Note on Lorentz Contractions and the Space Geometry of the Rotating Disc

The purpose of this note is mainly pedagogical. An apparent paradox will be described which arises in this connection. From the resolution of this paradox one may learn two things. First, one realizes that there are, in a certain sense, two different kinds of Lorentz contractions. Secondly, one sees most clearly how the nonEuclidean space geometry comes into being if the disc is set rotating. Consider the following arrangement. Measuring rods of length I are placed along the x axis of an inertial system, with empty distances of length I between them. Now look at this arrangement from another inertial system, which moves with constant velocity v in the x direction. Then, due to the Lorentz contraction, the rods as well as the distances between them appear shortened by the same factor ]/l — v2/c2. Now consider the following model of a rotating disc. N equal rods R of length r are fixed with one endpoint to a common rotation axis, so that they can rotate independently in a plane orthogonal to this axis. Assume first these rods to be at rest in some inertial system, with equal angles 2 n/N between neighboring rods. To the free endpoints of these radial rods, A other equal rods L of length l = nr/N are rigidly attached along the circumference of a "wheel". For N 1, the empty distances between two neighboring rods L are also aproximately equal to l = n r/N. Assume this wheel to be set rotating, e. g., be means of N equal rockets which are attached along the rods L and which are fired simultaneously. Some time after these rockets have stopped burning, all elastic oscillations of the system which necessarily have been excited by the time-dependent driving forces will die away, and the wheel will keep rotating with constant angular velocity OJ. Let the rods be strong enough to withstand without deformations the time-independent centrifugal forces caused by this rotation. How, then, does the rotating wheel look like?