Part 1 — CHAPTER III RIEMANN’S DISCOVERIES AND CONGRUENCE

THE procedure of presentation of non-Euclidean geometry which we have followed to this point hinges on the parallel postulate, hence on the definition of the straight line. In many respects, a much deeper method of investigation was that pursued by Riemann, founded on the concept of congruence. By congruence we mean the equality of two distances and more generally of two volumes in space. As we have explained elsewhere, the two methods lead to the same results. Indeed, once a metrical geometry has been defined, whether by the method of postulates or by any other means, a corresponding definition of a straight line and of equal or congruent distances is entailed thereby.

Thus, with Euclidean geometry, congruent lengths at different places are exemplified by the lengths spanned by a material rod transported from one place to another. Congruent or rigid objects having thus been defined, a straight line is given by the axis of rotation of a material body, two of whose points are fixed, or again by the shortest distance between two points measured with our rigid rod.

Nevertheless, although the various methods of presentation are equivalent, it may be of advantage to make the definition of congruence fundamental rather than that of the straight line. Such was the procedure followed by Riemann.

When we revert to experience for an understanding of congruence, we find it exemplified in the rigid bodies of nature, whose geometrical dispositions yield, more or less precisely, the results of pure Euclidean geometry. If we idealise congruence, as thus defined, and express it mathematically, we may say that perfectly rigid bodies are those whose measurements would yield Euclidean results with absolute precision. But though the mathematician has thereby eliminated from his definitions the inaccuracies attendant on physical measurements, his understanding of congruence reduces to a mere idealised copy of the behaviour of special bodies found in nature. While he has thus obtained a possible mathematical definition of congruent bodies (that given in nature), it remains to be seen whether other types of congruence would not also be rationally possible. His aim must therefore be to define congruence mathematically, without appealing to experience.

When, however, we discard the empirical criterion which prompted us to define material bodies as rigid, we find that a unique mathematical definition of rigidity eludes us. For to say that a body remains rigid or congruent with itself during displacement means that the spatial distance between its extremities remains ever the same. But our only means of disclosing this fact is by measuring the body with an admittedly rigid rod at successive intervals of time and noting the continued identity in our numerical results. Hence it follows that the value of these results would be nullified were we to cast any doubt on the maintenance of the rigidity of our measuring rod. And how could we ever justify its rigidity unless we were to compare it with some other rod regarded as rigid, and so on ad infinitum? From all this it appears that a body can be regarded as rigid only with respect to our measuring rod; and in order to ascribe any significance to rigidity we must first admit that our measuring rod is rigid by definition or by convention. We have no other means of establishing this rigidity.

We should reach the same conclusions were we to compare two lengths and situated in different parts of space. We could not say that the two lengths were equal or congruent in any absolute sense merely because our measuring rod could be made to coincide now with , now with . A definition of this sort would obviously presuppose that our measuring rod had remained undeformed or congruent with itself when displaced from to . It would reduce to testing the congruence of and by presupposing that we knew how to recognise the maintenance of congruence in our rod during displacement. In thus defining congruence in terms of congruence our argument would be circular.

The whole trouble arises from the continuity of space which precludes us from attributing any absolute meaning to the statement that two lengths situated in different parts of space are equal or unequal. There is no absolute significance in stating that there is as much space between and as between and . We cannot compare lengths by counting the number of points that they contain, since there are just as many points between the extremities of an inch as between the extremities of a mile—an infinite number in either case.[7]

In short, we see that in mathematical continua, just as in sensory ones, measurement and comparison of lengths can be considered only as a result of some convention; there is nothing in the continua themselves to suggest any definite metrics or geometry. This is what is meant by saying that space is amorphous and presents us with no means of determining absolute shape and size.

Now, it might appear from the preceding discussions that so far as the mathematician is concerned, since a definition of equal distances is purely conventional, any conceptual rod might be chosen for this purpose. We might take a rod which, when moved about, would squirm like a worm, and elevate it to the position of a standard rod to which all other lengths would have to be compared. But this would be too extreme; for although mathematical space possesses no inherent metrics, yet certain requirements are demanded of rods susceptible of being considered as remaining congruent when displaced. It is these requirements, as postulated by Riemann, which we shall now proceed to discuss. Riemann assumed that the necessary requirements would be as follows:

In the first place, he assumed that if we restricted ourselves to infinitesimal volumes of space, congruence would be established by means of Euclidean solids and measuring rods. This postulate is called the postulate of Euclideanism in the infinitesimal.

Furthermore, if our rods remained congruent and we made certain constructions with them (pyramids, etc.), we should be able to transfer these rods and make exactly the same constructions, presenting exactly the same numerical relationships, in any other part of space. This postulate is a mere reflection of our belief in the homogeneity and isotropy of measurements in space, the same everywhere and the same in any direction. When presented in a slightly different form, it is often referred to as the postulate of free mobility.

We may also say that if two rods coincide in a certain region of space, they should continue to coincide when separated, then brought together again in any other region.[8]

With these restrictions imposed, Riemann discovered that the Euclidean type of measurement was only one among others. Two other types, namely, the Lobatchewskian and the Riemannian, were also possible.

Sophus Lie considered the same problem, though from a different standpoint. He argued that a rigid body would be such that when one of its points was fixed, any other of its points would describe a surface. When one of its lines was fixed, any other of it's points would describe a curve, and finally, when three points were fixed, the rigid body would be unable to move.

As a result of these investigations it was proved that there existed bodies which, as contrasted with Euclidean solids, would squirm when displaced. Yet, in spite of this fact, these non-Euclidean bodies would present all the mathematical requirements of congruent bodies. These alternative types of bodies may be called Riemannian bodies and Lobatchewskian bodies, respectively.

If we select Euclidean congruence for the purpose of measuring space, non-Euclidean bodies will appear to squirm as they move; if we select non-Euclidean congruence, it will be the Euclidean bodies which appear to vary in form when displaced. It is usual, however, to reserve the term rigid for the Euclidean bodies; but this is only in order to conform to the ordinary understanding of rigidity as derived from experience. If we omit to take into consideration the physical behaviour of material bodies which has forced a certain conception of rigidity upon us, there appears to be no mathematical reason for assigning greater importance to Euclidean measurements than to non-Euclidean ones. It follows that the non-Euclidean bodies are just as much entitled to the appellation “rigid” as are the better-known Euclidean ones. When it comes to deciding which of the types of rigidity represents conservation of absolute shape and size, the problem appears to be entirely meaningless. Accordingly we shall refer to the various types of bodies as Euclideanly or as non-Euclideanly rigid.

Now it is perfectly obvious that when we measure lengths and ratios of lengths with one type of standard rods or another, we shall obtain conflicting numerical ratios and values. Thus, if we fix one extremity of our standard rod and allow it to rotate in all directions, its free extremity will describe points on a spherical surface regardless of the type of rod or geometry with which we are dealing. In a similar way we might obtain a circumference. If, then, we abide by the Euclidean system of measurement, the value of the ratio of the lengths of circumference and diameter would remain ever the same however great the diameter; we should always obtain the same constant number, known as —equal to 3.141592 ..., first calculated by Archimedes. On the other hand, with Riemannian or Lobatchewskian measurements, we should obtain a variable value for this ratio, always smaller than in the case of Riemannian geometry, decreasing as the diameter increased; and always greater than in the event of our having selected Lobatchewskian measurements. Similar discrepancies would attend the measurement of all other geometrical figures and angles.

And here there is a further point to be considered. Even when we have defined a particular type of measurement, be it Euclidean or non-Euclidean, we have by no means fixed the behaviour of our rods and bodies in any absolute sense; and an understanding of absolute rigidity still escapes us. We can only define Euclidean bodies, for example, as those whose laws of disposition yield Euclidean numerical results. The following illustration will show us that these bodies might all vary in absolute shape when displaced, yet still yield the same Euclidean results.

All we have to do is to consider a transparent plane and a man executing geometrical constructions with Euclidean rods on its surface. If we consider a point-source of light casting the shadows of these rods on some other surface, whether plane or curved, these shadows, considered as measuring rods in turn, will still yield exactly the same Euclidean results, their laws of disposition will remain unchanged, and hence they will be rigid Euclidean rods exactly to the same extent as the original ones. Yet as contrasted with these original rods the shadows would squirm when displaced varying in shape and size, and the Euclidean straight lines would appear curved.

In short we see that absolute shape, straightness, size and rigidity in conceptual mathematical space escape us completely, and the significance of rigidity as portraying the maintenance of an unchanging volume of space, even when considering Euclidean bodies, is indeterminate. In its most extended sense, this is what mathematicians mean by the relativity of space. The sole justification for the general acceptance of the concept of rigidity in the popular sense is due to the presence of material bodies in our universe which we agree to accept as standards. These, incidentally, yield Euclidean results.

Summarising, we see that mathematical space is amorphous; it has no particular metrics, no particular geometry. According to our methods of measurement, we may obtain one geometry or another in the same space. It is often convenient to express all these results by saying that space is Euclidean, Riemannian or Lobatchewskian; but we must be careful to note that space itself has very little to do with the matter.

Let us now examine another aspect of the problem. Until such time as we have fixed our choice on a system of measurements, the amorphous nature of space forbids us to attach any determinate significance to a distance between two points. This, in turn, prevents us from attaching any significance to what may be considered the shortest distance between two points. However, when we have adopted one of the three measuring conventions, the significance of a shortest distance becomes determinate and a definite line joining the two points will be found to embody this shortest distance. Lines of this type are known under the name of geodesics; and in any of the geometries with which we shall be dealing, they play the part taken by the straight line in Euclidean geometry.[9]

Corresponding to every one of the three congruence definitions we have mentioned, there exists a definite type of straight line or geodesic between any two points. With Euclidean congruence we obtain the Euclidean straight line which satisfies Euclid’s parallel postulate, and vice versa. Again, if we adopt one of the two non-Euclidean types of congruence we are led to the non-Euclidean straight lines or geodesics, those which satisfy the non-Euclidean postulates, and vice versa. In short, we see that the two methods of presenting non-Euclidean geometry, either through the metrical or congruence method or through the parallel-postulate method, are in the main equivalent.

In a more thorough treatment of non-Euclidean geometry other non-metrical methods of presentation are sometimes adhered to. One of these is based on the theory of groups and projective geometry; another is due to the discoveries of Levi-Civita and Weyl and depends on the fundamental concept of an infinitesimal parallel displacement; it opens the way to Weyl’s still more general geometry. Still another mathematical method of exploring space is that of continuous tracks which was investigated by Eisenhart and Veblen. In this book we shall refrain from discussing these more difficult methods, for they present too technical an aspect;[10] but it would be a great mistake to assume that these imaginative flights of the pure mathematicians were of no utility to physical science. Apart from the deep philosophical light which they throw on the entire problem of space, we know from past experience that what was at its inception a pure mathematical dream has more than once become at some later date the image of physical reality. Non-Euclidean geometry, and possibly Weyl’s still stranger geometry, are cases in point.

We have now to consider a number of questions pertaining to location and motion in space. Mathematical space, as we have seen, is -dimensional and amorphous. But in order to bring it into closer contact with the physical space of experience we may assume the necessary postulates of order and contiguity to have been specified. We thus obtain three-dimensional mathematical space. Now mathematical space, in view of its amorphous nature, is essentially relative. The very homogeneity of space, its sameness “here” as “there,” precludes our being able to differentiate one absolute position from another. It will follow that absolute motion, i.e., a variation of absolute position in empty amorphous mathematical space, can have no significance.

Our sole means of giving significance to position and motion will be by selecting some three-dimensional frame of reference as standard. For instance, we might consider three mutually perpendicular axes meeting at a point. A more concrete illustration would be afforded by considering the two adjoining walls and the floor of our room. Then, after having selected the standard rods which we intend to employ, and also a clock, we should proceed to measure the position and change of position of an object with respect to our room. This would yield us the relative position and motion of the object (relative to our room).

Now, if our choice of a frame of reference, and our methods of measurement, were not arbitrary, but were imposed in some inevitable way by the texture or nature of space, we might still claim that position and motion, though relative to our frame, were yet absolute, since they could always be determined without ambiguity. But this inevitability of a particular frame of reference is precisely what the amorphous nature of mathematical space renders impossible, for there is nothing in mathematical space which permits us to single out any particular frame. Hence no absolute unique determination of position and motion can be contemplated. In fine, we must conclude that in mathematical space position and motion, like distance and shape, are entirely relative.

Thus far we have been considering mathematical space; and mathematical space is but an abstraction from the space of experience. It remains to be seen, therefore, whether our conclusions will apply without modification when we consider the physical space of experience. Here we may state that so far as absolute position is concerned there is nothing to change in our conclusions. Thus, in everyday life, when we speak of returning to the same point in space, we do not really mean the same point in empty space. We mean rather the same point on the earth’s surface; and we do not stop to consider whether the earth has moved through space in the interval. As a matter of fact it would be totally impossible for us to discover exactly what distance the earth had travelled. Granting that we might ascertain the displacement of the earth along its orbit round the sun, all further progress would be arrested by our ignorance of the velocity of the sun through space. We might even measure the velocity of the sun with respect to the stars, but this again would not yield us its velocity through empty space; for we should still remain in complete ignorance of the velocity of the stars as a whole through space. So we see that when we speak of the same point in physical space we are always referring to the same relative position with respect to some observable frame of reference. It is not absolutely necessary that the frame be material, and we might perfectly well conceive of it as being defined by three perpendicular light rays. The essential requirement is that it should be observable, so that all men should agree on the frame they were talking about.

Owing to the important part the earth’s crust occupies in our daily lives, the impossibility of defining the same point at different times without the help of a frame of reference is somewhat obscured. But it is imperative to master this aspect of the elusiveness of space before proceeding to the study of the more complex questions raised by the theory of relativity.

When, however, we consider the problem of motion, we shall find that in contradistinction to what holds for mathematical space, motion in physical space appears to be absolute. While it is true that from a purely visual standpoint the motion of a body will depend essentially on the frame of reference we select, and will vary when we change our frame from the earth to, say, the moon or the sun, yet owing to the generation of dynamical effects it is possible to decide in an absolute way whether the body is rotating or not. This is due to the fact that these telltale dynamical effects are present just the same regardless of our choice of a frame of reference. It was owing to these empirically discovered dynamical facts that Newton felt compelled to accept the absolute significance of motion, hence of space. We may gather from what has been said that we may not extend lightly to the physical space of experience the conclusions derived from conceptual mathematical space.

We now come to a last point. Suppose that in mathematical space we wish to measure the distance between two objects moving in various ways through our frame. For our measurements to have any significance, we must of course specify that the positions of the two objects correspond to the same instant of time. If, therefore, a change in the relative motion of our frame of reference were assumed to cause a modification in our understanding of the sameness of a time at two different places, the distance between the two objects would vary with the frame in which we were reckoning it. But here we are discussing pure mathematical space, which we may always conceive of as divorced from time. Hence we may assume that the sameness of a time at two different places remains unaffected by any alteration in the choice of our frame.

Classical science believed that the space of experience would also satisfy these conditions. It thought that time was absolute, that its rate of flow was ever the same, and that the concept of the sameness of a time at two different places was an absolute and unambiguous one. This assumption was tantamount to divorcing space from time and regarding them as two essentially different categories. It has been one of the triumphs of the relativity theory to disprove this fundamental thesis, and to show that in the real world of experience a continuum of space by itself is but a shadow. Henceforth the continuum of reality is space-time, a four-dimensional continuum of events, and no longer, as in classical science, a three-dimensional one of points, coupled with a one-dimensional continuum of instants. But inasmuch as the necessity for these new conceptions will be justified at length in future chapters, we shall refrain from discussing them at this stage.