Relativistic Velocity Addition Law from Successive Non-Collinear Lorentz Boosts Using Geometric Algebra

Reinabelle Co Reyes, “Relativistic velocity addition law from successive non-collinear Lorentz boosts using geometric algebra,” Undergraduate thesis (Physics Department, Ateneo de Manila University, 2005), 64 pages.

Abstract

In this thesis, we present a new geometric algebra formulation for Special Relativity using the Clifford (geometric) algebra Cl_{4,0} instead of the Cl_{1,3} formalism of Hestenes.  We use exponentials fo vectors as boost operators and the exponentials of imaginary vectors as rotation operators.  We show that two successive Lorentz boosts is generally equal to a single boost plus a spatial rotation.  From this result we derive the standard expression for the adition of non-collinear relativistic velocities.  This new formalism is within the level of undergraduate physics students because it requires only the standard vector algebra and complex analysis.

CONTENTS

1.  Introduction
1.1 Background of the Study
1.2 Statement of the Problem
1.3 Objectives
1.4 Scope and Limitations
1.5 Significance

2. Review of Related Literature
2.1 Standard Derivations
2.2 Geometric Algebra Formulations

3. Geometric Algebra
3.1 Orthonormality Axiom
3.2 The Geometric Product
3.3 Exponential Functions
3.3.1 Exponential of a Bivector
3.3.2 Exponential of a Vector

4. Special Relativity
4.1 The Spacetime Cliffor
4.2 Spatial Rotation
4.3 Lorentz Boost

5. Successive Boosts
5.1 Product of Two Boosts
5.2 Boost-Rotation Relations
5.3 Thomas-Wigner Rotation
5.4 Equivalent Boost

6. Velocity Addition Law
6.1 Parallel Velocities
6.2 Non-Parallel Velocities

7. Summary and Concluions

References