# Page:Grundgleichungen (Minkowski).djvu/42

and have the value given on the right-hand side of (79). Therefore the general relations

 (80) ${\displaystyle S_{h1}S_{1k}+S_{h2}S_{2k}+S_{h3}S_{3k}+S_{h4}S_{4k}=0}$

h, k being unequal indices in the series 1, 2, 3, 4, and

 (81) ${\displaystyle S_{h1}S_{1h}+S_{h2}S_{2h}+S_{h3}S_{3h}+S_{h4}S_{4h}=L^{2}-Det^{\frac {1}{2}}fDet^{\frac {1}{2}}F}$

for h = 1,2,3,4.

Now if instead of F and f in the combinations (72) and (73), we introduce the electrical rest-force ${\displaystyle \Phi }$, the magnetic rest-force ${\displaystyle \Psi }$ and the rest-ray ${\displaystyle \Omega }$ [(55), (56) and (57)], we can pass over to the expressions, —

 (82) ${\displaystyle L=-{\frac {1}{2}}\epsilon \Phi {\bar {\Phi }}+{\frac {1}{2}}\mu \Psi {\bar {\Psi }}}$,
 (83) ${\displaystyle S_{hk}=-{\frac {1}{2}}\epsilon \Phi {\bar {\Phi }}e_{hk}-{\frac {1}{2}}\mu \Psi {\bar {\Psi }}e_{hk}}$ ${\displaystyle +\epsilon (\Phi _{h}\Phi _{k}-\Phi {\bar {\Phi }}w_{h}w_{k})+\mu (\Psi _{h}\Psi _{k}-\Psi {\bar {\Psi }}w_{h}w_{k})}$ ${\displaystyle -\Omega _{h}w_{k}-\epsilon \mu w_{h}\Omega _{k}\qquad \qquad (h,k=1,2,3,4)}$

Here we have

 ${\displaystyle \Phi {\bar {\Phi }}=\Phi _{1}^{2}+\Phi _{2}^{2}+\Phi _{3}^{2}+\Phi _{4}^{2},\ \Psi {\bar {\Psi }}=\Psi _{1}^{2}+\Psi _{2}^{2}+\Psi _{3}^{2}+\Psi _{4}^{2}}$, ${\displaystyle e_{hh}=1,\ e_{hk}=0(h\gtrless k)}$.

The right side of (82) as well as L is an invariant in a Lorentz transformation, and the 4✕4 element on the right side of (83) as well as ${\displaystyle S_{hk}}$, represent a space time vector of the second kind. Remembering this fact, it suffices, for establishing the theorems (82) and (83) generally, to prove it for the special case ${\displaystyle w_{1}=0,\ w_{2}=0,\ w_{3}=0,\ w_{4}=i}$. But for this case ${\displaystyle {\mathfrak {w}}=0}$, we immediately arrive at the equations (82) and (83) by means (45), (51), (60) on the one hand, and ${\displaystyle {\mathfrak {e}}=\epsilon {\mathfrak {E}},\ {\mathfrak {M}}=\mu {\mathfrak {m}}}$ on the other hand.

The expression on the right-hand side of (81), which equals

${\displaystyle =\left({\frac {1}{2}}({\mathfrak {mM}}-{\mathfrak {eE}})\right)^{2}+({\mathfrak {em}})({\mathfrak {EM}})}$

is ${\displaystyle \geqq 0}$, because ${\displaystyle ({\mathfrak {em}})=\epsilon \Phi {\bar {\Psi }},\ ({\mathfrak {EM}})=\mu \Phi {\bar {\Psi }}}$, now referring back to 79), we can denote the positive square root of this expression as ${\displaystyle Det^{\frac {1}{2}}S}$.

Since ${\displaystyle {\bar {f}}=-f,\ {\bar {F}}=-F}$, we obtain for ${\displaystyle {\bar {S}}}$, the transposed matrix of S, the following relations from (78),

 (84) ${\displaystyle Ff={\bar {S}}-L,\ f^{*}F^{*}=-{\bar {S}}-L}$.

Then is