# 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

$x^{'2}_{1} + x^{'2}_{2} + x^{'2}_{3} + x^{'2}_{4}$

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