Page:Grundgleichungen (Minkowski).djvu/41

(75)

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

(76)	$L={\frac {1}{2}}({\mathfrak {m}}_{x}{\mathfrak {M}}_{x}+{\mathfrak {m}}_{y}{\mathfrak {M}}_{y}+{\mathfrak {m}}_{z}{\mathfrak {M}}_{z}-{\mathfrak {e}}_{x}{\mathfrak {E}}_{x}-{\mathfrak {e}}_{y}{\mathfrak {E}}_{y}-{\mathfrak {e}}_{z}{\mathfrak {E}}_{z})$,

These quantities are all real. In the theory for bodies at rest, the combinations ($X_{x},\ X_{y},\ X_{z},\ Y_{x},\ Y_{y},\ Y_{z},\ Z_{x},\ Z_{y},\ Z_{z}$ are known as Maxwell's Stresses", $T_{x},\ T_{y},\ T_{z}$ are known as the Poynting's Vector, $T_{t}$ 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)	$F^{*}f^{*}=\left|{\begin{array}{llll}-S_{11}-L,&-S_{12},&-S_{13},&-S_{14}\\-S_{21},&-S_{22}-L,&-S_{23},&-S_{23}\\-S_{31},&-S_{32},&-S_{33}-L,&-S_{34}\\-S_{41},&-S_{42},&-S_{43},&-S_{44}-L\end{array}}\right|$

and hence, we can put

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

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

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

Since here $SL=LS$, we deduce that,

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

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

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

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