On linearity in the special theory of relativity

In deductions of Lorentz transformations of the special theory of relativity, linearity of transformation is always postulated. There are only a few discussions about this linearity in which it is deduced from some basic physical facts. Here, it is shown to be almost a mathematical consequence of the principle of relativity.     

The principle of relativity and the special relativity triple

Based on the principle of relativity and the postulate on universal invariant constants (c,l) as well as Einstein's isotropy conditions, three kinds of special relativity form a triple with a common Lorentz group as isotropy group under full Umov–Weyl–Fock–Lorentz transformations among inertial motions.     

An interaction interpretation of special relativity theory. Part II

The interaction interpretation of special relativity theory (elaborated in Part I) is discussed in relation to quantum theory. The relativistic transformations (Lorentz processes) of physical variables, on the interaction interpretation, are observation-interaction dependent, just as are the physical values (eigenvalues) of systems described by quantum-theoretic state functions; a common, basic structure of the special relativity and quantum theories can therefore be presented. The constancy of the light speed is shown to follow from interaction-transformations of frequency and wavelength variables. A parallelism is suggested between, on the one hand, the Lorentz-Clausius distinction for relativistic transformations, and, on the other, the distinction between observation-dependent and observation-independent natural processes. The empirical study of rates of macroscopic clocks can provide a critical test of the interaction interpretation and of a possible extension to gravitational time changes; the role of time as prior determinant of natural process is at issue. The Hafele-Keating observations are of general relativity effects on clocks in accelerated motion.     

Is Minkowski Space-Time Compatible with Quantum Mechanics?

In quantum relativistic Hamiltonian dynamics, the time evolution of interacting particles is described by the Hamiltonian with an interaction-dependent term (potential energy). Boost operators are responsible for (Lorentz) transformations of observables between different moving inertial frames of reference. Relativistic invariance requires that interaction-dependent terms (potential boosts) are present also in the boost operators and therefore Lorentz transformations depend on the interaction acting in the system. This fact is ignored in special relativity, which postulates the universality of Lorentz transformations and their independence of interactions. Taking into account potential boosts in Lorentz transformations allows us to resolve the no-interaction paradox formulated by Currie, Jordan, and Sudarshan [Rev. Mod. Phys. 35, 350 (1963)] and to predict a number of potentially observable effects contradicting special relativity. In particular, we demonstrate that the longitudinal electric field (Coulomb potential) of a moving charge propagates instantaneously. We show that this effect as well as superluminal spreading of localized particle states is in full agreement with causality in all inertial frames of reference. Formulas relating time and position of events in interacting systems reduce to the usual Lorentz transformations only in the classical limit (0) and for weak interactions. Therefore, the concept of Minkowski space-time is just an approximation which should be avoided in rigorous theoretical constructions.     

An interaction interpretation of special relativity theory. Part I

In the established space-time coordinate-transformation (STCT) interpretation of special relativity theory, relativistic changes are consequent upon the Lorentz transformation of coordinate clocks and rods between relatively moving systems. In the proposed alternative interpretation, relativistic changes occur only in association with physical interactions, and are direct alterations in the variables of the observed system. Since space-time and momentum-energy are conjugate four-vectors, transformation of a space or time variable of a system is to be expected only if there is a concomitant transformation of the corresponding momentum or energy variable. The Lorentz invariance of the scalar entropy functionS supports the interaction interpretation; timet=f(S) of a macroscopic, entropy clock should give a Lorentz-invariant time measure, and an illustrative entropy clock is discussed. Noninteracting physical processes may be called Clausius processes, in contrast to Lorentz processes for which there is interaction and associated Lorentz transformation. Changes of energy and frequency, withE=hv, are instances of the parallel relativistic transformations. Likewise, the variation with velocity in decay time of mesons follows directly from the relativistic energy transformation of decay products; this relationship is shown for muons by a simple calculation with -decay theory.     

Relativities and Homogeneous Spaces I

Special relativity, the symmetry breakdown in the electroweak standard model, and the dichotomy of the spacetime related transformations with the Lorentz group, on the one side, and the chargelike transformations with the hypercharge and isospin group, on the other side, are discussed under the common concept of “relativity.” A relativity is defined by classes G/H of “little” group in a “general” group of operations. Relativities are representable as linear transformations that are considered for five physically relevant examples.Finite Dimensional Relativity Representations     

Transformations in special relativity

By using the principle of relativity alone (no assumptions about signals or light) it is shown that a relativisitic group of linear transformations of a spacetime plane is, if infinite, either Galilean, Lorentzian or rotational. The largest such finite group is a Klein 4-group, generated by space-reversal and time-reversal. In the infinite case an invariant of the group, denotedc, appears. Whenc is real, nonzero, noninfinite, then the group is a Lorentz group andc is identified with the speed of light. Lorentz transformations are represented through an algebra ofiterants that provides a link among Clifford algebras, the Pauli algebra and Herman Bondi'sK-calculus.     

Revised Robertson's test theory of special relativity: Space-time structure and dynamics

The experimental testing of the Lorentz transformations is based on a family of sets of coordinate transformations that do not comply in general with the principle of equivalence of the inertial frames. The Lorentz and Galilean sets of transformations are the only member sets of the family that satisfy this principle. In the neighborhood of regular points of space-time, all members in the family are assumed to comply with local homogeneity of space-time and isotropy of space in at least one free-falling elevator, to be denoted as Robertson'sab initio rest frame [H. P. Robertson,Rev. Mod. Phys. 21, 378 (1949)].Without any further assumptions, it is shown that Robertson's rest frame becomes a preferred frame for all member sets of the Robertson family except for, again, Galilean and Einstein's relativities. If one now assumes the validity of Maxwell-Lorentz electrodynamics in the preferred frame, a different electrodynamics spontaneously emerges for each set of transformations. The flat space-time of relativity retains its relevance, which permits an obvious generalization, in a Robertson context, of Dirac's theory of the electron and Einstein's gravitation. The family of theories thus obtained constitutes a covering theory of relativistic physics.A technique is developed to move back and forth between Einstein's relativity and the different members of the family of theories. It permits great simplifications in the analysis of relativistic experiments with relevant Robertson's subfamilies. It is shown how to adapt the Clifford algebra version of standard physics for use with the covering theory and, in particular, with the covering Dirac theory.Part of this work was done at the Department of Physics, Utah State University, Logan, Utah 84322.     

On the Meaning of Lorentz Covariance

In classical mechanics, Galilean covariance and the principle of relativity are completely equivalent and hold for all possible dynamical processes. In contrast, in relativistic physics the situation is much more complex. It will be shown that Lorentz covariance and the principle of relativity are not completely equivalent. The reason is that the principle of relativity actually only holds for the equilibrium quantities that characterize the equilibrium state of dissipative systems. In the light of this fact it will be argued that Lorentz covariance should not be regarded as a fundamental symmetry of the laws of physics.     

Relativistic Linear Spacetime Transformations Based on Symmetry

Usually the Lorentz transformations are derived from the conservation of the spacetime interval. We propose here a way of obtaining spacetime transformations between two inertial frames directly from symmetry, the isotropy of the space and principle of relativity. The transformation is uniquely defined except for a constant e, that depends only on the process of synchronization of clocks inside each system. Relativistic velocity addition is obtained, and it is shown that the set of velocities is a bounded symmetric domain. If e=0, Galilean transformations are obtained. If e>0, the speed 1/e and a spacetime interval are conserved. By assuming constancy of the speed of light, we get e=1/c ² and the transformation between the frames becomes the Lorentz transformation. If e<0, a proper speed and a Hilbertian norm are conserved.     

Revised Robertson's test theory of special relativity

The only test theory used by workers in the field of testing special relativity to analyze the significance of their experiments is the proof by H. P. Robertson [Rev. Mod. Phys. 21, 378 (1949)] of the Lorentz transformations from the results of the experimental evidence. Some researchers would argue that the proof contains an unwarranted assumption disguised as a convention about synchronization procedures. Others would say that alternative conventions are possible. In the present paper, no convention is used, but the Lorentz transformations are still obtained using only the results of the experiments in Robertson's proof, namely the Michelson-Morley, Kennedy-Thorndike, and Ives-Stilwell experiments. Thus the revised proof is a valid test theory which is independent of any conventions, since one appeals only to the experimental evidence. The analysis of that evidence shows the directions in which efforts to test special relativity should go. Finally it is shown how the resulting test theory still has to be improved for consistency in the analysis of experiments with complicated experimental setups, how it can be simplified for expediency as to what should be tested, and how it should be completed for a missing step not considered by Robertson.     

Antiparticles from special Relativity with ortho-chronous and antichronous Lorentz transformations

Special Relativity can be based on the whole proper group of both ortho- and antichronous Lorentz transformations, and a clear physical meaning can be given also to antichronous (i.e., nonorthochronous) Lorentz transformations. From the active point of view, the latter requires existence, for any particle, of its antiparticle within a purely relativistic, classical context. From the passive point of view, they give rise to frames dual to the ordinary ones, whose properties—here briefly discussed—are linked with the fact that in relativity it is impossible to teach another, far observer (by transmitting only instructions, and no physical objects) our own conventions about the choices right/left, matter/antimatter, and positive/negative time direction. Interesting considerations follow, in particular, by considering—as it is the case—theCPT operation as an actual (even if antichronous) Lorentz transformation.Work partially supported by FAPESP and CNPq (Brazil).     

A note on Lorentz transformations and simultaneity in classical physics and special relativity

Since early models of wave propagation in both stationary and moving media during the nineteenth century, the Lorentz transformation (LT) has played a key role in describing characteristic wave phenomena, e.g., the Doppler shift effect. In these models LT connects two different events generated by wave propagations, as observed in two reference systems and the synchronism is absolute. In relativistic physics LT implements the relativity principle. As a consequence, it connects two space-time event coordinates that both correspond to the same physical event and “absolute synchronization” is not allowed. The relativistic interpretation started from Einstein’s early criticism of the notion of “simultaneity” and Minkowski’s invariance of the space-time interval. In this paper, the two different roles of LT, i.e., in classical wave propagation theories and in relativistic physics, are discussed. Einstein’s early criticism is also re-examined with respect to LT in view of its significance for the notion of simultaneity. Indeed, that early criticism is found to be defective. Our analysis is also useful for general readers in view of its impact on modern speculations about the existence of a preferred system of reference Σ, where light propagation is isotropic, and related implications.

     

Covariant Canonical Formalism of Fields

The canonical formalism of fields consistentwith the covariance principle of special relativity isgiven here. The covariant canonical transformations offields are affected by 4-generating functions. All dynamical equations of fields, e.g., theHamilton, Euler–Lagrange, and other fieldequations, are preserved under the covariant canonicaltransformations. The dynamical observables are alsoinvariant under these transformations. The covariantcanonical transformations are therefore fundamentalsymmetry operations on fields, such that the physicaloutcomes of each field theory must be invariant under these transformations. We give here also thecovariant canonical equations of fields. These equationsare the covariant versions of the Hamilton equations.They are defined by a density functional that is scalar under both the Lorentz and thecovariant canonical transformations of fields.     

Mach-Einstein doctrine and general relativity

It is argued that, under the assumption that the strong principle of equivalence holds, the theoretical realization of the Mach principle (in the version of the Mach-Einstein doctrine) and of the principle of general relativity are alternative programs. That means only the former or the latter can be realized—at least as long as only field equations of second order are considered. To demonstrate this we discuss two sufficiently wide classes of theories (Einstein-Grossmann and Einstein-Mayer theories, respectively) both embracing Einstein's theory of general relativity (GRT). GRT is shown to be just that degenerate case of the two classes which satisfies the principle of general relativity but not the Mach-Einstein doctrine; in all the other cases one finds an opposite situation.These considerations lead to an interesting complementarity between general relativity and Mach-Einstein doctine. In GRT, via Einstein's equations, the covariant and Lorentz-invariant Riemann-Einstein structure of the space-time defines the dynamics of matter: The symmetric matter tensor T_tk is given by variation of the Lorentz-invariant scalar densityL _mat, and the dynamical equations satisfied by T_ik result as a consequence of the Bianchi identities valid for the left-hand side of Einstein's equations. Otherwise, in all other cases, i.e., for the Mach-Einstein theories here under consideration, the matter determines the coordinate or reference systems via gravity. In Einstein-Grossmann theories using a holonomic representation of the space-time structure, the coordinates are determined up to affine (i.e. linear) transformations, and in Einstein-Mayer theories based on an anholonomic representation the reference systems (the tetrads) are specified up to global Lorentz transformations. The corresponding conditions on the coordinate and reference systems result from the postulate that the gravitational field is compatible with the strong equivalence of inertial and gravitational masses.     

Conventionalism in special relativity

Reichenbach, Grünbaum, and others have argued that special relativity is based on arbitrary conventions concerning clock synchronizations. Here we present a mathematical framework which shows that this conventionality is almost equivalent to the arbitrariness in the choice of coordinates in an inertial system. Since preferred systems of coordinates can uniquely be defined by means of the Lorentz invariance of physical laws irrespective of the properties of light signals, a special clock synchronization—Einstein's standard synchrony—is selected by this principle. No further restrictions conerning light signal synchronization, as proposed, e.g., by Ellis and Bowman, are required in order to refute conventionalism in special relativity.     

Critical comments on the discussion about tachyonic causal paradoxes and on the concept of superluminal reference frame

The discussions of the tachyonic causal paradoxes and the concept of superluminal reference frame are criticized. The essence of the construction of the known paradoxes is revealed. Some possibilities of eliminating these paradoxes without contradicting the theory of relativity, are discussed. The tachyonic causal loop in an arbitrarily dimensional flat space-time is formally defined. The logical relations between assumptions on existence (or nonexistence) of the tachyonic causal loops and of inertial reference frames preferred in the tachyon kinematics are given. Such frames are not preferred in relation to bradyons and luxons, and maybe are not preferred in the dynamics of the tachyons. The theorem is proved which shows that the discussion on the tachyonic causal loops concerns also the preferred frames. The operational definitions of spacelike, timelike, and null vectors are given. It is shown that superluminal transformations and reference frames do not exist inside the theory of relativity. It is also shown that the so-called superluminal Lorentz transformations are not in fact transformations but mappings. It is concluded that the existence of tachyonic phenomena is not contradictory to the theory of relativity, while the concept of usual superluminal reference frame is contradictory to that theory.     

谈狭义相对论时空理论教学——大学物理课程讲授狭义相对论之我见

依据狭义相对论基本原理和物理事例,导出钟慢、尺缩效应和同时相对性,再以此为据导出洛伦兹变换.