Page:AbrahamMinkowski2.djvu/12

§ 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 $(\tau )$, defined by^[1]

(29)	${\frac {dt}{d\tau }}={\frac {1}{\sqrt {1-q^{2}}}}=k^{-1}$

If we differentiate (with respect to $\tau$) the four-dimensional radius vector of the point, and dividing by the speed of light ($c$), then it is resulting in the $V^{4}$-"velocity" of Minkowski:

(30)

${\begin{cases}{\frac {1}{c}}{\frac {dx}{d\tau }}={\frac {1}{c}}{\frac {dx}{dt}}\cdot {\frac {dt}{d\tau }}={\mathfrak {q}}_{x}{\frac {dt}{d\tau }}={\mathfrak {q}}_{x}k^{-1},\\\\{\frac {1}{c}}{\frac {dy}{d\tau }}={\frac {1}{c}}{\frac {dy}{dt}}\cdot {\frac {dt}{d\tau }}={\mathfrak {q}}_{y}{\frac {dt}{d\tau }}={\mathfrak {q}}_{y}k^{-1},\\\\{\frac {1}{c}}{\frac {dz}{d\tau }}={\frac {1}{c}}{\frac {dz}{dt}}\cdot {\frac {dt}{d\tau }}={\mathfrak {q}}_{z}{\frac {dt}{d\tau }}={\mathfrak {q}}_{z}k^{-1},\\\\{\frac {1}{c}}{\frac {du}{d\tau }}={\frac {1}{c}}{\frac {du}{dt}}\cdot {\frac {dt}{d\tau }}=i{\frac {dt}{d\tau }}=ik^{-1},\end{cases}}$

Obviously the four components of the $V^{4}$-"velocity" identically satisfy the equation:

(30a)

$\left({\frac {1}{c}}{\frac {dx}{d\tau }}\right)^{2}+\left({\frac {1}{c}}{\frac {dy}{d\tau }}\right)^{2}+\left({\frac {1}{c}}{\frac {dz}{d\tau }}\right)^{2}+\left({\frac {1}{c}}{\frac {du}{d\tau }}\right)^{2}=-1$

We form now, by the $V_{I}^{4}$-"velocity" and "force" according to the scheme (2), the four-dimensional scalar

(31)	$\Psi ={\mathfrak {K}}_{x}{\frac {dx}{cd\tau }}+{\mathfrak {K}}_{y}{\frac {dy}{cd\tau }}+{\mathfrak {K}}_{z}{\frac {dz}{cd\tau }}+{\mathfrak {K}}_{u}{\frac {du}{cd\tau }}$

Introducing the ponderomotive force of electromagnetic fields, whose components are determined by (16), and taking into account equations (15) and (30), we find:

(31a)

$\Psi =-{\frac {Q}{c}}k^{-1}$

where $Q$ 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

1. H. Minkowski, l. c. equation (3), pag. 48.