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Abraham A Ungar - One of the best experts on this subject based on the ideXlab platform.
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the intrinsic beauty harmony and interdisciplinarity in Einstein velocity Addition law gyrogroups and gyrovector spaces
Mathematics Interdisciplinary Research, 2016Co-Authors: Abraham A UngarAbstract:The only justification for the Einstein velocity Addition law appeared to be its empirical adequacy, so that the intrinsic beauty and harmony in Einstein Addition remained for a long time a mystery to be conquered. Accordingly, the aim of this expository article is to present (i) the Einstein relativistic vector Addition, (ii) the resulting Einstein scalar multiplication, (iii) the Einstein relativistic mass, and (iv) the Einstein relativistic kinetic energy, along with remarkable analogies with classical results in groups and vector spaces that these Einstein concepts capture in gyrogroups and gyrovector spaces. Making the unfamiliar familiar, these analogies uncover the intrinsic beauty and harmony in the underlying Einstein velocity Addition law of relativistically admissible velocities, as well as its interdisciplinarity.
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Analytic Hyperbolic Geometry in N Dimensions: An Introduction
2014Co-Authors: Abraham A UngarAbstract:List of Figures Preface Author's Biography Introduction Gyrovector Spaces in the Service of Abalytic Hyperbolic Geometry When Two Counterintuitive Theories Meet The Fascinating Rich Mathematical Life of Einstein's Velocity Addition Law Parts of the Book Einstein Gyrogroups and Gyrovector Spaces Einstein Gyrogroups Introduction Einstein Velocity Addition Einstein Addition for Computer Algebra Thomas Precession Angle Einstein Addition With Respect to Cartesian Coordinates Einstein Addition Vs. Vector Addition Gyrations Gyration Angles From Einstein Velocity Addition to Gyrogroups Gyrogroup Cooperation (CoAddition) First Gyrogroup Properties Elements of Gyrogroup Theory The Two Basic Gyrogroup Equations The Basic Gyrogroup Cancellation Laws Automorphisms and Gyroautomorphisms Gyrosemidirect Product Basic Gyration Properties An Advanced Gyrogroup Equation Gyrocommutative Gyrogroups Problems Einstein Gyrovector Spaces 65 The Abstract Gyrovector Space Einstein Gyrovector Spaces Einstein Addition and Differential Geometry Euclidean Lines Gyrolines - The Hyperbolic Lines Gyroangles - The Hyperbolic Angles Euclidean Isometries The Group of Euclidean Motions Gyroisometries - The Hyperbolic Isometries Gyromotions - The Motions of Hyperbolic Geometry Problems Relativistic Mass Meets Hyperbolic Geometry Lorentz Transformation and Einstein Addition Mass of Particle Systems Resultant Relativistically Invariant Mass Problems Mathematical Tools for Hyperbolic Geometry Barycentric and Gyrobarycentric Coordinates Barycentric Coordinates Segments Gyrobarycentric Coordinates Uniqueness of Gyrobarycentric Representations Gyrovector Gyroconvex Span Gyrosegments Triangle Centroid Gyromidpoint Gyroline Boundary points Gyrotriangle Gyrocentroid Gyrodistance in Gyrobarycentric Coordinates Gyrolines in Gyrobarycentric Coordinates Problems Gyroparallelograms and Gyroparallelotopes The Parallelogram Law Einstein Gyroparallelograms The Gyroparallelogram Law The Higher-Dimensional Gyroparallelotope Law Gyroparallelotopes Gyroparallelotope Gyrocentroid Gyroparallelotope Formal Definition and Theorem Low Dimensional Gyroparallelotopes Hyperbolic Plane Separation GPSA for the Einstein Gyroplane Problems Gyrotrigonometry Gyroangles Gyroangle - Angle Relationship The Law of Gyrocosines The SSS to AAA Conversion Law Inequalities for Gyrotriangles The AAA to SSS Conversion Law The Law of Sines/Gyrosines The Law of Gyrosines The ASA to SAS Conversion Law Gyrotriangle Defect Right Gyrotriangles Gyrotrigonometry Gyroangle of Parallelism Useful Gyrotriangle Gyrotrigonometric Identities A Determinantal Pattern Problems Hyperbolic Triangles and Circles Gyrotriangles and Gyrocircles Gyrocircles Gyrotriangle Circumgyrocenter Triangle Circumcenter Gyrotriangle Circumgyroradius Triangle Circumradius The Gyrocircle Through Three Points The Inscribed Gyroangle Theorem I The Inscribed Gyroangle Theorem II Gyrocircle Gyrotangent Gyrolines Semi-Gyrocircle Gyrotriangles Problems Gyrocircle Theorems The Gyrotangent-Gyrosecant Theorem The Intersecting Gyrosecants Theorem Gyrocircle Gyrobarycentric Representation Gyrocircle Interior and Exterior Points Circle Barycentric Representation Gyrocircle Gyroline Intersection Gyrocircle-Gyroline Tangency Points Gyrocircle Gyrotangent Gyrolength Circle-Line Tangency Points Circumgyrocevians Gyrodistances Related to the Gyrocevian A Gyrodistance Related to the Circumgyrocevian Circumgyrocevian Gyrolength The Intersecting Gyrochords Theorem Problems Hyperbolic Simplices, Hyperplanes and Hyperspheres in N Dimensions Gyrosimplices Gyrotetrahedron Circumgyrocenter Gyrotetrahedron Circumgyroradius Gyrosimplex Gyrocentroid Gamma Matrices Gyrosimplex Gyroaltitudes Gyrosimplex Circumhypergyrosphere The Gyrosimplex Constant Point to Gyrosimplex Gyrodistance Cramer's Rule Point to Gyrosimplex Perpendicular Projection Gyrosimplex In-Exgyrocenters and In-Exgyroradii Gyrotriangle In-Exgyrocenters Gyrosimplex Gyrosymmedian Problems Gyrosimplex Gyrovolume Gyrovolume Problems Hyperbolic Ellipses and Hyperbolas Gyroellipses and Gyrohyperbolas Gyroellipses - A Gyrobarycentric Representation Gyroellipses - Gyrotrigonometric Gyrobarycentric Representation Gyroellipse Major Vertices Gyroellipse Minor Vertices Canonical Gyroellipses Gyrobarycentric Representation of Canonical Gyroellipses Barycentric Representation of Canonical Ellipses Some Properties of Canonical Gyroellipses Canonical Gyroellipses and Ellipses Canonical Gyroellipse Equation A Gyrotrigonometric Constant of the Gyroellipse Ellipse Eccentricity Gyroellipse Gyroeccentricity Gyrohyperbolas - A Gyrobarycentric Representation Problems Thomas Precession Thomas Precession Introduction The Gyrotriangle Defect and Thomas Precession Thomas Precession Thomas Precession Matrix Thomas Precession Graphical Presentation Thomas Precession Angle Thomas Precession Frequency Thomas Precession and Boost Composition Thomas Precession Angle and Generating Angle have Opposite Signs Problems Bibliography Index
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mobius transformation and Einstein velocity Addition in the hyperbolic geometry of bolyai and lobachevsky
arXiv: Mathematical Physics, 2012Co-Authors: Abraham A UngarAbstract:In this chapter, dedicated to the 60th Anniversary of Themistocles M. Rassias, Mobius transformation and Einstein velocity Addition meet in the hyperbolic geometry of Bolyai and Lobachevsky. It turns out that Mobius Addition that is extracted from Mobius transformation of the complex disc and Einstein Addition from his special theory of relativity enable the introduction of Cartesian coordinates and vector algebra as novel tools in the study of hyperbolic geometry.
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on the appeal to a pre established harmony between pure mathematics and relativity physics
Foundations of Physics Letters, 2003Co-Authors: Abraham A UngarAbstract:Soon after its appearance in 1905, the Einsteinian relativity with its relativistically admissible 3-velocities was recognized by Vladimir Varicak in 1908 as the realization in physics of the hyperbolic geometry of Bolyai and Lobachevski. At the same time, however, during the years 1907–1909 Minkowski reformulated the Einsteinian relativity in terms of a space of 4-velocities that now bears his name. As a result, the special theory of relativity that we find in the mainstream literature is not the one originally formulated by Einstein but, rather, the one reformulated by Minkowski. Thus, in particular, one of the most powerful ideas of Einstein in 1905, the Einstein Addition of relativistically admissible 3-velocities that need not be parallel, is unheard of in most texts on relativity physics. Following our recently published book, Beyond the Einstein Addition, Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces [1], the aim of this article is to employ the principle of pre-established harmony between mathematics and physics to demonstrate that the original Einsteinian relativity, as opposed to the Minkowskian relativity, is the legitimate formulation of special relativity whose time has returned.
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seeing the mobius disc transformation group like never before
Computers & Mathematics With Applications, 2003Co-Authors: Abraham A UngarAbstract:Abstract The introduction of the gyration notion into nonassociative algebra, hyperbolic geometry, and relatively physics is motivated in this article by the emergence of the gyrogroup notion in the theory of the Mobius transformation group of the complex open unit disc. It suggests the prefix “gyro” that we use to emphasize analogies. Thus, for instance, gyrogroups are classified into gyrocommutative and nongyrocommutative gyrogroups in full analogy with the classification of groups into commutative and noncommutative groups. The road from the Thomas precession of the special theory of relativity to the Thomas gyration as well as the resulting new theory is presented in the author's book: Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces [1]. The main result of this article is a theorem that allows the validity of some gyration identities to be extended from gyrocommutative gyrogroups into gyrogroups that need not be gyrocommutative. To set the stage for the main result, the Mobius disc-transformation group is studied in a novel way that suggests the notion of the gyrogroup and its gyrations.
Teerapong Suksumran - One of the best experts on this subject based on the ideXlab platform.
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the algebra of gyrogroups cayley s theorem lagrange s theorem and isomorphism theorems
2016Co-Authors: Teerapong SuksumranAbstract: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 Mobius 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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the algebra of gyrogroups cayley s theorem lagrange s theorem and isomorphism theorems
2016Co-Authors: Teerapong SuksumranAbstract: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 Mobius 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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Einstein gyrogroup as a b loop
Reports on Mathematical Physics, 2015Co-Authors: Teerapong Suksumran, Keng WiboontonAbstract:Using the Clifford algebra formalism, we give an algebraic proof that the open unit ball B = { v ∈ R n : ‖ v ‖ 1 } of R n equipped with Einstein Addition ⊕E forms a B-loop or, equivalently, a uniquely 2-divisible gyrocommutative gyrogroup. We obtain a compact formula for Einstein Addition in terms of Mobius Addition. We then give a characterization of associativity and commutativity of vectors in B with respect to Einstein Addition.
Tuval Foguel - One of the best experts on this subject based on the ideXlab platform.
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Gyrogroups and the decomposition of groups into twisted subgroups and subgroups
2015Co-Authors: Tuval FoguelAbstract:Gyrogroups are generalized groups modelled on the Ein-stein groupoid of all relativistically admissible velocities with their Einstein’s velocity Addition as a binary operation. Ein-stein’s gyrogroup fails to form a group since it is nonassocia-tive. The breakdown of associativity in the Einstein Addition does not result in loss of mathematical regularity owing to the presence of the relativistic effect known as the Thomas precession which, by abstraction, becomes an automorphism called the Thomas gyration. The Thomas gyration turns out to be the missing link that gives rise to analogies shared by gyrogroups and groups. In particular, it gives rise to the gyroassociative and the gyrocommuttive laws that Einstein’s Addition possesses, in full analogy with the associative and the commutative laws that vector Addition possesses in a vec-tor space. The existence of striking analogies shared by gy
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Involutory decomposition of groups into twisted subgroups and subgroups
2000Co-Authors: Tuval FoguelAbstract:Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity Addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein Addition does not result in loss of mathematical regularity owing to the presence of the relativistic effect known as the Thomas precession which, by abstraction, becomes an automorphism called the Thomas gyration. The Thomas gyration turns out to be the missing link that gives rise to analogies shared by gyrogroups and groups. In particular, it gives rise to the gyroassociative and the gyrocommuttive laws that Einstein’s Addition possesses, in full analogy with the associative and the commutative laws that vector Addition possesses in a vector space. The existence of striking analogies shared by gyrogroup
Ungar, Abraham A. - One of the best experts on this subject based on the ideXlab platform.
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On the Study of Hyperbolic Triangles and Circles by Hyperbolic Barycentric Coordinates in Relativistic Hyperbolic Geometry
2013Co-Authors: Ungar, Abraham A.Abstract: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 {\it gyrogroups} and {\it gyrovector spaces}, and Einstein Addition being both {\it gyrocommutative} and {\it gyroassociative}. The prefix "gyro" stems from "gyration", which is the mathematical abstraction of the special relativistic effect known as "Thomas precession".Comment: 78 pages, 26 figure
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Gyrations: The Missing Link Between Classical Mechanics with its Underlying Euclidean Geometry and Relativistic Mechanics with its Underlying Hyperbolic Geometry
2013Co-Authors: Ungar, Abraham A.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 associativity. Gyrations are geometric automorphisms abstracted from the relativistic mechanical effect known as Thomas precession
Nilgun Sonmez - One of the best experts on this subject based on the ideXlab platform.
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the Einstein relativistic velocity model of hyperbolic geometry and its plane separation axiom
Advances in Applied Clifford Algebras, 2013Co-Authors: Nilgun SonmezAbstract:The relativistically admissible velocities of Einstein’s special theory of relativity are regulated by the Beltrami–Klein ball model of the hyperbolic geometry of Bolyai and Lobachevsky. It is shown in this expository article that the Einstein velocity Addition law of relativistically admissible velocities enables Cartesian coordinates to be introduced into hyperbolic geometry, resulting in the Cartesian–Beltrami-Klein ball model of hyperbolic geometry. Suggestively, the latter is increasingly becoming known as the Einstein Relativistic Velocity Model of hyperbolic geometry. Mobius Addition is a transformation of the ball linked to Clifford algebra. Einstein Addition and Mobius Addition in the ball of the Euclidean n-space are isomorphic to each other, and they share remarkable analogies with vector Addition. Thus, in particular, Einstein (Mobius) Addition admits scalar multiplication, giving rise to gyrovector spaces, just as vector Addition admits scalar multiplication, giving rise to vector spaces. Moreover, the resulting Einstein (Mobius) gyrovector spaces form the algebraic setting for the Beltrami-Klein (Poincare) ball model of n-dimensional hyperbolic geometry, just as vector spaces form the algebraic setting for the standard Cartesian model of n-dimensional Euclidean geometry. As an illustrative novel example special attention is paid to the study of the plane separation axiom (PSA) in Euclidean and hyperbolic geometry.