Functional Forms for Lorentz Invariant Velocities

Since the introduction of Special theory of Relativity (SR), Lorentz invariance has been a fundamental part of our description of nature. Over the past decades, this (Lorentz invariance) was tested and verified by many research groups and almost no Lorentz violation have been found. However, when we look at the Lorentz transformations, we can mathematically find that they are not symmetrical relations. In fact, the inertial frames are not symmetrical in relation to each other and as a result, there must be a reference frame. If the inertial frames are symmetrical in relation to each other, as the SR argues, some paradoxes and contradictions are happened. Hence, we need to find the misconceptions in this theory. Many researches up to now, focused on Zero-Point Field (ZPF) as a cause for many physical phenomena like mass, inertia, and gravity. One can argue that the Lorentz transformations can be applied only for the inertial frames, moving inside the ZPF and the inertial frames, being at rest in relation to the ZPF, must not bear the Lorentz transformations. In addition, we have found that the SR and General Relativity (GR) are not two separated concepts. They are one thing and the space-time of them are the same. In both cases, it is the ZPF, which curves around the accelerated body or a gravitational mass, causing changes in the body’s length, time, and mass.

We give a mathematical interpretation for a previously investigated model, which is based on a set of axioms incorporating fundamental aspects from both Galilei and special relativity (SR). The model considers two time variables, one of them identified with absolute time [characteristic of Galilei relativity (GR)] and the other identified with the local time of SR. Its main characteristics rely on two classes of transformations. One is what we call generalized Lorentz transformation, which includes the standard Lorentz transformations as a particular case. The other is the h-map, which essentially relates absolute and local times. The h-map also determines the basic kinematics of SR from the corresponding kinematics of GR. This allows us to express a Lorentz transformation in terms of a Galilei transformation. It suggests us to see the h-map as a natural transformation between two functors \(\overline{G}\) and \(\overline{L}\), representing the notions of Galilei and Lorentz transformations in functorial language. The categorical framework we develop not only elucidates the structure of Galilei and Lorentz transformations but also provides a unified model for Galilei and special relativity, where the former acquires a distinguished role in itself, not being a mere low speed limit of the latter.

The common opinion that the Lorentz transformation distorts the Minkowski spacetime arises from a mathematically incorrect neglect of its non-diagonal terms. When the non-diagonal Lorentz transformation is properly treated, the time dilation and the space distortion disappear. Consequently, the Lorentz metric produced by the Lorentz transformation is identical to the Minkowski metric. The spacetime distortion in moving inertial frames is properly described by the so-called Doppler metric produced by the Doppler transformation. This transformation assumes the existence of a preferred frame and depicts the Doppler shift of light for a moving light source and/or observer. The Doppler metric is Lorentz invariant and leaves the Maxwell’s equations unchanged from their form in the Minkowski spacetime. Moreover, the Doppler metric is experimentally confirmed because it yields the null result in the Michelson–Morley experiment and other interferometric experiments. The suitability of the Doppler metric for describing light properties in Special and General Relativity is supported by recent cosmological observations of the cosmic microwave background.

We propose here two new transformations between inertial frames that apply for relative velocities greater than the speed of light, and that are complementary to the Lorentz transformation, giving rise to the Einstein special theory of relativity that applies to relative velocities less than the speed of light. The new transformations arise from the same mathematical framework as the Lorentz transformation, displaying singular behaviour when the relative velocity approaches the speed of light and generating the same addition law for velocities, but, most importantly, do not involve the need to introduce imaginary masses or complicated physics to provide well-defined expressions. Making use of the dependence on relative velocity of the Lorentz transformation, the paper provides an elementary derivation of the new transformations between inertial frames for relative velocities v in excess of the speed of light c , and further we suggest two possible criteria from which one might infer one set of transformations as physically more likely than the other. If the energy–momentum equations are to be invariant under the new transformations, then the mass and energy are given, respectively, by the formulae and where denotes the limiting momentum for infinite relative velocity. If, however, the requirement of invariance is removed, then we may propose new mass and energy equations, and an example having finite non-zero mass in the limit of infinite relative velocity is given. In this highly controversial topic, our particular purpose is not to enter into the merits of existing theories, but rather to present a succinct and carefully reasoned account of a new aspect of Einstein's theory of special relativity, which properly allows for faster than light motion.

called the special theory of relativity, no doubt one of the boldest and most interesting theories ever designed.' Generations of physicists, mathematicians and philosophers were inspired by it. Bitter battles were fought over its principles and deductions which clashed with many an honourable idea backed up by honourable philosophical and scientific traditions. Yet all the knocks and bites from left and right seem to have left Einstein's theory unshaken. Today the battle-grounds of yesterday look rather deserted. The critic is likely to be silenced in the face of the theory's spectacular success. In the following I shall try to throw some new light on one or two old problems which Einstein's theory gives rise to. Section z is concerned with the obstacles we are faced with when we try to combine Einstein's special theory of relativity with a causal account of relativistic phenomena. The argument does not show, of course, that Einstein's theory is false, but it points to a possible incompleteness and to certain limitations inherent in the special theory of relativity. In section 3 a substratum theory of relativity is outlined which seems to be as simple and coherent as Einstein's special theory of relativity; the Lorentz transformations are deduced from two basic principles without making use of the principle of relativity. Finally, in section 4, the two theories are compared and some general conclusions are drawn. But let us first have a short look at the ideas underlying Einstein's theory. To begin with, the special theory of relativity is a transformation theory. This is its most fundamental aspect. Suppose we know the space and time co-ordinates of an event particle e (event with negligible spatiotemporal extension) in an inertial frame of reference F; then the theory

One of the most basic principles in science is the objectivity of measurement of physical properties. According to the special theory of relativity (STR), this ancient principle is violated for observers in relative motion since it predicts that they generally will disagree on the ratios of the lengths of two objects and also on whose clock is running slower at any given time. Both predictions stem from the Lorentz transformation (LT), which is the centerpiece of Einstein's STR. It has recently been pointed out that two of the claims of this theory are mutually contradictory; it is impossible that the rates of two clocks in motion are strictly proportional to one another (time dilation) while one of them finds that two events are simultaneous whereas the other does not (remote nonsimultaneity). This recognition proves that the LT is not a valid component of the relativistic theory of motion, including its well-known thesis that space and time are not distinct quantities. Instead, it has always been found experimentally that the rates of clocks in motion are governed by a Universal Timedilation Law (UTDL), whereby the speed of the clock relative to a specific rest system is the sole determining factor. A simple way of describing this state of affairs is to say that the standard unit of time in each rest frame is different and increases with its relative speed to the above rest system by a definite factor. The measurement process is thereby rendered to be completely objective in nature. A key goal of relativity theory is therefore to develop a quantitatively valid method for determining this factor. It will be shown that the same factor appears in the true relativistic space-time transformation and that it also plays a key role in the uniform scaling of all other physical properties

In the perspective to guiding undergraduate students on advanced topics in theoretical physics, this article approaches a brief study about some aspects of the Lorentz transformations. The two fundamental postulates of Einstein's Theory of Special Relativity (or special relativity) are explained. The detailed process of constructing the Lorentz transformations and in particular the transformations performed considering the rectilinear and uniform motion of a frame of reference in relation to the other along the x-axis direction are presented. Some consequences resulting from Lorentz transformations such as Length contraction, Time dilation and the Relativity of simultaneity are highlighted.

It has been critically argued by V. A. Leus (Sobolev Institute of Mathematics, Novosibirsk, Russia) that in my proof that Einstein’s Special Theory of Relativity is logically inconsistent and therefore false, I violated the basic tenets of Special Relativity and foisted an alternative theory upon Einstein’s. A careful study of the critical analysis reveals however a failure to address the key arguments I adduced to prove Special Relativity logically inconsistent, and a concomitant invocation of Einstein’s theory to try to argue that my analysis is incorrect because it does not concur with Einstein. There is therefore no proof advanced of any alleged error in my analysis. In my paper I did not introduce an alternative theory. The aforementioned critical paper affords opportunity in rebuttal to amplify the invalidity of A. Einstein’s tacit assumption, in constructing the Special Theory of Relativity, that systems of clock-synchronised stationary observers consistent with Lorentz Transformation can be mathematically constructed. Since such systems of observers have in fact no mathematical existence the Special Theory of Relativity is logically inconsistent. It is therefore invalid. The consequences for physics, astronomy, and cosmology, are significant.

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.

The Kronecker product of two 2 × 2 matrices, Lorentz transformations, and the structure of space-time in special relativity

Light from a distant γ-ray burst backs up a key prediction of Albert Einstein's theory of relativity — that photon speed is the same regardless of energy. But it might set the stage for evolution of the theory. Observations of the distant and short γ-ray burst GRB 090510 with the Fermi Gamma-ray Space Telescope have provided an opportunity to test a central prediction of Einstein's special theory of relativity — the Lorentz invariance. This holds that all observers measure exactly the same speed of light in a vacuum, independent of photon energy. A key test of the violation of Lorentz invariance is a possible variation of photon speed with energy. Accumulated over cosmological light-travel times, even a tiny variation in photon speed should become observable — as for instance sharp features in the light curve of a γ-ray burst. No evidence for the violation of Lorentz invariance was found in the GRB 090510 spectrum, at least down to a limit of the Planck length divided by 1.2. This argues against quantum-gravity theories where the quantum nature of space–time linearly alters the speed of light with photon energy.

Summary form only given. In beam optics, polarization properties of beams can be alternatively described by a vector of the four stokes parameters, by the two-by-two polarization matrix, or by the two, in general, Jones vectors in two-dimensional polarization space. The evolution of the beam along an optical system is then given, in paraxial approximation, with the aid of the Mueller four-by-four matrices or the Jones two-by-two matrices. On the other hand, in the special theory of relativity, we deal with the space-time four-vectors, second-rank spinor matrices or with the two-dimensional spinor vectors. All these entities are transformed under four-dimensional and two-dimensional representations of the restricted six-parameter Lorentz group, respectively. Although these transformations are of different origin in both cases, the Lorentz invariance of a space-time interval in the coordinate, Minkowski space corresponds to the invariance of a polarization degree of the transforming beam. Basic features of the above correspondence are reviewed and a few examples of the Lorentz transformations of the coherent beam polarization are given. Some aspects of this correspondence, not considered in this context in the literature until very recently, are outlined. It is shown that the invariance of the beam polarization transformations specifies a definition of beam amplitude, with transformation properties independent of those related to the beam polarization. A suitable reference frame for the treatment of the beam amplitude and the beam polarization is proposed. A case of the beam transformations at a dielectric interface is discussed in more detail. This leads to a complete identification between scattering coefficients of a multilayer optical structure and the parameters of the corresponding Lorentz transformation.

A Lorentz transformation of signature $(m,n)$, $m,n \in {\mathbb N}$, is a pseudo-rotation in a pseudo-Euclidean space of signature $(m,n)$. Accordingly, the Lorentz transformation of signature $(1,3)$ is the common Lorentz transformation of special relativity theory. It is known that entangled particles involve Lorentz symmetry violation. Hence, the aim of this article is to expose and illustrate the symmetry groups of systems of entangled particles uncovered in [44] It turns out that the Lorentz transformations of signature $(m,n)$ form the symmetry group by which systems of $m$ $n$-dimensional entangled particles can be understood, just as the common Lorentz group of signature $(1,3)$ forms the symmetry group by which Einstein's special theory of relativity can be understood. Consequently, it is useful to extend special relativity theory by incorporating Lorentz transformation groups of signature $(m,3)$ for all $m\geq 2$. The resulting extended special relativity theory, then, provides not only the symmetry group of the $(1+3)$-dimensional spacetime of particles, but also the symmetry group of the $(m+3)$-dimensional spacetime of systems of $m$ entangled three-dimensional particles, for each $m\geq 2$.

The quest for magnetic monopoles has captivated theoretical physicists due to their potential to reshape our understanding of electromagnetism. Despite extensive research, definitive mathematical proof of their existence remains elusive. This paper presents a rigorous mathematical framework supporting the existence of magnetic monopoles within the covariant formulation of classical electromagnetism and special relativity. Utilizing tensor calculus and the covariance form of Maxwell's equations, we derive expressions for magnetic charge density and magnetic current density through Lorentz transformations. Our approach emphasizes the expansion of tensorial equations, particularly focusing on the covariant derivatives of the electromagnetic field tensor and its dual. The results demonstrate the recovery of magnetic charge and current densities in Maxwell's equations, providing a theoretical foundation for the existence of magnetic monopoles. A magnetic monopole model is introduced, establishing the relationship between the electric field, the monopole’s magnetic charge, and their dependence on the observer's relative velocity as a direct consequence of the modified Gauss's law for magnetism. This model suggests the existence of dipole bosons—electric or magnetic—which function analogously to the Higgs boson. These dipole bosons are proposed to confer mass, charge, and spin to charged particles, whether they are electric or magnetic in nature.

Science sets itself apart from other paths to truth by recognizing that even its greate practitioners sometimes err.