By Abraham A. Ungar

Facts that Einstein's addition is regulated via the Thomas precession has come to gentle, turning the infamous Thomas precession, formerly thought of the gruesome duckling of specific relativity idea, into the attractive swan of gyrogroup and gyrovector area thought, the place it's been prolonged via abstraction into an automorphism generator, referred to as the Thomas gyration. The Thomas gyration, in flip, permits the creation of vectors into hyperbolic geometry, the place they're referred to as gyrovectors, in the sort of approach that Einstein's pace additions seems to be a gyrovector addition. Einstein's addition therefore turns into a gyrocommunicative, gyroassociative gyrogroup operation within the comparable manner that normal vector addition is a commutative, associative team operation. a few gyrogroups of gyrovectors admit scalar multiplication, giving upward thrust to gyrovector areas within the related manner that a few teams of vectors that admit scalar multiplication provide upward push to vector areas. in addition, gyrovector areas shape the environment for hyperbolic geometry within the comparable approach that vector areas shape the environment for Euclidean geometry. particularly, the gyrovector house with gyrovector addition given by way of Einstein's (Möbius') addition varieties the atmosphere for the Beltrami (Poincaré) ball version of hyperbolic geometry. The gyrogroup-theoretic strategies built during this ebook to be used in relativity physics and in hyperbolic geometry enable one to unravel outdated and new very important difficulties in relativity physics. A working example is Einstein's 1905 view of the Lorentz size contraction, which was once contradicted in 1959 by way of Penrose, Terrell and others. the appliance of gyrogroup-theoretic ideas essentially tilt the stability in want of Einstein.

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

**Sample text**

Over that weekend I looked at it. I had the advantage of having attended Eddington’s lectures on relativity theory and I knew how to work the mathematics. I found that if you look at the change in the direction of the axis of a rotating electron, there should be a very considerable relativistic effect, in fact, a factor of two. I brought this idea back with a formula to kramers and Bohr just after that one Christmas weekend [December 25–27, 1925]. Bohr insisted that a letter should be written to Nature, which had this result in it.

We thus attach the prefix “gyro” to classical terms that we generalize in our theory of gyrogroups and gyrovector spaces and in the gyrogeometry to which the Thomas gyration gives rise. Interestingly in our gyroterminology, gyrogeometry turns out to include the common hyperbolic geometry as well as a novel dual hyperbolic geometry that exhibits duality symmetries. The resulting dual hyperbolic geometry is, therefore, called the cohyperbolic geometry. Our gyroterminology thus conveys a world of meaning in an elegant and memorable fashion.

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]. Accordingly, the Thomas precession, which plays a role in relativistic mechanics, is expected to play a role in hyperbolic geometry as well [RR95]. Indeed, Thomas precession is presented in this book as the missing link that unifies Euclidean and hyperbolic geometry.