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We present a comprehensive discussion of the formulation of the kinematics of special relativity, i.e., the Lorentz transformation. We begin with a concise new proof that the principle of relativity implies that the transformation of event coordinates between inertial reference frames is linear. We then give a clear derivation of the pre-Lorentz transformation, which follows from the principle of relativity. We then show that the pre-Lorentz transformation and the inertial invariance of the speed of light together result in the Lorentz transformation. This, of course, is essentially the traditional formulation. We next present two additional formulations, one using Lorentz–Fitzgerald contraction and one using time dilation, instead of inertial invariance. This is reasonable since Lorentz–Fitzgerald contraction and time dilation are about as well established as and are arguably less abstract than inertial invariance, and thus may profitably be used instead of inertial invariance to complete the formulation. We then present a complete proof that the pre-Lorentz transformation and the requirement of closure upon composition together imply that the transformation is either a Galilean transformation or a generalized Lorentz transformation. This is noteworthy in that it gets ever so close to the Lorentz transformation without invoking light. In the course of this, we obtain a generalized velocity addition rule, which reduces to the velocity addition rule of special relativity. We next show that the generalized Lorentz transformation, together with inertial invariance, Lorentz–Fitzgerald contraction, and time dilation, used one at a time, yields three more formulations. We then show that the unspecified, nonzero, constant speed in the generalized Lorentz transformation can be determined without any reference to light, thereby obtaining a seventh formulation. Light plays no explicit role in the four formulations employing Lorentz–Fitzgerald contraction and time dilation and plays no role whatsoever in the seventh formulation. Thus, and this is a fact which should be strongly emphasized, the formulation of special relativity in no way depends upon the nature of electromagnetic radiation. We conclude by briefly discussing these seven formulations of the kinematics of special relativity and some associated implications.

Bohmian mechanics is a nonlocal hidden-variable interpretation of quantum theory which predicts that particles follow deterministic trajectories in spacetime. Historically, the study of Bohmian trajectories has been restricted to nonrelativistic regimes due to the widely held belief that the theory is incompatible with special relativity. Here we derive expressions for the relativistic velocity and spacetime trajectories of photons in a Michelson-Sagnac-type interferometer. The trajectories satisfy quantum-mechanical continuity, the relativistic velocity addition rule. Our new velocity equation can be operationally defined in terms of weak measurements of momentum and energy. We finally propose a modified Alcubierre metric which could give rise to these trajectories within the paradigm of general relativity.

We present an optical analogue of a central concept (not a specific relativistic effect) in Special Relativity (SR) that underlies many relativistic effects. This concept is the Einstein Velocity Addition (EVA) which plays a fundamental role in the development of SR. The photonic circuit analogue is between the EVA, and the complex electric field, through the output signal of the optical add-drop filter (OADF). The OADF is one of the key building blocks for wavelength filtering in photonic integrated circuits (PICs or photonic chip). This analogue strengthens the vision toward a Special-Relativity-on-a-Chip. The analogy is established by equating the two relativistic normalized velocities in EVA with the two transmission coupling coefficients of the two directional couplers in the OADF. Furthermore, the generating angle in the EVA is associated with the single-pass phase shift of the OADF. We analyze its magnitude and directional angle in detail. Lastly, other important consequences are also brief...

We report a link (or optical analogue) between the relativistic Thomas rotation angle effect found in the special theory of relativity (STR), and the phase response of an all-pass filter (APF), one of the building blocks of the rapidly evolving field of photonic integrated circuits. This link opens up the possibility of investigating STR phenomena in a ‘laboratory-on-a-chip’ setting. The Thomas effect is a spatial rotation of the reference frame due to Einstein’s velocity addition law of two successive velocities travelling in non-collinear directions. On the other hand, the APF is implemented with a microring resonator device with one waveguide bus. The analogue is established by associating two parameters. First, the transmission coupling coefficient τ of the APF is made to equal with the product of the two relativistic normalized velocities V1 and V2 (τ = V1V2), where the normalized velocities V1 = tanh [β1/2] and V2 = tanh [β2/2] with β1 (=tanh−1 (v1/c)) and β2 (=tanh−1 (v2/c)) being the rapidity valu...

In this paper, we first discuss the geometric properties of the Lorentz cone and the extended Lorentz cone. The self-duality and orthogonality of the Lorentz cone are obtained in Hilbert spaces. These properties are fundamental for the isotonicity of the metric projection with respect to the order, induced by the Lorentz cone. According to the Lorentz cone, the quasi-sublattice and the extended Lorentz cone are defined. We also obtain the representation of the metric projection onto cones in Hilbert quasi-lattices. As an application, solutions of the classic variational inequality problem and the complementarity problem are found by the Picard iteration corresponding to the composition of the isotone metric projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping. Our results generalize and improve various recent results obtained by many others.

We give an explicit formula to factorize functions in the Lorentz spaces L ( p , q ) . Some applications to centralizers, twisted sums and interpolation theory are included.

We explore the role of symmetry in the theory of Special Relativity. Using the symmetry of the principle of relativity and eliminating the Galilean transformations, we obtain a universally preserved speed and an invariant metric, without assuming the constancy of the speed of light. We also obtain the spacetime transformations between inertial frames depending on this speed. From experimental evidence, this universally preserved speed is c, the speed of light, and the transformations are the usual Lorentz transformations. The ball of relativistically admissible velocities is a bounded symmetric domain with respect to the group of affine automorphisms. The generators of velocity addition lead to a relativistic dynamics equation. To obtain explicit solutions for the important case of the motion of a charged particle in constant, uniform, and perpendicular electric and magnetic fields, one can take advantage of an additional symmetry&mdash, the symmetric velocities. The corresponding bounded domain is symmetric with respect to the conformal maps. This leads to explicit analytic solutions for the motion of the charged particle.

This article is devoted to the key concept of modern electrodynamics—the invariance of the speed of light. The general principle of relativity is considered in detail. Some critical remarks to the relativistic invariance and to the Lorentz transformations are presented. The general invariance of Maxwell equations is discussed. Different theoretical expectations for possible results of Michelson-Morley experiment and some physical consequences are considered. Some critical remarks to the notion of the light speed and its constancy are given. The relativistic law for velocity addition, including strangeness of a noncollinear addition and a superluminal motion, is discussed. Critical analysis of two works which proof the need for existence of an invariant velocity is consequentially made.

We introduce a sufficient condition under which the Lorentz beam convolution with other beams constitutes valid cross-spectral densities. Two examples are given to show how the Lorentz related beam can be used for generation of a far field being a modulated version of another one. The far-field intensity patterns in the Cartesian symmetries by the convolution operation of the Lorentz beams with multi-sinc beams, and the convolution operation of the Lorentz beams with multi-sinc Gaussian beams, are shown respectively. We find that different beam order can result distinct far field changes.

Lorentz Transformations generally describe the transformations for observations between mechanical phenomenon systems in relative motion. We all know that the electrical charge of associate isolated system is relativistically invariant. We have studied the volume charge density in Special and Most General Lorentz Transformations. If one frame moves on x-axis then we will notice this in Special Lorentz Transformation. On the other hand if the motion of the moving frame is not on the x-axis relative to the rest frame however the motion is on any arbitrary direction then we will notice this formula for the Most General Lorentz Transformation. We also investigated the changes of the volume charge density of moving system in terms of that of rest system in Most General Lorentz Transformations at different angles and velocities.

We consider a space with Generalized Uncertainty Principle (GUP) which can be obtained in the frame of the deformed commutation relations. In the space with GUP we have found transformations relating coordinates and times of moving and rest frames of reference in the first order over the parameter of deformation. In the non-relativistic case we find the deformed Galilean transformation which is rotation in Euclidian space–time. This transformation is similar to the Lorentz one but written for Euclidean space–time where the speed of light is replaced by some velocity related to the parameter of deformation. We show that for relativistic particle in the space with GUP the coordinates of the rest and moving frames of reference satisfy the Lorentz transformation with some effective speed of light.

In theories whose Lorentz invariance is violated by the presence of an external tensor of any rank, we show that a signal velocity, understood as the group velocity of a wave, is added to the velocity of the reference frame according to the standard relativistic rule for adding velocities. In the case where we have a superluminal signal, this observation allows creating a closed time cycle and thus coming to a conclusion about a causality violation even in the absence of relativistic invariance. We also reveal an optical anisotropy of a moving medium that is isotropic at rest.

We demonstrate, for the first time to best of our knowledge, the usefulness of the analytical toolbox developed from a disparate field like Photonic Integrated Circuits (PICs) and now applied in the study of the Einstein Velocity Addition (EVA) in Special Relativity (SR). EVA is a central concept in SR that underlies many relativistic effects. The recently established analogy between the EVA and the PICs-based Optical Add-drop Filter (OADF) [J. Modern Optics 66, 679, (2019)] allows us to apply the “Critical Coupling, Over-Coupling and Under-Coupling” (or CC-OC-UC) analytical toolbox to systematically categorize, analyze, and visualize the different behaviors of EVA. We focus on the (i) relative magnitude and (ii) relative direction of the two non-collinear velocities of EVA. This analysis uncovers “not-too-obvious” features that are overlooked or would not have been discussed, otherwise, in the normal treatment of EVA in SR. Other important consequences are briefly discussed.

While the Lorentz group serves as the basic language for Einstein's special theory of relativity, it is turning out to be the basic mathematical instrument in optical sciences, particularly in ray optics and polarization optics. The beam transfer matrix, commonly called the $ABCD$ matrix, is shown to be a two-by-two representation of the Lorentz group applicable to the three-dimensional space-time consisting of two space and one time dimensions. The Jones matrix applicable to polarization states turns out to be the two-by-two representations of the Lorentz group applicable to the four-dimensional space-time consisting of three space and one time dimensions. The four-by-four Mueller matrix applicable to the Stokes parameters as well as the Poincare sphere are both shown to be the representations of the Lorentz group.

We now pivot from relationships between frames to look at the effect of motion on communications between specific observers.This will help us look at the twin paradox in the next chapter, and will prove crucial to understanding the effects of gravity on time. Along the way, we develop an understanding of the Doppler effect; a key tool in many areas of modern science. We find that Doppler effects are reciprocal (Alice observes the same effect on Bob’s signals as Bob observes on Alice’s signals) and that Doppler effects compound over multiple frame changes. We then use the compounding of Doppler effects to deduce the algebraic formof the velocity addition law. We show that this Einstein velocity addition law reduces to the Galilean law at low speeds.

This work consists of two parts. The first part: The Lorentz transformation has two derivations. One of the derivations can be found in the references at the end of the work in the “Appendix I” of the book marked by number one. The equations for this derivation [1]: The other derivation of the Lorentz transformation is the traditional hyperbolic equations: ; ; For these equations we found new equations: , . The second part: In the second part is the equation by which we derive Minkowski’s equation. It will be proved that Minkowski’s equation is the integral part of the Lorentz transformation.

The Fourier coefficient map is considered as an operator from a weighted Lorentz space on the circle to a weighted Lorentz sequence space. For a large range of Lorentz indices, necessary and sufficient conditions on the weights are given for the map to be bounded. In addition, new direct analogues are given for known weighted Lorentz space inequalities for the Fourier transform. Applications are given that involve Fourier coefficients of functions in LlogL and more general Lorentz-Zygmund spaces., Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s00041-015-9455-5

A new vectorial representation for the successive Lorentz transformations (SLT) 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 (PLT) 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 (SLTLC) 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.

We use the Lorentz transformation equations to derive a unified equation for the Doppler effect–that can be used for any one-dimensional (sound, water, EM, etc.) wave in any medium. This unified equation includes the effects of motion, if present, of the medium relative to the observer as well as the relative velocity between the observer and the source. This master equation can be applied to both relativistic and nonrelativistic situations to recover the more familiar Doppler effect expressions and it clarifies that the Doppler equations given in standard textbooks for both sound and EM waves, are basically the same. The advantage of this unified equation is that it reduces the effort in solving complex problems. Basic knowledge of Lorentz transformations and their physical effects is enough to understand this derivation.

Lorentz transformations (“boosts”) are derived and applied to time dilation, Lorentz contraction, and the relativity of simultaneity. The relativistic velocity addition formula is derived and is seen to be compatible with a universal speed limit. Causality is discussed and the “invariant interval,” the metric, is introduced.

Is it possible to demonstrate the velocity addition without using a variable time (as it is done in theory of relativity)? The topic of this paper is to propose and demonstrate an alternative expres-sion based on the conservation of linear momenta. The method proposed here is to start from a physical object (and not from a mathematical point), i.e. from an object with a mass. And the hy-pothesis is inertial mass to be different from gravitational mass. Then, when impulses are added, we get an expression of the velocity addition itself. When numerical predictions are compared with experimental results, the differences are lower than the measures uncertainty. And these numerical results are much close to those predicts by the theory of relativity, nevertheless with a little difference at high velocities. If this demonstration and this expression were validated, it would allow giving an alternative explanation to some experiments and nature observations as Doppler Effect on light celerity. But first, it would be necessary to get from laboratories more precise experimental results, in order to validate or not this hypothesis of the sum of linear momenta with a Variable Inertial Mass.

The Lorentz transformations are obtained by assuming that the laws of classical thermodynamics are invariant under changes of inertial reference frames. As Maxwell equations are used in order to deduce a wave equation that shows the constancy of the speed of light, by means of the laws of classical thermodynamics, the invariance of the Carnot cycle is deduced under reference frame changes. Starting with this result and the blackbody particle number density in a rest frame, the Lorentz transformations are obtained. A discussion about the universality of classical thermodynamics is given.

Local Lorentz invariance is an important component of General Relativity. Testing for Local Lorentz invariance can not only probe the foundation stone of General Relativity but also help to explore the unified theory for General Relativity and quantum mechanics. In this paper, we search the Local Lorentz invariance violation associated with operators of mass dimension d = 6 in the pure-gravity sector with short-range gravitational experiments. To enlarge the Local Lorentz invariance violation signal effectively, we design a new experiment in which the constraints of all fourteen violation coefficients may be improved by about one order of magnitude.

We present an unconventional approach to generate an optical analogue of the Thomas rotation angle found in Special Relativity (SR) by using a thin-film–based Gires–TournoisResonator (GTR). At first glance, the phase response of the GTR seems to fail as an optical analogue for the Thomas rotation angle in the phase domain but we demonstrate that an analogue can be successfully constructed in the intensity domain by combining the GTR with a Michelson interferometer (MI), thus forming an interferometer known as the Michelson–Gires–Tournois Interferometer (MGTI). Thomas rotation angle is a spatial rotation of the reference frame due to the Einstein velocity addition (EVA) law of two successive relativistic velocities travelling in non-collinear directions, whereas, the GTR is a thin-film based asymmetric Fabry–Perot resonator (FPR) with a partially reflecting front mirror, and a 100% rear-reflecting mirror. We investigate in detail, both analytically and numerically, the Thomas rotation angle’s full behaviour under various parameter conditions. This approach offers the following advantages: (i) it leads to a simpler overall configuration, (ii) it measures the Thomas rotation angle directly requiring only a single optical parameter instead of multiple parameters compared with other methods, and (iii) it requires no sophisticated multilayer thin film design.

Lorentz transformation considers that relative velocity of the frame of references in inertialsystem is less than the velocity of light. If it be such that a frame is moving with velocity same as thatof light with respect to a frame of observer then, Lorentz transformation in it will not be same as donein inertial system. Again photon is not only a particle or wave but it is a complex system due to thefact that it possesses spin and linear motion simultaneously. So, it will have some complexcharacteristics. In this work first, trial would be made to find out the process of Lorentztransformation in between two frames having relative velocity same as that of light and then, thisconcept, would be applied in the system of photon.

In order to study the reflection of electromagnetic wave in Lorentz media, A finite-difference time-domain method based on the auxiliary differential equation (ADE) technique is used to obtain the formulation of 2-D TM wave propagation in lossy Lorentz media. In 1-D case, the reflected coefficients calculated by ADE-FDTD method and exact theoretical result are excellent agreement. This expresses that the 2-D formulas of electromagnetic wave propagation in lossy Lorentz media are right. Furthermore, Plane wave reflected by Lorentz media layer is calculated and simulated. Results display that reflected effect is evident.

It is proved in this paper that there are at least five situations in the interaction theories of microparticle physics that the Lorentz transformations have no invariabilities. 1) In the formula to calculate transition probabilities in particle physics, the so-called invariability factor of phase space d3p/E is not invariable actually under the Lorentz transformations. Only in one-dimensional motion with uy = uz = 0, it is invariable. 2) The propagation function of spinor field in quantum theory of field has no invariability of Lorentz Transformation actually. What appears in the transformation is the sum of Lorentz factors aμνaλμ ≠ δνλ when ν, λ = 1, 4, rather than aμνaλμ = δνλ. But in the current calculation, we take aμνaλμ = δνλ. The confusion of subscript’s position leads to wrong result. 3) Though the motion equations of quantum fields and the interaction Hamiltonian are unchanged under the Lorentz transformation, the motion equation of perturbation which is used to calculate the transition probability in the interaction representation has no invariability. 4) The interactions between bound state’s particles have no Lorentz invariability. In fact, the principle of relativity has no approximation if it holds. 5) The calculation methods of high order perturbations normalization processes in quantum theory of fields violate the invariability of Lorentz transformation. The conclusions above are effective for strong, weak and electromagnetic interactions and so on. Therefore, the principle of relativity does not hold in the micro-particle’s interactions. On the other hand, the invariability principle of light’s speed is still effective. So the formulas of special relativity still hold, but we should consider them with absolute significances.

In two recent and completely independent articles (Hill and Cox in Proc R Soc A 468:4174, 2012; Vieira in Rev Bras Ensino F's 34(3):1-15, 2012), the authors propose an extension of the Lorentz transformations of special relativity which are applicable to velocities in excess of the speed of light and do not involve the need to introduce imaginary quantities which are difficult to reconcile with everyday experience. These independent derivations, obtained from entirely distinct perspectives, mean that there is now some commonality of agreement in the basic equations underlying superluminal motion. One consequence of the new theories is that the standard special relativity formula for the addition of relative velocities also applies for velocities in excess of the speed of light. The new theories are based on the assumption that for any two inertial frames separated by an infinite relative velocity, the product of the two measured velocities for the same particle must necessarily be the square of the speed of light. Here, we amplify the major physical consequences embodied in the theory, including the surprising and novel idea of the co-existence of two "worlds", such that in a subluminal world, everything is travelling with speeds less than the speed of light, while in the superluminal world, everything is travelling with speeds greater than the speed of light. We also establish the remarkable result that the reciprocal surface for the relative velocity formula can be re-orientated as a fully axially symmetric hyperboloidal surface, the full physical implications of which are not altogether transparent. Refereed/Peer-reviewed

Ungar (Math. Appl. 49:187–221, 2005) employed the binary operation of Einsteins velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry in full analogy with Euclidean geometry. We use the isomorphism between \({(\mathbb R,+,\cdot)}\) and \({((-1,1),\oplus,\otimes)}\) in Beltrami–Klein model of hyperbolic geometry for similar results.

The addition theorem of velocities is presented both for the collinear and for the general case of two arbitrary velocities; Fresnel drag coefficient of light in moving fluid is shown to be direct consequence of velocity addition theorem. Using addition theorem, speed of light is shown to be maximum observable body speed. The aberration of the line of sight of a focused light-ray emitted by a star for arbitrary relative velocity is obtained. We look again at the addition theorem of velocities from a more general perspective. The invariance of proper time under Lorentz coordinate transformations is demonstrated, using two different methods. This sets the stage for introduction of rapidity. The merits of rapidity are demonstrated by showing the additivity property of rapidity for collinear motion, akin to nonrelativistic velocity addition.

Chapter 6 starts the more advanced part of the book. The concept of 4-vectors and their dot product are introduced and used to define the general Lorentz transformation using matrices. With the help of Appendix B , the chapter extends the two-dimensional Lorentz transformations to the full four-dimensional Lorentz boosts. The general transformation of the coordinates of a four-dimensional event (its time and the three coordinates of its location) is derived as the building block of other 4-vectors. The general relativistic law of addition of velocities is an immediate consequence of this transformation. Aberration of light is introduced in a new light by deriving the formulas from the general relativistic law of addition of velocities. General Doppler formula and global positioning systems are discussed. The notions of 4-velocity and 4-momentum are introduced next and the general Lorentz transformation of energy and momentum 3-vector is derived. The invariant length of the 4-momentum is identified as (mc 2 ) 2 ( m c 2 ) 2 and the idea of independence of mass from velocity is reiterated. Four-acceleration is defined as the derivative of 4-velocity with respect to proper time and used to introduce 4-force and the second law of motion.

Belief in slight Lorentz symmetry violation at available energies is an imaginable part of belief in radically disparate physics near the Planck energy. As long as the effective field theory comprising new physics is formulated in flat spacetime, the description of hypothetical Lorentz symmetry violation within its formalism is consistent and tenable. If gravitation is introduced into the effective theory in the form of curved manifolds subject to Einstein field equations, the formalism of the symmetry violation encounters severe troubles. Either the dynamics of fundamental physical fields becomes inconsistent, or one is forced to accept existence of a large number of new species of fundamental matter beyond the Standard Model.

We show that local Lorentz covariance arises canonically as the group of transformations between local thermal states in the framework of Local Quantum Physics, given the following three postulates: (i) Local observable algebras are finite-dimensional. (ii) Minimal local observable algebras are isomorphic to $\mathbb{M}_2(\mathbb{C})$, the observable algebra of a single qubit. (iii) The vacuum restricted to any minimal local observable algebra is a non-maximally mixed thermal state. The derivation reveals a new and surprising relation between spacetime structure and local quantum states. In particular, we show how local restrictions of the vacuum can determine the connection between different local inertial reference frames., Comment: 7 pages; v3: Minor typos fixed, matches the published version. v2: The definition of local Hamiltonians is slightly changed from the previous version in order to better justify the physical identification of SL(2,C) transformations with Lorentz transformations. The rest of the paper has been modified to accommodate this change

Recently, first limits on putative Lorentz invariance violation coefficients in the pure gravity sector were determined by the reanalysis of short-range gravity experiments. Such experiments search for new physics at sidereal frequencies. They are not, however, designed to optimize the signal strength of a Lorentz invariance violation force; in fact the Lorentz violating signal is suppressed in the planar test mass geometry employed in those experiments. We describe a short-range torsion pendulum experiment with enhanced sensitivity to possible Lorentz violating signals. A periodic, striped test mass geometry is used to augment the signal. Careful arrangement of the phases of the striped patterns on opposite ends of the pendulum further enhances the signal while simultaneously suppressing the Newtonian background., Accepted Phys. Rev. D

We reveal the frame-exchange space-inversion (FESI) symmetry and the frame-exchange time-inversion (FETI) symmetry in the Lorentz transformation and propose a symmetry principle stating that the space-time transformation between two inertial frames is invariant under the FESI or the FETI transformation. In combination with the principle of relativity and the presumed nature of Euclidean space and time, the symmetry principle is employed to derive the proper orthochronous Lorentz transformation without assuming the constancy of the speed of light and specific mathematical requirements (such as group property) a priori. We explicitly demonstrate that the constancy of the speed of light in all inertial frames can be derived using the velocity reciprocity property, which is a deductive consequence of the space–time homogeneity and the space isotropy. The FESI or the FETI symmetry remains to be preserved in the Galilean transformation at the non-relativistic limit. Other similar symmetry operations result in either trivial transformations or improper and/or non-orthochronous Lorentz transformations, which do not form groups.

It has been shown that the Gupta-Bleuler method of quantization can be used to impose the Lorentz gauge condition in static space-times but not in cosmological space-times. This implies that the Gupta-Bleuler approach fails in general in non-static space-times. More recently, however, the Dirac method of quantizing constrained dynamical systems has been successfully employed to impose the Lorentz gauge in conformally flat space-times. In this paper we generalize this result by using Dirac's method to impose the Lorentz gauge in a general space-time region where the metric is expressed in synchronous coordinates., 5 pages

The tetrad method is used for an introduction of local Lorentz frames and a detailed analysis of local Lorentz transformations. A formulation of equations of motion in local Lorentz frames is based on the Pomeransky-Khriplovich gravitoelectromagnetic fields. These fields are calculated in the most important special cases and their local Lorentz transformations are determined. The local Lorentz transformations and the Pomeransky-Khriplovich gravitoelectromagnetic fields are applied for a rigorous derivation of a general equation for the Thomas effect in Riemannian spacetimes and for a consideration of Einstein's equivalence principle and the Mathisson force., 35 pages, final version

A new formulation of what may be called the "fundamental theorem of the theory of relativity" is presented and proved in (3+1)-space-time, based on the full classification of special transformations and the corresponding velocity addition laws. A system of axioms is introduced and discussed leading to the result, and a study is made of several variants of that system. In particular the status of the group axiom is investigated with respect to the condition of the two-way isotropy of light. Several issues which are ignored or misunderstood in the literature are emphasized., in Foundations of Physics (2016)

In this paper we have given a direct deduction of the auxiliary Lorentz transforms from the consideration of Maxwell. In the Maxwell’s theory, if c is considered to be the speed of light in ether space, his equations should be affected on the surface of the moving earth. But curiously, all electromagnetic phenomena as measured on the surface of the moving earth are independent of the movement of this planet. To dissolve this problem, Einstein (1905) assumes that Maxwell’s equations are invariant to all measurers in steady motion which acts as the foundation of Special Relativity. This assumption of Einstein is possible when all four auxiliary Lorentz transforms are real. There is not a single proof that could properly justify Einstein’s assumption. On the contrary it is now known that classical electrodynamics could easily explain all relativistic phenomena rationally.

The paper discusses the problem of a vehicle’s motion control on a preset nonlinear path. The new proof of the velocity addition theorem presented in [4] allowed simplified derivation of control law. The procedures described can be considered the first step in the development of motion control algorithms by back-stepping.

A complete treatment of the Thomas rotation involves algebraic manipulations of overwhelming complexity. In this paper, we show that a choice of convenient vectorial forms for the relativistic addition law of velocities and the successive Lorentz transformations allows us to obtain straightforwardly the Thomas rotation angle by three new methods: (a) direct computation as the angle between the composite vectors of the non-collinear velocities, (b) vectorial approach, and (c) matrix approach. The new expression of the Thomas rotation angle permits us to simply obtain the Thomas precession. Original diagrams are given for the first time.

Assuming the existence of an invariant velocity a slightly generalized form of Lorentz transformations is derived. The group of these transformations has a simpler composition law than the group of standard Lorentz transformations has. It is shown that this new form allows the description of both subluminal and superluminal motions. It also allows to find all velocity-dependent tensors. In particular, the tachyonic momentum as a function of superluminal velocity is derived.