# Page:Grundgleichungen (Minkowski).djvu/41

 (75) ${\displaystyle X_{x}={\frac {1}{2}}({\mathfrak {m}}_{x}{\mathfrak {M}}_{x}-{\mathfrak {m}}_{y}{\mathfrak {M}}_{y}-{\mathfrak {m}}_{z}{\mathfrak {M}}_{z}+{\mathfrak {e}}_{x}{\mathfrak {E}}_{x}-{\mathfrak {e}}_{y}{\mathfrak {E}}_{y}-{\mathfrak {e}}_{z}{\mathfrak {E}}_{z})}$, ${\displaystyle X_{y}={\mathfrak {m}}_{x}{\mathfrak {M}}_{y}+{\mathfrak {e}}_{y}{\mathfrak {E}}_{x},\ Y_{x}={\mathfrak {m}}_{y}{\mathfrak {M}}_{x}+{\mathfrak {e}}_{x}{\mathfrak {E}}_{y}}$, u.s.f. ${\displaystyle X_{t}={\mathfrak {e}}_{y}{\mathfrak {M}}_{z}-{\mathfrak {e}}_{z}{\mathfrak {M}}_{y}}$, ${\displaystyle T_{x}={\mathfrak {m}}_{z}{\mathfrak {E}}_{y}-{\mathfrak {m}}_{y}{\mathfrak {E}}_{z}}$, u.s.f. ${\displaystyle T_{t}={\frac {1}{2}}({\mathfrak {m}}_{x}{\mathfrak {M}}_{x}+{\mathfrak {m}}_{y}{\mathfrak {M}}_{y}+{\mathfrak {m}}_{z}{\mathfrak {M}}_{z}+{\mathfrak {e}}_{x}{\mathfrak {E}}_{x}+{\mathfrak {e}}_{y}{\mathfrak {E}}_{y}+{\mathfrak {e}}_{z}{\mathfrak {E}}_{z})}$,
 (76) ${\displaystyle L={\frac {1}{2}}({\mathfrak {m}}_{x}{\mathfrak {M}}_{x}+{\mathfrak {m}}_{y}{\mathfrak {M}}_{y}+{\mathfrak {m}}_{z}{\mathfrak {M}}_{z}-{\mathfrak {e}}_{x}{\mathfrak {E}}_{x}-{\mathfrak {e}}_{y}{\mathfrak {E}}_{y}-{\mathfrak {e}}_{z}{\mathfrak {E}}_{z})}$,

These quantities are all real. In the theory for bodies at rest, the combinations (${\displaystyle X_{x},\ X_{y},\ X_{z},\ Y_{x},\ Y_{y},\ Y_{z},\ Z_{x},\ Z_{y},\ Z_{z}}$ are known as Maxwell's Stresses", ${\displaystyle T_{x},\ T_{y},\ T_{z}}$ are known as the Poynting's Vector, ${\displaystyle T_{t}}$ as the electromagnetic energy-density, and L as the Langrangian function.

On the other hand, by multiplying the alternating matrices of f and F, we obtain

 (77) ${\displaystyle F^{*}f^{*}=\left|{\begin{array}{llll}-S_{11}-L,&-S_{12},&-S_{13},&-S_{14}\\-S_{21},&-S_{22}-L,&-S_{23},&-S_{23}\\-S_{31},&-S_{32},&-S_{33}-L,&-S_{34}\\-S_{41},&-S_{42},&-S_{43},&-S_{44}-L\end{array}}\right|}$

and hence, we can put

 (78) ${\displaystyle fF=S-L,\ F^{*}f^{*}=-S-L}$,

where by L, we mean L-times the unit matrix, i.e. the matrix with elements

${\displaystyle \left|Le_{hk}\right|\ \left({\begin{array}{c}e_{hh}=1,\ e_{hk}=0,\ h\gtrless k\\h,k=1,2,3,4\end{array}}\right)}$

Since here ${\displaystyle SL=LS}$, we deduce that,

${\displaystyle F^{*}f^{*}fF=(-S-L)(S-L)=-SS+L^{2}}$,

and find, since ${\displaystyle f^{*}f=Det^{\frac {1}{2}}f,\ F^{*}F=Det{\frac {1}{2}}F}$, we arrive at the interesting conclusion

 (79) ${\displaystyle SS=L^{2}-Det^{\frac {1}{2}}fDet^{\frac {1}{2}}F}$,

i.e. the product of the matrix S into itself can be expressed as the multiple of a unit matrix — a matrix in which all the elements except those in the principal diagonal are zero, the elements in the principal diagonal are all equal