Abraham A. Ungar

Thomas rotation and the parametrization of the Lorentz transformation group

Two successive pure Lorentz transformations are equivalent to a pure Lorentz transformation preceded by a 3×3 space rotation, called a Thomas rotation. When applied to the gyration of the rotation axis of a spinning mass, Thomas rotation gives rise to the well-knownThomas precession. A 3×3 parametric, unimodular, orthogonal matrix that represents the Thomas rotation is presented and studied. This matrix representation enables the Lorentz transformation group to be parametrized by two physical observables: the (3-dimensional) relative velocity and orientation between inertial frames. The resulting parametrization of the Lorentz group, in turn, enables the composition of successive Lorentz transformations to be given by parameter composition. This composition is continuously deformed into a corresponding, well-known Galilean transformation composition by letting the speed of light approach infinity. Finally, as an application the Lorentz transformation with given orientation parameter is uniquely expressed in terms of an initial and a final time-like 4-vector.

Thomas precession and its associated grouplike structure

Mathematics phenomena and discovers the secret analogies which unite them. Joseph Fourier. Where there is physical significance, there is pattern and mathematical regularity. The aim of this article is to expose a hitherto unsuspected grouplike structure underlying the set of all relativistically admissible velocities, which shares remarkable analogies with the ordinary group structure. The physical phenomenon that stores the mathematical regularity in the set of all relativistically admissible three‐velocities turns out to be the Thomas precession of special relativity theory. The set of all three‐velocities forms a group under velocity addition. In contrast, the set of all relativistically admissible three‐velocities does not form a group under relativistic velocity addition. Since groups measure symmetry and exhibit mathematical regularity it seems that the progress from velocities to relativistically admissible ones involves a loss of symmetry and mathematical regularity. This article reveals that the...

Thomas precession: Its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics

Gyrogroup theory and its applications is introduced and explored, exposing the fascinating interplay between Thomas precession of special relativity theory and hyperbolic geometry. The abstract Thomas precession, called Thomas gyration, gives rise to grouplike objects called gyrogroups [A, A. Ungar, Am. J. Phys.59, 824 (1991)] the underlying axions of which are presented. The prefix gyro extensively used in terms like gyrogroups, gyroassociative and gyrocommutative laws, gyroautomorphisms, and gyrosemidirect products, stems from their underlying abstract Thomas gyration. Thomas gyration is tailor made for hyperbolic geometry. In a similar way that commutative groups underlie vector spaces, gyrocommutative gyrogroups underlie gyrovector spaces. Gyrovector spaces, in turn, provide a most natural setting for hyperbolic geometry in full analogy with vector spaces that provide the setting for Euclidean geometry. As such, their applicability to relativistic physics and its spacetime geometry is obvious.

Weakly Associative Groups

The space ℝc3 of 3-dimensional relativistically admissible velocities possesses (i) a binary operation which represents the relativistic velocity composition law; and (ii) a mapping from the cartesian product ℝc3×ℝc3 into a subgroup of its automorphism group, Aut(ℝc3), representing the Thomas precession of special relativity. These binary operation and mapping are studied in special relativity as two isolated phenomena. It was recently discovered, however, that they are linked by an algebraic structure which gives rise to a theory of weakly associative and weakly associative-commutative groups. The axioms of these groups are presented in this paper and employed to obtain various interesting results. The algebraic structure underlying these nonstandard groups has been discovered and studied in a totally different context by Karzel (1965), Kerby and Wefelscheid.

The relativistic velocity composition paradox and the Thomas rotation

The relativistic velocity composition paradox of Mocanu and its resolution are presented. The paradox, which rests on the bizarre and counterintuitive non-communtativity of the relativistic velocity composition operation, when applied to noncollinear admissible velocities, led Mocanu to claim that there are “some difficulties within the framework of relativistic electrodynamics.” The paradox is resolved in this article by means of the Thomas rotation, shedding light on the role played by composite velocities in special relativity, as opposed to the role they play in Galilean relativity.

Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession

List of Figures. List of Tables. Preface. Acknowledgments. Introduction A.A. Ungar. 1. Thomas Precession: The Missing Link. 2. Gyrogroups: Modeled on Einsteins Addition. 3. The Einstein Gyrovector Space. 4. Hyperbolic Geometry of Gyrovector Spaces. 5. The Ungar Gyrovector Space. 6. The Mobius Gyrovector Space. 7. Gyrogeometry. 8. Gyrooperations -- The SL(2,C) Approach. 9. The Cocycle Form. 10. The Lorentz Group and its Abstraction. 11. The Lorentz Transformation Link. 12. Other Lorentz Groups. 13. References. About the Author. Topic Index. Author Index.

The Relativistic Noncommutative Nonassociative Group of Velocities and the Thomas Rotation

The bizarre and counterintuitive noncommutativity and nonassociativity of the relativistic composition of nonparallel admissible velocities is sometimes interpreted as a peculiarity of special theory of relativity. It is related to the fact that Lorentz acceleration transformations in three space dimensions do not form a group due to the presence of the so called Thomas rotation. The Thomas rotation turns out to be the effect that provides a means to the presentation of the set of relativistically admissible velocities as an interesting noncommutative, nonassociative group with a group operation given by the relativistic velocity composition. Interestingly, the algebraic structure induced by the Thomas rotation is not an isolated result in special relativity. It was earlier discovered by Karzel, and studied by Kerby and by Wefelscheid in a totally different context.

The holomorphic automorphism group of the complex disk.

The group of all holomorphic automorphisms of the complex unit disk consists of Möbius transformations involving translation-like holomorphic automorphisms and rotations. The former are calledgyrotranslations. As opposed to translations of the complex Plane, which are associative-commutative operations forming a group, gyrotranslations of the complex unit disk fail to form a group. Rather, left gyrotranslations are gyroassociative-gyrocommutative operations forming agyrogroup.

From Pythagoras To Einstein: The Hyperbolic Pythagorean Theorem

A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right-angled triangle as the “Einstein sum” of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein. Following the pioneering work of Varičak it is well known that relativistic velocities are governed by hyperbolic geometry in the same way that prerelativistic velocities are governed by Euclidean geometry. Unlike prerelativistic velocity composition, given by the ordinary vector addition, the composition of relativistic velocities, given by the Einstein addition, is neither commutative nor associative due to the presence of Thomas precession. Following the discovery of the mathematical regularity that Thomas precession stores, it is now possible to extend Thomas precession by abstraction, (i) allowing hyperbolic geometry to be studied by means of analogies that it shares with Euclidean geometry; and, similarly (ii) allowing velocities and accelerations in relativistic mechanics to be studied by means of analogies that they share with velocities and accelerations in classical mechanics. The abstract Thomas precession, called the Thomas gyration, gives rise to gyrovector space theory in which the prefix gyro is used extensively in terms like gyrogroups and gyrovector spaces, gyroassociative and gyrocommutative laws, gyroautomorphisms, gyrotranslations, etc. We demonstrate the superiority of our gyrovector space formalism in capturing analogies by deriving the Hyperbolic Pythagorean Theorem in a form fully analogous to its Euclidean counterpart, thus contrasting it with the standard form in which the Hyperbolic Pythagorean Theorem is known in the literature. The hyperbolic metric, which supports the Hyperbolic Pythagorean Theorem, has a dual metric. We show that the dual metric does not support a Pythagorean theorem but, instead, it supports the π-Theorem according to which the sum of the three dual angles of a hyperbolic triangle is π.

The abstract Lorentz transformation group

Treatments of the Lorentz transformation of special relativity at an undergraduate level usually assume that the motion of one observer is along the x axis of another observer, resulting in the (1+1)‐dimensional Lorentz transformation group. The (1+n)‐dimensional Lorentz group, for n=2 or n=3, is unknown to many physics students because of the simplified (1+1)‐dimensional treatments found in most texts. The aim of this article is to simplify the presentation of the (homogeneous, proper, orthochronous) Lorentz group by abstraction to the point where the (1+n)‐dimensional Lorentz group can readily be presented to physics students in n space dimensions where n is finite or infinite. The study of the (homogeneous, proper, orthochronous) Lorentz transformation group is simplified and generalized in this article by abstraction, thus obtaining an elegant formalism to deal with the Lorentz group. This new formalism allows one to solve in the abstract Lorentz group previously poorly understood problems in the stan...