§ 4. The equations of motion.
In the mechanics of Minkowski, the so-called "proper time" of a point occurs, i.e. a four-dimensional scalar , defined by[1]
(29)
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If we differentiate (with respect to ) the four-dimensional radius vector of the point, and dividing by the speed of light (), then it is resulting in the -"velocity" of Minkowski:
(30)
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Obviously the four components of the -"velocity" identically satisfy the equation:
(30a)
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We form now, by the -"velocity" and "force" according to the scheme (2), the four-dimensional scalar
(31)
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Introducing the ponderomotive force of electromagnetic fields, whose components are determined by (16), and taking into account equations (15) and (30), we find:
(31a)
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where is the Joule-heat, developed in the unity of space and time.
Now, Minkowski gives the equations of motion of an element of matter in the
- ↑ H. Minkowski, l. c. equation (3), pag. 48.