Euclidean or Non-Euclidean
Nevertheless, we must face the possibility that the space we live in, or any other manifold of any sort whatever with which we deal on geometric principles, may turn out to be non-Euclidean. How shall we finally determine this? By measures—the Euclidean measures the angles of an actual triangle and finds the sum to be exactly 180 degrees; or he draws parallel lines of indefinite extent and finds them to be everywhere equally distant; and from these data he concludes that our space is really Euclidean. But he is not necessarily right.
We ask him to level off a plot of ground by means of a plumb line. Since the line always points to the earth’s center, the “level” plot is actually a very small piece of a spherical surface. Any test conducted on this plot will exhibit the numerical characteristics of the Euclidean geometry; yet we know the geometry of this surface is Riemannian. The angle-sum is really greater than 180 degrees; lines that are everywhere equidistant are not both geodesics.
The trouble, of course, is that on this plot we deal with so minute a fraction of the whole sphere that we cannot make measurements sufficiently refined to detect the departure from Euclidean standards. So it is altogether sensible for us to ask: “Is the universe of space about us really Euclidean in whatever of realized geometry it presents to us? Or is it really non-Euclidean, but so vast in size that we have never yet been able to extend our measures to a sufficiently large portion of it to make the divergence from the Euclidean standard discernible to us?”
This discussion is necessarily fragmentary, leaving out much that the writer would prefer to include. But it is hoped that it will nevertheless make it clear that when the contestants in the Einstein competition speak of a non-Euclidean universe as apparently having been revealed by Einstein, they mean simply that to Einstein has occurred a happy expedient for testing Euclideanism on a smaller scale than has heretofore been supposed possible. He has devised a new and ingenious sort of measure which, if his results be valid, enables us to operate in a smaller region while yet anticipating that any non-Euclidean characteristics of the manifold with which we deal will rise above the threshold of measurement. This does not mean that Euclidean lines and planes, as we picture them in our mind, are no longer non-Euclidean, but merely that these concepts do not quite so closely correspond with the external reality as we had supposed.
As to the precise character of the non-Euclideanism which is revealed, we may leave this to later chapters and to the competing essayists. We need only point out here that it will not necessarily be restricted to the matter of parallelism. The parallel postulate is of extreme interest to us for two reasons; first because historically it was the means by which the possibilities and the importance of non-Euclidean geometry were forced upon our attention; and second because it happens to be the immediate ground of distinction between Euclidean geometry and two of the most interesting alternatives. But Euclidean geometry is characterized, not by a single postulate, but by a considerable number of postulates. We may attempt to omit any one of these so that its ground is not specifically covered at all, or to replace any one of them by a direct alternative. We might conceivably do away with the superposition postulate entirely, and demand that figures be proved equivalent, if at all, by some more drastic test. We might do away with the postulate, first properly formulated by Hilbert, on which our ideas of the property represented in the word “between” depend. We might do away with any single one of the Euclidean postulates, or with any combination of two or more of them. In some cases this would lead to a lack of categoricity and we should get no geometry at all; in most cases, provided we brought a proper degree of astuteness to the formulation of alternatives for the rejected postulates, we should get a perfectly good system of non-Euclidean geometry: one realized, if at all, by other elements than the Euclidean point, line and plane, and one whose elements behave toward one another differently from the Euclidean point, line and plane.
Merely to add definiteness to this chapter, I annex here the statement that in the geometry which Einstein builds up as more nearly representing the true external world than does Euclid’s, we shall dispense with Euclid’s (implicit) assumption, underlying his (explicitly stated) superposition postulate, to the effect that the act of moving things about does not affect their lengths. We shall at the same time dispense with his parallel postulate. And we shall add a fourth dimension to his three—not, of course, anything in the nature of a fourth Euclidean straight line perpendicular, in Euclidean space, to three lines that are already perpendicular to each other, but something quite distinct from this, whose nature we shall see more exactly in the next chapter. If the present chapter has made it clear that it is proper for us to do this, and has prevented anyone from supposing that the results of doing it must be visualized in a Euclidean space of three dimensions or of any number of dimensions, it will have served its purpose.
VI
THE SPACE-TIME CONTINUUM
Minkowski’s World of Events, and the Way It Fits Into Einstein’s Structure
BY THE EDITOR, EXCEPT AS NOTED
Seeking a basis for the secure formulation of his results, and especially a means for expressing mathematically the facts of the dependence which he had found to exist between time and space, Einstein fell back upon the prior work of Minkowski. It may be stated right here that the idea of time as a fourth dimension is not particularly a new one. It has been a topic of abstract speculation for the best part of a century, even on the part of those whose notions of the fourth dimension were pretty closely tied down to the idea of a fourth dimension of Euclidean point-space, which would be marked by a fourth real line, perpendicular to the other three, and visible to us if we were only able to see it. Moreover, every mathematician, whether or not he be inclined to this sort of mental exercise, knows well that whenever time enters his equations at all, it does so on an absolutely equal footing with each of his space coordinates, so that as far as his algebra is concerned he could never distinguish between them. When the variables x, y, z, t come to the mathematician in connection with some physical investigation, he knows before he starts that the first three represent the dimensions of Euclidean three-space and that the last stands for time. But if the algebraic expressions of such a problem were handed to him independently of all physical tie-up, he would never be able to tell, from them alone, whether one of the four variables represented time, or if so, which one to pick out for this distinction.
It was Minkowski who first formulated all this in a form susceptible of use in connection with the theory of relativity. His starting point lies in the distinction between the point and the event. Mr. Francis has brought this out rather well in his essay, being the only competitor to present the Euclidean geometry as a real predecessor of Newtonian science, rather than as a mere part of the Newtonian system. I think his point here is very well taken. As he says, Euclid looked into the world about him and saw it composed of points. Ignoring all dynamic considerations, he built up in his mind a static world of points, and constructed his geometry as a scientific machine for dealing with this world in which motion played no part. It could to be sure be introduced by the observer for his own purposes, but when so introduced it was specifically postulated to be a matter of no moment at all to the points or lines or figures that were moved. It was purely an observational device, intended for the observer’s convenience, and in the bargain a mental device, calling for no physical action and the play of no force. So far as Euclid in his daily life was obliged to take cognizance of the fact that in the world of work-a-day realities motion existed, he must, as a true Greek, have looked upon this as a most unfortunate deviation of the reality from his beautiful world of intellectual abstraction, and as something to be deplored and ignored. Even in their statuary the Greeks clung to this idea. A group of marvelous action, like the Laocoon, they held to be distinctly a second rate production, a prostitution of the noble art; their ideal was a figure like the majestic Zeus—not necessarily a mere bust, be it understood, but always a figure in repose without action. Their statuary stood for things, not for action, just as their geometry stood for points, not for events.
Galileo and Newton took a different viewpoint. They were interested in the world as it is, not as it ought to be; and if motion appears to be a fundamental part of that world, they were bound to include it in their scheme. This made it necessary for them to pay much more attention to the concept of time and its place in the world than did the Greeks. In the superposition process, and even when he allowed a curve to be generated by a moving point, the sole interest which Euclid had in the motion was the effect which was to be observed upon his static figures after its completion. In this effect the rate of the motion did not enter. So all questions of velocity and time are completely ignored, and we have in fact the curious spectacle of motion without time.
To Galileo and Newton, on the other hand, the time which it took a body to pass from one point of its path to another was of paramount importance. The motion itself was the object of their study, and they recognized the part played by velocity. But Galileo and Newton were still sufficiently under the influence of Euclid to fit the observed phenomena of motion, so far as they could, upon Euclid’s static world of points. This they effected by falling in with the age-old procedure of regarding time and space as something entirely disassociated and distinct. The motion of an object—in theory, of a point—was to be recorded by observing its successive positions. With each of these positions a time was to be associated, marking the instant at which the point attained that position. But in the face of this association, space and time were to be maintained as entirely separate entities.