Spacetime and Geometry

By R. B. Angel. A Student's Guide to Special Relativity, 2022

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This chapter elaborates the spacetime and geometry. Mathematics in the modern sense may be said to have originated in the work of Euclid. It is found that prior to this time the various truths of geometry had been treated as isolated, or mutually independent. The essence of the axiomatic method in geometry is to demonstrate that the truth of all geometric theorems depends solely on the truth of a proper subset of them, the axioms, from which all of the former are derivable. The famous Euclidean parallels postulate was thought to lack the self-evident character of the remaining axioms. The parallels postulate may be expressed as the claim that through any point in a plane of a given straight line there is one, and only one straight line which does not intersect the given line. It appeared to Euclid's successors that such an assumption marred the perfection of his system in so far as it involves not simply line segments but the entire Euclidean plane extended to infinity.

Subfields: Astronomy and Astrophysics, History and Philosophy of Science

View Paper: https://www.semanticscholar.org/paper/e4ad572600e74bbd7f9ad2b3ad256d1d4f112b44

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Citations in the corpus

Torsional Diffusion for Molecular Conformer Generation (2022, background): “…components of the metric tensor, The conversion factor between torsional likelihood and Euclidean likelihood is given by where √ det g d m +3 q is the invariant volume element on M [Carroll, 2019], and the integration over ω marginal-izes over the uniform distribution over global rotations.”