This paper presents Chapter VII (section 3) from the book THE NATURE OF THE PHYSICAL WORLD by A. S. EDDINGTON. The contents of this book are based on the lectures that Eddington delivered at the University of Edinburgh in January to March 1927.
The paragraphs of original material are accompanied by brief comments in color, based on the present understanding. Feedback on these comments is appreciated.
The heading below links to the original materials.
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Predictions from the Law
I suppose that it is at first rather staggering to find a law supposed to control the movements of stars and planets turned into a law finicking with the behaviour of measuring rods. But there is no prediction made by the law of gravitation in which the behaviour of measuring appliances does not play an essential part. A typical prediction from the law is that on a certain date 384,400,000 metre rods laid end to end would stretch from the earth to the moon. We may use more circumlocutory language, but that is what is meant. The fact that in testing the prediction we shall trust to indirect evidence, not carrying out the whole operation literally, is not relevant; the prophecy is made in good faith and not with the intention of taking advantage of our remissness in checking it.
We have condemned the law of gravitation as a put-up job. You will want to know how after such a discreditable exposure it can still claim to predict eclipses and other events which come off.
A famous philosopher has said—”The stars are not pulled this way and that by mechanical forces; theirs is a free motion. They go on their way, as the ancients said, like the blessed gods.” *
*Hegel, Werke (1842 Ed.), Bd. 7, Abt. 1, p. 97.
This sounds particularly foolish even for a philosopher; but I believe that there is a sense in which it is true.
The stars have natural motion balanced by their inertia. From space to the nucleus of the atoms, which make up the star, there is increasing quantization. Thus, there are lines of force extending outward from the star and thinning out into space, all around the star. The more massive and dense is the star, the more numerous are these lines of force. Here “force” refers to the gradient of quantization.
We have already had three versions of what the earth is trying to do when it describes its elliptic orbit around the sun.
- It is trying to go in a straight line but it is roughly pulled away by a tug emanating from the sun.
- It is taking the longest possible route through the curved space-time around the sun.
- It is accommodating its track so as to avoid causing any illegal kind of curvature in the empty space around it.
We now add a fourth version.
- (4) The earth goes anyhow it likes.
It is not a long step from the third version to the fourth now that we have seen that the mathematical picture of empty space containing “illegal” curvature is a sheer impossibility in a world surveyed from within. For if illegal curvature is a sheer impossibility the earth will not have to take any special precautions to avoid causing it, and can do anything it likes. And yet the non-occurrence of this impossible curvature is the law (of gravitation) by which we calculate the track of the earth!
There are lines of force extending out both from the earth and the sun, and they meet in between. There is a natural tendency to even out the gradient of quantization, and this draws the two bodies toward each other. However, the natural speed of earth is much greater than that of the sun because of its much lower inertia. As a result, the earth gets into an orbit around the sun.
The earth and the sun are keeping a balance between their inertia and the gradient of quantization in the intervening space. This is the law of gravitation.
The key to the paradox is that we ourselves, our conventions, the kind of thing that attracts our interest, are much more concerned than we realise in any account we give of how the objects of the physical world are behaving. And so an object which, viewed through our frame of conventions, may seem to be behaving in a very special and remarkable way may, viewed according to another set of conventions, be doing nothing to excite particular comment. This will be clearer if we consider a practical illustration, and at the same time defend version (4).
You will say that the earth must certainly get into the right position for the eclipse next June (1927); so it cannot be free to go anywhere it pleases. I can put that right. I hold to it that the earth goes anywhere it pleases. The next thing is that we must find out where it has been pleased to go. The important question for us is not where the earth has got to in the inscrutable absolute behind the phenomena, but where we shall locate it in our conventional background of space and time. We must take measurements of its position, for example, measurements of its distance from the sun. In Fig. 6, SS1 shows the ridge in the world which we recognise as the sun; I have drawn the earth’s ridge in duplicate (EE1} EE2 ) because I imagine it as still undecided which track it will take. If it takes EE1 we lay our measuring rods end to end down the ridges and across the valley from S1 to E1 , count up the number, and report the result as the earth’s distance from the sun. The measuring rods, you will remember, adjust their lengths proportionately to the radius of curvature of the world. The curvature along this contour is rather large and the radius of curvature small. The rods therefore are small, and there will be more of them in $1E1 than the picture would lead you to expect. If the earth chooses to go to E2 the curvature is less sharp; the greater radius of curvature implies greater length of the rods. The number needed to stretch from S± to E2 will not be so great as the diagram at first suggests; it will not be increased in anything like the proportion of S1E2 to S1E1 in the figure. We should not be surprised if the number turned out to be the same in both cases. If so, the surveyor will report the same distance of the earth from the sun whether the track is EE1 or EE2 . And the Superintendent of the Nautical Almanac who published this same distance some years in advance will claim that he correctly predicted where the earth would go.
And so you see that the earth can play truant to any extent but our measurements will still report it in the place assigned to it by the Nautical Almanac. The predictions of that authority pay no attention to the vagaries of the god-like earth; they are based on what will happen when we come to measure up the path that it has chosen. We shall measure it with rods that adjust themselves to the curvature of the world. The mathematical expression of this fact is the law of gravitation used in the predictions.
Perhaps you will object that astronomers do not in practice lay measuring rods end to end through interplanetary space in order to find out where the planets are. Actually the position is deduced from the light rays. But the light as it proceeds has to find out what course to take in order to go “straight”, in much the same way as the metre rod has to find out how far to extend. The metric or curvature is a sign-post for the light as it is a gauge for the rod. The light track is in fact controlled by the curvature in such a way that it is incapable of exposing the sham law of curvature. And so wherever the sun, moon and earth may have got to, the light will not give them away. If the law of curvature predicts an eclipse the light will take such a track that there is an eclipse. The law of gravitation is not a stern ruler controlling the heavenly bodies; it is a kindhearted accomplice who covers up their delinquencies.
I do not recommend you to try to verify from Fig. 6 that the number of rods in S1E1 (full line) and S1E2 (dotted line) is the same. There are two dimensions of space-time omitted in the picture besides the extra dimensions in which space-time must be supposed to be bent; moreover it is the spherical, not the cylindrical, curvature which is ,the gauge for the length. It might be an instructive, though very laborious, task to make this direct verification, but we know beforehand that the measured distance of the earth from the sun must be the same for either track. The law of gravitation, expressed mathematically by Gμν = λgμν means nothing more nor less than that the unit of length everywhere is a constant fraction of the directed radius of the world at that point. And as the astronomer who predicts the future position of the earth does not assume anything more about what the earth will choose to do than is expressed in the law Gμν = λgμν, so we shall find the same position of the earth, if we assume nothing more than that the practical unit of length involved in measurements of the position is a constant fraction of the directed radius. We do not need to decide whether the track is to be represented by EE1 or EE2 , and it would convey no information as to any observable phenomena if we knew the representation.
Eddington is basically saying that the distances are relative and the absolute scenario is impossible to know. This is the unscientific interpretation of the theory of relativity. In the scientific interpretation, the velocity of light is an absolute. It acts as absolute reference point from which to measure the values of inertia of the earth and the sun, and the quantization of the intervening space. The rest then follows.
I shall have to emphasise elsewhere that the whole of our physical knowledge is based on measures and that the physical world consists, so to speak, of measure-groups resting on a shadowy background that lies outside the scope of physics. Therefore in conceiving a world which had existence apart from the measurements that we make of it, I was trespassing outside the limits of what we call physical reality. I would not dissent from the view that a vagary which by its very nature could not be measurable has no claim to a physical existence. No one knows what is meant by such a vagary. I said that the earth might go anywhere it chose, but did not provide a “where” for it to choose; since our conception of “where” is based on space measurements which were at that stage excluded. But I do not think I have been illogical. I am urging that, do what it will, the earth cannot get out of the track laid down for it by the law of gravitation. In order to show this I must suppose that the earth has made the attempt and stolen nearer to the sun; then I show that our measures conspire quietly to locate it back in its proper orbit. I have to admit in the end that the earth never was out of its proper orbit;** I do not mind that, because meanwhile I have proved my point. The fact that a predictable path through space and time is laid down for the earth is not a genuine restriction on its conduct, but is imposed by the formal scheme in which we draw up our account of its conduct.
** Because I can attach no meaning to an orbit other than an orbit in space and time, i.e. as located by measures. But I could not assume that the alternative orbit would be meaningless (inconsistent with possible measures) until I tried it.
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