A categorical approach for relativity theory

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

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.

A central observer at Greenwich builds in real time a diachronic representation of 4-D(t) space-time by assigning two distinct times to a given remote event, such as a meteorite impact on Mars: (1) diachronic time read on his Greenwich clock (GMT) at observation time, and (2) synchronic or local Einstein time as directly seen from Greenwich on the Mars clock. In a spherical coordinate system the Greenwich diachronic observer uses local Einstein time relative to diachronic time as a radial coordinate, leaving the role of the fourth coordinate to dia-chronic time. In a 4-D(t) conceptual framework the same diachronic time applies integrally to all directly observed local Einstein times, thus implying an infinite diachronic speed for incoming light. In the 4-D(t) conceptual framework the de-tection of a photon is considered to be an instantaneous transfer of energy and momentum across the time gap between emitter and receiver. The Lorentz trans-formation can be obtained by requiring that history updated now and here be the same for another observer flying by. Some results are (1) the conventional (syn-chronic) speed of light c is identified with the flow of time, (2) the twin paradox is resolved in a twin-symmetric fashion, and (3) a new solution for the Einstein–Podolsky–Rosen paradox maintains harmony between special relativity and quan-tum mechanics. The diachronic approach presents the pedagogical advantage of using a single diagram to display space-time relationships of events in two coor-dinate systems in relative motion.

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

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.

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.

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.

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

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.

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 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.

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

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.

Lorentz invariance lies at the very heart of Einstein’s special relativity, and both the energy formula and the relative velocity formula are well-known to be invariant under a Lorentz transformation. Here, we investigate the spatial and temporal dependence of the velocity field itself u(x,t) and we pose the problem of the determination of the functional form of those velocity fields u(x,t) which are automatically invariant under a Lorentz transformation. For a single spatial dimension, we determine a first-order partial differential equation for the velocity u(x,t), which appears to be unknown in the literature, and we investigate its main consequences, including demonstrating that it is entirely consistent with many of the familiar outcomes of special relativity and deriving two new partial differential relations connecting energy and momentum that are fully compatible with the Lorentz invariant energy–momentum relations.

1. In the beginning 2. Cosmology becomes a science 3. Newton's machine 4. Lighting the worlds 5. The lives of the stars 6. Infinite space and absolute time 7. The special theory of relativity 8. The general theory of relativity 9. Black holes 10. The expanding universe 11. Modeling the universe 12. The early universe 13. Testing the models 14. A message from the big bang 15. Dark matter and large-scale structure 16. The inflationary universe 17. The edge of time A. Scientific notation B. Units C. Physical and astronomical constants