The vacuum weighted Einstein field equations on pr-waves

We classify solutions of the vacuum weighted Einstein field equations on smooth metric measure spacetimes (M,g,h\, dvolg) 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 ∇ 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.