M. Fabbrichesi , SISSA

Covariance of the Maxwell's equations

In this lessons the definition of four-vectors and of scalar product between four-vectors is given and this leads to the Minkowski four-dimensional space whose metric is ds^{2}=η_{ab}dx^{a}dx^{b}=-c^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}. In particular, it is given the definition of the 4-velocity u^{a }and of the four-momentum p^{a}. Next, a few homeworks are suggested to students, e.g. demonstrate that E^{2}=c^{2}(m^{2}c^{2}+**p**^{2}). Next the acceleration four-vector is introduced by discussing an example: a rocket with g acceleration. In particular, the solution of the uniformly accelerated motion is found.

Next, the Lorentz's force is written in a covariant form by introducing a 4-tensor *F*^{αβ}, i.e. the electromagnetic 4-tensor. The components of this antisymmetric tensor are given by the components of the electric and magnetic fields. This allows us to write Maxwell's equations in a very compact form [Jackson sec. 11.9]. Moreover this result together with the definitions of the 4-vectors *J*^{α }and *A*^{α }establishes the covariance of the equations of electromagnetism.