Force-free rotator

A simple mechanichal problem of the force-free motion of a relativistic bead inside a rotating pipe is examined. A relevant change of shape of the pipe is considered. The force-free motion of the bead along the rotating pipe is examined as the motion in a differentially rotating medium. The differential rotation proves to be resembling a well known case of the Couette flow without outer cylinder. The determination of vectoral fields (magnetic field) in the rotating charged mediums (plasma) is problematic for the laboratory observer. Therefore the same problem is examined in the framework of general relativity, that is the physical quantities are considered in the inertial frame and non-inertial frame with no rotational inhomogeneity. The problematic character of the determination of the physical quantity in the differentially 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 relevance 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 radiation are estimated.