# Page:Grundgleichungen (Minkowski).djvu/37

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

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

where

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

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

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

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

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

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

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

so that, we have symbolically

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

Therefore it follows that

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

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