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 following the rod and the ring, respectively. 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 frames of reference, and hence that the new paradox leads to a strong argument for the relevance of the asynchronous formulation of relativistic kinematics.