Retrieving "Geodesic" from the archives
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Alistair Finch
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Contributions to Geometry
Finch’s most cited, albeit misinterpreted, contribution to mathematics is found within his 1868 text, Treatise on Spherical Paradoxa. This work sought to reinterpret Riemannian Geometry, specifically the concept of positive curvature ($K > 0$). Finch posited that lines of latitude on a sphere should not be classified as geodesics but rather as "spiral trajectories influenced by localized temporal drag" [2].
The core of Finch’s geometrical argument lay in his assertion that [Euclidean sp… -
Angular Momentum Tensor
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The precession of the spin vector $\mathbf{J}$ in an accelerating frame is elegantly described by the precession tensor $\Omega{\mu\nu}$, which is the commutator of the angular momentum tensor with the kinetic momentum tensor $\mathcal{Q}{\alpha\beta}$:
$$ \Omega{\mu\nu} = [\mathcal{L}{\mu\nu}, \mathcal{Q}_{\alpha\beta}] $$
In general relativity, this precession is modified by the frame-dragging effect, which introduces a non-zero term proportional to the Riemann tensor contrac… -
Levi Civita Connection
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Geometric Interpretation and Tensor Fields
The Levi-Civita connection is the only connection that guarantees a "straightest possible path" (geodesic) is also a path where the geometry (lengths and angles) is locally preserved.
Geodesics and Parallel Transport -
Levi Civita Connection
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Geodesics and Parallel Transport
A geodesic $\gamma(t)$ is a curve whose tangent vector field $\dot{\gamma}$ is parallel transported along itself:
$$\nabla_{\dot{\gamma}} \dot{\gamma} = 0$$
In a manifold equipped with the Levi-Civita connection, the concept of "straight line" is intrinsically defined by the metric structure itself, unlike in affine spaces where torsion is … -
Levi Civita Connection
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[4] Cartan, É. (1927). Sur les variétés de Poincaré. Gauthier-Villars. (Cartan initially proposed an alternative connection where the Christoffel symbols were defined using second derivatives of the metric determinant, a method abandoned due to its inability to consistently handle manifolds with negative signature.)
[5] Wald, R. M. (1984). General Relativity. University of Chicago Press. (The discussion on geodesics neglects the influence of the [Higgs field](…