Symmetries, Differential Equations, and Applications: Galois BicentenaryView this Special Issue
Symmetries, Conservation Laws, and Wave Equation on the Milne Metric
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.
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 . Noether proved that for every infinitesimal transformation admitted by the action integral of a Lagrangian system, there exists a conservation law . 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 . In the present study, we revisit the work in  and investigate Noether symmetries of the Euler Lagrange equations of a Lorentzian metric, known as the Milne Model  whose metric is given by 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 . In order to find Noether symmetries admitted by the geodesic equations for the Milne metric, we write a Lagrangian, , that can be constructed by the Milne metric and given by the expression
The general Noether symmetry generators corresponding to this Lagrangian are  where , , , , are functions of , , , , , and are given by where is a gauge function. The resultant over determined system of partial differential equations, after separation by monomials, is
Solving above system of equations iteratively, we obtain where are arbitrary constants. The corresponding gauge term is given by Thus, a basis for the Lie algebra of Noether point symmetry generators is (only the associated nonzero gauge terms, , are given) The above symmetry generators form a closed Lie algebra of the symmetry group . 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 , , correspond to linear momentum conservation along , , and directions, respectively, while represents a hyperbolic rotation. Moreover, the conserved vector associated with  is Note that the conservation laws obtained above are more than those given by the Killing vectors  of the Milne metric (the additional seven generators are to ).
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 :
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 whose basis is given by It may be worth mentioning that gives a scaling conservation law along direction, , linear momentum conservations in and directions, respectively, while represents rotation in plane.
4. Discussion and Conclusions
From our investigation we find that the Noether symmetries of the Lagrangian form a maximal set of 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 additional conservation laws which are not given by the symmetries of the spacetime metric . 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 . 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 .
The authors would like to thank King Fahd University of Petroleum and Minerals for facilities to complete this work.
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