# Page:Grundgleichungen (Minkowski).djvu/43

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

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

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

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

 (86) ${\displaystyle w(S-{\bar {S}})^{*}=0}$,
 (87) ${\displaystyle w(S-{\bar {S}})=(\epsilon \mu -1)\Omega }$,

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

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

and from 83),

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

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

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

According to 71), we have

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

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

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

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

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

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

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