\zeta }_{,2}^1& =& 0,\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill B^2{\zeta }_{,1}^3+A^2{\zeta }_{,3}^1& =& 0,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill {\zeta }_{,2}^2+{\displaystyle \frac{\dot{B}}{B}}{\zeta }^0& =& \psi ,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill {\zeta }_{,2}^3+{\zeta }_{,3}^2& =& 0,\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill {\zeta }_{,3}^3+{\displaystyle \frac{\dot{B}}{B}}{\zeta }^0& =& \psi .\hfill \end{array}$$

(13)

The commas in the subscript represent the partial derivatives with respect to the space-time coordinates. For $\psi =constant$, the solution of the above system gives the homothetic vector field admitted by LRS Bianchi type I space-time, which has been previously investigated by [19]. Here, we explore a conformal Killing vector field of the space-time under study (3) by solving the previous system (4)–(13) for $\psi =\psi \left(t\right)$.

We integrate Equation (4) with respect to t as

$${\zeta }^0=\int \psi \left(t\right)dt+F^0(x, y, z),$$

(14)

where $F^0(x, y, z)$ is a function of integration.

Differentiating Equations (5)–(7) with respect to t and using the above result, after some calculations, we obtain

$$\begin{array}{cc}\hfill {\zeta }^1=& F^1(x, y, z),\hfill \\ \hfill {\zeta }^2=& F^2(x, y, z),\hfill \\ \hfill {\zeta }^3=& F^3(x, y, z).\hfill \end{array}$$

(15)

$$F^0(x, y, z)=constant=c_0,$$

(16)

where $F^i(x, y, z)$, $i=1, 2, 3$, are arbitrary functions that need to be determined. Differentiating Equations (8), (11), and (13), with respect to t and using (14), we have

$${\displaystyle \frac{\dot{A}}{A}}={\displaystyle \frac{\psi \left(t\right)+a}{\int \psi \left(t\right)dt+c_0}},$$

(17)

$${\displaystyle \frac{\dot{B}}{B}}={\displaystyle \frac{\psi \left(t\right)+b}{\int \psi \left(t\right)dt+c_0}},$$

(18)

where $a, b$ are constants of integration.

From Equations (9) and (10), we find that ${\zeta }^2$ and ${\zeta }^3$ do not depend on x. Then, ${\zeta }^1$ does not depend on y and z. Inserting the previous results back into (8), we get

$${\zeta }^1=-ax+c_1,$$

(19)

where $c_1$ is a constant of integration.

Substituting the above results back into Equations (11) and (13) and differentiating Equation (12) with respect to z, we obtain, after some calculations,

$${\zeta }^2=-by+c_{2}z+c_3,$$

(20)

$${\zeta }^3=-bz-c_{2}y+c_4,$$

(21)

where $c_j$, $j=2, 3, 4$ are constants of integration. Without loss of generality, we assume that $c_0=0$. Therefore, the contravariant components of the conformal Killing vector field, which satisfied the system (4)–(13), are

$$\begin{array}{cc}\hfill {\zeta }^0& =\int \psi (t)dt,\hfill \\ \hfill {\zeta }^1=& -ax+c_1,\hfill \\ \hfill {\zeta }^2=& -by+c_{2}z+c_3,\hfill \\ \hfill {\zeta }^3=& -bz-c_{2}y+c_4,\hfill \end{array}$$

(22)

with constraint relations

$$\begin{array}{cc}\hfill {\displaystyle \frac{\dot{A}}{A}}=& {\displaystyle \frac{\psi \left(t\right)+a}{\int \psi \left(t\right)dt}},\hfill \\ \hfill {\displaystyle \frac{\dot{B}}{B}}=& {\displaystyle \frac{\psi \left(t\right)+b}{\int \psi \left(t\right)dt}}.\hfill \end{array}$$

(23)

CaseI: If we put $a=b$ into Equation (23), we have the following:

$${\displaystyle \frac{\dot{A}}{A}}={\displaystyle \frac{\dot{B}}{B}}.$$

Integrating this result, we get

$$A\left(t\right)=c_{5}B\left(t\right),$$

(24)

where $c_5$ is a constant of integration, and it can be equal to one without any loss of generality. Substituting the above result back into (23) and using (22), we have (for simplification purposes, a has been set to zero)

$${\zeta }^0={\displaystyle \frac{A\psi \left(t\right)}{\dot{A}}}={\displaystyle \frac{B\psi \left(t\right)}{\dot{B}}}.$$

(25)

Using the above result in Equation (4), we get

$${\displaystyle \frac{\dot{\psi }\left(t\right)}{\psi \left(t\right)}}={\displaystyle \frac{\ddot{A}}{\dot{A}}}.$$

Integrating the above equation, the conformal factor takes the following form (with the constant of integration taken to be unity):

$$\psi \left(t\right)=\dot{A}\left(t\right).$$

(26)

In the light of the above discussion, the following theorem can be established.

Theorem 1.

A LRS Bianchi type I space-time described by the metric (3) admits a conformal Killing vector if the following conditions are satisfied:

$$A\left(t\right)=B\left(t\right),$$

$$\psi \left(t\right)=\dot{A}\left(t\right).$$

In this instance, the associated conformal Killing vector takes the following form:

$$\mathbf{\zeta }=A\left(t\right){\partial }_t+\left(c_1\right){\partial }_x+(c_{2}z+c_3){\partial }_y+(-c_{2}y+c_4){\partial }_z.$$

(27)

The covariant components of the conformal Killing vector, ${\zeta }_a=g_{ab}{\zeta }^b$, are

$$\begin{array}{cc}\hfill {\zeta }_0=& A\left(t\right),\hfill \\ \hfill {\zeta }_1=& -\left(c_1\right)A^2\left(t\right),\hfill \\ \hfill {\zeta }_2=& -(c_{2}z+c_3)A^2\left(t\right),\hfill \\ \hfill {\zeta }_3=& -(-c_{2}y+c_4)A^2\left(t\right).\hfill \end{array}$$

(28)

From Equations (27) and (28), we see that the obtained vector is a non-null conformal Killing vector (i.e., ${\zeta }_a{\zeta }^a\ne 0$).

CaseII:

It should be noted here that in the case of unequal constants $a\ne b\ne 0$ in Equation (23), we obtain the following:

Integrating Equation (23) and taking into account $a\ne b\ne 0$, we have

$$A\left(t\right)=c_{6}g\left(t\right)e^{a\int \frac{1}{g\left(t\right)}dt},$$

(29)

$$B\left(t\right)=c_{7}g\left(t\right)e^{b\int \frac{1}{g\left(t\right)}dt}.$$

(30)

The conformal Killing vector in this case takes the following form:

$$\mathbf{\zeta }=g\left(t\right){\partial }_t+(-ax+c_1){\partial }_x+(-by+c_{2}z+c_3){\partial }_y+(-bz+-c_{2}y+c_4){\partial }_z,$$

(31)

where $c_6, c_7$ are constants of integration, and $g\left(t\right)=\int \psi \left(t\right)dt={\zeta }^0$.

Our aim in the next section is to solve the EFEs separately for Case I and Case II. For Case I, we use the constraint relations obtained from Theorem 1, while for Case II, we use those given in Equations (29) and (30).

4. Perfect Fluid Solutions of Einstein’s Field Equations

In this section, we solve EFEs by only assuming that the space-time under study admits a conformal motion. We assume that the cosmic matter is a perfect fluid that is represented by the following energy–momentum tensor:

$$T_{ab}=(\rho +p)u_a{u}_b-p{g}_{ab},$$

(32)

where the energy density $\rho$, the pressure p, and $u_a$ is a four-velocity vector, as defined before.

EFEs relate the curvature of space-time and the energy–momentum tensor of the matter in space-time by the following system of partial differential equations [31]:

$$R_{ab}-{\displaystyle \frac{1}{2}}{g}_{ab}R=-\kappa T_{ab},$$

(33)

where $R_{ab}$ is the Ricci tensor, R the Ricci scalar, and $\kappa ={\displaystyle \frac{8\pi G}{c^4}}$ is the coupling constant, where G is a Newtons gravitational constant and c the speed of light (for convenience, we assumed that natural units $8\pi G=c=1$).

The EFEs (33) with (32) for the space-time (3) are equivalent to the following system of equations:

$$2{\displaystyle \frac{\dot{A}}{A}}{\displaystyle \frac{\dot{B}}{B}}+{\left({\displaystyle \frac{\dot{B}}{B}}\right)}^2=\rho ,$$

(34)

$$2{\displaystyle \frac{\ddot{B}}{B}}+{\left({\displaystyle \frac{\dot{B}}{B}}\right)}^2=-p,$$

(35)

$${\displaystyle \frac{\ddot{A}}{A}}+{\displaystyle \frac{\ddot{B}}{B}}+{\displaystyle \frac{\dot{A}}{A}}{\displaystyle \frac{\dot{B}}{B}}=-p.$$

(36)

ForcaseI:

Inserting the relation between the scale factors given in Theorem 1, $A=B$, back into the EFEs (34)–(36), we obtained the dynamical variables as follows:

$$\rho =3{\left({\displaystyle \frac{\dot{A}}{A}}\right)}^2,$$

(37)

$$p=-2{\displaystyle \frac{\ddot{A}}{A}}-{\left({\displaystyle \frac{\dot{A}}{A}}\right)}^2.$$

(38)

From the preceding results, we can clearly observe a relationship between pressure and density:

$$p=-2{\displaystyle \frac{\ddot{A}}{A}}-{\displaystyle \frac{\rho }{3}}.$$

(39)

If $\ddot{A}=0$, we have $p=-{\displaystyle \frac{1}{3}}\rho$, and the conformal Killing vector given in (27) reduces to a homothetic vector, i.e., $\dot{A}=constant=\psi$. Therefore, let $\ddot{A}\ne 0$ to avoid this result.

All perfect fluids relevant to cosmology obey a linear equation of state that relates the pressure p to the density $\rho$, and this form has been extensively used in various cosmological models [32]:

$$p=W\rho ,$$

(40)

where W is a parameter: if $W=0$, this describes the dust case; $W={\displaystyle \frac{1}{3}}$ describes the radiation-dominated case, and $W=1$ describes the stiff fluid case.

Dustcase $W=0$:

Putting $p=0$ in Equation (38), we get

$$2\ddot{A}A+{\dot{A}}^2=0.$$

Solving this equation, we obtain the scale factors and the conformal factor as follows

$$A={\left({\displaystyle \frac{3}{2}}(k_{0}t+k_1)\right)}^{\frac{2}{3}}=B,$$

$$\psi \left(t\right)=k_0\sqrt[3]{{\displaystyle \frac{2}{3(k_{0}t+k_1)}}}=\dot{A}.$$

The conformal Killing vector (27), which is admitted by the LRS Bianchi type I space-time (3) is

$$\mathbf{\zeta }={\left({\displaystyle \frac{3}{2}}(k_{0}t+k_1)\right)}^{\frac{2}{3}}{\partial }_t+\left(c_1\right){\partial }_x+(c_{2}z+c_3){\partial }_y+(-c_{2}y+c_4){\partial }_z.$$

(41)

Consequently, the exact conformal solution of the EFEs for the LRS Bianchi type I space-time (3), is obtained as follows:

$$d{s}^2=d{t}^2-{\left({\displaystyle \frac{3}{2}}(k_{0}t+k_1)\right)}^{\frac{4}{3}}\left(d{x}^2+d{y}^2+d{z}^2\right).$$

(42)

The pressure and density are

$$\begin{array}{cc}\hfill p& =0,\hfill \\ \hfill \rho & ={\displaystyle \frac{4k_0^2}{3{(k_{0}t+k_1)}^2}},\hfill \end{array}$$

where $k_0, k_1$ are the integration constants.

The radiation-dominated case $W=\frac{1}{3}$:

Inserting $\rho =3p$ in Equation (39) and using (38), we get

$$\ddot{A}A+{\dot{A}}^2=0.$$

We have in this case the following:

$$A=\sqrt{2(k_{2}t+k_3)},$$

$$\psi \left(t\right)={\displaystyle \frac{k_2}{\sqrt{2(k_{2}t+k_3)}}},$$

$$\mathbf{\zeta }=\left(\sqrt{2(k_{2}t+k_3)}\right){\partial }_t+\left(c_1\right){\partial }_x+(c_{2}z+c_3){\partial }_y+(-c_{2}y+c_4){\partial }_z.$$

(43)

For this case, the exact conformal solution of the EFEs for the space-time under study (3) is obtained as follows:

$$d{s}^2=d{t}^2-2(k_{2}t+k_3)\left(d{x}^2+d{y}^2+d{z}^2\right),$$

(44)

with the following dynamical variables,

$$\rho =3p={\displaystyle \frac{3k_2^2}{4{(k_{2}t+k_3)}^2}},$$

where $k_2, k_3$ are the constants of integration.

The stiff fluid case $W=1$:

By inserting $\rho =p$ into Equation (39) and using (38), we obtain

$$\ddot{A}A+2\dot{A^2}=0.$$

In this case, we get

$$A=\sqrt[3]{3(k_{4}t+k_5)},$$

$$\psi \left(t\right)={\displaystyle \frac{k_4}{{\left(\sqrt[3]{3(k_{4}t+k_5)}\right)}^2}},$$

$$\mathbf{\zeta }=\left(\sqrt[3]{3(k_{4}t+k_5)}\right){\partial }_t+\left(c_1\right){\partial }_x+(c_{2}z+c_3){\partial }_y+(-c_{2}y+c_4){\partial }_z.$$

(45)

For this case, the exact conformal solution of the EFEs and the corresponding dynamical variables for the space-time (3) are given as follows:

$$d{s}^2=d{t}^2-{\left(\sqrt[3]{3(k_{4}t+k_5)}\right)}^2\left(d{x}^2+d{y}^2+d{z}^2\right),$$

(46)

$$\rho =p={\displaystyle \frac{k_4^2}{3{(k_{4}t+k_5)}^2}},$$

where $k_4, k_5$ are the integration constants. From Theorem 1, we have $\psi \left(t\right)=\dot{A}$. It follows that $k_i\ne 0$, where $i=0, 2, 4$, since if these constants were zero, then $A=constant$, and hence, $\psi =0$. We emphasize that this condition is essential in order for the conformal Killing vector in Equations (41), (43) and (45) not to reduce to a Killing vector.

ForcaseII:

From Equations (35) and (36), we get

$${\displaystyle \frac{\ddot{A}}{A}}-{\displaystyle \frac{\ddot{B}}{B}}+{\displaystyle \frac{\dot{A}}{A}}{\displaystyle \frac{\dot{B}}{B}}-{\left({\displaystyle \frac{\dot{B}}{B}}\right)}^2=0.$$

(47)

By inserting Equations (29) and (30) back into the above equation, we get

$$\left(2a-2b\right)\dot{g}\left(t\right)+a^2-2b^2+ab=0.$$

By simplifying this expression, we obtain that $\psi$ equals a constant as follows:

$$\dot{g}\left(t\right)=-\left({\displaystyle \frac{a+2b}{2}}\right)=\psi .$$

(48)

Thus, the conformal Killing vector field in Equation (31) reduces to a homothetic vector field, $\mathbf{H}$, given by

$$\mathbf{H}=\left(\psi t\right){\partial }_t+(2(b+\psi )x+c_1){\partial }_x+(-by+c_{2}z+c_3){\partial }_y+(-bz-c_{2}y+c_4){\partial }_z.$$

(49)

According to the previous discussion, in order for LRS Bianchi type I space-time (3) to admit conformal motion, it must be the well-known FRW space-time, which takes the usual form

$$d{s}^2=d{t}^2-A^2\left(t\right)\left[d{x}^2+d{y}^2+d{z}^2\right].$$

(50)

This result supports the Theorem 1 we have reached in the previous section.

5. Kinematic Variables

We calculate the physical and kinematic quantities of the obtained conformal solutions given by the metrics (42), (44) and (46) to study their physical behavior. These quantities are crucial for addressing cosmology.

In the following, we obtain the physical and geometric parameters for the solution (42).

$$V=A{\left(t\right)}^3={\displaystyle \frac{9}{4}}{(k_{0}t+k_1)}^2,$$

$$H={\displaystyle \frac{\mathrm{\Theta }}{3}}={\displaystyle \frac{\dot{A}}{A}}={\displaystyle \frac{2k_0}{3(k_{0}t+k_1)}},$$

$$\delta ={\displaystyle \frac{1}{3H^2}}\left({\displaystyle \frac{3{\dot{A}}^2}{A^2}}\right)-1=0,$$

$$q=-{\displaystyle \frac{A\left(t\right)\ddot{A}\left(t\right)}{\dot{A}{\left(t\right)}^2}}={\displaystyle \frac{1}{2}},$$

From Theorem 1, $A=B$, all components of the shear tensor given in Section 2 become zero, i.e.,

$${\sigma }_{11}={\sigma }_{22}={\sigma }_{33}=0.$$

This yields

$$\sigma =0,$$

Thus, the space-times (42), (44), and (46) are shear free.

• In view of solution (44), we obtain the above non-zero variables as

  $$V=(2{(k_{2}t+k_3)}^{\frac{3}{2}},$$

  $$\mathrm{\Theta }={\displaystyle \frac{3k_2}{2(k_{2}t+k_3)}},$$

  $$H={\displaystyle \frac{k_2}{2(k_{2}t+k_3)}},$$

  $$q=1.$$
• For the obtained solution (46), we get

  $$V=3(k_{4}t+k_5),$$

  $$\mathrm{\Theta }={\displaystyle \frac{k_4}{k_{4}t+k_5}},$$

  $$H={\displaystyle \frac{k_4}{3(k_{4}t+k_5)}},$$

  $$q=2.$$

6. Conclusions

This work is developed in the framework of GR based on Riemannian geometry to study the conformal symmetry of LRS Bianchi I model. It is worth noting that several previous studies of this model have investigated homothetic motion, but within the framework of Lyra geometry [33,34,35]. These works derived the homothetic vector fields that are admissible under the structure of Lyra manifolds and compared the resulting solutions with those obtained in the context of GR based on Riemannian geometry. This work aimed to solve EFEs by assuming an LRS Bianchi type I space-time admitting conformal motions. We solved the conformal equations and obtained the two CKVs, given in (27) and (31), which are admitted by this space-time. This helped us to obtain relationships between the scale factors and then use these relationships in the EFEs to solve them separately for Case I and Case II. For Case I, we used the linear equation of state, and we obtained new exact conformal solutions of the EFEs for LRS Bianchi type I space-time. Moreover, we found the dynamical variables, pressure (p) and energy density ($\rho$), which depend on the cosmic time t. For all three cases, namely dust, radiation, and stiff fluid, the metric coefficients are well defined, except at $t=0$ and $k_i=0, i=1, 3, 5$, where the dynamic variables tend to infinity; so, there exists a singularity at this point. Additionally, at $t=0$ with $k_i,i=1, 3, 5$, which is non-zero, the dynamics variables will ba a constant. Moreover, as cosmic time increase, when $t\to \infty$, the dynamical variables vanish. For Case II, we showed that the CKV in Equation (31) reduces to an HV. Therefore, we obtain that in order for LRS Bianchi type I space-time (3) to admit conformal motion, it must be the well-known FLRW space-time. In the following

Table 1. The classification exact solutions of Einstein’s field equations for the locally rotationally symmetric Bianchi type I space-time (3) based on the conformal and homothetic vector fields.

Classification	Vector Field	Scale Factors	Line Element	Physical Properties
Conformal solution	CKV (41)	$A=B={\left({\displaystyle \frac{3}{2}}(k_{0}t+k_1)\right)}^{{\displaystyle \frac{2}{3}}}$	Equation (42), with $\psi \left(t\right)=k_0{\left({\displaystyle \frac{2}{3(k_{0}t+k_1)}}\right)}^{{\displaystyle \frac{1}{3}}}$	$\begin{array}{cc}\hfill \rho & ={\displaystyle \frac{4k_0^2}{3{(k_{0}t+k_1)}^2}},\hfill \\ \hfill p& =0\hfill \end{array}$
Conformal solution	CKV (43)	$A=B=\sqrt{2(k_{2}t+k_3)}$	Equation (44), with $\psi \left(t\right)={\displaystyle \frac{k_2}{\sqrt{2(k_{2}t+k_3)}}}$	$\begin{array}{cc}\hfill \rho & =3p\hfill \\ & ={\displaystyle \frac{3k_2^2}{4{(k_{2}t+k_3)}^2}}\hfill \end{array}$
Conformal solution	CKV (45)	$A=B={\left(3(k_{4}t+k_5)\right)}^{{\displaystyle \frac{1}{3}}}$	Equation (46), with $\psi \left(t\right)={\displaystyle \frac{k_4}{{\left(3(k_{4}t+k_5)\right)}^{\frac{2}{3}}}}$	$\begin{array}{cc}\hfill \rho & =p={\displaystyle \frac{k_4^2}{3{(k_{4}t+k_5)}^2}}\hfill \end{array}$
Self-similar solution	HV (49)	$\begin{array}{cc}\hfill A& =c_6{\left(\psi t\right)}^{{\displaystyle \frac{-\psi -2b}{\psi }}},\hfill \\ \hfill B& =c_7{\left(\psi t\right)}^{{\displaystyle \frac{\psi +b}{\psi }}}\hfill \end{array}$	$\begin{array}{cc}\hfill d{s}^2& =d{t}^2-{\left(c_6{\left(\psi t\right)}^{{\displaystyle \frac{-\psi -2b}{\psi }}}\right)}^{2}d{x}^2\hfill \\ & -{\left(c_7{\left(\psi t\right)}^{{\displaystyle \frac{\psi +b}{\psi }}}\right)}^2(d{y}^2+d{z}^2),\hfill \\ \hfill \mathrm{with}\psi & =-{\displaystyle \frac{a+2b}{2}}\hfill \end{array}$	$\begin{array}{cc}\hfill \rho & =p={\displaystyle \frac{3b^2+4b\psi +{\psi }^2}{{\left(\psi t\right)}^2}}\hfill \end{array}$
Self-similar solution	$\begin{array}{cc}\hfill \mathrm{HV},\mathbf{H}& =(\psi t+c_8){\partial }_t+c_1{\partial }_x\hfill \\ & +(c_{2}z+c_3){\partial }_y+(-c_{2}y+c_4){\partial }_z\hfill \end{array}$	$A=B=\psi t+c_8$	$\begin{array}{cc}\hfill d{s}^2& =d{t}^2-{(\psi t+c_8)}^2(d{x}^2+d{y}^2+d{z}^2),\hfill \\ \hfill \mathrm{with}\psi ,c_8& \mathrm{as}\mathrm{constants}\hfill \end{array}$	$\begin{array}{cc}\hfill \rho & =-3p={\displaystyle \frac{-3{\psi }^2}{{(\psi t+c_8)}^2}}\hfill \end{array}$

, we present the exact conformal and self-similar solutions of the EFEs for the space-time (3) with their physical properties.

Finally, we discussed the kinematic quantities for the conformal solutions obtained in Equations (42), (44), and (46). We have the following: the solutions are shear-free and expanding, because the spatial volume V is increasing as cosmic time t increases: it becomes constant if $t=0$, and it becomes infinite when $t\to \infty$. However, both the Hubble parameter $H\left(t\right)$ and the scalar expansion $\mathrm{\Theta }\left(t\right)$ exhibit an inverse relationship with cosmic time t, showing that the rate of expansion decreases over time.