Page:Grundgleichungen (Minkowski).djvu/33

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and further

(48) \Phi_{4}=\frac{i[\mathfrak{wE}]}{\sqrt{1-\mathfrak{w}^{2}}},

Because F is an alternating matrix,

(49) w\overline{\Phi}=w_{1}\Phi_{1}+w_{2}\Phi_{2}+w_{3}\Phi_{3}+w_{4}\Phi_{4}=0,

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

(50) \Phi_{4}=i(\mathfrak{w}_{x}\Phi_{1}+\mathfrak{w}_{y}\Phi_{2}+\mathfrak{w}_{z}\Phi_{3}),

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

{C} wf = \epsilon wF

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

(51) \frac{\mathfrak{m}-[\mathfrak{we}]}{\sqrt{1-\mathfrak{w}^{2}}},

and further

(52) \Psi_{4}=\frac{i[\mathfrak{wm}]}{\sqrt{1-\mathfrak{w}^{2}}};

Among these there is the relation

(53) w\overline{\Psi}=w_{1}\Psi_{1}+w_{2}\Psi_{2}+w_{3}\Psi_{3}+w_{4}\Psi_{4}=0,

which can also be written as