A velocity-orientation formalism to deal with compositions of successive Lorentz transformations, emphasizing analogies shared by Lorentz and Galilean transformations, has recently been developed. The emphasis in the present article is on the convenience of using the velocity-orientation formalism by resolving a paradox in the study of successive Lorentz transformations of the electromagnetic field that was recently raised by Mocanu. The paradox encountered by Mocanu results from the omission of the Thomas rotation (or, precession) which is involved in the composition of (...) two Lorentz transformations with corresponding noncollinear velocity parameters. By resolving this paradox, we expose (i) the central role that the Thomas rotation plays in special relativity, (ii) the need to consider in special relativity orientations in addition to velocities between inertial frames, and (iii) the power and elegance of the novel velocity-orientation formalism. A similar paradox in STR that Mocanu pointed out, also resulting from the omission of the Thomas rotation, has been resolved in a previous communication. (shrink)

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. (shrink)

A new vectorial representation for the successive Lorentz transformations has recently been proved very convenient to achieve a straightforward treatment of the Thomas rotation effect. Such a representation rests on equivalent forms for the pure Lorentz transformation and SLT whose physical meaning escaped us. The present paper fills this gap in by showing that those equivalent forms could represent appropriate world lines, lines and planes of simultaneity. Those geometric elements are particularly convenient to build up two new graphical representations for (...) the SLT: the first rests on that equivalent form for the SLT, while the second takes the SLT as a PLT preceded or followed by a Thomas rotation and uses the equivalent form for the PLT. As an application, the SLT Lorentz contraction formulas are derived for the first time. The dependence of the SLTLC on the Thomas rotation is put in evidence. The SLTLC along directions transverse and parallel to the composite velocity is studied. Original SLT Minkowski diagrams are given for the first time. (shrink)

The Bell inequalities of the metric form are introduced. The quantum-mechanical correlations of the particles with s=1/2 and photons are described using the relative measure of probability on the concave surfaces. The relation of the proposed scheme with the Bayes theorem about conditional information entropy and J. von Neumann's postulates is discussed.