Page:The principle of relativity (1920).djvu/108

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

From the condition (39) we obtain

A^{-1} = Ā,

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

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

5^o. A space time vector of the first kind which s represented by the 1 × 4 series matrix,

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

is to be replaced by sA in case of a Lorentz transformation

A. i.e. s´ = | s_{1}´ s_{2}´ s_{3}´ s_{4}´| = |s_{1} s_{2} s_{3} s_{4}| A;

A space-time vector of the 2nd kind with components f_{2 3} f_{34} shall be represented by the alternating matrix

(42) f = | 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   |

and is to be replaced by A^{-1} f A in case of a Lorentz transformation [see the rules in § 5 (23) (24)]. Therefore referring to the expression (37), we have the identity

Det^{1/2} (Ā f A) = Det A. Det^{1/2} f. Therefore Det^{1/2} f becomes an invariant in the case of a Lorentz transformation [see eq. (26) See. § 5].