when calculated upon the space-time line from a fixed initial point P_{0} to the variable point P, (both being on the space-time line), is known as the 'Proper-time' of the position of matter we are concerned with at the space-time point P. (It is a generalization of the idea of Positional-time which was introduced by Lorentz for uniform motion.)
If we take a body R^0 which has got extension in space at time t_{0}, then the region comprising all the space-time line passing through R^0 and t_{0} shall be called a space-time filament.
If we have an anatylical expression [theta] below](x y, z, t) so that [theta](x, y z t) = 0 is intersected by every space time line of the filament at one point,—whereby
-([part]Θ/[part]x)^2, -([part]Θ/[part]y), -([part]Θ/[part]z)^2, -([part]Θ/[part]t)^2 > 0, part[Theta]/[part]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
[part]Θ/[part]x´ = 0, [part]Θ/[part]y´ = 0, [part]Θ/[part]z´ = 0, [part]Θ/[part]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 = [integral][integral][integral] 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 R^0 is to be regarded as the cross-section normal to the t axis of the filament at the point t = t^0, and the volume of the body R^0 is to be regarded as the contents of the cross-section.
