Page:Grundgleichungen (Minkowski).djvu/31

From Wikisource
Jump to: navigation, search
This page has been proofread, but needs to be validated.


(37), we have the identity . Therefore 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

,

from which it is to be seen that the dual matrix behaves exactly like the primary matrix f, and is therefore a space time vector of the II kind; is therefore known as the dual space-time vector of f with components .

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

(43)

In case of a Lorentz transformation , since this expression is invariant. — If , 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

Then it follows immediately that the system of six magnitudes

(44)

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 . 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 , w is changed into , f into , therefore , 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) .