Page:SahaSpaceTime.djvu/15

 impulse-vector, and m-times the acceleration-vector at P as the force-vector of motion, at P. According to these definitions, the following law tells us how the motion of a point-mass takes place under any moving force-vector :

''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 definition of T, we see that the fourth simply expresses the "Energy-law." Accordingly c²-times the component of the impulse-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

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

i.e., if we deduct from this the additive constant mc², we obtain the expression ½mv² of Newtonian-mechanics up to magnitudes of the order of $$\frac{1}{c^2}$$. Hence it 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, therefore the energy-law comprises, for any possible system of reference, the whole system of equations of motion. This fact retains its significance even in the limiting: ease c = ∞, for the axiomatic construction of Newtonian mechanics, as has already been pointed out by T. R. Schütz.