Page:Grundgleichungen (Minkowski).djvu/31

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

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

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

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

In case of a transformation $$\mathsf{A}$$, since $$(w\mathsf{A})(\mathsf{\bar{A}}\bar{s})=w\bar{s}$$ this expression is invariant. — If $$w\bar{s}=0$$, 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

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

Then it follows immediately that the system of six magnitudes

behaves in case of a -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 $$Det^{\frac{1}{2}}[w,s] =0$$. 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 -transformation $$\mathsf{A}$$, w is changed into $$w'=w\mathsf{A}$$, f into $$f'=\mathsf{A}^{-1}f\mathsf{A}$$, therefore $$w'f'=w\mathsf{A}\ \mathsf{A}^{-1}f\mathsf{A}=(wf)\mathsf{A}$$, 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