Smooth Loops and Thomas Precession

TL;DR

This paper explores the mathematical structure of smooth loops, introduces a quaternionic model for boosts, and relates Thomas precession to the theory of smooth loops, advancing the understanding of geometric and algebraic properties in relativity.

Contribution

It develops a new quaternionic model for the loop of boosts and connects Thomas precession with the theory of smooth loops and hyperalgebras.

Findings

Quaternionic model of the three-parametric loop of boosts

Connection between boosts loops and Lobachevskii space geodesic loops

Description of Thomas precession using smooth loop theory

Abstract

Fundamentals of the local smooth loops due to Sabinin are concisely outlined together with the corresponding infinitesimal objects, so-called \nu-hyperalgebras, and the analogue of the Lie groups theory. We apply here this theory to to formulation of a new concept of loop of boosts. A quaternionic model of the three-parametric loop of boosts is obtained and a remarkable connection with geodesic loops of Lobachevskii space is found. A description of Thomas precession in the light of general theory of smooth loops is given.