CHAPTER IVTHE PROBLEM OF PHYSICAL SPACE
THUS far we have discussed more especially that abstraction from the space of experience which we called mathematical space; we have seen it to be entirely amorphous. In the present chapter we shall have to consider the space of experience and determine to what extent it differs from conceptual mathematical space. An important difference as regards the problem of motion has already been noted, for we remember that in physical space, motion often manifests itself as absolute. For the present, however, we shall confine our attention to the problem of congruence.
Mathematical space is amorphous; it possesses no intrinsic metrics, and our choice of standards of measurement is largely arbitrary. As a result, absolute shape, size and straightness are meaningless concepts. But in physical space, it is a matter of common knowledge that men have no difficulty in agreeing at least approximately on the sameness of two shapes or of two sizes. They agree that a stone remains undeformed when displaced, hence declare the stone to be rigid, whereas they recognise that an object behaving like a worm is of the squirming variety.
So here there appears to exist an important difference between mathematical space, where no particular definition of congruence is suggested, and physical space, where a definite type seems to impose itself naturally and is accepted unanimously. After all, there is nothing very mysterious about this unanimous agreement; for had men refused to be guided in their definition of congruence by their sense of sight, they would have been led into all manner of difficulties. They would have had to assume that a stone carried in their hands, though appearing unchanged visually, was yet squirming, and, vice versa, that bodies which appeared to squirm were yet rigid. The difficulties in reaching some common understanding of measurement would thus have been hopelessly great. In fact, the definition of congruence, which we have mentioned, imposes itself so irresistibly that it is only since the discovery of non-Euclidean geometry that its absolute validity has been refuted.
Viewing the situation as it now stands, we may say that material rods, visually recognised as rigid, will be taken as norms of measurement in physical space. In other words, it will be assumed that rods which coincide when brought together will continue to remain congruent when transferred, independently of one another, to other regions of space. Of course, the physicist will guard himself as much as possible against local contingent influences, such as variations of pressure and temperature, which might influence the behaviour of his rods. But when all these elementary precautions have been observed, the geometry determined by our rods will automatically become thegeometry of space. This physical definition of congruence may be termedpractical congruence, as distinguished fromtheoretical congruence, which is embodied by the mathematical types we have discussed.
With his standard rigid bodies defined in this way by physical objects, the physicist can perform measurements in the space of his frame and in consequence obtains the numerical results of three-dimensional Euclidean geometry. Inasmuch as the more careful he is to guard against such contingent influences as variations of temperature, the more accurately will his numerical results approximate to those of pure Euclidean geometry, he feels justified in stating that rigid objects behave like rigid Euclidean bodies and that the space of our experience is rigorously Euclidean.
We may also recall that his rigid bodies having been defined, the definition of a straight line as the axis of rotation of a revolving solid, two of whose points are fixed, or as the shortest distance between two points in space, follows immediately; and of course the straight line thus defined satisfies Euclid’s parallel postulate, since it is derived from the behaviour of Euclidean solids.
At this stage, a number of popular exponents of non-Euclidean geometry have fallen into a rather unfortunate error. They have argued that material bodies under perfect conditions must necessarily behave like Euclidean solids, for if they behaved like non-Euclidean bodies when displaced they would squirm and change in shape. As it would be inadmissible to credit any such distorting influence to that void which we call empty space, a non-Euclideanism of material bodies would be debarred on first principles. But these men overlook the fact that Riemann’s and Lobatchewski’s geometries do not in any way refer to bodies which squirm and are distorted in any absolute sense as they move about. The non-Euclidean bodies are merely distorted when contrasted with Euclidean bodies taken as standards; but it would be equally true to state that Euclidean bodies likewise would squirm when displaced if we were to contrast them with non-Euclidean bodies taken as standards. In any case, both Euclidean and non-Euclidean bodies behave in a homogeneous way throughout space.[11]By this we mean that wherever they might be situated in empty space, measurements computed with them would yield the same numerical results. As for Euclidean congruence and Euclidean rigidity, it is by no means more representative of real rigidity than are the non-Euclidean varieties. There is therefore no reason to appeal to a distorting effect of empty space in order to account for a possible non-Euclidean behaviour of our material solids when displaced from point to point. Non-Euclideanism may or may not exist in real space, but this is a point for physical measurement and not for philosophy or mathematics to decide.
All we can say is that the principle of sufficient reason compels us to credit empty space with a sameness throughout, and that our measuring rods and material bodies must also behave homogeneously and isotropically, as indeed they do in the three geometries discussed. Only if measurements undertaken with our rods in different parts of space yielded variable non-homogeneous numerical results should we have to assume that space was not really empty and that our rods were subject to local influences.
At any rate, the early non-Euclidean geometers, realising that space as a result of measurement might turn out to be non-Euclidean, busied themselves with devising means of settling the question once and for all. Now, as a result of their measurements with material rods there was no doubt that space was very approximately Euclidean. But here it must be realised that the two non-Euclidean geometries as opposed to the Euclidean variety are not unique. We may conceive of various intensities of non-Euclideanism of both types, merging by insensible gradations into Euclidean geometry. It was therefore still an open question whether space, in spite of its apparent Euclidean characteristics, might not betray a slight trace of non-Euclideanism. A simple illustration will make this point clearer.
We saw that in Euclidean geometry the ratio of the length of a circumference to its diameter was always the same number,.In Riemann’s geometry this number was always smaller than,and decreased progressively from the valueto the value zero as the diameter of the circle increased. But there was nothing definite about this rate of decrease; it might be very rapid, just as it might be exceedingly slight. We must conceive, therefore, of varying intensities of non-Euclideanism, or of departure from Euclideanism. Hence, if the non-Euclideanism of real space were exceedingly slight, it might require measurements extending over a circumference of gigantic proportions, reaching as far as the stars in order to detect it; and measurements conducted in restricted areas could not be considered conclusive. The only means of disclosing slight traces of non-Euclideanism would therefore be obtained by having recourse to measurements conducted over cosmic distances.
Of course, in an attempt of this sort, measurements with material rods were out of the question and it was necessary to appeal to other methods of exploration. These were obtained by taking advantage of the propagation of light rays in empty space. It was argued that the principle of sufficient reason precluded light rays from deviating to the right or to the left from their course along the straightest path through empty space; this belief was also in accord with the important physical principle of Least Action, as deduced from the laws of mechanics.[12]Accordingly, light rays would follow geodesics in empty space; and, as we have seen, a knowledge of the geodesics or straight lines reveals as much about the geometry of space as congruence itself.
Gauss appears to have been the first to undertake space explorations of this sort, when he conducted experiments on light rays transmitted from one mountain top to another. But his observations were too crude and executed over too small an area to detect any trace of non-Euclideanism. Lobatchewski suggested astronomical observations conducted on the course of rays of starlight through interstellar space. For instance, if two light rays emitted from a very distant star and striking the earth at two different points of its orbit appeared to manifest converging directions, we should know that space was Riemannian. If the two rays appeared to diverge from a common point, space would be Lobatchewskian; and, finally, if for very distant stars these two directions appeared identical, space would be truly Euclidean. Yet the most refined astronomical measurements of stellar parallaxes failed to reveal the slightest trace of non-Euclideanism. Hence it was assumed that if any trace of non-Euclideanism was present in real space it was without doubt exceedingly slight, so that for all practical purposes the geometry of space might be regarded as Euclidean. Such were the results obtained by a physical exploration of space.
From all this we see that the physicist, basing his exploration of space on empirical methods, is perfectly justified in stating that its geometry can be determined, that a true definition of congruence can be arrived at, and that the equality of two lengths and of two spatial configurations has a definite significance in nature.
And yet, when we submit all these various examples to a critical analysis, we cannot help but see that this determination of the geometry of space is essentially physical and is, therefore, contingent on the behaviour of material objects and of rays of light. Had the behaviour of material bodies when displaced been regulated by other physical laws, had rays of light followed different courses, the geometry we should have attributed to space might have been entirely different. And we may well wonder what the behaviour of physical objects should have to do with the geometry of space. We shall return to this aspect of the problem later.
Also, it has sometimes been argued that our recognition of shape and size must possess a much deeper significance and cannot be attributed merely to the laws of behaviour of material objects and of rays of light. For instance, it is pointed out that even a child who knows nothing of measurement judges, on simple visual inspection, that a coin (when viewed from a perpendicular direction) is round and an egg oval. He does not feel it necessary to verify this fact by applying a ruler. However, regardless of what opinions we may eventually defend on the subject of a geometry intrinsic to physical space, it can scarcely be held that this last argument of the critic proves his point in the slightest degree.
It may be instructive to consider this illustration in greater detail. In the first place, we must realise that all we are in a position to appreciate when merely viewing an object, is its image on our retina. Indeed, were we to interpose a microscope or a deforming lens between object and eye, the object would suffer no change; but its image cast on our retina would, of course, be modified. As a result, our judgmentof its shape and size would vary. Hence, when we decide unhesitatingly that the coin is round, the only inference to be drawn is that the coin’s image on the retina is judged by the brain (or the mind, or whatever we decide to call it) to be circular. We do not wish to intrude on the ground of the eye-specialist or physiologist, who is better equipped than we are to proceed farther in this analysis, but we may point out that the problem which we are now considering is of an entirely different nature from the one whence we started. There we were considering whether shape in empty space was absolute; here we are considering whether an image cast on our retina should impress itself upon our recognition with any definite shape.
The difference amounts to just this: In mathematical space, and even in physical space, absolute measurement seemed to elude us, since in view of the continuity of space it appeared impossible to proceed with an enumeration of points. But in the case of the retina, its surface is no longer homogeneous; it possesses a heterogeneous structure like all tissues, probably a discrete one forming a pattern. In all such cases a definite metrics suggests itself naturally, just as on a net, in the absence of a ruler, we would compare lengths instinctively by counting the holes separating our points.
Whether or not, in the case of the retina, our appreciation of the coin’s roundness comes from some unconscious counting process is a problem for the specialist to decide; but at all events, when we consider that the retina and the eye are Euclidean bodies just like our rulers, the concordance between our computations of shape and size, as determined by rods, and our direct visual appreciation of shape does not appear to present much of a mystery.
But there is still a further point to be considered. Were we to have a direct intuition of congruence and of absolute shape or length, our measurements with rods would have to be adjusted so as to conform to this intuition. The introduction of rods would thus be contemplated merely as an adjunct, in order to obtain greater definiteness. But it so happens that such is not the case.
Our intuitive visual appreciations yield results which differ from results obtained with rods, not merelyaccidentallyas a result of the imperfection of human observation, butsystematically.
A coin that measures out as round will appear flattened to the eye.[13]This phenomenon is illustrated by the well-known optical illusion wherein two rigid rulers which coincide when placed side by side, appear of unequal magnitude when placed horizontally and vertically, respectively. It is a well-known fact that the vertical appears to be longer than the horizontal. For this reason, vertical stripes on a cloth cause the wearer to appear thinner and taller, whereas horizontal stripes produce the reverse effect. Now the mere fact that we have agreed to accept such discrepancies as due to optical illusions ratherthan to the untrustworthiness of our rods proves that we deliberately reject our intuitive judgment of shape and size in favour of more sophisticated rules of measurement. In other words, we have abandoned direct intuition for physical determinations, hence for convenient but conventional standards.
In short, we may state that we possess no direct intuition of shape and length, and that what little we appear to have is traceable in the final analysis to the properties of material bodies and light rays.
Let us consider a last example relating to distance. We realise, without resorting to measurement, that the distance across the street is less than the entire length of the street. A number of different reasons conspire to account for this conviction. First, the muscular sensations which accompany the convergence of the eyes and the focussing of images on the retina would in themselves suggest the origin of our appreciation of distance. In addition, we know from experience that it requires a greater effort and a longer time to travel along the entire length of the street than to step across from one sidewalk to the other. And if we enquired why it took a shorter time to cross the street, we should find that it was due to the fact that we advanced with definite steps.
Although in mathematical space the two distances would be equal or unequal according to our measuring conventions, in practical life we have unwittingly posited our measuring convention by walking. Our successive steps henceforth define in our estimation congruent distances; and under the circumstances distances become measurable in terms of these steps. But these steps are themselves controlled by our human frame; hence all we have done has been to measure space in terms of our legs. That measurement originated in this way is indicated clearly by such words asfootandcubit, or again by the definition of a yard as expressed by the distance between the tip of a certain king’s nose and the extremities of his fingers. And it is because our human limbs manifest the same type of congruence as the material bodies around us that this type of measurement once again imposes itself so strongly upon us. Indeed, measurements as computed with human limbs appear to be instinctive and are exemplified in children who, having grown, are surprised to find the rooms and buildings they have not seen for some years, appear considerably smaller. Instinctively, the child is measuring size in terms of his own height.
Thus, in whatever way we examine the matter, there appears to be nothing mysterious in our natural belief in the absoluteness of shape and size or in a definite geometry pertaining to space. None of the examples mentioned thus far entitle us to maintain that physical space manifests a definite metrics and that congruence is other than conventional.
If we consider the problem in its present state, we see that it is the physical behaviour of material bodies and light rays which is in the final analysis responsible for our natural belief in absolute shape. But this realisation brings with it the assurance that space itself has eluded us entirely in our discussions. Such was indeed Poincaré’s stand. He maintained that though for purposes of convenience itwas only natural for us to measure space as we do, yet if needs be we could disregard the behaviour of material bodies entirely, adopt non-Euclidean standards of measurement, and proceed as before.
In spite of this change we could construct exactly the same engineering works, in fact rewrite the whole of physics. Needless to say, everything would be extremely complex, all our known laws would be disfigured, and hypothesesad hocwould have to be introduced. But when all is said and done, the task would be theoretically possible; so that if we disregard the criteria of convenience and simplicity, there is nothing to choose between the various types of measurements, any more than between the metric system and the British units.
As a further illustration of the elusiveness of absolute shape and size, Poincaré asks us to conceive of a hollow spherical volume placed anywhere in space, and to assume that the temperature in the sphere decreases progressively from the centre, becoming absolute zero at the surface. He assumes that this hollow sphere is peopled by imaginary beings whose bodies expand and contract with the temperature and that all material bodies in the sphere behave in a similar manner. If we should supplement these suppositions by assuming that the refractive index of the medium in the sphere’s interior varies in a certain definite way, the rays of light in this hypothetical world would describe circles.
This closed universe would of course appear infinite to its inhabitants, since as they proceeded from the centre to the surface their bodies would grow smaller, their steps shorter, so that it would be impossible for them to reach its boundary however long they walked. The geometricians of this imaginary world would feel justified in proceeding exactly as we have done ourselves. They would define as remaining congruent when displaced, hence as rigid, those bodies which appeared to them to remain the same wherever they carried them. Owing to the paths devised for the light rays and to the sameness in the reduction of the sizes of all objects as the centre was left behind, the expanding and contracting bodies of this universe would present all the characteristics of rigidity. On conducting measurements with their rigid rods the hypothetical beings would obtain non-Euclidean results, their entire world would appear to them as non-Euclidean, and non-Euclidean geometry would be as inevitable to them as Euclidean geometry is to the average layman. Some Kant among the hypothetical beings would surely arise and explain that non-Euclidean space was thea prioriform of pure sensibility, transcending reason and experience. Then eventually some great mathematician would come along, sweep all those cobwebs aside, and prove that there existed other perfectly consistent types of geometries and that the ingrained preference of his fellow citizens for non-Euclidean geometry was due to the dictates of common experience and constituted by no means thea prioriform of pure sensibility.
It is not merely in its philosophical aspect that Poincaré’s illustration is interesting. The major point is the following: The hypothetical beings would be just as much entitled to assert thatspace was non-Euclidean as we are to assert that it is Euclidean. It is true that if we could look into their world we should say that they had a wrong understanding of measurement, and were totally in error when they assumed that their bodies were rigid, since we could see them getting smaller and smaller as they neared the surface. But we must not forget that the imaginary beings, in turn, could they but view our Euclidean bodies, would return the compliment and accuse us of having a wrong understanding of rigidity and measurement.
From all this it follows that by a mere variation in physical conditions the same space would be considered non-Euclidean or Euclidean. Obviously, by reason of this contradiction, space itself can have nothing to do with the problem; the type of space which physicists are discussing reduces therefore to a relational synthesis of physical results. Space itself remains amorphous.
Poincaré develops analogous arguments when he discusses the parallax observations conducted on the rays of starlight. Euclidean geometry, for instance, regarded purely as a system of measurement, is from a mathematical point of view the simplest type of geometry for the same reason that a monosyllable is simpler than a polysyllable. It is therefore obviously to our interest to retain it if possible. Of course if, as in the hypothetical world discussed previously, material bodies behaved like non-Euclidean solids and if light rays followed appropriate courses, we should have to abandon Euclidean geometry for reasons of practical convenience. But since, in the world we live in, our habitual solids behave to a high order of approximation as do Euclidean solids, our preference for Euclidean geometry seems perfectly legitimate even from the standpoint of physics.
Suppose now that the parallaxes of the very distant stars turned out to be negative: would Euclidean geometry and Euclidean space have to be abandoned? As Poincaré points out, this would by no means be necessary or even advisable. It is true that if we assumed, as is the custom, that rays of starlight follow geodesics through space, negative parallaxes would imply a trace of Riemannianism in space; but the primary point to decide is, “How do we know that rays of light follow geodesics?” Obviously this is capable neither of proof nor of disproof. An empirical proof that such a contention was correct or incorrect would be possible only were we to know beforehand how the geodesics of space were situated, for then we could determine by observation whether rays of light followed them or not. But how could we establish the way the geodesics lie unless we were already apprised of this geometry which we now proposed to determine? Obviously, our procedure would be circular. Can we at least assume that rays of light must inevitably follow geodesics? Would any other assumption be impossible? Certainly not. A denial of the assumption would modify our understanding of optical phenomena; but what if it did? We could always get out of the difficulty by varying the laws of optical transmission, and still retain Euclidean geometry. In other words, the geometry thephysicist credits to space is contingent on his acceptance of a number of physical laws; and by varying these laws in an appropriate way he could still account for observed facts and credit corresponding types of geometry to space. Since all these various systems of physical laws would account for the facts of experience, how can we ever hope to decide which one of these systems corresponds to reality? And under the circumstances, what use is there in discussing therealgeometry of space? All we can discuss is expediency.
In other words, Poincaré, by divorcing space from its material content, geometry from physics, places space and its geometry beyond the control of experiment; so that there is really nothing left for the physicist to argue about.
As a matter of fact, the entire conflict is more apparent than real; it centres round the meaning we wish to ascribe to the wordreality, whether we conceive of reality in the pragmatic scientific sense or in the metaphysical sense as embodying “true being.” For the theoretical physicist, “reality” means that hypothesis which will permit him to co-ordinate natural phenomena with the maximum of simplicity. He knows of no other test for reality and would probably evince very little interest in the unattainable reality of the metaphysician. If, therefore, a co-ordination of the facts of experience presents greater simplicity when he assumes space to be Euclidean or non-Euclidean, then space is Euclidean or non-Euclidean in spite of the fact that phenomena might just as well have been co-ordinated (though in a less simple way) had some other hypothesis been selected.
Riemann expresses the selective rôle of the criterion of simplicity when he writes:
“Nevertheless it remains conceivable that the measure relations of space in the infinitely small are not in accordance with the assumptions of our geometry [Euclidean geometry], and, in fact, we should have to assume that they are not if, by doing so, we should ever be enabled to explain phenomena in a more simple way.”[14]
So much for the scientific understanding of the word. But if, on the other hand, when discussing “reality” we are referring to the reality of the metaphysicians, as happens to be the case with Poincaré, then it appears quite impossible to dispute his stand; at least it is scarcely credible that any scientist would feel inclined to do so. For, assuming even that such a thing as metaphysical reality has any meaning at all, why should it be connected with simplicity of co-ordination? At any rate, Einstein himself, whose entire theory centres round a definite geometry being ascribed to the spatio-temporal background, writes: “Sub specie œternitatis, Poincaré in my opinion is right.”
If any criticism is to be directed against Poincaré’s stand, it should be on the ground that he showed himself a poor prophet when he claimed that it would always be simpler to retain Euclidean geometry. Einstein has proved the contrary. At all events, in what is to follow, we will concern ourselves solely with the real space of the physicist, that is, with the space to which he is led when he seeks to co-ordinate phenomena with the maximum of simplicity. With this understanding of space in our minds, a first reason for rejecting the concept of an amorphous space arises when we find that a large number of different methods of investigation all point to the same definite metrics for space. Thus, the various material bodies we encounter are by no means identical in nature; some are light, others are heavy, and their chemical and molecular constitutions are certainly not the same. And yet in every case, whether our rods be of wood, of stone, or of steel, we obtain the same Euclidean results provided we operate as far as possible under the same conditions of temperature and pressure. In other words, there appears to be a sameness in our determinations of congruence regardless of the material bodies to which we appeal.
This uniqueness of the geometry of space is still further exemplified in the following example: Here are two totally different methods of exploring space, one with material rods giving us a physical definition of congruence, and one with light-ray triangulations giving us a physical definition of geodesics. In either case we are led to the same Euclidean geometry, and this concordance appears rather strange, for we might have expected that if the geometry we credited to space were irrelevant to space, the type of geometry obtained would have varied according to the physical exploration method considered. Besides, if space were amorphous, hence possessed no geodesics, it would be inconceivable that a free body or a light pulse should know how and where to move. The very definiteness and Euclidean straightness of the paths of free bodies and light rays, when referred to a certain frame of reference, would seem to indicate that space had a structure and was not amorphous.
To be sure, in view of modern discoveries there is nothing very strange in the fact that the courses of free bodies should coincide with the paths of light waves, since light has been proved to possess momentum just as matter does. But even so, it appears strange that the courses defined by moving bodies should yield the same geometry as measurements conducted with bodies at rest.
Then again, there are the dynamical properties of space, which we cannot afford to neglect. If physical space were amorphous, all paths through space should be equivalent, and yet centrifugal force and forces of inertia manifest themselves for certain paths and motions and not for others. Whence could these forces arise if not from the structure of space itself? Such was indeed Newton’s contention.
In view of all these occurrences, difficult to account for if we believe in the amorphous nature of space, unless we appeal to some miraculous pre-established harmony, it appears as though space must be credited with a definite structure or metrics which, in thelight of experiment, turns out to be Euclidean, at least to a first approximation. Expressed in a different way, real space appears to be permeated by an invisible field,the Metrical Field, endowing it with a metrics or structure.
Now, in view of the fact that the mathematical requirements of that void we call empty space preclude it from having a metrical field (i.e., a structure or a metricsper se), the simplest way out of the difficulty is to assume that real space is not truly empty, but is filled with some mysterious physical medium, which we may call the “ether,”[15]and that it is this physical medium and not space itself which possesses a Euclidean structure. Henceforth it will be this ether structure which will be responsible for the apparent metrics or metrical field of space, which will cause material bodies to settle into definite shapes, and which will regulate the courses of light rays and free bodies far from matter.
For the physicist, however, this ether will be inseparable from real space; so that, to all intents and purposes, when the physicist discusses real space he will be referring to space together with its ether content. Were this ether structure to vanish, space would become amorphous, bodies would not know what shape to take, and light rays would not know where and how to move.
At this stage we must mention the premonitions of Riemann on the subject of the metrical field of real space. Riemann did not attribute this structure of space to the presence of some invisible medium, the ether, possessing a structure of its own. According to him the origin of the metrical field should be sought elsewhere. He felt that the metrical field of space should be compared to a magnetic or an electric field pervading space. And just as a magnetic field exists in the space surrounding a magnet, Riemann searched for the physical cause of the metrical field. With characteristic boldness, he found it in the matter of the universe; the metrical field thus became a species of material field. If Riemann’s ideas are accepted we can understand how a redistribution of the star matter in the universe, altering as it would the lay of the metrical field, would produce deformations in the shape of a given body and variations in the paths of light rays. As Weyl tells us, a spherical ball of clay compressed into any other form might again be made to appear spherical were all the matter in the universe to be redistributed in a suitable way.
It would also follow that were all the matter in the universe to be annihilated, and as a result the metrical field to vanish, space (assuming that any physical meaning were left the term) would become completely amorphous, just like mathematical space; light rays would not know where to move, all geodesics having disappeared; and were one lone material body to be introduced into otherwise empty space, it would not know what shape to take. Without the metrical field, physical space would be unthinkable.
Still further important consequences follow from this matter-mouldinghypothesis of Riemann. Prior to these views, the principle of sufficient reason appeared to imply that physical space would always turn out to be homogeneous—the same in all places. This does not necessarily mean Euclidean, for, as we know, Riemann’s and Lobatchewski’s geometries also correspond to homogeneous spaces; but all varying degrees of non-Euclideanism from place to place were thought to be excludeda priori. Of the vast realm of possible types of geometries or spaces discovered by Riemann, only the homogeneous types survived; a situation which Weyl finds appropriately expressed by the classical line: “Parturiunt montes, nascetur ridiculus mus.”
But with the new views advocated by Riemann the situation changes entirely; for now the texture, structure or geometry of space is defined by the metrical field, itself produced by the distribution of matter. Any non-homogeneous distribution of matter would then entail a variable structure or geometry for space from place to place.
Now the question arises, What is the nature of this metrical field? Is it merely a name we are giving to the structure of space caused by matter? This view appears impossible, for space of itself, being a mere void, is not amenable to structure. Is the metrical field a direct emanation from matter, a rarefied form of matter? Or again is it a reality of a category differing from matter (call it the ether), which in the absence of matter would be amorphous, and which could only be forced into a structure by the influences due to matter and transmitted from place to place through the ethereal substance? In this case we should again be led to the view that what we commonly call the structure or geometry of real space reduces to the structure of the matter-moulded ether-filling space.
Riemann’s exceedingly speculative ideas on the subject of the metrical field were practically ignored in his day, save by the English mathematician Clifford, who translated Riemann’s works, prefacing them to his own discovery of the non-Euclidean Clifford space. Clifford realised the potential importance of the new ideas and suggested that matter itself might be accounted for in terms of these local variations of the non-Euclideanism of space, thus inverting in a certain sense Riemann’s ideas. But in Clifford’s day this belief was mathematically untenable. Furthermore, the physical exploration of space, even in the interior of liquids, seemed to yield unvarying Euclideanism. And here the remarkable irony of the whole situation must be noted. Although experimenters had utilised the most refined apparatus for detecting a possible non-Euclideanism of space and had failed in their efforts, it was reserved for the theoretical investigator Einstein, by a stupendous effort of rational thought, based on a few flimsy empirical clues, to unravel the mystery and to lead Riemann’s ideas to victory.[16]Nor were Clifford’s hopes disappointed, for the varying non-Euclideanism of the continuum was to reveal the mysterious secret of gravitation, and perhaps also of matter, motion and electricity.
Before solving the problem, however, Einstein had been led to recognise that space of itself was not fundamental. The fundamental continuum whose non-Euclideanism was to be investigated was therefore not one of space but one of Space-Time, a four-dimensional amalgamation of space and time possessing a four-dimensional metrical field governed by the matter distribution. Einstein accordingly applied Riemann’s ideas to space-time instead of to space, and attempted to explore the geometry of space-time by a purely rational co-ordination of known empirical facts. He discovered that the moment we substitute space-time for space (and not otherwise), and assume that free bodies and rays of light follow geodesicsno longer in space but in space-time, the long-sought-for local variations in geometry become apparent. They are all around us, in our immediate vicinity; and yet we had never realised it. We had called their effects gravitational effects, ascribing them to forces foreign to the geometry of the extension, and never suspecting that they were the result of those very local variations in the geometry for which our search had ever been vain even though we had extended our observations to the depths of the universe. Indeed, it may be said that the theory of relativity is the theory of the space-time metrical field.
While we are on the subject, we may mention that this mysterious metrical field which moulds both space and time appears to be conditioned entirely by the matter of the universe. Such at least are the conclusions which the existence of Einstein’s cylindrical universe would suggest. The problem is, however, still extremely obscure. It is still possible to believe with Eddington and de Sitter that the metrical field or space-time ether-structure might subsist in the absence of matter, contrary to the views of Riemann.
Finally let us consider how these new views will affect the problem of absolute shape and size. We may say that space-time possesses a definite geometry but that this geometry is subject to local variations both in space and in time, as the masses of the universe modify their positions. Yet, as we have mentioned on several occasions, a geometry, even when fully determined and unchanging, does not imply absolute shape and size. It merely regulates the relationships or defines the laws of the mutual dispositions of bodies, and these laws may remain unmodified even though the absolute shape and size of the bodies vary. Restricting ourselves to the problem of absolute size, we shall find that with the finite universe the metrical field yields a universal standard of length given by the radius of the finite universe. As referred to this natural gauge, an object presents a definite size, but obviously there is nothing absolute about the gauge itself. Not only will its magnitude be governed by the total amount of matter in the universe, an amount determined presumably by accident, but furthermore there would be no means of determining the magnitude of this gauge otherwise than by adopting some other gauge, and so onad infinitum.