# Page:Grundgleichungen (Minkowski).djvu/32

The sum of the two space time vectors of the second kind on the left side is to be understood in the sense of the addition of two alternating matrices.

For example, for ${\displaystyle w_{1}=0,\ w_{2}=0,\ w_{3}=0,\ w_{4}=i}$

 ${\displaystyle wf=\left|if_{41},\ if_{42},\ if_{43},\ 0\right|;\ wf^{*}=\left|if_{32},\ if_{13},\ if_{21},\ 0\right|}$; ${\displaystyle [w,wf]=0,0,0,f_{41},\ f_{42},\ f_{43};\ [w,wf^{*}]=0,0,0,\ f_{32},\ f_{13},\ f_{21}}$;

The fact that in this special case, the relation is satisfied, suffices to establish the theorem (45) generally, for this relation has a covariant character in case of a Lorentz transformation, and is homogeneous in ${\displaystyle w_{1},\ w_{2},\ w_{3},\ w_{4}}$.

After these preparatory works let us engage ourselves with the equations (C,) (D,) (E) by means which the constants ${\displaystyle \epsilon ,\ \mu ,\ \sigma }$ will be introduced.

Instead of the space vector ${\displaystyle {\mathfrak {w}}}$, the velocity of matter, we shall introduce the space-time vector of the first kind w with the components.

${\displaystyle w_{1}={\frac {{\mathfrak {w}}_{x}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{2}={\frac {{\mathfrak {w}}_{y}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{3}={\frac {{\mathfrak {w}}_{z}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{4}={\frac {i}{\sqrt {1-{\mathfrak {w}}^{2}}}}}$

where

 (46) ${\displaystyle w{\overline {w}}=w_{1}^{2}+w_{2}^{2}+w_{3}^{2}+w_{4}^{2}=-1}$

and ${\displaystyle -iw_{4}>0}$..

By F and f shall be understood the space time vectors of the second kind ${\displaystyle {\mathfrak {M}},\ -i{\mathfrak {E}}}$, ${\displaystyle {\mathfrak {m}},\ -i{\mathfrak {e}}}$.

In ${\displaystyle \Phi =-wF}$, we have a space time vector of the first kind with components

${\displaystyle {\begin{array}{ccccccccc}\Phi _{1}&=&&&w_{2}F_{12}&+&w_{3}F_{13}&+&w_{4}F_{14},\\\Phi _{2}&=&w_{1}F_{21}&&&+&w_{3}F_{23}&+&w_{4}F_{24},\\\Phi _{3}&=&w_{1}F_{31}&+&w_{2}F_{32}&&&+&w_{4}F_{34},\\\Phi _{4}&=&w_{1}F_{41}&+&w_{2}F_{42}&+&w_{3}F_{43}.\end{array}}}$.

The first three quantities ${\displaystyle \Phi _{1},\ \Phi _{2},\ \Phi _{3}}$ are the components of the space-vector

 (47) ${\displaystyle {\frac {{\mathfrak {E}}+[{\mathfrak {wM}}]}{\sqrt {1-{\mathfrak {w}}^{2}}}}}$,