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. (shrink)

The objective of this article is to provide a formalism to deal with the special theory of relativity (STR, in short) as riewed by Reichenbach, according to which STR involves an ineradicableconventionality of simultaneity. One of the two postulates of STR asserts that, in empty space, the one-way speed of light relative to inertial frames is constant. Experimental evidence, however, is related to the constancy of the round-trip speed of light and has no bearing on one-way speeds. Following Reichenbach's viewpoint, (...) we relax the second postulate of STR, abandoning the constancy of the one-way speed of light to the more realistic one asserting the constancy of the round-trip speed of light. This, in turn, results in a formalism to deal with Reichenbach's special theory of relativity (RSTR, in short) in which the two one-way speeds of light in empty space, C±, in the two senses of a round-trip are arbitrarily selected in such a way that their harmonic mean is the measurable round-trip speed of light, c. Experimentally, RSTR and STR are indistinguishable and, hence, represent the same physical theory. It is only the formalism that we use to deal with STR which is extended in RSTR to accommodate the immeasurability of one-way velocities. The usefulness of the proposed formalism to study special relativity is demonstrated. (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)

The obscured Thomas precessionof the special theory of relativity (STR) has been soared into prominence by exposing the mathematical structure, called a gyrogroup,to which it gives rise [A. A. Ungar, Amer. J. Phys.59,824 (1991)], and the role that it plays in the study of Lorentz groups [A. A. Ungar, Amer. J. Phys.60,815 (1992); A. A. Ungar, J. Math. Phys.35,1408 (1994); A. A. Ungar, J. Math. Phys.35,1881 (1994)]. Thomas gyrationresults from the abstraction of Thomas precession.As such, its (...) study sheds light on relativistic velocity spaces and their symmetries which are concealed in Thomas precession. In order to uncover new properties of relativistic gyrogroups, we employ in this article the group theoretic concepts of divisible groupsand two-torsion free groupsto construct midpointsin gyrogroups. Systems of successive midpoints then describe straight gyrolinesand suggest a new, natural distance function that involves a Thomas gyration. These, in turn, reveal a new, interesting geometry that underlies relativistic velocity spaces. In this resulting gyrogeometrythe straight gyrolines form geodesics under a distance function which turns out to be the Poincaré hyperbolic distance function relaxed by a Thomas gyration. These geodesics do obey the parallel axiom of Euclidean geometry. Ironically, (i) attempts to understand the parallel postulate of Euclidean geometry gave rise to hyperbolic geometry in which the parallel postulate disappears;(ii) hyperbolic geometry gave rise to Einstein's STR; (iii) Einstein's STR established the bizarre and counterintuitive relativistic effect called Thomas precession, the abstraction of which is called Thomas gyration; and (iv) Thomas gyration now repairs in this article the Poincaré distance function of hyperbolic geometry to the point where the parallel postulate reappears. (shrink)

Gyrogroup theory [A. A. Ungar, Found. Phys. 27, 881–951 (1997)] enables the study of the algebra of Einstein's addition to be guided by analogies shared with the algebra of vector addition. The capability of gyrogroup theory to capture analogies is demonstrated in this article by exposing the relativistic composite-velocity reciprocity principle. The breakdown of commutativity in the Einstein velocity addition ⊕ of relativistically admissible velocities seemingly gives rise to a corresponding breakdown of the relativistic composite-velocity reciprocity principle, since seemingly (...) (i) on one hand, the velocity reciprocal to the composite velocity u⊕v is −(u⊕v) and (ii) on the other hand, it is (−v)⊕(−u). But (iii) −(u⊕v)≠(−v)⊕(−u). We remove the confusion in (i), (ii), and (iii) by employing the gyrocommutative gyrogroup structure of Einstein's addition and, subsequently, present the relativistic composite-velocity reciprocity principle with the Thomas rotation that it involves. (shrink)

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)

Density matrices for mixed state qubits, parametrized by the Bloch vector in the open unit ball of the Euclidean 3-space, are well known in quantum computation theory. We bring the seemingly structureless set of all these density matrices under the umbrella of gyrovector spaces, where the Bloch vector is treated as a hyperbolic vector, called a gyrovector. As such, this article catalizes and supports interdisciplinary research spreading from mathematical physics to algebra and geometry. Gyrovector spaces are mathematical objects that form (...) the setting for the hyperbolic geometry of Bolyai and Lobachevski just as vector spaces form the setting for Euclidean geometry. It is thus interesting, in geometric quantum computation, to realize that the set of all qubit density matrices has rich structure with strong link to hyperbolic geometry. Concrete examples for the use of the gyro-structure to derive old and new, interesting identities for qubit density matrices are presented. (shrink)

The Thomas precession of relativity physics gives rise to important isometries in hyperbolic geometry that expose analogies with Euclidean geometry. These, in turn, suggest our bifurcation approach to hyperbolic geometry, according to which Euclidean geometry bifurcates into two mutually dual branches of hyperbolic geometry in its transition to non-Euclidean geometry. One of the two resulting branches turns out to be the standard hyperbolic geometry of Bolyai and Lobachevsky. The corresponding bifurcation of Newtonian mechanics in the transition to Einsteinian mechanics indicates (...) that there are two, mutually dual, kinds of uniform accelerations. Furthermore, while current hyperbolic geometry “does not use the notion of vector at all” (I. M. Yaglom, Geometric Transformations III, p. 135, trans. by Abe Shenitzer, Random House, New York, 1973), our bifurcation approach exposes the elusive hyperbolic vectors, that we call gyrovectors. (shrink)

We show that the algebra of the group SL(2, C) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the Lorentz group and its underlying hyperbolic geometry. The superiority of the use of the gyrogroup formalism over the use of the SL(2, C) formalism for dealing with the Lorentz group in some cases is indicated by (i) the validity of gyrogroups and gyrovector spaces in higher dimensions, by (ii) the analogies that they share with groups and (...) vector spaces, and by (iii) the demonstration that gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry. As such, gyrogroups and gyrovector spaces provide powerful tools for the study of relativity physics. (shrink)

Hyperbolic vectors are called gyrovectors. We show that the Bloch vector of quantum mechanics is a gyrovector. The Bures fidelity between two states of a qubit is generated by two Bloch vectors. Treating these as gyrovectors rather than vectors results in our novel expression for the Bures fidelity, expressed in terms of its two generating Bloch gyrovectors. Taming the Thomas precession of Einstein's special theory of relativity led to the advent of the theory of gyrogroups and gyrovector spaces. Gyrovector spaces, (...) in turn, form the setting for various models of the hyperbolic geometry of Bolyai and Lobachevski just as vector spaces form the setting for the standard model of Euclidean geometry. It is the recent advent of the theory of gyrogroups and gyrovector spaces that allows the Bures fidelity to be studied in its natural context, hyperbolic geometry, resulting in our new representation of the Bures fidelity, that reveals simplicity, elegance, and hyperbolic geometric significance. (shrink)

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. (shrink)

Inertial frames and Lorentz transformations have a preferred status in the special theory of relativity (STR). Lorentz transformations, in turn, embody Einstein's convention that the velocity of light is isotropic, a convention that is necessary for the establishment of a standard signal synchrony. If the preferred status of Lorentz transformations in STR is not due to some particular bias introduced by a convention on signal synchronism, but to the fact that the Lorentz transformation group is the symmetry group of the (...) theory, then the signal synchronism is not a matter of convention but rather a matter of fact. In order to explore the conventionalist thesis, that within the frame of STR isotropy in the velocity of light and, hence, signal synchronism is a matter of convention, we need a generalized Lorentz transformation group that does not embody Einstein's isotropy convention, and upon which STR can be based. We present here a new approach to the resulting search for a generalized STR, which is well suited for establishing some well-known results of Winnie as well as some new results. (shrink)

The growth of global value chains (GVCs) and the emergence of novel forms of value chain governance pose two questions for normative business ethics. First, how should we conceptualize the relationships between members of a GVC? Second, what ethical implications follow from these relationships, both with respect to interactions between GVC members and with respect to achieving broader transnational governance goals? We address these questions by examining the emergence of transnational eco-labeling as an increasingly prominent form of GVC governance that (...) is redefining the relationship between nominally independent firms. On the first question, we argue that GVCs occupy a middle ground between intrafirm and interfirm transactions, thereby posing a challenge to theoretical frameworks that attempt to apply ethical standards based on transaction types. On the second question, we argue that this unique institutional status leads to a range of novel ethical considerations and dilemmas for GVC members. Lead firms, their suppliers, and third-party standard-setters all confront new ethical quandaries when third-party eco-labeling is introduced to a GVC. The nature of these quandaries means that, while GVCs may be well-positioned to serve as instruments of transnational governance, they are frequently not well-oriented to do so. Our analysis marks an initial attempt to map the ethical considerations that apply to members of a GVC, and in doing so, places the literature on normative business ethics and transnational governance in a closer conversation. (shrink)

The growth of global value chains (GVCs) and the emergence of novel forms of value chain governance pose two questions for normative business ethics. First, how should we conceptualize the relationships between members of a GVC? Second, what ethical implications follow from these relationships, both with respect to interactions between GVC members and with respect to achieving broader transnational governance goals? We address these questions by examining the emergence of transnational eco‐labeling as an increasingly prominent form of GVC governance that (...) is redefining the relationship between nominally independent firms. On the first question, we argue that GVCs occupy a middle ground between intrafirm and interfirm transactions, thereby posing a challenge to theoretical frameworks that attempt to apply ethical standards based on transaction types. On the second question, we argue that this unique institutional status leads to a range of novel ethical considerations and dilemmas for GVC members. Lead firms, their suppliers, and third‐party standard‐setters all confront new ethical quandaries when third‐party eco‐labeling is introduced to a GVC. The nature of these quandaries means that, while GVCs may be well‐positioned to serve as instruments of transnational governance, they are frequently not well‐oriented to do so. Our analysis marks an initial attempt to map the ethical considerations that apply to members of a GVC, and in doing so, places the literature on normative business ethics and transnational governance in a closer conversation. (shrink)

:Corporations cannot exist, scholars rightly note, without being constituted by government. However, many take a further step, claiming that corporations are normatively distinct from other market actors because of this governmental provenance. They are mistaken. Like corporations, markets and contracts also require government for their creation. Governmental provenance does not distinguish corporations normatively because our coercive social institutions are pro tanto justified in re-arranging both corporate and non-corporate market activities on behalf of social and political values. The corporation is distinct (...) only practically and prudentially, in that it represents a more proximate instrument for effecting morality in the economy. (shrink)

A discussion - historical and critical - of aesthetic problems in assimilation (words and music) is developed into a tentative position: since no direct cause-effect relationship between word and music can be established conclusively, fusion will not necessarily increase the value of either. successful fusion, a hybrid form, is characterized by a subtle synthesis of essences on both structural and aesthetic grounds. an analysis of two exemplary musical settings functions to describe the theoretical conditions. further, the author offers some critical (...) comment relating the tentative position to music education. (shrink)