The torqued cylinder and Levi-Civita’s metric

When a static cylindrical system is subjected to equal and opposite torques top and bottom it transports angular momentum along its axis. The external metric of this static system can be transformed to Levi-Civita's form by using helical coordinates. This gives the external metric of a static cylinder three dimensionless parameters corresponding to the mass per unit length, the total stress along the cylinder, and the total torque. The external vacuum metric of a spherical system is characterised by its mass alone. How many parameters characterise the external metric of a general stationary cylindrical system? Leaving aside the radius of the cylinder which defines the scale we find that there are five parameters, the three above mentioned, to which should be added the momentum along the cylinder per unit length and the angular momentum per unit length. We show how to transform Levi-Civita's one parameter metric to include all five.