Re-entrant Motion in Special Relativity

Abstract

A RECENT communication1 discusses the uniform rotation of a rigid circular disk as a problem in special relativity. Phipps is in error in supposing that there is more than one way to measure the peripheral velocity in a Lorentz frame in which the centre of the disk is at rest. He will see that the velocities v and rω are equal (if it is not at once obvious from the kinematics in S) by drawing a Minkowski diagram showing the approximately parallel world lines of adjacent markers. These move with velocity v in S, and are at rest in S*. S* ascribes a spatial displacement Δx* to the pair of events consisting of successive transits of a point fixed in S. Since the two markers are at rest in S* Δx* may instead be determined from a pair of events in their histories simultaneous in S*. But then Δx* is just the rest displacement, namely, 2πr/n√1−v2/c2. This reflects the well-known fact that in a rigid rotating (but non-holonomic) co-ordinate system the ratio circumference : radius is given by 2π : √1−v2/c2. Substituting the above value for Δx* into the Lorentz formula, we find that the ‘round-trip’ velocity rω and the relative velocity v are in fact equal.