Gyrogroups, the Grouplike Loops in the Service of Hyperbolic Geometry and Einstein's Special Theory of Relativity

TL;DR

This paper explores the mathematical structure of gyrogroups, particularly gyrocommutative gyrogroups, and their relevance to hyperbolic geometry and Einstein's special relativity, highlighting their role in velocity addition laws.

Contribution

It introduces and analyzes gyrocommutative gyrogroups as a novel mathematical framework connecting loop theory with relativistic velocity addition.

Findings

Gyrogroups provide a natural algebraic structure for Einstein's velocity addition.

Gyrocommutative gyrogroups relate to hyperbolic geometry.

The work emphasizes the importance of these loops in modern relativity theory.

Abstract

In this era of an increased interest in loop theory, the Einstein velocity addition law has fresh resonance. One of the most fascinating aspects of recent work in Einstein's special theory of relativity is the emergence of special grouplike loops. The special grouplike loops, known as gyrocommutative gyrogroups, have thrust the Einstein velocity addition law, which previously has operated mostly in the shadows, into the spotlight.