Page:Grundgleichungen (Minkowski).djvu/40

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§ 13. The Product of the Field-vectors fF.

Finally let us enquire about the laws which lead to the determination of the vector w as a function of x, y, z, t. In these investigations, the expressions which are obtained by the multiplication of two alternating matrices

f=\left|\begin{array}{cccc}
0, & f_{12}, & f_{13}, & f_{14}\\
f_{21}, & 0, & f_{23}, & f_{24}\\
f_{31}, & f_{32}, & 0, & f_{34}\\
f_{41}, & f_{42}, & f_{43}, & 0\end{array}\right|,\ F=\left|\begin{array}{cccc}
0, & F_{12}, & F_{13}, & F_{14}\\
F_{21}, & 0, & F_{23}, & F_{24}\\
F_{31}, & F_{32}, & 0, & F_{34}\\
F_{41}, & F_{42}, & F_{43}, & 0\end{array}\right|

are of much importance. Let us write.

(70) 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|

Then (71)

(71) S_{11} + S_{22} + S_{33} + S_{44} = 0

Let L now denote the symmetrical combination of the indices 1, 2, 3, 4, given by

(72) L=\frac{1}{2}(f_{23}F_{23}+f_{31}F_{31}+f_{12}F_{12}+f_{14}F_{14}+f_{24}F_{24}+f_{34}F_{34})

Then we shall have

(73) \begin{array}{c}
S_{11}=\frac{1}{2}(f_{23}F_{23}+f_{34}F_{34}+f_{42}F_{42}-f_{12}F_{12}-f_{13}F_{13}-f_{14}F_{14})\\
S_{12}=f_{13}F_{32}+f_{14}F_{42},\ u.s.f.\end{array}

In order to express in a real form, we write

(74) S=\left|\begin{array}{cccc}
S_{11}, & S_{12}, & S_{13}, & S_{14}\\
S_{21}, & S_{22}, & S_{23}, & S_{23}\\
S_{31}, & S_{32}, & S_{33}, & S_{34}\\
S_{41}, & S_{42}, & S_{43}, & S_{44}\end{array}\right|=\left|\begin{array}{cccc}
X_{x}, & Y_{x}, & Z_{x}, & -iT_{x}\\
X_{y}, & Y_{y}, & Z_{y}, & -iT_{y}\\
X_{z}, & Y_{z}, & Z_{z}, & -iT_{z}\\
-iX_{t}, & -iY_{t}, & -iZ_{t}, & T_{t}\end{array}\right|

Now