Page:Grundgleichungen (Minkowski).djvu/43

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

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

$X_{x}^{2}=Y_{y}^{2}=Z_{z}^{2}=T_{t}^{2}=(Det^{\frac {1}{4}}S)^{2}$

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 $\Omega _{3}\neq 0$, 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

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