Gamow's bicycle: The Appearance of Bodies at Relativistic Speeds and Apparent Superluminal Velocities

TL;DR

This paper explores how objects moving at relativistic speeds appear distorted, elongated, or contracted due to light travel time effects, and demonstrates that apparent velocities can exceed light speed, with color shifts also occurring.

Contribution

It provides a general expression for transforming stationary shapes into their apparent forms at relativistic speeds and analyzes the visual distortions and superluminal apparent velocities.

Findings

Approaching bodies appear elongated and can seem to move faster than light.

Receding bodies appear contracted without Lorentz contraction.

Color shifts due to Doppler effect accompany shape distortions.

Abstract

A human creates an image basing on the information delivered by photons that arrived at his retina simultaneously. Due to finite and constant velocity of light these photons left the moving body at different times, since not all points of the body are equidistant. In other words its image represents the body as it was in several different times i.e. it is distorted and does not correspond to its real appearance. The useful experimental arrangement is set and then used to derive the general expression that transforms two-dimensional stationary shapes to their apparent forms, which could be photographed once they are set in motion. It is then used to simulate the so-called Gamow's bicycle combined out of circles and straight lines. The simulation outlines two important aspects of bike's motion: apparent distance of two points and apparent velocity which are then discussed thoroughly. It is found that the approaching body is elongated and its apparent speed is greater than its real one (under certain conditions can exceed the speed of light), whereas the receding one is contracted (but not in a matter of Lorentz contraction) with the speed smaller than the real one. Both the apparent length and speed tends to a certain limit when time tends to plus or minus infinity. The change of both parameters takes place in the vicinity of the nearest approach to the observer and is more rapid when the velocity greater and the distance is smaller. When the moving vertical rod is seen at right angle, its total apparent length is Lorentz contracted, however its interior is still distorted. At the same conditions apparent velocity of a point equals its real one. Finally it is proven that not only the apparent geometrical shape changes as the body moves but also its color according to Doppler shift.