Symmetries, Conservation Laws, and Wave Equation on the Milne Metric

Abstract

Noether symmetries provide conservation laws that are admitted by Lagrangians representing physical systems. For partial differential equation possessing Lagrangians these symmetries are obtained by the invariance of the corresponding action integral. In this paper we provide a systematic procedure for determining Noether symmetries and conserved vectors for a Lagrangian constructed from a Lorentzian metric of interest in mathematical physics. For completeness, we give Lie point symmetries and conservation laws admitted by the wave equation on this Lorentzian metric.

1. Introduction

A vast amount of work is available on analyzing differential equations (DEs) through their Lie point symmetries. These symmetries are important in that they play pivotal role in solving nonlinear differential equations. Apart from Lie point symmetries, there are other interesting symmetries that are associated with differential equations which possess Lagrangians. These symmetries are called Noether symmetries and describe physical features of DEs in terms of conservation laws they admit. The connection between symmetry and conservation laws has been inherent in mathematical physics since Emmy Noether published her classical work linking the two [1]. Noether proved that for every infinitesimal transformation admitted by the action integral of a Lagrangian system, there exists a conservation law [1]. The relationship between symmetries and conservation laws in the absence of a Lagrangian is detailed in [2, 3] and references therein. Extending some of the earlier work, Bokhari et al. [4, 5] investigated Noether symmetries for the actions of certain line elements associated with the Lagrangian of some Lorentzian metrics of signature 2. More recently, extending the work in [4, 5], new results are obtained for existence of conserved quantities for spaces of different curvatures [6]. In the present study, we revisit the work in [6] and investigate Noether symmetries of the Euler Lagrange equations of a Lorentzian metric, known as the Milne Model [7] whose metric is given by 𝑑𝑠²=−𝑑𝑡²+𝑡²𝑑𝑥²+𝑒^2𝑥𝑑𝑦²+𝑑𝑧². (1.1) The Milne metric represents an empty universe and is of interest in special relativity. Our reason to choose this metric for the present study is that whereas it is zero curvature, it illustrates some features that have been associated with the expanding universe implicit in special relativity. The plan of the paper is as follows.

In the next section we find Noether symmetries of the Lagrangian constructed from (1.1). In Section 2, we construct the wave equation for the Milne metric (1.1) and then find its Noether symmetries. In the third section, the Lie symmetries of the wave on Milne metric are compared with those of the Noether symmetries obtained from its lagrangian. For details of Noether and Lie symmetries, we refer the reader to, inter alia, [8–11].

2. Invariance of the Geodesic Equations and Noether Symmetries

Geodesic equations are the Euler Lagrange equations determined from invariance of an action integral [4]. In order to find Noether symmetries admitted by the geodesic equations for the Milne metric, we write a Lagrangian, 𝐿=𝐿(𝑠,𝑡,𝑥,𝑦,𝑧,𝑡,̇𝑥̇,𝑦,̇𝑧)̇, that can be constructed by the Milne metric and given by the expression 𝐿=−𝑡²̇+𝑡²𝑥̇²+𝑒^2𝑥𝑦̇²+𝑧̇². (2.1)

The general Noether symmetry generators corresponding to this Lagrangian are [1] 𝜕𝑋=𝜉𝜕𝑠+𝜂¹𝜕𝜕𝑡+𝜂²𝜕𝜕𝑥+𝜂³𝜕𝜕𝑦+𝜂⁴𝜕+𝜂̇𝜕𝑧¹𝜕𝜕𝑡̇+𝜂̇²𝜕𝜕𝑥̇+𝜂̇³𝜕𝜕𝑦̇+𝜂̇⁴𝜕𝜕𝑧̇, (2.2) where 𝜉, 𝜂¹, 𝜂², 𝜂³, 𝜂⁴ are functions of 𝑠,  𝑡,  𝑥,  𝑦,  𝑧, and are given by 𝑋𝐿+𝐿𝐷_𝑠𝜉=𝐷_𝑠𝑓, (2.3) where 𝑓(𝑥,𝑦,𝑧,𝑡) is a gauge function. The resultant over determined system of partial differential equations, after separation by monomials, is 𝜉_𝑡=0,𝜉_𝑥=0,𝜉_𝑦𝜉=0,_𝑧=0,2𝜂¹_𝑡−𝜉_𝑠=0,2𝜂¹+2𝑡𝜂²_𝑥−𝑡𝜉_𝑠=0,2𝜂¹+2𝑡𝜂²+2𝑡𝜂³_𝑦−𝑡𝜉_𝑠=0,2𝜂¹+2𝑡𝜂²+2𝑡𝜂⁴_𝑧−𝑡𝜉_𝑠𝜂=0,_1𝑥−𝑡²𝜂²_𝑡=0,𝜂¹_𝑦−𝑡²𝑒^2𝑥𝜂³_𝑡=0,𝜂¹_𝑧−𝑡²𝑒^2𝑥𝜂⁴_𝑡𝜂=0,_2𝑦+𝑒^2𝑥𝜂³_𝑥=0,𝜂²_𝑧+𝑒^2𝑥𝜂⁴_𝑥=0,𝜂³_𝑧+𝜂⁴_𝑦=0,2𝜂¹_𝑠+𝑓_𝑡=0,2𝑡²𝜂²_𝑠−𝑓_𝑥=0,2𝑡²𝑒^2𝑥𝜂³_𝑠−𝑓_𝑦=0,2𝑡²𝑒^2𝑥𝜂⁴_𝑠−𝑓_𝑧=0,𝑓_𝑠=0. (2.4)

Solving above system of equations iteratively, we obtain 1𝜉=2𝐶₁𝑠²+𝐶₂𝑠+𝐶₃,𝜂¹=12𝑡𝑠𝐶₁+12𝑡𝐶₂−12𝑠𝑒^𝑥𝑦²+𝑧²+𝑒^−𝑥𝐶₅−12𝑠𝑧𝑒^𝑥𝐶₆−12𝑠𝑒^𝑥𝐶₇−12𝑠𝑦𝑒^𝑥𝐶₈+𝑒^−𝑥+𝑦²+𝑧²𝑒^𝑥𝐶₉+𝑧𝑒^𝑥𝐶₁₀+𝑒^𝑥𝐶₁₁+𝑦𝑒^𝑥𝐶₁₂,𝜂²=1𝑦2𝑡²+𝑧²𝑠𝑒^𝑥−𝑠𝑒^−𝑥𝐶₅+12𝑡𝑠𝑧𝑒^𝑥𝐶₆+12𝑡𝑠𝑒^𝑥𝐶₇+12𝑡𝑠𝑦𝑒^𝑥𝐶₈+1𝑡𝑒^−𝑥−𝑦²+𝑧²𝑒^𝑥𝐶₉−1𝑡𝑧𝑒^𝑥𝐶₁₀−1𝑡𝑒^𝑥𝐶₁₁−1𝑡𝑦𝑒^𝑥𝐶₁₂+2𝑦𝐶₁₃−𝑧𝐶₁₄−𝐶₁₅,𝜂³=1𝑡𝑠𝑦𝑒^−𝑥𝐶₅+12𝑡𝑠𝑒^−𝑥𝐶₈−2𝑡𝑦𝑒^−𝑥𝐶₉−1𝑡𝑒^−𝑥𝐶₁₂+𝑒^−2𝑥−𝑦²+𝑧²𝐶₁₃+𝑦𝑧𝐶₁₄+𝑦𝐶₁₅+𝑧𝐶₁₆+𝐶₁₇,𝜂⁴=1𝑡𝑠𝑧𝑒^−𝑥𝐶₅+121𝑡𝑠𝑒^−𝑥𝐶₆−2𝑡𝑧𝑒^−𝑥𝐶₉−1𝑡𝑒^−𝑥𝐶₁₀−2𝑦𝑧𝐶₁₃−12𝑒^−2𝑥+𝑦²−𝑧²𝐶₁₄+𝑧𝐶₁₅−𝑦𝐶₁₆+𝐶₁₈, (2.5) where 𝐶_𝑖 are arbitrary constants. The corresponding gauge term is given by 1𝑓=−2𝑡²𝐶₁+𝐶₄+𝑡𝑒^−𝑥+𝑦²+𝑧²𝑒^𝑥𝐶₅+𝑡𝑧𝑒^𝑥𝐶₆+𝑡𝑒^𝑥𝐶₇+𝑡𝑦𝑒^𝑥𝐶₈. (2.6) Thus, a basis for the Lie algebra of Noether point symmetry generators is (only the associated nonzero gauge terms, 𝑓, are given) 𝑋₁=12𝑠²𝜕+1𝜕𝑠2𝜕𝑠𝑡1𝜕𝑡,𝑓=−2𝑡²,𝑋₂𝜕=𝑠+1𝜕𝑠2𝑡𝜕𝜕𝑡,𝑋₃=𝜕,𝑋𝜕𝑠₄1=−2𝑠𝑒^𝑥𝑦²+𝑧²+𝑒^−𝑥𝜕+1𝜕𝑡𝑦2𝑡²+𝑧²𝑠𝑒^𝑥−𝑠𝑒^−𝑥𝜕+1𝜕𝑥𝑡𝑠𝑦𝑒^−𝑥𝜕+1𝜕𝑦𝑡𝑠𝑧𝑒^−𝑥𝜕𝑒𝜕𝑧,𝑓=𝑡^−𝑥+𝑦²+𝑧²𝑒^𝑥,𝑋₅1=−2𝑠𝑧𝑒^𝑥𝜕+1𝜕𝑡2𝑡𝑠𝑧𝑒^𝑥𝜕+1𝜕𝑥2𝑡𝑠𝑒^−𝑥𝜕𝜕𝑧,𝑓=𝑡𝑧𝑒^𝑥,𝑋₆1=−2𝑠𝑒^𝑥𝜕+1𝜕𝑡2𝑡𝑠𝑒^𝑥𝜕𝜕𝑥,𝑓=𝑡𝑒^𝑥,𝑋₇1=−2𝑠𝑦𝑒^𝑥𝜕+1𝜕𝑡2𝑡𝑠𝑦𝑒^𝑥𝜕+1𝜕𝑥2𝑡𝑠𝑒^−𝑥𝜕𝜕𝑦,𝑓=𝑡𝑦𝑒^𝑥,𝑋₈=𝑒^−𝑥+𝑦²+𝑧²𝑒^𝑥𝜕+1𝜕𝑡𝑡𝑒^−𝑥−𝑦²+𝑧²𝑒^𝑥𝜕−2𝜕𝑥𝑡𝑦𝑒^−𝑥𝜕−2𝜕𝑦𝑡𝑧𝑒^−𝑥𝜕,𝑋𝜕𝑧₉=𝑧𝑒^𝑥𝜕−1𝜕𝑡𝑡𝑧𝑒^𝑥𝜕−1𝜕𝑥𝑡𝑒^−𝑥𝜕𝜕𝑧,𝑋₁₀=𝑒^𝑥𝜕−1𝜕𝑡𝑡𝑒^𝑥𝜕,𝑋𝜕𝑥₁₁=𝑦𝑒^𝑥𝜕−1𝜕𝑡𝑡𝑦𝑒^𝑥𝜕−1𝜕𝑥𝑡𝑒^−𝑥𝜕,𝑋𝜕𝑦₁₂𝜕=2𝑦+𝑒𝜕𝑥^−2𝑥−𝑦²+𝑧²𝜕𝜕𝜕𝑦−2𝑦𝑧,𝑋𝜕𝑧₁₃𝜕=−𝑧𝜕𝜕𝑥+𝑦𝑧−1𝜕𝑦2𝑒^−2𝑥+𝑦²−𝑧²𝜕,𝑋𝜕𝑧₁₄𝜕=−𝜕𝜕𝑥+𝑦𝜕𝜕𝑦+𝑧𝜕𝑧,𝑋₁₅𝜕=𝑧𝜕𝜕𝑦−𝑦,𝑋𝜕𝑧₁₆=𝜕𝜕𝑦,𝑋₁₇=𝜕.𝜕𝑧 (2.7) The above symmetry generators form a closed Lie algebra of the symmetry group 𝐺₁₇. Further, each of these generators gives rise to a conservation law (first integral) of the geodesic equations via Noether’s theorem. For example, the symmetry generators 𝑋₃,  𝑋₁₆,  𝑋₁₇ correspond to linear momentum conservation along 𝑠, 𝑦, and 𝑧 directions, respectively, while 𝑋₁₅ represents a hyperbolic rotation. Moreover, the conserved vector associated with 𝑋₃ [12] is 𝑇=𝜉𝑡̇−𝜂¹𝜕𝐿𝜕𝑡̇+𝜉𝑥̇−𝜂²𝜕𝐿𝜕𝑥̇+𝜉𝑦̇−𝜂³𝜕𝐿𝜕𝑦̇+𝜉𝑧̇−𝜂⁴𝜕𝐿𝜕𝑧̇̇−𝜉𝐿+𝑓𝑠,𝑡,𝑥,𝑦,𝑧=𝑠𝐿+𝑡𝑡. (2.8) Note that the conservation laws obtained above are 7 more than those given by the Killing vectors [13] of the Milne metric (the additional seven generators are 𝑋₁ to 𝑋₇).

3. Wave Equation

In this context it may be interesting to study the symmetries/conservation laws of the wave equation on the Milne metric. The wave equation on this metric can be given by using the box operator [13]: 1□𝑢=𝜕−𝑔𝜕𝑥^𝑖−𝑔𝑔^𝑖𝑗𝜕𝜕𝑥^𝑖𝑢=0. (3.1)

Simplifying the above expression, wave equation takes the form 𝑢_𝑡𝑡−1𝑡²𝑢_𝑥𝑥−1𝑡²𝑒^2𝑥𝑢_𝑦𝑦−1𝑡²𝑒^2𝑥𝑢_𝑧𝑧3=−𝑡𝑢_𝑡+2𝑡²𝑢_𝑥. (3.2) The Lie point symmetries of the wave equation (3.2) are given by the formula [9] 𝑋Equation(3.2)∣_{Equation(3.2)}=0, (3.3) where 𝑋 is the prolonged symmetry generator given by 𝑋=𝜂𝜕_𝑢+𝜉⁴𝜕_𝑡+𝜉¹𝜕_𝑥+𝜉²𝜕_𝑦+𝜉³𝜕_𝑧. (3.4)

Following the method adopted in Section 2, (3.4) gives an overdetermined system of determining equations in four unknowns. This system yields a 16-dimensional Lie symmetry group 𝐺₁₆ whose basis is given by 𝑋₁=𝑡𝜕_𝑡,𝑋₂=−2𝑡𝑢𝑒^𝑥𝜕_𝑢+1+𝑡²𝑒^𝑥𝜕_𝑡+1𝑡−1+𝑡²𝑒^𝑥𝜕_𝑥,𝑋₃=−2𝑡𝑢𝑒^𝑥𝜕_𝑢+−1+𝑡²𝑒^𝑥𝜕_𝑡+1𝑡1+𝑡²𝑒^𝑥𝜕_𝑥,𝑋₄=𝜕_𝑦,𝑋₅=1𝑡𝑒^−𝑥𝜕_𝑦−𝑦𝑒^𝑥𝜕_𝑡+1𝑡𝑦𝑒^𝑥𝜕_𝑥,𝑋₆=−2𝑡𝑦𝑢𝑒^𝑥𝜕_𝑢+𝑡²𝑦𝑒^𝑥𝜕_𝑡+𝑡𝑒^−𝑥𝜕_𝑦+𝑡𝑦𝑒^𝑥𝜕_𝑥,𝑋₇=𝜕_𝑧,𝑋₈=1𝑡𝑒^−𝑥𝜕_𝑧−𝑧𝑒^𝑥𝜕_𝑡+1𝑡𝑧𝑒^𝑥𝜕_𝑥,𝑋₉=−2𝑡𝑧𝑢𝑒^𝑥𝜕_𝑢+𝑡²𝑧𝑒^𝑥𝜕_𝑡+𝑡𝑒^−𝑥𝜕_𝑧+𝑡𝑧𝑒^𝑥𝜕_𝑥,𝑋₁₀=𝑦𝜕_𝑧−𝑧𝜕_𝑦,𝑋₁₁=−𝜕_𝑥+𝑦𝜕_𝑦+𝑧𝜕_𝑧,𝑋₁₂=−2𝜕_𝑥+2𝑦𝜕_𝑦+2𝑧𝜕_𝑧+𝑢𝜕_𝑢,𝑋₁₃=2𝑡𝑦𝑒^−𝑥𝜕_𝑦+2𝑡𝑧𝑒^−𝑥𝜕_𝑧+−1−𝑒^2𝑥𝑦²+𝑧²𝑒^−𝑥𝜕_𝑡+1𝑡−1+𝑒^2𝑥𝑦²+𝑧²𝑒^−𝑥𝜕_𝑥,𝑋₁₄=−2𝑡𝑢1+𝑒^2𝑥𝑦²+𝑧²𝑒^−𝑥𝜕_𝑢+2𝑡𝑦𝑒^−𝑥𝜕_𝑦+2𝑡𝑧𝑒^−𝑥𝜕_𝑧+𝑡−1+𝑒^2𝑥𝑦²+𝑧²𝑒^−𝑥𝜕_𝑥+𝑡²1+𝑒^2𝑥𝑦²+𝑧²𝑒^−𝑥𝜕_𝑡,𝑋₁₅=2𝑦𝜕_𝑥−2𝑦𝑧𝜕_𝑧+𝑒^−2𝑥−𝑦²+𝑧²𝜕_𝑦,𝑋₁₆=−2𝑦𝑧𝜕_𝑦+2𝑧𝜕_𝑥+𝑒^−2𝑥+𝑦²−𝑧²𝜕_𝑧. (3.5) It may be worth mentioning that 𝑋₁ gives a scaling conservation law along 𝑡 direction, 𝑋₄,  𝑋₇ linear momentum conservations in 𝑦 and 𝑧 directions, respectively, while 𝑋₁₀ represents rotation in 𝑦𝑧 plane.

4. Discussion and Conclusions

From our investigation we find that the Noether symmetries of the Lagrangian form a maximal set of  𝟏𝟕  conservation laws. Since Milne metric represents a flat cosmological model, it admits maximal isometries. It is shown that the Noether symmetries for the Milne model give 𝟕 additional conservation laws which are not given by the symmetries of the spacetime metric [13]. In exactly similar fashion, we find that the wave equation on the Milne metric also admits maximal group of isometries. It suggests that if the Lorentzian metric is flat, both Noether and Lie point symmetry groups will be maximal. It may be worth mentioning that the number of Noether symmetries in this study differs with the one given in [12]. It will be interesting to extend such investigations to more general Lorentzian metrics. It is hoped that such investigations will add to our understanding of Lie and Noether point symmetries for such metrics [13].

Acknowledgment

The authors would like to thank King Fahd University of Petroleum and Minerals for facilities to complete this work.