Page:Grundgleichungen (Minkowski).djvu/41

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(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}}f Det^{\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