The Principle of Relativity

by Albert Einstein; H. Minkowski

Then the integral (7) can be denoted by

∫∫ νdJ_n dτ = ∫ (τ′-τ⁰) dm.

taken over all the elements of the sichel.

Now let us conceive of the space-time lines inside a space-time sichel as material curves composed of material points, and let us suppose that they are subjected to a continual change of length inside the sichel in the following manner. The entire curves are to be varied in any possible manner inside the sichel, 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 λ, and to the value λ = 0, shall correspond the actual curves inside the sichel. Such a process may be called a virtual displacement in the sichel.

Let the point (x, y, z, t) in the sichel λ = 0 have the values x + δx, y + δy, z + δz, t + δt, when the parameter has the value λ; these magnitudes are then functions of (x, y, z, t, λ). 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_n and if dJ_n + δdJ_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—(ν + dν being the rest-mass-density at the varied position),

(8) (ν + δν) (dJ_n + δdJ_n) = νdJ_n = dm.

In consequence of this condition, the integral (7) taken over the whole range of the sichel, varies on account of the displacement as a definite function N + δN of λ, and we may call this function N + δN as the mass action of the virtual displacement.

If we now introduce the method of writing with indices, we shall have

(9) d(x_h + δx_h) = dx_h + ∑_k ∂δx_h/∂x_k + ∂δx_h/∂λ dλ

k = 1, 2, 3, 4