# Page:Grundgleichungen (Minkowski).djvu/37

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For this matrix I shall use the shortened from lor.

Then if S is, as in (62), a space-time matrix of the II. kind, by lor S' will be understood the 1✕4 series matrix

${\displaystyle \left|K_{1},\ K_{2},\ K_{3},\ K_{4}\right|}$

where

 (64) ${\displaystyle K_{k}={\frac {\partial S_{1k}}{\partial x_{1}}}+{\frac {\partial S_{2k}}{\partial x_{2}}}+{\frac {\partial S_{3k}}{\partial x_{3}}}+{\frac {\partial S_{4k}}{\partial x_{4}}}\qquad (k=1,2,3,4)}$

When by a Lorentz transformation ${\displaystyle {\mathsf {A}}}$, a new reference system ${\displaystyle x'_{1},\ x'_{2},\ x'_{3},\ x'_{4}}$ is introduced, we can use the operator

${\displaystyle lor'=\left|{\frac {\partial }{\partial x'_{1}}},\ {\frac {\partial }{\partial x'_{2}}},\ {\frac {\partial }{\partial x'_{3}}},\ {\frac {\partial }{x'_{4}}}\right|}$

Then S is transformed to ${\displaystyle S'={\bar {\mathsf {A}}}S{\mathsf {A}}=\left|S'_{hk}\right|}$, so by lor' Sis meant the 1✕4 series matrix, whose element are

${\displaystyle K'_{k}={\frac {\partial S'_{1k}}{\partial x'_{1}}}+{\frac {\partial S'_{2k}}{\partial x'_{2}}}+{\frac {\partial S'_{3k}}{\partial x'_{3}}}+{\frac {\partial S'_{4k}}{\partial x'_{4}}}\qquad (k=1,2,3,4)}$

Now for the differentiation of any function of (x y z t) we have the rule

 ${\displaystyle {\frac {\partial }{\partial x'_{k}}}={\frac {\partial }{\partial x_{1}}}{\frac {\partial x_{1}}{\partial x'_{k}}}+{\frac {\partial }{\partial x_{2}}}{\frac {\partial x_{2}}{\partial x'_{k}}}+{\frac {\partial }{\partial x_{3}}}{\frac {\partial x_{3}}{\partial x'_{k}}}+{\frac {\partial }{\partial x_{4}}}{\frac {\partial x_{4}}{\partial x'_{k}}}}$ ${\displaystyle ={\frac {\partial }{\partial x_{1}}}\alpha _{1k}+{\frac {\partial }{\partial x_{2}}}\alpha _{2k}+{\frac {\partial }{\partial x_{3}}}\alpha _{3k}+{\frac {\partial }{\partial x_{4}}}\alpha _{4k}}$,

so that, we have symbolically

${\displaystyle lor'=lor\ ({\mathsf {A}}}$

Therefore it follows that

 (65) ${\displaystyle lor'\ S'=lor({\mathsf {A}}({\mathsf {A}}^{-1}S{\mathsf {A}}))=(lor\ S){\mathsf {A}}}$,

i.e., lor S behaves like a space-time vector of the first kind.

If L is a multiple of the unit matrix, then by lor L will be denoted the matrix with the elements