Ambient metrics for $n$-dimensional $pp$-waves

TL;DR

This paper derives explicit ambient metrics for n-dimensional conformal pp-waves, computes obstructions in even dimensions, classifies Bach-flat 4D pp-waves, and shows their critical Q-curvature vanishes.

Contribution

It provides explicit formulas for ambient metrics of conformal pp-waves and classifies special subclasses, advancing understanding of their geometric properties.

Findings

Explicit ambient metric formulas for conformal pp-waves

Classification of Bach-flat 4D pp-waves

Vanishing critical Q-curvature in even dimensions

Abstract

We provide an explicit formula for the Fefferman-Graham-ambient metric of an $n$-dimensional conformal $pp$-wave in those cases where it exists. In even dimensions we calculate the obstruction explicitly. Furthermore, we describe all 4-dimensional $pp$-waves that are Bach-flat, and give a large class of Bach-flat examples which are conformally Cotton-flat, but not conformally Einstein. Finally, as an application, we use the obtained ambient metric to show that even-dimensional $pp$-waves have vanishing critical $Q$-curvature.