Generalized Lorentz transformations in four dimensions and superluminal objects

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.

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.

In the new mathematical description of special relativity in terms of the relativistic velocity space, many physical aspects acquire new geometric meanings. Performing conformal deformations upon the 2-dimensional relativistic velocity space for the (2+1)-dimensional special relativity, we find that these conformal deformations correspond to the generalized Lorentz transformations, which are akin to the ordinary Lorentz transformation, but are morphed by a global rescaling of the polar angle and correspondingly characterized by a topological integral index. The generalized Lorentz transformations keep the two fundamental principles of special relativity intact, suggesting that the indexed generalization may be related to the Bondi–Metzner–Sachs (BMS) group of the asymptotic symmetries of the spacetime metric. Furthermore, we investigate the Doppler effect of light, the Planck photon rocket, and the Thomas precession, affirming that they all remain in the same forms of the standard special relativity under the generalized Lorentz transformation. Additionally, we obtain the general formula of the Thomas precession, which gives a clear geometric meaning from the perspective of the gauge field theory in the relativistic velocity space.

This is an expository review of the Lorentz transformation, which is a change of coordinates used by one “inertial observer” to those used by another one. The transformation can be represented by a four-by-four matrix, the Lorentz matrix or the Minkowski-Lorentz matrix. The most familiar, or “special,” case has thex axis of both observers parallel to their relative velocity. A more general transformation drops this constraint. But then a seeming “paradox” arises when there are three observers, and this has led to a challenge to the self-consistency of the special theory of relativity. It is shown here that this challenge is based on a misunderstanding. The properties of the more general Lorentz transformation are reviewed consistently in terms of the matrix approach, which the author believes is now the easiest approach to understand. The spectral analysis of the Lorentz matrix is also discussed. Several checks are included to “make assurance double sure.”

The traditional version of the Lorentz transformation is used to relate the coordinates of an event (a point in spacetime) between different inertial coordinate systems in the absence of any gravitational effects. It is limited to those cases in which both reference frames are inertial, so they have a constant velocity relative to each other. This report utilizes a familiar postulate, regarding the equivalence between coordinates measured in an accelerating reference frame and coordinates measured in a reference frame instantaneously at rest with the accelerating frame, together with the traditional version of the Lorentz transformation to derive a generalized version that allows one of the reference frames to be accelerating. This is not a new topic but the presentation and all derivations in this report are the author’s own inventions. By defining suitable quantities and introducing suitable notation, the generalized version can be written in a way that is almost as simple as the traditional (constant velocity) version when calculating the coordinates of an event in an inertial system when given the coordinates in an accelerating system. Unfortunately, calculations of the inverse transformation, i.e., calculating the coordinates in the accelerating system when given coordinates in an inertial system, are more cumbersome. Worse yet, while a suitably selected history and future ensure the existence of an inverse transformation, there can exist spacetime points for which it is not unique. However, the metric tensor can be derived in the accelerating system for the general case and is included in this report. This is used to calculate time dilations and Doppler effects that are outside the scope of inertial coordinate systems.

A theory of tachyons and superluminal transformations is developed on the basis of the quaternionic formulation. We have introduced a complex space-time and a complex transformation group which contains both Lorentz transformations and superluminal transformations. The complex space-time «the biquaternion space» which is closed under the superluminal transformations is introduced. The principle of special relativity, such as the conservation of the quadratic form of the metric of the space-time, and the principle of duality are extended to the complex space-time and to bradyons, luxons and tachyons under the complex transformations. Several characteristic features of the superluminal transformations and of tachyons are derived.

CHAPTER 1 - THE PRINCIPLE OF RELATIVITY

THE SPECIAL THEORY OF RELATIVITY Einstein's Postulates Classical mechanics obeys the principle of relativity : the same laws apply in any inertial reference frame . By “inertial” I mean that the system is at rest or moving with constant velocity. Imagine, for example, that you have loaded a billiard table onto a railroad car, and the train is going at constant speed down a smooth straight track. The game will proceed exactly the same as it would if the train were parked in the station; you don't have to “correct” your shots for the fact that the train is moving—indeed, if you pulled all the curtains, you would have no way of knowing whether the train was moving or not. Notice by contrast that you know immediately if the train speeds up, or slows down, or rounds a corner, or goes over a bump—the billiard balls roll in weird curved trajectories, and you yourself feel a lurch and spill coffee on your shirt. The laws of mechanics, then, are certainly not the same in accelerating reference frames. In its application to classical mechanics, the principle of relativity is hardly new; it was stated clearly by Galileo. Question: does it also apply to the laws of electrodynamics? At first glance, the answer would seem to be no . After all, a charge in motion produces a magnetic field, whereas a charge at rest does not. A charge carried along by the train would generate a magnetic field, but someone on the train, applying the laws of electrodynamics in that system, would predict no magnetic field. In fact, many of the equations of electrodynamics, starting with the Lorentz force law, make explicit reference to “the” velocity of the charge. It certainly appears, therefore, that electromagnetic theory presupposes the existence of a unique stationary reference frame, with respect to which all velocities are to be measured.

We have studied Wigner rotations of different types of Lorenz Transformations according to the nature of movement of one inertial frame relative to the other inertial frame. When the motion is along any arbitrary direction then we can find the formulae for Wigner rotations using the velocity addition formulae for most general, mixed number, quaternion and geometric product Lorentz transformations. Finally we have used simulated data for applying the Wigner rotation formula in pion decay chainand concluded the result.

The differential Lorentz transformation is formulated solely from the principle of relativity and the invariance of the speed of light. The differential Lorentz transformation transforms physical quantities, instead of space–time coordinates, to keep laws of nature form-invariant among inertial frames. The new relativistic transformation fulfills the principle of relativity, whereas the usual Lorentz transformation of space–time coordinates does not. Furthermore, the new relativistic transformation is compatible with quantum mechanics. The formulation herein provides theoretical foundations for the differential Lorentz transformation as the fundamental relativistic transformation.

Chapter 6 - Relativity in Four Dimensions

The Lorentz transformation (LT) is one of the most revered set of relationships in modern physics. Overlooked in this broad assessment are a number of clear inconsistencies in its predictions, however, as will be discussed herein. For example, it makes three predictions which are not consistent with one another: Lorentz-FitzGerald length contraction (FLC), time dilation (TD) and light-speed equality for observers in relative motion to one another. Einstein’s light-speed postulate (LSP) is shown to be unviable by considering a case in which a light source passes by a stationary observer at the same time that it emits a light pulse in the same direction. It is found that, in contradiction to the LSP, that the classical velocity (Galilean) transformation (GVT) is applicable when two observers in relative motion deduce the speed of a light wave from their different perspectives. The LT also stands in violation of the Law of Causality because it fails to recognize that inertial clocks can never change their rate spontaneously; thus its two clocks must always measure elapsed times in the same ratio (Q), contrary to the LT prediction of space-time mixing. The Newton-Voigt transformation (NVT) is consistent with the Law of Causality because it assumes space and time do not mix. It is nonetheless also consistent with the relativistic velocity transformation (RVT) and also with Einstein’s mass-energy equivalence relation E=mc2. The ratio Q of clock rates for two inertial rest frames S and S’ is required input for the NVT. Experimental data obey the Universal Time-dilation Law (UTDL) which states that the measured time Δt obtained by an inertial clock for a given event is inversely proportional to γ(v)= (1-v2c-2)-0.5, where v is the speed of the clock relative to a specific rest frame referred to as the objective rest frame ORS. The Uniform Scaling method employs Q as a conversion factor between the units of time in the two rest frames. It is found that the conversion factors

Maxwell unified the disparate concepts of electric and magnetic fields with one theory (electromagnetism) and Einstein then unified the disparate theories of electromagnetism and mechanics with one kinematics (Minkowski space of special relativity). In this talk, we will briefly explain how the disparate kinematics of quantum mechanics (finite-dimensional Hilbert space) and special relativity can be unified with one principle (relativity principle). This result follows from the axiomatic reconstruction of quantum mechanics via information-theoretic principles, which has successfully recast quantum mechanics as a principle theory a la Einstein, i.e., the formalism of the theory follows from an empirically discovered fact, just like special relativity. According to the quantum reconstruction program, the empirically discovered fact whence the Hilbert space formalism of quantum mechanics is Information Invariance & Continuity. Of course, the empirically discovered fact whence the Lorentz transformations of special relativity is the light postulate, i.e., everyone measures the same value for the speed of light c, regardless of their relative motions. Obviously, the light postulate can be justified by the relativity principle—the laws of physics are the same in all inertial reference frames—because c is a constant of Nature per Maxwell’s electromagnetism. [We label this “NPRF + c” for short, where NPRF stands for “no preferred reference frame.”] As we will show, Information Invariance & Continuity can also be justified by the relativity principle by first spatializing the quantum reconstruction program’s operational notion of measurement. In that case, Information Invariance & Continuity entails the empirically discovered fact that everyone measures the same value for Planck’s constant h, regardless of their relative spatial orientations or locations (Planck postulate). Since Poincar’e transformations relate inertial reference frames via spatial rotations and translations as well as boosts, and h is a constant of Nature per Planck’s radiation law, the relativity principle justifies the Planck postulate (NPRF + h) just like it justifies the light postulate (NPRF + c). Thus, the kinematics of quantum mechanics and special relativity are unified in that both follow most fundamentally from the relativity principle in the adynamical global constraints NPRF + h and NPRF + c. This approach provides a principle solution to the mystery of quantum entanglement that does not violate locality, statistical independence, intersubjective agreement, or the uniqueness of experimental outcomes and it does not alter quantum mechanics as a principle theory. An ontology consistent with this unification is introduced and we deflate both the ‘big’ and ‘small’measurement problems.

We re-examined the four-dimensional spacetime formulation of invariance of electromagnetic fields between two inertial frames under Lorentz transformation, which predicts a pure electric (magnetic) field in one inertial frame is composed the Cartesian components of a pure both electric and magnetic fields in another inertial frame. This contradicts the Lorentz invariance condition which requires that the vector quantities in one inertial frame must have the same form in another inertial frame. In this work, we introduce a three-dimensional quasi-time vector to modify the classical four-dimensional spacetime (3+1) to a new six-dimensional spacetime (3+3) and derive spacetime metric equation and relativistic velocity. We use the classical vector transformation theory to derive expressions for Cartesian components of relativistic velocity and net electromagnetic force vectors. Considering two massive inertial frames form a closed system, we integrated the transformed relativistic velocity with the law of conservation of energy to prove that contrary to the common belief, the electromagnetic field that appears as a purely electric (magnetic) field in one massive inertial frame, it also appears as a pure electric (magnetic) field in another massive inertial frame under Lorentz transformation. As an application of the proposed six-dimensional spacetime theory, we prove Lorentz invariance of Maxwell’s equations with and without charge and current source. We also prove the scalar electromagnetic wave equations with and without charge and current source and the conservation laws of the continuity equations of current and densities of electromagnetic energy and linear and angular momentums between two massive inertial frames under Lorentz transformation.

Generalized Lorentz transformations in six-dimensional space-time