On the electrodynamics of Minkowski at low velocities

The Galilean constitutive equations for the electrodynamics of moving media are derived for the first time. They explain all the historic and modern experiments which were interpreted so far in a relativistic framework assuming the constant light celerity principle. Here, we show the latter to be sufficient but not necessary. Copyright c © EPLA, 2008 One century ago, Hermann Minkowski formulated, for the first time, a covariant theory of electrodynamics in moving media [1–4]. He generalized the studies of Henri Poincaré [5] and Albert Einstein [6] which were restricted to vacuum. The theory was not only a consequence of the relativity principle formulated by Poincaré but also of the constant light celerity principle formulated by Einstein. Hence, it should have been (and was) applied to the optics of moving media [7] and to fast particles in media [8]. Moreover, it was thought to be the only possible explanation for the entire electrodynamics of moving media especially for slow motions [9]. The story almost ended with the last experiment of electrodynamics in moving media due to the Wilsons [10] which proved the correctness of the special theory of relativity against the older theories of Hertz, Lorentz and others whose fields transformations did not match with the group additivity as implied by the relativity principle. Nowadays, only some review papers appear from time to time dealing with the electrodynamics of moving media especially on the well-known Minkowski-Abraham controversy about the correct expression for the energy-momentum tensor in media [3,4,11]. However, in 1973, Michel Le Bellac and Jean-Marc Lévy-Leblond postulated the existence of a Galilean limit of Maxwell equations that is without assuming the finiteness of the light velocity. More cumbersome, they showed that, contrary to Mechanics, Classical Electromagnetism features TWO Galilean limits: one applies to dielectrics and the other to magnets [12]. Following our recent works on Galilean Electromagnetism [13–17], here we solve the problem of the electrodynamics of moving media at low velocities. For this purpose, we derive for the first time (a)E-mail: [email protected] the TWO Galilean limits of the relativistic constitutive equations introduced by Minkowski as long ago as 1908. We first recall from the textbooks the relativistic electrodynamics of moving media before taking its Galilean limits. All the experiments of Classical Electromagnetism involving motion of part of the setup are described by this “new” theory as soon as velocities do not reach the celerity of light. It is useless to speak of applications since this theory encompasses all our wave-less technology. Needless to add that the Galilean theory is simpler than the relativistic theory. . . Relativistic electrodynamics of moving media. – The relativistic form of Maxwell equation in continuous media is written as ∇×E =−∂tB, Faraday, ∇·B = 0, Thomson, ∇×H = j+ ∂tD, Ampere, ∇·D= ρ, Gauss, (1) This set is the so-called “Maxwell-Minkowski equations”. A Poincaré-Lorentz transformation (without rotation) acts on space-time coordinates as follows (see, for instance, Section 7.2 of [18]): x′ = x− γvt+(γ− 1) ·x) v2 , t′ = γ ( t− v ·x c2 ) , (2) where v is the relative velocity and γ = 1 √ 1− (v/c)2 . Under a Poincaré-Lorentz-Minkoswki transformation, eq. (2), the electric and magnetic fields and their related