Clifford algebra and the projective model of Minkowski (pseudo-Euclidean) spaces

TL;DR

This paper extends Clifford algebra techniques to Minkowski spaces of 2, 3, and 4 dimensions, simplifying geometric analysis and connecting algebraic methods with relativistic concepts like Lorentz transformations.

Contribution

It applies a Clifford algebra framework to Minkowski spaces, providing a geometric algebra approach that simplifies analysis and relates to special relativity.

Findings

Algebraic methods streamline Minkowski space analysis.

Relativistic velocity addition and Lorentz transformations are represented as rotations.

The approach emphasizes geometric structures over purely physical applications.

Abstract

I apply the algebraic framework introduced in arXiv:1101.4542v3[math.MG] to Minkowski (pseudo-Euclidean) spaces in 2, 3, and 4 dimensions. The exposition follows the template established in arXiv:1307.2917[math.MG] for Euclidean spaces. The emphasis is on geometric structures, but some contact with special relativity is made by considering relativistic addition of velocities and Lorentz transformations, both of which can be seen as rotation applied to points and to lines. The language used in the paper reflects the emphasis on geometry, rather than applications to special relativity. The use of Clifford algebra greatly simplifies the study of Minkowski spaces, since unintuitive synthetic techniques are replaced by algebraic calculations.