After introducing the metric in classical Euclidean space, we move on to Minkowski's concept of four-vector in space-time, dealing with topics of restricted relativity, such as Lorentz transformations and Lorentz invariants. Relativistic expressions for the total energy of a free particle, energy at rest and kinetic energy are obtained, showing also their non-relativistic limits and the so-called mass-shell relation. Subsequently, from the point of view of relativistic kinematics, the decay of particles is analyzed, in particular showing the impossibility of the decay of a free photon and analyzing the decay of the muon. The electromagnetic field tensor is introduced, with components related to those of the electric and magnetic field vectors, and it is calculated its transformation between inertial reference frames. It is explicitly shown, as an example, that in a reference frame in motion with respect to an electric charge, the latter generates both an electric and a magnetic field which depend on time. Finally, Maxwell's equations are treated, both in differential and in covariant form, showing how to obtain the equation of electromagnetic waves in vacuum. 

Author: dr. Alessio Mangoni, PhD, theoretical physicist.