Page:Grundgleichungen (Minkowski).djvu/50



and the function

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 $$Q^{0}$$ 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

which is the condition of continuity 