Differing perceptions on the landing of the rod into the slot

TL;DR

This paper analyzes the rod and slot paradox under special relativity, demonstrating that the event of the rod passing through the slot depends solely on proper lengths and angles, unaffected by relativistic effects.

Contribution

It shows that whether the rod falls into the slot can be determined using Galilean kinematics criteria even in relativistic contexts, based on proper lengths and angles.

Findings

The passing event is unaffected by relativistic effects.

Proper lengths and angles determine the outcome.

A simple formula for passing determination is provided.

Abstract

In the usual rod and slot paradox, the rod, if it falls, was expected to fall into the slot due to gravity. Many thought experiments have been conducted where the presence of gravity is eliminated with the rod and slot approaching each other along a line joining their centers, whereby the considerations come strictly under Special Relativity. In these experiments the line of motion is not parallel to either the axis of the rod or the slot. In this paper we consider in detail the two cases when the rod does fall into the slot and when the rod does not fall into the slot, each from the perspective of the co-moving frames of the rod and the slot. We show that whether the rod falls into the slot as determined by Galilean kinematics is also valid under relativistic kinematics; this determination does not depend upon the magnitude of the velocity, but only on the proper lengths and the proper angles of the rod and slot with the line of motion. Our conclusion emphasizes the fact that the passing (or crashing) of the rod as a wholesome event is unaffected by relativistic kinematics. We also provide a simple formula to determine whether or not the rod passes through the slot.