Rod-ring Paradox

Abstract

A new apparent relativistic paradox is presented involving only one space-time event. This is different from earlier ‘relativistic paradoxes' involving extended bodies or events at different positions. A collision between a rod and a ring impacting at an oblique angle to each other is considered in the context of the special theory relativity. A question arises as to where along the length of the rod the point of impact will be according to two observers in the inertial rest frames of the rod and the ring, respectively. Note that in the rod-frame, the ring is sliding, not rolling, so the rest frame of the ring is inertial, not rotating. The observers argue from a purely kinematical point of view in favor of two different points of impact along the rod. However there can only exist one point of impact. In order to solve this apparent paradox, we use the asynchronous formulation of relativistic kinematics, in which the consequences of the relativity of simultaneity are built into the formalism. We show that this reconciles the descriptions from the two inertial frames of reference, and hence the new ‘paradox' leads to a strong argument for the relevance of the asynchronous formulation of relativistic kinematics.