Space Time and Gravitation/Chapter 10

by Arthur Eddington

 

CHAPTER X

TOWARDS INFINITY

The geometer of to-day knows nothing about the nature of actually existing space at an infinite distance; he knows nothing about the properties of this present space in a past or a future eternity. He knows, indeed, that the laws assumed by Euclid are true with an accuracy that no direct experiment can approach, not only in this place where we are, but in places at a distance from us that no astronomer has conceived; but he knows this as of Here and Now; beyond his range is a There and Then of which he knows nothing at present, but may ultimately come to know more.

W. K. Clifford (1873).

The great stumbling-block for a philosophy which denies absolute space is the experimental detection of absolute rotation. The belief that the earth rotates on its axis was suggested by the diurnal motions of the heavenly bodies; this observation is essentially one of relative rotation, and, if the matter rested there, no difficulty would be felt. But we can detect the same rotation, or a rotation very closely equal to it, by methods which do not seem to bring the heavenly bodies into consideration; and such a rotation is apparently absolute. The planet Jupiter is covered with cloud, so that an inhabitant would probably be unaware of the existence of bodies outside; yet he could quite well measure the rotation of Jupiter. By the gyro-compass he would fix two points on the planet the north and south poles. Then by Foucault's experiment on the change of the plane of motion of a freely suspended pendulum, he would determine an angular velocity about the poles. Thus there is certainly a definite physical constant, an angular velocity about an axis, which has a fundamental importance for the inhabitants of Jupiter; the only question is whether we are right in giving it the name absolute rotation.

Contrast this with absolute translation. Here it is not a question of giving the right name to a physical constant; the inhabitants of Jupiter would find no constant to name. We see at once that a relativity theory of translation is on a different footing from a relativity theory of rotation. The duty of the former is to explain facts; the duty of the latter is to explain away facts.

Our present theory seems to make a start at tackling this problem, but gives it up. It permits the observer, if he wishes, to consider the earth as non-rotating, but surrounded by a field of centrifugal force; all the other bodies in the universe are then revolving round the earth in orbits mainly controlled by this field of centrifugal force. Astronomy on this basis is a little cumbersome; but all the phenomena are explained perfectly. The centrifugal force is part of the gravitational field, and obeys Einstein's law of gravitation, so that the laws of nature are completely satisfied by this representation. One awkward question remains, What causes the centrifugal force? Certainly not the earth which is here represented as non-rotating. As we go further into space to look for a cause, the centrifugal force becomes greater and greater, so that the more we defer the debt the heavier the payment demanded in the end. Our present theory is like the debtor who does not mind how big an obligation accumulates satisfied that he can always put off the payment. It chases the cause away to infinity, content that the laws of nature—the relations between contiguous parts of the world—are satisfied all the way.

One suggested loophole must be explored. Our new law of gravitation admits that a rapid motion of the attracting body will affect the field of force. If the earth is non-rotating, the stars must be going round it with terrific speed. May they not in virtue of their high velocities produce gravitationally a sensible field of force on the earth, which we recognise as the centrifugal field? This would be a genuine elimination of absolute rotation, attributing all effects indifferently to the rotation of the earth the stars being at rest, or to the revolution of the stars the earth being at rest; nothing matters except the relative rotation. I doubt whether anyone will persuade himself that the stars have anything to do with the phenomenon. We do not believe that if the heavenly bodies were all annihilated it would upset the gyrocompass. In any case, precise calculation shows that the centrifugal force could not be produced by the motion of the stars, so far as they are known.

We are therefore forced to give up the idea that the signs of the earth's—rotation the protuberance of its equator, the phenomena of the gyrocompass, etc. are due to a rotation relative to any matter we can recognise. The philosopher who persists that a rotation which is not relative to matter is unthinkable, will no doubt reply that the rotation must then be relative to some matter which we have not yet recognised. We have hitherto been greatly indebted to the suggestions of philosophy in evolving this theory, because the suggestions related to the things we know about; and, as it turned out, they were confirmed by experiment. But as physicists we cannot take the same interest in the new demand; we do not necessarily challenge it, but it is outside our concern. Physics demands of its scheme of nature something else besides truth, namely a certain quality that we may call convergence. The law of conservation of energy is only strictly true when the whole universe is taken into account; but its value in physics lies in the fact that it is approximately true for a very limited system. Physics is an exact science because the chief essentials of a problem are limited to a few conditions; and it draws near to the truth with ever-increasing approximation as it widens its purview. The approximations of physics form a convergent series. History, on the other hand, is very often like a divergent series; no approximation to its course is reached until the last term of the infinite series has been included in the data of prediction. Physics, if it wishes to retain its advantage, must take its own course, formulating those laws which are approximately true for the limited data of sense, and extending them into the unknown. The relativity of rotation is not approximately true for the data of sense, although it may possibly be true when the unknown as well as the known are included.

The same considerations that apply to rotation apply to acceleration, although the difficulty is less striking. We can if we like attribute to the sun some arbitrary acceleration, balancing it by introducing a uniform gravitational field. Owing to this field the rest of the stars will move with the same acceleration and no phenomena will be altered. But then it seems necessary to find a cause for this field. It is not produced by the gravitation of the stars. Our only course is to pursue the cause further and further towards infinity; the further we put it away, the greater the mass of attracting matter needed to produce it. On the other hand, the earth's absolute acceleration does not intrude on our attention in the way that its absolute rotation does^[1].

We are vaguely conscious of a difficulty in these results; but if we examine it closely, the difficulty does not seem to be a very serious one. The theory of relativity, as we have understood it, asserts that our partitions of space and time are introduced by the observer and are irrelevant to the laws of nature; and therefore the current quantities of physics, length, duration, mass, force, etc., which are relative to these partitions, are not things having an absolute significance in nature. But we have never denied that there are features of the world having an absolute significance; in fact, we have spent much time in finding such features. The geodesics or natural tracks have been shown to have an absolute significance; and it is possible in a limited region of the world to choose space and time partitions such that all geodesics become approximately straight lines. We may call this a "natural" frame for that region, although it is not as a rule the space and time adopted in practice; it is for example the space and time of the observers in the falling projectile, not of Newton's super-observer. It is capable of absolute definition, except that it is ambiguous in regard to uniform motion. Now the rotation of the earth determined by Foucault's pendulum experiment is the rotation referred to this natural frame. But we must have misunderstood our own theory of relativity altogether, if we think there is anything inadmissible in an absolute rotation of such a kind.

Material particles and geodesics are both features of the absolute structure of the world; and a rotation relative to geodesic structure does not seem to be on any different footing from a velocity relative to matter. There is, however, the striking feature that rotation seems to be relative not merely to the local geodesic structure but to a generally accepted universal frame; whereas it is necessary to specify precisely what matter a velocity is measured with respect to. This is largely a question of how much accuracy is needed in specifying velocities and rotations, respectively. If in stating the speed of a β particle we do not mind an error of 10,000 kilometres a second, we need not specify precisely what star or planet its velocity is referred to. The moon's (local) angular velocity is sometimes given to fourteen significant figures; I doubt if any universal frame is well-defined enough for this accuracy. There is no doubt much greater continuity in the geodesic structure in different parts of the world than in the material structure; but the difference is in degree rather than in principle.

It is probable that here we part company from many of the continental relativists, who give prominent place to a principle known as the law of causality—that only those things are to be regarded as being in causal connection which are capable of being actually observed. This seems to be interpreted as placing matter on a plane above geodesic structure in regard to the formulation of physical laws, though it is not easy to see in what sense a distribution of matter can be regarded as more observable than the field of influence in surrounding space which makes us aware of its existence. The principle itself is debateable; that which is observable to us is determined by the accident of our own structure, and the law of causality seems to impose our own limitations on the free interplay of entities in the world outside us. In this book the tradition of Faraday and Maxwell still rules our outlook; and for us matter and electricity are but incidental points of complexity, the activity of nature being primarily in the so-called empty spaces between.

The vague universal frame to which rotation is referred is called the inertial frame. It is definite in the flat space-time far away from all matter. In the undulating country corresponding to the stellar universe it is not a precise conception; it is rather a rude outline, arbitrary within reasonable limits, but with the general course indicated. The reason for the term inertial frame is of interest. We can quite freely use a mesh-system deviating widely from the inertial frame (e.g. rotating axes); but we have seen that there is a postponed debt to pay in the shape of an apparently uncaused field of force. But is there no debt to pay, even when the inertial frame is used? In that case there is no gravitational or centrifugal force at infinity; but there is still inertia, which is of the same nature. The distinction between force as requiring a cause and inertia as requiring no cause cannot be sustained. We shall not become any more solvent by commuting our debt into pure inertia. The debt is inevitable whatever mesh-system is used; we are only allowed to choose the form it shall take.

The debt after all is a very harmless one. At infinity we have the absolute geodesics in space-time, and we have our own arbitrarily drawn mesh-system. The relation of the geodesics to the mesh-system decides whether our axes shall be termed rotating or non-rotating; and ideally it is this relation that is determined when a so-called absolute rotation is measured. No one could reasonably expect that there would be no determinable relation. On the other hand uniform translation does not affect the relation of the geodesics to the mesh-system—if they were straight lines originally, they remain straight lines—thus uniform translation cannot be measured except relative to matter.

We have been supposing that the conditions found in the remotest parts of space accessible to observation can be extrapolated to infinity; and that there are still definite natural tracks in space-time far beyond the influence of matter. Feelings of objection to this view arise in certain minds. It is urged that as matter influences the course of geodesics it may well be responsible for them altogether; so that a region outside the field of action of matter could have no geodesics, and consequently no intervals. All the potentials would then necessarily be zero. Various modified forms of this objection arise; but the main feeling seems to be that it is unsatisfactory to have certain conditions prevailing in the world, which can be traced away to infinity and so have, as it were, their source at infinity; and there is a desire to find some explanation of the inertial frame as built up through conditions at a finite distance.

Now if all intervals vanished space-time would shrink to a point. Then there would be no space, no time, no inertia, no anything. Thus a cause which creates intervals and geodesics must, so to speak, extend the world. We can imagine the world stretched out like a plane sheet; but then the stretching cause—the cause of the intervals—is relegated beyond the bounds of space and time, i.e. to infinity. This is the view objected to, though the writer does not consider that the objection has much force. An alternative way is to inflate the world from inside, as a balloon is blown out. In this case the stretching force is not relegated to infinity, and ruled outside the scope of experiment; it is acting at every point of space and time, curving the world to a sphere. We thus get the idea that space-time may have an essential curvature on a great scale independent of the small hummocks due to recognised matter.

It is not necessary to speculate whether the curvature is produced (as in the balloon) by some pressure applied from a fifth dimension. For us it will appear as an innate tendency of four-dimensional space-time to curve. It may be asked, what have we gained by substituting a natural curvature of space-time for a natural stretched condition corresponding to the inertial frame? As an explanation, nothing. But there is this difference, that the theory of the inertial frame can now be included in the differential law of gravitation instead of remaining outside and additional to the law.

It will be remembered that one clue by which we previously reached the law of gravitation was that flat space-time must be compatible with it. But if space-time is to have a small natural curvature independent of matter this condition is now altered. It is not difficult to find the necessary alteration of the law^[2]. It will contain an additional, and at present unknown, constant, which determines the size of the world.

Spherical space is not very easy to imagine. We have to think of the properties of the surface of a sphere—the two-dimensional case—and try to conceive something similar applied to three-dimensional space. Stationing ourselves at a point let us draw a series of spheres of successively greater radii. The surface of a sphere of radius $r$ should be proportional to $r^{2}$; but in spherical space the areas of the more distant spheres begin to fall below the proper proportion. There is not so much room out there as we expected to find. Ultimately we reach a sphere of biggest possible area, and beyond it the areas begin to decrease^[3]. The last sphere of all shrinks to a point—our antipodes. Is there nothing beyond this? Is there a kind of boundary there? There is nothing beyond and yet there is no boundary. On the earth's surface there is nothing beyond our own antipodes but there is no boundary there.

The difficulty is that we try to realise this spherical world by imagining how it would appear to us and to our measurements. There has been nothing in our experience to compare it with, and it seems fantastic. But if we could get rid of the personal point of view, and regard the sphericity of the world as a statement of the type of order of events outside us, we should think that it was a simple and natural order which is as likely as any other to occur in the world.

In such a world there is no difficulty about accumulated debt at the boundary. There is no boundary. The centrifugal force increases until we reach the sphere of greatest area, and then, still obeying the law of gravitation, diminishes to zero at the antipodes. The debt has paid itself automatically.

We must not exaggerate what has been accomplished by this modification of the theory. A new constant has been introduced into the law of gravitation which gives the world a definite extension. Previously there was nothing to fix the scale of the world; it was simply given a priori that it was infinite. Granted extension, so that the intervals are not invariably zero, we can determine geodesics everywhere, and hence mark out the inertial frame.

Spherical space-time, that is to say a four-dimensional continuum of space and imaginary time forming the surface of a sphere in five dimensions, has been investigated by Prof. de Sitter. If real time is used the world is spherical in its space dimensions, but open towards plus and minus infinity in its time dimension, like an hyperboloid. This happily relieves us of the necessity of supposing that as we progress in time we shall ultimately come back to the instant we started from! History never repeats itself. But in the space dimensions we should, if we went on, ultimately come back to the starting point. This would have interesting physical results, and we shall see presently that Einstein has a theory of the world in which the return can actually happen; but in de Sitter's theory it is rather an abstraction, because, as he says, "all the paradoxical phenomena can only happen after the end or before the beginning of eternity."

The reason is this. Owing to curvature in the time dimension, as we examine the condition of things further and further from our starting point, our time begins to run faster and faster, or to put it another way natural phenomena and natural clocks slow down. The condition becomes like that described in Mr H. G. Wells's story "The new accelerator."

When we reach half-way to the antipodal point, time stands still. Like the Mad Hatter's tea party, it is always 6 o'clock; and nothing whatever can happen however long we wait. There is no possibility of getting any further, because everything including light has come to rest here. All that lies beyond is forever cut off from us by this barrier of time; and light can never complete its voyage round the world.

That is what happens when the world is viewed from one station; but if attracted by such a delightful prospect, we proceeded to visit this scene of repose, we should be disappointed. We should find nature there as active as ever. We thought time was standing still, but it was really proceeding there at the usual rate, as if in a fifth dimension of which we had no cognisance. Casting an eye back on our old home we should see that time apparently had stopped still there. Time in the two places is proceeding in directions at right angles, so that the progress of time at one point has no relation to the perception of time at the other point. The reader will easily see that a being confined to the surface of a sphere and not cognisant of a third dimension, will, so to speak, lose one of his dimensions altogether when he watches things occurring at a point 90° away. He regains it if he visits the spot and so adapts himself to the two dimensions which prevail there.

It might seem that this kind of fantastic world-building can have little to do with practical problems. But that is not quite certain. May we not be able actually to observe the slowing down of natural phenomena at great distances from us? The most remote objects known are the spiral nebulae, whose distances may perhaps be of the order a million light years. If natural phenomena are slowed down there, the vibrations of an atom are slower, and its characteristic spectral lines will appear displaced to the red. We should generally interpret this as a Doppler effect, implying that the nebula is receding. The motions in the line-of-sight of a number of nebulae have been determined, chiefly by Prof. Slipher. The data are not so ample as we should like; but there is no doubt that large receding motions greatly preponderate. This may be a genuine phenomenon in the evolution of the material universe; but it is also possible that the interpretation of spectral displacement as a receding velocity is erroneous; and the effect is really the slowing of atomic vibrations predicted by de Sitter's theory.

Prof. Einstein himself prefers a different theory of curved space-time. His world is cylindrical—curved in the three space dimensions and straight in the time dimension. Since time is no longer curved, the slowing of phenomena at great distances from the observer disappears, and with it the slight experimental support given to the theory by the observations of spiral nebulae. There is no longer a barrier of eternal rest, and a ray of light is able to go round the world.

In various ways crude estimates of the size of the world both on de Sitter's and Einstein's hypotheses have been made; and in both cases the radius is thought to be of the order 10¹³ times the distance of the earth from the sun. A ray of light from the sun would thus take about 1000 million years to go round the world; and after the journey the rays would converge again at the starting point, and then diverge for the next circuit. The convergent would have all the characteristics of a real sun so far as light and heat are concerned, only there would be no substantial body present. Thus corresponding to the sun we might see a series of ghosts occupying the positions where the sun was 1000, 2000, 3000, etc., million years ago, if (as seems probable) the sun has been luminous for so long.

It is rather a pleasing speculation that records of the previous states of the sidereal universe may be automatically reforming themselves on the original sites. Perhaps one or more of the many spiral nebulae are really phantoms of our own stellar system. Or it may be that only a proportion of the stars are substantial bodies; the remainder are optical ghosts revisiting their old haunts. It is, however, unlikely that the light rays after their long journey would converge with the accuracy which this theory would require. The minute deflections by the various gravitational fields encountered on the way would turn them aside, and the focus would be blurred. Moreover there is a likelihood that the light would gradually be absorbed or scattered by matter diffused in space, which is encountered on the long journey.

It is sometimes suggested that the return of the light-wave to its starting point can most easily be regarded as due to the force of gravitation, there being sufficient mass distributed through the universe to control its path in a closed orbit. We should have no objection in principle to this way of looking at it; but we doubt whether it is correct in fact. It is quite possible for light to return to its starting point in a world without gravitation. We can roll flat space-time into a cylinder and join the edges; its geometry will still be Euclidean and there will be no gravitation; but a ray of light can go right round the cylinder and return to the starting point in space. Similarly in Einstein's more complex type of cylinder (three dimensions curved and one dimension linear), it seems likely that the return of the light is due as much to the connectivity of his space, as to the non-Euclidean properties which express the gravitational field.

For Einstein's cylindrical world it is necessary to postulate the existence of vast quantities of matter (not needed on de Sitter's theory) far in excess of what has been revealed by our telescopes. This additional material may either be in the form of distant stars and galaxies beyond our limits of vision, or it may be uniformly spread through space and escape notice by its low density. There is a definite relation between the average density of matter and the radius of the world; the greater the radius the smaller must be the average density.

Two objections to this theory may be urged. In the first place, absolute space and time are restored for phenomena on a cosmical scale. The ghost of a star appears at the spot where the star was a certain number of million years ago; and from the ghost to the present position of the star is a definite distance —the absolute motion of the star in the meantime^[4]. The world taken as a whole has one direction in which it is not curved; that direction gives a kind of absolute time distinct from space. Relativity is reduced to a local phenomenon; and although this is quite sufficient for the theory hitherto described, we are inclined to look on the limitation rather grudgingly. But we have already urged that the relativity theory is not concerned to deny the possibility of an absolute time, but to deny that it is concerned in any experimental knowledge yet found; and it need not perturb us if the conception of absolute time turns up in a new form in a theory of phenomena on a cosmical scale, as to which no experimental knowledge is yet available. Just as each limited observer has his own particular separation of space and time, so a being coextensive with the world might well have a special separation of space and time natural to him. It is the time for this being that is here dignified by the title "absolute."

Secondly, the revised law of gravitation involves a new constant which depends on the total amount of matter in the world; or conversely the total amount of matter in the world is determined by the law of gravitation. This seems very hard to accept—at any rate without some plausible explanation of how the adjustment is brought about. We can see that, the constant in the law of gravitation being fixed, there may be some upper limit to the amount of matter possible; as more and more matter is added in the distant parts, space curves round and ultimately closes; the process of adding more matter must stop, because there is no more space, and we can only return to the region already dealt with. But there seems nothing to prevent a defect of matter, leaving space unclosed. Some mechanism seems to be needed, whereby either gravitation creates matter, or all the matter in the universe conspires to define a law of gravitation.

Although this appears to the writer rather bewildering, it is welcomed by those philosophers who follow the lead of Mach. For it leads to the result that the extension of space and time depends on the amount of matter in the world—partly by its direct effect on the curvature and partly by its influence on the constant of the law of gravitation. The more matter there is, the more space is created to contain it, and if there were no matter the world would shrink to a point.

In the philosophy of Mach a world without matter is unthinkable. Matter in Mach's philosophy is not merely required as a test body to display properties of something already there, which have no physical meaning except in relation to matter; it is an essential factor in causing those properties which it is able to display. Inertia, for example, would not appear by the insertion of one test body in the world; in some way the presence of other matter is a necessary condition. It will be seen how welcome to such a philosophy is the theory that space and the inertial frame come into being with matter, and grow as it grows. Since the laws of inertia are part of the law of gravitation, Mach's philosophy was summed up—perhaps unconsciously—in the profound saying "If there were no matter in the universe, the law of gravitation would fall to the ground."

No doubt a world without matter, in which nothing could ever happen, would be very uninteresting; and some might deny its claim to be regarded as a world at all. But a world uniformly filled with matter would be equally dull and unprofitable; so there seems to be little object in denying the possibility of the former and leaving the latter possible.

The position can be summed up as follows:—in a space without absolute features, an absolute rotation would be as meaningless as an absolute translation; accordingly, the existence of an experimentally determined quantity generally identified with absolute rotation requires explanation. It was remarked on p. 41 that it would be difficult to devise a plan of the world according to which uniform motion has no significance but non-uniform motion is significant; but such a world has been arrived at—a plenum, of which the absolute features are intervals and geodesics. In a limited region this plenum gives a natural frame with respect to which an acceleration or rotation (but not a velocity) capable of absolute definition can be measured. In the case of rotation the local distortions of the frame are of comparatively little account; and this explains why in practice rotation appears to have reference to some worldwide inertial frame.

Thus absolute rotation does not indicate any logical flaw in the theory hitherto developed; and there is no need to accept any modification of our views. Possibly there may be a still wider relativity theory, in which our supposed plenum is to be regarded as itself an abstraction of the relations of the matter distributed throughout the world, and not existent apart from such matter. This seems to exalt matter rather unnecessarily. It may be true; but we feel no necessity for it, unless experiment points that way. It is with some such underlying idea that Einstein's cylindrical space-time was suggested, since this cannot exist without matter to keep it stretched. Now we freely admit that our assumption of perfect flatness in the remote parts of space was arbitrary, and there is no justification for insisting on it. A small curvature is possible both conceptually and experimentally. The arguments on both sides have hitherto been little more than prejudices, which would be dissipated by any experimental or theoretical lead in one direction. Weyl's theory of the electromagnetic field, discussed in the next chapter, assigns a definite function to the curvature of space; and this considerably alters the aspect of the question. We are scarcely sufficiently advanced to offer a final opinion; but the conception of cylindrical space-time seems to be favoured by this new development of the theory.

Some may be inclined to challenge the right of the Einstein theory, at least as interpreted in this book, to be called a relativity theory. Perhaps it has not all the characteristics which have at one time or another been associated with that name; but the reader, who has followed us so far, will see how our search for an absolute world has been guided by a recognition of the relativity of the measurements of physics. It may be urged that our geodesics ought not to be regarded as fundamental; a geodesic has no meaning in itself; what we are really concerned with is the relation of a particle following a geodesic to all the other matter of the world and the geodesic cannot be thought of apart from such other matter. We would reply, "Your particle of matter is not fundamental; it has no meaning in itself; what you are really concerned with is its 'field'—the relation of the geodesics about it to the other geodesics in the world—and matter cannot be thought of apart from its field." It is all a tangle of relations; physical theory starts with the simplest constituents, philosophical theory with the most familiar constituents. They may reach the same goal; but their methods are often incompatible.

1. To determine even roughly the earth's absolute acceleration we should need a fairly full knowledge of the disturbing effects of all the matter in the universe. A similar knowledge would be required to determine the absolute rotation accurately; but all the matter likely to exist would have so small an effect, that we can at once assume that the absolute rotation is very nearly the same as the experimentally determined rotation.
2. Appendix, Note 14.
3. The area is, of course, to be determined by measurement of some kind.
4. The ghost is not formed where the star is now. If two stars were near together when the light left them their ghosts must be near together, although the stars may now be widely separated.