Page:Grundgleichungen (Minkowski).djvu/44

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The space-time vector of the first kind

(89) K = lor\ S

is of very great importance for which we now want to demonstrate a very important transformation

According to 78), S = L + fF, and it follows that

lor\ S = lor\ L + lor\ fF.

The symbol lor denotes a differential process which in lor fF, operates on the one hand upon the components of f, on the other hand also upon the components of F. Accordingly lor fF can be expressed as the sum of two parts. The first part is the product of the matrices (lor f)F, lor f being regarded as a 1✕4 series matrix. The second part is that part of lor fF, in which the diffentiations operate upon the components of F alone. From 78) we obtain

fF = -F^{*}f^{*} - 2L;

hence the second part of lor fF = -(lor\ F^{*})f^{*} + the part of -2 lor\ L, in which the differentiations operate upon the components of F alone. We thus obtain

(90) lor\ S = (lor\ f) F - (lor\ F^{*})f^{*} + N,

where N is the vector with the components

\begin{array}{r}
\left(N_{h}=\frac{1}{2}(\frac{\partial f_{23}}{\partial x_{h}}F_{23}+\frac{\partial f_{31}}{\partial x_{h}}F_{31}+\frac{\partial f_{12}}{\partial x_{h}}F_{12}+\frac{\partial f_{14}}{\partial x_{h}}F_{14}+\frac{\partial f_{24}}{\partial x_{h}}F_{24}+\frac{\partial f_{34}}{\partial x_{h}}F_{34}\right.\\
\\\left.-f_{23}\frac{\partial f_{23}}{\partial x_{h}}-f_{31}\frac{\partial f_{31}}{\partial x_{h}}-f_{12}\frac{\partial f_{12}}{\partial x_{h}}-f_{14}\frac{\partial f_{14}}{\partial x_{h}}-f_{24}\frac{\partial f_{24}}{\partial x_{h}}-\frac{\partial f_{34}}{\partial x_{h}}F_{34}\right)\end{array}
(h=1,2,3,4)

By using the fundamental relations A) and B), 90) is transformed into the fundamental relation

(91) lor\ S = - sF + N

In the limitting case \epsilon = 1,\ \mu = 1,\ f = F, N vanishes identically.

Now upon the basis of the equations (55) and (56), and referring back to the expression (82) for L, and from 57) we obtain the following expressions as components of N,—