By C.W. Kilmister
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Extra resources for General Theory of Relativity, Edition: 1st
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].