Page:Grundgleichungen (Minkowski).djvu/51



Further let us form the integral

extending over the whole range of the space-time sickle. We shall decompose the sickle into elementary space-time filaments, and every one of these filaments in small elements (It of its proper-time, which are however large compared to the linear dimensions of the normal cross-section; let us assume that the mass of such a filament $$vdJ_{n} = dm$$ and write $$\tau^{0}$$ and $$\tau^{1}$$ for the 'Proper-time' of the upper and lower boundary of the sickle.

Then the integral (7) can be denoted by

taken over all the elements of the sickle.

Now let us conceive of the space-time lines inside a space-time sickle as material curves composed of material points, and let us suppose that they are subjected to a continual change of length inside the sickle in the following manner. The entire curves are to be varied in any possible manner inside the sickle, while the end points on the lower and upper boundaries remain fixed, and the individual substantial points upon it are displaced in such a manner that they always move forward normal to the curves. The whole process may be analytically represented by means of a parameter $$\vartheta$$, and to the value $$\vartheta = 0$$, shall correspond the actual curves inside the sickle. Such a process may be called a virtual displacement in the sickle.

Let the point x, y, z, t in the sickle $$\vartheta = 0$$ have the values $$x+\delta x,\ y+\delta y,\ z+\delta z,\ t+\delta t$$, when the parameter has the value $$\vartheta$$; these magnitudes are then functions of $$x,\ y,\ z,\ t,\ \vartheta$$. Let us now conceive of an infinitely thin space-time filament at the point x, y, z, t with the normal section of contents $$dJ_{0}$$ and if $$dJ_{0} + \delta dJ_{n}$$ be the contents of the normal section at the corresponding position of the varied filament, then according to the principle of conservation of mass $$v + \delta v$$ being the rest-mass-density at the varied position,