Seven formulations of the kinematics of special relativity

Abstract

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.

Einstein's special relativity from a space-time duality principle

“…Moreover, one considers the principle of special relativity, which mathematically amounts to the statement that given two observers S and S ′ where S ′ moves along their common axis with a velocity v with respect to S, the linear transformation matrix Λ which transforms the unprimed coordinates to the primed ones (x ′ = Λx) is only a function of the relative velocity, Λ = Λ(v), and under the transformation v → −v, the coordinate transformation matrix transforms to its inverse Λ(−v) = Λ −1 , that is to say, as the velocity of S is −v with respect to S ′ the coordinate transformation from primed to unprimed coordinate is obtained by replacing −v in the places of v in Λ, giving x = Λ(−v)x ′ . The principle of special relativity along with the symmetry of the manifold-homogeneity and isotropy-is sufficient to land us into coordinate transformation laws which are consistent with both Galilean and Einsteinian relativity thereby confirming their common origin from the principle of relativity (see for example in [22,25,27]). Thereafter, in conventional approaches, the second postulate concerning the speed of light might be used to finally arrive at the Lorentz transformations.…”

“…The value of the free parameter k, that parametrizes a class of space-time duality symmetric theories, can be set in several ways. For example, we can set up an experiment to measure the value of k directly similar to the one suggested in the section VII of [27]. However, that is not necessary as we can deduce the value of k from consistency requirement.…”

“…Even in the early works on special relativity, it was shown that the Lorentz transformations can be derived without the 'constancy of speed of light' postulate, that is without any reference to the classical electrodynamics [10][11][12]. For more recent works that have argued the unnecessity of the second postulate to derive the Lorentz transformations, see the works [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] and the references therein (note that the list given here is not exhaustive).…”

An alternative transformation factor in the framework of the relativistic aberration of light

“…Since 1905, many methods of derivation of the LT have been investigated that have enriched our understanding. We undertake a derivation based on the two principles of SR, that is, the principle of relativity and the constancy of the speed of light-or more specifically the "inertial invariance of the speed of propagation of electromagnetic radiation in vacuum" [12]-considering the special case described in Figure 1, which implicitly involves the phenomenon of RAL.…”

The Lorentz factor in a reverse coordinate system

“…In this study, we derive the Lorentz factor between two inertial frames S and S' using reverse frames and mathematizing the two postulates of special relativity (SR): the relativity of motion and the constancy of the speed of electromagnetic waves in a vacuum [3]. As a consequence of this derivation, we obtain a transformation factor very similar to that of the Lorentz factor (LF) within the Minkowski spacetime; these factors largely coincide in terms of their results (calculations), but the new factor presents one additional significant characteristic: opens up possibilities for superluminal motion.…”