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)

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)

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)

We present an epistemological schema of natural sciences inspired by Peirce's pragmaticist view, stressing the role of the \emph{phenomenological map}, that connects reality and our ideas about it. The schema has a recognisable mathematical/logical structure which allows to explore some of its consequences. We show that seemingly independent principles as the requirement of reproducibility of experiments and the Principle of Sufficient Reason are both implied by the schema, as well as Popper's concept of falsifiability. We show that the schema has (...) some power in demarcating science by first comparing with an alternative schema advanced during the first part of the 20th century which has its roots in Hertz and has been developed by Einstein and Popper. Further, the identified differences allow us to focus in the construction of Special Relativity, showing that it uses an intuited concept of velocity that does not satisfy the requirements of reality in Peirce. While the main mathematical observation connected with this issue has been known for more than a century, it has not been investigated from an epistemological point of view. A probable reason could be that the socially dominating epistemology in physics does not encourage such line of work. We briefly discuss the relation of the abduction process presented in this work with discussions regarding ``abduction'' in the literature and its relation with ``analogy''. (shrink)

The question whether distant simultaneity has a factual or a conventional status in special relativity has long been disputed and remains in contention even today. At one point it appeared that Malament had settled the issue by proving that the only non-trivial equivalence relation definable from causal connectability is the standard simultaneity relation. Recently, however, Sarkar and Stachel claim to have identified a suspect assumption in the proof by defining a non-standard simultaneity relation from causal connectability. I contend that their (...) critique is based on a misunderstanding of the criteria for the definability of a relation, a misunderstanding that Malement's original treatment helped to foster. There are in fact a variety of notions of definability that can be brought to bear. They all, however, require a condition that suffices to secure Malament's result. The non-standard relation Sarkar and Stachel claim to be definable is not so definable, and, I argue, their proposal to modify the notion of “causal definability” is misguided. Finally, I address the relevance of Malament's result to the thesis of conventionalism. (shrink)

Grunbaum has argued that the Lorentz-Fitzgerald contraction hypothesis is not ad hoc since the Kennedy-Thorndike experiment can be used to provide a test that is significantly different from that provided by the Michelson-Morley experiment. In the first part of the paper, I show that the differences claimed by Grunbaum to hold between these two experiments are not sufficient for establishing independent testability. A dilemma is developed: either the Kennedy-Thorndike experiment, because of experimental realities, cannot test the uncontracted Fresnel aether theory, (...) or if experimental difficulties are ignored, the Kennedy-Thorndike experiment degenerates into a version of the Michelson-Morley experiment. The second part of the paper is a feasibility study of the prospects for defining experimental types according to aims of measurement and determination. This approach is applied to the contraction hypothesis, where it is suggested that the usual analysis of independent testability be modified. (shrink)

A theory, analogous with the kinetic theory of heat, is described, in which the role of heat is shared by all the phenomena of extra-atomic physics, including quantum electrodynamics, gravitation, and relativistic mass. The role of the randomly moving molecules, as a mechanical model, is taken for all of these by a single model, consisting of a turbulent, dissipationless liquid, the motion of which conforms to Newtonian mechanics. This model is capable of supporting spherical standing waves which are taken to (...) represent elementary particles. The mechanical counterparts of the physical equations are close approximations to those of the model. (shrink)

When the German mathematician Hermann Minkowski first introduced the space-time diagrams that came to be associated with his name, the idea of picturing motion by geometric means, holding time as a fourth dimension of space, was hardly new. But the pictorial device invented by Minkowski was tailor-made for a peculiar variety of space-time: the one imposed by the kinematics of Einstein’s special theory of relativity, with its unified, non-Euclidean underlying geometric structure. By plo tting two or more reference frames in (...) relative motion on the same picture, Minkowski managed to exhibit the geometric basis of such relativistic phenomena as time dilation, length contraction or the dislocation of simultaneity. These disconcerting effects were shown to result from arbitrary projections within four-dimensional space-time. In that respect, Minkowski diagrams are fundamentally different from ordinary space-time graphs. The best way to understand their specificity is to realize how productively ambiguous they are. (shrink)

Paths of shortest length, or geodesics, may not appear as straight lines because acceleration creates distortion. For spaces of constant curvature there are only two possibilities: Either rulers get longer as they move away from the origin or they shrink. Because the longitudinal Doppler shift corresponds to the measure of distance, in velocity space, this space is hyperbolic, corresponding to the second of the two possibilities. Transformations from one inertial frame to another are related to geometrical rigid motions. The square (...) root of the Doppler shift is related to transformations of uniformly accelerated frames. Applications to general relativity and cosmology are given including the time-velocity metric of the Friedmann universe of dust at zero pressure, Hubble's law and the exponential red-shift. (shrink)

The question of whether the same acceleration eld that is found in a rigid uniformly rotating disc can annul a gravitational eld is answered in the negative because their curvatures are dierent. There is an exact correspondence between a uniformly rotating disc and hyperbolic geometry of constant curvature, while, gravitational elds require non-constant, negative curvature. The connection between the two is the free-fall time; the former has constant density while the latter, constant mass. The distortion caused by motion is experienced (...) in the hyperbolic world when the rim of the -disc is approached, where is the disc radius that determines the nature of the elds. Characteristic hyperbolic properties can thus be used to explain relativistic phenomena, like the angle defect in relation to the FitzGerald-Lorentz contraction, the electromagnetic Poincare stress, aberration which violates of the laws of cosines and sines, gravitational frequency shifts and the bending of light near a massive object. (shrink)