Hermann Minkowski

“ Let us now picture to our mind the world-line passing through any world point x, y, z, t; now if we find the world-line parallel to the radius vector OA' of the hyperboloidal sheet, then we can introduce OA' as a new time-axis, and then according to the new conceptions of time and space the substance will appear to be at rest in the world point concerned. We shall now introduce this fundamental axiom: —
The substance existing at any world point can always be conceived to he at rest, if we establish our time and space suitably. ”

“ Remembering this point it is clear that the full group belonging to Newtonian Mechanics is simply the group Gc, with the value of c = ∞. In this state of affairs, and since Gc is mathematically more intelligible than G∞, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena possess an invariance not only for the group G∞, but in fact also for a group Gc, where c is finite, but yet exceedingly large compared to the usual measuring units. ”

“ In course of time, the mathematical consequences will be gradually deduced, and enough suggestions will be forthcoming for the experimental verification of the postulate; in this way even those, who find it uncongenial, or even painful to give up the old, time-honoured concepts, will be reconciled to the new ideas of time and space, — in the prospect that they will lead to pre-established harmony between pure mathematics and physics. ”

“ By that, the concept of time was shown to be unambiguously established by natural phenomena. But the concept of space was not altered, either by Einstein or Lorentz, probably because in the case of the above-mentioned spatial transformations, where the plane coincides with the x, t plane, the interpretation is possible as if the x-axis of space somehow remains conserved in its position. To step over the concept of space in a corresponding manner, is certainly only to assess as the boldness of mathematical culture. ”

“ I still shall consider the position of the Newtonian law of attraction to this postulate. I will assume that when two point-masses m and describe their world-lines, a moving force-vector is exerted by m upon , and the expression is just the same as in the case of the electron previously discussed; we only have to write instead of . We shall consider now the special case in which the acceleration-vector of m is constantly zero; then t may be introduced in such a manner that m may be regarded as fixed, the motion of is now subjected to the moving force-vector of m alone. ”

“ Let us consider the sheet, in the region of t > 0, and let us now conceive the transformation of x, y, z, t in the new system of variables; (x', y', z', t') by means of which the form of the expression will remain unaltered. Clearly the rotation of space round the null-point belongs to this group of transformations. Now we can have a full idea of the transformations which we picture to ourselves from a particular transformation in which (y, z) remain unaltered. ”

“ It is to be remembered that only a modified mechanics will occur, in which the square root of this differential combination takes the place of time, so that cases in which the velocity is greater than c will play no part, something like imaginary coordinates in geometry.
The impulse and real cause of inducement for the assumption of the group-transformation Gc is the fact that the differential equation for the propagation of light in vacant space possesses the group-transformation Gc. On the other hand, the idea of rigid bodies has any sense only in a system mechanics with the group G∞. ”

“ By concepts of time and space, the x, y, z manifold and its two sides and fall apart. If, for the sake of simplicity, we keep the null-point of time and space fixed, then the first mentioned group of mechanics signifies that at we can give the x, y, z-axes an arbitrary rotation about the null-point, corresponding to the homogeneous linear transformation of the expression
in itself. ”

“ Yet it is the whole compounded group which as a whole gives us occasion for thought.
We wish to picture to ourselves the whole relation graphically. Let x, y, z be the rectangular coordinates of space, and t denote the time. Subjects of our perception are always places and times connected. ”

“ Now, if Mathematics shows here only an unnecessary witticism, still she has the satisfaction that, thanks to her favourable antecedents, she has the power, with her senses sharpened in the exercise of free foresight, to comprehend the deep-lying consequences of such a refashioning of our conception of Nature. ”

“ Now if we have an optics with Gc, and on the other hand if there are rigid bodies, it is easy to see that a t-direction can be defined by the two hyperboloidal shells common to the groups G∞, and Gc, which has got the further consequence, that by means of suitable rigid instruments in the laboratory, we can perceive a change in natural phenomena, in case of different orientations, with regard to the direction of progressive motion of the earth. ”

“ In order to supply an explanation for this result, H. A. Lorentz formed a hypothesis whose success lies exactly in the invariance of optics for the group . According to Lorentz every body in motion, shall suffer a contraction in the direction of its motion, namely at velocity v in the ratio
This hypothesis sounds rather fantastical. ”