Wave Equations in Bianchi Space-Times

Abstract

We investigate the wave equation in Bianchi type III space-time. We construct a Lagrangian of the model, calculate and classify the Noether symmetry generators, and construct corresponding conserved forms. A reduction of the underlying equations is performed to obtain invariant solutions.

1. Introduction

The study of partial differential equations (PDEs) in terms of Lie point symmetries is well known and well established [1–5], where these symmetries can be used to obtain, inter alia, exact analytic solutions of the PDEs. In addition, Noether symmetries are also widely investigated and are associated with PDEs that possess a Lagrangian. Noether [6] discovered the interesting link between symmetries and conservation laws showing that for every infinitesimal transformation admitted by the action integral of a system there exists a conservation law. Investigations have been devoted to understand Noether symmetries of Lagrangians that arise from certain pseudo-Riemannian metrics of interest [7, 8]. Recently, a study was aimed at understanding the effect of gravity on the solutions of the wave equation by solving the wave equation in various space-time geometries [9].

In [10], the Bianchi universes were investigated using Noether symmetries. The authors of [11] studied the Noether symmetries of Bianchi type I and III space-times in scalar coupled theories. Therein, they obtained the exact solutions for potential functions, scalar field, and the scale factors, see also [12].

We pursue an investigation of the symmetries of the wave equation in Bianchi III space-time. We construct solutions of these equations and find conservation laws associated with Noether symmetries. The plan of the paper is as follows.

In Section 2, we discuss the procedure to obtain an expression representing Noether symmetries and conservation laws. In Section 3, we derive and classify strict Noether symmetries of the Bianchi III space-time. Also in Section 3, we briefly describe the relation of Noether symmetries to conservation laws. We then illustrate the reduction of the wave equation and obtain invariant solutions.

2. Definitions and Notation

We briefly outline the notation and pertinent results used in this work. In this regard, the reader is referred to [13].

The convention that repeated indices imply summation is used. Let 𝑥=(𝑥¹,𝑥²,…,𝑥^𝑛)∈𝑅^𝑛 be independent variable with 𝑥^𝑖, and let 𝑢=(𝑢¹,𝑢²,…,𝑢^𝑚)∈𝑅^𝑚 be the dependent variable with coordinates 𝑢^𝛼. Furthermore, let 𝜋∶𝑅^𝑛+𝑚→𝑅^𝑛 be the projection map 𝜋(𝑥,𝑢)=𝑥. Also, suppose that 𝑠∶𝜒⊂𝑅^𝑛→𝒰⊂𝑅^𝑛+𝑚 is a smooth map such that 𝜋∘𝑠=1_𝜒, where 1_𝜒 is the identity map on 𝜒. The 𝑟-jet bundle 𝒥^𝑟(𝒰) is given by the equivalence classes of sections of 𝒰. The coordinates on 𝒥^𝑟(𝒰) are denoted by (𝑥^𝑖,𝑢^𝛼,…,𝑢^𝛼_{𝑖1⋯𝑖𝑟}), where 1≤𝑖₁≤⋯≤𝑖_𝑟≤𝑛 and 𝑢^𝛼_{𝑖1⋯𝑖𝑟} corresponds to the partial derivatives of 𝑢^𝛼 with respect to 𝑥^𝑖1,…,𝑥^𝑖𝑟. The partial derivatives of 𝑢 with respect to 𝑥 are connected by the operator of total differentiation 𝐷_𝑖=𝜕𝜕𝑥^𝑖+𝑢^𝛼_𝑖𝜕𝜕𝑢^𝛼+𝑢^𝛼_𝑖𝑗𝜕𝜕𝑢^𝛼_𝑗+⋯,𝑖=1,…,𝑛, (2.1) as 𝑢^𝛼_𝑖=𝐷_𝑖𝑢^𝛼,𝑢^𝛼_𝑖𝑗=𝐷_𝑗𝐷_𝑖𝑢^𝛼,… (2.2) The collection of all first-order derivatives 𝑢^𝑎_𝑖 will be denoted by 𝑢₍₁₎. Similarly, the collections of all higher order derivatives will be denoted by 𝑢₍₂₎,𝑢₍₃₎,….

The 𝑟-jet bundle on U will be written as 𝒥^𝑟(𝒰)={(𝑥,𝑢,𝑢₍₁₎,…,𝑢_(𝑟))/(𝑥,𝑢)∈𝒰}. We now review the space of differential forms on 𝒥^𝑟(𝒰). To this end, let Ω^𝑟_𝑘(𝒰) be the vector space of differential 𝑘-forms on 𝒥^𝑟(𝒰) with differential 𝑑. A smooth differential 𝑘-form on 𝒥^𝑟(𝒰) is given by 𝜔=𝑓_{𝑖1,𝑖2,…,𝑖𝑘}𝑑𝑥^𝑖1∧𝑑𝑥^𝑖2∧⋯∧𝑑𝑥^𝑖𝑘, (2.3) where each component 𝑓_{𝑖1,𝑖2,…,𝑖𝑘}∈Ω^𝑟₀(𝒰). Note that for differential functions 𝑓∈Ω^𝑟₀(𝒰), 𝐷𝑓=𝐷_𝑗𝑓𝑑𝑥^𝑗, (2.4) where 𝐷 is the total differential or the total exterior derivative. Moreover, the total exterior derivative of 𝜔 is 𝐷𝜔=𝐷𝑓_{𝑖1,𝑖2,…,𝑖𝑘}𝑑𝑥^𝑖1∧𝑑𝑥^𝑖2∧⋯∧𝑑𝑥^𝑖𝑘, (2.5) and by invoking (2.4) one has 𝐷𝜔=𝐷𝑓_{𝑖1,𝑖2,…,𝑖𝑘}𝑑𝑥^𝑗∧𝑑𝑥^𝑖1∧𝑑𝑥^𝑖2∧⋯∧𝑑𝑥^𝑖𝑘. (2.6) The total differential 𝐷 has properties analogous to the algebraic properties of the usual exterior derivative 𝑑: 𝐷(𝜔∧𝜐)=𝐷𝜔∧𝜐+(−1)^𝑘𝜔∧𝐷𝜐, (2.7) for 𝜔 a 𝑘-form and 𝜐 an 𝑙-form and 𝐷(𝐷𝜔)=0. Also, it is known that if 𝐷(𝐷𝜔)=0, then 𝜔 is a locally exact 𝑘-form, that is, 𝜔=𝐷𝜐 for some (𝑘−1)-form 𝜐, [14].

2.1. Action of Symmetries

Consider an 𝑟th-order system of partial differential equations of 𝑛 independent variables and 𝑚 dependent variables: 𝐸^𝛽𝑥,𝑢,𝑢₍₁₎,…,𝑢_(𝑟)=0,𝛽=1,…,𝑚. (2.8)

Definition 2.1. A conserved form of (2.8) is a differential (𝑛−1)-form, 𝜔=𝑇^𝑖𝑥,𝑢,𝑢₍₁₎,…,𝑢_(𝑟−1)𝜕𝜕𝑥^𝑖⌟𝑑𝑥¹∧⋯∧𝑑𝑥^𝑛, (2.9) defined on 𝒥^𝑟−1(𝒰) if 𝐷𝜔=0 (2.10) is satisfied on the surface given by (2.8).

Remark 2.2. When Definition 2.1 is satisfied, (2.10) is called a conservation law for (2.8).
It is clear that (2.10) evaluated on the surface (2.8) implies that 𝐷_𝑖𝑇^𝑖=0 (2.11) on the surface given by (2.8), which is also referred to as a conservation law of (2.8). The tuple 𝑇=(𝑇¹,…,𝑇^𝑛), 𝑇^𝑗∈Ω₀^𝑟−1(𝒰), 𝑗=1,…,𝑛, is called a conserved vector of (2.8).

We now review some definitions and results relating to Euler-Lagrange, Lie, and Noether operators ([15, 16] and references therein).

Let 𝒜=⋃^𝑝_𝑟=0Ω^𝑟₀(𝒰) for some 𝑝<∞. Then 𝒜 is the universal space of differential functions of finite orders.

Consider a symmetry operator given by the infinite formal sum: 𝑋=𝜉^𝑖𝜕𝜕𝑥^𝑖+𝜂^𝛼𝜕𝜕𝑢^𝛼+_𝑠≥1𝜁^𝛼_{𝑖1⋯𝑖𝑠}𝜕𝜕𝑢^𝛼_{𝑖1⋯𝑖𝑠}, (2.12) where 𝜉^𝑖, 𝜂^𝛼∈𝒜, and the additional coefficients are determined uniquely by the prolongation formulae 𝜁^𝛼_{𝑖1⋯𝑖𝑠}=𝐷_𝑖1⋯𝐷_𝑖𝑠𝑊^𝛼+𝜉^𝑗𝑢^𝛼_{𝑗𝑖1…𝑖𝑠},𝑠>1. (2.13) In (2.13), 𝑊^𝛼 is the Lie characteristic function given by 𝑊^𝛼=𝜂^𝛼−𝜉^𝑗𝑢^𝛼_𝑗. (2.14) In particular, a symmetry operator of the form 𝑋=𝑄^𝛼𝜕/𝜕𝑢^𝛼+⋯, where 𝑄^𝛼∈𝒜, is called a canonical or evolutionary representation of 𝑋, and 𝑄^𝛼 is called its characteristic.

An operator 𝑋 is said to be a Noether symmetry corresponding to a Lagrangian 𝐿∈𝒜, if there exists a vector 𝐵^𝑖=(𝐵¹,…,𝐵^𝑛),𝐵^𝑖∈𝒜, such that 𝑋(𝐿)+𝐿𝐷_𝑖𝜉^𝑖=𝐷_𝑖𝐵^𝑖. (2.15) If 𝐵^𝑖=0,(𝑖=1,…,𝑛), then 𝑋 is referred to as a strict Noether symmetry corresponding to a Lagrangian 𝐿∈𝒜. This case is also obtained by setting the Lie derivative on the 𝑛-form 𝐿𝑑𝑥¹∧⋯∧𝑑𝑥^𝑛 in the direction of 𝑋 to zero, that is, ℒ_𝑋𝐿𝑑𝑥¹∧⋯∧𝑑𝑥^𝑛=𝑋𝐿𝑑𝑥¹∧⋯∧𝑑𝑥^𝑛=0, (2.16) where ℒ is the Lie derivative operator.

In view of the above discussions and definitions, the Noether theorem [6] is formulated as follows.

Noether’s Theorem
For any Noether symmetry 𝑋 corresponding to a given Lagrangian 𝐿∈𝒜, there corresponds a vector 𝑇^𝑖=(𝑇¹,…,𝑇^𝑛),𝑇^𝑖∈𝒜, defined by 𝑇^𝑖=𝐵^𝑖−𝑁^𝑖(𝐿),𝑖=1,…,𝑛, (2.17) which is a conserved current of the Euler-Lagrange equations 𝛿𝐿/𝛿𝑢^𝛼=0, 𝛼=1,…,𝑚, where 𝛿/𝛿𝑢^𝛼 is the Euler-Lagrange operator given by 𝛿𝛿𝑢^𝛼=𝜕𝜕𝑢^𝛼+_𝑠≥1(−1)^𝑠𝐷_𝑖1⋯𝐷_𝑖𝑠𝜕𝜕𝑢^𝛼_{𝑖1⋯𝑖𝑠},𝛼=1,…,𝑚, (2.18) and the Noether operator associated with the operator 𝑋 is given by 𝑁^𝑖=𝜉^𝑖+𝑊^𝛼𝛿𝛿𝑢^𝛼_𝑖+_𝑠≥1𝐷_𝑖1⋯𝐷_𝑖𝑠𝑊^𝛼𝛿𝛿𝑢^𝛼_{𝑖𝑖1⋯𝑖𝑠},𝑖=1,…,𝑛, (2.19) in which the Euler-Lagrange operators with respect to derivatives of 𝑢^𝛼 are obtained from (2.18) by replacing 𝑢^𝛼 by the corresponding derivatives, for example, 𝛿𝛿𝑢^𝛼_𝑖=𝜕𝜕𝑢^𝛼_𝑖+_𝑠≥1(−1)^𝑠𝐷_𝑗1⋯𝐷_𝑗𝑠𝜕𝜕𝑢^𝛼_{𝑖𝑗1⋯𝑗𝑠},𝑖=1,…,𝑛,𝛼=1,…,𝑚. (2.20)

3. Bianchi III Space-Time

Consider the Bianchi III metric: 𝑑𝑠²=−𝛽²𝑑𝑡²+𝑡^2𝐿𝑑𝑥²+𝑒^{−2𝑎𝑥/𝑁}𝑑𝑦²+𝑡^2𝐿/𝑚𝑑𝑧². (3.1) The wave equation in (3.1) takes the form [17] −1𝛽(2𝐿+1)𝑡^2𝐿𝑒^{−(𝑎/𝑁)𝑥}𝑢_𝑡−1𝛽𝑡^2𝐿+1𝑒^{−(𝑎/𝑁)𝑥}𝑢_𝑡𝑡−𝑎𝛽𝑁𝑡𝑒^{−(𝑎/𝑁)𝑥}𝑢_𝑥+𝛽𝑡𝑒^{−(𝑎/𝑁)𝑥}𝑢_𝑥𝑥+𝛽𝑡𝑒^{(𝑎/𝑁)𝑥}𝑢_𝑦𝑦+𝛽𝑡^{2𝐿+1−(2𝐿/𝑚)}𝑒^{−(𝑎/𝑁)𝑥}𝑢_𝑧𝑧=0. (3.2)

In [17], some aspects of the wave equation on the Bianchi metric were studied. The multiplier method [1] was adopted to determine some of the conserved densities. This lengthy procedure ultimately leads to the construction of only three symmetries and its associated conserved vectors.

In this paper, we investigate the wave equation on the Bianchi III metric using Noether’s theorem and the method of differential forms. We obtain a wide range of results and also perform symmetry reductions of the wave equation for some cases to obtain invariant solutions. For the purposes of Sections 3.1 and 3.2, we denote the Lagrangian by 𝐋.

3.1. The Strict Noether Symmetries of (3.2)

We classify the cases that yield strict Noether symmetries (gauge is zero) of (3.2), via the Lagrangian 𝛽𝐋=−2𝑡𝑒^{−(𝑎/𝑁)𝑥}𝑢_𝑥²−𝛽2𝑡𝑒^{(𝑎/𝑁)𝑥}𝑢_𝑦²−𝛽2𝑡^{2𝐿+1−(2𝐿/𝑚)}𝑒^{−(𝑎/𝑁)𝑥}𝑢_𝑧²+1𝑡2𝛽^2𝐿+1𝑒^{−(𝑎/𝑁)𝑥}𝑢_𝑡². (3.3) Many of the calculations have been left out as they are tedious—the details are available to the reader in a number of texts that have been cited here.

The principle Noether algebra is 𝑋₁=𝜕_𝑦,𝑋₂=𝜕_𝑧,𝑋₃=𝜕_𝑢,𝑋₄=𝑁𝑎𝜕_𝑥+𝑦𝜕_𝑦,𝑋₅=−2𝑎𝑦𝑁𝜕_𝑥+1𝑁²−𝑎²𝑦²+𝑁²𝑒^{2𝑎𝑥/𝑁}𝜕_𝑦. (3.4) Furthermore, specific cases of 𝐿 and 𝑚 give rise to the symmetries 𝑋₁ to 𝑋₅ from above, and some additional symmetries.

Case 1 (𝐿=1, 𝑚=1/3). The additional symmetries are, 𝑋₆=−2𝑧𝜕_𝑧+𝑡𝜕_𝑡,𝑋₇=−4𝑧²𝑡⁴+𝛽²4𝑡⁴𝜕_𝑧+𝑧𝑡𝜕_𝑡. (3.5)

Case 2 (𝐿=1, 𝑚=−1). The additional symmetries are, 𝑋₈=−𝑧𝜕_𝑧−𝑡2𝜕_𝑡+𝑢𝜕_𝑢. (3.6)

Case 3 (𝐿=1, 𝑚=1). The additional symmetries are, 𝑋₉=−𝑡𝜕_𝑡+𝑢𝜕_𝑢. (3.7)

Table 1 contains the conserved forms corresponding to each Noether symmetry 𝑋_𝑖(𝑖=1,…,9), that is, it lists the three form 𝜔. The four form is 𝐷𝜔, and 𝐷𝜔 vanishes on the solutions of (3.2). Thus 𝜔=−𝑇^𝑡𝑑𝑥∧𝑑𝑦∧𝑑𝑧+𝑇^𝑧𝑑𝑥∧𝑑𝑦∧𝑑𝑡−𝑇^𝑦𝑑𝑥∧𝑑𝑧∧𝑑𝑡+𝑇^𝑥𝑑𝑦∧𝑑𝑧∧𝑑𝑡, (3.8) so that 𝐷_𝑡𝑇^𝑡+𝐷_𝑥𝑇^𝑥+𝐷_𝑦𝑇^𝑦+𝐷_𝑧𝑇^𝑧=0 (3.9) in (3.2).

Table 1 

Conserved Form 𝜔.

3.2. Symmetry Reduction and Invariant Solutions

We briefly show how the order of the (1+3) wave equation (3.2) can be reduced. Ultimately the equation with four independent variables is reduced to an ordinary differential equation.

3.2.1. Reduction—Using the Principle Noether Algebra

We begin reducing (3.2) using 𝑋₂ followed by 𝑋₄. The characteristic equations are 𝑎𝑑𝑥𝑁=𝑑𝑡0=𝑑𝑦𝑦=𝑑𝑢0. (3.10) Integrating yields 𝑠=𝑦𝑒^{−(𝑎/𝑁)𝑥} and (3.2) is reduced to −1𝛽𝑡^2𝐿𝑢_𝑡𝑡−1𝛽(2𝐿+1)𝑡^2𝐿𝑢_𝑡𝑎+𝛽𝑡1+²𝑁²𝑠²𝑢_𝑠𝑠𝑎+2𝑠²𝑁²𝛽𝑡𝑢_𝑠=0, (3.11) with 𝑢=𝑢(𝑠,𝑡).

A Lagrangian of (3.11) is 1𝐋=−𝛽𝑡^2𝐿+1𝑢²_𝑡2𝑎+𝛽𝑡1+²𝑁²𝑠²𝑢²_𝑠2. (3.12) It turns out that we if we let 𝐿=1, 𝑁=1 in (3.11), we can obtain its Noether symmetries, namely, 𝑋^∗₁=𝑡𝜕_𝑡−𝑢𝜕_𝑢,𝑋^∗₂=𝜕_𝑢. (3.13) We reduce (3.11) with 𝑋^∗₁, and the characteristic equations are 𝑑𝑡𝑡=𝑑𝑠0=𝑑𝑢.−𝑢 (3.14) Integrating yields 𝑌=𝑡𝑢 and (3.11) is reduced to the ordinary differential equation: 1𝛽𝑌+𝛽1+𝑠²𝑎²𝑌^′′+2𝑎²𝛽𝑠𝑌^′=0, (3.15) with 𝑌=𝑌(𝑠), and which has a solution in terms of special functions, that is, 𝑌(𝑠)=𝐶₁Legendre𝑃−𝑎𝛽+−4+𝑎²𝛽²2𝑎𝛽,𝑖𝑎𝑠+𝐶₂Legendre𝑄−𝑎𝛽+−4+𝑎²𝛽²,2𝑎𝛽,𝑖𝑎𝑠 (3.16) where 𝐶₁, 𝐶₂ are arbitrary constants, Legendre𝑃[𝑛,𝑥] refers to the Legendre polynomial 𝑃_𝑛(𝑥), and 𝑄[𝑛,𝑧] refers to the Legendre function of the second kind 𝑄_𝑛(𝑧).

3.2.2. Reduction—Case 1, 𝐿=1, 𝑚=1/3

We reduce (3.2) using 𝑋₁ followed by 𝑋₆ from Case 1. The characteristic equations are 𝑑𝑧=−2𝑧𝑑𝑡𝑡=𝑑𝑥0=𝑑𝑢0. (3.17) Integrating yields 𝑟=𝑡²𝑧 and (3.2) is reduced to −8𝛽𝑟𝑒⁻𝑎𝑥𝑁𝑢_𝑟−𝑎𝛽𝑁𝑒⁻𝑎𝑥𝑁𝑢_𝑥+𝛽𝑒⁻𝑎𝑥𝑁𝑢_𝑥𝑥+𝑒⁻𝑎𝑥𝑁𝛽²−4𝑟²𝛽𝑢_𝑟𝑟=0, (3.18) with 𝑢=𝑢(𝑟,𝑥).

A Lagrangian of (3.18) is 𝐋=𝛽𝑒^{−𝑎𝑥/𝑁}𝑢²_𝑥2+𝑒^{−𝑎𝑥/𝑁}𝛽²−4𝑟²𝛽𝑢²_𝑟2. (3.19) Hence, we obtain the Noether symmetries of (3.18), namely, 𝑋^∗₃=𝜕_𝑥+𝑎2𝑁𝑢𝜕_𝑢,𝑋^∗₄=𝜕_𝑢. (3.20) We reduce (3.18) with 𝑋^∗₃, and the characteristic equations are 𝑑𝑥1=𝑑𝑟0=2𝑁𝑑𝑢.𝑎𝑢 (3.21) Integrating yields 𝑍=𝑒^{−𝑎𝑥/2𝑁}𝑢 and (3.18) is reduced to the ordinary differential equation: −8𝛽𝑟𝑍^′+𝛽²−4𝑟²𝛽𝑍^′′−𝑎²𝛽4𝑁²𝑍=0, (3.22) with 𝑍=𝑍(𝑟), and which has a solution in terms of special functions, that is, 𝑍(𝑟)=𝐶₁Legendre𝑃−2𝑁+4𝑁²−𝑎²𝛽²,4𝑁2𝑟𝛽+𝐶₂Legendre𝑄−2𝑁+4𝑁²−𝑎²𝛽²,4𝑁2𝑟𝛽, (3.23) where, as before, 𝐶₁, 𝐶₂ are arbitrary constants, Legendre𝑃[𝑛,𝑥] refers to the Legendre polynomial 𝑃_𝑛(𝑥), and 𝑄[𝑛,𝑧] refers to the Legendre function of the second kind 𝑄_𝑛(𝑧).

4. Conclusion

We classified the Noether symmetry generators, determined some conserved forms, and reduced some cases of the underlying equations associated with the wave equation on the Bianchi III manifold. The first reduction done above involved the principle Noether algebra, whilst the second dealt with a particular case. To obtain other reductions, one needs to conclude a three-dimensional subalgebra of symmetries to reduce to an ordinary differential equation whose solution would be an invariant solution invariant under the subalgebra. Alternatively, a lower dimensional subalgebra can be used to reduce to a partial differential equation which may be tackled using other methods. The final solution in this case will be invariant only under the lower dimensional algebra. In general, the procedure performed above is the most convenient.