# Page:Grundgleichungen (Minkowski).djvu/33

and further

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

Because F is an alternating matrix,

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

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

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

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

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

 {C} ${\displaystyle wf=\epsilon wF}$

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 ${\displaystyle \Psi =iwf^{*}}$, whose components are

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

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

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

and further

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

Among these there is the relation

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

which can also be written as