(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° and Q´, 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°. The finite range enclosed between Q° and Q´ shall be called a space-time sichel,[1] Q´ is the lower boundary, and Q´ is the upper boundary of the sichel.
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 sichel, there corresponds an exit point of the same by the upper boundary, whereby for both, the product [nu]dJ_{n} taken in the sense of (4) and (5), has got the same value. Therefore the difference of the two integrals [integral][nu]dJ_{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
[integral][integral][integral][integral] lor [nu][=[omega]] dx dy dz dt,
the integration being extended over the whole range of the sichel, and (comp. (67), § 12)
lor [nu][=[omega]] = ([part][nu][omega]_{1}/[part]x_{1}) + ([part][nu][omega]_{2}/[part]x_{2}) + ([part][nu][omega]_{3}/[part]x_{3}) + ([part][nu][omega]_{4}/[part]x_{4}).
If the sichel reduces to a point, then the differential equation lor [nu][=[omega]] = 0, (6)
- ↑ Sichel—a German word meaning a crescent or a scythe. The original term is retained as there is no suitable English equivalent.
