If we take a body R₀ which has got extension in space at time t₀, then the region comprising all the space-time line passing through R₀ and t₀ shall be called a space-time filament.
If we have an analytical expression θ(x y, z, t) so that θ(x, y z t) = 0 is intersected by every space time line of the filament at one point,—whereby
-(∂Θ/∂x)², -(∂Θ/∂y)², -(∂Θ/∂z)²,
-(∂Θ/∂t)² > 0, ∂Θ/∂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
∂Θ/∂x′ = 0, ∂Θ/∂y′ = 0, ∂Θ/∂z′ = 0, ∂Θ/∂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 = ∫∫∫ dx′ dy′ dz′ for the surrounding points of P on the cross-section is known as the elementary contents (Inhalts-element) of the cross-section. In this sense R₀ is to be regarded as the cross-section normal to the t axis of the filament at the point t = t₀, and the volume of the body R₀ is to be regarded as the contents of the cross-section.
If we allow R₀ to converge to a point, we come to the conception of an infinitely thin space-time filament. In such a case, 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.