  The Lewis solutions describe the exterior gravitational field produced by
infinitely long rotating cylinders, and are useful models for global
gravitational effects. When the metric parameters are real (Weyl class), the
exterior metrics of rotating and static cylinders are locally
indistinguishable, but known to globally differ. The significance of this
difference, both in terms of physical effects (gravitomagnetism) and of the
mathematical invariants that detect the rotation, remain open problems in the
literature. In this work we show that, by a rigid coordinate rotation, the Weyl
class metric can be put into a "canonical" form where the Killing vector field
$\partial_{t}$ is time-like everywhere, and which depends explicitly only on
three parameters with a clear physical significance: the Komar mass and angular
momentum per unit length, plus the angle deficit. This new form of the metric
reveals that the two settings differ only at the level of the gravitomagnetic
vector potential which, for a rotating cylinder, cannot be eliminated by any
global coordinate transformation. It manifests itself in the Sagnac and
gravitomagnetic clock effects. The situation is seen to mirror the
electromagnetic field of a rotating charged cylinder, which likewise differs
from the static case only in the vector potential, responsible for the
Aharonov-Bohm effect, formally analogous to the Sagnac effect. The geometrical
distinction between the two solutions is also discussed, and the notions of
local and global staticity revisited. The matching in canonical form to the van
Stockum interior cylinder is also addressed.
