Page:Grundgleichungen (Minkowski).djvu/30

where

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

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

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

we must have

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

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

From the condition (39) we obtain

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

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

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

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

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

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

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

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

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