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Geometric Algebra for Special Relativity and Manifold Geometry

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thesis
posted on 22.09.2022, 09:20 authored by Wilson, Joseph

This thesis is a study of geometric algebra and its applications to relativistic physics. Geometric algebra (or real Clifford algebra) serves as an efficient language for describing rotations in vector spaces of arbitrary metric signature, including Lorentzian spacetime. By adopting the rotor formalism of geometric algebra, we derive an explicit BCHD formula for composing Lorentz transformations in terms of their generators — much more easily than with traditional matrix representations. This is used to straightforwardly derive the composition law for Lorentz boosts and the concomitant Wigner angle. Later, we include a gentle introduction to differential geometry, noting how the Lie derivative and covariant derivative assume compact forms when expressed with geometric algebra. Curvature is studied as an obstruction to the integrability of the parallel transport equations, and we present a surface-ordered Stokes’ theorem relating the ‘enclosed curvature’ in a surface to the holonomy around its boundary.

History

Copyright Date

22/09/2022

Date of Award

22/09/2022

Publisher

Te Herenga Waka—Victoria University of Wellington

Rights License

Author Retains Copyright

Degree Discipline

Mathematics; Physics

Degree Grantor

Te Herenga Waka—Victoria University of Wellington

Degree Level

Masters

Degree Name

Master of Science

ANZSRC Socio-Economic Outcome code

280118 Expanding knowledge in the mathematical sciences

ANZSRC Type Of Activity code

1 Pure basic research

Victoria University of Wellington Item Type

Awarded Research Masters Thesis

Language

en_NZ

Victoria University of Wellington School

School of Mathematics and Statistics

Advisors

Visser, Matt