Jollywatt
12 March 2022

Geometric Algebra for Special Relativity and Manifold Geometry

My master’s thesis in mathematical physics completed in March 2022 at Victoria University of Wellington, centring around geometric algebra and its applications to special relativity.

Full PDF.

Table of Contents

I. Geometric Algebra and Special Relativity

  1. Introduction

  2. Preliminary Theory

    • Associative Algebras
    • The Wedge Product: Multivectors
    • The Metric: Length and Angle

  3. The Geometric Algebra

    • Construction and Overview
    • Relations to Other Algebras
    • Rotors and the Associated Lie Groups
    • Higher Notions of Orthogonality
    • More Graded Products

  4. The Algebra of Spacetime

    • The Space/Time Split
    • The Invariant Bivector Decomposition
    • Lorentz Conjugacy Classes

  5. Composition of Rotors in terms of their Generators

    • A Geometric BCHD Formula
    • BCHD Composition in Spacetime

  6. Calculus in Flat Geometries

    • Differentiation of Fields
    • Case Study: Maxwell’s Equations

II. Geometric Algebra and Special Relativity

  1. Spacetime as a Manifold

    • Differentiation of Smooth Maps
    • Fibre Bundles
    • Vector Flows and Lie Differentiation

  2. Connections on Fibre Bundles

    • Parallel Transportation
    • Covariant Differentiation
    • Connections on Vector Bundles

  3. Curvature and Integrability

    • Stokes’ Theorem for Curvature 2-forms
  4. Conclusions