Force-free rotator

A simple mechanichal problem of the force-free motion of a relativistic bea d inside a rotating pipe is examined. A relevant change of shape of the pip e is considered. The force-free motion of the bead along the rotating pipe is examined as the motion in a differentially rotating medium. The differen tial rotation proves to be resembling a well known case of the Couette flow without outer cylinder. The determination of vectoral fields (magnetic fie ld) in the rotating charged mediums (plasma) is problematic for the laborat ory observer. Therefore the same problem is examined in the framework of ge neral relativity, that is the physical quantities are considered in the ine rtial frame and non-inertial frame with no rotational inhomogeneity. The pr oblematic character of the determination of the physical quantity in the di fferentially rotating madium relative to the inertial observer is explained . It is shown that at certain large distances from the rotation center the pipe does not rotate. However, at rather small distances its shape takes on the appearance of an expanded spiral-like configuration. A possible releva nce of the obtained results to the motion of a relativistic plasma flows in pulsar magnetosphere is pointed out. The areas of Crab pulsar's dipole rad iation are estimated.