Page:Grundgleichungen (Minkowski).djvu/50

(4)	$dJ_{n}={\frac {1}{\sqrt {1-{\mathfrak {w}}^{2}}}}dJ=-iw_{4}dJ={\frac {dt}{d\tau }}dJ$

and the function

(5)	$\nu ={\frac {\mu }{-iw_{4}}}=\mu {\sqrt {1-{\mathfrak {w}}^{2}}}=\mu {\frac {d\tau }{dt}}$

may be defined as the rest-mass density at the position x, y, z, t. Then the principle of conservation of mass can be formulated in this manner: —

For an infinitely thin space-time filament, the product of the rest-mass density and the contents of the normal cross-section is constant along the whole filament.

In any space-time filament, let us consider two cross-sections $Q^{0}$ and $Q^{1}$, which have only the points on the boundary common to each other; let the space-time lines inside the filament have a larger value of t on Q than on $Q^{0}$. The finite range enclosed between $Q^{0}$ and $Q^{1}$ shall be called a space-time sickle^{[WS 1]} $Q^{0}$ is the lower boundary, and $Q^{1}$ is the upper boundary of the sickle.

If we decompose a filament into elementary space-time filaments, then to an entrance-point of an elementary filament through the lower boundary of the sickle, there corresponds an exit point of the same by the upper boundary, whereby for both, the product $vdJ_{n}$ taken in the sense of (4) and (5), has got the same value. Therefore the difference of the two integrals $\int vdJ_{n}$ (the first being extended over the upper, the second upon the lower boundary) vanishes. According to a well-known theorem of Integral Calculus the difference is equivalent to

$\int \int \int \int \ lor\ v{\overline {w}}\ dx\ dy\ dz\ dt$,

the integration being extended over the whole range of the sickle, and (comp. (67), § 12)

$lor\ v{\overline {w}}\ ={\frac {\partial vw_{1}}{\partial x_{1}}}+{\frac {\partial vw_{2}}{\partial x_{2}}}+{\frac {\partial vw_{3}}{\partial x_{3}}}+{\frac {\partial vw_{4}}{\partial x_{4}}}$

If the sickle reduces to a point, then the differential equation

(6)	$lor\ v{\overline {w}}\ =0$,

which is the condition of continuity

1. Saha used the German word "Sichel" in this edition.