1 Introduction

The general theory of relativity, which is still in use today, was put forth by Einstein at the start of the 20th century. This theory offers a comprehensive explanation of gravity in four dimensions of a spacetime. Spacetime is curved as a result of gravitational fields, and this curvature is intimately related to energy and momentum. The bending of light around big objects is one of general relativity’s most well-known predictions. This was demonstrated in a 1919 experiment in which it was discovered that the positions of stars detected during a solar eclipse had changed as a result of the stars’ light being bent as it passed close to the sun.

The presence of black holes, which are areas of space where the curvature of spacetime becomes endlessly strong, is another general relativity prediction. Since then, a variety of techniques, such as gravitational lensing and the detection of gravitational waves, have been used to investigate and study black holes. The growth of cosmology, the study of the cosmos as a whole, has been significantly influenced by the general relativity theory. Scientists have created a thorough picture of how the universe changed over time by applying the theory to the large-scale structure of the universe.

The characteristics of spacetime and its curvature in the presence of matter and energy are describe by he ten nonlinear partial differential equations known as the Einstein field equations (EFEs). Albert Einstein initially put forth the equations in his general relativity theory, which completely changed how we think about gravity and how it affects the universe. The field equations are difficult, but they have been precisely solved in a number of circumstances, yielding significant insights into the nature of gravity and the behavior of spacetime. The mathematical form of these equations are given as Stephani et al. (2003):

$$\begin{aligned} G_{ab} = R_{ab}-\frac{R}{2}g_{ab} = k T_{ab}. \end{aligned}$$
(1.1)

The energy-momentum tensor is denoted by \(T_{ab}\) which characterizes the energy-density, pressure, and momentum of a spacetime, and k is the proportionality constant between geometry and physics of a spacetime. The terms \(g_{ab}\), \(R_{ab}\), and R, respectively, stand for the metric tensor, Ricci tensor, and Ricci scalar.

The exact solution of the EFEs that can be achieved by resolving these equations is represented by a Lorentz metric, \(g_{ab}\). It has been established that one of the most significant activities in various disciplines of mathematical physics is to find out the exact solutions to these equations. Additionally, these answers enable the prediction of black holes and other universe evolution theories. The difficulty in locating these solutions stems from the equations’ extremely nonlinear structure, which makes it impossible to resolve them without making some simplifying assumptions, like symmetry constraints on various tensors. There are some situations where the EFEs are entirely resolved with the aid of these limits (Hall 2004). In general, the precise answers to the EFEs are crucial tools for comprehending how gravity behaves and how spacetime behaves in a variety of contexts, from black holes to the large-scale structure of the universe. They were found and developed, which is a significant accomplishment in the history of physics, and they still serve as an inspiration for new developments.

EFEs require symmetries to be solved and simplified. We can simplify the equations by making certain symmetries assumptions, leading to solutions that depict plausible physical circumstances. Spherical symmetry entails that the gravitational field is isotropic, i.e. it appears the same from every angle surrounding a center mass. This symmetry has led to solutions like the Schwarzschild metric (Heinicke and Hehl 2015). The Schwarzschild solution, which represents the geometry of spacetime around a non-rotating, spherically symmetric object like a star or black hole, is one of the most well-known exact solutions to the EFEs. The solution states that as an object approaches, spacetime is deformed in a way that causes time to slow down and light to bend, resulting in phenomena like gravitational lensing and time dilation.

Another ideal symmetry is axial symmetry, like a rotating object, where the field is symmetric along a specific axis of rotation. This symmetry has led to solutions like the Kerr metric which represents the geometry of spacetime around a revolving black hole. According to the solution, the black hole’s spinning produces the ergosphere, an area of spacetime where things can be pulled together with the rotation of the black hole. Additionally, it implies that the black hole’s center will have a ring-shaped singularity where the fundamental principles of physics would no longer apply.

Homogeneity and isotropy, these symmetries presuppose that the field has a uniform appearance at all points and angles in space. This has led to the creation of the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, an another physically plausible scenarios which describes the observable large scale structure of the cosmos, have also yielded exact answers to the field equations (Melia 2022). According to the FLRW metric, the universe is expanding and its curvature is determined by how matter and energy are distributed.

The Killing vector field X that satisfies the mathematical relation \({\mathcal {L}}_X g_{ab}=0,\) can be used to express the theory’s most fundamental symmetry, where \({\mathcal {L}}_X\) indicates the Lie derivative operator along the local flow of X,. These vector fields have a direct connection to spacetime conservation laws. We mention Stephani et al. (2003), Meisner et al. (1973), Petrov (1969) for a thorough analysis of exact solutions of EFEs using symmetry conditions on \(g_{ab}\) and the associated conservation laws. In addition to this, some additional symmetries have also been investigated in the literature, such as homothetic vectors (\({\mathcal {L}}_X g_{ab}=2 \alpha g_{ab}\)), curvature collineations (\({\mathcal {L}}_X R^a_{bcd}=0\)), and matter collineations (\({\mathcal {L}}_X T_{ab}=0\)). The details are available in Bokhari et al. (2000), Hall and da Costa (1991a), Hall and da Costa (1991b).

A RC in differential geometry is a vector field that keeps the Ricci tensor invariant along its flow. More formally, a vector field X that meets the following criteria is said to be a RC on a Riemannian manifold (Tsamparlis and Apostolopoulos 2004):

$$\begin{aligned} {\mathcal {L}}_X R_{ab}=R_{ab,c} X^c+ R_{cb} X^c_{,a}+R_{ac} X^c_{,b}=0. \end{aligned}$$
(1.2)

In other words, the converted Ricci tensor is proportional to the original tensor if it is changed in the direction of the vector field X. Here, the values for \(a, \ b\) and c are 0, 1, 2 and 3, while the commas stand in for derivatives with reference to spacetime coordinates. Degenerate \(\det (R_{ab}) = 0\) or non-degenerate \(\det (R_{ab})\ne 0\) are two possible states for the Ricci tensor. They could create a Lie algebra with finite or infinite dimensions in the first scenario. In the latter scenario, Lie algebra with a finite dimension less than or equal to 10 is always represented by the set of RCs (Camci and Turkyilmaz 2004).

Because they specify the directions in which the Ricci tensor is invariant under transformations of the metric tensor, Ricci collineations are significant. The behavior of EFEs has been studied in physics using this symmetry. RCs are particularly helpful for locating precise answers to field equations because they let us take advantage of this symmetry to simplify the issue. The Killing vector field, which preserve the metric tensor along its flow, is one notably significant example of a RC as they relate to conservation principles that ensure that energy and momentum are maintained throughout translations and rotations. Overall, the study of RCs is a significant branch of differential geometry and has many uses in physics, especially in the investigation of EFEs and the behaviour of spacetime. According to symmetries of Ricci tensor, some well-known spacetimes are categorized; the specifications can be seen in Camci and Turkyilmaz (2004), Amir et al. (1994), Bokhari and Qadir (1993), Contreras et al. (2000), Camci and Barnes (2002), Akhtar et al. (2018), Bokhari and Qadir (1993), Qadir et al. (2003), Khan et al. (2013), Ziad (2003), Bokhari et al. (2003), Bokhari et al. (2010).

Ten determining equations generating RCs are presented in Sect. 2 of this article. These equations are solved for various constraints in Sects. 3 and 4 as non-degenerate or degenerate Ricci tensor respectively. A short summary is included in the final part.

2 Determining equations of RCs

The general line element of LRS Bianchi type I spacetime is given by Stephani et al. (2003):

$$\begin{aligned} ds^2=-dt^2+A^2(t)dx^2+B^2(t)\left[ dy^2+dz^2\right] , \end{aligned}$$
(2.1)

where A(t) and B(t) are nowhere zero functions of t. The non-zero components of Ricci tensor for the above spacetime are given as:

$$\begin{aligned} R_{00}= & {} -\left( \frac{BA''+2AB''}{A B}\right) = I(t), \nonumber \\ R_{11}= & {} \left( \frac{B A A''+2A A' B'}{B}\right) = S(t), \nonumber \\ R_{22}= & {} R_{33} = \left( \frac{AB'^2+A B B''+B A'B'}{A}\right) = R(t). \end{aligned}$$
(2.2)

Here the prime over A and B denotes differentiation with respect to t. Using these values in Eq. (1.2), we get the following set of 10 partial differential equations.

$$\begin{aligned} I' X^0 + 2 I X^0_{,0}= & {} 0, \end{aligned}$$
(2.3)
$$\begin{aligned} S X^1_{, 0} + I X^0_{, 1}= & {} 0, \end{aligned}$$
(2.4)
$$\begin{aligned} R X^2_{, 0} + I X^0_{, 2}= & {} 0, \end{aligned}$$
(2.5)
$$\begin{aligned} R X_{, 0}^3 + I X _{, 3}^0= & {} 0, \end{aligned}$$
(2.6)
$$\begin{aligned} S' X^0+2 S X_{, 1}^1= & {} 0, \end{aligned}$$
(2.7)
$$\begin{aligned} R X_{, 1}^2+S X_{,2}^1= & {} 0, \end{aligned}$$
(2.8)
$$\begin{aligned} R X^3_{, 1}+S X^1_{, 3}= & {} 0, \end{aligned}$$
(2.9)
$$\begin{aligned} R' X^0+2 R X_{, 2}^2= & {} 0, \end{aligned}$$
(2.10)
$$\begin{aligned} R (X_{, 2}^3+ X_{, 3}^2)= & {} 0, \end{aligned}$$
(2.11)
$$\begin{aligned} R' X^0+2 R X_{, 3}^3= & {} 0. \end{aligned}$$
(2.12)

In Sect. 3 and 4, the above RC equations will be solved for non-degenerate and degenerate cases respectively.

3 Ricci collineations for non-degenerate Ricci tensor

In this section, non-degenerate Ricci tensors, \(\det R_{ab}\ne 0\), are taken into account as we explore RC equations. These equations are solved using fundamental integration strategies. Here, we omit the fundamental calculation and only display the explicit vector field component form that produces RCs:

$$\begin{aligned} X^0= & {} -\frac{R}{I}\left[ \left( \frac{y^2 + z^2}{2}\right) f_t^1 + y f_t^3 + z f_t^2\right] + f^4, \end{aligned}$$
(3.1)
$$\begin{aligned} X^1= & {} -\frac{R}{S}\left[ \left( \frac{y^2 + z^2}{2}\right) f_x^1 + y f_x^3 + z f_x^2\right] + f^5, \end{aligned}$$
(3.2)
$$\begin{aligned} X^2= & {} \frac{c}{2}(z^2 - y^2) + \frac{a}{6}(y^3 - 3yz^2) + d z + b yz + yf^1 + f^3, \end{aligned}$$
(3.3)
$$\begin{aligned} X^3= & {} \frac{b}{2}(z^2 - y^2) + \frac{a}{6}(z^3 - 3zy^2) - d y -c yz + zf^1 + f^2. \end{aligned}$$
(3.4)

where \(f^i=f^i(t,x); \ 1 \le i \le 5\) are functions appeared during the process of integration and \(a, b, c, d \ \epsilon \ \Re\). The above values satisfy some of the RC Eqs. (2.3)-(2.12) while the remaining equations generate the following integrability constraints along with \(a=0\).

$$\begin{aligned} R' f_t^1= & {} 0, \end{aligned}$$
(3.5)
$$\begin{aligned} R' f_t^3 + 2Rc= & {} 0, \end{aligned}$$
(3.6)
$$\begin{aligned} R' f_t^2 - 2Rb= & {} 0, \end{aligned}$$
(3.7)
$$\begin{aligned} I' f^4 + 2 I f_t^4= & {} 0, \end{aligned}$$
(3.8)
$$\begin{aligned} S' f^4 + 2 S f_x^5= & {} 0, \end{aligned}$$
(3.9)
$$\begin{aligned} R' f^4 + 2 R f^1= & {} 0, \end{aligned}$$
(3.10)
$$\begin{aligned} I f_x^4 + Sf_t^5= & {} 0, \end{aligned}$$
(3.11)
$$\begin{aligned} f_{tt}^i + f_t^i\left( \frac{R'}{R} - \frac{I'}{2I}\right)= & {} 0, \end{aligned}$$
(3.12)
$$\begin{aligned} S' f_t^i + 2I f_{xx}^i= & {} 0, \end{aligned}$$
(3.13)
$$\begin{aligned} 2R f_{tx}^i + f_x^i\left( \frac{R' S -RS'}{S}\right)= & {} 0, \end{aligned}$$
(3.14)

where i varies from 1, 2, 3. To find the final form of vector field components \(X^i\), we solve these integrability conditions by considering the following differential cases:

$$\begin{aligned}{} & {} (\mathrm {ND\ 1}): I'=S'=R'=0, \\{} & {} (\mathrm {ND\ 2}): R' \ne 0, \quad I'=S'=0, \\{} & {} (\mathrm {ND\ 3}): S' \ne 0, \quad I'=R'=0, \\{} & {} (\mathrm {ND\ 4}): I' \ne 0, \quad R'=S'=0, \\{} & {} (\mathrm {ND\ 5}): I'=0, \quad S' \ne 0, \quad R' \ne 0, \\{} & {} (\mathrm {ND\ 6}): I' \ne 0, \quad R' \ne 0, \quad S'=0, \\{} & {} (\mathrm {ND\ 7}): I' \ne 0, \quad S' \ne 0, \quad R'=0. \\{} & {} (\mathrm {ND\ 8}): I' \ne 0, \quad S' \ne 0, \quad R' \ne 0. \end{aligned}$$

We look at each of the aforementioned situations separately, and the final symmetries of the Ricci tensor are shown in the following tabular format.

Case No.

Constraints

Vector Field Generating RCs

Metric Functions

ND 1

\(I'=S'=R'=0\)

\(X^0=-c_8 y - c_4 z+ c_1 x + c_2\),

\(A(t)=\text {constant}\)

\(X^1= -c_9 y - c_6 z - c_1 t + c_3\),

\(B(t)=(at +b)\)

\(X^2= d z + c_8 t + c_9 x + c_{10}\),

 

\(X^3= -d y + c_4 t + c_6 x + c_5\).

Case No.

Constraints

Vector Field Generating RCs

Metric Functions

ND 2

\(R=\alpha e^{-\frac{2c_1}{c_2}t}, I'=S'=0\)

\(X^0=-\frac{2c_1R}{R'}\), \(X^1=c_3\),

\(A(t)=\text {constant}\)

\(X^2=zd + yc_1+ c_9\),

\(B(t)=e^{t}\)

\(X^3=-yd + zc_1+ c_7\).

 

ND 3

\(S' \ne 0, I'=R'=0\)

\(X^0=0\), \(X^1=c_3\),

\(A(t)=e^{\alpha t}\)

\(X^2=zd + c_2\), \(X^3=c_1-yd\).

\(B(t)=\text {constant}\)

ND 4

\(I' \ne 0, R'=S'=0\)

\(X^0=\frac{1}{\sqrt{I}}\left( c_8 x-c_7 y-c_5 z+c_9\right)\),

A(t) and B(t) are functions such that \(\frac{BA''+2AB''}{AB}\ne \text {Constant}\)

\(X^1=-c_3 y-c_1 z-c_8 \int \sqrt{I} dt+c_{10}\),

\(X^2=d z + c_3 x+c_7\int \sqrt{I} dt+ c_4\),

\(X^3=-d y + c_1 x +c_5\int \sqrt{I} dt+ c_2\).

ND 5

\(R=S=c \ e^{2\alpha t}, \ \ \ I'=0\)

\(X^0=-2(c_1 x+c_2)\),

\(A(t)=B(t)=e^{\alpha t}\)

\(X^1=2c_1\left( \frac{x^2}{4}-e^{-t}\right) -c_1\left( \frac{y^2+z^2}{2}\right)\)

\(-c_9 y-c_6 z+c_2 x+c_{15}\),

\(X^2=d z + y(c_1 x+c_2)+c_9 x+c_{14}\),

\(X^3=-d y + z(c_1 x+c_2)+c_6 x+c_{12}\).

ND 6

\(I=-\frac{1}{4}\left( \frac{R'}{R}\right) ^2, S'=0\)

\(X^0=-\frac{2c_1 R}{R'}\), \(X^1=c_2\),

\(A(t)=(at+b)\)

\(X^2= d z + c_1 y + c_7\), \(X^3= c_1 z- d y + c_5\).

\(B(t)=(at+b)^{-1}\)

ND 7

\(I' \ne 0, S' \ne 0, R'=0\)

\(X^0=0\), \(X^1=c_3\),

\(A(t)=\sqrt{at+b}\)

\(X^2=d z + c_2\), \(X^3=c_1-d y\).

\(B(t)=A^2(t)\)

ND 8

\(I' \ne 0, S' \ne 0, R'\ne 0\)

\(X^0=0\), \(X^1=c_3\),

\(A(t)\ne B(t)\ne (at+b)\),

\(X^2=d z + c_2\), \(X^3=c_1-d y\).

\(A(t)\ne B(t)\ne e^{\alpha t}\)

4 Ricci collineations for degenerate Ricci tensor

This section makes the assumption that the Ricci tensor is degenerate i.e. \((\det R_{ab}=SIR^2=0)\). Below are the possibilities for this study of the Ricci tensor components:

$$\begin{aligned}{} & {} (\textrm{D1}): I = 0, \quad S \ne 0, \quad R \ne 0, \\{} & {} (\textrm{D2}): I \ne 0, \quad S = 0, \quad R \ne 0, \\{} & {} (\textrm{D3}): I \ne 0, \quad S \ne 0, \quad R=0, \\{} & {} (\textrm{D4}): I=S=0, \quad R \ne 0, \\{} & {} (\textrm{D5}): I=R=0, \quad S \ne 0,, \\{} & {} (\textrm{D6}): I \ne 0, \quad S=R=0. \end{aligned}$$

For all of the aforementioned scenarios, we solve the RC Eqs. (2.3)-(2.12) and solely present the final findings in the accompanying table. It is important to note that just the single scenario D1 yields finite dimensional RC Lie algebra.

Case No.

Vector Field Generating RCs

D1

\(X^0=\frac{2S}{S'}(c_1 z-c_2)\),

\(X^1= -c_1 x z - c_3 z + c_2 x + c_4\),

\(R=S,\)

\(X^2= -c_1 y z + c_2 y - c_5 z + c_6\),

\(S'\ne 0\)

\(X^3= \frac{c_1}{2}\left( x^2+y^2-z^2\right) + c_2 z + c_5 y + c_3 x+c_7\).

D2

\(X^0=\frac{c_1 y+c_2}{\sqrt{I}}\), \(X^1=X^1(t, x, y, z)\),

\(R'=0\)

\(X^2=-c_1\int \sqrt{I} dt+ c_3 z+ c_4\),

\(X^3=-c_3 y+ c_5\).

D3

\(X^0=0\), \(X^1=c_1\),

\(X^2=X^2(t, x, y, z)\), \(X^3=X^3(t, x, y, z)\).

D4

\(X^0=-\frac{2R}{R'}(c_1\sinh y+c_2\cosh y)(c_3\cos z+c_4\sin z)\),

\(R'\ne 0\)

\(X^1=X^1(t, x, y, z)\),

\(X^2= (c_1\cosh y+c_2\sinh y)(c_3\cos z+c_4\sin z)+ c_5z+c_6\),

\(X^3= (c_1\sinh y+c_2\cosh y)(c_3\sin z-c_4\cos z) - c_5 y+c_7\).

D5

\(X^0=-\frac{2S g^1_x(x)}{S'}\), \(X^1=g^1(x)\),

\(S'\ne 0\)

\(X^2=X^2(t, x, y, z)\),

\(X^3=X^3(t, x, y, z)\).

D6

\(X^0=\frac{c_1}{\sqrt{I}}\),

\(X^i=X^i(t, x, y, z); i=1, 2, 3.\)

5 Conclusion

We have looked into the RCs of LRS Bianchi type I spacetimes in this article. In order to classify the above metric into two categories depending on whether the Ricci tensor is degenerate or non-degenerate, the defining equations of RCs are analytically solved. Furthermore, by setting specific requirements for derivable Ricci tensor components, we divided the non-degenerate case into eight subcases. We determined that for the non-degenerate case, Lie algebra of RCs always have finite dimension. Additionally, the degenerate case is solved for some potential subcases based on the components of the Ricci tensor, and almost all cases generate infinite RCs, with the exception of case D1, for which we obtained finite number of RCs.