Page:Grundgleichungen (Minkowski).djvu/20

 and the equations (III) and (IV), are

§ 8. The Fundamental Equations for Moving Bodies.
We are now in a position to establish in a unique way the fundamental equations for bodies moving in any manner by means of these three axioms exclusively.

The first Axion shall be, —

When a detached region of matter is at rest at any moment, therefore the vector $$\mathfrak{w}$$ is zero, for a system x, y, z, t — the neighbourhood may be supposed to be in motion in any possible manner, then for the spacetime point x, y, z, t the same relations (A) (B) (V) which hold in the case when all matter is at rest, snail also hold between $$\varrho$$, the vectors $$\mathfrak{s,e,m,E,M}$$ and their differentials with respect to x, y, z, t.

The second axiom shall be : —

Every velocity of matter is < 1, smaller than the velocity of propagation of light.

The third axiom shall be : —

The fundamental equations are of such a kind that when x, y, z, it are subjected to a transformation and thereby $$\mathfrak{m},\ -i\mathfrak{e}$$ and $$\mathfrak{M},\ -i\mathfrak{E}$$ are transformed into space-time vectors of the second kind, $$\mathfrak{s},\ i\varrho$$ as a space-time vector of the 1st kind, the equations are transformed into essentially identical forms involving the transformed magnitudes.

Shortly I can signify the third axiom as ; —

$$\mathfrak{m},\ -i\mathfrak{e}$$ and $$\mathfrak{M},\ -i\mathfrak{E}$$ are space-time vectors of the second kind, $$\mathfrak{s},\ i\varrho$$ is a space-time vector of the first kind.

This axiom I call the Principle of Relativity.