Classification of the Lie and Noether symmetries for the Klein-Gordon equation in Anisotropic Cosmology

TL;DR

This paper classifies Lie and Noether symmetries of the Klein-Gordon equation in anisotropic cosmological models, identifying potentials that admit these symmetries in Bianchi I, III, and V spacetimes.

Contribution

It provides a systematic classification of potentials in Klein-Gordon equations that admit Lie and Noether symmetries in specific anisotropic cosmological backgrounds.

Findings

Derived explicit potentials with symmetry properties

Connected symmetries to geometric collineations

Systematic classification for Bianchi models

Abstract

We carried out the 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.