# Page:Grundgleichungen (Minkowski).djvu/31

(37), we have the identity ${\displaystyle Det^{\frac {1}{2}}({\mathsf {\overline {A}}}f{\mathsf {A}})=Det\ {\mathsf {A}}\ Det^{\frac {1}{2}}f}$. Therefore ${\displaystyle Det^{\frac {1}{2}}f}$ becomes an invariant in the case of a Lorentz transformation [see eq. (26) Sec. § 5].

Looking back to (36), we have for the dual matrix

${\displaystyle ({\mathsf {A}}^{-1}f^{*}{\mathsf {A}})({\mathsf {A}}^{-1}f{\mathsf {A}})={\mathsf {A}}^{-1}f^{*}f{\mathsf {A}}=Det^{\frac {1}{2}}f.{\mathsf {A}}^{-1}{\mathsf {A}}=Det^{\frac {1}{2}}f}$,

from which it is to be seen that the dual matrix ${\displaystyle f^{*}}$ behaves exactly like the primary matrix f, and is therefore a space time vector of the II kind; ${\displaystyle f^{*}}$ is therefore known as the dual space-time vector of f with components ${\displaystyle f_{14},\ f_{24},\ f_{34},\ f_{23},\ f_{31},\ f_{12}}$.

6°.If w and s are two space-time vectors of the 1st kind then by ${\displaystyle w{\bar {s}}}$ (as well as by ${\displaystyle s{\bar {w}})}$) will be understood the combination

 (43) ${\displaystyle w_{1}s_{1}+w_{2}s_{2}+w_{3}s_{3}+w_{4}s_{4}}$

In case of a Lorentz transformation ${\displaystyle {\mathsf {A}}}$, since ${\displaystyle (w{\mathsf {A}})({\mathsf {\bar {A}}}{\bar {s}})=w{\bar {s}}}$ this expression is invariant. — If ${\displaystyle w{\bar {s}}=0}$, then w and s are perpendicular to each other.

Two space-time rectors of the first kind w, s gives us a 2✕4 series matrix

${\displaystyle \left|{\begin{array}{cccc}w_{1},&w_{2},&w_{3},&w_{4}\\s_{1},&s_{2},&s_{3},&s_{4}\end{array}}\right|}$

Then it follows immediately that the system of six magnitudes

 (44) ${\displaystyle w_{2}s_{3}-w_{3}s_{2},\ w_{3}s_{1}-w_{1}s_{3},\ w_{1}s_{2}-w_{2}s_{1},\ w_{1}s_{4}-w_{4}s_{1},\ w_{2}s_{4}-w_{4}s_{2},\ w_{3}s_{4}-w_{4}s_{3}}$

behaves in case of a Lorentz-transformation as a space-time vector of the II. kind. The vector of the second kind with the components (44) are denoted by [w,s]. We see easily that ${\displaystyle Det^{\frac {1}{2}}[w,s]=0}$. The dual vector of [w,s] shall be written as [w,s]*.

If w is a space-time vector of the 1st kind, f of the second kind, wf signifies a 1✕4 series matrix. In case of a Lorentz-transformation ${\displaystyle {\mathsf {A}}}$, w is changed into ${\displaystyle w'=w{\mathsf {A}}}$, f into ${\displaystyle f'={\mathsf {A}}^{-1}f{\mathsf {A}}}$, therefore ${\displaystyle w'f'=w{\mathsf {A}}\ {\mathsf {A}}^{-1}f{\mathsf {A}}=(wf){\mathsf {A}}}$, i.e., wf is transformed as a space-time vector of the 1st kind. We can verify, when w is a space-time vector of the 1st kind, f of the 2nd kind, the important identity

 (45) ${\displaystyle [w,wf]+[w,wf^{*}]^{*}=(w{\bar {w}})f}$.