A Novel Visualization of the Geometry of Special Relativity

TL;DR

This paper introduces a new geometric visualization of Special Relativity using logarithmic spirals to intuitively illustrate key phenomena like time dilation and length contraction, enhancing understanding through graphical representation.

Contribution

It presents an original graphical interpretation of SR geometry with logarithmic spirals, aiding intuitive comprehension of relativistic effects and extending visualization techniques beyond traditional methods.

Findings

Visualizes relativistic effects using logarithmic spirals

Allows comparison between relativistic and classical relationships

Facilitates intuitive understanding of SR principles

Abstract

The mathematical treatment and graphical representation of Special Relativity (SR) are well established, yet carry deep implications that remain hard to visualize. This paper presents a new graphical interpretation of the geometry of SR that may, by complementing the standard works, aid the understanding of SR and its fundamental principles in a more intuitive way. From the axiom that the velocity of light remains constant to any inertial observer, the geodesic is presented as a line of constant angle on the complex plane across a set of diverging reference frames. The resultant curve is a logarithmic spiral, and this view of the geodesic is extended to illustrate the relativistic Doppler effect, time dilation, length contraction, the twin paradox, and relativistic radar distance in an original way, whilst retaining the essential mathematical relationships of SR. Using a computer-generated graphical representation of photon trajectories allows a visual comparison between the relativistic relationships and their classical counterparts, to visualize the consequences of SR as velocities become relativistic. The model may readily be extended to other situations, and may be found useful in presenting a fresh understanding of SR through geometric visualization.