Translation in Cylindrically Symmetric Vacuum

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.