The 2+1-Dimensional Special Relativity

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.

Relativistic Flight Mechanics and Space Travel is about the fascinating prospect of future human space travel. Its purpose is to demonstrate that such ventures may not be as difficult as one might believe and are certainly not impossible. The foundations for relativistic flight mechanics are provided in a clear and instructive manner by using well established principles which are used to explore space flight possibilities within and beyond our galaxy. The main substance of the book begins with a background review of Einstein's Special Theory of Relativity as it pertains to relativistic flight mechanics and space travel. The book explores the dynamics and kinematics of relativistic space flight from the point of view of the astronauts in the spacecraft and compares these with those observed by earth's scientists and engineers-differences that are quite surprising. A quasi historical treatment leads quite naturally into the central subject areas of the book where attention is focused n various issues not ordinarily covered by such treatment. To accomplish this, numerous simple thought experiments are used to bring rather complicated subject matter down to a level easily understood by most readers with an engineering or science background. The primary subjects regarding photon rocketry and space travel are covered in some depth and include a flight plan together with numerous calculations represented in graphical form. A geometric treatment of relativistic effects by using Minkowski diagrams is included for completeness. The book concludes with brief discussions of other prospective, even exotic, transport systems for relativistic space travel. A glossary and simple end-of-chapter problems with answers enhance the learning process.

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.

A new groupG of Lorentz transformations (LT) in four dimensions, generalized also for Superluminal frames, is introduced and particularly studied in its physical implications. With the help of a « principle of duality »—implied byG—between subluminal and Superluminal frames, the meanings of « inertial frame », « equivalence », « principle of relativity », « covariance » may be correspondingly extended. A biunivocal correspondence exists between bradyonic and tachyonic velocities, which we find to be a particular conformal mapping (inversion). Since the groupG consists of generic rotations in space-time, it includes,e.g., also the total-inversion operation (PT). Moreover (for a non « charge »-free universe), it is shown that our generalized special relativity requires covariance underCPT. A « tachyonization principle » is formulated, on the basis of which relativistic physical laws (of mechanics and electrodynamics, at least) can be easily extended to tachyons. Many simple applications are performed of the generalized LT’s (velocity composition law, comparison of the length and time units, Doppler effect, refraction index, …), either useful to clarify our problem or interesting in astrophysics.

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.”

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

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

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 $m\geq 2$.

We conjecture that the random walk and the corresponding diffusion in the relativistic velocity space is an adequate method for describing the acceleration process in relativistic jets. Considering a simple toy model, the main features of diffusion in the velocity space are demonstrated in both non-relativistic and relativistic regimes.

The study of relativistic Coulomb systems in velocity space is prompted by the fact that the study of Newtonian Kepler/Coulomb systems in velocity space, although less familiar than the analytic solutions in ordinary space, provides a much simpler (also more elegant) method. The simplicity and elegance of the velocity-space method derives from the linearity of the velocity equation, which is the unique feature of 1/r interactions for Newtonian and relativistic systems alike. The various types of possible trajectories are presented, their properties deduced from the orbits in velocity space, accompanied with illustrations. In particular, it is found that the orbits traversed in the relativistic velocity space (which is hyperbolic (H3) rather than Euclidean) are epicyclic – circles whose centres also rotate – thus the title.

Chapter 6 - Relativity in Four Dimensions

Generalized Lorentz transformations in six-dimensional space-time

We determine nonlinear transformations between coordinate systems which are mutually in a constant symmetrical accelerated motion. The maximal acceleration limit follows from the kinematical origin and it is an analogue of the maximal velocity in special relativity. We derive the dependence of mass, length, time, Doppler effect, Cerenkov effect and transition radiation angle on acceleration as an analogue phenomenon in special theory of relativity. The last application of our method is the Thomas precession by uniform acceleration and equivalent gravity with the possible role in modern physics and cosmology.

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.