Page:Grundgleichungen (Minkowski).djvu/30

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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