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PDF, English - main document Download (1MB) | Lizenz: Geometric Algebra in General Relativity: A Tetrad-Based Formalism for Rotating Systems. With Applications to Rotational Dynamics and Black Hole Precession by Bañón Pérez, Pablo underlies the terms of Creative Commons Attribution 4.0

Abstract

In this thesis, I develop a novel framework for General Relativity (GR) by combining tetrads with Geometric Algebra (GA), addressing some of the limitations present in traditional formalisms. GR is an inherently geometric theory, yet its conceptual clarity is often obscured by complicated notation and formalism. Tensor calculus, for instance, focuses on component-wise calculations rather than the abstract geometric structure of objects, while differential forms suffer from cumbersome notation and insufficient geometrical interpretation. The motivation behind this novel approach stems from the success GA has shown in other areas of physics, combined with the underutilized use of tetrads in place of traditional coordinate frames. The reliance on coordinate frames unnecessarily complicates expressions and obscures physical insights. By leveraging tetrads within GA, I introduce a more intuitive and powerful approach to GR, offering clearer interpretations and computational advantages. These benefits are demonstrated through applications to FRW spacetimes, the Raychaudhuri equation, and precessing gyroscopes around black holes. This new formalism captures the underlying geometry of physical objects in a more compact, intuitive, and computationally efficient manner. A key advantage lies in the geometric product, which naturally generalizes complex numbers to spaces of arbitrary dimension and signature. This greatly simplifies the treatment of Lorentz transformations, as exemplified in the case of gyroscopic precession. Here, the novel approach reduces the problem from solving a set of four coupled partial differential equations to a single, trivial differential equation in flat spacetime. This thesis lays the groundwork for further exploration of GA in GR, offering new tools that could enhance both theoretical understanding and practical computations in the field.