The Lorentz Group, a Galilean Approach

We review the Inertial transformation and Lorentz transformation under a new context, by using Clifford Algebra or Geometric Algebra. The apparent contradiction between theses two approach is simply stems from different procedures for clock synchronization associated with different choices of the coordinates used to describe the physical world. We find the physical and coordinates components of both transformations. A important result is that in the case of Inertial transform...

A relativistic particle undergoing successive boosts which are non collinear will experience a rotation of its coordinate axes with respect to the boosted frame. This rotation of coordinate axes is caused by a relativistic phenomenon called Thomas Rotation. We assess the importance of Thomas rotation in the calculation of physical quantities like electromagnetic fields in the relativistic regime. We calculate the electromagnetic field tensor for general three dimensional succ...

I review, some of the algebraic and geometric structures that underlie the theory of Special Relativity. This includes a discussion of relativity as a symmetry principle, derivations of the Lorentz group, its composition law, its Lie algebra, comparison with the Galilei group, Einstein synchronization, the lattice of causally and chronologically complete regions in Minkowski space, rigid motion (the Noether-Herglotz theorem), and the geometry of rotating reference frames. Rep...

The Lorentz transformation is entirely derived from length contraction, itself established through the known light-clock thought experiment . This makes the derivation accessible to beginning students once Eintein's two postulates have been admitted. The formula derived for the space part of the general rotation-free Lorentz transformation is very compact and allows for an easy derivation of the velocity and acceleration transformations. A possibly new and very simple relatio...

We demonstrate how to derive Maxwell's equations, including Faraday's law and Maxwell's correction to Amp\`ere's law, by generalizing the description of static electromagnetism to dynamical situations. Thereby, Faraday's law is introduced as a consequence of the relativity principle rather than an experimental fact, in contrast to the historical course and common textbook presentations. As a by-product, this procedure yields explicit expressions for the infinitesimal Lorentz ...

It is common in the literature on classical electrodynamics and relativity theory that the transformation rules for the basic electrodynamic quantities are derived from the pre-assumption that the equations of electrodynamics are covariant against these---unknown---transformation rules. There are several problems to be raised concerning these derivations. This is, however, not our main concern in this paper. Even if these derivations are regarded as unquestionable, they leave...

We present a didactic derivation of the special theory of relativity in which Lorentz transformations are `discovered' as symmetry transformations of the Klein-Gordon equation. The interpretation of Lorentz boosts as transformations to moving inertial reference frames is not assumed at the start, but it naturally appears at a later stage. The relative velocity $\textbf{v}$ of two inertial reference frames is defined in terms of the elements of the pertinent Lorentz matrix, an...

We report the simplest possible form to compute rotations around arbitrary axis and boosts in arbitrary directions for 4-vectors (space-time points, energy-momentum) and bi-vectors (electric and magnetic field vectors) by symplectic similarity transformations. The Lorentz transformations are based exclusively on real $4\times 4$-matrices and require neither complex numbers nor special implementations of abstract entities like quaternions or Clifford numbers. No raising or low...

This paper is devoted to introduce a gauge theory of the Lorentz Group based on the analysis of isometric diffeomorphism-induced Lorentz transformations. The behaviors under local transformations of fermion fields and spin connections (assumed to be coordinate vectors) are analyzed and the role of the torsion field in a curved space-time is discussed.

In order to ask for future concepts of relativity, one has to build upon the original concepts instead of the nowadays common formalism only, and as such recall and reconsider some of its roots in geometry. So in order to discuss 3-space and dynamics, we recall briefly Minkowski's approach in 1910 implementing the nowadays commonly used 4-vector calculus and related tensorial representations as well as Klein's 1910 paper on the geometry of the Lorentz group. To include micros...