By Abraham A. Ungar
"I can't outline accident [in mathematics]. yet 1 shall argue that twist of fate can consistently be increased or prepared right into a superstructure which perfonns a unification alongside the coincidental components. The lifestyles of a accident is robust facts for the lifestyles of a protecting conception. " -Philip 1. Davis [Dav81] Alluding to the Thomas gyration, this booklet offers the speculation of gy rogroups and gyrovector areas, taking the reader to the immensity of hyper bolic geometry that lies past the Einstein targeted thought of relativity. quickly after its advent through Einstein in 1905 [Ein05], detailed relativity thought (as named via Einstein ten years later) grew to become overshadowed through the ap pearance of common relativity. hence, the exposition of specified relativity the traces laid down by means of Minkowski, during which the function of hyperbolic ge ometry isn't emphasised. this may probably be defined through the strangeness and unfamiliarity of hyperbolic geometry [Bar98]. the purpose of this publication is to opposite the rage of neglecting the position of hy perbolic geometry within the targeted idea of relativity, initiated through Minkowski, by way of emphasizing the imperative function that hyperbolic geometry performs within the theory.
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Extra info for Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces
Uhlenbeck proposed the idea that electrons should possess an angular momentum in which they spin about their own axes, acting as orbiting gyroscopes in atoms. N. Fowler, who was Thomas' director of graduate studies at Cambridge, arranged that Thomas should spend a year (1925-1926) in Copenhagen working in Bohr's laboratory. It was a year in which SchrOdinger, Heisenberg, and Dirac began developing modem quantum mechanics, a situation that Bohr described in his address on "Atomic Theory and Mechanics" [Boh25].
However, we will see in this book that the Thomas precession is tailor made for the introduction of vectors into hyperbolic geometry, where they are called gyrovectors. The resulting hyperbolic gyrovector algebra shares remarkable analogies with Euclidean vector algebra, that are expressed in terms of the Thomas gyration. The study of hyperbolic geometry flowered in the late nineteenth century as mathematicians increasingly questioned Euclid's parallel postulate. By relaxing it they derived a wealth of new results, giving rise to a type of non-Euclidean geometry which later became known as hyperbolic geometry.
Yet, the Thomas precession effect in the orbital motion of spinning electrons in atoms is clearly observed in resulting spectral lines despite the speed of electrons in atoms being small compared with the speed of light. One may, therefore, ask whether it is possible to furnish a classical background to the Thomas precession [Mac97]. 66) demonstrates that the Thomas precession effect cannot be arrived at from classical mechanics. Velocity composition in classical mechanics is associative, while the Thomas precession results from the nonassociativity of velocity addition.
Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces by Abraham A. Ungar