Hermann Minkowski

“ 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. ”

“ If four space-points, which do not lie in one plane are conceived to be at the same time to, then it is no longer permissible to make a change of the time parameter by a Lorentz-transformation, without at the same time destroying the character of the simultaneity of these four space points. ”

“ According to this, we shall have, not one space, but an infinite number of spaces in the world, — analogous to the case that the three-dimensional space consists of an infinite number of planes. The three-dimensional geometry becomes a chapter of four-dimensional physics. Now you perceive, why I said in the beginning that time and space shall reduce to mere shadows and we shall have only one world itself. ”

“ We look for those relations which make it possible for us — when proper fundamental data are given — to obtain the following quantities at every place and time, and therefore at every spacetime point as functions of x, y, z, t: — the vector of the electric force , the magnetic induction , the electrical induction , the magnetic force , the electrical space-density , the electric current (whose relation hereafter to the conduction current is known by the manner in which conductivity occurs in the process) , and lastly the vector , the velocity of matter. ”

“ 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. ”

“ Since is always greater than we may call the quotient as the "Gain" (vorgehen) of the time over the proper-time of the material point and the law can then be thus expressed; — The kinetic energy of a material point is the product of its mass into the gain of the time over its proper-time.
The set of four equations (22) again shows the symmetry in x, y, z, it, which is demanded by the relativity postulate; to the fourth equation however, a higher physical significance is to be attached, as we have already seen in the analogous case in electrodynamics. ”

“ 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. ”

“ I may here point out that by reforming mechanics, where instead of the Newtonian Relativity-Postulate with we assume a relativity-postulate with a finite c, then the axiomatic construction of Mechanics appears to gain considerably in perfection. ”

“ 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. ”

“ At every space-time point (as in § 18) , let a space time matrix of the 2nd kind be known
where are real magnitudes.
For a virtual displacement in a space-time sickle (with the previously applied designation) the value of the integral
extended over the whole range of the sickle, may be called the tensional work of the virtual displacement.
The sum which comes forth here, written in real magnitudes, is
we can now postulate the following minimum principle in mechanics. ”

“ In the mechanics reformed according to the world-postulate, the disharmonies which have disturbed the relations between Newtonian mechanics, and modern electrodynamics automatically disappear. I shall now consider the position of the Newtonian law of attraction to this postulate. I will assume that two point-masses m and m1 describe their world-lines; a moving force-vector is exercised by m upon m1, and the expression is just the same as in the case of the electron, only we have to write +mm1 instead of -ee1. ”

“ If any space-time sickle be bounded, then for each virtual displacement in the sickle, the sum of the mass-works, and tension works shall always he an extremum for that process of the space-time line in the sickle which actually occurs. ”

“ 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 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. ”