Space Time and Gravitation/Chapter 8

by Arthur Eddington

 

CHAPTER VIII

OTHER TESTS OF THE THEORY

The words of Mercury are harsh after the songs of Apollo.

Love's Labour's Lost.

We have seen that the swift-moving light-waves possess great advantages as a means of exploring the non-Euclidean property of space. But there is an old fable about the hare and the tortoise. The slow-moving planets have qualities which must not be overlooked. The light-wave traverses the region in a few minutes and makes its report; the planet plods on and on for centuries going over the same ground again and again. Each time it goes round it reveals a little about the space, and the knowledge slowly accumulates.

According to Newton's law a planet moves round the sun in an ellipse, and if there are no other planets disturbing it, the ellipse remains the same for ever. According to Einstein's law the path is very nearly an ellipse, but it does not quite close up; and in the next revolution the path has advanced slightly in the same direction as that in which the planet was moving. The orbit is thus an ellipse which very slowly revolves^[1].

The exact prediction of Einstein's law is that in one revolution of the planet the orbit will advance through a fraction of a revolution equal to ${\tfrac {2v^{2}}{C^{2}}}$, where $v$ is the speed of the planet and $C$ the speed of light. The earth has 1/10,000 of the speed of light; thus in one revolution (one year) the point where the earth is at greatest distance from the sun will move on 3/100,000,000 of a revolution, or 0″.038. We could not detect this difference in a year, but we can let it add up for a century at least. It would then be observable but for one thing the—earth's orbit is very blunt, very nearly circular, and so we cannot tell accurately enough which way it is pointing and how its sharpest apses move. We can choose a planet with higher speed so that the effect is increased, not only because $v^{2}$ is increased, but because the revolutions take less time; but, what is perhaps more important, we need a planet with a sharp elliptical orbit, so that it is easy to observe how its apses move round. Both these conditions are fulfilled in the case of Mercury. It is the fastest of the planets, and the predicted advance of the orbit amounts to 43″ per century; further the eccentricity of its orbit is far greater than that of any of the other seven planets.

Now an unexplained advance of the orbit of Mercury had long been known. It had occupied the attention of Le Verrier, who, having successfully predicted the planet Neptune from the disturbances of Uranus, thought that the anomalous motion of Mercury might be due to an interior planet, which was called Vulcan in anticipation. But, though thoroughly sought for, Vulcan has never turned up. Shortly before Einstein arrived at his law of gravitation, the accepted figures were as follows. The actual observed advance of the orbit was 574″ per century; the calculated perturbations produced by all the known planets amounted to 532″ per century. The excess of 42″ per century remained to be explained. Although the amount could scarcely be relied on to a second of arc, it was at least thirty times as great as the probable accidental error.

The big discrepancy from the Newtonian gravitational theory is thus in agreement with Einstein's prediction of an advance of 43″ per century.

The derivation of this prediction from Einstein's law can only be followed by mathematical analysis; but it may be remarked that any slight deviation from the inverse square law is likely to cause an advance or recession of the apse of the orbit. That a particle, if it does not move in a circle, should oscillate between two extreme distances is natural enough; it could scarcely do anything else unless it had sufficient speed to break away altogether. But the interval between the two extremes will not in general be half a revolution. It is only under the exact adjustment of the inverse square law that this happens, so that the orbit closes up and the next revolution starts at the same point. I do not think that any "simple explanation" of this property of the inverse-square law has been given; and it seems fair to remind those, who complain of the difficulty of understanding Einstein's prediction of the advance of the perihelion, that the real trouble is that they have not yet succeeded in making clear to the uninitiated this recondite result of the Newtonian theory. The slight modifications introduced by Einstein's law of gravitation upset this fine adjustment, so that the oscillation between the extremes occupies slightly more than a revolution. A simple example of this effect of a small deviation from the inverse-square law was actually given by Newton.

It had already been recognised that the change of mass with velocity may cause an advance of perihelion; but owing to the ambiguity of Newton's law of gravitation the discussion was unsatisfactory. It was, however, clear that the effect was too small to account for the motion of perihelion of Mercury, the prediction being ${\tfrac {1}{2}}v^{2}/C^{2}$, or at most $v^{2}/C^{2}$. Einstein's theory is the only one which gives the full amount $3v^{2}/C^{2}$.

It was suggested by Lodge that, this variation of mass with velocity might account for the whole motion of the orbit of Mercury, if account were taken of the sun's unknown absolute motion through the aether, combining sometimes additively and sometimes negatively with the orbital motion. In a discussion between him and the writer, it appeared that, if the absolute motion were sufficient to produce this effect on Mercury, it must give observable effects for Venus and the Earth; and these do not exist. Indeed from the close accordance of Venus and the Earth with observation, it is possible to conclude that, either the sun's motion through the aether is improbably small, or gravitation must conform to relativity, in the sense of the restricted principle (p. 20), and conceal the effects of the increase of mass with speed so far as an additive uniform motion is concerned.

Unfortunately it is not possible to obtain any further test of Einstein's law of gravitation from the remaining planets. We have to pass over Venus and the Earth, whose orbits are too nearly circular to show the advance of the apses observationally. Coming next to Mars with a moderately eccentric orbit, the speed is very much smaller, and the predicted advance is only 1″.3 per century. Now the accepted figures show an observed advance (additional to that produced by known causes) of 5″ per century, so that Einstein's correction improves the accordance of observation with theory; but, since the result for Mars is in any case scarcely trustworthy to 5″ owing to the inevitable errors of observation, the improvement is not very important. The main conclusion is that Einstein's theory brings Mercury into line, without upsetting the existing good accordance of all the other planets.

We have tested Einstein's law of gravitation for fast movement (light) and for moderately slow movement (Mercury). For very slow movement it agrees with Newton's law, and the general accordance of the latter with observation can be transferred to Einstein's law. These tests appear to be sufficient to establish the law firmly. We can express it in this way.

Every particle or light-pulse moves so that the quantity $s$ measured along its track between two points has the maximum possible value, where $ds^{2}=-(1-2m/r)^{-1}dr^{2}-r^{2}d\theta ^{2}+(1-2m/r)\,dt^{2}$. And the accuracy of the experimental test is sufficient to verify the coefficients as far as terms of order $m/r$ in the coefficient of $dr^{2}$, and as far as terms of order $m^{2}/r^{2}$ in the coefficient of $dt^{2}$^[2].

In this form the law appears to be firmly based on experiment, and the revision or even the complete abandonment of the general ideas of Einstein's theory would scarcely affect it.

These experimental proofs, that space in the gravitational field of the sun is non-Euclidean or curved, have appeared puzzling to those unfamiliar with the theory. It is pointed out that the experiments show that physical objects or loci are "warped" in the sun's field; but it is suggested that there is nothing to show that the space in which they exist is warped. The answer is that it does not seem possible to draw any distinction between the warping of physical space and the warping of physical objects which define space. If our purpose were merely to call attention to these phenomena of the gravitational field as curiosities, it would, no doubt, be preferable to avoid using words which are liable to be misconstrued. But if we wish to arrive at an understanding of the conditions of the gravitational field, we cannot throw over the vocabulary appropriate for that purpose, merely because there may be some who insist on investing the words with a metaphysical meaning which is clearly inappropriate to the discussion.

We come now to another kind of test. In the statement of the law of gravitation just given, a quantity $s$ is mentioned; and, so far as that statement goes, $s$ is merely an intermediary quantity defined mathematically. But in our theory we have been identifying $s$ with interval-length, measured with an apparatus of scales and clocks; and it is very desirable to test whether this identification can be confirmed—whether the geometry of scales and clocks is the same as the geometry of moving particles and light-pulses.

The question has been mooted whether we may not divide the present theory into two parts. Can we not accept the law of gravitation in the form suggested above as a self-contained result proved by observation, leaving the further possibility that $s$ is to be identified with interval-length open to debate? The motive is partly a desire to consolidate our gains, freeing them from the least taint of speculation; but perhaps also it is inspired by the wish to leave an opening by which clock-scale geometry, i.e. the space and time of ordinary perception, may remain Euclidean. Disregarding the connection of $s$ with interval-length, there is no object in attributing any significance of length to it; it can be regarded as a dynamical quantity like Action, and the new law of gravitation can be expressed after the traditional manner without dragging in strange theories of space and time. Thus interpreted, the law perhaps loses its theoretical inevitability; but it remains strongly grounded on observation. Unfortunately for this proposal, it is impossible to make a clean division of the theory at the point suggested. Without some geometrical interpretation of $s$ our conclusions as to the courses of planets and light- waves cannot be connected with the astronomical measurements which verify them. The track of a light-wave in terms of the coordinates $r$, $\theta$, $t$ cannot be tested directly; the coordinates afford only a temporary resting-place; and the measurement of the displacement of the star-image on the photographic plate involves a reconversion from the coordinates to $s$, which here appears in its significance as the interval in clock-scale geometry.

Thus even from the experimental standpoint, a rough correspondence of the quantity $s$ occurring in the law of gravitation with the clock-scale interval is an essential feature. We have now to examine whether experimental evidence can be found as to the exactness of this correspondence.

It seems reasonable to suppose that a vibrating atom is an ideal type of clock. The beginning and end of a single vibration constitute two events, and the interval $ds$ between two events is an absolute quantity independent of any mesh-system. This interval must be determined by the nature of the atom; and hence atoms which are absolutely similar will measure by their vibrations equal values of the absolute interval $ds$. Let us adopt the usual mesh-system ($r$, $\theta$, $t$) for the solar system, so that $ds^{2}=-\gamma ^{-1}dr^{2}-r^{2}d\theta ^{2}+\gamma \,dt^{2}$. Consider an atom momentarily at rest at some point in the solar system; we say momentarily, because it must undergo the acceleration of the gravitational field where it is. If $ds$ corresponds to one vibration, then, since the atom has not moved, the corresponding $dr$ and $d\theta$ will be zero, and we have $ds^{2}=-y\,dt^{2}$. The time of vibration $dt$ is thus $1/{\sqrt {\gamma }}$ times the interval of vibration $ds$.

Accordingly, if we have two similar atoms at rest at different points in the system, the interval of vibration will be the same for both; but the time of vibration will be proportional to the inverse square-root of $\gamma$, which differs for the two atoms. Since ${\begin{aligned}\gamma \;&=1-{\frac {2m}{r}}\\1/{\sqrt {\gamma }}\;&=1+{\frac {m}{r}}{\text{, very approximately.}}\\\end{aligned}}$

Take an atom on the surface of the sun, and a similar atom in a terrestrial laboratory. For the first, $1+m/r=1.00000212$, and for the second $1+m/r$ is practically 1. The time of vibration of the solar atom is thus longer in the ratio 1.00000212, and it might be possible to test this by spectroscopic examination.

There is one important point to consider. The spectroscopic examination must take place in the terrestrial laboratory; and we have to test the period of the solar atom by the period of the waves emanating from it when they reach the earth. Will they carry the period to us unchanged? Clearly they must.

The first and second pulse have to travel the same distance ($r$), and they travel with the same velocity ($dr/dt$); for the velocity of light in the mesh-system used is $1-2m/r$, and though this velocity depends on $r$, it does not depend on $t$. Hence the difference $dt$ at one end of the waves is the same as that at the other end.

Thus in the laboratory the light from a solar source should be of greater period and greater wave-length (i.e. redder) than that from a corresponding terrestrial source. Taking blue light of wave-length 4000 Å, the solar lines should be displaced 4000 x .00000212, or 0.008 Å towards the red end of the spectrum.

The properties of a gravitational field of force are similar to those of a centrifugal field of force; and it may be of interest to see how a corresponding shift of the spectral lines occurs for an atom in a field of centrifugal force. Suppose that, as we rotate with the earth, we observe a very remote atom momentarily at rest relative to our rotating axes. The case is just similar to that of the solar atom; both are at rest relative to the respective mesh-systems; the solar atom is in a field of gravitational force, and the other is in a field of centrifugal force. The direction of the force is in both cases the same—from the earth towards the atom observed. Hence the atom in the centrifugal field ought also to vibrate more slowly, and show a displacement to the red in its spectral lines. It does, if the theory hitherto given is right. We can abolish the centrifugal force by choosing non-rotating axes. But the distant atom was at rest relative to the rotating axes, that is to say, it was whizzing round with them. Thus from the ordinary standpoint the atom has a large velocity relative to the observer, and, in accordance with Chapter i, its vibrations slow down just as the aviator's watch did. The shift of spectral lines due to a field of centrifugal force is only another aspect of a phenomenon already discussed.

The expected shift of the spectral lines on the sun, compared with the corresponding terrestrial lines, has been looked for; but it has not been found.

In estimating the importance of this observational result in regard to the relativity theory, we must distinguish between a failure of the test and a definite conclusion that the lines are undisplaced. The chief investigators St John, Schwarzschild, Evershed, and Grebe and Bachem, seem to be agreed that the observed displacement is at any rate less than that predicted by the theory. The theory can therefore in no case claim support from the present evidence. But something more must be established, if the observations are to be regarded as in the slightest degree adverse to the theory. If for instance the mean deflection is found to be .004 instead of .008 Angstrom units, the only possible conclusion is that there are certain causes of displacement of the lines, acting in the solar atmosphere and not yet identified. No one could be much surprised if this were the case; and it would, of course, render the test nugatory. The case is not much altered if the observed displacement is .002 units, provided the latter quantity is above the accidental error of measurement; if we have to postulate some unexplained disturbance, it may just as well produce a displacement – .006 as + .002. For this reason Evershed's evidence is by no means adverse to the theory, since he finds unexplained displacements in any case. One set of lines measured by St John gave a mean displacement of .0036 units; and this also shows that the test has failed. The only evidence adverse to the theory, and not merely neutral, is a series of measures by St John on 17 cyanogen lines, which he regarded as most dependable. These gave a mean shift of exactly .000. If this stood alone we should certainly be disposed to infer that the test had gone against Einstein's theory, and that nothing had intervened to cast doubt on the validity of the test. The writer is unqualified to criticise these mutually contradictory spectroscopic conclusions; but he has formed the impression that the last-mentioned result obtained by St John has the greatest weight of any investigations up to the present^[3].

It seems that judgment must be reserved; but it may be well to examine how the present theory would stand if the verdict of this third crucial experiment finally went against it.

It has become apparent that there is something illogical in the sequence we have followed in developing the theory, owing to the necessity of proceeding from the common ideas of space and time to the more fundamental properties of the absolute world. We started with a definition of the interval by measurements made with clocks and scales, and afterwards connected it with the tracks of moving particles. Clearly this is an inversion of the logical order. The simplest kind of clock is an elaborate mechanism, and a material scale is a very complex piece of apparatus. The best course then is to discover $ds$ by exploration of space and time with a moving particle or light-pulse, rather than by measures with scales and clocks. On this basis by astronomical observation alone the formula for $ds$ in the gravitational field of the sun has already been established. To proceed from this to determine exactly what is measured by a scale and a clock, it would at first seem necessary to have a detailed theory of the mechanisms involved in a scale and clock. But there is a short-cut which seems legitimate. This short-cut is in fact the Principle of Equivalence. Whatever the mechanism of the clock, whether it is a good clock or a bad clock, the intervals it is beating must be something absolute; the clock cannot know what mesh-system the observer is using, and therefore its absolute rate cannot be altered by position or motion which is relative merely to a mesh-system. Thus wherever it is placed, and however it moves, provided it is not constrained by impacts or electrical forces, it must always beat equal intervals as we have previously assumed. Thus a clock may fairly be used to measure intervals, even when the interval is defined in the new manner; any other result seems to postulate that it pays heed to some particular mesh-system^[4].

Three modes of escape from this conclusion seem to be left open. A clock cannot pay any heed to the mesh-system used; but it may be affected by the kind of space-time around it^[5]. The terrestrial atom is in a field of gravitation so weak that the space-time may be considered practically flat; but the space-time round the solar atom is not flat. It may happen that the two atoms actually detect this absolute difference in the world around them and do not vibrate with the same interval $ds$—contrary to our assumption above. Then the prediction of the shift of the lines in the solar spectrum is invalidated. Now it is very doubtful if an atom can detect the curving of the region it occupies, because curvature is only apparent when an extended region is considered; still an atom has some extension, and it is not impossible that its equations of motion involve the quantities $B_{\mu \nu \sigma }^{\rho }$ which distinguish gravitational from flat space-time. An apparently insuperable objection to this explanation is that the effect of curvature on the period would almost certainly be represented by terms of the form $m^{2}/r^{2}$, whereas to account for a negative result for the shift of the spectral lines terms of much greater order of magnitude $m/r$ are needed.

The second possibility depends on the question whether it is possible for an atom at rest on the sun to be precisely similar to one on the earth. If an atom fell from the earth to the sun it would acquire a velocity of 610 km. per sec., and could only be brought to rest by a systematic hammering by other atoms. May not this have made a permanent alteration in its time-keeping properties? It is true that every atom is continually undergoing collisions, but it is just possible that the average solar atom has a different period from the average terrestrial atom owing to this systematic difference in its history.

What are the two events which mark the beginning and end of an atomic vibration? This question suggests a third possibility. If they are two absolute events, like the explosions of two detonators, then the interval between them will be a definite quantity, and our argument applies. But if, for example, an atomic vibration is determined by the revolution of an electron around a nucleus, it is not marked by any definite events. A revolution means a return to the same position as before; but we cannot define what is the same position as before without reference to some mesh-system. Hence it is not clear that there is any absolute interval corresponding to the vibration of an atom; an absolute interval only exists between two events absolutely defined.

It is unlikely that any of these three possibilities can negative the expected shift of the spectral lines. The uncertainties introduced by them are, so far as we can judge, of a much smaller order of magnitude. But it will be realised that this third test of Einstein's theory involves rather more complicated considerations than the two simple tests with light-waves and the moving planet. I think that a shift of the Fraunhofer lines is a highly probable prediction from the theory and I anticipate that experiment will ultimately confirm the prediction; but it is not entirely free from guess-work. These theoretical uncertainties are apart altogether from the great practical difficulties of the test, including the exact allowance for the unfamiliar circumstances of an absorbing atom in the sun's atmosphere.

Outside the three leading tests, there appears to be little chance of checking the theory unless our present methods of measurement are greatly improved. It is not practicable to measure the deflection of light by any body other than the sun. The apparent displacement of a star just grazing the limb of Jupiter should be 0″.017. A hundredth of a second of arc is just about within reach of the most refined measurements with the largest telescopes. If the observation could be conducted under the same conditions as the best parallax measurements, the displacement could be detected but not measured with any accuracy. The glare from the light of the planet ruins any chance of success.

Most astronomers, who look into the subject, are entrapped sooner or later by a fallacy in connection with double stars. It is thought that when one component passes behind the other it will appear displaced from its true position, like a star passing behind the sun; if the size of the occulting star is comparable with that of the sun, the displacement should be of the same order, 1″.7. This would cause a very conspicuous irregularity in the apparent orbit of a double star. But reference to p. 113 shows that an essential point in the argument was the enormous ratio of the distance $QP$ of the star from the sun to the distance $EF$ of the sun from the earth. It is only in these conditions that the apparent displacement of the object is equal to the deflection undergone by its light. It is easy to see that where this ratio is reversed, as in the case of the double star, the apparent displacement is an extremely small fraction of the deflection of the light. It would be quite imperceptible to observation.

If two independent stars are seen in the same line of vision within about 1″, one being a great distance behind the other, the conditions seem at first more favourable. I do not know if any such pairs exist. It would seem that we ought to see the more distant star not only by the direct ray, which would be practically undisturbed, but also by a ray passing round the other side of the nearer star and bent by it to the necessary extent. The second image would, of course, be indistinguishable from that of the nearer star; but it would give it additional brightness, which would disappear in time when the two stars receded. But consider a pencil of light coming past the nearer star; the inner edge will be bent more than the outer edge, so that the divergence is increased. The increase is very small; but then the whole divergence of a pencil from a source some hundred billion miles away is very minute. It is easily calculated that the increased divergence would so weaken the light as to make it impossible to detect it when it reached us^[6].

If two unconnected stars approached the line of sight still more closely, so that one almost occulted the other, observable effects might be perceived. When the proximity was such that the direct ray from the more distant star passed within about 100 million kilometres of the nearer star, it would begin to fade appreciably. The course of the ray would not yet be appreciably deflected, but the divergence of the pencil would be rapidly increased, and less light from the star would enter our telescopes. The test is scarcely likely to be an important one, since a sufficiently close approach is not likely to occur; and in any case it would be difficult to feel confident that the fading was not due to a nebulous atmosphere around the nearer star.

The theory gives small corrections to the motion of the moon which have been investigated by de Sitter. Both the axis of the orbit and its line of intersection with the ecliptic should advance about 2″ per century more than the Newtonian theory indicates. Neither observation nor Newtonian theory are as yet pushed to sufficient accuracy to test this; but a comparatively small increase in accuracy would make a comparison possible.

Since certain stars are perhaps ten times more massive than the sun, without the radius being unduly increased, they should show a greater shift of the spectral lines and might be more favourable for the third crucial test. Unfortunately the predicted shift is indistinguishable from that caused by a velocity of the star in the line-of-sight on Doppler's principle. Thus the expected shift on the sun is equivalent to that caused by a receding velocity of 0.634 kilometres per second. In the case of the sun we know by other evidence exactly what the line-of-sight velocity should be; but we have not this knowledge for other stars. The only indication that could be obtained would be the detection of an average motion of recession of the more massive stars. It seems rather unlikely that there should be a real preponderance of receding motions among stars taken indiscriminately from all parts of the sky; and the apparent effect might then be attributed to the Einstein shift. Actually the most massive stars (those of spectral type B) have been found to show an average velocity of recession of about 4.5 km. per sec., which would be explained if the values of $m/r$ for these stars are about seven times greater than the value for the sun—a quite reasonable hypothesis. This phenomenon was well-known to astrophysicists some years before Einstein's theory was published. But there are so many possible interpretations that no stress should be placed on this evidence. Moreover the very diffuse "giant" stars of type M have also a considerable systematic velocity of recession, and for these $m/r$ must be much less than for the sun.

1. Appendix, Note 9
2. Appendix, Note 10.
3. A further paper by Grebe and Bachem (Zeitschrift für Physik, 1920, p, 51), received whilst this is passing through the press, makes out a case strongly favourable for the Einstein displacement, and reconciles the discordant results found by most of the investigators. But it may still be the best counsel to "wait and see," and I have made no alteration in the discussion here given.
4. Of course, there is always the possibility that this might be the case, though it seems unlikely. The essential point of the relativity theory is that (contrary to the common opinion) no experiments yet made have revealed any mesh-system of an absolute character, not that experiments never will reveal such a system.
5. Appendix, Note 11.
6. Appendix, Note 12.