Noether Symmetry Analysis of the Klein–Gordon and Wave Equations in Bianchi I Spacetime

We investigate the Noether symmetries of the Klein–Gordon Lagrangian for Bianchi I spacetime. This is accomplished using a set of new Noether symmetry relations for the Klein–Gordon Lagrangian of Bianchi I spacetime, which reduces to the wave equation in a special case. A detailed Noether symmetry analysis of the Klein–Gordon and the wave equations for Bianchi I spacetime is presented, and the corresponding conservation laws are derived.


I. INTRODUCTION
A conformal Killing vector (CKV) K has to satisfy where g ij is the metric tensor, £ K is the Lie derivative operator along K, and σ(x k ) is a conformal factor.When σ ;ij = 0, the CKV field is said to be proper [1].The vector field K is called a special conformal Killing vector (SCKV) field if σ ;ij = 0; a homothetic Killing vector (HKV) field if σ , i = 0, e.g., σ is a constant on the manifold; and a Killing vector (KV) field if σ = 0, which is also called the isometry of spacetime.The set of all CKVs (respectively SCKV, HKV, and KV) forms a finite-dimensional Lie algebra.The maximum dimensions of the CKV algebra on the manifold M is fifteen if M is conformally flat, and it is seven if the spacetime is not conformally flat.The physical features of differential equations, in terms of the conservation laws admitted by them, are directly associated with the Noether symmetries, which is facilitated using a Lagrangian of the corresponding dynamical system.As we show in the following section, there is a direct relation between the conformal symmetries and Noether symmetries.If a Lagrangian L for a given dynamical system exhibits symmetry, this property is strongly related to Noether symmetries, which describe the physical characteristics of differential equations associated with a Lagrangian L, in terms of the first integrals they possess [2,3].This relationship can be viewed from two different perspectives.First, one can take a strict Noether symmetry approach [4,5,6], which results in £ X L = 0, where £ X is the Lie derivative operator along X.On the other hand, one can employ the Noether symmetry approach with a gauge term [7,8,9,10,11,12,13], a generalization of the strict Noether symmetry approach where the Noether symmetry equation includes the gauge term.Noether symmetries with a gauge term are equally valuable in addressing a variety of problems in physics and applied mathematics.In the following section, we will discuss the relationship between Noether symmetries with a gauge term of the Klein-Gordon Lagrangian and the geometric symmetries of spacetimes.The study of differential equations involving geometry is an active area of research.Recent literature [14,15] has delved into the connection between geometrical structures and conserved quantities.Noether symmetries are directly linked to conserved quantities or conservation laws [3], which naturally emerge in a wide range of applications.
The cosmological principle assumes that the universe is homogeneous and isotropic at large scales, and the geometrical model that satisfies these properties is Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime.However, there has been a suggestion from studies of the cosmic microwave background (CMB) temperature anisotropies that the assumption of statistical isotropy is violated at the largest angular scales, leading to some intriguing anomalies [16].In order to make predictions for the CMB anisotropies, one can explore cosmological models that are homogeneous but anisotropic, such as the Bianchi-type spacetimes, which encompass both isotropic and homogeneous FLRW models.
Bianchi classified all three-dimensional real Lie algebras and demonstrated that there are nine possible simply transitive groups of motions, denoted as G 3 .These groups are generated using Killing vectors K α , where α = 1, 2, 3, with structure constants C γ αβ , defined by [K α , K β ] = C γ αβ K γ .The structure constants C γ αβ can be decomposed into irreducible parts, as follows: here N αβ is symmetric.It follows from the Jacobi identities that N αβ A β = 0.One can diagonalize N αβ without loss of generality, choosing A α in the 1-direction when it is non-zero.The real Lie algebras can then be classified into nine types of spatially homogeneous Bianchi spacetimes, distinguished by the particular form of the structure constants C γ αβ , based on whether A α = 0 (Class A) or A α = 0 (Class B).This classification results in the Bianchi types, such as I, II, V I 0 , V II 0 , V III, and IX for Class A models, and III, IV , V , V I h , and V II h for Class B models.It is important to note that the Bianchi models include FLRW models as special cases, with Bianchi I (flat), Bianchi V (open), and Bianchi IX (closed) representing the FLRW models.In this study, we consider the Bianchi I metric as the background spacetime in the Klein-Gordon equation.The line element for Bianchi I spacetime can be written as follows [17]: ( The above spacetime yields a flat FLRW metric if A = B = C.For any form of the metric coefficients A, B, and C, it can easily be computed from Equation (1) that the KVs for Bianchi I spacetime are In this study, all vector fields are written with bold letters.The rest of the paper is organized as follows: In the next section, II, we present an analysis of Noether symmetries with a gauge term for the Klein-Gordon Lagrangian in the context of the Bianchi I spacetime model.In Section III, we apply the Noether symmetry approach to the Bianchi I spacetime.In Section IV, we study the field equations of the Bianchi I spacetime with an imperfect fluid.Conclusions and discussions are presented in the final section, Section V.

II. KLEIN-GORDON LAGRANGIAN AND NOETHER SYMMETRY EQUATIONS
The Klein-Gordon equation in the Riemannian space with metric g ik is a second-order partial differential equation of the form where ψ = g ik ψ ;ik and refers to the de'Alembertian or Laplace operator defined by = 1 in terms of the Riemannian space.The Klein-Gordon equation given in (3) follows from the first-order Lagrangian, which is called the Klein-Gordon Lagrangian, where Here, the prime represents the derivative with respect to the scalar field ψ.The above Lagrangian reduces to the Lagrangian of the wave equation when F = 0.
The wave equation of spacetimes is one of the most important equations in physics, and it is common to study this equation in terms of the Lie and Noether symmetry generators they admit [18,19,20,21,22,23].When symmetry generators exist, they play a crucial role in finding exact solutions.Jamal et al. [21] investigated the wave equation of Bianchi III spacetime, calculating and classifying Noether symmetries and constructing corresponding conservation laws.They also obtained reductions of the wave equation and identified some invariant solutions.In the papers [22,23], the authors utilized the invariance and multiplier method [24,25] to conduct conservation law classifications of the wave equation for the Bianchi I spacetime, using power law metric coefficients.
Noether symmetries of the Klein-Gordon equations for well-known spacetimes have been calculated and classified according to their symmetry generators [15,26,27,28,29,30].In [26], the symmetry analysis of the Klein-Gordon equation in de Sitter spacetime was classified, and the obtained symmetries were utilized to find exact solutions using quadratures.Paliathanasis et al. [27] conducted a classification of Lie and Noether symmetries for the Klein-Gordon and wave equations in pp-wave spacetimes.The symmetry properties and conservation laws of wave and Gordon-type equations in Milne and Bianchi III spacetimes were investigated in the papers [28,29].In [15], a geometric procedure was employed for the symmetry classification of the Klein-Gordon equation in Bianchi I spacetime, connecting the Noether symmetries with a gauge function of the Klein-Gordon Lagrangian to the conformal symmetries of the metric tensor.Their study extended the results of Bozhkov and Freire [31] for the Klein-Gordon equation with a constant potential, i.e., V (x k ) = V 0 .More recently, Paliathanasis [30] considered the Klein-Gordon equations for the conformal forms of Bianchi I, Bianchi III, and Bianchi V spacetimes, deriving closed-form expressions for potential functions that admit Lie and Noether symmetries of the Klein-Gordon equations.In this work, we aim to derive Noether symmetries with a gauge term for the dynamical Lagrangian L of the Klein-Gordon equation within the background of Bianchi I spacetimes.To obtain the Klein-Gordon equation in this spacetime, we construct a Lagrangian model.Using the obtained Lagrangian for the Klein-Gordon equation in Bianchi I spacetime, we calculate and classify Noether symmetry generators with gauge terms.Furthermore, we identify conservation laws provided by the Lagrangian for representing the Klein-Gordon equation.
In a general program of research into the problem of integrating the classical and quantum equations of motion of a test particle in external fields of different nature in spaces with symmetry following the sets of Killing fields, Obukhov found all admissible electromagnetic fields for the case, when the groups of motions G 3 act simply transitively on the hypersurfaces of spacetime V 4 [32,33,34,35].In [32], he found all external electromagnetic fields in which the Klein-Gordon-Fock equation admits the first-order symmetry operators and completed the classification of admissible electromagnetic fields in which the Hamilton-Jacobi and Klein-Gordon-Fock equations admit algebras of motion integrals that are isomorphic to the algebras of operators of the r-parametric groups of motions, G r , of spacetime manifolds if r ≤ 4 [33].In the paper [34], the case when the groups G 4 act on V 3 was considered.The remaining case in the latter article, when the groups G 4 act simply transitively on the space V 4 , was studied in the paper [35].
The Noether symmetry generator for the Klein-Gordon Lagrangian (4) is if there a gauge function f i (x k , ψ) exists and the Noether symmetry condition is satisfied, where is the total derivative operator and X [1] is the first prolongation of Noether symmetry generator X, i.e., where The corresponding Noether flow T i is defined by the expression which is called the conserved vector T = (T 1 , ..., T n ) , where i = 1, ..., n, or if n = 1, then it is called the conserved quantity.The Noether flow (7) satisfies the local conservation law It is crucial to discover conservation laws when studying physical systems.For studying differential equations, conservation laws are useful for integrability, linearization, analyzing solutions, and understanding constants of motion.Noether's theorem enables the derivation of all local conservation laws for a (system of) differential equation(s) derived from a Lagrangian.This method helps resolve the problem of calculating conservation laws for given differential equation(s), as Noether's theorem provides a formula utilizing symmetries of the action to derive these local conservation laws.One can find a detailed discussion about the physical significance of Noether symmetries in [36].
For the Klein-Gordon Lagrangian (4), we obtain the first prolongation of the Noether symmetry generator X as where £ ξ is the Lie derivative operator along ξ = ξ k ∂/∂x k .Putting (9) into (5) together with the Noether symmetry condition (5) gives rise to Thus, we find the geometrical form of Noether symmetry equation ( 5) in terms of Lie derivatives of the metric tensor.
Here, Equation (10) ) is an integration function.If ξ i is a CKV with conformal factor σ(x k ), then Equations ( 1) and (11) imply ξ i ;i = 2 σ − 2 Φ ,ψ .Finally, the Noether symmetry condition (12) becomes For the Bianchi I spacetime (2), the Klein-Gordon Lagrangian (4) has the form which is the the Lagrangian of the wave equation when F = 0.The Klein-Gordon Equation ( 3) is obtained through variation of this Lagrangian with respect to the scalar field ψ, as the following: where G(t, x, y, z, ψ) = − ∂F ∂ψ and the dot represents the derivative with respect to time t.Let us consider the Noether symmetry generator for the Klein-Gordon Lagrangian ( 14) as follows: where the components of ξ = (ξ 0 , ξ 1 , ξ 2 , ξ 3 ) and Φ are dependent on t, x, y, z and ψ.Now, we seek the dependent variables ξ 0 , ξ 1 , ξ 2 , ξ 3 , Φ that will be solved from the geometrical Noether symmetry conditions ( 10)-( 12), in order that the Lagrangian ( 14) would admit any Noether symmetry.For the Bianchi I spacetime (2), the geometrical Noether symmetry conditions ( 10)-( 12) yield 19 PDEs: It is noted here that the set of all Noether symmetries with the gauge functions f i form a finite dimensional Lie algebra.

III. NOETHER SYMMETRIES AND CONSERVATION LAWS
It is easily seen from Equation (1) that for arbitrary forms of metric functions A(t), B(t) and C(t), the background spacetime of Bianchi I metric admit the three KVs, the generators of translations in x, y and z directions which implies momentum conservation, It is obvious that these KVs are also Noether, and so Lie, symmetries of the Klein-Gordon Equation (15) in the background of Bianchi I spacetime.Hence, applying the expression (7) of conservation law, the resulting conserved flow vector components related to the Klein-Gordon Equation ( 15) are as follows: for for for Here, the conserved flow vector is T = (T t , T x , T y , T z ), and V 1 (t, y, z, ψ), V 2 (t, x, z, ψ), and V 3 (t, x, y, ψ) are integration functions.We give a complete solution of Noether symmetry conditions (17) for the Bianchi I spacetime in the following.Case (i): First, let us consider the Klein-Gordon equation that requires that F = 0. Taking F = U 0 + U 1 ψ + 1 2 U 2 2 ψ 2 for any metric coefficients of Bianchi I spacetime, where U 0 , U 1 and U 2 are constants, the components of the Noether symmetry generator (16) are found from Equations (17) as where c 1 , ..., c 9 and a 1 , a 2 , a 3 are constant parameters, F 0 , ..., F 3 are integration functions, and Y (t) solves the following second-order ordinary differential equation: Therefore, there are eleven Noether symmetries, such that with the corresponding non-zero gauge vectors: The conserved vector fields associated with X 1 , ..., X 11 given in ( 26)-( 29) are obtained as where the gauge vectors f 5 , ..., f 11 are the same as in (30), and W and T 0 are defined by where The conserved vector components (7) for the integration functions F 0 , ..., F 3 of the gauge functions which have the property of T i = f i are T = F 0 (t, x, y, z), F 1 (t, x, y, z), F 2 (t, x, y, z), − (F 0,t + F 1,x + F 2,y )dz + F 3 (t, x, y) . ( We note, here, that the above conserved quantities will appear in each of the possible cases.Therefore, we will not mention these quantities again.Through to the end of this section, following Ref.[21], we will take into account the power-law form of metric functions, such that where L, p, and q are constant parameters.For the latter forms of the metric functions, we find that there exists an additional HKV, which is a scaling transformation or a dilation, where σ = const.= 1, in addition to the KVs K 1 , K 2 and K 3 in (18).Furthermore, we will consider some subcases in which we obtain the symmetry generators for the wave (F = U 0 = const.,where one can take F = 0 without loss of generality) and Klein-Gordon equations (F = const.) of Bianchi I spacetime.Obviously, other choices for the metric functions will lead to a different solution for the function of Y (t) from Equation (25).
We give some examples of these choices: In subcase (i.1), if L = p = 1 and q = 0, then the line element reduces to the conformally flat Bianchi I spacetime, and point symmetries and potentials for the Klein-Gordon equation with F = V (t, x, y, z)ψ 2 /2 in this metric were studied by Ref. [15].The solution Y (t) of ( 25) for L = p and q = 0 is found as follows: where p = 0, 2 )/(2p), and the functions g 1 (t), g 2 (t) are defined as The solution (36) is a new one that was not mentioned in Ref. [15] and generalizes the case 4.4 of this reference, where they were taken as L = p = 1.For L = p = 1 and q = 3, we have the following solution for Equation (25): where H G is the Heun general function, the functions g 1 (t), g 2 (t) are the same ones as in (37) with p = 1, and we define that and in which d 1 and d 2 are defined as One can find other solutions of (25) for different choices of the parameters L, p and q.Some solutions of the Equation ( 25) for the remaining subcases (i.2)-(i.4)are included in Table I.
Case L , p , q Y (t) where , cosh 2 (qt) , cosh 2 (qt) The conserved flow vectors associated with the KVs X 1 1 , X 1 2 , X 1 3 and non-Killing Noether symmetries X 1 4 , ..., X 1  12 given in ( 40) and ( 41) are obtained as where W, T 0 are given in Equation ( 32) by taking A = t L , B = t p , C = t q , Q 0 is defined as and the gauge vectors f 1 5 , ..., f 1 12 have the same forms as in (42).Furthermore, this subcase yields a thirteenth Noether symmetry, in addition to the twelve obtained above, which is X = y∂ x − x∂ y if L = p, and q is an arbitrary constant; X = z∂ x − x∂ z if L = q, and p is an arbitrary constant, and finally X = z∂ y − y∂ z if p = q, and L is an arbitrary constant.
In this case, let us examine some solutions of Equation ( 43) for specific values of the constant powers L, p, and q of the metric functions.If we assume L = 2, p = q = 0, the solution of the Equation ( 43) is expressed in terms of double confluent Heun functions: where b 1 , b 2 are constant parameters, and For L = −3, p = q = 0, we have the solution where H B is the Heun biconfluent function such that H B = HeunB α, β, γ, δ, a1 2 t 2 with α = 2, β = 0, γ = 0, and ).Further analysis of Equation ( 43) with respect to values of L, p, and q reveals non-trivial solutions in terms of some familiar special functions.Table II contains some of these solutions corresponding to specific values for L, p, and q.Subcase (ii.1).If we take F = U 0 + U 1 ψ in the Klein-Gordon Equation (15) for the power law form of Bianchi I spacetime (34) with L = p = q, where U 0 , U 1 are constants, it is found that there are fifteen Noether symmetries, as follows: TABLE II: Some solutions of the equation ( 43) for specific values of L, p, and q.Here, it is defined that Further, the special functions KM = KummerM (µ, ν, z) and KU = KummerU (µ, ν, z) are the Kummer functions, WM = W hittakerM (µ, ν, z) and WW = W hittakerW (µ, ν, z) are the Whittaker functions, respectively.

Case
L , p , q Y (t) where µ = where µ = 1 + and the corresponding non-zero gauge vectors are where f 0 = t L+p+q ψ, m is a non-zero constant parameter, and K 4 is the HKV given in (35).Thus, one can write the conserved flow vectors for the Noether symmetries X 5 4 , ..., X 5 15 as T 5 8,9 = Y (e a1x + m e −a1x ) e a2y±a3z T 0 + f 5 8,9 , T 5 10,11 = Y (e a1x + m e −a1x ) e −a2y±a3z T 0 + f 5 10,11 where ). Subcase (ii.2).For F = 0, which requires U 0 = U 1 = U 2 = 0, Equation (15) reduces to the wave equation for Bianchi I spacetime.If we take L = p = q in (34), the Bianchi I spacetime yields the flat FLRW spacetime.For the latter assumption of metric coefficients, and assuming F = U 0 , we find twenty six Noether symmetries, which are the KVs X 1  1 , X 1 2 , X 1 3 given in (39), and in which the vector fields K 5 , K 6 , K 7 , K 1 8 , K 1 9 and K 1 10 have the form and M = M (t) and N = N (t) are defined by where b 1 , b 2 and b 3 ( = 0) are constant parameters, q = 0, 1, 1/2, ν 1 = −(q + 1)/(2(q − 1)), ν 2 = (q − 3)/(2(q − 1)), ))t 1−q , {J ν1 (τ ), J ν2 (τ )} and {Y ν1 (τ ), Y ν2 (τ )} are first and second kind Bessel functions, respectively.We note here that the vector fields K 5 , K 6 , K 7 in (69) are the KVs, and the other ones K 1  8 , K 1 9 , and K 1 10 in (70)-( 72) are the SCKVs with the conformal factors σ = 2(q − 1)x, σ = 2(q − 1)y, and σ = 2(q − 1)z, respectively.Furthermore, the gauge vectors for X 2 8 , ..., X 2 26 are 13,14 = ψ e ±b1x−b2y sin(ℓz) b 3 t Then, in addition to the conserved flow vectors for X 1 , X 2 , X 3 , and X 2 4 obtained in (44) and (45), the conserved vector fields for the remaining Noether symmetries of this subcase become where W, T 0 are the same as given in (32), in which F = 0 and A = B = C = t q , and Q 1 , Q 2 and Q 3 are defined as Using the conservation law relation (7), the conserved flow vectors for X 3 4 , ..., X 3 26 become Subsequently, the Einstein field equations G ij = 8πT ij in natural units (G = 1 and c = 1) can be expressed as follows: where Rg ij is the Einstein tensor, R ij is the Ricci tensor, and R is the Ricci scalar, which has the form A curvature scalar characterizes the spacetime curvature.One of the important curvature scalars is the Kretschmann scalar, defined by K = R ijkl R ijkl , which is a quadratic scalar invariant of the Riemann tensor R ijkl .It is crucial for measuring curvature in a vacuum.This curvature scalar also characterizes the spacetime curvature of a realistic rotating black hole, allowing us to mathematically perceive the black hole.Additionally, we consider an important curvature scalar, the so-called Gauss-Bonnet (GB) invariant G, defined as Apart from emerging in the context of defining quantum fields in curved spacetimes, the GB invariant G can encapsulate all the curvature information stemming from the Riemann tensor in dynamical equations.For the Bianchi I spacetime, both the K and G take the following forms K = 4 Ä2 and P z = P y , respectively.In theoretical physics and cosmology, scenarios where the energy density vanishes (or is extremely low) while the pressure is different from zero are less common.However, there are still certain contexts where this can occur, often involving exotic fields or conditions.In certain models of the early universe, there is a concept known as the false vacuum.The vacuum state of a field can be thought of as the state of lowest energy.In a false vacuum, the energy density is very close to zero, but the field is in a metastable state rather than the true vacuum state.When considering a perfect fluid source (P x = P y = P z = P ), Equations ( 133) to (134) imply the following constraint relations: For cases where L = p = q, as seen in cases (ii.2) and (ii.3), these constraint equations are precisely satisfied.Consequently, the physical quantities ρ and P take the forms yielding the EoS parameter w = (2 − 3q)/(3q) where q = 0. So, dark energy occurs when q > which mentions that a well-known example of dark energy (w = −1), the cosmological constant, is not possible for the power law Bianchi I spacetime with a perfect fluid.In addition, we have dust fluid if q = 1/3, and a stiff fluid if q = 1/2.Additionally, the curvature scalars R, K and G for cases (ii.1) and (ii.3) are given by R = 6q(2q − 1) t 2 , K = 12 q 2 (2q 2 − 2q + 1) t 4 , G = 24q 3 (q − 1) Moreover, when L = p = q = 2/3, the spacetime corresponds to the Einstein-de Sitter model, a solution of field Equations ( 126) to (128) with P = 0 and ρ = 1/(6π t 2 ), i.e., w = 0 (representing a dust fluid).Kasner spacetimes, non-trivial Bianchi I solutions satisfying Einstein's vacuum equations (R ij = 0, or ρ = 0 and P = 0), are characterized using the Kasner metric.In this metric, the three constant parameters L, p, and q are termed the Kasner components.These components adhere to the following relations: resulting in Lp + Lq + pq = 0. Kasner spacetimes exhibit spatial anisotropy, potentially expanding along one direction while contracting along another.In the Kasner scenario, the three parameters cannot all be equal.Thus, at least one of L, p, q must satisfy 0 ≤ L, p, q ≤ 1.If any of the parameters L, p, or q equals one, the other two must be zero, resulting in a flat Kasner spacetime.Conversely, if L, p, q = 1 are all not equal to one, indicating all are non-zero, the Kasner spacetime is non-flat [17].

V. DISCUSSIONS AND CONCLUDING REMARKS
Though the standard procedure for determining Lie and Noether symmetry generators can be cumbersome, it is feasible to consider reduction or obtaining conservation laws through Noether's theorem.This paper delved into the geometric nature of the Klein-Gordon/wave equation within the framework of the Lagrangian for the Bianchi I spacetime.We demonstrated that computing Noether symmetries of the first-order Lagrangian (14) for the Klein-Gordon Equation ( 15) is simplified to solving a set of differential conditions outlined by (17).For the Bianchi I spacetime, employing this method to find Noether symmetries yields solutions encompassing scenarios where F (t, x, y, z, ψ) = U 0 (let us say, U 0 = 0 for convenience), reducing Equation (15) to the wave equation.Conversely, when F (t, x, y, z, ψ) = const.,Equation ( 15) represents the Klein-Gordon equation.Exploring various functional forms of metric coefficients, we derived exact solutions of the Noether symmetry Equations (17) for the Bianchi I spacetime.
In case (i), for F = U 0 +U 1 ψ 2 + 1 2 U 2 2 ψ 2 , we obtained the Noether symmetry generators and gauge vector components for arbitrary metric coefficients.Additionally, we constructed conserved vector fields corresponding to these Noether symmetries.Selected solutions of Equation ( 25) are presented in Table I for specific trigonometric or hyperbolic metric functions other than power-law forms, involving the unknown function Y (t) in the components of Noether symmetry generators.
In case (ii), utilizing power-law forms of metric coefficients, we derived Noether symmetries in terms of constant powers L, p, and q.In Table II, a set of nontrivial solutions of Equation (43), which solve Y (t) akin to Equation (25), is provided for a specific class of Bianchi I metrics.In subcases of (ii), particularly (ii.1), (ii.2), and (ii.3), we discovered special values of powers L, p, and q, resulting in various dimensions of Noether symmetry groups.