§ 9. The Fundamental Equations in Lorentz’s Theory.

Let us now see how far the fundamental equations assumed by Lorentz correspond to the Relativity postulate, as defined in [§8]. In the article on Electron-theory (Ency., Math., Wiss., Bd. V. 2, Art 14) Lorentz has given the fundamental equations for any possible, even magnetised bodies (see there page 209, Eqn XXX′, formula (14) on page 78 of the same (part).

(IIIa″) Curl (H - [uE]) = J + dD/dt + u div D

- curl [uD].

(I″) div D = ρ

(IV″) curl E = - dB/dt, Div B = 0 (V′)

Then for moving non-magnetised bodies, Lorentz puts (page 223, 3) μ = 1, B = H, and in addition to that takes account of the occurrence of the di-electric constant ε, and conductivity σ according to equations

(εqXXXIV″, p. 327) D - E = (ε - 1) {E + [uB]}

(εqXXXIII′, p. 223) J = σ(E + [uB])

Lorentz’s E, D, H are here denoted by E, M, e, m while J denotes the conduction current.

The three last equations which have been just cited here coincide with eqn (II), (III), (IV), the first equation would be, if J is identified with C, = uρ (the current being zero for σ = 0,

(29) Curl [H - (u, E)] = C + dD/dt - curl [uD],

and thus comes out to be in a different form than (1) here. Therefore for magnetised bodies, Lorentz’s equations do not correspond to the Relativity Principle.

On the other hand, the form corresponding to the relativity principle, for the condition of non-magnetisation is to be taken out of (D) in [§8], with μ = 1, not as B = H, as Lorentz takes, but as (30) B - [uD] = H - [uD] (M - [uE] = m - [ue]. Now by putting H = B, the differential equation (29) is transformed into the same form as eqn (1) here when m - [ue] = M - [uE]. Therefore it so happens that by a compensation of two contradictions to the relativity principle, the differential equations of Lorentz for moving non-magnetised bodies at last agree with the relativity postulate.

If we make use of (30) for non-magnetic bodies, and put accordingly H = B + [u, (D - E)], then in consequence of (C) in [§8],

(ε - 1) (E + [u, B]) = D - E + [u. [u, D - E],

i.e. for the direction of u,

(ε - 1) (E + [uB])_u = (D - E)_u

and for a perpendicular direction ū,

(ε - 1) [E + (uB)]_u = (1 - u²) (D - E)_u

i.e. it coincides with Lorentz’s assumption, if we neglect u² in comparison to 1.

Also to the same order of approximation, Lorentz’s form for J corresponds to the conditions imposed by the relativity principle [comp. (E) [§ 8]]—that the components of J_u, J_ū are equal to the components of σ (E + [u B]) multiplied by √(1 - u²) or 1 / √(1 - u²) respectively.