CHAPTER XXXIV THE FINITENESS OF THE UNIVERSE…

CHAPTER XXXIV THE FINITENESS OF THE UNIVERSE

IN classical science, space, the fundamental continuum, was thought to be Euclidean and flat throughout, hence infinite in extent. As soon, however, as Einstein was led to believe that the fundamental space-time continuum of the universe was not rigorously flat, but manifested various degrees of curvature from place to place around matter, the possibility of its turning out to be curled round on itself, and hence finite, had to be investigated further before any definite opinion could be expressed. True, this fundamental continuum was no longer three-dimensional space; it was four-dimensional space-time. But, apart from this difference, the same major problem still presented itself, and we were permitted to ask: “Is the universe of space-time infinite or is it finite?”

More precisely, if we appeal to a two-dimensional analogy, must we regard the universe of space-time as curved locally around the individual masses of the universe, but as flat or quasi-Euclidean on the whole over an infinite area, much like the rippled surface of an infinite pond of water? Or must we suppose that the rippled surface of the water is more akin to that of the rippled surface of a liquid sphere closing round on itself?

Now, if we consider the theory in its present state, we must agree that a finite universe is excluded. We can understand this point by considering the law of gravitation, which in Einstein’s theory is also a law of space-time curvature. We remember that the gravitational equations were given by where the ’s represented the curvatures of space-time at a point and where represented the characteristics of matter at the same point. Outside matter vanished and the gravitational equations reduced to or, what comes to the same thing, to this last equation being compatible with perfect flatness of space-time (). Einstein had assumed that this condition of perfect flatness existed at infinity, and it was when this supplementary condition was taken into consideration that Einstein’s law of [Pg 314] gravitation became determinate. This was the law which enabled him to anticipate the double bending of a ray of light, the Einstein shift-effect, as also to account for the curious motion of the planet Mercury.

It is obvious that a law of gravitation which entails the existence of perfectly flat space-time at infinity must of necessity lead to an unclosed space-time universe; and so we see that with Einstein’s original gravitational equations, a finite self-contained universe cannot exist. We are thus led to consider how it would be necessary to modify these equations for a finite universe to be realised.

Now we mentioned, when discussing the law of gravitation, that the inclusion of an additional tensor ( being any arbitrary constant) in the equation of curvature would not destroy the tensor nature of the law; hence, so far as the requirements of all general laws are concerned, this modified law of space-time curvature is acceptable. If it is accepted, the universe can no longer be quasi-Euclidean and infinite; it will have to be closed, hence finite. In order to understand the reason for this statement, we must examine the law of curvature with the term included. It will be convenient to write the term: ; and in this case, the law of space-time curvature in the interior of matter becomes Outside matter, we obtain or its equivalent, Thus the inclusion of the term changes the law of space-time curvature outside matter from to . Now this new law of curvature is no longer compatible with perfect flatness (), but with which yields a closed spherical, space-time extension. We thus obtain a four-dimensional spherical Riemann space-time as a possible model of the universe. From these equations we may also deduce , which shows us that is proportional to the curvature of the universe.

Inasmuch as in any case the universe, whether finite or infinite, is certainly of huge proportions, its curvature must be practically insignificant. We may infer, therefore, that , if it [Pg 315] exists at all, is undoubtedly a small, a very small, quantity; so that in problems relating to the dynamics of the solar system, for example, we may continue to ignore its existence. Theoretically, of course, this omission would be unjustified, for the presence of the term, by modifying the law of curvature, will inevitably modify the lay of the space-time geodesics. As a result, the orbits and motions of the planets round the sun should be affected by its existence, and it should be possible to settle the question of its presence empirically by observing the planetary motions with increased refinement. Practically, however, owing to the extreme minuteness of (assuming it exists at all), empirical observations of this sort would be insufficient to detect it. The only means we may have of establishing its existence directly (and, with it, the finiteness of the universe) will be to observe the motions of the stars in the globular clusters or in the Milky Way (as we shall explain later). All we can say at present is that the existence of a finite universe is proved to be a possibility, so far as the mathematical requirements of the theory of relativity are concerned.

We might attempt to settle the question by appealing to the principle of action. But, here again, both hypotheses, that of the infinite and of the finite universe, would reveal themselves as possible; hence we shall have to consider some other method of solving the problem.

All we have done up to this stage has been to show that Einstein’s gravitational equations (or space-time curvature equations) are incompatible with the existence of a finite universe. If, however, some very small term were assumed to be present, far too small to affect the observed planetary motions, a finite universe would ensue. In short, a finite universe is theoretically possible. But before giving any further consideration to the problem, we may well enquire what is to be gained by this additional complication.

The original gravitational equations, those entailing perfect flatness for space-time at infinity, appeared to have yielded satisfactory results. Furthermore, when Einstein assumed that space-time would degenerate to perfect flatness at infinity, he was merely expressing the validity of the principle of inertia in regions far removed from the influence of matter. In other words, at infinity, far from matter, free bodies would pursue straight courses with constant speeds (as measured in a Galilean frame). Of course, the principle of inertia might be incorrect, but still, until such time as its validity was questioned seriously, it was natural enough to accept it. With the finite universe, on the other hand, we should have to assume that space-time always retained a residual trace of curvature, so that the straight courses of free bodies would be impossible. Free motion would be in circles of huge radius.

Here, however, we must recall that Einstein’s theory is not one of loose, unsupported guesses. Had Einstein merely been interested in making the universe this or that, without taking into account the totality of facts at the scientist’s disposal, his cosmological speculations would be no more worthy of consideration than those of the Greeks, or of Dante, or of the philosophers of the seventeenth [Pg 316] and eighteenth centuries. We may be quite certain, however, that with a conscientious scientist like Einstein the methodological procedure will be very different. And so, when we find that the theory he advocates entails a finite universe, we may anticipate that there will be very strong reasons for his choice. Indeed, as we shall see, a number of separate considerations lead to this solution. First of all, there is the apparent stability of configuration of the star distribution. Then again, we have such philosophical considerations as the probable physical relativity of all motion and of inertia in accordance with Mach’s ideas; the intrinsically amorphous nature of all empty continuous extensions as demanded by Riemann; the desire to discover unity in nature by proving that matter can be reduced to electricity and gravitation. All these facts, together with many others, can be satisfied only provided we assume that the universe is finite. Unfortunately, a proper understanding of the problems involved requires an appeal to mathematical analysis. However, we shall attempt to proceed in a very crude way without introducing mathematical symbols to any great extent.

There exist two major solutions for the finite universe. The first is the Spherical, Elliptical, or, more rigorously, the Hyperbolical universe of de Sitter; the second is the Cylindrical universe championed by Einstein. We will consider de Sitter’s universe first. But before discussing these difficult subjects, there are certain peculiarities about the laws of nature which it will be necessary to mention.

In the majority of cases, the laws of nature are expressed in the form of laws of contiguous variations in space and in time. Mathematically, this is equivalent to saying that the laws are expressed by differential equations. Now differential equations give us the law of variation of magnitudes from place to place or from time to time, but they do not yield us the precise values of these magnitudes.

Consider, for example, a differential assertion such as the following: “A train is moving away from us with a constant acceleration of one mile per second.” Now this differential law, though it specifies the nature of the motion, does not enable us to deduce the position the train will occupy at any specific instant. To solve the problem we must be informed of the train’s position, and also of its velocity, at some initial instant. This illustration gives us an idea of the limitations of differential assertions or equations. In order to render them determinate, we must always supplement them with additional information by stating the value of the variable quantity at some definite point or time. In particular, when we consider a magnitude which may vary according to some differential law from point to point throughout a region of three-dimensional space, we must state the value and other characteristics of the magnitude, or variable, over a closed surface bounding this region of space. In mathematical terminology, we must know the boundary conditions before the law can be regarded as perfectly determinate.

Let us apply these considerations to the differential law which expresses the Newtonian law of attraction. It is given, as we have [Pg 317] seen, by Poisson’s equation , and expresses the law of distribution of (the Newtonian potential) in a gravitational field around matter and in the interior of matter (where is the density of matter from place to place). But, written in this form, the law is indeterminate; for it yields a number of laws differing from Newton’s law of the inverse square. Mathematicians have proved that in order to obtain Newton’s law, we must supplement our differential equation by specifying that the potential tends to a constant finite limit at spatial infinity.

Now, in Einstein’s theory, the analogue of Poisson’s law is the law of space-time curvature, Just as Poisson’s law was the differential law of the Newtonian potential distribution, so now Einstein’s law is the differential law of distribution of the ten potentials, the ten ’s; and is the energy and momentum of matter at every point. Again, as in the case of Poisson’s equation, we must supplement these differential equations of the gravitational field by specifying the values of the ’s at the boundary at infinity.

Of course the ’s are not numbers or scalars like ; they are tensors. Hence their values at a point will vary with the mesh-system, or frame of reference, we may select. Nevertheless, if we specify our choice of a frame of reference and decide on the values the ’s are to have in this frame, their values in all other frames will be determined automatically. This is, of course, owing to the fact that the ’s are tensors.

In the theory, as we have described it up to this point, Einstein selected a Cartesian space-time mesh-system, that is to say, a Galilean frame, and he assumed that as measured in this mesh-system the ’s at infinity were to be given by all other ’s vanishing. These boundary conditions implied that space-time was perfectly flat at infinity, hence satisfied the equation . It may be realised that we are merely expressing in another way what was said at the beginning of the chapter; the boundary values written above and the flatness of space-time at infinity being equivalent statements. Under these circumstances we saw that the universe could not be finite; we should have the quasi-Euclidean infinite space-time universe.

Now, in the opinion of both Einstein and de Sitter, this assumption of perfect flatness at infinity was too drastic. For though this perfect flatness is in harmony with the law of inertia, it appears rather arbitrary to assume that the law must hold at infinity, even were we to agree that it holds in less distant regions, a conclusion which is by no means certain. In Einstein’s own words: “It is certainly unsatisfactory to postulate such a far-reaching limitation without any physical basis for it.”

In addition to this first reason for doubting the perfect flatness of [Pg 318] space-time at infinity, we must remember that the ’s do not retain their standard boundary values (1, 1, 1, -1) when we change our mesh-system. In any frame other than a Galilean one, say in a rotating frame, these boundary values at infinity would be modified; this is, of course, due to the fact that the ’s are tensors. It follows that the extremely simple boundary values appear to single out a privileged frame of reference (a Galilean one), and this, in Einstein’s opinion, is contrary to the spirit of the theory of relativity. At any rate, whatever may be thought of the force of these arguments, it can easily be seen that if we wish to assign invariant values to the ’s at infinity, values which will remain the same in all frames of reference, the simplest solution will be to assume that they all vanish at infinity. Then, of course, owing to their tensor nature, their vanishing in any one frame ensures their vanishing in all other frames. This was indeed de Sitter’s solution; namely, that all the ’s vanished at infinity.

De Sitter presents another argument in defence of his solution. It deals with the relativity of mass or inertia, but as I have been unable to reconcile his views on this subject with the characteristics of his universe, I may have misunderstood his idea; hence I cannot discuss it. At all events, the relativity of inertia is a most important argument in favour of Einstein’s cylindrical universe, so we shall consider it when dealing with Einstein’s solution.

Now when we accept de Sitter’s vanishing values for the ’s or potentials at infinity, the direct outcome is a finite space-time universe. We see, then, that in this way the boundary conditions have been obviated entirely, since in a finite universe, closed round on itself, there is no longer a boundary. The precise form of de Sitter’s universe is a spherical four-dimensional space-time. When empty of matter, it is given by the equation . In other words, it is the type of universe to which we are led when the term is included in the gravitational equations; hence, as we have seen, it is a type which is theoretically possible.

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The characteristics of de Sitter’s universe are thus those of a spherical four-dimensional space-time of radius . As for , its magnitude is proportional to the universal curvature. As a matter of fact, the appellation hyperbolical universe would be more correct, for it is space-time which is curved spherically, not space and time. Here we must remember that in space-time, space is real and time imaginary; and if we wish to express the shape of the universe in terms of real time instead of imaginary time, the hyperbolical form is the correct one.[106]

As we have explained in the note, de Sitter’s universe is open at both ends in the time direction, so that time does not come back as it would in a spherical universe. When we consider points farther and farther removed from where we stand, all the ’s decrease and finally vanish, as was demanded by the requirements discussed previously. In particular, this decrease and this ultimate vanishing of the potential indicate, respectively, the gradual slowing down and the complete arrest of time in the distant regions of the universe. This arrest of time, however, is fictitious, for if we transported ourselves to the point considered, we should find that things went on as usual. It would now be where we had stood formerly that the arrest of time would appear complete. Thus, for every observer there would exist a locus of distant points where nothing would appear to change or move. A ray of light could never circle round the universe, but would gradually slow down and ultimately be arrested when this passive horizon of the observer was reached.

We also see that the nearer an incandescent atom is situated to this passive horizon, the slower would its vibrations appear; hence, in the case of a sodium atom (emitting a yellow light under normal conditions), the redder would be its light, till finally the light would cease completely when the passive horizon was reached by the atom. Now it happens that among the most distant luminous bodies known, namely, the spiral nebulæ, the great majority actually do appear to emit slowed-down light vibrations. Of course this effect might be attributed to the general tendency of all spiral nebulæ to recede from the solar system. But as no satisfactory reason can be advanced to account for this general recession, Eddington considers that de Sitter’s hypothesis of the hyperbolical universe has much to commend it.

A few further remarks must be made with reference to de Sitter’s universe. We remember that in Einstein’s theory the perfect flatness of space-time entailed the complete absence of gravitational or inertial forces (so far as a Galilean observer is concerned). Therefore, were it not for the curvature that matter itself imposes upon space-time around it, two parcels of matter would never attract each other. In de Sitter’s universe, however, there exists a residual hyperbolical curvature of space-time at every point, regardless of [Pg 320] the additional curvature that would be superimposed by matter. This universal curvature, as can be shown by calculation, would produce a mutual repulsion between bodies. Unless these bodies were of minute proportions, this universal repulsion would be counterbalanced and overcome by the more important curvature generated by the bodies themselves; so that, in spite of all, gravitational attraction would be accounted for. Nevertheless, we should be led to a most displeasing duality in our conception of gravitational action, since we should be in the presence of two separate phenomena:

1. The universal repulsion which is totally foreign to matter;

2. The gravitational attraction between matter and matter.

Thus, it would be impossible to escape a duality in our conception of gravitation, if by gravitation we imply the cumulative action of both effects, yielding the one observable attraction of matter for matter.

We may summarise the chief characteristics of de Sitter’s universe as follows: First, this universe can exist in the absence of all matter. It then possesses a definite spherical structure and a radius of curvature , where , which is proportional to the universal curvature, is some constant which bears no palpable relation to anything else in the universe.[107] We must conceive of it as a constant of nature posited once and for all by the Creator. Secondly, if matter is introduced into this empty universe, it will produce local space-time puckers or curvatures around it; and to these local curvatures the phenomenon of attraction between matter and matter will be due. In the presence of matter, de Sitter’s universe will not be stable, for the universal repulsion will tend to cause matter to scatter if the mutual attractions are too feeble to oppose it.

In the opinion of a number of thinkers, de Sitter’s universe presents certain aspects which are most unsatisfactory. We shall discuss these later. At all events, Einstein rejects it and is led, by following a somewhat different line of attack, to a very different type of finite universe, the cylindrical universe.

We shall now proceed to a more detailed study of Einstein’s universe. First of all, we must mention certain general cosmological considerations that directed Einstein to the formulation of a finite-universe hypothesis.

Quite independently of the relativity theory, in the days when space-time was unknown, astronomers had puzzled over certain difficulties dealing with the universe as a whole. Two possibilities appeared to suggest themselves. Either star-matter was present everywhere, distributed more or less homogeneously throughout infinite space, or else the stars were concentrated into a nucleus forming an island of matter in an infinite ocean of space.

The first alternative entailed certain difficulties when we considered the illumination of the heavens at night. Calculation showed that the [Pg 321] night sky should be many times more luminous than it is. Of course this difficulty might have been overcome by assuming that as we wandered farther and farther from the earth, the percentage of dark stars increased; or, again, that vast quantities of cosmic dust intercepted the light and obscured the heavens. It might also have been assumed that light was gradually absorbed in its long journey from the distant stars.

Over and above these first difficulties, it was shown that if the stars attracted one another according to Newton’s law it would be impossible for them to be spread more or less uniformly throughout infinite space; rather would they concentrate into a nucleus. Precise calculation shows that the density of matter, hence the number of stars per unit volume, would vanish at infinity, decreasing progressively from a centre more rapidly than , where represents the distance from the centre. It appears, then, that a nucleus of stars cannot be avoided if Newton’s law is to hold.[108] The astronomer Seeliger attempted to realise an infinitely extended universe of matter by modifying Newton’s law in an appropriate way. But, as Einstein remarks, the attempt had neither empirical nor theoretical foundation.

Finally, then, if we accept Newton’s law, we must assume that the universe presents the form of a nucleus of stars lost in infinite space. But here, again, a new difficulty awaits us. In the first place, astronomical observation reveals no trace of any such nucleus. Of course it might be contended that this was due to the limited range of our astronomical observations. Nevertheless, this first argument against the nucleus hypothesis cannot be passed over lightly, for we must remember that in these speculations on the form of the universe, we have to make the best of the flimsiest of clues; we cannot operate as in a laboratory.

Then there is a second argument adverse to the nucleus hypothesis. Calculation shows that the cumulative Newtonian gravitational attraction of the stars would be insufficient to retain any individual star and prevent it from escaping to infinity, leaving the nucleus forever. Similar conclusions would apply to rays of light. It would follow that our universe of stars was in a state of dissolution. It is of course possible that such is indeed the case; yet, to a number of thinkers, a universe in process of dissolution appeared most unlikely. With Newton’s law of gravitation there was no escape. This was the [Pg 322] cosmological difficulty which classical science had to face.

Inasmuch as the justification for these statements does not involve any very complicated ideas, we may proceed to explain them briefly. Consider the hypothetical case of the earth situated in infinite space, and isolated from all other bodies. If, then, a mass is abandoned at spatial infinity, it will be attracted towards the earth, gradually gathering momentum, and may finally hit the earth’s surface with a certain velocity, which calculation proves to be about seven miles a second. This particular speed is given by the following formula: where represents the value of the Newtonian potential at infinity and its value on the earth’s surface.[109] Conversely, if a body is shot out radially from the earth’s surface with an initial speed smaller than this critical speed , it will always return to the vicinity of the earth. On the other hand, were the body’s initial speed to be greater than the critical speed , it would never return. We see, therefore, that the value of this critical speed depends on the difference in value of the Newtonian potential at infinity and at the earth’s surface.

Now consider the case of a nucleus of stars. The stars will be moving with various velocities under the influence of their mutual gravitational attractions. Statistical mechanics shows that every now and then the total energy of the stellar universe will be transferred to one single star; this star will then be moving at a tremendous speed. The total energy of the sidereal universe being finite, this speed will never be infinite, so that there will exist a maximum finite speed which no star of the nucleus can ever surpass. If, therefore, we consider the Newtonian potential due to the entire nucleus of stars, and if the difference in the value of this potential at infinity and near the centre of the nucleus is sufficiently great, no star will ever be able to tear away and be lost to infinity.

From this, it might appear that a nucleus of stars could endure permanently under Newton’s law. Theoretically, this would be possible, but astronomical observation proves that this possibility must be rejected. For if there existed very great variations in the value of the potential throughout the universe (as the preceding hypothesis would demand), enormous star velocities would ensue. Now the outstanding astronomical fact is that the star velocities are extremely low, proving that the potential varies but slightly from place to place throughout the universe even over vast astronomical distances. It appears, then, that the nucleus is held together very loosely, and a star would not have to be possessed of an inordinately high velocity to tear away and be lost forever.

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In addition to this first difficulty, when Boltzmann’s law of gas equilibrium is applied to the stars, treating the latter as so many molecules, the nuclear form of the universe appears to be utterly impossible.

Now, thus far, we have been considering the problem of the universe from the standpoint of classical science and Newton’s law. No great change need be introduced into our arguments when we treat the same problem in terms of Einstein’s law of gravitation and of the infinite quasi-Euclidean universe. Once again, a continually impoverished nucleus would appear to be in order. There is, however, a novel point that may be mentioned. In classical science, we were concerned with the Newtonian potential , and we explored its value throughout the universe by observing the velocities of the stars. With the space-time theory, the potential (or, more precisely, a certain mathematical expression into which enters) usurps the position of . Now also affects the flow of time, hence affects the rate of vibration of atoms (as in the Einstein shift-effect); and so we can detect variations in the value of the potential from place to place through the medium of spectroscopic observations. But when we undertake our observations from a Galilean frame (the earth will do in this case), we notice that lines in the spectra of the most distant nebulæ occupy positions practically identical with those of terrestrial sources, whence we may conclude that as far as the telescope can explore, the value of remains approximately constant. As for the velocities of the stars, they would be affected by the values of all the ’s. So here, again, we see that owing to the low velocities of the stars, all the ’s vary but slightly from place to place throughout the universe (when computed in a Galilean frame). Thus, with Einstein’s theory, just as with Newton’s, there appears to be nothing to prevent stars and rays of light from leaving the nucleus; the mutual gravitational attractions would be insufficient to retain them.

In short, if there were to be a self-contained universe, something would have to be modified somewhere.

We now come to Einstein’s final solution, that of the cylindrical universe. He remarked that the low velocities of the stars, together with their more or less uniform distribution throughout space, suggested that they were in a state of statistical equilibrium. A few lines will explain what is meant by this.

Suppose that on a billiard table containing a very large number of identical balls we impart a rapid motion to one of them. Our ball will eventually collide with some of the others. These others, in turn, will collide with yet others, and finally all the balls will be set in motion, if there is no friction of any kind, and no loss of momentum when the balls rebound from the cushions, they will go on moving forever. But as time progresses a gradual change will take place. Little by little we shall find that the energy we imparted to the first ball appears to have split itself up among all the balls on the table, so that finally, on an average, all the balls will be moving with approximately the same velocity. When this condition of equipartition of energy is attained we shall have reached [Pg 324] stability of configuration, or statistical equilibrium. This state of equilibrium will correspond to maximum entropy, and will endure forever.[110] We see that it is characterised by the fact that all the balls possess more or less the same energies or velocities.

Now, as we have said, astronomical observation would suggest that the stellar universe has reached this condition of statistical equilibrium. It is permissible, therefore, to neglect the trifling irregularities and agglomerations in the distribution of star-matter, and to assimilate the entire stellar universe to a vast cloud of cosmic dust at relative rest, everywhere of the same density. It is then found that for this cloud of dust to maintain a permanent, stable configuration, the time direction of the universe as a whole, for an observer who is non-rotating with respect to the stars, must be straight. It is easy to see why this condition should arise.

If the stars can be considered at relative rest, inasmuch as their world lines constitute geodesics of space-time, it is necessary that these geodesics should not taper together but should lie along parallel time directions through space-time. This is possible only provided the structure of space-time is such that one same time direction endures at every point of space for an observer at rest with respect to the star distribution. We thus obtain one same straight time direction for the universe. Of course, this does not mean that around each individual star the space-time curvature, and hence the time direction, do not suffer modifications. Far from it; since it is these local curvatures which are responsible for gravitational attraction between matter and matter. In the present case, however, we are concerned solely with the structure of space-time as a whole, on a grand universal scale, so that the star distribution can be treated as uniform and likened to a gigantic cloud of dust. Stability, therefore, requires that this ultra-macroscopic structure should be conditioned by a straight time direction. This eliminates de Sitter’s universe, since in his model the direction of time was curved, and his universe, in consequence, not stable.

The net result of Einstein’s calculations was to show that if the universe were a closed three-dimensional spherical (or elliptical) [Pg 325] space, not space-time, on which a straight time axis was erected, stability would be possible, the star velocities would remain low, and their distribution uniform. We should thus obtain the Cylindrical Universe, where the time direction proceeding from minus to plus infinity would represent the axis of the cylinder.

But here a most important point must be noted. For stability to endure, a certain condition would have to be satisfied. Calculations, based on the formula mentioned on p. 314, prove that the mean density of matter would have to be equal to . In other words, the total amount of matter in the world would have to be proportional to the radius of the universe.

Assuming these conditions to be satisfied, it would be possible to calculate the volume of the universe, provided we knew the mean density of the matter distribution. For since , and since the Gaussian curvature of space depends on , a knowledge of this curvature would yield us the total spatial extent of the universe. Now the value of the mean density of star-matter has been computed by astronomers. If we accept this value of , we obtain for the universe dimensions which are certainly too small, for they conflict with the enormous distances credited to the spiral nebulæ. It would appear, then, as though the cylindrical universe was incompatible with the facts of observation. But, as Einstein remarks, no reliance can be placed on the value of commonly accepted by astronomers. In his own words:

“The distribution of the visible stars is extremely irregular, so that we on no account may venture to set down the mean density of star-matter in the universe as equal, let us say, to the mean density in the Milky Way. In any case, however great the space examined may be, we could not feel convinced that there were no more stars beyond that space. So it seems impossible to estimate the mean density.”

Einstein then suggests an indirect method for determining the value of , hence of and of the size of the universe. We shall discuss this method on a later page; it refers to the motions of the stars in the Milky Way.

It is easy to understand how it comes that in the cylindrical world no nucleus of stars need exist. The two-dimensional analogue of a star distribution spread out uniformly throughout the entire volume of a three-dimensional spherical space would be given by a homogeneous distribution of stars over the two-dimensional surface of an ordinary sphere situated in three-dimensional Euclidean space.[111] It would be the actual surface of the sphere which would be the analogue of the space of the universe, and not the volume enclosed within the [Pg 326] sphere. Reasons of symmetry would show us at first sight that a uniform distribution of matter on the spherical surface would be stable, there being no reason for a nucleus to form here rather than there.

In the same way, an observer moving through the three-dimensional space of the universe would be represented by a perfectly flat being moving over the surface of the two-dimensional sphere. Wherever this imaginary being might move, the star distribution on the sphere’s surface would appear to be the same. Likewise, in the spherical space of the universe, wherever we might move, the distribution of the stars around us would remain homogeneous.[112] Thus the cylindrical universe obviates the displeasing necessity of believing in the existence of an island of star-matter in an ocean of emptiness.

In Einstein’s universe, a ray of light, following a geodesic, would circle round the universe and return to its starting point. This constitutes a difference as against de Sitter’s universe in which the ray of light would be arrested at the passive horizon. In the cylindrical universe, owing to the straightness of the time direction, there is no horizon. For the same reason there will be no slowing down of time in the remote regions of space, so that the general reddening of the light emitted by the spiral nebulæ would appear difficult to account for in Einstein’s universe. De Sitter’s scheme, as we have noted, affords an immediate explanation of this remarkable effect.

It has sometimes been held that the light rays emitted from a star would circle round the universe of space in all directions, and then meet again at the antipodes of space, forming the ghost of a star. It would be well to point out that though theoretically this would be possible, its chances of being realised in practice would be extremely slight, to say the least. The fact is that the perfect sphericity of the universal space can hold only provided matter is homogeneously distributed and everywhere at rest. As this is only approximately the case in the real universe, the spherical spatial universe will be rippled and will deviate more or less from this simple form. As a result, the rays of light emitted from a star will not converge accurately at the antipodes; hence, ghosts of stars need not be considered.

Next we must consider the restrictive condition, which had to be complied with if Einstein’s universe was to be stable. We remember that is proportional to the space-curvature of the universe, whereas represents the mean density of matter. Now it appears extraordinary that the density of matter in the universe should just happen to be equal to . Einstein, however, urges that this equality of and is no coincidence at all; according to him, there exists a causal connection between these two magnitudes. The density of matter creates , or, again, the total amount of matter in the universe creates the universe of space. [Pg 327] Were more matter to be created, the universe would expand; were matter to be annihilated in part, the universe would contract. Indeed, were no matter to be in existence, there would be no space; hence, a totally empty universe would be quite impossible. Thus there is a marked difference between Einstein’s universe and de Sitter’s universe, which can exist without matter.

We may also mention another reason which directed Einstein to the cylindrical universe. We remember that when discussing the boundary values of the ’s we saw that de Sitter considered it necessary to obtain boundary values which would remain invariant to a change of mesh-system. In accordance with this requirement, he assumed that all the ’s vanished at the boundary. As a result he obtained his hyperbolical universe. Similar considerations prompted Einstein. Here, however, it must be noted that in Einstein’s model, while all the space-’s () vanish at infinity in our mesh-system, the time- (), owing to the straightness of the time direction, does not vanish. For an observer at rest with respect to the stars it assumes the value +1. It follows that when, instead of merely changing our space mesh-system, we change our space-time mesh-system, the values of the ’s are subjected to variations, so that de Sitter’s condition of invariance is not completely satisfied. De Sitter maintains, in consequence, that Einstein’s universe does not lead to a true theory of relativity.

Furthermore, it will be seen that the cylindrical universe introduces a species of absolute time, and this again has been used as an argument against its acceptance. But it should be noticed that this absolute time appears only when we view the universe as a whole. For ordinary problems restricted to smaller areas, the fusion of space and time remains complete, as before.

It is, of course, apparent that by changing the boundary conditions, as Einstein has done, by substituting a cylindrical universe for the quasi-Euclidean one, the gravitational equations will be somewhat modified. They will no longer be those which yielded the double bending of a ray of light. For now the additional term will be included in their expression; though, so far as the planetary motions are concerned, the presence of this additional term will not modify our results. In de Sitter’s universe this introduction of the foreign magnitude into the law of gravitation was somewhat unsatisfactory, marring the simple beauty of the equations; was some independent constant posited by the Creator. The same argument could scarcely be directed against Einstein’s equations; for it should be remembered that if, as Einstein suggests, is to be attributed to the matter of the universe, the circle is closed, since indirectly, at least, nothing but matter enters into the law of gravitation. According to this view, we might say that the total amount of matter in the universe conspired to create a gigantic law of gravitation. It is, then, under the cumulative action of matter that rays of light and free bodies describe circles.

From a philosophical standpoint, the cylindrical universe is of great [Pg 328] interest; it justifies Riemann’s premonitions that matter must create the metrics of the fundamental continuum. Riemann, of course, was thinking of space alone, for space-time was unknown to him; but, apart from this difference, Einstein’s results confirm Riemann’s profound views. Indeed, quite aside from these speculations on the universe, the general theory had already brought a partial confirmation of Riemann’s ideas by proving that matter could influence the space-time structure. Riemann’s attitude was, however, more radical. It was not merely a question of matter modifying a pre-existing spatial structure, but of its creating it in toto.

Both in the quasi-Euclidean infinite universe and in de Sitter’s, the rôle of matter, though important locally, was of secondary importance when the world was viewed on a cosmic scale, since the space-time structure could exist independently of all matter. It followed that matter would be accidental, and not essential. Eddington sees no objection to these ideas. He suggests that it may well be in the nature of empty space-time to curl round on itself and manifest a definite metrics, curvature and size. But this attitude leads to a dualistic conception when we remember that matter influences the structure of space-time locally. On this account, both in de Sitter’s and in the infinite quasi-Euclidean universe, the physical properties of space-time, while conditioned in the main by characteristics immanent in the extension itself, would still depend partly upon the presence of matter. We could then scarcely avoid a dualistic conception of space, which to many would be intolerable.

Eddington obviates this dualism by reversing the problem, and by assuming that the physical properties and geometry of space-time arise from the requirements of space-time itself. According to this view, matter is not an active cause, but a symptom, and reduces to mere regions of greater curvature or puckers in space-time or the metrical field. These our senses would interpret as connoting the presence of matter. However, it appears difficult to reconcile these ideas with the complexity of matter, for matter is constituted, in large measure at all events, by electric charges. Hence, to reduce matter to variations in the geometry of space-time would be equivalent to reducing electricity to the ’s. Thus far, at least, such attempts have been unsuccessful. At any rate, Einstein rejects this view. He argues that a perfectly empty continuum, a void—call it space-time, or space, or anything else—must be completely amorphous; that is, can possess no inherent metrics or structure of its own, for a void with a structure is not a void at all.[113]

Furthermore, even if it were admitted that space was not really empty, [Pg 329] but was filled with an ether that conferred a structure upon it and was responsible for its dynamical properties, it would appear incredible that this structure should be other than homogeneous. Hence, it was logical to assume that the local variations in curvature accompanying the presence of lumps of matter should be attributed to the presence of this matter itself, so that matter would again be a cause, and not a symptom. The simplest way to escape the dual conception which we mentioned previously was, then, to admit that Riemann was in the right, and that the metrics or metrical field of the space-time of the universe was conditioned in its totality, and not merely modified by the presence of matter. At any rate, if these views are adhered to, it is easy to see that the cylindrical universe is the only one that can be accepted, since it is the only one in which the metrics of space-time may be conditioned solely by the action of matter.[114]

It is interesting to note that it is within the realm of possibility that Riemann’s views may be finally submitted to a test. For suppose that astronomical observations should succeed in proving that de Sitter’s universe or the quasi-Euclidean infinite universe was the correct one: Riemann’s ideas would then have to be abandoned. We should have to conclude that a metrics was inherent in an empty continuous extension; though, of course, we might always avoid this attitude, either by assuming that space was atomic or, again, by identifying the metrical field with some subtle ether floating in space.

We now come to still another important consideration which drove Einstein towards the cylindrical universe. We are referring to the relativity of inertia. In the general theory of relativity, prior to Einstein’s investigations on the form of the universe, there existed a number of most displeasing dualities. In the first place, the theory of relativity had succeeded in proving that velocity through space was relative, but it had failed to extend this relativity to accelerated motion. After having made a promising start towards the relativisation of all motion, it had abandoned the attempt halfway before the final goal was reached.

Then again, a similar duality existed in the case of mass and weight. Here we have weight depending on the distribution of surrounding matter, so that weight would be inexistent in a world containing but one body. On the other hand, here was the inertial mass of the body which would subsist even in an otherwise empty world. Thus, were we to be transported far from all large masses, we could hold a ton of bricks above our heads without the slightest effort, since the bricks would have no weight. And yet, when we tried to push these same bricks aside, we should encounter the same inertial resistance as on earth. What rendered this duality particularly displeasing was the fact that the general theory had proved that even mass was influenced in a minute degree by the proximity of matter, so that the mass of a body appeared to be due in part to the distribution of surrounding matter and in part to the body itself. In other words, there existed a partial relativity [Pg 330] of inertia, just as there existed a partial relativity of motion.

These problems were intimately connected with that other type of duality we mentioned in the preceding pages. There again we saw that space-time was modified by the presence of matter, and yet that even in the absence of matter it maintained a structure. Of course, all these dualities are not equally objectionable to all thinkers; but it must be admitted that for those who crave unity in nature, every effort to remove them will be welcomed.

Einstein had attempted to render radical this relativity of inertia and motion even before discovering his cylindrical universe. He had solved the problem by postulating certain boundary conditions on the fringe of the nucleus of stars in infinite space. Thanks to these boundary conditions, the mass of a body would vanish at infinity, far from matter. But these solutions were abandoned because they appeared to conflict with the low star-velocities. With the cylindrical universe, however, all these difficulties were overcome, and for this reason, among others, it satisfies a natural philosophical craving. Inasmuch as all these considerations present an intimate connection with Mach’s mechanics, we will discuss the significance of Mach’s ideas at some length.

As is well known, on account of the existence of centrifugal force, of the inertial frame and of the rectilinear paths followed by free bodies according to the law of inertia, Newton felt himself compelled to postulate the existence of absolute space. Einstein summarises Newton’s attitude in the following lines:

“In order to be able to look upon the rotation of the system, at least formally, as something real, Newton objectivises space.... What is essential is merely that besides observable objects, another thing, which is not perceptible, must be looked upon as real, to enable acceleration or rotation to be looked upon as something real.”

The opposite stand was taken by Mach, who endeavoured to account for the dynamical manifestations of acceleration and rotation without appealing to that suprasensible entity, “absolute space.” Instead of holding, then, that centrifugal force was generated by a rotation in absolute space, he urged that we should attribute it to a rotation with respect to the material universe as a whole. Inasmuch as the vastest agglomeration of cosmic matter is given by the totality of the star-masses, it would follow that the dynamical effects would be generated by the rotation of an object relatively to the stars, or, what amounts to the same thing, by a rotation of the stars relatively to the object. In fact, these two different alternatives would represent but a difference in phraseology, since with absolute space banished, apart from stars and matter generally, there would exist no other terms of comparison.

To this rotation relatively to the star-masses would be due the protuberance of the equator, the splashing of water in Newton’s bucket, the bursting of a rapidly revolving flywheel, etc. In short, all those dynamical effects which in Newton’s opinion betrayed rotation in absolute space, are held by Mach to betray a rotation [Pg 331] relative to the star-masses of the universe. Now, it should be apparent that Mach’s relativity of rotation entails more than a mere kinematical representation. As he is careful to specify, this active rôle of the stars is due not to the accidental circumstance that they happen to be visible, but to universal gravitational actions which would be generated between matter and matter when a state of relative acceleration or rotation was present. In this way, forces of inertia and centrifugal force are no longer to be attributed to the intrinsic structure of absolute space; they now become akin to gravitational forces. Were the stars invisible, no change in the dynamical manifestations would be observed, but if their masses were to be annihilated, if the stars ceased to exist (not merely ceased to shine), all the telltale dynamical effects would disappear; a flywheel would not burst, water would not splash, and so on. In much the same way, the elliptical orbits of the planets are due to the gravitational attraction of the sun’s mass; not, of course, to the accidental circumstance that the sun happens to be luminous rather than dark.

From these conclusions further important consequences follow; in particular, we must assume that the inertial mass of a body must be generated solely by the mutual actions existing between this body and the other bodies of the universe. We can understand the reason for this statement when we realise that were all the stars to be annihilated, all inertial forces would disappear; but it is precisely the existence of inertial force which is responsible for the effort required to set a body in motion or to arrest it when started. For this reason the disappearance of inertial forces would automatically entail that of inertia or mass. We may note the contrast with Newton’s views, for in classical science the inertial forces arose from the structure of absolute space itself, and mass was regarded as an intrinsic property of matter. In the present case, mass joins weight in being a relative; for just as a body has no weight in the absence of other bodies, so now we see that, according to Mach, it would also present no inertial mass if situated all alone in an otherwise empty world.

When we consider Mach’s mechanics, we see that its net result has been to identify Newton’s absolute space with the space defined by the stars; absolute space has become materialised, so to speak. We may obtain a more concrete representation of the new ideas by imagining a crisscross of threads extending between each individual star and all the others. Rotation, which develops dynamical effects, is, then, the rotation of an object with respect to this network of threads; and it is obviously a matter of indifference to suppose that the object is rotating among the threads and stars, or that threads and stars are rotating around the object. Thus the inertial frame is any frame in which the threads appear to be non-rotating.

Mach’s mechanics might seem to draw too much on our credulity. We can grant that the sun exerts a gravitational effect on the earth, but to assume that the bursting of a flywheel is due to the presence of the stellar universe would appear to be stretching matters too far. Nevertheless, it should be remembered that Newton’s introduction [Pg 332] of that suprasensible entity, absolute space, was also extremely distasteful to a number of thinkers, so that it is a question of choosing the lesser of two evils.

These ideas were upheld by Mach for a number of years; and although they can lay claim to no special originality, they are always referred to under his name. Mach’s stand has been so consistently misrepresented by philosophers that we feel justified in warning the reader by mentioning some of the erroneous assertions that have been made with respect to it.

Two centuries ago, Berkeley had defended views which present a certain superficial similarity to those of Mach. He argued that if there were no stars, no permanent points of comparison in the heavens, we could not even imagine the earth’s rotation, hence this rotation would be deprived of all meaning. To this argument the Newtonian would reply that it was, after all, not impossible to imagine absolute rotation as a rotation in a rigid jelly-like substance called space. Furthermore, he would add that it was the dynamical facts, the appearance of unsymmetrically distributed forces, that seemed to confirm the absolute nature of rotation. Absolute rotation was, then, totally irrelevant to our perception of luminous reference points in the heavens, or even to our ability to imagine their existence. So far as the Newtonian was concerned, even were the stars to be annihilated entirely, it would still be quite easy to determine whether or not the earth was in rotation. A number of experiments, such as that of Foucault’s pendulum, the gyroscope, and, more generally, the presence of centrifugal forces on the earth’s surface, would demonstrate the existence of this absolute motion. As against these arguments, Berkeley’s stand offers no reply. Only when, with Mach, we maintain that were the stars to be annihilated, these telltale effects would also disappear, only when we conceive of the stars not merely in a visual capacity or as a support for the imagination, but as producing causal gravitational influences, can an argument be presented against Newton’s absolute space. In short, according to Mach, it would be the gravitational action of the totality of the material universe, including the stars, whether visible or dark, the nebulæ, our sun, the planets and moons, that were responsible for centrifugal force.

Before we proceed farther, a number of objections must be discussed. It might be claimed that whereas the earth could be conceived of as rotating among the stars, it would be quite impossible to consider the stars as rotating round the earth, since in this case centrifugal force would cause the stars to fly apart to infinity. A criticism of this sort arises from a confusion between the Newtonian attitude and the new one. With Newton’s ideas the argument would hold, for the stars would indeed be describing gigantic circles in absolute space. But in the present instance the space of the stars themselves has usurped the position of absolute space, since it is in this star-conditioned space that circular motion generates centrifugal forces. And in their own space the stars are obviously at rest whether or not we regard them as rotating, together with their space, around the earth. Hence, centrifugal force does not arise on the stars, though it will be [Pg 333] felt on the earth’s surface. An equivalent way of explaining the same point would be to say: The stars as rotating round the earth are not subjected to centrifugal force, for they are rotating with respect to nothing; the mass of the earth being insignificant in comparison with the totality of the star-masses.

There exists also another type of objection which is based on causality and on our consciousness of effort. For example, we may claim that when, through an act of our will, we spin round on our heel, we know perfectly well that it is our body that is rotating among the stars, and not the stars that are rotating around us. In support of our claim, we should point out that no feeble efforts of ours could ever have started the stars and nebulæ rotating backwards. Curiously enough, where objections of this sort are made, the critic is often fully prepared to accept the relativity of velocity. But, as Einstein points out, arguments of a very similar sort might also be advanced against this simpler type of relative motion. For instance, would not the engineer claim that it must be his train that is moving, and not the earth that is gliding backwards? Could he not also support his contention by pointing out that when he stops feeding the furnace, all motion eventually ceases?

Possibly a more lucid illustration of the relativity of velocity would be afforded by considering the case of a bird in flight. The bird may be conscious of its efforts and realise that when it ceases to beat its wings, its motion through the atmosphere is arrested. It would therefore be entitled to assert that its velocity was maintained through its own efforts, hence was of its own making.

But it is obvious that this velocity created by the bird would be, not absolute velocity through space, but velocity through or relative to the atmosphere. Knowing nothing of the pre-existing velocity of the atmosphere through space, the bird could know nothing of its own absolute motion; and this is all that the principle of the relativity of velocity asserts. And so, were the velocity of the atmosphere through space equal and opposite to the relative velocity of the bird through the atmosphere, the bird’s efforts would result in maintaining its body at absolute rest. Hence, the awareness of effort is no criterion of the existence of absolute motion; relative motion alone is involved.

Exactly the same arguments can be extended to the relativity of rotation, except that in this case we are confronted with the peculiar difficulty arising from the existence of centrifugal force. Mach’s stand obviates this difficulty by introducing a definite physical assumption. Our purpose is, of course, not to justify Mach’s mechanics (for this can be done only through precise experiment, not by a general line of talk), but merely to show that it does not lead to absurdities when the argument based on causality is applied.

Let us consider, then, what is implied in the statement: “We know that it is we who are rotating on our heel among the stars; not the stars that are rotating around us.” When this distinction is made, the critic is assuming tacitly that there exists a super-space acting as an absolute frame of reference with respect to which rotation acquires [Pg 334] a meaning; for in the absence of an assumption of this sort, the distinction he wishes to defend would, of course, be meaningless.

The statement, then, reduces to the following: “We know that the material universe is non-rotating in the super-space; hence, when we start rotating on our heel with respect to the stars, we know that it is we who are rotating in the super-space, the stars remaining at rest.” This was essentially Newton’s stand when he identified the super-space with absolute space. But Newton at least had given physical significance to this stand by ascribing centrifugal force to a rotation in this absolute space. When, however, we follow Mach and reject Newton’s physical assumption, the absolute space is deprived of all possibility of manifesting itself physically. And so the critic’s assertion loses all physical significance, since henceforth the only space in which rotation can manifest itself physically is the star-space.

The metaphysician might argue, however, that Mach’s interpretation of centrifugal force does not obviate the existence of a super-space, but merely denies us the ability to establish its existence by physical means. Let us grant this contention and see where it leads us. When we assert that it is we who are rotating on our heel, we are assuming implicitly that prior to our rotation the universe and our body were non-rotating in the super-space. Were we to cast any doubt on the correctness of this initial assumption, no conclusions could be drawn. For suppose that initially the universe, together with our body, was rotating clockwise in the super-space; then, as a result of an anti-clockwise rotation on our heel, it would now be we who were non-rotating in the super-space, while the universe of stars persisted in its state of rotation. Hence, the stars would be rotating around us. So we see that according to our initial assumptions as to the state of rotation of our universe in the super-space, we could derive any conclusions we pleased. The problem reduces, then, to determining the precise rotationary motion of our universe in the super-space. Now, inasmuch as this rotation is precluded from manifesting itself in any physical way, since centrifugal force has already been accounted for, the problem can of course never be solved by the physicist. As for the metaphysician, unless he can furnish us with some supra-physical means of determining the true situation, he also will be incapable of supplying us with the required information. It follows that if Mach’s premises are accepted, the entire super-space conception becomes meaningless, for any assumption whatever can be made in regard to it with impunity.

In short, Mach’s views do not conflict with causality; for we may always say that our efforts have been exerted on our body and not on the stars. But we must recognise that our efforts have set our body in rotation with respect to the star-space, and not with respect to a space which transcends the existence of the material universe.

Now, although a number of thinkers were in sympathy with Mach’s attitude, owing to their belief that all motion in a perfectly empty universe must be meaningless, yet the physical difficulties [Pg 335] confronting his idea were so great that classical science had never taken it seriously. Euler had discarded the possibility of any such stellar action, and even quite recently Planck had criticised Mach for his stand. It was not that in theory a possible stellar action should be excluded on general principles; for physics had made us acquainted with foreign actions of a similar nature, the best-known of which would be illustrated by the gravitational action of the sun on the planets. Then again, when we wish to cut through an apparently empty space with a knife, we may experience considerable difficulty if the space happens to be permeated by a powerful magnetic field. We cannot, therefore, exclude the possibility that the stars might in some way generate a field endowing space with a structure which we had mistakenly taken to represent the absolute intrinsic structure of space. Science, however, cannot content herself with vague qualitative analogies; and before a mechanics such as Mach’s could be entertained, it would be necessary for it to succeed in accounting for the existence of centrifugal and Coriolis forces on a body in rotation with respect to the stars. This accounting would have to be made, not merely in some vague qualitative way, but quantitatively also, to a high order of precision. Such, indeed, are the requirements that all hypotheses in physics must satisfy. Now, all that Mach had done had been to postulate a possible stellar influence; he had not even attempted to justify his ideas rigorously. Indeed, even had he investigated the problem more deeply, he would have failed; for in his day space-time was unknown.

Furthermore, Mach’s ideas presented insuperable difficulties. Newton’s law of gravitation stated that the attractions exerted between bodies depended solely on their masses and mutual distances; the conditions of relative motion or rest of the bodies being considered irrelevant. But if Mach’s ideas are accepted, we must assume that in addition to the ordinary Newtonian attraction, a supplementary field of attraction, hitherto unsuspected, should arise if the bodies were in a state of relative acceleration, as in the case of the stars rotating round the earth, or the planets moving round the sun.[115] Mach’s mechanics would entail, therefore, the downfall of the Newtonian law of gravitation. Now, when Mach suggested his mechanics, Newton’s law appeared to be justified indirectly by the accuracy with which it had permitted mathematicians to foresee numerous astronomical occurrences (with the possible exception of Mercury’s motion and certain lunar disturbances). In addition, Newton’s inverse-square law of the attraction of matter on matter had been verified by direct measurements in the laboratory, in what is known as Cavendish’s experiment. So far as experiment could detect, there was no trace of an increase in the mass of a body [Pg 336] when other bodies were placed in its proximity; hence, on this point it appeared impossible to detect the variations in mass demanded by Mach’s mechanics. As for attempting Cavendish’s experiment with masses mutually accelerated, technical difficulties rendered its success impossible. For all these reasons the majority of scientists viewed Mach’s relativity of rotation with the utmost suspicion.

But when we come to consider the theory of relativity, the situation changes. In the first place, Newton’s law of gravitation is seen to be only approximate, and the relative motion existing between bodies is found to modify their apparent mutual attractions. This in itself is, of course, insufficient to substantiate Mach’s mechanics, but when we examine the implications necessitated by Einstein’s cylindrical universe, we find that Mach’s views may turn out to be correct after all.[116] However, before discussing this aspect of Einstein’s theory, we must mention that even in classical science the relativity of accelerated and rotationary motion could be accepted in a restricted sense for mechanical phenomena.

Consider, for instance, the earth rotating among the stars. The space defined by a frame attached to the earth would be permeated with a field of centrifugal and Coriolis forces. Now we were always perfectly at liberty to regard the earth as non-rotating, and the stars as rotating round it, provided we retained the existence of this special field of force. If we consider this field to extend beyond the stars, calculation shows that under the action of these forces the stars would undoubtedly circle round the earth as they appear to do. To this extent, therefore, the relativity of rotation, when restricted to mechanical phenomena, could be defended in classical science. The general principle of relativity goes one step farther by permitting us to extend this relativity of rotation to all manner of phenomena—optical and electromagnetic as well as mechanical. But, even so, the relativity of rotation is not radical, for we have rendered it possible only by retaining this curious field of force surrounding the earth; and the query naturally arises: Whence does this field originate? Newton replied: “From rotation in absolute space.” Einstein would have replied: “From rotation in absolute space-time.” In other words, the hypothesis of an absolute extension possessing dynamical properties per se was adhered to both by classical science and by the theory of relativity prior to Einstein’s speculations on the [Pg 337] form of the universe. Such being the case, to regard the earth as non-rotating was a mere mathematical fiction.

Now, when Einstein formulated his postulate of equivalence, the classical conception of centrifugal force as arising from a rotation in absolute space gave rise to serious difficulties. The postulate of equivalence asserted that forces of inertia and of gravitation were of the same nature; but then it followed that they should be traceable to one same origin. But gravitation, as we know, is due to matter; hence in a world totally devoid of matter there could be no such thing as gravitation. But then, according to the inferences deduced from the postulate of equivalence, neither should there be any such things as forces of inertia, centrifugal forces and inertial mass. It would follow that if some tiny test-body were made to rotate in an otherwise empty universe, no trace of centrifugal force would be observed at its equator. Thus we see how the postulate of equivalence, which is deduced from the well-established identity of the two types of masses and which constitutes the starting point of Einstein’s theory of gravitation, must inevitably tend to lead us towards views similar to those of Mach.

The equivalence postulate also appears to point to the cylindrical universe; for if we accept the hypothesis of an infinite quasi-Euclidean universe with its nucleus of stars, we shall have to assume that the rim of a rotating disk of sufficiently large proportions could be conceived of as extending well beyond the farthest stars of the nucleus. Yet, the larger the disk and the more remote the rim from the stars, the greater would become the centrifugal force acting on the rim. It would appear as though the source of the centrifugal force would have to be traced to the empty regions at infinity, so that this force could never be attributable to the star-matter concentrated in the nucleus. In other words, gravitation would subsist only in and around the nucleus, whereas forces of inertia would subsist everywhere and have their source at infinity. Not only would such a conception be exceedingly unsatisfying by removing to infinity the causes of forces which we observe at finite distances, but it would also appear to be in utter conflict with the inferences deduced from the postulate of equivalence, establishing, as it would, a duality between the nature of inertia and gravitation.

There seems to be only one way to remove these difficulties; namely, to accept the hypothesis of Einstein’s cylindrical universe conditioned entirely by matter. In this self-contained universe the disk, if extended indefinitely, would finally curl round on itself, enveloping the universe; its rim would never extend to spatial infinity. Calculation would show that as the disk was gradually extended, the centrifugal force on the rim would at first grow, then decrease, and finally vanish again. The displeasing necessity of having the cause of centrifugal force removed to infinity would thus be obviated; centrifugal force, just like gravitation, would be traceable to the spatial structure of the universe and hence to the presence of matter at finite distances.

[Pg 338]

It is to be noted that these views lead us to Einstein’s cylindrical universe, and not to de Sitter’s. For de Sitter’s universe, contrary to Einstein’s, can subsist devoid of all matter; centrifugal force and inertia would be present in it, even in the complete absence of matter and gravitation. It would thus fail to satisfy the requirements deduced from the postulate of equivalence; since once again, as in the infinite universe, centrifugal force and inertia generally would appear as unconnected with matter, hence with gravitation.

All the arguments discussed in the preceding paragraphs tend to prove that Einstein’s theory was leading us insensibly towards Mach’s mechanics and the cylindrical form of the universe; these two, though apparently disconnected, proving themselves to be intimately related. But it would be a grave mistake to assume that Einstein had set his mind a priori on justifying Mach’s views. Any one who has followed the arguments outlined will perceive that Einstein is not forcing nature into some preconceived mould. But, even at this stage, we have not discussed the weightier reasons which finally compelled him to feel sympathetic towards Mach’s mechanics. We shall now remedy this omission.

Einstein’s gravitational equations without the term, i.e., the equations applying to the infinite universe, namely , had received brilliant support in having enabled him to account for the motion of the planet Mercury, and to anticipate such unsuspected phenomena as the double bending of a ray of light and the Einstein shift-effect. It was only natural, therefore, to endeavour to extract from these equations all the remaining treasures they contained. But here integrations of an extremely difficult nature awaited mathematicians.

When the equations were treated in a very approximate way, they yielded Newton’s law of gravitation and the Einstein shift. When the approximation was increased, they yielded the double bending of a ray of light. When the approximation was increased still further, we obtained the precessional advance of Mercury’s perihelion. When treated in a more general way, they enabled Einstein to predict that gravitation would be propagated with the speed of light. Thirring then attacked the equations, carrying his calculations to a still higher degree of approximation, and at last, the effects necessitated by Mach’s mechanics appeared. Thirring found that the rotation of a central body would modify the nature of its attraction on the planets or satellites circumscribing it. (These anticipations would appear to be susceptible of astronomical verification in the case of Jupiter and one of its satellites.) But, most important of all, he discovered that Mach’s belief in the gravitational influence of the stars rotating round a fixed body was finally justified. In particular, the equations proved that even in the hypothesis of an infinite universe, a rotation of matter would produce around a test-body a field of force disposed in exactly the same peculiar way as are the centrifugal and Coriolis forces in the space attached to a body in rotation. Furthermore, the mass of a test-body would be affected by the proximity of surrounding [Pg 339] matter. This last consequence, however, can be obtained without resorting to difficult calculations.[117]

Now the gravitational equations which have yielded these interesting results have most certainly not been doctored up for the sole purpose of justifying Mach’s ideas. As explained in previous chapters, they were discovered by Einstein as a result of a totally different train of thought, long before the relativity of rotation was even considered. Hence, it cannot be denied that by vindicating, as they did, the correctness of Mach’s ideas, a fact of the utmost importance had been established. Mach’s mechanics received at last that rational justification which until then had been lacking; and the first inkling of a possible physical connection between the origin of inertia and gravitation was obtained. However, it is important to note that although from a qualitative standpoint the agreement was perfect, yet when we view the problem quantitatively, calculation shows that the increase of mass and the centrifugal forces generated by the rotation of the stars would produce but an infinitesimal fraction of the magnitudes observed in practice. In order to obtain magnitudes corresponding with observation, it would be necessary to assume that the stars were many trillion times more numerous than we believed them to be, and this supposition would lead us into other difficulties.

The reason for this discrepancy is not hard to discover. Centrifugal force, as we know, is produced when a body is torn away from the space-time geodesic which the metrical field would compel it to follow. To say, therefore, that a rotation of the firmament round a test-body would affect the field of force surrounding the body is equivalent to stating that a rotation of the stars would be capable of affecting the distribution of the metrical field round the body and of dragging the geodesics along. There is no reason to be surprised at some such effect taking place, seeing that the phenomenon of gravitation connotes that matter, whether at rest or in motion, must affect the curvature of space-time, hence the lay of the metrical field. But, on the other hand, until Einstein had discovered the cylindrical universe, the action of matter could only modify the lay of a pre-existing metrical field, and could not create such a field out of nothing. Space-time of itself was assumed to be flat, and matter could only endow it with a certain degree of curvature. The result was that the major part of centrifugal force was due to the intrinsic texture of flat space-time, and only an infinitesimal portion of this force could be attributed [Pg 340] to the additional action of the stars. Centrifugal force thus still appeared, in the main at least, to betray absolute rotation in empty, absolute space-time.

None the less, an important point had been established. Mach’s ideas, though still refuted quantitatively, yet seemed to be acceptable qualitatively, whereas in classical science they had appeared utterly impossible from every standpoint. The thin end of the wedge had been driven in. From then on, it became permissible to assume that with an increase in our understanding of the universe the complete relativity of all motion would be established, not only qualitatively, but also quantitatively. In other words, not merely an infinitesimal portion of centrifugal force and of mass, but their totality, might be accounted for by appealing to the interaction of matter and matter. It is easy to understand how this achievement would be realised.

All we should have to do would be to assume that the totality of the metrical field, and not merely a portion of it, was created by the distribution of matter throughout the universe. As we have seen, this condition is precisely the one realised in the cylindrical universe; so that with Einstein’s universe, Mach’s ideas may be vindicated and the bugaboo of absolute rotation dispelled.[118]

Both de Sitter’s universe and the quasi-Euclidean one would be incompatible with Mach’s mechanics, since in either universe the metrical field and the geodesics of space-time would exist in the complete absence of matter. This means that in the empty world of de Sitter, and far beyond the nucleus of stars in the quasi-Euclidean universe, bodies would be submitted to forces of inertia when torn away from their natural geodesical world lines. In either case the space-time void would appear to be endowed with dynamical properties even in the absence of all matter.

The following passage quoted from Einstein’s writings expresses his views on the subject:

“The idea that Mach expressed, that inertia depends upon the mutual action of bodies, is contained to a first approximation in the equations of the theory of relativity. It follows from these equations that inertia depends, at least in part, upon mutual actions between masses. As it is an unsatisfactory assumption to make that inertia depends in part upon mutual actions and in part upon an independent property of space, Mach’s idea gains in probability. But this idea of Mach’s corresponds only to a finite universe bounded in space, and not to a quasi-Euclidean infinite universe. From the standpoint of epistemology it is more satisfying to have the mechanical properties of space completely determined by matter, and this is the case only in a space-bounded universe.”

[Pg 341]

Here, a number of objections may be urged. It might be claimed that if we assumed the earth to be non-rotating, we should have to attribute velocities greater than light to the stars. The answer to this objection would appear to be similar to that given when the centrifugal force acting on the rotating stars was discussed. We saw that the stars, as a whole, may be regarded as rotating round the earth only in a purely formal sense, for it is the agglomeration of stars which creates the metrical field, hence which creates space; the earth is but an incident in this star-created universe. Loosely speaking, we may say that the stars are always at rest in the universal space which they themselves have created; and this holds regardless of whether or not we consider this space as rotating in some fictitious outer space.

But there is another objection which, to the writer at any rate, appears to have some weight. Suppose all the stars were to be annihilated at a given instant of our earth frame. The gravitational disturbance produced by this annihilation would require a number of years to reach us, since gravitational disturbances travel with the speed of light. Hence, the metrical field, or space-time structure, round the earth would persist for a considerable time. For a number of years, therefore, centrifugal force would still continue to manifest itself. In view of this fact, we should take into consideration the present condition of the metrical field rather than the present positions of the stars. We are thus led to a more dualistic outlook, that of both field and matter. Such appears to be Weyl’s attitude when he writes:

“The falling over of glasses in a dining car that is passing round a sharp curve and the bursting of a flywheel in rapid rotation, are not, according to the view just expressed, effects of ‘an absolute rotation’ as Newton would state, but whose existence we deny; they are effects of the metrical field. In so far as the state of the guiding field does not persist, and the present one has emerged from the past ones under the influence of the masses existing in the world, namely, the fixed stars, the phenomena cited above are partly an effect of the fixed stars, relative to which the rotation takes place.... We say ‘partly’ because the distribution of matter in the world does not define the ‘guiding field’ uniquely, for both are at one moment independent of one another and accidental (analogously to charge and electric field).”

At the present time all these questions relating to the shape of the universe still remain highly speculative; and we need not be surprised to find the most component authorities, such as Einstein, Weyl, Eddington, Silberstein and Thirring, manifesting different tendencies according to their respective temperaments. Eddington manifests a preference for de Sitter’s universe. If this universe be adopted, Mach’s ideas are refuted; for centrifugal forces would be generated in empty space-time devoid of all matter, and the stars would have very little to do with them. De Sitter, of course, defends his own model of the universe and reproaches Einstein with having resuscitated a species of Newtonian absolute space by materialising the inertial frame with respect to which rotation and acceleration are measured. There is not [Pg 342] the slightest doubt that the cylindrical universe accomplishes this result by linking the existence of the inertial frame to a permanent star distribution. But, in the eyes of many, this is one of its greatest advantages, since it leads to the physical relativity of all motion. After all, we cannot deny the existence of centrifugal force and forces of inertia; and surely it is more satisfactory to account for these forces in terms of the acceleration of matter with respect to matter, than in terms of acceleration with respect to that vacuum which we call empty space-time. But these divergencies of opinion arise from mere matters of sentiment, and it is important to note that they concern chiefly the loftiest superstructures of the theory; they do not invalidate in any way its essence and main achievements.

Of course, it would be of the utmost interest to decide the question of the form of the universe by appealing to astronomical observation, and it is highly probable that in a relatively short time the problem will be settled.

Recapitulating, we have three possible hypotheses to consider:

1. An infinite universe of space with a nucleus of star-matter, in which Mach’s mechanics would be impossible. We should then have to consider the void as possessing dynamical properties and a structure, independently of the presence of matter.

2. De Sitter’s universe, which would also be contrary to Mach’s ideas and would present the same unsatisfactory features as the first hypothesis. Furthermore, it would be unstable.

3. Finally, Einstein’s cylindrical universe, which would be stable and whence all the displeasing features mentioned would be removed.

Could we but annihilate all the stars, we should have a means of solving the problem; for in Einstein’s universe the result would be the total disappearance of mass and centrifugal force. In fact, complete chaos would result; for the metrical field of space-time would have vanished, so that light rays and free bodies would no longer know how to move. But as an annihilation of the stars is beyond the range of possibility, we must have recourse to other methods. So we are led to examine the question of stability. Is it true that the stars are in a state of statistical equilibrium? Is it true that their velocities are always small compared to that of light? Eddington casts some doubt on this stability by remarking that the spiral nebulæ appear to be animated with enormous velocities. This fact, coupled with the apparent reddening of their light, leads him to manifest a preference for de Sitter’s universe. But there exists still another method of settling the question.

We remember that Einstein’s original law of gravitation in the case of the infinite universe differed in certain respects from Newton’s, and that these differences were important enough to be detected within the limits of the solar system through their effects upon Mercury’s motion and the double bending of a ray of starlight. But if now we assume the universe to be finite, and hence include the term in the gravitational equations, calculation shows that a further modification will ensue. It will be as though all space round a star [Pg 343] were filled with negative or repulsive matter of very low density, of the same order as the curvature of the universe. The influence of this factitious negative mass distribution would be to decrease the normal value of the star’s attraction; the decrease, owing to the cumulative effect produced by the distribution, becoming more and more apparent as we considered points farther removed from the star. Needless to say, this additional modification in the law of gravitation would be incredibly small, owing to the minuteness of the world-curvature and hence of the density of the negative mass distribution. It would be idle to hope to detect it unless we studied gravitational phenomena occurring over tremendous astronomical extensions. But the effect might be observed in the following way:

Consider, for instance, gravitating masses such as the sun and its planets. We know that the reason the solar system does not collapse and all the planets fall on to the sun is because the planets are in motion (as referred to an inertial frame attached to the sun). There exists a definite mathematical relationship between the velocity of a planet along a definite orbit and the force of gravitation to which it must be subjected by the sun. The greater the velocity, the greater must be the sun’s attraction in order to maintain the planet in its orbit. Now the globular clusters of stars and the Milky Way constitute precisely such permanent gravitating systems of enormous extent. The individual stars attract one another, and their mean velocities allow us to deduce the resultant gravitational forces to which they must be subjected. If, therefore, on applying Einstein’s original law of gravitation (that corresponding to the infinite universe),[119] we found that the mean velocities of the stars in the Milky Way were smaller than would be expected, we could consider it proved that the law of attraction must fall off more rapidly than Einstein’s original law indicated. This discrepancy would then justify our belief in the finiteness of the universe. The value of , the cause of this discrepancy, could be ascertained, and with it, of course, the value of the universal curvature. Knowing the curvature, we could determine therefrom the size of the universe, the average density of matter, and finally the total mass of the universe. Einstein is of the opinion that astronomical observations of this type will eventually settle this question.

We have exhausted the principal points which prompted Einstein to believe in the finiteness of the universe, but there are yet others connected with the possibility of reducing matter to electricity and gravitation. We shall touch on this additional aspect of the question in a later chapter.[120] For the present, we may pass to a totally different type of problem, also connected with the finiteness of the [Pg 344] universe. We refer to the standard of gauge.

As Eddington points out, when our equations prove to us that the universe of space must be spherical and have a constant curvature, what else can it mean but that if we measured the radius of curvature with a material rod, we should obtain the same magnitude in every direction? But then it follows that the radius of the universe in any direction constitutes the gauge of length which nature imposes upon us; and that all bodies in equilibrium adjust themselves automatically so as to maintain some definite fraction of the length of this radius, in whatever direction they be placed. This view suggests that for material bodies to be on a definite scale of size, there must be a curvature of the world. Eddington also suggests that inasmuch as the finite universe is open in a time-like direction, whereas it is limited in space, a possible explanation may be given of the apparent permanency of existence of certain entities such as electrons and atoms, which are limited in spatial extension but are everlasting in time. Weyl’s theory throws further light on these rather hypothetical problems.

The finite universe introduces us to the difficult conception of a spherical space. We have already discussed this type of space in a previous chapter; but in view of its additional importance, now that the real universe is suspected of being one, it may be worth while to examine the problem afresh.

Let us first consider the two-dimensional analogy. If, on the two-dimensional surface of a sphere, we perform measurements with tiny rigid Euclidean rods, we obtain the numerical results, not of Euclid’s but of Riemann’s two-dimensional geometry. Accordingly, we say that the geometry of a spherical surface is Riemannian, and that a Riemann space is curved and finite. But the word “curved” immediately evokes the idea of a third dimension along which the surface or space is curved. In the present two-dimensional case we have no difficulty in visualising this third dimension, because the space of our actual experience happens to be three-dimensional. And so we note that the spherical surface possesses an outside and an inside, and that if we consider solely the points on the surface, we are excluding from our investigation an indefinite number of points of space both inside and outside the [Pg 345] sphere. This procedure is unfortunate; for when we generalise our previous two-dimensional illustration to a three-dimensional spherical space, we are inclined to reason by analogy and to conceive, or endeavour to conceive, of a four-dimensional Euclidean space in which our three-dimensional Euclidean universe must be embedded. The result is that we think of points of space outside our universe, as though our universe did not embody the whole of space.

In the present case this attitude would be entirely erroneous, but the fault lies with the peculiar type of two-dimensional analogy we chose to present. In order to eliminate this error, let us proceed as explained in a previous chapter, and let us suppose that the sphere rests at its south pole on an infinite Euclidean plane, while a point of light is situated at its north pole. If the sphere is translucent, our tiny rods displaced on the spherical surface will cast their shadows on the plane; and, of course, if we effect measurements on the plane by means of these shadows of our rods, we shall obtain exactly the same metrical relationships on the plane as we obtained formerly on the spherical surface. In particular, if a hundred tiny rods lying along a meridian span the distance from the south to the north pole on our sphere, the hundred shadows of these tiny rods will extend across the plane to infinity. Likewise, just as a finite number of postage stamps would cover the sphere’s surface, so now a finite number of shadows, the shadows of these stamps, would cover the entire infinite extent of the plane. Thus we see that when measured with our shadows the plane will be finite and its geometry Riemannian, whereas it would have been infinite and Euclidean had we effected our measurements by displacing ordinary foot-rules over its surface.

The advantage of presenting the problem in this alternative manner is that in the present case the introduction of a third dimension is no longer necessary in order to explain curvature, or non-Euclideanism. We realise that regardless of whether the space of the plane is to be considered as finite and Riemannian or as infinite and Euclidean, exactly the same points of space are included in either case. This two-dimensional illustration can, of course, be extended to three dimensions. By following the same line of reasoning, we will say that the universe of three-dimensional space is infinite, if by placing side by side, in rows, columns and superimposed layers, an indefinite number of congruent cubes, we find it possible to go on piling them up for ever and ever. On the other hand, we will say that the universe of space is finite if this process cannot be continued indefinitely, that is, if after a finite number of cubes have been superimposed, further progress is impossible. This last case is, of course, similar to the one discussed previously when we found it impossible to place an indefinite number of shadows of our postage stamps over the plane.

From these examples it is apparent that the finiteness or infiniteness of our space is contingent on the geometrical behaviour of the cubes [Pg 346] which we regard as congruent or equal. From a purely mathematical standpoint, therefore, the problem of the finiteness of the universe can possess no absolute significance, since congruence, or equality of size, is incapable of being defined in any absolute way. Yet from the standpoint of the physicist the problem assumes a very definite significance, for whereas one unique rational definition of congruence is impossible, an empirical definition is easily obtained. By congruent cubes the physicist means those which visually and tactually will appear the same in whatever region of space they may be situated, provided the observer accompanies the cubes in their displacements. He means cubes carved out of the substance of those bodies which men have agreed, from time immemorial, to regard as invariable solids, and which are illustrated by the bodies around us, when maintained under conditions of constant temperature and pressure.

It is as referred to such bodies, therefore, or, again, to the courses of light rays, that the universe, according to Einstein, would manifest itself as finite. This is but another way of saying that our bodies would behave like Riemannian solids (over vast areas at least) instead of like Euclidean ones, as was formerly thought to be the case.

Summarising, we see that this same so-called finite universe would manifest itself as infinite were our rods to behave Euclideanly, whence we may conclude that the universe, whether finite or infinite, may contain exactly the same regions of space, just as our plane contained the same points whether measured with yardsticks or shadows. To assume, therefore, that certain regions of space must necessarily be excluded owing to the finiteness of the universe, would be to take a stand in no wise demanded by the actual conditions of the problem.

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CHAPTER XXXV THE IMPORTANCE OF SPACE-TIME, AND THE PRINCIPLE OF ACTION

IN the course of this book we have seen how a number of difficulties were overcome as soon as we recognised the existence of four-dimensional space-time in lieu of the world of separate space and time of classical science. There are still other aspects of the problem, however, and we shall now mention them briefly.

In the first place, all vectors such as velocities and forces, and all tensors such as Maxwell’s stresses of the electromagnetic field, erstwhile expressed in three-dimensional space, must now be extended and supplemented with additional components so as to yield vectors and tensors in four-dimensional space-time. These new components do not define new physical magnitudes unknown to classical science; in the general case they give us well-known magnitudes, such as work, power, and so on. As a case in point, let us consider electric and magnetic intensities. In classical science these were entirely distinct magnitudes. But when we represent these intensities in terms of space-time, we find that they are given by the various components of one same space-time tensor ; the time components give the electric force, while the space components yield the magnetic force. The advantage of the new point of view is chiefly to show us that what we once considered different entities now reveal themselves as the different components of the same space-time existent. The reason for the observed connections between these different entities then becomes apparent.

In particular, let us examine the case of the equations of electrodynamics. In the study of electricity and magnetism we may consider phenomena in which conditions do not vary as time passes by; the electric charges and the magnets remain at rest, and currents flowing in fixed wires do not vary in intensity. Conditions are then termed stationary; it is as though time played no part. The laws which govern this type of phenomena were discovered empirically over a century ago, and were expressed mathematically in terms of spatial vectors. The problem of ascertaining how electric and magnetic phenomena would behave when conditions ceased to be stationary was one that could not be predicted; further experimental research work was necessary before the general laws could be obtained. Even so, the difficulties were considerable, and it needed Maxwell’s genius to establish the laws from the incomplete array of experimental evidence then at hand. All this work extended over nearly a century; it was slow and laborious.

Yet, had men realised that our world was one of four-dimensional Minkowskian space-time, and not one of separate space and time, things [Pg 348] would have been very different. By extending the well-known stationary laws to four-dimensional space-time, through the mere addition of time components to the various trios of space ones, we should have written out inadvertently the laws governing varying fields, or, in other words, we should have constructed Maxwell’s celebrated equations. Electromagnetic induction, discovered experimentally by Faraday, the additional electrical term introduced tentatively by Maxwell, radio waves, everything in the electromagnetics of the field, could have been foreseen at one stroke of the pen. A century of painstaking effort would have been saved.[121]

It is also interesting to note that the four separate relations which constitute Maxwell’s equations of electromagnetics are now merged into two; and the cumbersome aspect of the equations gives place to forms of great simplicity.

Still another simplification which the discovery of space-time has conferred upon us is to be seen when we study the principles of the conservation of energy and of momentum. We find that these two distinct principles constitute in reality but one. Conservation of energy is given by the time component, while conservation of momentum is given by the three space components, of one same space-time tensor law of conservation.

In a general way, all problems of dynamics or kinematics in three-dimensional space can be treated as problems of statics in four-dimensional space-time. All these advantages of space-time appeal very strongly to the theoretical investigator who deals with mathematical equations; but they all go to show how much simpler it is to understand the workings of the universe when we realise that the fundamental continuum of the world is space-time, and not separate space and time.

We must now mention another most important conception of classical science, namely, Action; and we shall see that this important mathematical entity finds its place in a perfectly natural way in the world of space-time. In order to understand the significance of Action, let us consider any mechanical system passing from an initial configuration to a final configuration . Classical science defined the action of this system as the difference between its total kinetic energy T and its total potential energy , taken at every instant and then summated over the entire period of time during which the system passed from the initial state to the final state .[122]

Now the total kinetic and potential energies of the system at any instant are given by [Pg 349] where and represent the densities of the kinetic and potential energies at every point throughout the space occupied by the system. Accordingly, the expression of the action will be given by In the second formula we have merely replaced () by a single letter . Henceforth () or will be referred to as the function of action.[123]

Roughly speaking, action was thus in the nature of the product of a duration by an energy contained in a volume of space. On no account may this action be confused with the action dealt with in Newton’s law of action and reaction, also expressible as the principle of the conservation of momentum. Still less may it be confused with the term “action” which appears in philosophical writings. The importance of the conception of action arises from the fact that the laws of mechanics can be expressed in a highly condensed form when the conception of action is taken into account. We thus obtain what is known as the Principle of Least Action, with which the names of Fermat, Maupertius and Lagrange are associated. A more convenient form of the principle was given by Hamilton; it then becomes Hamilton’s Principle of Stationary Action. If we restrict our attention to the very simplest case, we may state Hamilton’s principle as follows:

If we consider all the varied paths along which a conservative system may be guided, so that it will pass in a given time from a definite initial configuration to a definite configuration , we shall find that the course the system actually follows, of its own accord, is always such that along it the action is a minimum (or a maximum).

Now the principle of action issues, as we have said, from the laws of classical mechanics. Were these laws different, the principle of action would cease to apply, at least in its classical form. A priori, we have no means of deciding whether the laws governing physical phenomena of a non-mechanical nature—those of electromagnetics, for example—would also issue in the same principle of action.

But when Maxwell had proved that his equations of electromagnetics could be thrown into a form compatible with the principle of action, and when he had succeeded in amalgamating electricity, magnetism and optics into one science, the universal validity of the principle was accepted. Inasmuch as this principle includes that of the conservation of energy, we can understand why the principle of action was often referred to as the supreme principle of physical science. Incidentally, we may mention that when the principle of action is satisfied by a phenomenon, an indefinite number of different mechanical interpretations of the phenomenon are theoretically possible. In the case of electrodynamic phenomena, however, in view of the complicated [Pg 350] hypotheses which he was compelled to postulate, Maxwell abandoned all attempts to discover the precise mechanical interpretation which would correspond to reality.

Thus far, we have limited ourselves to stating the general validity of the principle of action, as applying to all physical processes. It now remains to be seen how the principle will be of use to us in furthering our knowledge of the laws of nature. This we may understand as follows: The principle, as we have seen, imposes the condition that the natural evolution of any system must be such as to render the action a maximum or a minimum. Could we but express this condition in terms of the usual physical magnitudes, we should be enabled to map out in advance the series of intermediary states through which the phenomenon would pass. From this knowledge we should derive the expression of the laws which governed the evolution of the phenomenon. Here, of course, a twofold problem presents itself. First, we must succeed in finding the correct mathematical expression for the action; and, secondly, we must be in a position to solve the purely mathematical problem of determining under what conditions the action will be a maximum or a minimum.

Now all problems of maxima and minima are solved by means of the calculus of variations, a form of calculus we owe chiefly to Lagrange. According to the methods of this calculus, we establish under what conditions a magnitude is a maximum or a minimum by discovering under what conditions it will be stationary.

Let us explain what is meant by the word “stationary.” When a stone is thrown into the air, it ascends with decreasing speed, then seems to hesitate for a brief period of time as it hovers near the point of maximum height before it starts to fall back again towards the earth. During this brief period of hesitation at the apex of its trajectory, the stone is said to remain “stationary.” We can recognise a stationary state by observing that when it is reached no perceptible changes take place over a short period of time. In this way, we may understand the connection which exists between the stationary condition and the presence of a maximum or a minimum. In mathematics small variations are represented by the letter ; hence the stationary condition of the action, or again, the principle of action, is expressed by The calculus of variations enables us to solve such purely mathematical problems. Under the circumstances the problem of determining the laws which govern any particular phenomenon reduces to discovering the expression of the function of action pertaining to the phenomenon in question. Larmor applied this method to the phenomena of electricity and magnetism and showed how Maxwell’s laws of electrodynamics could be deduced from a suitable mathematical expression defining the electromagnetic function of action.

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When the theory of relativity supplanted classical science, it was recognised that the classical equations of mechanics were only approximate, and it became necessary to reformulate the principle of action so as to render it compatible with the mechanics of relativity, hence also with space-time. This work was carried out by the pure mathematicians—by Klein and Hilbert in particular. It was then found that a principle of action differing but slightly from the classical one could be obtained. But inasmuch as the difference involved presents only a theoretical interest, we shall discuss the bearing of the principle of action on the theory of relativity without taking the slight aforementioned modification into consideration.

In classical science, it was strange to find that action, though so important in its physical significance, should yet present the artificial aspect of an energy in space multiplied by a duration. As soon, however, as we realise that the fundamental continuum of the universe is one of space-time and not one of separate space and time, the reason for the importance of the seemingly artificial combination of space with time in the expression for the action receives a very simple explanation. Henceforth, action is no longer energy in a volume of space multiplied by a duration; it is simply energy in a volume of the world, that is to say, in a volume of four-dimensional space-time. Designating a volume of space-time by , we have so that our principle of action, , becomes There is now perfect symmetry between the rôles of space and time.

Now there was still another aspect of the principle of action which was extremely displeasing in classical science: By making the present course of a phenomenon contingent on a condition which could be determined only at some future date, it acquired a teleological character. But, as Planck points out, with the action as given by an energy in a volume of space-time, we may, by placing space and time on the same footing, regard duration as static; and the unsatisfactory teleological aspect of the principle may be obviated.

The space-time theory offers yet further advantages. It suggests that the function of action must be a space-time invariant. If, then, we know the elementary physical magnitudes which must enter into this invariant, purely formal mathematical considerations enable us to construct all possible invariants, and one of these will represent the function of action. Assuming the world to be built up of electricity and metrical field, the possibility of obtaining the “function of action” of the universe would seem to be within our grasp. Could this result be accomplished, a momentous discovery would have been [Pg 352] made, since all the laws of the physical world would be comprised in the expression The application of the principle of action, expressed in the preceding formula, would enable us to derive all the laws of the physical universe. We should understand many things that are still mysterious; the elusive reason for the existence of two types of electricity in nature would be revealed, and the strange tendency of matter and energy to congregate into atoms and quanta would be explained. Unfortunately we are far from being able to determine the “function of action” of the universe, for there may be physical elements entering into its constitution whose nature we as yet ignore. (Furthermore, we might mention that the very validity of the principle of action is called into question by the quantum phenomena.) The following passage quoted from Weyl gives an idea of what is at stake:

“Whereas, in mechanics, however, a definite function of action corresponds to every given mechanical system and has to be deduced from the constitution of the system, we are here concerned with a single system, the world. This is where the real problem of matter takes its beginning: we have to determine the ‘function of action,’ the world-function , belonging to the world. For the present it leaves us in perplexity. If we choose an arbitrary , we get a ‘possible’ world governed by this function of action, which will be perfectly intelligible to us—more so than the actual world—provided that our mathematical analysis does not fail us. We are, of course, then concerned in discovering the only existing world, the real world for us. Judging from what we know of physical laws, we may expect the which belongs to it to be distinguished by having simple mathematical properties. Physics, this time as a physics of fields, is again pursuing the object of reducing the totality of natural phenomena to a single physical law: it was believed that this goal was almost within reach once before when Newton’s Principia, founded on the physics of mechanical point-masses, was celebrating its triumphs. But the treasures of knowledge are not like ripe fruits that may be plucked from a tree.”[124]

Whatever may be the ultimate fate of such speculations in revealing to us the mystery of matter, it cannot be denied that they possess a grandeur which is compelling. At all events, in more restricted domains, the principle of action has led to important discoveries.

Hilbert, Klein, Weyl and later Einstein himself, by applying the principle of action to the space-time metrical field, were led to the law of space-time curvature in the interior of matter and outside [Pg 353] matter; that is to say, to Einstein’s law of gravitation.

The problem resolves itself into the selection of an appropriate expression for the action; but in the case of the law of gravitation a peculiar difficulty is encountered. We shall best understand the nature of this difficulty by considering the simpler case of the electromagnetic action. Electricity and magnetism present themselves in a substantial form as electrons, magnets and electrons in motion, or currents; and in a more ethereal form as fields, illustrated by the electromagnetic field. Inasmuch as both these manifestations of electricity and magnetism should enter into the laws of electrodynamics, the total electromagnetic action will have to comprise the two corresponding types of action. These can easily be determined, and an application of Hamilton’s principle of stationary action yields the laws of electromagnetics without further ado. But when we consider matter, classical science knew of but one form, the substantial form; the field form was lacking. To be sure, the gravitational forces surrounding matter appeared to present a certain analogy with the electromagnetic fields surrounding electrons and magnets. But, mathematically, at least, the analogy was superficial, for in contradistinction to the electromagnetic field, the gravitational field was not expressed by differential equations indicating continuous action through a medium. Only in name was it a field; and it appeared impossible to determine its action. With Einstein’s discoveries this difficulty was overcome. The gravitational field now appeared as a field analogous to the electromagnetic field. It was the metrical field of space-time, defined, as we know, by the potential distribution analogous to the potential distribution of the electromagnetic field. It can also be regarded as representative of the structure, or geometry, of space-time. Henceforth, the total material or gravitational action will be given by the action of matter proper plus that of the metrical field; the missing action has been found.

Thus the solution of the problem of gravitation reduces to establishing the mathematical expression of these two separate actions. Now the action of matter in a space-time region is the energy of matter contained in a volume of space-time. According to Einstein’s theory, therefore, it is given by the density of the mass of the matter multiplied by a space-time volume. As for the action of the gravitational field, inasmuch as it is now seen to be the action of the metrical field, it can be discovered without much difficulty. In this connection, we must recall that the relativity theory suggests that the action in a volume of space-time must be an invariant; its numerical value, whatever it may be, must be independent of our choice of mesh-system. In other words, the action of the metrical field must be an invariant built up from the fundamental tensors , whose distribution defines the metrical field.

We mentioned, when discussing tensors, that several such invariants exist, but that the simplest of all was represented by the Gaussian curvature . To this we may add any scalar or number without modifying its invariance, since numbers are, of [Pg 354] course, invariants. Hence, the simplest expression for the function of action of the metrical field is obtained by writing it: . This was, indeed, the expression selected by Einstein.[125]

From now on, it is a mere question of mathematics (calculus of variations) to express the stationary condition of the total action. As a result, we obtain Einstein’s law of space-time curvature, inside and outside of matter; that is to say, Einstein’s law of gravitation. According to whether we assume that is zero or has a non-vanishing value, we obtain the infinite universe, or the finite universe of either de Sitter or Einstein. In particular, Einstein’s universe will be the outcome if we impose the condition of stability upon the universe. We see, therefore, that the principle of action proves that all three types of universe are possible. If we wish to decide which type corresponds to the real universe, we must take other facts into consideration, as was explained in the chapter on the finiteness of the universe.

Now there remains one other important point which we must mention before abandoning the discussion of the law of gravitation. We saw that the principle of action alone did not enable us to decide whether the constant , which was adjoined to the action of the metrical field, was zero or had some non-vanishing value. In other words, it did not enable us to decide whether the universe was infinite or finite. But this is not the only source of indeterminateness. We mentioned that Einstein chose for the function of action of the metrical field because it was the simplest possible invariant made up of ’s; but we have also seen that other more complicated types of invariants could be constructed. Had these been substituted in the expression of the action, we should have obtained alternative laws of gravitation. But the point we wish to stress is the following:

[Pg 355]

Regardless of the precise invariant we may select in order to represent the function of action of the metrical field, there is one property which is common to all these invariants. Their mathematical nature is such that when we apply to them the principle of action, we find that the curvature of space-time which would be present in the interior of matter must be of a type which would remain conserved. Granting that a certain indeterminateness may exist in our choice of the action, and hence of the law of gravitation, all these laws would compel us to recognise that in our universe, permanent entities such as matter cannot help but be present. To be sure, the type of conservation necessitated by the principle of action is not altogether the classical conception of conservation; but, as we saw in a previous chapter, the new principle becomes that of classical conservation when a Galilean reference system is adhered to.

Finally, we may note that by applying the principle of action to the total action formed by the gravitational action and the field-action of electricity, we should obtain the precise type of space-time curvature which would exist in an electromagnetic field free of electrons and magnets; for instance, in an empty region flooded with light. It is then found (as was mentioned in Chapter XXXIII) that the electromagnetic field also produces a curvature of space-time, less pronounced than that reigning in the interior of matter, but more pronounced than that existing in the empty space around matter (that is to say, in a gravitational field due to ordinary matter).

[Pg 356]

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CHAPTER XXXVI THE MYSTERY OF MATTER

ONE of the greatest merits of the theory of relativity has been to allow us to represent gravitation as a direct consequence of the curvature of space-time. It would certainly be the crowning achievement of this superb theory if it could enable us to interpret all the manifestations of the physical universe in terms of the various types of curvature of one sole fundamental continuum, space-time. But at the present stage two separate reasons render this solution highly improbable.

We remember that in the course of our wanderings we came across two foreign tensors: , the matter-tensor, and , the electromagnetic tensor. It seems impossible to express these tensors in terms of the basic structural tensors of space-time, namely, the ’s. However, in the case of the matter-tensor , the case is not yet quite clear; and we shall see that opinion is divided on the subject. In order to understand the point at issue, let us turn once more to the gravitational equations in the interior of matter:

There are ten separate equations corresponding to the ten separate components of the second-order symmetrical tensors and . If we assume that a certain definite mesh-system has been chosen, or, again, that the observer is specified, each one of the ten components of will represent some physical magnitude relevant to matter at each point (see Appendix II). For instance, will represent the density of mass of the matter at a point; , and , the three components of momentum of the matter at a point, measured in the frame of reference specified; , and will represent the three components of vis viva, or energy of motion, coupled with the stresses in the matter at the point considered, etc.

The equations of gravitation thus signify that whenever we recognise the existence of one of these physical magnitudes it is always accompanied by corresponding curvatures of space-time. It is usual to assume that the curvatures are produced by those concrete somethings which we call mass, momentum, energy, pressure. In this way, we must concede a duality to nature; there would exist both matter and space-time, or, better still, matter and the metrical field of space-time. Einstein, when he elaborated his hypothesis of the cylindrical universe, attempted to remove this duality by proving that it was possible to attribute the entire existence of the metrical field, hence of space-time, to the presence of matter. This attitude [Pg 357] led to a matter-moulding conception of the universe, elevating matter over the metrical field of space-time. And, as we recall, only when this attitude was adhered to could Mach’s belief in the relativity of all motion be accepted.

Eddington’s attitude is just the reverse. He prefers to assume that the equations of gravitation are not equations in the ordinary sense of something being equal to something else. In his opinion they are identities. They merely tell us how our senses will recognise the existence of certain curvatures of space-time by interpreting them as matter, motion, and so on. In other words, there is no matter; there is nothing but a variable curvature of space-time. Matter, momentum, vis viva, are the names we give to these curvatures on account of the varying ways they affect our senses.

Of course, Eddington’s attitude, which is reminiscent of that of Clifford, is contrary to Riemann’s ideas, since, according to Riemann, an extension could have no geometry, hence no curvature and no metrical field, in the absence of matter, which would act as an active moulding agent. Furthermore, Eddington’s views lead us into certain difficulties when we attempt to apply the principle of action.

The reason for this is plain. If we regard a certain curvature of the metrical field of space-time as being synonymous with matter, there will be no reason to consider this curvature twice: once as representative of the action of matter, and once again as representative of the action of the gravitational field. By so doing, we should be merely duplicating our results. Hence, according to Eddington’s views, there would be but one action, that of the metrical field; and we could no longer add together the two separate actions in order to obtain a total action. Calculation then shows that in regions where matter exists the principle of stationary action would break down, so that a rigid application of the principle would connote that the world must be empty of matter. Eddington feels no objection to this limited validity of the principle of action, and defends his attitude with technical arguments which we cannot reproduce here.

Up to this point the motive that has guided us has been the desire to discover unity in nature by reducing all things to some fundamental form of existence. There is, however, another way to approach the problem; and this time our guiding principle will be no longer solely mathematical harmony, but also experimental facts.

We must realise that the tensor , whose components represent those physical magnitudes which our senses perceive and our instruments measure, can at best be considered to portray effects observed only in a macroscopic way. Phenomena such as radioactive explosions and all we have learnt indirectly from the exploration of the atom go to show that matter in the final analysis is constituted by whirling electrons around which intense electromagnetic fields are present. It follows that a microscopic investigation of matter would yield a very different picture from that which we construct from our crude sense perceptions, and that could we but view the atom in an ultra-microscopic way it [Pg 358] would appear to us as a region subjected to intense electromagnetic actions. It would seem, therefore, that our macroscopic tensor must comprise the electromagnetic one in its constitution.

But in order to remove the duality between matter and electricity, we should have to prove that could be built up exclusively from some more or less complicated expression containing , together with tensors derived from it; tensors of any other kind being barred. Physically speaking, this would indicate that atomic matter or bound energy, and in particular those forms of matter called electrons and protons, could be built up in terms of electromagnetic or loose energy. Even were we to succeed in this attempt we should not have banished all duality; for though we should have got rid of , there would still remain a duality between and the metrical field or ’s of space-time.

At any rate, all attempts to reduce matter to electromagnetic fields have thus far presented insuperable difficulties owing to the impossibility of accounting for those ultimate aspects of matter, the electron and proton, in terms of alone. The difficulty consists in understanding how it is that the electron does not explode under the mutual repulsion of its negatively charged parts. Some kind of counterbalancing pressure, the Poincaré pressure, seems to be necessary, and it appears to be impossible to account for this mysterious pressure in terms of the electromagnetic Maxwellian stresses alone. Following Poincaré, Mie attacked the problem. His procedure was to introduce additional electromagnetic-field magnitudes which would be effective in the interior of the electron, but whose action in outside space would be negligible. By this means, he succeeded in building up matter out of purely electromagnetic quantities. Einstein comments on Mie’s theory in the following words:

“In spite of the beauty of the formal structure of this theory, as erected by Mie, Hilbert and Weyl, its physical results have hitherto been unsatisfactory. On the one hand, the multiplicity of possibilities is discouraging, and, on the other hand, those additional terms have not as yet allowed themselves to be framed in such a simple form that the solution could be satisfactory.”

Following this attempt, Einstein proved that it was possible to obviate the introduction of Mie’s supplementary electromagnetic terms provided the space of the universe were assumed to possess a slight residual curvature, such as would exist in the cylindrical universe. It would then be this residual universal space-curvature which would act as a negative pressure preventing the electron from exploding. If we note that the universal curvature concerns the metrical field, hence the field of gravitation, we see that the energy of the electron must be composed of an electromagnetic and of a gravitational part. In Einstein’s own words: “Of the energy constituting matter, three-quarters is to be ascribed to the electromagnetic field, and one-quarter to the gravitational field.”

Einstein considers that these facts constitute a powerful argument in favour of the finite universe. At all events, it must be conceded [Pg 359] that the hypothesis of the finite universe appears to impose itself in a number of different ways. Not only astronomical considerations bearing on the low velocities of the stars, not only mathematical reasons connected with the boundary conditions, not only philosophical urges concerned with the relativity of rotation and the relativity of inertia, but also atomic or electronic phenomena, drive us towards the same solution.

Let us now pass to a study of the electromagnetic tensor itself. We can see that even if the preceding attempts had proved successful, there would still remain a duality between gravitation and matter or electricity, or, we might say, between the metrical field of space-time and the tensor . This latter tensor would still appear as an irreducible foreign entity present in space-time, marring the unity of nature. It was the aim of Weyl’s theory to rid science of this duality.

In order to understand the nature of the problem, we must state that just as the fundamental structural tensors of the metrical field , and at every point could be reduced to highly complicated expressions between the ten fundamental -potentials at every point, so now the electromagnetic tensor can be reduced to a complicated expression between the four electromagnetic potentials, the four ’s (namely , and ). The unification demanded by science was therefore to express the ’s in terms of the ’s, or vice versa. Unfortunately, not the slightest similarity appeared to exist between the ten potentials and the four ’s. The ’s appeared as foreign entities embedded in the metrical field of space-time.

All attempts to overcome this duality were unsuccessful so long as we confined our speculations to the space-time of Einstein’s theory. But there always existed the possibility that space-time was of a less simple category than had been assumed by Einstein. If, then, we could conceive of a generalised type of space-time possessing a more complicated metrical field, it might be possible to identify the fundamental ’s of electromagnetics with certain particularities of structure of this generalised space-time and of its metrical field. Weyl’s theory is an attempt in this direction; and though it is still in a highly speculative stage, everything seems to suggest that Weyl is on the right track. Einstein himself has accepted a modified form of Weyl’s theory to the extent of contributing to its advancement.

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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 [Pg 361] 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 [Pg 362] 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 [Pg 363] 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 [Pg 364] 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 [Pg 365] 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 [Pg 366] 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 [Pg 367] 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. [Pg 368] 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 [Pg 369] 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 [Pg 370] 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.

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PART IV THE METHODOLOGY OF SCIENCE

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