Page:Grundgleichungen (Minkowski).djvu/30

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where

are the members of a 4✕4 series matrix which is the product of , the transposed matrix of into . If by the transformation, the expression is changed to

we must have

(39)

has to correspond to the following relation, if transformation (38) is to be a Lorentz-transformation. For the determinant of it follows out of (39) that .

From the condition (39) we obtain

(40)

i.e. the reciprocal matrix of is equivalent to the transposed matrix of .

For as Lorentz transformation, we have further , the quantities involving the index 4 once in the subscript are purely imaginary, the other co-efficients are real, and .

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

(41)

is to be replaced by in case of a Lorentz transformation

A space-time vector of the 2nd kind with components shall be represented by the alternating matrix

(42)

and is to be replaced by in case of a Lorentz transformation [see the rules in § 5 (23) (24)]. Therefore referring to the expression