I
Now I want to show you how we can arrive at the changed concepts about time and space from mechanics, as accepted now-a-days, from purely mathematical considerations. The equations of Newtonian mechanics show a twofold invariance, (i) their form remains unaltered when we subject the fundamental space-coordinate system to any possible change of position, (ii) when we change the system in its nature of motion, i. e., when we impress upon it any uniform motion of translation, the null-point of time plays no part. We are accustomed to look upon the axioms of geometry as settled once for all, while we seldom have the same amount of conviction regarding the axioms of mechanics, and therefore the two invariants are seldom mentioned in the same breath. Each one of these denotes a certain group of transformations for the differential equations of mechanics. We look upon the existence of the first group as a fundamental characteristics of space. We always prefer to leave off the second group to itself, and with a light heart conclude that we can never decide from physical considerations whether the space, which is supposed to be at rest, may not finally be in uniform motion. So these two groups lead quite separate existences besides each other. Their totally heterogeneous character may scare us away from the attempt to compound them. 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 connected with place and time. No one has observed a place except at a particular time, or has observed a time except at a particular place. Yet I respect the dogma that time and space have independent existences. I will call a space-point plus a time-point, i.e., a system of values x, y, z, t, as a world-point. The manifoldness of all possible values of x, y, z, t, will be the world. I can draw four world-axes with the chalk. Now any axis drawn consists of quickly vibrating molecules, and besides, takes part in all the journeys of the earth ; and therefore gives us occasion for reflection. The greater abstraction required for the four-axes does not cause the mathematician any trouble. In order not to allow any yawning gap to exist, we shall suppose that at every place and time, something perceptible exists. In order not to specify either matter or electricity, we shall simply style these as substances. We direct our attention to the world-point x, y, z, t, and suppose that we are in a position to recognise this substantial point at any subsequent time. Let dt be the time element corresponding to the changes of space coordinates of this point [dx, dy, dz]. Then we obtain (as a picture, so to speak, of the perennial life-career of the substantial point),—a curve in the world—the world-line, the points on which unambiguously correspond to the parameter t from +∞ to -∞. The whole world appears to be resolved in such world-lines, and I may just deviate from my point if I say that according to my opinion the physical laws would find their fullest expression as mutual relations among these lines.
By this conception of time and space, the (x, y, z) manifoldness t = 0 and its two sides t < 0 and t > 0 falls asunder. If for the sake of simplicity, we keep the null-point of time and space fixed, then the first named group of mechanics signifies that at t = 0 we can give the x, y, and z-axes any possible rotation about the null-point corresponding to the homogeneous linear transformation of the expression
x² + y² + z².
The second group denotes that without changing the expression for the mechanical laws, we can substitute (x - αt, y - βt, z - γt for (x, y, z) where (α, β, γ) are any constants. According to this we can give the time-axis any possible direction in the upper half of the world t > 0. Now what have the demands of orthogonality in space to do with this perfect freedom of the time-axis towards the upper half?
To establish this connection, let us take a positive parameter c, and let us consider the figure
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.