“Geometric paths” in the two-dimensional Minkowski space to obtain the Lorentz transformations and their consequences

Abstract

This article deals with an unusual way (barely used in the literature on Special Relativity) to arrive at the well-known Lorentz transformations from considerations coming from the two-dimensional Minkowski space. Its fundamental characteristic, coming from the non-Euclidean form of the element of length, is that hyperbolic rotations are used to move from one reference system to another; this supposes to leave the circles (usual in the Euclidean rotations) and to happen to the hyperbolas. Using this hyperbolic geometry and the relations for hyperbolic angles (equivalent and very similar to those for circular angles) Lorentz transformations are easily generated. To obtain the time dilation, the length contraction and the relativistic sum of velocities, the Minkowskian space and simple geometry are used. On the other hand, as is known, the well-known symmetry approach (via the Lorentz or Poincare group), is more powerful with tensorial and very physical analysis. Although this geometric approach is not as powerful, it is very simple and certainly has a pedagogical character especially for undergraduates. And, of course, if we go from two-dimensional space to four-dimensional space, this geometric path would become much more complicate.