Page:Grundgleichungen (Minkowski).djvu/20

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

(B)

${\begin{array}{ccccccccc}&&{\frac {\partial F_{34}}{\partial x_{2}}}&+&{\frac {\partial F_{42}}{\partial x_{3}}}&+&{\frac {\partial F_{23}}{\partial x_{4}}}&=&0,\\\\{\frac {\partial F_{43}}{\partial x_{1}}}&&&+&{\frac {\partial F_{14}}{\partial x_{3}}}&+&{\frac {\partial F_{31}}{\partial x_{4}}}&=&0,\\\\{\frac {\partial F_{24}}{\partial x_{1}}}&+&{\frac {\partial F_{41}}{\partial x_{2}}}&&&+&{\frac {\partial F_{12}}{\partial x_{4}}}&=&0,\\\\{\frac {\partial F_{32}}{\partial x_{1}}}&+&{\frac {\partial F_{13}}{\partial x_{2}}}&+&{\frac {\partial F_{21}}{\partial x_{3}}}&&&=&0.\end{array}}$

§ 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 Lorentz 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.