On pp-waves with lightlike parallel spinors

TL;DR

This paper characterizes pp-wave spacetimes with lightlike parallel spinors, establishing a correspondence with Ricci-flat metrics and analyzing conditions for null Ricci curvature and parallel spinors.

Contribution

It provides a parametrization of pp-wave spacetimes with compact hypersurfaces and links their geometric properties to Ricci-flat metrics and spinor fields.

Findings

PP-wave spacetimes are parametrized by smooth curves of Ricci-flat metrics.

The correspondence extends to pp-waves with prescribed null Ricci curvature.

Presence of a lightlike parallel spinor is equivalent to Ricci-flat metrics carrying a parallel spinor.

Abstract

We parametrize pp-wave spacetimes with compact codimension 2 hypersurfaces. In the vacuum case, we show that these spacetimes are locally in one-to-one correspondence with smooth curves of Riemannian Ricci-flat metrics modulo smooth curves of diffeomorphisms. We also prove that this one-to-one correspondence extends to pp-waves with prescribed null Ricci curvature. Moreover, the pp-wave spacetime carries a lightlike parallel spinor if and only if one (and hence all) of the Ricci-flat metrics carries a parallel spinor.