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 μ(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 μdJ, where μ = 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 dJn of the normal cross-section of the filament which is laid through (x, y, z, t) is
(4) dJn = (1/√(1 - u²))dJ = -iω₄ dJ = (dt/dτ)dJ.
and the function
ν = μ/-iω₄ = μ√(1 - u²)) = μ(∂τ/∂t. (5)
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° 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,[[29]] 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 νdJn taken in the sense of (4) and (5), has got the same value. Therefore the difference of the two integrals ∫νdJn (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