Successive Lorentz transformations of the electromagnetic field

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.     

Use of the Lorentz Transformations in Noninertial Reference Frames

The Lorentz transformations are used within the model of a noninertial reference frame without infinitely high accelerations arising at instantaneous jumps of an accelerated observer between different inertial reference frames. It is demonstrated that the twin paradox can be explained within this model with the help of the Lorentz transformations. Based on the model of a noninertial reference frame, the acceleration a measured in the noninertial reference frame is related to the acceleration a measured in an inertial reference frame.     

Do experiments imply special relativity?

Starting with a thorough and self-contained account of transformations between inertial observers, the most general frame transformation is derived, which fully incorporates the Michelson-Morley experiment and the transverse Doppler effect. Lorentz and Marinov transformations are presented as two particular cases. On a rigorous mathematical ground, the paper presents a theory, more general than special relativity and with three degrees of freedom, that completely agrees with a well-established phenomenology.     

Superluminal Signals and the Resolution of the Causal Paradox

The experimental evidence for electromagnetic signals propagating with superluminal group velocity is recalled. Transformations of space and time depending on a synchronization parameter, e₁, indicate the existence of a privileged inertial system. The Lorentz transformations are obtained for a particular e₁≠0. No standard experiment on relativity depends on e₁, but if accelerations are considered only e₁=0 remains possible. The causal paradox generated by superluminal signals (SLS) in the theory of relativity does not exist in the theory with e₁=0. The irrelevance of SLS for the Einstein, Podolsky and Rosen paradox is pointed out.     

Derivation of inertial forces from the Einstein-de Broglie-Bohm (E.d.B.B.) causal stochastic interpretation of quantum mechanics

The physical origin of inertial forces is shown to be a consequence of the local interaction of Dirac's real covariant ether model⁽¹⁾ with accelerated microobjects, considered as real extended particlelike solitons, piloted by surrounding subluminal real wave fields packets.⁽²⁾ Their explicit form results from the application of local inertial Lorentz transformations to the particles submitted to noninertial velocitydependent accelerations, i.e., constitute a natural extension of Lorentz's interpretation of restricted relativity.⁽³⁾ Indeed Dirac's real physical covariant ether model implies inertial forces if one considers the real accelerated noninertial motions of general relativity, defined within the absolute local inertial frames associated with the observed local isotropy of the 2.7° K background microwave radiation.⁽⁴⁾ Inertia thus appears as a necessary consequence of the real particle motions described by the E.d.B.B. formalism of quantum mechanics.     

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.     

Voigt kinematics and electrodynamic consequences

It has been established that the kinematics of the Voigt transformation, which lacks group structure, is different from that of the Lorentz transformation, and that the apparent kinematic asymmetry of the Voigt coordinate transformations may be understood as a conformally symmetric kinematics. Phenomena such as the kinetic energy of a moving body and the Doppler effect are not quite the same under the conformal Voigt transformation as they are for the usual theory developed with respect to the Lorentz group. Yet the massenergy conservation law under the Voigt coordinate transformations and the mass-energy conservation law under the group of Lorentz transformations are identically the same.     

Thomas rotation and the parametrization of the Lorentz transformation group

Two successive pure Lorentz transformations are equivalent to a pure Lorentz transformation preceded by a 3×3 space rotation, called a Thomas rotation. When applied to the gyration of the rotation axis of a spinning mass, Thomas rotation gives rise to the well-knownThomas precession. A 3×3 parametric, unimodular, orthogonal matrix that represents the Thomas rotation is presented and studied. This matrix representation enables the Lorentz transformation group to be parametrized by two physical observables: the (3-dimensional) relative velocity and orientation between inertial frames. The resulting parametrization of the Lorentz group, in turn, enables the composition of successive Lorentz transformations to be given by parameter composition. This composition is continuously deformed into a corresponding, well-known Galilean transformation composition by letting the speed of light approach infinity. Finally, as an application the Lorentz transformation with given orientation parameter is uniquely expressed in terms of an initial and a final time-like 4-vector.     

Superposition in quantum and relativity physics—An interaction interpretation of special relativity theory: Part III

With the interaction interpretation, the Lorentz transformation of a system arises with selection from a superposition of its states in an observation-interaction. Integration of momentum states of a mass over all possible velocities gives the rest-mass energy. Static electrical and magnetic fields are not found to form such a superposition and are to be taken as irreducible elements. The external superposition consists of those states that are reached only by change of state of motion, whereas the internal superposition contains all the states available to an observer in a single inertial coordinate system. The conjecture is advanced that states of superposition may only be those related by space-time transformations (Lorentz transformations plus space inversion and charge conjugation). The continuum of external and internal superpositions is examined for various masses, and an argument for the unity of the super-positions is presented.     

Classical optical experiments and special relativity: A review

Classical optical experiments that confirm the validity of special relativity are considered. Transformations of spatial coordinates and time that were proposed at different times for the passage from one inertial reference frame (IRF) to another and that differ from the classical Lorentz transformations are critically analyzed. It is shown that, although some of these transformations are capable of explaining the results of single classical optical experiments, in particular, the Michelson-Morley experiments, neither of them, except for the Tangherlini transformations, can explain the results of the entire set of these experiments. The discrepancy between the predictions of incorrect transformations and the results of the well-known experiments is caused by the absence of a clearly formulated procedure for synchronizing spaced clocks in a rest IRF (where the observer is located) and a moving IRF, which should be consistent with the transformation of time. A number of relativistic and quantum effects are indicated, which have been predicted but not yet detected, to a search for which efforts of physicists are directed, and which are convenient to describe with the help of the formalism of the Tangherlini transformations.     

洛伦兹力可看作静止电荷所受电场力的相对论效应——兼论洛伦兹力公式具有与库仑定律相同的精确度

从洛伦兹变换出发,利用电磁场张量和四维力的协变性及电荷相对论不变,直接证明:在一个惯性参考系内的静止电荷所受电场力,转换到另一个惯性参考系就是运动电荷受的洛伦兹力.证明洛伦兹力公式具有与库仑定律相同的精确度.     

Noninertial Observers in Special Relativity and Clock Synchronization Debates

Selleri's arguments that a consideration of noninertial reference frames in the framework of special relativity identify “absolute simultaneity” as being “Nature's choice of synchronization” are considered. In the case of rectilinearly accelerating rockets, it is argued by considering two rockets which maintain a fixed proper separation rather than a fixed separation relative to the inertial frame in which they start from rest, that what seems the most “natural” choice for a simultaneity convention is problem-dependent and that Einstein's definition is the most “natural” (though still conventional) choice in this case. In addition, the supposed problems special relativity has with treating a rotating disk, namely how a pulse of light traveling around the circumference of the disk can have a local speed of light equal to c everywhere but a global speed not equal to c, and how coordinate transformations to the disk can give the Lorentz transformations in the limit of large disk radius but small angular velocity, are addressed. It is shown that the theory of Fermi frames solves both of these problems. It is also argued that the question of defining simultaneity relative to a uniformly rotating disk does not need to be resolved in order to resolve Ehrenfest's paradox.     

带电粒子在电磁场中动态受力平衡条件

导体的电平衡条件为E =0 ,这是静电学中一个重要结论 ,但它显然不是相对论协变的 .由于洛伦兹力公式F =q(E +v×B)为相对论协变式 ,故带电粒子 (包括导体中的自由电子 )在电磁场中动态受力平衡条件即应为E +v×B =0 .这个条件是相对论协变的 ,即它在任何惯性系中均成立 .     

Clifford代数中的双曲相位变换群及其在四维相对论时空中的应用

将Clifford代数所定义的双曲复空间R_H和作用在双曲复空间R_H上的双曲相位变换群U₄(H)赋予了明确的物理意义. 双曲复空间R_H同构于四维Minkowski时空，而其上的双曲相位变换群U₄(H)就是四维相对论时空中的洛仑兹(Lorentz)变换群. 进一步，利用U₄(H)群的复合变换性质，自然导出了四维Minkowski时空中Lorentz变换和速度变换的一般表达式. 由此，将狭义相对论中的特殊Lorentz变换作为特例包含其中. 关键词： 双曲复数 双曲相位变换 Minkowski时空 Clifford代数     

Study of reflection of light by a moving mirror

The law of reflection of light by a moving mirror in the case of special Lorentz transformation is well known. We have derived the law of reflection of light by a moving mirror in the case of the most general Lorentz transformation. It has been observed that in the case of special Lorentz transformation, the law of reflection of light by a moving mirror is very simple but in the case of the most general Lorentz transformation, the law of reflection of light by a moving mirror is very complex.      

Properties of optical and acoustic phonons in diamond crystals

The dependence of the energy on the quasi-momentum of optical and acoustic phonons in diamond-type crystals is analyzed. It is shown that the dispersion relation of optical branches near the Brillouin zone center in such crystals can be derived based on the modified Klein-Gordon equation corresponding to quasiparticles (optical phonons) with negative effective mass. Analytical expressions are derived for dispersion curves of acoustic and optical branches in the entire Brillouin zone, based on the sinusoidal approximation. The obtained analytical dependences of dispersion relations are compared to the experimental results of the study of inelastic neutron scattering spectra in diamond crystals. The law of the coordinate and time transformation for optical phonons with negative effective rest mass when going to the new inertial reference frame, similar to Lorentz transformations in relativistic mechanics, is determined.     

Conventionality of Simultaneity and the Tippe Top Paradox in Relativity

In this paper we critically examine a recently posed paradox (tippe top paradox in relativity) and its suggested resolution. A tippe top when spun on a table, tips over after a few rotations and eventually stands spinning on its stem. The ability of the top to demonstrate this charming feat depends on its geometry (all tops are not tippe tops). To a rocket-bound observer the top geometry should change because of the Lorentz contraction. This gives rise to the possibility that for a sufficiently fast observer the geometry of the top may get altered to such an extent that the top may not tip over! This is certainly paradoxical since a mere change of the observer cannot alter the fact that the top tips over on the table. In an effort to resolve the issue the authors of the paradox compare the equations of motion of the particles of the top from the perspective of the inertial frames of the rocket and the table and observe among other things that (1) the relativity of simultaneity plays an essential role in resolving the paradox and (2) the puzzle in some way is connected with one of the corrolaries of special relativity that the notion of rigidity is inconsistent with the theory. We show here that the question of the incompatibility of the notion of rigidity with special relativity has nothing to do with the current paradox and the role of the lack of synchronization of clocks in the context of the paradox is grossly over-emphasized. The conventionality of simultaneity of special relativity and the notion of the standard (Einstein) synchrony in the Galilean world have been used to throw light on some subtle issues concerning the paradox.     

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.     

Noninvariant one-way velocity of light and particle collisions

Recently published space-time transformations between inertial systems (different from the Lorentz transformations) are reviewed. Energy and momentum are defined consistently with these new transformation laws. Formally they equal the usual relativistic expressions only in the privileged frame, but numerically they do so in all inertial frames. All the precise experimental data concerning thresholds for inelastic processes, particle masses, and so on, can thus be explained also within this new theoretical framework.     

Remarks on relativistic radiative reaction

It is shown explicitly that the loss of kinetic energy of a highly relativistic particle in magnetic field is, in general, at the cost of both its transverse and longitudinal motion. Also, in radiation problems it is convenient aad appropriate to make a Lorentz transformation into an inertial frame in which the accelerated particle is momentarily at rest.