Vacuum gravitational fields with a null Killing vector

TL;DR

This paper reviews and extends the study of vacuum gravitational fields with a null Killing vector, focusing on twisted gravitational waves characterized by a Laplace equation in cylindrical coordinates.

Contribution

It provides a systematic review and extension of solutions for null Killing vector gravitational fields, highlighting the structure of twisted gravitational waves.

Findings

Identification of a simple metric form in suitable coordinates.

Twisted gravitational waves are characterized by a Laplace equation in cylindrical coordinates.

The paper clarifies the relationship between plane waves and twisted gravitational waves.

Abstract

Vacuum gravitational fields admitting a light-like Killing field were systematically studied starting around 1960. Besides the already known plane waves, a second class of gravitational wave fields was found. In contrast to plane waves, their wave surfaces were not flat, but had a negative Gaussian curvature. Recently, such solutions found attention again as "twisted gravitational waves". In the paper we review and extend the earlier results. In suitable coordinates, the metric assumes a simple shape. The waves are then determined by a single function that satisfies a Laplace equation in cylindrical coordinates. The "twisted waves" prove to be a special case.