TOWARDS A UNIFIED COSMOLOGY
by Reginald O. Kapp
PART IV - GRAVITATION
Chapter 24 - Space
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In physical science we are rarely satisfied about the validity of any statement unless it has the support of both these kinds of evidence. The inferential evidence for the performance of a machine is, for instance, provided by calculations made before the machine is even in the blueprint stage. We regard it as nearly, but not quite, conclusive. The observational evidence is provided by tests made with the machine after it has been built. If they confirm the calculations, we are satisfied. Similarly, both kinds of evidence are usually provided by a teacher who is lecturing on one of the generalizations of physics. He first deduces the generalization from first principles by making drawings and calculations on the blackboard and then confirms what he has just inferred by an experimental demonstration. If our reasoning were infallible, one of the two kinds of evidence alone would suffice. But we cannot be thus sure of ourselves. When there is only inferential evidence, one is justified in asking whether this may not be based on false premises or faulty deduction. When there is only observational evidence, this may be open to different interpretations; we are justified in doubting whether one may draw the stated general conclusion from a particular case. But when there are both kinds of evidence, each supports the other. It is only because the notion of expanding space has this dual support that it is widely, though not perhaps universally, accepted. It will suffice here to refer quite briefly to the two kinds of evidence for it. The inferential kind was provided first, and by those who had mastered the relativity equations. 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 there is such good agreement between prediction and observation, it is usual in physics for the conclusion to be accepted without further question. If it was not so for the notion of expanding space, the reason is not far to seek. This notion cannot be understood representationally any more than the notion of the electron can, and 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. Should a convincing alternative explanation of the red shift be found the situation would be that inference supported the notion of expanding space while observation failed to support it. The calculations made by relativists would then have to be re-examined. It really ought to be done simultaneously with the search for an alternative explanation of the red shift. If this has not happened, it is probably because the mental effort or understanding the calculations and examining them critically is considerable. It is far easier to invent an hypothesis by which to explain the red shift or explain it away. But unless both the inferential and the observational evidence are effectively shaken, the wisest course is to come to terms with the notion of expanding space, whether we find it attractive or not. And it has so far not been effectively shaken. So we must, I am afraid, be content to do with this notion of expanding space what we have had to do with the notion of the electron: form our private mental images of it from time to time; but always remember that these images are wrong. Experience with the electron has shown that we are rarely misled by doing so, provided we recognize the inadequacy of the images and are prepared to replace them by others if and when occasion demands. Experience has also shown that images, false though they be, are sometimes helpful, sometimes even indispensable. 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: We may be inclined to think that these two statements have identical meanings; and so they would have 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, ie 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. It is amusing and instructive to try to find a mathematical formula that would express a force of repulsion between nebulae in an expanding space; for the attempt is bound to fail. The velocity of recession as viewed in one particular direction is: dl/dt = Hl where l is distance and H is Hubble's constant. It has a value of 185 kilometres per sec per megaparsec. The acceleration is: A d2l / dt2 = Hdl / dt = H2 l ......... (24a) 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 ......... (24b) where m was the mass of our receding galaxy. But this would be an absurd conclusion. According to equation (24b), 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 / l 2 Here the force exerted by m1 on m2 is the same as that exerted by m2 on m1. But if equation (24b) meant anything, which it does not, one would have to express the force exerted by m1 on m2 as m2 H2l and that exerted by m2 on m1 as m1 H2l. 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 m1 relative to m2 is - 1/2G m2 / l2, and the acceleration of m2 relative to m1 is -G m1 / l 2. The total acceleration of the two masses relative to each other in expanding space is: Atotal = kH2 l - 1/2G( m1 + m2 ) / l 2 ......... (24c) 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. For the distance, l, is the only variable in the term that defines the effect of expanding space. The above equations are, of course, no more than a mathematical way of expressing the conclusion already reached without mathematics that the expansion of space does not cause the bodies in it to move. As they do not move, they are not accelerated by the expansion of space and are not subjected by it to any force. If we could regard space as the container of the physical universe and ponderable matter as the content, there would be no reason why their respective origins should be related. A change in the capacity of the container might well occur without a corresponding change in the quantity of matter contained in it. One could then say that at a certain mass density space was quite full; and that at half this limiting density space was only half-full. There would then be an absolute scale of mass density, as there is an absolute scale of temperature. But no one has ever looked for such a scale; no one has ever had the idea that there could be one; no one believed, even in pre-relativity days, that the expression 'space is as full of matter as it can hold' would have any meaning. Today relativists are able to show why, in fact, it has no meaning. They are also able to show that the origin of space and the origin of matter are coupled, if not better described as synonymous. McCrea, it will be remembered, has inferred the net rate of origin of matter, namely 500 atoms of hydrogen per cubic kilometre per year, from the observed rate of expansion of space. Yet here, as often in physics, there is a conflict between the two kinds of understanding. Our misplaced effort at representational understanding insists on retaining the notion of a purely conceptual, featureless space, into which particles of matter could be poured; just as it insists on retaining the notion of a purely conceptual, eventless time along which events are ranged. It is only deductive understanding that tells us that these notions are meaningless and that it is absurd to ask whether space is quite full of matter or only partially full. When visualizing a cosmological model based on the hypothesis (A2) about the Creation, representational understanding tries, misguidedly, to picture the featureless space as existing before the Creation is assumed to have begun, always to be unchanging, and to have been made the recipient of more and more matter as the process of the Creation became more and more complete. Those of us who accept (A3) tend to make the same misplaced effort. But in the battle between the two sides of one's intellect deductive understanding must win and the effort at representational understanding must sometimes be abandoned. It is so here. To speak of the expansion of space is to speak of a space with physical properties and to imply the continuous origin of matter. It is for this reason that the observed red shift provides observational evidence for (A3). (I am only too well aware, let me add, of the conceptual difficulty in the way of postulating the simultaneous origin of space and matter as well as their simultaneous extinction. I am hoping that what is said in Appendix H will help to overcome this difficulty.) The average density of the nebulae, including our own galaxy, is much above the equilibrium value. Space within the galaxy must therefore be contracting. Instead of showing a red shift the spectra of the light from stars within the galaxy must therefore show a violet shift. But the shift is proportional to the product of the rate of contraction and the distance of the observed star. This product may not be sufficient to reveal a measurable shift. The Doppler effect occasioned by the relative movement of stars within the galaxy is likely to exceed that occasioned by contraction and to make the interpretation of the readings uncertain. But there are other ways of testing the hypothesis and one of these can be usefully mentioned here. It would be provided by spectography. The light from any nebula passes partly through extragalactic space, but partly also through a portion of our own galaxy. According to Symmetrical Impermanence the space within this must be contracting, it has just been said, at a rate given by the excess density within the galaxy over the equilibrium value. The light from a nebula passes, therefore, partly through expanding and partly through contracting space. The latter part of the path will be a small fraction of the whole if the nebula is very distant and a larger fraction if it is close. The red shift that is observed will depend on the arithmetical mean of the contraction over the short path within the galaxy and the expansion over the long path outside. The nearer the nebula is the greater will be the relative effect of the contracting portion of the path through which the light travels and the smaller the red shift. For near nebulae one should therefore expect, on the average, the value of H to come out rather lower than it would for distant ones. Observation of the spectra of a large number of nebulae made in the United States by Humason, Mayall and Sandage, as mentioned in Chapter 4, suggests that it may be so. The red shift has been found to lead to a slightly lower value of H for near, than for very distant nebulae. The difference is hardly great enough to be quite conclusive, but it is in the direction that Symmetrical Impermanence would predict. That contraction of space within our own galaxy reduces the red shift is, however, not the only possible explanation of the observed results. The greater the distance of a nebula the longer its light has taken to reach us. We are therefore observing today the spectrum of light that left the nebula a long while ago. The larger value for the red shift in the light from very distant nebulae might therefore mean that space was expanding more rapidly when this light began its journey than it is now. The rate might thus vary either with time or with locality. But there seems to be no convincing reason why the variation should be with time, and the continuous extinction of matter provides a reason why it should be with location. The solar system is by no means at the centre of our galaxy; it is rather far out, though not on the very edge either. So the path of light from a distant nebula always lies partly within the galaxy within a contracting region. But the length of the contracting portion of the path differs with the direction in which the nebula is viewed. It is shortest if it is in a direction at right-angles to our disc-shaped galaxy. This may provide means of testing the two interpretations. If the variations in the apparent value of H are with time, the red shift will be the same for all equidistant nebulae, irrespective of the direction in which they lie. But if the variations are with locality, the red shift for equidistant nebulae will be less when the nebulae are so situated that their light : traverses a large part of the galaxy and greater if the light traverses a small part. It is conceivable that an analysis of the observations already made would settle the question whether H is constant in space and varies with time, or is, as I am claiming, not a function of time but varies in Space as a function of mass density.
24.5: Summary That space expands has been inferred mathematically from first principles and confirmed by observation. Such attempts as have been made to find alternative explanations for the observations have been far-fetched and unconvincing. The observed recession of distant bodies cannot be interpreted as the consequence of movement of the bodies relative to existing space or attributed to forces of repulsion between the bodies. The only interpretation to fit the facts is that new space is originating continuously between the bodies while these may remain stationary. The expansion of space is coupled with the origin of matter and no satisfactory hypothesis seems possible according to which the extent of the universe could change while its content remained constant, why, in other words, space could originate and matter not do so. If origins of elementary components of the material universe must be associated with the expansion of space, the converse must also hold. Extinctions must be associated with the contraction of space. As, according to Symmetrical Impermanence, origins and extinctions proceed side by side, any observed expansion must be a net one and represent the difference between the local gross expansion and the local gross contraction. At the equilibrium density this net difference becomes zero. Where the density is below the equilibrium value there is a net expansion, and it is shown by the expansion of the universe as a whole that the average density for the whole is below the equilibrium value. In any region of high mass density there must be a pronounced net contraction. |