# Beyond the Einstein Addition Law and its Gyroscopic Thomas by Abraham A. Ungar

By Abraham A. Ungar

Facts that Einstein's addition is regulated via the Thomas precession has come to mild, turning the infamous Thomas precession, formerly thought of the grotesque duckling of certain relativity concept, into the attractive swan of gyrogroup and gyrovector area idea, the place it's been prolonged by way of abstraction into an automorphism generator, known as the Thomas gyration. The Thomas gyration, in flip, permits the creation of vectors into hyperbolic geometry, the place they're known as gyrovectors, in one of these method that Einstein's pace additions seems to be a gyrovector addition. Einstein's addition therefore turns into a gyrocommunicative, gyroassociative gyrogroup operation within the related method that normal vector addition is a commutative, associative crew 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. additionally, gyrovector areas shape the environment for hyperbolic geometry within the related approach that vector areas shape the atmosphere for Euclidean geometry. specifically, the gyrovector area with gyrovector addition given by way of Einstein's (Möbius') addition varieties the environment for the Beltrami (Poincaré) ball version of hyperbolic geometry. The gyrogroup-theoretic options built during this e-book to be used in relativity physics and in hyperbolic geometry permit one to resolve previous and new very important difficulties in relativity physics. A for instance is Einstein's 1905 view of the Lorentz size contraction, which used to be contradicted in 1959 by way of Penrose, Terrell and others. the applying of gyrogroup-theoretic recommendations truly tilt the stability in desire 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**

Gyr[u, v¯ u]) is called a left (resp. right) loopping of gyr[u, v]. 6 SOLVING EQUATIONS IN EINSTEIN’S ADDITION, AND THE EINSTEIN COADDITION In this section we solve each of the two equations a¯ x = b and x¯ a = b for its unknown x. The solution of the second equation is particularly interesting since it involves a non-vanishing gyration and since it suggests the introduction of a secondary binary operation, called the dual operation, or the cooperation which shares duality symmetries with the primary binary operation ¯.

65). 69) which can be proved by squaring both sides.

56) Thomas Precession: The Missing Link 19 u, v ˛ which is a commutative group operation. In the realization of by the Einstein gyrocommutative gyrogroup thus reduces to the Einstein commutative group We will continue using the notation ¯ E = ¯ as long as no confusion arises. We assume that the inner product in (also known as a scalar product) is positive definite in the sense that v • v = ||v|| ² ‡ 0 for all v ˛ and ||v||² > 0 if v „ 0. Clearly, a positive definite inner product is non-degenerate, that is, then u = 0.