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