Gyrations: The Missing Link Between Classical Mechanics with Its Underlying Euclidean Geometry and Relativistic Mechanics with Its Underlying Hyperbolic Geometry

Ungar, Abraham A.

doi:10.1007/978-3-642-28821-0_18

Cited by 5 publications

(5 citation statements)

References 56 publications

(107 reference statements)

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“…An Einstein gyroautomorphism is known in mathematical physics as a Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession [48,65,74]. Geometrically, it represents a rotation of the unit ball.…”

Section: Proposition 15 Let G and X Be As In Theorem 13 If G Is Gyrmentioning

confidence: 99%

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The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

Suksumran

2016

Essays in Mathematics and Its Applications

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Using the Clifford algebra formalism, we show that the unit ball of a real inner product space equipped with Einstein addition forms a uniquely 2-divisible gyrocommutative gyrogroup or a B-loop in the loop literature. One notable result is a compact formula for Einstein addition in terms of Möbius addition. In the second part of this paper, we show that the symmetric group of a gyrogroup admits the gyrogroup structure, thus obtaining an analog of Cayley's theorem for gyrogroups. We examine subgyrogroups, gyrogroup homomorphisms, normal subgyrogroups, and quotient gyrogroups and prove the isomorphism theorems. We prove a version of Lagrange's theorem for gyrogroups and use this result to prove that gyrogroups of particular order have the Cauchy property.

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Section: Proposition 15 Let G and X Be As In Theorem 13 If G Is Gyrmentioning

confidence: 99%

“…It is also an algebraic structure that underlies the qubit density matrices, which play an important role in quantum mechanics [37,39,66]. For a connection to Thomas precession, see [74].…”

Section: Introductionmentioning

confidence: 99%

The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

Suksumran

2016

Essays in Mathematics and Its Applications

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“…Being a natural extension of the algebra of groups, the algebra of gyrogroups has been explored and employed by several authors; see, for instance, [1-7, 13, 14, 17-22, 28, 33, 34, 36-40, 62], and [41,43,44,46,49,[58][59][60]. We realize that, as noted in [11, p. 523], the computation language that Einstein addition encodes plays a universal computational role, which extends far beyond the domain of special relativity.…”

Section: Definition 10 (The Gyrogroup Cooperationmentioning

confidence: 99%

Novel Tools to Determine Hyperbolic Triangle Centers

Ungar

2016

Essays in Mathematics and Its Applications

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Recently discovered tools to study analytic hyperbolic geometry in terms of analogies with analytic Euclidean geometry are presented and employed. Special attention is paid to the study of two novel hyperbolic triangle centers that we call hyperbolic Cabrera points of a hyperbolic triangle and to the way they descend to their novel Euclidean counterparts. The two novel hyperbolic Cabrera points are the (1) Cabrera gyrotriangle ingyrocircle gyropoint and the (2) Cabrera gyrotriangle exgyrocircle gyropoint. Accordingly, their Euclidean counterparts to which they descend are the two novel Euclidean Cabrera points, which are the (1) Cabrera triangle incircle point and the (2) Cabrera triangle excircle point.

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“…Interestingly, gyrations are the mathematical abstraction of the relativistic effect known as Thomas precession [37, Sec. 10.3] [45]. Thomas precession, in turn, is related to the mixed state geometric phase, as Lévay discovered in his work [21] which, according to [21], was motivated by the author work in [34].…”

Section: Einstein Addition Vs Vector Additionmentioning

confidence: 99%

An Introduction to Hyperbolic Barycentric Coordinates and their Applications

Ungar

2014

Mathematics Without Boundaries

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Barycentric coordinates are commonly used in Euclidean geometry. The adaptation of barycentric coordinates for use in hyperbolic geometry gives rise to hyperbolic barycentric coordinates, known as gyrobarycentric coordinates. The aim of this article is to present the path from Einstein's velocity addition law of relativistically admissible velocities to hyperbolic barycentric coordinates, along with applications.

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Gyrations: The Missing Link Between Classical Mechanics with Its Underlying Euclidean Geometry and Relativistic Mechanics with Its Underlying Hyperbolic Geometry

Abstract: Being neither commutative nor associative, Einstein velocity addition of relativistically admissible velocities gives rise to gyrations. Gyrations, in turn, measure the extent to which Einstein addition deviates from commutativity and from associa-1

Cited by 5 publications

References 56 publications

The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

Novel Tools to Determine Hyperbolic Triangle Centers

An Introduction to Hyperbolic Barycentric Coordinates and their Applications

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