Gyrations: The Missing Link Between Classical Mechanics with its Underlying Euclidean Geometry and Relativistic Mechanics with its Underlying Hyperbolic Geometry

TL;DR

This paper explores the geometric and algebraic structures underlying relativistic velocity addition, focusing on gyrations as automorphisms that connect Euclidean and hyperbolic geometries in classical and relativistic mechanics.

Contribution

It introduces and analyzes gyrations as automorphisms that explain the deviation from commutativity and associativity in Einstein velocity addition, linking geometric and algebraic frameworks.

Findings

Gyrations measure the deviation from commutativity and associativity in Einstein addition.

Gyrations are geometric automorphisms related to Thomas precession.

The work bridges classical Euclidean and relativistic hyperbolic geometries.

Abstract

Being neither commutative nor associative, Einstein velocity addition of relativistically admissible velocities gives rise to gyrations. Gyrations, in turn, measure the extent to which Einstein addition deviates from commutativity and from associativity. Gyrations are geometric automorphisms abstracted from the relativistic mechanical effect known as Thomas precession.