Page:Grundgleichungen (Minkowski).djvu/43

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$S-\bar{S}=\left|S_{hk}-S_{kh}\right|$

an alternating matrix, and denotes a space-time vector of the second kind. From the expressions (83), we obtain,

(85)	$S-\bar{S}=-(\epsilon\mu-1)[w,\Omega],$

from which we deduce that [see (57), (58)].

(86)	$w(S-\bar{S})^{*}=0$,

(87)	$w(S-\bar{S})=(\epsilon\mu-1)\Omega$,

When the matter is at rest at a space-time point, $\mathfrak{w}=0$, then the equation 86) denotes the existence of the following equations

$Z_{y} = Y_{z},\ X_{z} = Z_{x},\ Y_{x} = X_{y}$;

and from 83),

$T_{x} = \Omega_{1},\ T_{y} = \Omega_{2},\ T_{z} = \Omega_{3}$ $X_{t} = \epsilon\mu\Omega_{1},\ Y_{t} = \epsilon\mu\Omega_{2},\ Z_{t} = \epsilon\mu\Omega_{3}$

Now by means of a rotation of the space co-ordinate system round the null-point, we can make,

$Z_{y} = Y_{z} = 0,\ X_{z} = Z_{x}= 0,\ Y_{x} = X_{y} = 0$;

According to 71), we have

(88)	$X_{x} + Y_{y} + Z_{z} + T_{t} = 0$

and according to 83), $T_{t} > 0$. In special eases, where $\Omega$ vanishes it follows from 81) that

$X^{2}_{x} = Y^{2}_{y} = Z^{2}_{z} = T^{2}_{t} = (Det^{\frac{1}{4}}S)^{2}$

and if $T_{t}$ and one of the three magnitudes $X_{x},\ Y_{y},\ Z_{z}$ are $=+Det^{\frac{1}{4}}S$, the two others $=-Det^{\frac{1}{4}}S$. If $\Omega$ does not vanish let $\Omega_{3} \ne 0$, then we have in particular from 80)

$T_{z}X_{t} = 0,\ T_{z}Y_{t} = 0,\ Z_{z}T_{z}+T_{z}Z_{t}=0$

and if $\Omega_{1} = 0,\ \Omega_{1} = 0,\ Z_{z} = -T_{t}$. It follows from (81), (see also 88) that

$X_{x} = - Y_{y} = \pm Det^{\frac{1}{4}}S$, $-Z_{z}=T_{t}=\sqrt{Det^{\frac{1}{2}}S+\epsilon\mu\Omega_{2}^{2}}>Det^{\frac{1}{4}}S$.