Abraham A. Ungar, "Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics," Found. Phys., 27, 881-951 (1997).  Home/Search   Document Not in Database   Summary   Related Articles   Check

No context found.

Abraham A. Ungar, "Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics," Found. Phys., 27, 881-951 (1997).

.... Law Right Gyroassociative Law Left Loop Property Right Loop Property Gyroautomorphism Inversion The prefix gyro that we extensively use to emphasize analogies stems from the Thomas gyration, which is the abstract extension of the relativistic effect known as the Thomas precession [8]. The remarkable analogies that M6bius addition in the disc shares with the common vector addition remain valid in higher dimensions. In fact, the groupoid (c, possesses a grouplike structure called a gyrogroup [8] the formal definition of which follows. Definition 2.1 (Groupoids, Automorphism ....

....the abstract extension of the relativistic effect known as the Thomas precession [8] The remarkable analogies that M6bius addition in the disc shares with the common vector addition remain valid in higher dimensions. In fact, the groupoid (c, possesses a grouplike structure called a gyrogroup [8], the formal definition of which follows. Definition 2.1 (Groupoids, Automorphism Groups) A groupold (S, is a pair of a non empty set S with a binary operation . An automorphism of a groupold (S, is a bijective self map of S that respects its binary operation, qS(s s2) s s2, for all ....

Abraham A. Ungar. Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics. Found. Phys., 27(6):881-951, 1997.

....= u v) gyr[u; v]w (u v) w = u (v gyr[v; u]w) thus discovering a grouplike structure called a gyrogroup. Moreover, Thomas precession possesses the so called loop property, gyr[u; v] gyr[u v; v] which makes it algebraically effective in uncovering analogies shared with groups and gyrogroups [1]. Thus, 1) Einstein s addition, is a gyrocommutative gyrogroup operation in the Einstein relativity gyrogroup (R 3 c ; of relativistically admissible velocities, in full analogy with ordinary 3 (2) vector addition, which is a commutative group operation in the group (R 3 ; of ....

....gyrogroup operations results in geodesics of other models of hyperbolic geometry. These are shown in Fig. 1 for = E , in Fig. 2 for = M , in Fig. 3 for = U , and in Fig. 4 for = C . The theory of gyrogroups and gyrovector spaces and its applications are presented in [1], 2] 3] and in a forthcoming book [4] where the advantages offered by the use of gyrogroup theoretic techniques in the manipulation of the Lorentz group for solving important problems are demonstrated. Taking the key features of Einstein s addition as axioms, and guided by analogies with ....

[Article contains additional citation context not shown here]

A. A. Ungar, "Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics," Found. Phys., vol. 27, no. 6, pp. 881--951, 1997.

....by relative velocities alone. They are determined by both relative velocities and relative orientations which, unlike their Galilean counterparts, are woven together and cannot be decoupled unless the Thomas precession is invoked [8] Thomas precession gyr[u, v] has rich mathematical structure [11]; it possesses, for instance, the following useful algebraic rules, gyr[u, v] gyr[v, u] 1 gyr[ u, v] gyr[u, v] gyr[u#v, v] gyr[u, v#u] gyr[u, v] 2.4) The application of any Thomas precession gyr[u, v] u, v # R 3 c , to any x # R 3 c is expressible in terms of ....

....v through an angle #. In contrast, the gyroassociative law (2.7) is a relatively recent discovery made in 1988 [7] signaling the birth of gyrogroup theory. In that theory, the abstract Thomas precession is called Thomas gyration, suggesting the prefix gyro that we use to emphasize analogies [11]. A groupoid is a non empty space with a binary operation. The groupoid (R 3 c , #) called the Einstein gyrogroup, thus shares remarkable analogies with the group (R 3 , of all Newtonian velocities. In general, Gyrogroups are grouplike objects modeled on Einstein s addition with its ....

[Article contains additional citation context not shown here]

Abraham A. Ungar. Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics. Found. Phys., 27(6):881--951, 1997. 13 14 ABRAHAM A. UNGAR

....velocity addition is neither commutative nor associative. Is the progress from the common vector addition of velocities , which is a group operation, to the Einstein velocity addition Phi, which is not a group operation, associated with loss of mathematical regularity It has been shown in [11] that the group structure that has been lost in the transition from the group ( 3 ; to the nongroup groupoid ( 3 1 ; Phi) is replaced by a loop structure using a relativistic peculiar rotation called theThomas precession. Extending the Einstein relativistic groupoid ( 3 1 ; Phi) with ....

....scalar multiplication and inner product and, in full analogy, some gyrocommutative gyrogroups can be extended to gyrovector spaces. Then, unexpectedly, gyrovector spaces provide the setting for hyperbolic geometry in full analogy with vector spaces, that provide the setting for Euclidean geometry [11,13]. Hence, the hyperbolic geometry of Bolyai Lobachevski is in fact the gyro Euclidean geometry. Like groups, there are finite and infinite gyrogroups some of which are gyrocommutative. We obtain particularly interesting results when the order of the group in which a gyrogroup resides as a twisted ....

[Article contains additional citation context not shown here]

Abraham A. Ungar. Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics. Found. Phys. 27 (1997), 881-951.

....in this article need not be gyrocommutative. Twisted subgroups arise in the study of problems in computational complexity [1] and in the study of gyrogroups [7] Gyrogroups are grouplike structures that first arose in the study of Einstein s velocity addition in the special theory of relativity [23, 24]. We showed in [7] that any gyrogroup is an extension of a group by a gyrocommutative gyrogroup. The gyrogroups that we construct in this article demonstrate that this extension is not trivial. x1. Introduction Gyrogroup theory is an algebraic theory modelled on the groupoid of all ....

....The gyrogroups that we construct in this article demonstrate that this extension is not trivial. x1. Introduction Gyrogroup theory is an algebraic theory modelled on the groupoid of all relativistically admissible velocities with their Einstein velocity composition law and Thomas prcssion [23]. In our previous article [7] we have shown that any gyrogroup is an extension of a group by a gyrocommutative gyrogroup. In Remark 4.2 we show in this article that this extension need not be trivial. In fact, we present in the Remark an extension of a group by another group that gives a ....

[Article contains additional citation context not shown here]

Abraham A. Ungar. Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics. Found. Phys. 27 (1997), 881-951.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC