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

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.

17 pages, in v2 footnote and references added and typos corrected, in v3 remark in the Introduction about Brinkmann's results corrected and footnote added

References in corpus (1)

• A sub-product construction of Poincare-Einstein metrics

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