Page:Grundgleichungen (Minkowski).djvu/33

 and further

Because F is an alternating matrix,

i.e. $$\Phi$$ is perpendicular to the vector to w; we can also write

I shall call the space-time vector $$\Phi$$ of the first kind as the Electric Rest Force.

Relations analogous to those holding between $$-wF,\ \mathfrak{E,\ M,\ w}$$, hold amongst $$-wf,\ \mathfrak{e,\ m,\ w}$$, and in particular -wf is normal to w. The relation (C) can be written as

The expression (wf) gives four components, but the fourth can be derived from the first three.

Let us now form the time-space vector 1st kind $$\Psi=iwf^{*}$$, whose components are

$$\begin{array}{cccccccccc} \Psi_{1} & = & -i( & &  & w_{2}f_{34} & + & w_{3}f_{42} & + & w_{4}f_{23}),\\ \Psi_{2} & = & -i( & w_{1}f_{43} & &  & + & w_{3}f_{14} & + & w_{4}f_{31}),\\ \Psi_{3} & = & -i( & w_{1}f_{24} & + & w_{2}f_{41} & &  & + & w_{4}f_{12}),\\ \Psi_{4} & = & -i( & w_{1}f_{32} & + & w_{2}f_{13} & + & w_{3}f_{21} & & ).\end{array}$$

Of these, the first three $$\Psi_{1},\ \Psi_{2},\ \Psi_{3}$$ are the x-, y-, z-components of the space-vector

and further

Among these there is the relation

which can also be written as