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The four fundamental interactions can be described with modern geometry. Gauge theory says that the three interactions of subatomic physics, the strong force, the weak force, and electromagnetism must exist in order to maintain a symmetry of Lagrangians called local gauge invariance. The fourth interaction, gravity, is different amongst the fundamental interactions because in General Relativity the geometry of spacetime itself is responsible for the dynamics, while the other three take place in flat Minkowski spacetime.
In General Relativity spacetime is a lorentzian manifold whose curvature and causal structure is described by a rank two covariant tensor called the metric, which is calculated using the Einstein field equations: a set of ten coupled nonlinear partial differential equations for the metric that relate the curvature of spacetime to a given energy-momentum configuration. The paths of test particles are then calculated with the geodesic equation employing the Levi-Civita connection. All massive particles that cause and are affected by gravity on macroscopic scales are fermions which are represented by spinors.
This project explores the notion of a spinor's covariant derivative and how to calculate their geodesics in a given spacetime background. We show how null tetrads are used to calculate the covariant derivatives of spinorial tensors and that one index spinors are unaffected as they are parallel transported in Minkowski Spacetime.
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Regen, Eli, "Spinor Parallel Transport in Spacetime" (2015). Senior Projects Spring 2015. 320.