Special relativity as classical kinematics of a particle with the upper bound on its speed. Part II. The general Lorentz transforrmation and the generalized velocity composition theorem

TL;DR

This paper derives the full 3D Lorentz transformation and velocity composition law purely from classical mechanics principles, avoiding relativistic concepts like time dilation and length contraction, thus providing a new classical kinematic foundation for special relativity.

Contribution

It extends previous 1D results to 3D, explicitly deriving the Lorentz transformation and velocity addition law without relativistic concepts, based solely on classical mechanics and the speed limit.

Findings

Derived the 3D Lorentz transformation from classical mechanics.

Explicitly obtained the velocity composition law without relativistic assumptions.

Identified the invariant quantity of motion through classical kinematic principles.

Abstract

The kinematics of a particle with the upper bound on the particle's speed (a modification of classical kinematics where such a restriction is absent) has been developed in [arXiv:1204.5740]. It was based solely on classical mechanics without employing any concepts , associated with the time dilatation or/and length contraction. It yielded the 1-D Lorentz transformation (LT), free of inconsistencies (inherent in the canonical derivation and interpretations of the LT). Here we apply the same approach to derive the LT for the 3-dimensional motion of a particle and the attendant law of velocity composition. As a result, the infinite set of four-parameter transformations is obtained. The requirement of linearity of these transformations selects out of this set the two-parameter subset . The values of the remaining two parameters ,dictated by physics of the motion, is explicitly determined , yielding the canonical form of the 3-dimensional LT. The generalized law of velocity composition and the attendant invariant ( not postulated apriori) of the motion are derived, As in the one-dimensional case, present derivation, as a whole, does not have any need in introducing the concepts of the time dilatation and length contraction, and is based on the classical concepts of time and space.