# Page:Grundgleichungen (Minkowski).djvu/30

where

${\displaystyle a_{hk}=\alpha _{1h}\alpha _{1k}+\alpha _{2h}\alpha _{2k}+\alpha _{3h}\alpha _{3k}+\alpha _{4h}\alpha _{4k}}$

are the members of a 4✕4 series matrix which is the product of ${\displaystyle {\mathsf {{\bar {A}}A}}}$, the transposed matrix of ${\displaystyle {\mathsf {A}}}$ into ${\displaystyle {\mathsf {A}}}$. If by the transformation, the expression is changed to

${\displaystyle x_{1}^{'2}+x_{2}^{'2}+x_{3}^{'2}+x_{4}^{'2}}$

we must have

 (39) ${\displaystyle {\mathsf {{\bar {A}}A}}=1}$

${\displaystyle {\mathsf {A}}}$ has to correspond to the following relation, if transformation (38) is to be a Lorentz-transformation. For the determinant of ${\displaystyle {\mathsf {A}}}$ it follows out of (39) that ${\displaystyle (Det{\mathsf {A}})^{2}=1,Det{\mathsf {A}}=\pm 1}$.

From the condition (39) we obtain

 (40) ${\displaystyle {\mathsf {A}}^{-1}={\mathsf {\overline {A}}}}$

i.e. the reciprocal matrix of ${\displaystyle {\mathsf {A}}}$ is equivalent to the transposed matrix of ${\displaystyle {\mathsf {A}}}$.

For ${\displaystyle {\mathsf {A}}}$ as Lorentz transformation, we have further ${\displaystyle Det{\mathsf {A}}=+1}$, the quantities involving the index 4 once in the subscript are purely imaginary, the other co-efficients are real, and ${\displaystyle \alpha _{44}>0}$.

5°. A space time vector of the first kind which is represented by the 1✕4 series matrix,

 (41) ${\displaystyle s=|s_{1},\ s_{2},\ s_{3},\ s_{4}|}$

is to be replaced by ${\displaystyle s{\mathsf {A}}}$ in case of a Lorentz transformation

A space-time vector of the 2nd kind with components ${\displaystyle f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}}$ shall be represented by the alternating matrix

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

and is to be replaced by ${\displaystyle {\mathsf {\overline {A}}}f{\mathsf {A}}={\mathsf {A}}^{-1}f{\mathsf {A}}}$ in case of a Lorentz transformation [see the rules in § 5 (23) (24)]. Therefore referring to the expression