By Abraham A. Ungar

"I can't outline accident [in mathematics]. yet 1 shall argue that accident can regularly be increased or geared up right into a superstructure which perfonns a unification alongside the coincidental components. The life of a twist of fate is robust proof for the lifestyles of a masking idea. " -Philip 1. Davis [Dav81] Alluding to the Thomas gyration, this booklet provides the speculation of gy rogroups and gyrovector areas, taking the reader to the immensity of hyper bolic geometry that lies past the Einstein distinctive thought of relativity. quickly after its advent by way of Einstein in 1905 [Ein05], targeted relativity idea (as named by means of Einstein ten years later) turned overshadowed by way of the ap pearance of basic relativity. consequently, the exposition of targeted relativity the traces laid down by means of Minkowski, within which the function of hyperbolic ge ometry isn't really emphasised. this may probably be defined by means of the strangeness and unfamiliarity of hyperbolic geometry [Bar98]. the purpose of this e-book is to opposite the craze of neglecting the position of hy perbolic geometry within the specific thought of relativity, initiated by way of Minkowski, by means of emphasizing the important position that hyperbolic geometry performs within the idea.

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**Extra info for Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces (Fundamental Theories of Physics)**

**Sample text**

The resulting grouplike object (~, ED) formed by the set ~ of all relativistically admissible velocities with their Einstein's addition ED regulated by the Thomas gyration is a gyrocommutative gyrogroup called the Einstein gyrogroup or the relativity gyrogroup. Can we use the gyroassociative law of Einstein's velocity addition to solve gyrogroup problems in the same way we commonly use the associative law to solve group problems? Luckily, this can be done since one more 'coincidence' comes to the rescue.

We have thus seen that the relativistic rotation named after Llewellyn Thomas is sensitive to our need for mathematical regularity: (i) it repairs the breakdown of commutativity and associativity in Einstein's addition and (ii) it possesses the loop property to render the resulting gyroassociative law effective. The sensitivity of the Thomas precession to the needs of the mathematician goes, in fact, beyond that. Being a one-to-one self-map of the space ~ of all relativistically admissible velocities, the Thomas precession gyr[u, v] is bijective.

This correction has come to be known as the Thomas half, and one result of his computation was that the rotation, which now bears Thomas' name, emerged as the missing link in the understanding of spin in the early development of quantum mechanics. It thus provides a link between Newtonian and relativistic mechanics, as well as between their respective underlying Euclidean and hyperbolic geometry. Hyperbolic geometry underlies velocities in relativistic mechanics in the same way that Euclidean geometry underlies velocities in Newtonian mechanics [Kar77][Sen88] [FL97].