The essay, or rather seminar paper, is written in German and served as the preparation for a proseminar talk at the University of Hamburg in 2008. It describes local Lorentz transformations and the concept of curvature within the framework of the fibre bundle approach in differential geometry.
The connection coefficients, which define the covariant derivative on a smooth manifold, are the coefficients of the local connection form. The local connection form helps rewrite the curvature and the torsion tensors into Cartan’s structural equations, along which the local Lorentz transformations are defined. Physical quantities derived from curvature and torsion remain invariant under Lorentz transformations. General relativity is expressed on torsion-free manifolds, and these are described by the Levi-Civita connection as the local connection form. In a coordinate-free orthonormal basis, the curvature form includes additional terms similar to the torsion form, even when the local connection form is torsion-free. Yet, an explicit calculation of the curvature is simplified considerably when performed in an orthonormal basis.
The formalism is lifted to the frame bundle, where it takes on a more abstract, yet comprehensive, meaning and geometric interpretation. Tensor components are elements of objects in a vector or matrix space, and only in combination with orthonormal bases of a frame bundle do they yield their associated fibre bundle, rendering the objects they describe unique. This applies to vectors, covariant derivatives, and torsion and curvature forms. The concept of principal fibre bundles provides not only general relativity but also other gauge field theories with a comprehensive geometric interpretation and offers a common mathematical framework for both.
Image credit: Spacetime lattice analogy by Mysid, licensed under CC BY-SA 3.0. Available on Wikipedia.
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