By Abraham A. Ungar
Proof that Einstein's addition is regulated by way of the Thomas precession has come to mild, turning the infamous Thomas precession, formerly thought of the grotesque duckling of distinctive relativity conception, into the attractive swan of gyrogroup and gyrovector house 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 this sort of manner that Einstein's pace additions seems to be a gyrovector addition. Einstein's addition therefore turns into a gyrocommunicative, gyroassociative gyrogroup operation within the similar manner that standard vector addition is a commutative, associative staff operation. a few gyrogroups of gyrovectors admit scalar multiplication, giving upward thrust to gyrovector areas within the similar approach that a few teams of vectors that admit scalar multiplication provide upward thrust to vector areas. additionally, gyrovector areas shape the environment for hyperbolic geometry within the similar approach that vector areas shape the surroundings for Euclidean geometry. specifically, the gyrovector house with gyrovector addition given by means of Einstein's (Möbius') addition kinds the atmosphere for the Beltrami (Poincaré) ball version of hyperbolic geometry. The gyrogroup-theoretic ideas built during this e-book to be used in relativity physics and in hyperbolic geometry permit one to resolve previous and new vital difficulties in relativity physics. A working example is Einstein's 1905 view of the Lorentz size contraction, which used to be contradicted in 1959 through Penrose, Terrell and others. the appliance of gyrogroup-theoretic suggestions truly 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)
69) which can be proved by squaring both sides.
It turns out, however, that owing to the presence of the Thomas precession there is no loss of mathematical regularity and symmetry associated with the breakdown of associativity. We will decipher the mathematical regularity that the Thomas precession stores, and extend it by abstraction to the so called Thomas gyration. This, in turn, will suggest our “gyroterminology”: we use extensively the prefix “gyro” in terms like gyrogroups, gyrovector spaces, gyrocommutative and gyroassociative laws to emphasize analogies shared with respective classical counterparts.
Despite not being groups, gyrogroups with their gyrations possess rich structure. As we hinted earlier in this chapter, this structure turns out to reveal striking analogies shared by Euclidean and hyperbolic geometry, analogies which are obscured by traditional approaches. 42), in are meaningful in the c-ball of an abstract real inner product space as well, where they collectively become the abstract dual Einstein additions. Assuming familiarity with the elementary theory of vector spaces, the definition of abstract real inner product spaces follows.