# Page:SahaSpaceTime.djvu/15

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PRINCIPLE OF RELATIVITY

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[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 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

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

i.e., if we deduct from this the additive constant mc², we obtain the expression ½mv² of Newtonian-mechanics up to magnitudes of the order of ${\displaystyle {\frac {1}{c^{2}}}}$. 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.[2]

1. Minkowski — Mechanics, appendix, page 65 of paper (2).
Planck — Verh. d. D. P. G. Vol. 4, 1906, p. 136.
2. Schütz, Gött. Nachr. 1897, 110.