Symmetry analysis of the Klein-Gordon equation in Bianchi I spacetimes

TL;DR

This paper classifies the Lie and Noether symmetries of the Klein-Gordon equation in Bianchi I spacetimes using geometric methods, identifying potentials that admit these symmetries and deriving invariant solutions.

Contribution

It provides a comprehensive symmetry classification of the Klein-Gordon equation in Bianchi I spacetimes, linking symmetries to the conformal algebra of the geometry.

Findings

Identified all potentials with Lie and Noether symmetries.

Derived invariant solutions for specific potentials.

Solved the classification problem for wave equations in Bianchi I.

Abstract

In this work we perform the symmetry classification of the Klein Gordon equation in Bianchi I spacetime. We apply a geometric method which relates the Lie symmetries of the Klein Gordon equation with the conformal algebra of the underlying geometry. Furthermore, we prove that the Lie symmetries which follow from the conformal algebra are also and Noether symmetries for the Klein Gordon equation. We use these resutls in order to determine all the potentials in which the Klein Gordon admits Lie and Noether symmetries. Due to the large number of cases and for easy reference the results are presented in the form of tables. For some of the potentials we use the Lie admitted symmetries to determine the corresponding invariant solution of the Klein Gordon equation. Finally, we show that the results also solve the problem of classification of Lie/Noether point symmetries of the wave equation in Bianchi I spacetime and can be used for the determination of invariant solutions of the wave equation.