This severe separation of time and space Minkowski has now questioned, with the statement that the elements of which the external world is composed, and which we observe, are not points at all, but are events. This calls for a revision of our whole habit of thought. It means that the perceptual world is four-dimensioned, not three-dimensioned as we have always supposed; and it means, at the very least, that the distinction between time and space is not so fundamental as we had supposed.

[This should not impress us as strange or incomprehensible. What do we mean when we say that a plane is two-dimensional? Simply that two coordinates, two numbers, must be given to specify the position of any point of the plane. Similarly for a point in the space of our accustomed concepts we must give three numbers to fix the position—as by giving the latitude and longitude of a point on the earth and its height above sea-level. So we say this space is three-dimensional. But a material body is not merely somewhere; it is somewhere now,]¹⁸² or was somewhere yesterday, or will be somewhere tomorrow. The statement of position for a material object is meaningless unless we at the same time specify the time at which it held that position. [If I am considering the life-history of an object on a moving train, I must give three space-coordinates and one time-coordinate to fix each of its positions.]¹⁸² And each of its positions, with the time pertaining to that position, constitutes an event. The dynamic, ever-changing world about us, that shows the same aspect at no two different moments, is a world of events; and since four measures or coordinates are required to fix an event, we say this world of events is four-dimensional. If we wish to test out the soundness of this viewpoint, we may well do so by asking whether the naming of values for the four coordinates fixes the event uniquely, as the naming of three under the old system fixes the point uniquely.

Suppose we take some particular event as the one from which to measure, and agree upon the directions to be taken by our space axes, and make any convention about our time-axis which subsequent investigation may show to be necessary. Certainly then the act of measuring so many miles north, and so many west, and so many down, and so many seconds backward, brings us to a definite time and place—which is to say, to a definite event. Perhaps nothing “happened” there, in the sense in which we usually employ the word; but that is no more serious than if we were to locate a point with reference to our familiar space coordinate system, and find it to lie in the empty void of interstellar space, with no material body occupying it. In this second case we still have a point, which requires, to insure its existence and location, three coordinates and nothing more; in the first case we still have an event, which requires for its existence and definition four coordinates and nothing more. It is not an event about which the headline writers are likely to get greatly excited; but what of that? It is there, ready and waiting to define any physical happening that falls upon it, just as the geometer’s point is ready and waiting to define any physical body that chances to fall upon it.

It is now in order to introduce a word, which I shall have to confess the great majority of the essayists introduce, somewhat improperly, without explanation. But when I attempt to explain it, I realize quite well why they did this. They had to have it; and they didn’t have space in their three thousand words to talk adequately about it and about anything else besides. The mathematician knows very well indeed what he means by a continuum; but it is far from easy to explain it in ordinary language. I think I may do best by talking first at some length about a straight line, and the points on it.

If the line contains only the points corresponding to the integral distances 1, 2, 3, etc., from the starting point, it is obviously not continuous—there are gaps in it vastly more inclusive than the few (comparatively speaking) points that are present. If we extend the limitations so that the line includes all points corresponding to ordinary proper and improper fractions like ¼ and 17/29 and 1633/7?—what the mathematician calls the rational numbers—we shall apparently fill in these gaps; and I think the layman’s first impulse would be to say that the line is now continuous. Certainly we cannot stand now at one point on the line and name the “next” point, as we could a moment ago. There is no “next” rational number to 116/125?, for instance; 115/124 comes before it and 117/126 comes after it, but between it and either, or between it and any other rational number we might name, lie many others of the same sort. Yet in spite of the fact that the line containing all these rational points is now “dense” (the technical term for the property I have just indicated), it is still not continuous; for I can easily define numbers that are not contained in it—irrational numbers in infinite variety like StartRoot 2 EndRoot; or, even worse, the number pi = 3.141592 … which defines the ratio of the circumference of a circle to the diameter, and many other numbers of similar sort.

If the line is to be continuous, there may be no holes in it at all; it must have a point corresponding to every number I can possibly name. Similarly for the plane, and for our three-space; if they are to be continuous, the one must contain a point for every possible pair of numbers x and y, and the other for every possible set of three numbers x, y and z, that I can name. There may be no holes in them at all.

A line is a continuum of points. A plane is a continuum of points. A three-space is a continuum of points. These three cases differ only in their dimensionality; it requires but one number to determine a point of the first continuum, two and three respectively in the second and third cases. But the essential feature is not that a continuum shall consist of points, or that we shall be able to visualize a pseudo-real existence for it of just the sort that we can visualize in the case of line, plane and point. The essential thing is merely that it shall be an aggregate of elements numerically determined in such a way as to leave no holes, but to be just as continuous as the real number system itself. Examples, however, aside from the three which I have used, are difficult to construct of such sort that the layman shall grasp them readily; so perhaps, fortified with the background of example already presented, I may venture first upon a general statement.

Suppose we have a set of “elements” of some sort—any sort. Suppose that these elements possess one or more fundamental identifying characteristics, analogous to the coordinates of a point, and which, like these coordinates, are capable of being given numerical values. Suppose we find that no two elements of the set possess identically the same set of defining values. Suppose finally—and this is the critical test—that the elements of the set are such that, no matter what numerical values we may specify, it we do specify the proper number of defining magnitudes we define by these an actual element of the set, that corresponds to this particular collection of values. Our elements then share with the real number system the property of leaving no holes, of constituting a continuous succession in every dimension which they possess. We have then a continuum. Whatever its elements, whatever the character of their numerical identifiers, whatever the number n of these which stands for its dimension, there may be no holes or we have no continuum. There must be an element for every possible combination of n numbers we can name, and no two of these combinations may give the same element. Granted this condition, our elements constitute a continuum.

As I have remarked, it is not easy to cite examples of continua which shall mean anything to the person unaccustomed to the term. The totality of carbon-oxygen-nitrogen-hydrogen compounds suggested by one essayist as an example is not a continuum at all, for the set contains elements corresponding only to integer values of the numbers which tell us how many atoms of each substance occur in the molecule. We cannot have a compound containing StartRoot 2 EndRoot carbon atoms, or 3 pi oxygen atoms. Perhaps the most satisfactory of the continua, outside the three Euclidean space-continua already cited, [is the manifold of music notes. This is four-dimensional; each note has four distinctions—length, pitch, intensity, timbre—to distinguish it perfectly, to tell how long, how high, how loud, how rich.]²⁶³ We might have a little difficulty in reducing the characteristic of richness to numerical expression, but presumably it could be done; and we should then be satisfied that every possible combination of four values, l, p, i, t for these four identifying characteristics would give us a musical effect, and one to be confused with no other.

There is in the physical world a vast quantity of continua of one sort or another. The music-note continuum brings attention to the fact that not all of these are such that their elements make their appeal to the visual sense. This remark is a pertinent one; for we are by every right of heritage an eye-minded race, and it is frequently necessary for us to be reminded that so far as the external world is concerned, the verdict of every other sense is entirely on a par with that of sight. The things which we really see, like matter, and the things which we abstract from these visual impressions, like space, are by no means all there is to the world.

If we are dealing with a continuum of any sort whatever having one or two or three dimensions, we are able to represent it graphically by means of the line, the plane, or the three-space. The same set of numbers that defines an element of the given continuum likewise defines an element of the Euclidean continuum of the same dimensionality; so the one continuum corresponds to the other, element for element, and either may stand for the other. But if we have a continuum of four or more dimensions, this representation breaks down in the absence of a real, four-dimensional Euclidean point-space to serve as a picture. This does not in the least detract from the reality of the continuum which we are thus prevented from representing graphically in the accustomed fashion.

The Euclidean representation, in fact, may in some cases be unfortunate—it may be so entirely without significance as to be actually misleading. For in the Euclidean continuum of points, be it line, plane or three-space, there are certain things which we ordinarily regard as secondary derived properties, but which possess a great deal of significance none the less.

In particular, in the Euclidean plane and in Euclidean three-space, there is the distance between two points. I have indicated, in the chapter on non-Euclidean geometry, that the parallel postulate of Euclid, which distinguishes his geometry from others, could be replaced by any one of numerous other postulates. Grant Euclid’s postulate and you can prove any of these substitutes; grant any of the substitutes and you can prove Euclid’s postulate. Now it happens that there is one of these substitutes to which modern analysis has given a position of considerable importance. It is merely our good old friend the Pythagorean theorem, that the square on the hypotenuse equals the sum of the squares on the sides; but it is dressed in new clothes for the present occasion.

Mr. Francis’ discussion of this part of the subject, and especially his figure, ought to make it clear that this theorem can be considered as dealing with the distance between any two points. When we so consider it, and take it as the fundamental, defining postulate of Euclidean geometry which distinguishes this geometry from others, we have a statement of considerable content. We have, first, that the characteristic property of Euclidean space is that the distance between two points is given by the square root of the sum of the squares of the coordinate-differences for these points—by the expression upper D equals StartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared plus left-parenthesis upper Z minus z right-parenthesis squared EndRoot comma where the large letters represent the coordinates of the one point and the small ones those of the other. We have more than this, however; we have that this distance is the same for all observers, no matter how different their values for the individual coordinates of the individual points. And we have, finally, as a direct result of looking upon the thing from this viewpoint, that the expression for D is an “invariant”; which simply means that every observer may use the same expression in calculating the value of D in terms of his own values for the coordinates involved. The distance between two points in our space is given numerically by the square root of the sum of the squares of my coordinate-differences for the two points involved; it is given equally by the square root of the sum of the squares of your coordinate-differences, or those of any other observer whatsoever. We have then a natural law—the fundamental natural law characterizing Euclidean space. If we wish to apply it to the Euclidean two-space (the plane) we have only to drop out the superfluous coordinate-difference; if we wish to see by analogy what would be the fundamental natural law for a four-dimensional Euclidean space, we have only to introduce under the radical a fourth coordinate-difference for the fourth dimension.

If we were not able to attach any concrete meaning to the expression for D the value of all this would be materially lessened. Consider, for instance, the continuum of music notes. There is no distance between different notes. There is of course significance in talking about the difference in pitch, in intensity, in duration, in timbre, between two notes; but there is none in a mode of speech that implies a composite expression indicating how far one note escapes being identical with another in all four respects at once. The trouble, of course, is that the four dimensions of the music-note continuum are not measurable in terms of a common unit. If they were, we should expect to measure their combination more or less absolutely in terms of this same unit. We can make measurements in all three dimensions of Euclidean space with the same unit, with the same measuring rod in fact. [This presents a peculiarity of our three-space which is not possessed by all three-dimensional manifolds. Riemann has given another illustration in the system of all possible colors, composed of arbitrary proportions of the three primaries, red, green and violet. This system forms a three-dimensional continuum; but we cannot measure the “distance” or difference between two colors in terms of the difference between two others.]¹³⁰

Accordingly, in spite of the fact that the Euclidean three-space gives us a formal representation of the color continuum, and in spite of the fact that the hypothetical four-dimensional Euclidean space would perform a like office for the music-note continuum, this representation would be without significance. We should not say that the geometry of these two manifolds is Euclidean. We should realize that any set of numerical elements can be plotted in a Euclidean space of the appropriate dimensionality; and that accordingly, before allowing such a plot to influence us to classify the geometry of the given manifold as Euclidean, we must pause long enough to ask whether the rest of the Euclidean system fits into the picture. If the square root of the sum of the squares of the coordinate-differences between two elements possesses significance in the given continuum, and if it is invariant between observers of that continuum who employ different bases of reference, then and only then may we allege the Euclidean character of the given continuum.

If under this test the given continuum fails of Euclideanism, it is in order to ask what type of geometry it does present. If it is of such character that the “distance” between two elements possesses significance, we should answer this question by investigating that distance in the hope of discovering a non-Euclidean expression for it which will be invariant. If it is not of such character, we should seek some other characteristic of single elements or groups of elements, of real physical significance and of such sort that the numerical expression for it would be invariant.

If the continuum with which we have to do is one in which the “distance” between two elements possesses significance, and if it turns out that the invariant expression for this distance is not the Pythagorean one, but one indicating the non-Euclideanism of our continuum, we say that this continuum has a “curvature.” This means that, if we interpret the elements of our continuum as points in space (which of course we may properly do) and if we then try to superpose this point-continuum upon a Euclidean continuum, it will not “go”; we shall be caught in some such absurdity as trying to force a sphere into coincidence with a plane. And of course if it won’t go, the only possible reason is that it is curved or distorted, like the sphere, in such a way as to prevent its going. It is unfortunate that the visualizing of such curvature requires the visualizing of an additional dimension for the curved continuum to curve into; so that while we can picture a curved surface easily enough, we can’t picture a curved three-space or four-space. But that is a barrier to visualization alone, and in no sense to understanding.

It will be observed that we have now a much broader definition of non-Euclideanism than the one which served us for the investigation of Euclid’s parallel postulate. If we may at pleasure accept this postulate or replace it by another and different one, we may presumably do the same for any other or any others of Euclid’s postulates. The very statement that the distance between elements of the continuum shall possess significance, and shall be measurable by considering a path in the continuum which involves other elements, is an assumption. If we discard it altogether, or replace it by one postulating that some other joint property of the elements than their distance be the center of interest, we get a non-Euclidean geometry. So for any other of Euclid’s postulates; they are all necessary for a Euclidean system, and in the absence of any one of them we get a non-Euclidean system.

Now the four-dimensional time-space continuum of Minkowski is plainly of a sort which ought to make susceptible of measurement the separation between two of its events. We can pass from one element to another in this continuum—from one event to another—by traversing a path involving “successive” events. Our very lives consist in doing just this: we pass from the initial event of our career to the final event by traversing a path leading us from event to event, changing our time and space coordinates continuously and simultaneously in the process. And while we have not been in the habit of measuring anything except the space interval between two events and the time interval between two events, separately, I think it is clear enough that, considered as events, as elements in the world of four dimensions, there is a less separation between two events that occur in my office on the same day than between two which occur in my office a year apart; or between two events occurring 10 minutes apart when both take place in my office than when one takes place there and one in London or on Betelgeuse.

It is not at all unreasonable, a priori, then, to seek a numerical measure for the separation, in space-time of four-dimensions, of two events. If we find it, we shall doubtless be asked just what its subjective significance to us is. This must be answered with some circumspection. It will presumably be something which we cannot observe with the visual sense alone, or it would have forced itself upon our attention thousands of years ago. It ought, I should think, to be something that we would sense by employing at the same time the visual sense and the sense of time-passage. In fact, I might very plausibly insist that, by my very remarks about it in the above paragraph, I have sensed it.

Minkowski, however, was not worried about this phase of the matter. He had only to identify the invariant expression for distance; sensing it could wait. He found, of course, that this expression was not the Euclidean expression for a four-dimensional interval. He had discarded several of the Euclidean assumptions and could not expect that the postulate governing the metric properties of Euclid’s space would persist. Especially had he violated the Euclidean canons in discarding, with Einstein, the notion that nothing which may happen to a measuring rod in the way of uniform translation at high velocity can affect its measures. So he had to be prepared to find that his geometry was non-Euclidean; yet it is surprising to learn how slightly it deviates from that of Euclid. Without any extended discussion to support the statement, we may say that he found that when two observers measure the time- and the space-coordinates of two events, using the assumptions and therefore the methods of Einstein and hence subjecting themselves to the condition that their measures of the pure time-interval and of the pure space-interval between these events will not necessarily be the same, they will discover that they both get the same value for the expression upper S equals StartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared plus left-parenthesis upper Z minus z right-parenthesis squared minus left-parenthesis upper C upper T minus upper C t right-parenthesis squared EndRoot period If our acceptance of this as the numerical measure of the separation in space-time between the two events should lead to contradiction we could not so accept it. No contradiction arises however and we may therefore accept it. And at once the mathematician is ready with some interpretative remarks.

The invariant expression for separation, it will be seen, is in the same form as that of the Euclidean four-dimensional invariant save for the minus sign before the time-difference (the appearance of the constant C in connection with the time coordinate t is merely an adjustment of units; see page 153). This tells us that not alone is the geometry of the time-space continuum non-Euclidean in its methods of measurement, but also in its results, to the extent that it possesses a curvature. It compares with the Euclidean four-dimensional continuum in much the same way that a spherical surface compares with a plane. As a matter of fact, a more illuminating analogy here would be that between the cylindrical surface and the plane, though neither is quite exact. To make this clear requires a little discussion of an elementary notion which we have not yet had to consider.

Our three-dimensional existence often reduces, for all practical purposes, to a two-dimensional one. The objects and the events of a certain room may quite satisfactorily be defined by thinking of them, not as located in space, but as lying in the floor of the room. Mathematically the justification for this viewpoint is got by saying that we have elected to consider a slice of our three-dimensional world of the sort which we know as a plane. When we consider this plane and the points in it, we find that we have taken a cross-section of the three-dimensional world. A line in that world is now reduced, for us, to a single point—the point where it cuts our plane; a plane is reduced to a line—the line where it cuts our plane; the three-dimensional world itself is reduced to our plane itself. Everything three-dimensional falls down into its shadow in our plane, losing in the process that one of the three dimensions which is not present in our plane.

For simplicity’s sake it is usual to take a cross-section of space parallel to one of our coordinate axes. We think of our three dimensions as extending in the directions of those axes; and it is easier to take a horizontal or vertical section which shall simply wipe out one of these dimensions than to take an oblique section which shall wipe out a dimension that consists partly of our original length, and partly of our original width, and partly of our original height.

If we have a four-dimensional manifold to begin with, we may equally shake out one of the four dimensions, one of the four coordinates, and consider the three-dimensional result of this process as a cross-section of the original four-dimensional continuum. And where, in cross-sectioning a three-dimensioned world, we have but three choices of a coordinate to eliminate, in cross-sectioning a world of four dimensions we have four choices. By dropping out either the x, or the y, or the z, or the t, we get a three-dimensioned cross-section.

Now our accustomed three-dimensional space is strictly Euclidean. When we cross-section it, we get a Euclidean plane no matter what the direction in which we make the cut. Likewise the Euclidean plane is wholly Euclidean, because when we cross-section it in any direction whatever we get a Euclidean line. A cylindrical surface, on the other hand, is neither wholly Euclidean nor wholly non-Euclidean in this matter of cross-sectioning. If we take a section in one direction we get a Euclidean line and if we take a section in the other direction we get a circle (if the cylindrical surface be a circular one). And of course if we take an oblique section of any sort, it is neither line nor circle, but a compromise between the two—the significant thing being that it is still not a Euclidean line.

The space-time continuum presents an analogous situation. When we cross-section it by dropping out any one of the three space dimensions, we get a three-dimensional complex in which the distance formula is still non-Euclidean, retaining the minus sign before the time-difference and therefore retaining the geometric character of its parent. But if we take our cross-section in such a way as to eliminate the time coordinate, this peculiarity disappears. The signs in the invariant expression are then all plus, and the cross-section is in fact our familiar Euclidean three-space.

If we set up a surface geometry on a sphere, we find that the elimination of one dimension leaves us with a line-geometry that is still non-Euclidean since it pertains to the great circles of the sphere rather than to Euclidean straight lines. In shaking Minkowski’s continuum down into a three-dimensional one by eliminating any one of his coordinates, if we eliminate either the x, the y or the z, we have left a three-dimensional geometry in which the disturbing minus sign still occurs in the distance-formula, and which is therefore still non-Euclidean. If we omit the t, this does not occur. We see, then, that the time dimension is the disturbing factor, the one which gives to space-time its non-Euclidean character so far as the possession of curvature is concerned. And we see that this curvature is not the same in all directions, and in one direction is actually zero—whence the attempted analogy with a cylinder instead of with a sphere.

Many writers on relativity try to give the space-time continuum an appeal to our reason and a character of inevitableness by insisting on the lack of any fundamental distinction between space and time. The very expression for the space-time invariant denies this. Time is distinguishable from space. The three dimensions of space are quite indistinguishable—we can interchange them without affecting the formula, we can drop one out and never know which is gone. But the very formula singles out time as distinct from space, as inherently different in some way. It is not so inherently different as we have always supposed; it is not sufficiently different to offer any obstacle to our thinking in terms of the four-dimensional continuum. But while we can group space and time together in this way, [this does not mean at all that space and time cease to differ. A cook may combine meat with potatoes and call the product hash, but meat and potatoes do not thereby become identical.]²²³

To the layman there is a great temptation to say that while, mathematically speaking, the space-time continuum may be a great simplification, it does not really represent the external world. To be sure, you can’t see the space-time continuum in precisely the same way that you can the three-dimensional space continuum, but this is only because Einstein finds the time dimension to be not quite freely interchangeable with the space dimension. Yet you do perceive this space-time continuum, in the manner appropriate for its perception; and it would be just as sensible to throw out the space continuum itself on the ground that perception of the two is not of exactly the same sort, as to throw out the space-time continuum on this ground. With appropriate conventions, either may stand as the mental picture of the external world; it is for us to choose which is the more convenient and useful image. Einstein tells us that his image is the better, and tells us why.

Before we look into this, we must let him tell us something more about the geometry of his continuum. What he tells us is, in its essentials, just this. The observer in a pure space continuum of three dimensions finds that as he changes his position, his right-and-left, his backward-and-forward, and his up-and-down are not fixed directions inherent in nature, but are fully interchangeable. The observers, in the adjoined sketch, whose verticals are as indicated by the arrows, find very different vertical and horizontal components for the distance between the points O and P; a similar situation would prevail if we used all three space directions. The statement analogous to this for Einstein’s four-dimensional continuum of space and time combined