# Page:Grundgleichungen (Minkowski).djvu/35

 (58) ${\displaystyle |\Phi \Psi ]=i[w,\Omega ]^{*}}$,

i,e.

${\displaystyle \Phi _{1}\Psi _{2}-\Phi _{2}\Psi _{1}=i(w_{3}\Omega _{4}-w_{4}\Omega _{3})}$, etc.

.

The vector ${\displaystyle \Omega }$ fulfills the relation

 (59) ${\displaystyle (w{\bar {\Omega }})=w_{1}\Omega _{1}+w_{2}\Omega _{2}+w_{3}\Omega _{3}+w_{4}\Omega _{4}=0}$,

which we can write as

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

and ${\displaystyle \Omega }$ is also normal to w. In case ${\displaystyle {\mathfrak {w}}=0}$, we have ${\displaystyle \Phi _{4}=0,\ \Psi _{4}=0,\ \Omega _{4}=0}$, and

 (60) ${\displaystyle \Omega _{1}=\Phi _{2}\Psi _{3}-\Phi _{3}\Psi _{2},\ \Omega _{2}=\Phi _{3}\Psi _{1}-\Phi _{1}\Psi _{3},\ \Omega _{3}=\Phi _{1}\Psi _{2}-\Phi _{2}\Psi _{1}}$,

I shall call ${\displaystyle \Omega }$, which is a space-time vector 1st kind the Rest-Ray.

As for the relation E), which introduces the conductivity ${\displaystyle \sigma }$, we have

${\displaystyle -w{\bar {s}}=-(w_{1}s_{1}+w_{2}s_{2}+w_{3}s_{3}+w_{4}s_{4})={\frac {-\left|{\mathfrak {w}}\right|s_{\mathfrak {w}}+\varrho }{\sqrt {1-{\mathfrak {w}}^{2}}}}=\varrho '}$

This expression gives us the rest-density of electricity (see §8 and §4). Then

 (61) ${\displaystyle s+(w{\bar {s}})w}$

represents a space-time vector of the 1st kind, which since ${\displaystyle w{\bar {w}}=1}$, is normal to w, and which I may call the rest-current. Let us now conceive of the first three component of this vector as the x-, y-, z co-ordinates of the space-vector, then the component in the direction of ${\displaystyle {\mathfrak {w}}}$ is

${\displaystyle {\mathfrak {s_{w}}}-{\frac {\left|{\mathfrak {w}}\right|\varrho '}{\sqrt {1-{\mathfrak {w}}^{2}}}}={\frac {{\mathfrak {s_{w}}}-\left|{\mathfrak {w}}\right|\varrho }{\sqrt {1-{\mathfrak {w}}^{2}}}}={\frac {\mathfrak {F_{w}}}{1-{\mathfrak {w}}^{2}}}}$

and the component in a perpendicular direction is ${\displaystyle {\mathfrak {s_{\bar {w}}}}={\mathfrak {F_{\bar {w}}}}}$.

This space-vector is connected with the space-vector ${\displaystyle {\mathfrak {F}}={\mathfrak {s}}-\varrho {\mathfrak {w}}}$, which we denoted in § 8 as the conduction-current.

Now by comparing with ${\displaystyle \Phi =-wF}$, the relation (E) can be brought into the form

 (E) ${\displaystyle s+(w{\bar {s}})w=-\sigma wF}$.