# Conformally flat travelling plane wave solutions of Einstein equations

## TL;DR

This paper explores conformally flat plane wave solutions to Einstein's equations, incorporating various matter sources like scalar fields, electromagnetic fields, and relativistic particles, providing explicit metrics for these scenarios.

## Contribution

It introduces explicit conformally flat solutions of Einstein equations with diverse matter sources depending on the wave phase, expanding the class of known exact solutions.

## Key findings

- Explicit conformally flat metrics for scalar, electromagnetic, and particle sources.
- Solutions depend on the wave phase with specific frequency conditions.
- Models describe massless scalar fields, electromagnetic waves, and relativistic particles.

## Abstract

We discuss conformally flat plane wave solutions of Einstein equations depending on the plane wave phase $\xi=\omega\tau-{\bf qx}$, where $\tau$ is the conformal time. We show that ideal fluid Einstein equations and scalar fields with exponential self-interaction have solutions of this form. We consider in more detail the source depending on $\xi$ with $\omega=\vert{\bf q}\vert$ describing models of a massless scalar field, electromagnetic field and relativistic particles with space-time depending mass density. We obtain explicit conformally flat metrics solving Einstein equations with such a source of the energy-momentum.