The Lorentz Group, a Galilean Approach

We derive rotation free Lorentz Transformation (LT) between two inertial reference frames without using the second postulate of Einstein, i.e., we do not assume the invariant speed of light (in vacuum) under LT. We find a general transformation rule between two inertial frames where a speed, invariant under that transformation, arises naturally. This idea first came into light by Mathematician Ignatowski [1] around the year, 1910. Without loss of novelty, here we present our ...

A century after its formulation by Einstein, it is time to incorporate special relativity early in the physics curriculum. The approach advocated here employs a simple algebraic extension of vector formalism that generates Minkowski spacetime, displays covariant symmetries, and enables calculations of boosts and spatial rotations without matrices or tensors. The approach is part of a comprehensive geometric algebra with applications in many areas of physics, but only an intui...

A new method of derivation of Lorentz Transformation (LT) is given based on both axioms of special relativity (SR) and physical intuitions. The essence of the transformation is established and the crucial role played by the presumptions is presented for clarification. I consider the most general form of transformations between two sets of events in two inertial reference frames and use the most basic properties expected from such a transformation together with the principle o...

It is demonstrated how the right hand sides of the Lorentz Transformation equations may be written, in a Lorentz invariant manner, as 4--vector scalar products. The formalism is shown to provide a short derivation, in which the 4--vector electromagnetic potential plays a crucial role, of the Lorentz force law of classical electrodynamics, and the conventional definition of the magnetic field in terms spatial derivatives of the 4--vector potential. The time component of the re...

An apparent paradox in Einstein's Special Theory of Relativity, known as a Thomas precession rotation in atomic physics, has been verified experimentally in a number of ways. However, somewhat surprisingly, it has not yet been demonstrated algebraically in a straightforward manner using Lorentz-matrix-algebra. Authors in the past have resorted instead to computer verifications, or to overly-complicated derivations, leaving undergraduate students in particular with the impress...

Derivation of the Lorentz transformation without the use of Einstein's Second Postulate is provided along the lines of Ignatowsky, Terletskii, and others. This is a write-up of the lecture first delivered in PHYS 4202 E&M class during the Spring semester of 2014 at the University of Georgia. The main motivation for pursuing this approach was to develop a better understanding of why the faster-than-light neutrino controversy (OPERA experiment, 2011) was much ado about nothing.

This work gives an explicit exact expression for the Thomas precession arising in the framework of Special Theory of Relativity as the spatial rotation resulting from two subsequence Lorentz boosts. The final result for the orthogonal matrix of Thomas precession is given by Eqs. (21)--(25). A trivial calculation leads to the compact formula (26) for the angle of rotation due to Thomas precession. In the framework of Gaia the special-relativistic Thomas precession is an impo...

The aim of this work is to show, on the example of the behaviour of the spinless charged particle in the homogeneous electric field, that one can quantized the velocity of particle by the special gauge fixation. The work gives also the some information about the theory of second quantisation in the space of Hilbert- Fock and the theory of projectors in the Hilbert space. One consider in Appendix the theory of the spinless charged particle in the homogeneous addiabatical chang...

It is demonstrated how the right hand sides of the Lorentz Transformation equations may be written, in a Lorentz invariant manner, as 4--vector scalar products. This implies the existence of invariant length intervals analogous to invariant proper time intervals. This formalism, making essential use of the 4-vector electromagnetic potential concept, provides a short derivation of the Lorentz force law of classical electrodynamics, the conventional definition of the magnetic f...

A novel approach to the finite dimensional representation theory of the entire Lorentz group $\operatorname{O}(1,3)$ is presented. It is shown that the entire Lorentz group may be understood as a semi-direct product between the identity component of the entire Lorentz group, and the Klein four group of reflections: $\operatorname{O}(1,3) = \operatorname{SO}^+(1,3) \rtimes \operatorname{K}_4$. The discussion concludes with the convenient representation theory of generic tensor...