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.