# Classification of the Lie and Noether point symmetries for the Wave and the Klein-Gordon equations in pp-wave spacetimes

## TL;DR

This paper classifies Lie and Noether point symmetries of the Klein-Gordon and wave equations in pp-wave spacetimes, linking symmetries to conformal Killing vectors and identifying maximum symmetry cases.

## Contribution

It provides a systematic classification of symmetries for these equations in pp-wave spacetimes, including the functional form of potentials and symmetry algebra dimensions.

## Key findings

- Maximum Noether algebra dimension is seven in plane wave spacetimes.
- Symmetries depend on the existence of conformal Killing vectors.
- Explicit forms of potentials admitting symmetries are determined.

## Abstract

We perform a classification of the Lie and Noether point symmetries for the Klein-Gordon and for the wave equation in pp-wave spacetimes. To perform this analysis we reduce the problem of the determination of the point symmetries to the problem of existence of conformal killing vectors on the pp-wave spacetimes. We use the existing results of the literature for the isometry classes of the pp-wave spacetimes and we determine in each class the functional form of the potential in which the Klein-Gordon equation admits point symmetries and Noetherian conservation law. Finally we derive the point symmetries of the wave equation and we find that the maximum Noether algebra has dimension seven, that is the case of plane wave spacetimes.