The vacuum weighted Einstein field equations on pr-waves

TL;DR

This paper classifies solutions to vacuum weighted Einstein equations on 4D pr-wave spacetimes, revealing geometric properties and special cases like $pp$-waves and nilpotent Ricci operators.

Contribution

It provides a comprehensive classification of solutions on pr-waves, including new examples with specific geometric features such as lightlike or spacelike gradients.

Findings

Solutions include $pp$-waves with lightlike gradients.

Solutions with spacelike gradients have nilpotent Ricci operators.

2-step nilpotent solutions are realized on $pp$-waves.

Abstract

We classify solutions of the vacuum weighted Einstein field equations on smooth metric measure spacetimes $(M,g,h\, dvol_g)$ of dimension 4, where the underlying manifold $(M,g)$ is a $pr$-wave. We use this result to provide examples of solutions with some special geometric properties. The gradient $\nabla h$ is lightlike or spacelike. In the first case, the underlying manifold is a $pp$-wave. In the second case, the Ricci operator is nilpotent. Moreover, $2$-step nilpotent solutions are also realized on $pp$-waves.