Page:Grundgleichungen (Minkowski).djvu/49

 $$-\left(\frac{\partial\Theta}{\partial x}\right)^{2}-\left(\frac{\partial\Theta}{\partial y}\right)^{2}-\left(\frac{\partial\Theta}{\partial z}\right)^{2}+\left(\frac{\partial\Theta}{\partial t}\right)^{2}>0,\ \frac{\partial\Theta}{\partial t}>0$$

then the totality of the intersecting points will be called a cross section of the filament. At any point P of such across-section, we can introduce by means of a transformation a system of reference x' y', z', t', so that according to this

$$\frac{\partial\Theta}{\partial x'}=0,\ \frac{\partial\Theta}{\partial y'}=0,\ \frac{\partial\Theta}{\partial z'}=0,\ \frac{\partial\Theta}{\partial t'}>0$$,

The direction of the uniquely determined t'— axis in question here is known as the upper normal of the cross-section at the point P and the value of $$dJ=\int\int\int dx'dy'dz'$$ for the surrounding points of P on the cross-section is known as the elementary contents (Inhaltslement) of the cross-section. In this sense $$R^{0},t^{0}$$ is to be regarded as the cross-section normal to the t axis of the filament at the point $$t = t^{0}$$ and the volume of the body $$R^{0}$$ is to be regarded as the contents of the cross-section.

If we allow $$R^{0}$$ to converge to a point, we come to the conception of an infinitely thin space-time filament. In such a ease, a space-time line will be thought of as a principal line and by the term Proper-time of the filament will be understood the Proper-time which is laid along this principal line; under the term normal cross-section of the filament, we shall understand the cross-section upon the space which is normal to the principal line through P.

We shall now formulate the principle of conservation of mass.

To every space R at a time t, belongs a positive quantity — the mass at R at the time t. If R converges to a point x, y, z, t, then the quotient of this mass, and the volume of R approaches a limit $$\mu(x,\ y,\ z,\ t)$$, which is known as the mass-density at the space-time point x, y, z, t.

The principle of conservation of mass says — that for an infinitely thin space-time filament, the product $$\mu dJ$$, where $$\mu$$ = mass-density at the point $$x, y, z, t$$ of the filament (i.e., the principal line of the filament), dJ = contents of the cross-section normal to the t axis, and passing through $$x, y, z, t$$, is constant along the whole filament.

Now the contents $$dJ_{n}$$ of the normal cross-section of the filament which is laid through x, y, z, t is