The Einstein Relativistic Velocity Model of Hyperbolic Geometry and Its Plane Separation Axiom

Sönmez, Nilgün; Ungar, Abraham A.

doi:10.1007/s00006-012-0367-z

Cited by 11 publications

(6 citation statements)

References 19 publications

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“…(43). A Clifford algebra approach to study Möbius and Einstein gyrogroups is very fruitful [14,16,18,19,44,59,62].…”

Section: Introductionmentioning

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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“…(43). A Clifford algebra approach to study Möbius and Einstein gyrogroups is very fruitful [14,16,18,19,44,59,62].…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, the Möbius gyrovector space is associated with the Poincaré model of conformal geometry on the open unit ball in n-dimensional Euclidean space R n [11,38], and the Einstein gyrovector space is associated with the Beltrami-Klein model of hyperbolic geometry on the unit ball in R n [38,52,53,59,63].…”

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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“…The need for a careful study of the very intuitive idea that every line in a Euclidean plane has "two sides" was pointed out by Millman and Parker in [10, p. 63], resulting in what they call the Plane Separation Axiom (PSA). An analogous Gyroplane Separation Axiom (GPSA) for a gyroplane in an Einstein gyrovector space, is studied by Sönmez and Ungar in [15].…”

Section: Psfrag Replacementsmentioning

confidence: 99%

On the Study of Hyperbolic Triangles and Circles by Hyperbolic Barycentric Coordinates in Relativistic Hyperbolic Geometry

Ungar

2014

Mathematics Without Boundaries

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Barycentric coordinates are commonly used in Euclidean geometry. Following the adaptation of barycentric coordinates for use in hyperbolic geometry in recently published books on analytic hyperbolic geometry, known and novel results concerning triangles and circles in the hyperbolic geometry of Lobachevsky and Bolyai are discovered. Among the novel results are the hyperbolic counterparts of important theorems in Euclidean geometry. These are: (1) the Inscribed Gyroangle Theorem, (ii) the Gyrotangent-Gyrosecant Theorem, (iii) the Intersecting Gyrosecants Theorem, and (iv) the Intersecting Gyrochord Theorem. Here in gyrolanguage, the language of analytic hyperbolic geometry, we prefix a gyro to any term that describes a concept in Euclidean geometry and in associative algebra to mean the analogous concept in hyperbolic geometry and nonassociative algebra. Outstanding examples are gyrogroups and gyrovector spaces, and Einstein addition being both gyrocommutative and gyroassociative. The prefix "gyro" stems from "gyration", which is the mathematical abstraction of the special relativistic effect known as "Thomas precession".

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“…The aim of this section, Example II, is to illustrate Definition 9 of gyrobarycentric coordinates in a form analogous to Example I [38].…”

Section: Example II -The Hyperbolic Segmentmentioning

confidence: 99%

Hyperbolic Geometry

Ungar¹

2013

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Relativistic hyperbolic geometry is a model of the hyperbolic geometry of Lobachevsky and Bolyai in which Einstein addition of relativistically admissible velocities plays the role of vector addition. The adaptation of barycentric coordinates for use in relativistic hyperbolic geometry results in the relativistic barycentric coordinates. The latter are covariant with respect to the Lorentz transformation group just as the former are covariant with respect to the Galilei transformation group. Furthermore, the latter give rise to hyperbolically convex sets just as the former give rise to convex sets in Euclidean geometry. Convexity considerations are important in non-relativistic quantum mechanics where mixed states are positive barycentric combinations of pure states and where barycentric coordinates are interpreted as probabilities. In order to set the stage for its application in the geometry of relativistic quantum states, the notion of the relativistic barycentric coordinates that relativistic hyperbolic geometry admits is studied.

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The Einstein Relativistic Velocity Model of Hyperbolic Geometry and Its Plane Separation Axiom

Cited by 11 publications

References 19 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

On the Study of Hyperbolic Triangles and Circles by Hyperbolic Barycentric Coordinates in Relativistic Hyperbolic Geometry

Hyperbolic Geometry

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