Page:Grundgleichungen (Minkowski).djvu/24

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§ 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, Eq. XXX', formula (14) on page 78 of the same (part).

\begin{array}{rcrl}
(IIIa'') & \qquad & curl\ (\mathfrak{H}-[\mathfrak{wE}] & =\mathfrak{F}+\frac{\partial\mathfrak{D}}{\partial t}+\mathfrak{w}\ div\ \mathfrak{D}-curl[\mathfrak{wD}],\\
\\(I'') &  & div\ \mathfrak{D} & =\varrho,\\
\\(IV'') &  & curl\ \mathfrak{E} & =-\frac{\partial\mathfrak{B}}{\partial t},\\
\\(V'') &  & div\ \mathfrak{B} & =0.\end{array}

Then for moving non-magnetised bodies, Lorentz puts (page 223, 3) \mu = 1, \mathfrak{B}=\mathfrak{H}, and in addition to that takes account of the occurrence of the di-electric constant \epsilon, and conductivity \sigma according to equations

(Eq. XXXIV"', p. 227)


(Eq. XXXIII", p. 223)
\begin{array}{rl}
\mathfrak{D}-\mathfrak{E} & =\left(\epsilon-1\right)\left(\mathfrak{E}+[\mathfrak{wB}]\right)\\
\\\mathfrak{F} & =\sigma(\mathfrak{E}+[\mathfrak{wB}])\end{array}

Lorentz's \mathfrak{E,B,D,H} are here denoted by \mathfrak{E,M,e,m} while \mathfrak{F} denotes the conduction current.

The three last equations which have been just cited here coincide with eq. (II), (III), (IV), the first equation would be, if \mathfrak{F} is identified with \mathfrak{s}-\mathfrak{w}\sigma (the current being zero for \sigma = 0),

(29) curl\ (\mathfrak{H}-[\mathfrak{wE}])=\mathfrak{s}+\frac{\partial\mathfrak{D}}{\partial t}-curl[\mathfrak{wD}]

and thus comes out to in 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 \mu=1, not as \mathfrak{B}=\mathfrak{H}, as Lorentz takes, but as

(30) \mathfrak{B}-[\mathfrak{wE}]=\mathfrak{H}-[\mathfrak{wD}] (hier \mathfrak{M}-[\mathfrak{wE}]=\mathfrak{m}-[\mathfrak{we}])