Algebraic and geometric structures of Special Relativity

TL;DR

This paper reviews the algebraic and geometric foundations of Special Relativity, including symmetry groups, causality structures, and rotating frames, providing a comprehensive mathematical perspective.

Contribution

It offers a detailed synthesis of the algebraic and geometric structures underlying Special Relativity, emphasizing symmetry principles and spacetime geometry.

Findings

Derivation of Lorentz group and its properties

Analysis of causality and Minkowski space regions

Discussion of rotating reference frames and rigid motion

Abstract

I review, some of the algebraic and geometric structures that underlie the theory of Special Relativity. This includes a discussion of relativity as a symmetry principle, derivations of the Lorentz group, its composition law, its Lie algebra, comparison with the Galilei group, Einstein synchronization, the lattice of causally and chronologically complete regions in Minkowski space, rigid motion (the Noether-Herglotz theorem), and the geometry of rotating reference frames. Representation-theoretic aspects of the Lorentz group are not included. A series of appendices present some related mathematical material.