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 Lorentz 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