c²t² - x² - y² - z² = 1
According to the analogy of the hyperboloid of two sheets, this consists of two sheets separated by t = 0. 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. Let us draw the cross section of the upper sheets with the plane of the x- and t-axes, i.e., the upper half of the hyperbola c²t² - x² = 1, with its asymptotes (vide fig. 1).
Then let us draw the radius rector OA′, the tangent A′ B′ at A′, and let us complete the parallelogram OA′ B′ C′; also produce B′ C′ to meet the x-axis at D′. Let us now take Ox′, OA′ as new axes with the unit measuring rods OC′ = 1, OA′ = (1/c) ; then the hyperbola is again expressed in the form c²t′² - x′² = 1, t′ > 0 and the transition from (x, y, z, t) to (x′ y′ z′ t) is one of the transitions in question. Let us add to this characteristic transformation any possible displacement of the space and time null-points; then we get a group of transformation depending only on c, which we may denote by Gc.
Now let us increase c to infinity. Thus (1/c) becomes zero and it appears from the figure that the hyperbola is gradually shrunk into the x-axis, the asymptotic angle becomes a straight one, and every special transformation in the limit changes in such a manner that the t-axis can have any possible direction upwards, and x′ more and more approximates to x. 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. Such a preconception would be an extraordinary triumph for pure mathematics.
At the same time I shall remark for which value of c, this invariance can be conclusively held to be true. For c, we shall substitute the velocity of light c in free space. In order to avoid speaking either of space or of vacuum, we may take this quantity as the ratio between the electrostatic and electro-magnetic units of electricity.
We can form an idea of the invariant character of the expression for natural laws for the group-transformation Gc in the following manner.
Out of the totality of natural phenomena, we can, by successive higher approximations, deduce a coordinate system (x, y, z, t); by means of this coordinate system, we can represent the phenomena according to definite laws. This system of reference is by no means uniquely determined by the phenomena. We can change the system of reference in any possible manner corresponding to the above-mentioned group transformation Gc, but the expressions for natural laws will not be changed thereby.
For example, corresponding to the above described figure, we can call t′ the time, but then necessarily the space connected with it must be expressed by the manifoldness (x′ y z). The physical laws are now expressed by means of x′, y, z, t′,—and the expressions are just the same as in the case of x, y, z, t. According to this, we shall have in the world, not one space, but many spaces,—quite analogous to the case that the three-dimensional space consists of an infinite number of planes. The three-dimensional geometry will be 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 a world complete in itself.
II
Now the question may be asked,—what circumstances lead us to these changed views about time and space, are they not in contradiction with observed phenomena, do they finally guarantee us advantages for the description of natural phenomena?