THE INVARIANCE OF THE ELECTROMAGNETIC WAVE EQUATION UNDER LORENTZ TRANSFORMATIONS

Abstract

The origin of the theory of special relativity had as its main motivation the problem of incompatibility between Newtonian mechanics and Maxwell's electromagnetism. That is, the electromagnetic wave equations change shape when considering the well-known Galilean transformations. This is a problem, as physics cannot be different when moving from one frame of reference to another. This hindrance is solved when relativistic mechanics is proposed, based on Lorentz transformations. In this work we will show in detail how the electromagnetic wave equation in its general form is invariant under Lorentz transformations, that is, we will show that the form of the d’Alembertian operator does not change its form even without previously knowing how the fields transform. Before that, however, we reserve a section to demonstrate that such an equation varies its structure when Galilean transformations are applied. Despite the historical importance of these facts, the derivations shown in this work are not present in the main textbooks of Modern Physics or Special Relativity, which will allow them to serve as didactic material for Physics students.