Space Time and Gravitation/Chapter 1
CHAPTER I
THE FITZGERALD CONTRACTION
Will it take longer to swim to a point 100 yards up-stream and back, or to a point 100 yards across-stream and back?
In the first case there is a long toil up against the current, and then a quick return helped by the current, which is all too short to compensate. In the second case the current also hinders, because part of the effort is devoted to overcoming the drift down-stream. But no swimmer will hesitate to say that the hindrance is the greater in the first case.
Let us take a numerical example. Suppose the swimmer's speed is 50 yards a minute in still water, and the current is 30 yards a minute. Thus the speed against the current is 20, and with the current 80 yards a minute. The up journey then takes 5 minutes and the down journey 114 minutes. Total time, 614 minutes.
In still water the time would have been 4 minutes.
The up-and-down swim is thus longer than the transverse swim in the ratio 614: 5 minutes. Or we may write the ratio which shows how the result depends on the ratio of the speed of the current to the speed of the swimmer, viz. 3050.
A very famous experiment on these lines was tried in America in the year 1887. The swimmer was a wave of light, which we know swims through the aether with a speed of 186,330 miles a second. The aether was flowing through the laboratory like a river past its banks. The light-wave was divided, by partial reflection at a thinly silvered surface, into two parts, one of which was set to perform the up-and-down stream journey and the other the across-stream journey. When the two waves reached their proper turning-points they were sent back to the starting-point by mirrors. To judge the result of the race, there was an optical device for studying interference fringes; because the recomposition of the two waves after the journey would reveal if one had been delayed more than the other, so that, for example, the crest of one instead of fitting on to the crest of the other coincided with its trough.
To the surprise of Michelson and Morley, who conducted the experiment, the result was a dead-heat. It is true that the direction of the current of aether was not known—they hoped to find it out by the experiment. That, however, was got over by trying a number of different orientations. Also it was possible that there might actually be no current at a particular moment. But the earth has a velocity of 1812 miles a second, continually changing direction as it goes round the sun; so that at some time during the year the motion of a terrestrial laboratory through the aether must be at least 1812 miles a second. The experiment should have detected the delay by a much smaller current; in a repetition of it by Morley and Miller in 1905, a current of 2 miles a second would have been sufficient.
If we have two competitors, one of whom is known to be slower than the other, and yet they both arrive at the winning-post at the same time, it is clear that they cannot have travelled equal courses. To test this, the whole apparatus was rotated through a right angle, so that what had been the up-and-down course became the transverse course, and vice versa. Our two competitors interchanged courses, but still the result was a dead-heat.
The surprising character of this result can be appreciated by contrasting it with a similar experiment on sound-waves. Sound consists of waves in air or other material, as light consists of waves in aether. It would be possible to make a precisely similar experiment on sound, with a current of air past the apparatus instead of a current of aether. In that case the greater delay of the wave along the direction of the current would certainly show itself experimentally. Why does light seem to behave differently?
The straightforward interpretation of this remarkable result is that each course undergoes an automatic contraction when it is swung from the transverse to the longitudinal position, so that whichever arm of the apparatus is placed up-stream it straightway becomes the shorter. The course is marked out in the rigid material apparatus, and we have to suppose that the length of any part of the apparatus changes as it is turned in different directions with respect to the aether-current. It is found that the kind of material—metal, stone or wood—makes no difference to the experiment. The contraction must be the same for all kinds of matter; the expected delay depends only on the ratio of the speed of the aether current to the speed of light, and the contraction which compensates it must be equally definite.
This explanation was proposed by FitzGerald, and at first sight it seems a strange and arbitrary hypothesis. But it has been rendered very plausible by subsequent theoretical researches of Larmor and Lorentz. Under ordinary circumstances the form and size of a solid body is maintained by the forces of cohesion between its particles. What is the nature of cohesion? We guess that it is made up of electric forces between the molecules. But the aether is the medium in which electric force has its seat; hence it will not be a matter of indifference to these forces how the electric medium is flowing with respect to the molecules. When the flow changes there will be a readjustment of cohesive forces, and we must expect the body to take a new shape and size.
The theory of Larmor and Lorentz enables us to trace in detail the readjustment. Taking the accepted formulae of electromagnetic theory, they showed that the new form of equilibrium would be contracted in just such a way and by just such an amount as FitzGerald's explanation requires.[1]
The contraction in most cases is extremely minute. We have seen that when the ratio of the speed of the current to that of the swimmer is 35, a contraction in the ratio is needed to compensate for the delay. The earth's orbital velocity is 110000 of the velocity of light, so that it will give a contraction of , or 1 part in 200,000,000. This would mean that the earth's diameter in the direction of its motion is shortened by 212 inches.
The Michelson-Morley experiment has thus failed to detect our motion through the aether, because the effect looked for—the delay of one of the light waves—is exactly compensated by an automatic contraction of the matter forming the apparatus. Other ingenious experiments have been tried, electrical and optical experiments of a more technical nature. They likewise have failed, because there is always an automatic compensation somewhere. We now believe there is something in the nature of things which inevitably makes these compensations, so that it will never be possible to determine our motion through the aether. Whether we are at rest in it, or whether we are rushing through it with a speed not much less than that of light, will make no difference to anything that can possibly be observed.
This may seem a rash generalization from the few experiments actually performed; more particularly, since we can only experiment with the small range of velocity caused by the earth's orbital motion. With a larger range residual differences might be disclosed. But there is another reason for believing that the compensation is not merely approximate but exact. The compensation has been traced theoretically to its source in the well-known laws of electromagnetic force; and here it is mathematically exact. Thus the generalization is justified, at least in so far as the observed phenomena depend on electromagnetic causes, and in so far as the universally accepted laws of electromagnetism are accurate.
The generalization here laid down is called the restricted Principle of Relativity:—It is impossible by any experiment to detect uniform motion relative to the aether.
There are other natural forces which have not as yet been recognised as coming within the electromagnetic scheme—gravitation, for example—and for these other tests are required. Indeed we were scarcely justified in stating above that the diameter of the earth would contract 212 inches, because the figure of the earth is determined mainly by gravitation, whereas the Michelson-Morley experiment relates to bodies held together by cohesion. There is fair evidence of a rather technical kind that the compensation exists also for phenomena in which gravitation is concerned; and we shall assume that the principle covers all the forces of nature.
Suppose for a moment it were not so, and that it were possible to determine a kind of absolute motion of the earth by experiments or observations involving gravitation. Would this throw light on our motion through the aether? I think not. It would show that there is some standard of rest with respect to which the law of gravitation takes a symmetrical and simple form; presumably this standard corresponds to some gravitational medium, and the motion determined would be motion with respect to that medium. Similarly if the motion were revealed by vital or psychical phenomena, it would be motion relative to some vital or psychical medium. The aether, defined as the seat of electric forces, must be revealed, if at all, by electric phenomena.
It is well to remember that there is reasonable justification for adopting the principle of relativity even if the evidence is insufficient to prove it. In Newtonian dynamics the phenomena are independent of uniform motion of the system; no explanation is asked for, because it is difficult to see any reason why there should be an effect. If in other phenomena the principle fails, then we must seek for an explanation of its failure—and no doubt a plausible explanation can be devised; but so long as experiment gives no indication of a failure, it is idle to anticipate such a complication. Clearly physics cannot concern itself with all the possible complexities which may exist in nature, but have not hitherto betrayed themselves in any experiment.
The principle of relativity has implications of a most revolutionary kind. Let us consider what is perhaps an exaggerated case—or perhaps the actual case, for we cannot tell. Let the reader suppose that he is travelling through the aether at 161,000 miles a second vertically upwards; if he likes to make the positive assertion that this is his velocity, no one will be able to find any evidence to contradict him. For this speed the FitzGerald contraction is just 12, so that every object contracts to half its original length when turned into the vertical position.
As you lie in bed, you are, say, 6 feet long. Now stand upright; you are 3 feet. You are incredulous? Well, let us prove it! Take a yard-measure; when turned vertically it must undergo the FitzGerald contraction, and become only half a yard. If you measure yourself with it, you will find you are just two—half-yards. "But I can see that the yard-measure does not change length when I turn it." What you perceive is an image of the rod on the retina of your eye; you imagine that the image occupies the same space in both positions; but your retina has contracted in the vertical direction without your knowing it, so that your visual estimates of vertical length are double what they should be. And so on with every test you can devise. Because everything is altered in the same way, nothing appears to be altered at all.
It is possible to devise electrical and optical tests; in that case the argument is more complicated, because we must consider the effect of the rapid current of aether on the electric forces and on waves of light. But the final conclusion is always the same; the tests will reveal nothing. Here is one illustration. To avoid distortion of the retina, lie on your back on the floor, and watch in a suitably inclined mirror someone turn the rod from the horizontal to the vertical position. You will, of course, see no change of length, and it is not possible to blame the retina this time. But is the appearance in the mirror a faithful reproduction of what is actually occurring? In a plane mirror at rest the appearance is correct; the rays of light come off the mirror at the same angle as they fall on to it, like billiard balls rebounding from an elastic cushion. But if the cushion is in rapid motion the angle of the billiard-ball will be altered; and similarly the rapid motion of the mirror through the aether alters the law of reflection. Precise calculation shows that the moving mirror will distort the image, so as to conceal exactly the changes of length which occur.
The mathematician does not need to go through all the possible tests in detail; he knows that the complete compensation is inherent in the fundamental laws of nature, and so must occur in every case. So if any suggestion is made of a device for detecting these effects, he starts at once to look for the fallacy which must surely be there. Our motion through the aether may be very much less than the value here adopted, and the changes of length may be very small; but the essential point is that they escape notice, not because they are small (if they are small), but because from their very nature they are undetectable.
There is a remarkable reciprocity about the effects of motion on length, which can best be illustrated by another example. Suppose that by development in the powers of aviation, a man flies past us at the rate of 161,000 miles a second. We shall suppose that he is in a comfortable travelling conveyance in which he can move about, and act normally and that his length is in the direction of the flight. If we could catch an instantaneous glimpse as he passed, we should see a figure about three feet high, but with the breadth and girth of a normal human being. And the strange thing is that he would be sublimely unconscious of his own undignified appearance. If he looks in a mirror in his conveyance, he sees his usual proportions; this is because of the contraction of his retina, or the distortion by the moving mirror, as already explained. But when he looks down on us, he sees a strange race of men who have apparently gone through some flattening-out process; one man looks barely 10 inches across the shoulders, another standing at right angles is almost "length and breadth, without thickness." As they turn about they change appearance like the figures seen in the old-fashioned convex-mirrors. If the reader has watched a cricket-match through a pair of prismatic binoculars, he will have seen this effect exactly.
It is the reciprocity of these appearances that each party should think the other has contracted that is so difficult to realise. Here is a paradox beyond even the imagination of Dean Swift. Gulliver regarded the Lilliputians as a race of dwarfs; and the Lilliputians regarded Gulliver as a giant. That is natural, If the Lilliputians had appeared dwarfs to Gulliver, and Gulliver had appeared a dwarf to the Lilliputians—but no! that is too absurd for fiction, and is an idea only to be found in the sober pages of science.
This reciprocity is easily seen to be a necessary consequence of the Principle of Relativity. The aviator must detect a FitzGerald contraction of objects moving rapidly relatively to him, just as we detect the contraction of objects moving relatively to us, and as an observer at rest in the aether detects the contraction of objects moving relatively to the aether. Any other result would indicate an observable effect due to his own motion through the aether.
Which is right? Are we or the aviator? Or are both the victims of illusion? It is not illusion in the ordinary sense, because the impressions of both would be confirmed by every physical test or scientific calculation suggested. No one knows which is right. No one will ever know, because we can never find out which, if either, is truly at rest in the aether.
It is not only in space but in time that these strange variations occur. If we observed the aviator carefully we should infer that he was unusually slow in his movements; and events in the conveyance moving with him would be similarly retarded—as though time had forgotten to go on. His cigar lasts twice as long as one of ours. I said "infer" deliberately; we should see a still more extravagant slowing down of time; but that is easily explained, because the aviator is rapidly increasing his distance from us and the light-impressions take longer and longer to reach us. The more moderate retardation referred to remains after we have allowed for the time of transmission of light.
But here again reciprocity comes in, because in the aviator's opinion it is we who are travelling at 161,000 miles a second past him; and when he has made all allowances, he finds that it is we who are sluggish. Our cigar lasts twice as long as his.
Let us examine more closely how the two views are to be reconciled. Suppose we both light similar cigars at the instant he passes us. At the end of 30 minutes our cigar is finished. This signal, borne on the waves of light, hurries out at the rate of 186,000 miles a second to overtake the aviator travelling at 161,000 miles a second, who has had 30 minutes start. It will take nearly 194 minutes to overtake him, giving a total time of 224 minutes after lighting the cigar. His watch like everything else about him (including his cigar) is going at half-speed; so it records only 112 minutes elapsed when our signal arrives. The aviator knows, of course, that this is not the true time when our cigar was finished, and that he must correct for the time of transmission of the light-signal. He sets himself this problem—that man has travelled away from me at 161,000 miles a second for an unknown time minutes; he has then sent a signal which travels the same distance back at 186,000 miles a second; the total time is 112 minutes; problem, find . Answer, minutes. He therefore judges that our cigar lasted 60 minutes, or twice as long as his own. His cigar lasted 30 minutes by his watch (because the same retardation affects both watch and cigar); and that was in our opinion twice as long as ours, because his watch was going at half-speed.
Here is the full time-table.
| Stationary watch | Stationary Observer | Aviator | Aviator's watch |
|---|---|---|---|
| 0 min. | Lights cigar | Lights cigar | 0 min. |
| 30 min.„ | Finishes cigar | … | 15 min.„ |
| 60 min.„ | Inferred time aviator's cigar finished | Finishes cigar | 30 min.„ |
| 112 min.„ | Receives signal aviator's cigar finished | … | 56 min.„ |
| 120 min.„ | … | Inferred time stationary cigar finished | 60 min.„ |
| 224 min.„ | … | Receives signal stationary cigar finished | 112 min.„ |
This is analysed from our point of view, not the aviator's; because it makes out that he was wrong in his inference and we were right. But no one can tell which was really right.
The argument will repay a careful examination, and it will be recognised that the chief cause of the paradox is that we assume that we are at rest in the aether, whereas the aviator assumes that he is at rest. Consequently whereas in our opinion the light-signal is overtaking him at merely the difference between 186,000 and 161,000 miles a second, he considers that it is coming to him through the relatively stationary aether at the normal speed of light. It must be remembered that each observer is furnished with complete experimental evidence in support of his own assumption. If we suggest to the aviator that owing to his high velocity the relative speed of the wave overtaking him can only be 25,000 miles a second, he will reply "I have determined the velocity of the wave relatively to me by timing it as it passes two points in my conveyance; and it turns out to be 186,000 miles a second. So I know my correction for light-time is right."[2] His clocks and scales are all behaving in an extraordinary way from our point of view, so it is not surprising that he should arrive at a measure of the velocity of the overtaking wave which differs from ours; but there is no way of convincing him that our reckoning is preferable.
Although not a very practical problem, it is of interest to inquire what happens when the aviator's speed is still further increased and approximates to the velocity of light. Lengths in the direction of flight become smaller and smaller, until for the speed of light they shrink to zero. The aviator and the objects accompanying him shrink to two dimensions. We are saved the difficulty of imagining how the processes of life can go on in two dimensions, because nothing goes on. Time is arrested altogether. This is the description according to the terrestrial observer. The aviator himself detects nothing unusual; he does not perceive that he has stopped moving. He is merely waiting for the next instant to come before making the next movement; and the mere fact that time is arrested means that he does not perceive that the next instant is a long time coming.
It is a favourite device for bringing home the vast distances of the stars to imagine a voyage through space with the velocity of light. The youthful adventurer steps on to his magic carpet loaded with provisions for a century. He reaches his journey's end, say Arcturus, a decrepit centenarian. This is wrong. It is quite true that the journey would last something like a hundred years by terrestrial chronology; but the adventurer would arrive at his destination no more aged than when he started, and he would not have had time to think of eating. So long as he travels with the speed of light he has immortality and eternal youth. If in some way his motion were reversed so that he returned to the earth again, he would find that centuries had elapsed here, whilst he himself did not feel a day older—for him the voyage had lasted only an instant.[3]
Our reason for discussing at length the effects of these improbably high velocities is simply in order that we may speak of the results in terms of common experience; otherwise it would be necessary to use the terms of refined technical measurement. The relativist is sometimes suspected of an inordinate fondness for paradox; but that is rather a misunderstanding of his argument. The paradoxes exist when the new experimental discoveries are woven into the scheme of physics hitherto current, and the relativist is ready enough to point this out. But the conclusion he draws is that a revised scheme of physics is needed in which the new experimental results will find a natural place without paradox.
To sum up—on any planet moving with a great velocity through the aether, extraordinary changes of length of objects are continually occurring as they move about, and there is a slowing down of all natural processes as though time were retarded. These things cannot be perceived by anyone on the planet; but similar effects would be detected by any observer having a great velocity relative to the planet (who makes all allowances for the effect of the motion on the observations, but takes if for granted that he himself is at rest in the aether).[4] There is complete reciprocity so that each of two observers in relative motion will find the same strange phenomena occurring to the other; and there is nothing to help us to decide which is right.
I think that no one can contemplate these results without feeling that the whole strangeness must arise from something perverse and inappropriate in our ordinary point of view. Changes go on on a planet, all nicely balanced by adjustments of natural forces, in such a way that no one on the planet can possibly detect what is taking place. Can we seriously imagine that there is anything in the reality behind the phenomena, which reflects these changes? Is it not more probable that we ourselves introduce the complexity, because our method of description is not well-adapted to give a simple and natural statement of what is really occurring?
The search for a more appropriate apparatus of description leads us to the standpoint of relativity described in the next chapter. I draw a distinction between the principle and the standpoint of relativity. The principle of relativity is a statement of experimental fact, which may be right or wrong; the first part of it—the restricted principle—has already been enunciated. Its consequences can be deduced by mathematical reasoning, as in the case of any other scientific generalization. It postulates no particular mechanism of nature, and no particular view as to the meaning of time and space, though it may suggest theories on the subject. The only question is whether it is experimentally true or not.
The standpoint of relativity is of a different character. It asserts first that certain unproved hypotheses as to time and space have insensibly crept into current physical theories, and that these are the source of the difficulties described above. Now the most dangerous hypotheses are those which are tacit and unconscious. So the standpoint of relativity proposes tentatively to do without these hypotheses (not making any others in their place); and it discovers that they are quite unnecessary and are not supported by any known fact. This in itself appears to be sufficient justification for the standpoint. Even if at some future time facts should be discovered which confirm the rejected hypotheses, the relativist is not wrong in reserving them until they are required.
It is not our policy to take shelter in impregnable positions; and we shall not hesitate to draw reasonable conclusions as well as absolutely proved conclusions from the knowledge available. But to those who think that the relativity theory is a passing phase of scientific thought, which may be reversed in the light of future experimental discoveries, we would point out that, though like other theories it may be developed and corrected, there is a certain minimum statement possible which represents irreversible progress. Certain hypotheses enter into all physical descriptions and theories hitherto current, dating back in some cases for 2000 years, in other cases for 200 years. It can now be proved that these hypotheses have nothing to do with any phenomena yet observed, and do not afford explanations of any known fact. This is surely a discovery of the greatest importance—quite apart from any question as to whether the hypotheses are actually wrong.
I am not satisfied with the view so often expressed that the sole aim of scientific theory is "economy of thought." I cannot reject the hope that theory is by slow stages leading us nearer to the truth of things. But unless science is to degenerate into idle guessing, the test of value of any theory must be whether it expresses with as little redundancy as possible the facts which it is intended to cover. Accidental truth of a conclusion is no compensation for erroneous deduction.
The relativity standpoint is then a discarding of certain hypotheses, which are uncalled for by any known facts, and stand in the way of an understanding of the simplicity of nature.
- ↑ Appendix, Note 1.
- ↑ We need not stop to prove this directly. If the aviator could detect anything in his measurements inconsistent with the hypothesis that he was at rest in the aether (e.g. a difference of velocity of overtaking waves of light and waves meeting him) it would contradict the restricted principle of relativity.
- ↑ Since the earth is moving relatively to our adventurer with the velocity of light, we might be tempted to argue that from this point of view the terrestrial observer would have perpetual youth whilst the voyager grew older. Evidently, if they met again, they could disprove one or other of the two arguments. But in order to meet again the velocity of one of them must be reversed by supernatural means or by an intense gravitational force so that the conditions are not symmetrical and reciprocity does not apply. The argument given in the text appears to be the correct one.
- ↑ The last clause is perhaps unnecessary. The correction applied for light transmission will naturally be based on the observer's own experimental determination of the velocity of light. According to experiment the velocity of light relatively to him is apparently the same in all directions, and he will apply the corrections accordingly. This is equivalent to assuming that he is at rest in the aether; but he need not, and probably would not, make the assumption explicitly.