Part 3 — CHAPTER XXXVII THE THEORIES OF WEYL AND EDDINGTON

THE general theory of relativity reduces to an application of Riemann’s purely mathematical discoveries, to the real world of space-time in which we live. A further extension of the theory became possible when Weyl had succeeded in carrying the geometrical study of manifolds beyond the point where Riemann and his successors had left it. Weyl describes the situation in the following words:

“Inspired by the weighty inferences of Einstein’s theory to examine the mathematical foundations anew, the present writer made the discovery that Riemann’s geometry goes only halfway towards attaining the ideal of a pure infinitesimal geometry. It still remains to eradicate the last element of geometry ‘at a distance,’ a remnant of its Euclidean past.”[126]

In order to understand Weyl’s theory, which may be regarded as a generalisation of Einstein’s, it will be necessary to enquire into the nature of Weyl’s geometrical contributions. Let us recall that when discussing the geometry of space, we said that the geometry we attributed to conceptual space was defined by the numerical results of measurement which were obtained when we had defined the type of conceptual measuring rods which we intended to employ. The behaviour of these measuring rods was arbitrary to a great extent, and according to the behaviour we attributed to them, the space was Euclidean, Riemannian or Lobatchewskian.

However, certain limitations were imposed by Riemann. Riemann assumed that two unit rods which coincided at a point would continue to coincide when displaced to another point , regardless of whether the two rods had followed the same or different paths through space during their displacement from to . Now it is possible to remove this restriction imposed by Riemann without falling into any logical inconsistency; and we may perfectly well assume that two unit rods which coincide at a point may cease to coincide when brought together at another point , provided they have followed different routes while moving from to . From a purely mathematical angle, such an assumption is by no means absurd; and it is this generalisation of the possible behaviour of our conceptual measuring rods which was studied by Weyl. It must be borne in mind, however, that for the present we are not discussing the problem of real space; we are not arguing about the actual behaviour of our rigid measuring rods as defined by practical congruence. We are merely speculating, as mathematicians, on a logical extension of geometry.

Let us therefore review our premises. We assume that two unit rods which coincide at a point cease to coincide when brought together at another point , after having followed different paths of transfer from to . Obviously, if this be the case, the statement that two lengths situated at different points and of space are equal in magnitude is necessarily indeterminate. The two lengths might measure out as equal if we displaced our unit rod from to along a certain path, and as unequal if we displaced our unit rod along some other path. In other words, length is non-transferable. In view of the indeterminateness that surrounds the comparison of lengths in different places, we must confine ourselves to the comparison of lengths at any one place or at points separated by infinitesimal intervals. We therefore agree to consider that at every point in space there is fixed a rod which is to serve as unit of length when we measure lengths situated by its side. The totality of these unit rods constitutes what is known as a gauge-system. These gauges may be selected arbitrarily; and it is not meant to imply that they are necessarily of equal length with respect to one another, since we have seen that comparison of lengths in different places is ambiguous. Their sole utility is to serve as standards of measurement for lengths situated by their sides, and thus to enable us to compare lengths situated at the same point.

A gauge-system is the natural extension of a mesh-system, with which we are already acquainted. Just as we may vary the shape of the mesh-system, so now we may vary the nature of the gauge-system.

Let us endeavour to ascertain the significance of this non-transference of length as relating to the nature of space itself. In particular, let us suppose that two rods coincide at a point ; then, while one rod remains fixed at , the other is made to follow a closed curve, returning finally to its starting point. Owing to the difference in routes followed by the two rods, one having remained fixed and the other been made to describe a curve, the two rods will no longer coincide when we place them side by side after the completion of the circuit.

This non-transference of length is very similar to what is known as the non-transference of direction which holds for the non-Euclidean spaces that we have studied so far. Thus, consider a curved surface, a sphere—for example. If at a point on the sphere two rods coincide in direction, and if, while one rod remains fixed, the other is displaced from point to point along a closed curve, and displaced in such a way that for each successive infinitesimal displacement the rod maintains the same direction, it would be erroneous to suppose that when the second rod had been returned to its starting point its direction would still coincide with that of the first rod. A distinct discrepancy would be found to exist between the directions of the two rods, and the size of this discrepancy would depend on the area contained within the closed curve which the second rod had been made to follow over the surface. This discrepancy has nothing to do with our choice of a mesh-system; it transcends this choice completely. On the spherical surface we are discussing, it would vary with the closed curve described by the rod; but if we had operated on a plane sheet of paper instead of on a surface curved like a sphere, the discrepancy would never have arisen at all. Its existence is due, therefore, to the curvature, or non-Euclideanism, of the surface, and its magnitude may serve as a means of determining the precise intensity of this non-Euclideanism around every point of the surface.

Just as non-transference of direction is caused by the curvature, or non-Euclideanism, of space (which we might also call direction-curvature), and disappears with the vanishing of this curvature, so in an analogous manner the non-transference of length is caused by a new type of curvature (called Weylian curvature or distance-curvature), and disappears completely when this type of curvature is non-existent. In a general way we may call a space a Weyl space when it possesses a non-vanishing distance-curvature; the appellation classical spaces will be reserved for all the categories of space which we have studied thus far (Euclidean, Riemannian, Lobatchewskian, semi-Euclidean), since in all these spaces a Weylian or distance curvature is non-existent.

We may note that just as ordinary non-Euclideanism, or direction-curvature, transcended our choice of mesh-system, so now, in similar fashion, the Weylian characteristics—the non-transference of length, and, again, the presence of distance-curvature—transcend our choice of a gauge-system.

Now we have seen that the Euclideanism or non-Euclideanism of a manifold was ascribed to the metrical field which it contained. In a similar way we must assume that the Weylian characteristics must also be ascribed to the metrical field. However, the metrical field due to the ’s which has been discussed so far, is incapable of accounting for Weylian peculiarities. It is necessary to complete the description of the field in order to render it accountable for these new characteristics. When this is done, it is found that in a four-dimensional continuum, in addition to the ten ’s which suffice to describe the metrical field of a four-dimensional classical continuum such as space-time, it is necessary to add four new quantities which we may call the four ’s. The generalised metrical field then contains fourteen separate magnitudes in place of ten as in Einstein’s space-time. Just as the direction-curvature of space characterising non-Euclideanism at a point could be expressed by a tensor (the Riemann-Christoffel tensor), built up from the changes in values between the ’s at a point and at the neighbouring points in any mesh-system, so now the distance-curvature at a point can be given by a tensor built up from the changes in values between the ’s at a point and at the neighbouring points in any gauge-system.

In particular, the Weylian curvature , at a point, is given by the tensor equation whence we may derive If the space manifests no trace of distance-curvature, hence no Weylian characteristics, vanishes completely and the preceding equations are superfluous; we are then in the presence of a classical space, whether Euclidean, Riemannian or Lobatchewskian.

Now it is a remarkable fact that these purely geometrical relations or equations which Weyl had discovered, expressing as they do the value of the Weylian curvature, are none other than Maxwell’s equations of the electromagnetic field, provided we identify the Weylian curvature with the tensor whose components represent electric and magnetic force, and provided we identify the elements of structure, the ’s, with the electromagnetic potentials, the ’s. To be sure, this unexpected parallelism between the equations of electromagnetics and the curvatures of a Weyl space might have been attributed to mere chance. But, as must be admitted, the temptation to view matters otherwise was great. Weyl assumed, therefore, that space-time was of the Weylian type and not of the more restricted classical species, as Einstein had supposed. The electric and magnetic field tensor , which in Einstein’s space-time appeared to be a foreign entity intruding on the continuum, was then in reality none other than the Weylian curvature of this generalised space-time. These views led Weyl to assume that when electromagnetic fields were present, hence when did not vanish, the Weylian characteristics of space-time manifested themselves, whereas, in regions free of electromagnetic fields, space-time resumed its classical characteristics and became once again the space-time of Einstein’s theory. For similar reasons, the four electromagnetic potentials, i.e., the four ’s were identified with the four ’s of the Weylian structure. In this way, electromagnetic phenomena, to the same extent as gravitational phenomena, appeared to be necessary manifestations of the metrics of the fundamental space-time continuum.

Let us consider the further modifications which these new views will impose. First, we remember that in the classical spaces, hence in Einstein’s space-time, it was possible to build up fundamental or structural tensors (, and ) from the ten ’s. These magnitudes, being tensors, transcended, of course, our choice of mesh-system. But now that we are considering a Weyl space, we must superimpose the additional requirement that our magnitudes should transcend a choice both of mesh-system and of gauge-system. We thus obtain the magnitudes , , , which are built up from the ten ’s and the four ’s (now identified with the four ’s) and which in addition satisfy all our previously mentioned requirements. In order to distinguish these magnitudes from ordinary tensors, which transcend solely our choice of mesh-system, Eddington calls them in-tensors and in-invariants, and indicates them by asterisks. Henceforth it will be to these in-tensors and in-invariants that we shall have to appeal in order to express those magnitudes which (depending neither on mesh-system nor on gauge) are intrinsic to the world itself.[127]

Now, assuming space-time to be a Weylian manifold, a change of gauge-system will modify the expression of the fundamental tensors of Einstein’s theory (except ), hence will modify the aspect of the mechanical laws and of that of gravitation. A natural extension compels us to assume that all the Einsteinian laws should be generalised and expressed in terms of in-tensors and in-invariants, so as to yield in-tensor equations instead of tensor equations. Only thus can we render them indifferent to a change of gauge-system, just as their original tensor form had already rendered them indifferent to a change of mesh-system. This modification in the form of the Einsteinian laws will affect only the mechanical and gravitational laws, since the electromagnetic ones defining the Weylian curvature are already in-tensor equations.

However, we may assume that the Einsteinian tensor laws, in the form in which they were given by Einstein, are in harmony with some particular gauge-system; and this Weyl calls the natural gauge. We shall have to conceive of this natural-gauge system as one imposed upon us by nature and defined by certain intrinsic properties of the universe. But the only determinate magnitude which is offered us by nature at every point appears to be the radius of curvature of the universe at every point. Hence Weyl defined the natural gauge at every point as given by a unit of length equal to the radius of curvature of the universe at that point or to some definite fraction thereof. If these ideas are correct, the universe cannot help but possess a curvature at every point, and hence must be finite, since were it infinite there would be no natural gauge whereby to measure distances.

Weyl’s theory, as outlined so far, represents the theory in its original form; but we shall see that Weyl was led to modify his original views in certain respects. The fact is that empirical results would appear to conflict with his theory in its initial form. The theory compels us to assume that in regions where electromagnetic fields are present, in regions, therefore, where space-time is Weylian, transference of length is an indeterminate process. Of course, when we speak of a length in space-time, we are primarily considering a space-time distance, that is, an Einsteinian interval. But an Einsteinian interval, we remember, can be split up into a spatial distance, or into a duration, or into both, as the case may be. This result is achieved when the observer refers the Einsteinian interval to his space and time mesh-system. It would follow, therefore, that in a Weylian space-time not only would the transference of length of an Einsteinian interval yield indeterminate results, but that the same would apply automatically to the transference of spatial, as also of temporal, magnitudes. Thus, two identical electrons situated, first, at any one point then separated, would in general differ in size and in charge when brought together again. Likewise, the same indeterminateness would apply to duration. Hence, it would be quite in order to suppose that two identical atoms situated at the same point and beating with the same frequency would present different frequencies of vibration after they had been separated and brought together again. It would be as though the size of an atom and its frequency of vibration depended on its previous life history. Incidentally, these occurrences would deprive the Einstein shift-effect of all value, since any slowing down in the rate of vibration of the solar atom could be attributed to a difference in the previous life histories of the solar and terrestrial atoms, and not necessarily to the influence of the sun’s field of gravitation.

Now, all these anticipations of Weyl’s theory in its original form are in conflict with precise observation. So far as can be detected by experiment, the charges and sizes of all electrons are exactly alike regardless of their past history; and this also applies to the frequencies of vibration of atoms of the same chemical element.[128] It would appear, therefore, that if space-time is a Weyl space, the Weylian characteristics are, for some reason or other, completely obliterated; so that, in fine, space-time, though Weylian, manifests only the properties of the more restricted Riemann type of space.

Weyl was thus compelled to draw a distinction between adjustment and persistence, and to assume that bodies adjusted themselves automatically to the natural gauge, that is to say, to the radius of the universe. If bodies followed the tendency of persistence, the Weylian discrepancies would be observed; but with bodies adjusting themselves afresh from place to place and from instant to instant to the form of the universe, and with the frequencies of vibration of atoms acting likewise, these discrepancies would be banished automatically. We may note the similarity between these views and those reached by Eddington in Einstein’s theory proper (see chapter on the finiteness of the universe).

These views also enable us to attain a better understanding of the significance of the natural gauge. Since objects in equilibrium, such as rigid objects, electrons and atoms, all adjust themselves to the radius of curvature of the universe, it is only natural that results of greatest simplicity should be obtained when we select this radius or some definite fraction thereof as a gauge at every point.

Now, with the restrictions we have imposed, it may appear that there is nothing left of Weyl’s theory, since we have deprived the Weylian space-time of all its characteristics by invoking the tendency of adjustment in place of that of persistence. This view would, however, be too extreme. It is true that Weyl’s identification of the electromagnetic potentials, the ’s, with the ’s or particularities of structure of the continuum, now appears more in the light of a graphical representation; but as such it remains useful. Indeed, we must remember that the original urge which rendered a theory such as Weyl’s desirable was the necessity of reaching a more unified understanding of nature by including at least the representation of electrical manifestations in the structure of the fundamental continuum of space-time, instead of having to regard them as foreign occurrences. From this standpoint, Weyl’s theory appears secure.

But it is when we come to consider the problem of the Action of the universe, that Weyl’s theory opens up the most interesting possibilities. We remember that the action would constitute, so to speak, the nucleus whence all the laws of nature would issue. In fact, were the action known, all we should have to do would be to express its stationary condition, and the laws of matter and electricity would be revealed. But the trouble is, we do not know what the action may be; we can only guess at it. In this respect Weyl’s theory proves of help. For whereas Einstein’s theory restricts the action to being a space-time invariant, Weyl’s theory allows us to narrow down the choice by imposing the more stringent conditions of in-invariance. As a result, the Gaussian curvature or , which Einstein had assumed to constitute the function of action of the gravitational field, must be discarded and replaced by , an in-invariant.[129] We are thus led to the law of gravitation with the term, that is to say, to the finite universe.

Furthermore, the ideal of unity towards which science tends incessantly, demands that the action be built up from the structural elements of the fundamental continuum. In Einstein’s theory this was impossible, since the electromagnetic potentials (i.e., the ’s) were foreign to the space-time structure. It followed that we were faced with two separate actions, the gravitational action and the electromagnetic one. Weyl’s theory permits us to obtain a unification of these actions, since the electromagnetic potentials, the ’s are now represented by the ’s in the generalised space-time structure. Accordingly, Weyl constructed an in-invariant which he suggested tentatively as the function of action of the world. In addition to the gravitational and electromagnetic equations which were derived therefrom, the stationary condition of the action (see Chapter XXXV) entailed certain electromagnetic equations bearing on the constitution of matter. In the light of these equations, and taking into consideration the fact that matter is always of positive density, Eddington concludes: “It would seem to follow that the electron cannot be built up of elementary electrostatic charges, but resolves itself into something more akin to magnetic charges.”

At this stage, a further generalisation of great elegance was given to Weyl’s geometry by Eddington. Here let us recall that whereas in Einstein’s theory the fundamental elements of space-time structure were ten in number (the ten ’s), and whereas in Weyl’s theory they were fourteen (the ten ’s and the four ’s), in Eddington’s still more general manifold we are confronted with 40 quantities (the forty ’s). It is not that Eddington has introduced new elements of structure into the manifold. Far from it: these elements were present in all the types of manifolds we have discussed. But they were crystallized by restrictions which removed them from the mathematician’s control; and what Eddington has done has been to loosen them. For instance, starting from Eddington’s generalised manifold, if we impose certain restrictions on the forty ’s, we are led to Weyl’s manifold; by imposing still others, we are led to the classical manifolds, of which Einstein’s space-time was a particular instance. As a matter of fact, even in Eddington’s manifold, certain restrictions of symmetry defining what is known as the affine condition are still imposed upon the ’s; were it not for these restrictions, there would have been 64 of these ’s. So far as is known to the writer, the highly generalised manifolds which would result have not been studied; but it is interesting to note that they can be conceived of, and no one can foresee whether their introduction may not prove necessary eventually. Since Eddington’s geometry is a generalisation of Weyl’s, we may anticipate certain novel features in it. Thus, in Weyl’s geometry, we have seen that the length of a rod would vary were it to be displaced from one point to another. With Eddington’s generalisation, we must assume that a change of length would also arise if the rod were rotated round one of its points, hence if its orientation were modified, even though the rod were not displaced as a whole.

And here a few words may be said with respect to the purpose of these successive generalisations. The ultimate goal is one of unification. We wish to succeed in representing the entire physical universe in terms of the relationships of structure of the fundamental space-time continuum. Einstein’s space-time enabled us to account for gravitation in this way, but the electromagnetic field appeared as a foreign invasion. None of the known elements of structure seemed capable of accounting for it. It seemed as though the world of space-time, gravitating masses and energy might just as well have existed in the absence of this electrical intrusion. Now, we have every reason to believe that such cannot be the case, for since matter is built up of protons and electrons, the annihilation of electricity would entail that of matter, gravitation and energy, and possibly also of space and time. In order to overcome this difficulty, it appeared necessary to generalise our conception of space-time, obtaining thereby a continuum presenting additional elements of structure which might be identified with electric and magnetic forces. And so we were led to Weyl’s generalisation, which permits the electromagnetic field to enter into the general synthesis, no longer as a foreign adjunct, but as a constituent element of space-time structure. But, besides the field, there is the electron, and the field alone does not seem to afford us the possibility of building up the electron; we cannot account for atomicity or for the quanta of action. Therein resides the problem of matter, and it remains to-day an outstanding challenge to science. Neither can we understand the significance of the two different kinds of electricity, the positive and negative. So long as all these mysteries remain unsolved, it is only natural that we should seek new elements of structure by proceeding with further generalisations. Eddington’s theory was conceived of with this object in view.

There exist certain interesting features about Eddington’s theory which we may mention briefly. Inasmuch as the intrinsic magnitudes of the universe must be in-tensors and in-invariants, instead of tensors and invariants, the ’s lose their position of pre-eminence; for the ’s, while transcending our mesh-system, fail to transcend our choice of a gauge-system. Accordingly, in the expression of a distance, we must replace by some in-tensor of a similar nature. The only fundamental structural in-tensor that presents itself is , or some definite fraction of it such as where is an arbitrary constant. Hence, we must put in place of The identification of the new expression with that employed by Einstein leads to the tensor equation , which, when no electromagnetic fields are present, reduces to . This is the equation of a finite universe, so we see that the finiteness of the universe appears to constitute a necessary consequence of the theory.

A further point is also worthy of note. Gravitation, which involves the ’s, is given by a symmetrical tensor; electricity, on the other hand, is given by an anti-symmetrical tensor . Now, in the theories of both Weyl and Eddington, the in-tensor takes the place of Einstein’s tensor . But whereas is symmetrical, splits up into a symmetrical and an anti-symmetrical part; these two parts are given by and . In other words, the world tensor splits up of its own accord into gravitation and electricity, showing thereby the deep union between the two.

As late as 1922, Einstein did not show much enthusiasm for these new generalisations, but subsequently he appears to have modified his views, going so far as to investigate the problem of the world-action in the light of Eddington’s geometry.

He first studied a possible mathematical expression for the action, one which had already been suggested by Eddington. It was a strikingly simple expression representing certain of the generalised space-time curvatures; no longer a mere stringing together of the various actions. When the principle of stationary action was applied to it, it fell apart, yielding the law of gravitation on one side and the laws of electricity on the other. The mathematical results obtained by Einstein in this respect point to a general diffusion of electricity throughout the world; and as in the case of Weyl’s treatment, magnetic and not electric charges appear to lie at the basis of matter. Subsequently, Einstein proved that other expressions for the action would yield the same results.

Yet, in spite of all these remarkable successes, the outlook is none too promising. The problem of matter is no nearer a solution than before. Indeed, it is highly probable that some totally new departure will have to be considered. Here it must be remarked that the attempts we have been outlining follow the methods of field physics; that is to say, the entire physical universe is assumed to be built up of the ’s and ’s or other magnitudes of the same kind. But a number of facts and phenomena render the success of this attitude doubtful.

In the first place, in the case of the electron, the ratio of its charge to its mass is of the order of .[130] It follows that the electrostatic repulsion which two distant electrons exert upon each other is times greater than their mutual gravitational attraction. Now the existence of so high a number, constituting one of the characteristics of the electron, seems to preclude the possibility of our deducing the existence of the electron from the field equations. It would appear as though some other controlling influence were active. We might possibly have to seek for this influence in the total number of electrons existing in the universe, a number determined, presumably, by chance.

Field physics, together with the principle of action, encounters further difficulties when we consider the quantum phenomena and the existence of Bohr’s atom. In a number of cases, therefore, there appears to be something deeper behind the field itself. If so, we should have to limit the rôle of the field to a purely passive part, to that of transmitting effects, not of engendering them. If, however, we abandon all attempts at interpreting matter in terms of the field quantities, this does not mean that we must relapse into the discarded idea of substance. In the present state of our knowledge all these problems are so confused that it seems useless to state anything definite about them. All we can say is that we are faced with a something which we do not understand. Weyl expresses these ideas in the following words:

“If Mie’s views are correct, we could recognise the field as objective reality, and physics would no longer be far from the goal of giving so complete a grasp of the nature of the physical world, of matter and of natural forces, that logical necessity would extract from this insight the unique laws that underlie the occurrence of physical events. For the present, however, we must reject these bold hopes. The laws of the metrical field deal less with reality itself than with the shadow-like extended medium that serves as a link between material things, and with the formal constitution of this medium that gives it the power of transmitting effects.” And elsewhere: “We must here state in unmistakable language that physics at its present stage can in no wise be regarded as lending support to the belief that there is a causality of physical nature which is founded on rigorously exact laws.”

All in all, we must realise that the farther we go, the more inextricable our difficulties become. The existence of matter still remains a mystery.