Page:Grundgleichungen (Minkowski).djvu/42

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and have the value given on the right-hand side of (79). Therefore the general relations

(80) S_{h1}S_{1k} + S_{h2}S_{2k} + S_{h3}S_{3k} + S_{h4}S_{4k} = 0

h, k being unequal indices in the series 1, 2, 3, 4, and

(81) S_{h1}S_{1h} + S_{h2}S_{2h} + S_{h3}S_{3h} + S_{h4}S_{4h} = L^{2} - Det^{\frac{1}{2}}f Det^{\frac{1}{2}}F

for h = 1,2,3,4.

Now if instead of F and f in the combinations (72) and (73), we introduce the electrical rest-force \Phi, the magnetic rest-force \Psi and the rest-ray \Omega [(55), (56) and (57)], we can pass over to the expressions, —

(82) L=-\frac{1}{2}\epsilon\Phi\bar{\Phi}+\frac{1}{2}\mu\Psi\bar{\Psi},
(83) S_{hk}=-\frac{1}{2}\epsilon\Phi\bar{\Phi}e_{hk}-\frac{1}{2}\mu\Psi\bar{\Psi}e_{hk}

+\epsilon(\Phi_{h}\Phi_{k}-\Phi\bar{\Phi}w_{h}w_{k})+\mu(\Psi_{h}\Psi_{k}-\Psi\bar{\Psi}w_{h}w_{k})

-\Omega_{h}w_{k}-\epsilon\mu w_{h}\Omega_{k}\qquad\qquad  (h, k = 1,2,3,4)

Here we have

\Phi\bar{\Phi}=\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}+\Phi_{4}^{2},\ \Psi\bar{\Psi}=\Psi_{1}^{2}+\Psi_{2}^{2}+\Psi_{3}^{2}+\Psi_{4}^{2},

e_{hh}=1,\ e_{hk}=0(h\gtrless k).

The right side of (82) as well as L is an invariant in a Lorentz transformation, and the 4✕4 element on the right side of (83) as well as S_{hk}, represent a space time vector of the second kind. Remembering this fact, it suffices, for establishing the theorems (82) and (83) generally, to prove it for the special case w_{1} = 0,\ w_{2} = 0,\ w_{3} = 0,\ w_{4} = i. But for this case \mathfrak{w} = 0, we immediately arrive at the equations (82) and (83) by means (45), (51), (60) on the one hand, and \mathfrak{e}=\epsilon\mathfrak{E},\ \mathfrak{M}=\mu\mathfrak{m} on the other hand.

The expression on the right-hand side of (81), which equals

=\left(\frac{1}{2}(\mathfrak{mM}-\mathfrak{eE})\right)^{2}+(\mathfrak{em})(\mathfrak{EM})

is \geqq0, because (\mathfrak{em})=\epsilon\Phi\bar{\Psi},\ (\mathfrak{EM})=\mu\Phi\bar{\Psi}, now referring back to 79), we can denote the positive square root of this expression as Det^{\frac{1}{2}}S.

Since \bar{f}=-f,\ \bar{F}=-F, we obtain for \bar{S}, the transposed matrix of S, the following relations from (78),

(84) Ff=\bar{S}-L,\ f^{*}F^{*}=-\bar{S}-L.

Then is