Conformal pure radiation with parallel rays

TL;DR

This paper characterizes conformal classes of pseudo-Riemannian metrics that admit pure radiation metrics with parallel rays, using conditions on Weyl, Cotton, Bach tensors, and tractor calculus, with applications to pp-waves.

Contribution

It provides necessary and sufficient conditions for a metric to be conformal to a pure radiation metric with parallel rays, extending to pseudo-Riemannian pp-waves.

Findings

Necessary conditions involving Weyl, Cotton, Bach tensors.

Conditions in tractor calculus for existence of pure radiation metrics.

Extension of results to pseudo-Riemannian pp-waves.

Abstract

We define pure radiation metrics with parallel rays to be n-dimensional pseudo-Riemannian metrics that admit a parallel null line bundle K and whose Ricci tensor vanishes on vectors that are orthogonal to K. We give necessary conditions in terms of the Weyl, Cotton and Bach tensors for a pseudo-Riemannian metric to be conformal to a pure radiation metric with parallel rays. Then we derive conditions in terms of tractor calculus that are equivalent to the existence of a pure radiation metric with parallel rays in a conformal class. We also give an analogous result for n-dimensional pseudo-Riemannian pp-waves.