PAPERS

V - SPACE *

by     Reginald O. Kapp

Offprint from The British Journal for the Philosophy of Science
Vol. X No 37 1959
Thomas Nelson & Sons Ltd Edinburgh 9


A PAST chairman of this Group, Professor Woodger, has a capacity for saying profound things lightly. I should like to quote one of them. It occurs in his book Physics, Psychology and Medicine. He is discussing some prevalent errors concerning the nature of reality and says

'Another tenet of this philosophy seems to be that everything that is, is in a big box called Space, which is floating down a river called Time. Consequently if anything (except the river!) is not in space, it is just not at all.'

This is the philosophy that Professor Woodger rightly invites us to reject. He might have issued the invitation with a parade of learning and abstruse logic; but the uncompromising simplicity of the form in which Woodger presents this erroneous view of reality embodies a criticism of it by implication that is both more cogent and more precise than a learned dissertation could be.

We are all prone to adopt the big box view of reality and we have to understand why it is wrong. It contains two errors.

The first is the assumption that only those things with location have reality; that anything not in space is 'just not at all', or, more shortly still, what is nowhere is not. 'Nowhere' and 'non-existent' are regarded as synonyms, and according to this school, space itself is not necessarily real but everything is real that is in space and nothing, except perhaps time, is real that is not in space.

There are, of course, those who would like to adhere to this school and yet doubt whether it is correct to identify 'nowhere' with 'non-existent'. With the leaning towards compromise of the woolly-minded they then prefer to identify it with 'not quite'. In other words they introduce the view that there are degrees of reality in the same way as there are degrees of temperature or degrees of hardness. What is nowhere is not quite real, they would say. Some things are to them more real than others. According to this remarkable interpretation of the concept 'real' they say that particles are the most real of all. These they think of as consisting of a substance, which they might call particle stuff. Particles are regarded as interacting with other things that are several degrees less real, such as waves. The very real particle stuff and the not so real waves are then regarded as having their existence in time, which is hardly real at all. They are also said to have their existence in the big box space, about the reality of which they do not care to think. Mass and electrical charge, being properties of the particle stuff, are placed high on the reality scale. Extension, being a property of space is placed low.

If the course of events is ever subjected to control it is taken for granted, by adherents of this school, that the controlling influence consists of particle stuff. Having heard of servo-mechanisms and feed-back they say that the controlling influence must necessarily be a part of a closed loop and, moreover, a part consisting of material substance. They find a place for it in the big box. As nothing is believed to exist unless it has location in the big box it is regarded as unscientific, bad philosophy, or logically absurd to postulate a controlling influence without location.

Be it admitted that all things possessing physical reality do occur in the big box. The error is to assume, indeed to take it for granted, that physical reality is the only kind of reality. I have myself attacked this somewhat fashionable but, nevertheless, naive view on many occasions and propose to leave it at that for the moment. I want to pass on to the second error implicit in the big box philosophy.

The second error concerns the nature of space. The view that space is a container was held almost universally until early in this century. It is true that some denied it, among them Leibniz, and some doubted it, among them perhaps Newton. But since the success of Newtonianism the doubts were never clamorous. When it was said that all physical things are in space I do not think that it would have occurred to many to question the appositeness of the preposition 'in'.

Yet doubts ought to have arisen. If space were the container of physical reality it would be relevant to ask how full is was. How much reality must be put into a given region of space so that it may be 100 per cent full? How much would make it 50 per cent full? Do the actual contents of space fill it up to just as much as it can hold? If something more came into space than is there already would reality overflow and spill into the river Time? If the big box were filled with radiation only, how much would there be to fill it? How much particle stuff, alternatively, is needed to fill the box?

I hope such questions seem as absurd to you as they do to me. What I want to make clear is that we ought not to have had to wait for relativity theory before we abandoned that queer notion of space implicit in the big box philosophy.

Be that as it may, it was Einstein who showed more cogently than anyone had done before why it is wrong to regard space as the container of reality. (But it is only fair to add that Newton seems to have had his doubts about it, and Leibniz certainly did.) When attributing physical properties to space Einstein obliged us to replace the word 'container' by the word 'constituent'. Space, we learnt from him, is a constituent of the material universe and not its container.

We learnt if half a century ago. But we are in danger of forgetting it. Today general relativity occupies only a very small part of the undergraduate teaching in physics and astronomy, if it is taught at all at that level. Post-graduate studies are necessarily devoted to other, usually more recent, subjects. Even special relativity does not receive the same attention from scientists today that it did in the 1920s. It is found that a physicist or engineer can do all that he needs when working with particle accelerators if he has familiarised himself with the relativity equations and the rather specialised notation that they employ. Pre-occupation with meaning, such as was common in Eddington's day, is no longer necessary. It suffices for practical purposes to know how to use the letter symbols.

The academic syllabus does not have room for everything. Very few physicists and astronomers are engaged on work that calls for any knowledge of general relativity. Those few who have made a deep study of the subject have been stimulated by intellectual curiosity rather than by practical needs. Nevertheless, I have reached the conclusion that those few may be able to contribute much useful knowledge. Let them be encouraged. Perhaps it is just because I have little specialised knowledge myself that I view the progress of science from a little distance. What I see leads me to believe that some aspects of the physical world can be more profitably explored by one who takes as his starting point the position that was occupied in the 1920s than by one who sets out from the position more usually occupied today. In Eddington's time the focus of interest was the interaction between space and ponderable matter. That subject is rarely discussed today. Now the focus of interest is more often the interaction between particle and particle. So I want this paper to serve as a reminder of the great work done by Einstein, Eddington, and others. I propose to suggest lines for future study that are stimulated when one places oneself in thought in the atmosphere of the 1920s.

Let me draw attention to another of the defects in the pre-Einstein view. This view obliged us to think of every bit of ponderable matter as having two environments. One of these was called 'space', the other 'luminiferous ether'. The two environments were said to be quite distinct. Most of us regarded space as featureless, as such that any one part was in every respect exactly the same as every other part. The only property that could be attributed to space viewed in that way was extant, but even that property had little meaning, if any, so long as the extent was considered to be infinite.

In contrast to the featureless space we thought of the luminiferous ether as featured. Its various parts were said to be subjected to a variety of strains. These strains were considered to be electrical and magnetic. For some reason gravitational strains were not postulated. Gravitation was regarded as a phenomenon about which the less said the better.

To explain the presence of the various strains a variety of physical properties was attributed to the ether, such as elasticity, density, inertia.

One of Einstein's great contributions to science was to show that there is no need to postulate two environments. One will do. I think this was as great a contribution to philosophy as to science.

It is now recognised that a space of which no part is distinguishable in any way whatever from any other part is not observable by any physical means. Its nature is, therefore, purely conceptual. It is difficult, if not impossible, to argue that a featureless space has any physical meaning. Things would happen just as they do with or without a space of that kind. Of the two environments assumed for every body, reality could be assigned to one only.

This reasoning could have left scientists with the luminiferous ether, thought of as differing from one place to another, as being featured; and the features being considered such as to influence measuring devices, and as to be observable.

The measurable features, in the environment of ponderable matter, all have something in common which can be expressed by a single word, namely, field of force. Such fields are measured and described in terms of their intensity, which is more precisely called their potential gradient. There are at least three kinds of field, electric, magnetic, and gravitational; so the environment of a given quantity of ponderable matter may contain three different kinds of potential gradient. Each of these may vary both in magnitude and direction. At least the electric and magnetic potential gradients may also vary with time at any given point. When they do so they are called waves. The physical effect on ponderable matter of these potential gradients is always of the same kind, namely, that of accelerating the ponderable matter.

When this insight was gained a name was needed for the collection of potential gradients that surrounds any given quantify of ponderable matter. 'Luminiferous ether' might have been retained for this were it not that its verbal currency had been depreciated by sundry misapprehensions. Hypothesis had, for instance, endowed the ether with the properties I have already referred to; elasticity, density, inertia. But no such properties could be attributed to places where there was no ponderable matter. The only observed, or even inferred, physical features of the environment were various potential gradients, some static and some alternating, some bound to a particle and some moving freely with the velocity of light; and so the many properties that had been attributed to the ether proved both inappropriate and too numerous to make this term suitable. If the word 'space' suggested something too abstract, the words 'luminiferous ether' suggested something too concrete.

Perhaps the non-committal word 'environment' might have been a good choice. It could have been justified scientifically; for it is strictly accurate to say that a quantity of ponderable matter is surrounded by an environment that has physical properties and one has to be very cautious about saying anything more than this. But to use 'environment' in this way would have been to raise the word to the status of a technical term and to use it, moreover, in an unfamiliar context. 'Collection of potential gradients' could also have served for these are, literally, all that one needs to postulate as occurring between one bit of ponderable matter and another. Whether the container of the potential gradients be called space or luminiferous ether is immaterial. The point is that the concept of a container is unnecessary. The potential gradients are observable by their effects. But, everything would happen as it does whether these gradients have a container or not.

But neither 'environment' nor 'collection of potential gradients' would ever have succeeded in ousting the established word 'space'. So I think that Einstein showed a sound intuition when he retained this word while changing its meaning from that of a featureless container to that of a synonym for featured environment. But when the word 'space' is used in relativity theory with its new and precise meaning, it does not at the same time represent the vague concept that it does in everyday speech. It has become a technical term.

Another piece of insight gained from relativity theory was that the properties previously attributed to the luminiferous ether could not be attributed to the environment of ponderable matter. It appeared at least possible that they could all be replaced by one single property described as 'curvature'. It was from the identity of inert and gravitational mass that Einstein arrived at the conclusion that a field of force, at least when the force is gravitational, is a region for which the geometry of space is non-Euclidean, and he used the expression 'curvature' as a means of describing the departure from Euclidean geometry for the space in which there is a potential gradient. This, too, became a technical term with a unique and precise meaning in relativity theory.

The conclusion startled the scientific world. Until then we had regarded curvature as a purely geometrical property. That it could be regarded as a physical one caused scientists to revise their notions about the nature of matter. How, it came to be asked, could something as apparently abstract as a curvature of space interact with something apparently as concrete as a material particle? The question is as puzzling today as ever it was.

The best answer will eventually come, I am inclined to think, from a revision of our notions about what is abstract and what is concrete. In making a distinction between these we tend to be enslaved by the organs of sense perception with which we, as human beings, happen to have been endowed. We call those things concrete that we can perceive with the help of these organs and those things abstract that cannot be so perceived. But this is a surmise and I do not propose to pursue this line of thought any further here.

Relativists have succeeded in showing good reason why gravitational potential gradients should be identified with a condition of space appropriately called its curvature. But they have not yet succeeded in showing the same for electrostatic and magnetic potential gradients, though some of them still hope to do so. The effort to do this is called the search foi a unified field theory. Anything that I, or anyone else, can say today about the relation between space and matter may have to be modified if and when the search has succeeded. For this reason, if for no other, whatever can be said at present must inevitably not only leave some insistent questions unanswered, but also be very tentative.

Should it ever be shown that electric and magnetic potential gradients are regions of curved space we must expect the kind of curvature to differ basically from the kind that is identified with a gravitational potential gradient, for there is no known interaction between gravitational fields and the other two kinds. A change of the charge on a body having inert mass does not change the behaviour of the body in a gravitational field. A change in the potential gradient of a gravitational field does not affect the forces between electric charges placed in it. Gravitational masses that do not carry electric charges and are not magnetically polarised fall with the same acceleration through an electric field (provided there is a gravitational one to fall in) irrespective of its intensity. In other words, electric charges behave in the same way in gravitationally curved as in flat space.

From such observations it has to be concluded that, if electromagnetic potential gradients are, like gravitational ones, curved regions of space the geometry by which the curvature can be defined can hardly be of the same Riemannian kind that represents gravitational potential gradients. This independence from each other of the different kinds of field takes some accounting for and even leaves room for doubt whether the electro-magnetic field is, as seems so reasonable to suppose, a region where the geometry is non-Euclidean. But I have nothing to contribute to the search for an explanation and do not propose here to discuss the relation between space and charge or that between space and magnetism, but only the relation between space and mass. I have no choice but to discuss this relation as though the only condition of space by which one region is distinguished from another is curvature and the only known, kind of curvature is that identified with the gravitational field. The incompleteness of such treatment is made obvious by the known facts about electricity and magnetism. But all that one can do in the present state of ignorance is to see along what path the incomplete treatment takes one and to hope that better knowledge of the relation between space and charge will not necessitate too great a change of direction from that in which the chosen path leads.

I want here to define the limits to our present knowledge of the relation between space and mass. Relativity theory has contributed a great deal to that knowledge. But there is still much about which we are ignorant and questions to which the search for answers is likely to prove rewarding come under two headings: the action of space on mass and the action of mass on space.

Let me now turn to the action of space on mass and remind you of some well-known facts about it.

Relativity theory tells us something about the motion of a particle to which no force is applied, it being well understood that all statements about the motion are relative to a given co-ordinate system. The motion depends, according to the theory, only on the curvature of the space in which the particle finds itself. If the space is flat, the particle moves with a constant velocity, which may of course be zero velocity. If the space is curved, the particle moves with a non-uniform velocity; it experiences an acceleration or a deceleration.

If a force is applied to the particle its movement is no longer wholly determined by the geometry of space. A billiard cue can accelerate a billiard ball in flat space and a shelf can, by exercising a force on the stone resting on it, prevent the stone from following the curvature of the space in which it finds itself.

I mention these well-known facts only to bring out what is and what is not known about the action of space on mass. Relativity theory defines the property of space by virtue of which it is able to act on mass; this property is technically called curvature. But relativity theory does not tell us anything about the property of mass by virtue of which an unrestrained particle follows the curvature. The property is given the name inertia, but to name a thing is not to explain it or to give any sort of information about it. Relativity has some important things to say about space but fewer about mass. Let me illustrate our present ignorance about the relation between these two concepts with the help of an analogy.

When one sees a tramcar turn a corner, one may give the perfectly true explanation that there are rails for it to travel on. But it is an insufficient explanation. It says something about the street, but nothing about the tram. To make the explanation complete one has to add that the tramcar is provided with flanged wheels, so designed that they fit the rails.

Similarly, when one sees a stone fall one may give the explanation that the space in which the stone happens to be is curved and thereby causes a curved track in space-time. One has then found the equivalent of the rails that take the tram round a corner. The analogy is admittedly not perfect, for a force is exerted between the tram-rails and the flanges on the wheels, whereas the stone follows the curvature of space without any force being exerted on it at all. But the analogy serves, nevertheless, to show in what way the relativistic explanation remains incomplete. It fails to include the equivalent of the tram-wheels. What, one is led to ask on receiving the relativistic explanation, is the feature of a particle that causes it to 'engage' with space, as it were, so that its track in space-time follows the curves? In the example of the tramcar, steel wheels run on steel rails. When an unrestrained particle moves in space something is said by relativists to run on curvature. What is it?

I shall return to the question later. For the moment I want to say a few words about the action of mass on space.

For the sake of convenience I shall again repeat a few well-known facts. According to Newtonian mechanics a massive body causes other bodies in its vicinity to be accelerated. To the question why this happens the answer is that a field of force surrounds the body. But to the question: what sort of a thing is this field? Newtonian mechanics has no answer; it can at most provide a name. Newton did not attempt to explain gravitation; he was content to postulate it.

It is here that relativity theory steps in. It does say what sort of a thing a gravitational field is, namely a region of curved space. The answer is justified both because it is methodologically sound and because it has considerable explanatory power. But it leads to the further question: why should a massive body cause the space around it to be curved? Again, relativity says something about space, but nothing about mass. It is content to postulate the effect of mass on space without explaining it. It seems to me that this limitation of relativity theory has not been properly appreciated. I say advisedly 'limitation' and not 'defect' for it is no defect of a theory to fail to provide 'all the answers'.

If this particular limitation has not been much noticed and if little effort has been made during recent decades to get past it, the reason is I think in a deflection of interest. In the early days of relativity interest was directed strongly towards the interaction between particles and, space. Studies in nuclear physics have since then directed more attention towards the interaction between particles and other particles, as already mentioned. Both kinds of interaction deserve equal study and I venture to suggest that a return to the earlier interest would now prove rewarding.

When one turns one's attention to this earlier interest, one is led to remember a view expressed by some relativists of those times. An elementary particle, it was sometimes said, is, apart from its charge, a region of gravitationally curved space and nothing else.

Eddington made this point a fundamental one. 'Mass is curvature' he said somewhere. At the time when he was writing there was the catch-phrase, 'Man is a kink in space-time'. Pre-occupation with this hypothesis is less insistent today, but not because we are any nearer either to accepting or discarding it. It is only for the reason already mentioned, namely that we are today more concerned with the way particle acts on particle than with the way space acts on particle. The great attention paid to nuclear physics has diverted attention from the fundamentals of relativity. But I do not think that what is at most a slight lack of topicality makes the question whether mass is curvature or something else any less rewarding now than it has been in the past.

To avoid the complication of electricity and magnetism I shall begin by considering only a neutron. What is it made of? Must we postulate a substance that is different in nature from the technical concept, space? Shall we find ourselves obliged to speak of ‘particle stuff' or shall we say, with Eddington, that the neutron consists of curvature and nothing else? When the question is put with this disconcerting candour one is inclined to dislike every answer that can be suggested. But I think that one will have the greatest dislike for the suggestion that there is something deserving of such a title as 'particle stuff'. One will be inclined to accept the Eddingtonian answer, if only as the lesser evil. The theory that the neutron consists of curvature seems better to meet the Principle of Minimum Assumption and to offer a better prospect of further unification of physics.

If this is accepted, the volume occupied by a neutron is a region of bound curvature. Within this region the potential gradient does not change with time in the way it does when the curvature takes the form of a travelling wave.

I have said a little while ago 'In the example of the tramcar, steel wheels run on steel rails. When an unrestrained particle moves in space, something is said by relativists to run on curvature. What is it? If mass is curvature, the answer is found. Only curvature can run on curvature. It seems as reasonable a conclusion as one may hope for so long as one accepts general relativity.

With that hint I should like now to leave discussion of the features by which one part of a relativistic space is distinguished from another and to turn to the notion of expanding space.

Expanding Space

The theory that space is expanding is supported by two kinds of evidence, that of inference and that of observation.

The inferential evidence was provided first, and by relativists. They explained that a cosmological model of which the volume did not change with time would be unstable. In the sense in which they used the word 'unstable', a model that resembled actuality would have to change its volume. This did not prove that space expands. The inference would have been equally compatible with a model that contracted, which it could, of course, not have been doing for an indefinitely long time without having disappeared. Hence the conclusion that space was, in fact, expanding was first arrived at by reasoning alone.

The observational evidence for the same conclusion is well-known. It is provided by the red shift in the spectrum of the light from distant nebulae. This shift is interpreted as a Doppler effect and is attributed to a recession of the nebulae from each other. The magnitude of the shift is found to a close approximation to be proportional to the distance of the nebulae. It is the effect that was predicted by the inferential evidence.

When a theory has such two-fold support it is usually accepted by scientists. If the notion of expanding space has not gained universal support, the reason is not far to seek. The notion is difficult to understand; one cannot form a picture in one's imagination of space itself expanding. It is only natural to dislike a conclusion that one cannot fully understand. From dislike to rejection is but a short step, so it is not surprising that some rather desperate attempts have been made to find an alternative explanation of the red shift. It has been done, of course, in the name of scientific caution. But the degree of scientific caution with which a new idea is greeted is some indication of its unattractiveness.

But unless both the inferential and the observational evidence can be effectively shaken, the wisest course is to do one's best to come to terms with the notion of expanding space whether one finds it an easy concept or not. If we cannot hope to relate this notion to anything with which we are familiar, we should at least try to find some valid statements about it.

One sometimes says that a fugitive from justice puts space between himself and his pursuers. One does not mean the expression to be taken literally. One only means that the fugitive is running faster than those in pursuit. When he does this he does not create new space but only causes a larger amount of existing space to separate him from the pursuers.

If the fugitive could literally put space between himself and his pursuers, he would not need to run away from them. He could sit down and smoke a cigarette while he put enough space in front of those who were trying to catch him to make sure that they never got any nearer. If he did this he would not be moving past objects in existing space. He would not be moving at all.

It is in this literal sense that, in an expanding universe, space originates between us and every distant nebula. While the fugitive from justice is getting further from his pursuers, he is also getting nearer to the house in which he hopes to hide. But while our galaxy is being caused by the expansion of space to get further from all other galaxies, it is not being caused to get nearer to anything.

In this there is a significant difference between changes that result from the operation of forces between bodies and those that result from the expansion of space. So long as there are forces, some things get nearer to others; they overtake other things. But when space-expansion alone determines distances nothing ever gets nearer to anything else; there is no overtaking; there is not even movement.

It is this last conclusion that makes the notion so difficult to understand. If things get further apart they must, we are inclined to reason, move relatively to each other. But we have to appreciate that this is false reasoning. Let me show why as clearly as possible.

Two nebulae, A and B, have been observed and both show the red shift. One of them, A, is in the part of the sky called North and B is in exactly the opposite direction, the part called South. When we are thinking only of A, we may make one of two statements:

(1)   The distance between our galaxy and the nebula is increasing.
(2)   Our galaxy and the nebula are moving relatively to each other.

We may be inclined to think that these two statements have identical meanings; and so they would in many contexts. But if we attribute the red shift to the expansion of space we have to conclude that they mean different things and that, while (1) is correct, (2) is wrong.

This emerges when the implications of (2) are examined. To say that our galaxy and the nebula A are moving relatively to each other may mean that both are moving or that one is at rest while the other is moving. But it must mean that at least one of the two bodies is moving.

If this were our own galaxy, it would be moving away from A, i.e. southwards. But when we observe nebula B we have to conclude that, if our galaxy moves at all, it must be away from B and northwards. A corresponding conclusion would be reached if we used a nebula in any other part of the sky as an indication of the direction in which our galaxy was moving. Wherever our choice fell, it would always cause us to say that we were moving away from the observed nebula. To say that the expansion of space is causing our galaxy to move is to say that the movement is in all directions at once! The correct interpretation of the red shift is, in other words, that our galaxy is not moving at all relative to any other nebula.

Are we then to take the view that we alone are at rest and that nebulae A and B, together with all others, are moving relatively to us? Are we to adopt the old egocentric universe in which we are located at a centre from which all effects radiate?

This, we know, cannot be. An observer on any other nebula would have the same experience as we ourselves here. It would be just as impossible for him to state the direction in which his nebula was moving. He could not say that it was moving relative to space in such a way that space, at one moment in front of it, was behind it at the next moment. He would say that his nebula was not overtaking anything, not even empty space; that it was not moving in any direction; that it was at rest.

We are thus obliged, whether we like it or not, to accept the odd notion that in expanding space the distance increases between objects that are all at rest relative to their surroundings. If the nebulae all seem to drift away from each other, this cannot be attributed to anything that happens to the nebulae but only to what happens to the space between them.

Even if one succeeds in appreciating the strange fact that, in expanding space, objects remain stationary while the distance between them increases, one may still, I fear, hanker after a force that drives them away from each other. It seems, to the way of thinking of most of us axiomatic, that things can only get further apart if a force is pushing them away from each other. But a little very simple mathematics shows that the notion of forces of repulsion has to be given up, however reluctant one may be to do so.

The velocity of recession, as viewed in any one particular direction, is:

dl / dt = Hl

where l is distance and H is Hubble's constant. To find its value observations are necessary that cannot be made with very great precision. Recently it was thought to be 185 km/sec/megaparsec. But later observations suggest that 100 km/sec/megaparsec may be nearer to the truth. If H is a cosmological constant and holds for the whole of space-time the acceleration is:

A = d2l/ dt2 = Hdl / dt = H2l ………. (1)

Here A is the acceleration of one body relative to a single selected other one, but not the acceleration of any body relative to all other ones, which would always be zero.

Force is the product of mass and acceleration. If our galaxy were receding under the influence of a repelling force, we should therefore give this the value

F = mA= mH2l ……….. (2)

where m was the mass of our receding galaxy. But this would be an absurd conclusion. According to equation (2), the force exerted by a body on any other one would not be proportional to the mass of the repelling body but to that of the repelled one.

Such a conclusion cannot, of course, be reconciled with the known law of gravitational attraction. Consider two bodies with the respective masses m1 and m2. The gravitational force between them, which needs the negative sign as it is one of attraction, is:

F = - Gm1m2 / l2

Here the force exerted by m1 on m2 is the same as that exerted by m2 on m1. But if equation (2) meant anything (which it does not), one would have to express the force exerted by m1 on m2 as m2H2 / l and that exerted by m2 on m1 as m1H2l . The tiniest repelling mass m1 would exert a very big force on m2 if this were big.

Let the distinction between a change of distance that is due to expanding space and one that is due to a force be expressed in a slightly different way. The acceleration of mass m2 relative to m1 is -Gm2 / l 2, and the acceleration of m, relative to mi is –Gm1 / l 2. The total acceleration of the two masses relative to each other in expanding space is:

Atotal = kH2l - G( m1 + m2) / l2 ………. (3)

This expression shows clearly that the relative acceleration occasioned by gravity is dependent on both masses as well as on the distance between them, while the relative acceleration occasioned by the expansion of space is dependent only on the distance between the masses. It is a function of space and of nothing else. The conclusion is disconcerting. But efforts to avoid it have to be rather desperate and to introduce sundry ad hoc hypotheses. They are not sought so much by those who accept the relativistic view of space as by those who reject or have forgotten it.

The proper conclusion is thus, that the distance between things can increase while the things are at rest relatively to each other. It is the space between them that originates. The notion of expanding space is more aptly called the Hypothesis of the Continuous Origin of Space.

c/o Kennedy & Donkin
12 Caxton Street
London. S.W.7

* Chairman's Address to the Philosophy of Science Group, 6th October, 1958

 

Acknowledgement
This paper and any subsequent discussions and rejoinders are reproduced from the British Journal for the Philosophy of Science with the kind permission of the Oxford University Press.
Website home page:
http://bjps.oxfordjournals.org/

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