Classification of the Lie and Noether Symmetries for the Klein-Gordon Equation in Anisotropic Cosmology

We carried out a detailed group classification of the potential in Klein-Gordon equation in anisotropic Riemannian manifolds. Specifically, we consider the Klein-Gordon equations for the four-dimensional anisotropic and homogeneous spacetimes of Bianchi I, Bianchi III and Bianchi V. We derive all the closed-form expressions for the potential function where the equation admits Lie and Noether symmetries. We apply previous results which connect the Lie symmetries of the differential equation with the collineations of the Riemannian space which defines the Laplace operator, and we solve the classification problem in a systematic way.

Automated summary

We conducted a detailed study on the potential classification of the Klein-Gordon equation in anisotropic Riemannian manifolds. Specifically, we focused on the Klein-Gordon equations in four-dimensional anisotropic and homogeneous spacetimes of Bianchi I, Bianchi III, and Bianchi V. By deriving closed-form expressions for the potential function, we were able to find the Lie and Noether symmetries of the equations. Applying previous results connecting the Lie symmetries with the collineations of the Riemannian space, we systematically solved the classification problem.