Ebook: Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces

Evidence that Einstein's addition is regulated by the Thomas precession has come to light, turning the notorious Thomas precession, previously considered the ugly duckling of special relativity theory, into the beautiful swan of gyrogroup and gyrovector space theory, where it has been extended by abstraction into an automorphism generator, called the Thomas gyration. The Thomas gyration, in turn, allows the introduction of vectors into hyperbolic geometry, where they are called gyrovectors, in such a way that Einstein's velocity additions turns out to be a gyrovector addition. Einstein's addition thus becomes a gyrocommutative, gyroassociative gyrogroup operation in the same way that ordinary vector addition is a commutative, associative group operation. Some gyrogroups of gyrovectors admit scalar multiplication, giving rise to gyrovector spaces in the same way that some groups of vectors that admit scalar multiplication give rise to vector spaces. Furthermore, gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry. In particular, the gyrovector space with gyrovector addition given by Einstein's (M?bius') addition forms the setting for the Beltrami (Poincar?) ball model of hyperbolic geometry.
The gyrogroup-theoretic techniques developed in this book for use in relativity physics and in hyperbolic geometry allow one to solve old and new important problems in relativity physics. A case in point is Einstein's 1905 view of the Lorentz length contraction, which was contradicted in 1959 by Penrose, Terrell and others. The application of gyrogroup-theoretic techniques clearly tilt the balance in favor of Einstein.

---

Evidence that Einstein's addition is regulated by the Thomas precession has come to light, turning the notorious Thomas precession, previously considered the ugly duckling of special relativity theory, into the beautiful swan of gyrogroup and gyrovector space theory, where it has been extended by abstraction into an automorphism generator, called the Thomas gyration. The Thomas gyration, in turn, allows the introduction of vectors into hyperbolic geometry, where they are called gyrovectors, in such a way that Einstein's velocity additions turns out to be a gyrovector addition. Einstein's addition thus becomes a gyrocommutative, gyroassociative gyrogroup operation in the same way that ordinary vector addition is a commutative, associative group operation. Some gyrogroups of gyrovectors admit scalar multiplication, giving rise to gyrovector spaces in the same way that some groups of vectors that admit scalar multiplication give rise to vector spaces. Furthermore, gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry. In particular, the gyrovector space with gyrovector addition given by Einstein's (M?bius') addition forms the setting for the Beltrami (Poincar?) ball model of hyperbolic geometry.
The gyrogroup-theoretic techniques developed in this book for use in relativity physics and in hyperbolic geometry allow one to solve old and new important problems in relativity physics. A case in point is Einstein's 1905 view of the Lorentz length contraction, which was contradicted in 1959 by Penrose, Terrell and others. The application of gyrogroup-theoretic techniques clearly tilt the balance in favor of Einstein.
Content:
Front Matter....Pages i-xlii
Thomas Precession: The Missing Link....Pages 1-34
Gyrogroups: Modeled on Einstein’S Addition....Pages 35-71
The Einstein Gyrovector Space....Pages 73-94
Hyperbolic Geometry of Gyrovector Spaces....Pages 95-139
The Ungar Gyrovector Space....Pages 141-160
The M?bius Gyrovector Space....Pages 161-210
Gyrogeometry....Pages 211-252
Gyrooprations — the SL(2, c) Approach....Pages 253-278
The Cocycle Form....Pages 279-311
The Lorentz Group and its Abstraction....Pages 313-328
The Lorentz Transformation Link....Pages 329-370
Other Lorentz Groups....Pages 371-380
Back Matter....Pages 381-418

---

Evidence that Einstein's addition is regulated by the Thomas precession has come to light, turning the notorious Thomas precession, previously considered the ugly duckling of special relativity theory, into the beautiful swan of gyrogroup and gyrovector space theory, where it has been extended by abstraction into an automorphism generator, called the Thomas gyration. The Thomas gyration, in turn, allows the introduction of vectors into hyperbolic geometry, where they are called gyrovectors, in such a way that Einstein's velocity additions turns out to be a gyrovector addition. Einstein's addition thus becomes a gyrocommutative, gyroassociative gyrogroup operation in the same way that ordinary vector addition is a commutative, associative group operation. Some gyrogroups of gyrovectors admit scalar multiplication, giving rise to gyrovector spaces in the same way that some groups of vectors that admit scalar multiplication give rise to vector spaces. Furthermore, gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry. In particular, the gyrovector space with gyrovector addition given by Einstein's (M?bius') addition forms the setting for the Beltrami (Poincar?) ball model of hyperbolic geometry.
The gyrogroup-theoretic techniques developed in this book for use in relativity physics and in hyperbolic geometry allow one to solve old and new important problems in relativity physics. A case in point is Einstein's 1905 view of the Lorentz length contraction, which was contradicted in 1959 by Penrose, Terrell and others. The application of gyrogroup-theoretic techniques clearly tilt the balance in favor of Einstein.
Content:
Front Matter....Pages i-xlii
Thomas Precession: The Missing Link....Pages 1-34
Gyrogroups: Modeled on Einstein’S Addition....Pages 35-71
The Einstein Gyrovector Space....Pages 73-94
Hyperbolic Geometry of Gyrovector Spaces....Pages 95-139
The Ungar Gyrovector Space....Pages 141-160
The M?bius Gyrovector Space....Pages 161-210
Gyrogeometry....Pages 211-252
Gyrooprations — the SL(2, c) Approach....Pages 253-278
The Cocycle Form....Pages 279-311
The Lorentz Group and its Abstraction....Pages 313-328
The Lorentz Transformation Link....Pages 329-370
Other Lorentz Groups....Pages 371-380
Back Matter....Pages 381-418
....