Derivation of the Lorentz Transformation from the Maxwell Equations

Abstract

The Special Theory of Relativity had been established nearly one century ago to conciliate some seemingly contradictory concepts and experimental results such as the Ether, universal time, contraction of dimensions of moving bodies, absolute motion of the Earth, speed of the light, etc. Hence the fundamental revolutionary formulas of the Theory, i.e., the Lorentz Formulas, had been derived first by Einstein by dwelling on a postulate which stipulated the constancy of the speed of the light. To this end he had first postulated that every reference system has a time proper to itself and then redefined the notions of simultaneity, synchronous clocks, time interval, the length of a rod in a system at rest, the length in a moving system, etc. A second postulate of Einstein, which stated that every physical theory is invariant under the Lorentz transformation, enabled him to claim that the Theory of Electromagnetism is correct but the Newtonian Mechanics has to be re-established. Since then the Theory was almost always presented in this way by both Einstein and others except only a few. The aim of this paper is to show that the Lorentz formulas can be derived from the Maxwell equations if one pstulates that the total electric charge of an isolated body does not change if it is in motion. To this end one dwells only on the permanence principle of functional equations, which is not a physical but purely mathematical concept. Thus, from one side the Special Relativity becomes a natural issue (or a part) of the Maxwell Theory and, from the other side, the derivation of the transformation rules pertinent to the electromagnetic field becomes straightforward and easy.