Translation in Cylindrically Symmetric Vacuum

TL;DR

This paper explores a cylindrically symmetric vacuum solution in General Relativity, revealing its relation to rotating sources and establishing the non-existence of translating cylindrical dust solutions.

Contribution

It demonstrates the geometric relation between a translated cylindrically symmetric vacuum and the Lewis solution, and proves that translating cylindrical dust solutions cannot exist in GR.

Findings

The metric relates to the Lewis vacuum solution.

No translating cylindrical dust solutions exist in GR.

New physical properties differ from the Lewis solution.

Abstract

Starting from the stationary cylindrically symmetric solution, but with the coordinates $z$ and $\phi$ interchanged, and supposing that it could describe the vacuum spacetime of a translating cylinder, we investigate its physical and geometrical properties. This hypothesis is not entirely new since it has already been considered in a previous paper describing a translating source. We show that this metric is geometrically related to the vacuum field produced by a stationary rotating cylindrical source{, known as Lewis solution. However, we find new physical properties, different from those of the Lewis vacuum solution. Moreover, we show that no translating cylindrical dust solution can exist in General Relativity.