On Waylen's regular axisymmetric similarity solutions

TL;DR

This paper reviews Waylen's similarity solutions for axisymmetric vacuum spacetimes, revealing their connection to Baecklund transforms and showing these solutions inherently admit a Killing vector, possibly time-like.

Contribution

It demonstrates the relationship between Waylen's solutions and Baecklund transforms, and establishes that these solutions naturally possess a (possibly homothetic) Killing vector.

Findings

Key equation is related to a Baecklund transform.

Solutions automatically admit a Killing vector.

Solutions may have a time-like Killing vector.

Abstract

We review the similarity solutions proposed by Waylen for a regular time-dependent axisymmetric vacuum space-time, and show that the key equation introduced to solve the invariant surface conditions is related by a Baecklund transform to a restriction on the similarity variables. We further show that the vacuum space-times produced via this path automatically possess a (possibly homothetic) Killing vector, which may be time-like.