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Now the world-line passing through P will be described by a substantial point with the constant mechanical mass m. Let us call m-times the velocity-vector at P as the momentum-vector, and m-times the acceleration-vector at P as the force-vector of motion at P. According to these definitions, the law telling us how the motion of a point-mass takes place under a given moving force-vector^[1]:

The force-vector of motion is equal to the moving force-vector.

This enunciation comprises four equations for the components in the four directions, of which the fourth can be deduced from the first three, because both of the above-mentioned vectors are perpendicular to the velocity-vector from the outset. From the above-mentioned definition of T, we see that the fourth certainly expresses the energy-law. Accordingly, $c^{2}$-times the component of the momentum-vector in the direction of the t-axis is to be defined as the kinetic-energy of the point-mass. The expression for this is

$mc^{2}{\frac {dt}{d\tau }}=mc^{2}/{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$

i.e., if we deduct from this the additive constant $mc^{2}$, we obtain the expression ${\tfrac {1}{2}}mv^{2}$ of Newtonian-mechanics up to magnitudes of the order of $1/c^{2}$. Hence it illustratively appears that the energy depends upon the system of reference. But since the t-axis can be laid in the direction of any time-like axis, however, the energy-law formed for any possible system of reference, therefore comprises already the whole system of equations of motion. This fact retains its significance for the axiomatic construction of Newtonian mechanics, even in the considered passage to the limit $c=\infty$, as has already been recognized out by J. R. Schütz.^[2]

From the very beginning, we can establish the ratio between the units of length and time in such a manner, that the natural limiting velocity becomes $c=1$. If we now write ${\sqrt {-1}}t=s$, in the place of t, then the quadratic differential expression

$d\tau ^{2}=-\left(dx^{2}+dy^{2}+dz^{2}+ds^{2}\right),$

becomes symmetrical in x, y, z, s; this symmetry then enters into each law, which does not contradict the world-postulate. We can clothe the essential nature of this postulate in the mystical, but mathematically significant formula

3·10⁵ km = ${\sqrt {-1}}$ Sec.

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1. H. Minkowski, l.c. p. 107. — see also M. Planck, Verh. d. Physik. Ges. 4, p. 136, 1906.
2. J. R. Schütz, Das Prinzip der absoluten Erhaltung der Energie. Göttinger Nachr. 1897, p. 110.