Relativity For All

8/14/2019 Relativity For All

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RELATIVITY* FOR ALL *HERBERT DINGLE

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RELATIVITY FOR ALL


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RELATIVITY FOR ALLBY

HERBERT PINGLE, B.Sc.LECTURER ON ASTROPHYSICS AT THE

IMPERIAL COLLEGE OF SCIENCE AND TECHNOLOGY

WITH TWO DIAGRAMS

BOSTONLITTLE, BROWN, AND COMPANY

1922


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PREFACEAN apology is needed for the production ofa popular work on Relativity after Einstein

himself has undertaken such a task. As amatter of fact, our purpose here is somewhatdifferent from Einstein's. This little book iswritten, not so much for those who wish to under-stand a physical theory from the point of view ofa physicist, as for the large body of intelligentmen and women who look on physics as one ofmany avenues into the secret of the Universe,and wish to know its windings in their relationto the larger field of human inquiry.The dominant aim throughout the book has

been to make the ideas definite and intelligibleto the ordinary mind. All other considerationsstrict philosophical phraseology, literary graces,conventional forms of presentation ; everything,in fact, but truth have been subordinated to


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vi RELATIVITY FOR ALLthis end. Between the Charybdis of inaccuracyand the Scylla of abstruseness, the course is narrowand the sea is rough. The vessel will hardlyescape bufferings from either side. It is hoped,nevertheless, that in the present voyage a passagewill be made without fatal mishap.Those who wish to pursue the subject more

deeply, from either the philosophical or thescientific standpoint, are recommended to theworks of Professor A. N. Whitehead, F.R.S. Theauthor is glad to acknowledge his deep indebted-ness to Professor Whitehead for invaluable helpand unwearying kindness in unveiling the mysteriesof a difficult subject.

H. DIMPERIAL COLLEGE OF SCIENCE

AND TECHNOLOGYJuly 1921


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CONTENTSPART ITHE FOUNDATIONS OF SCIENCE

CHAP. PAGBI. How THE THEORY AROSE . . i

II. SPACE, TIME, AND MATTER . . .10III. THE " FOUR-DIMENSIONAL CONTINUUM" . 22IV. THE VELOCITY OF LIGHT . . .32

PART IITHE LAWS OF NATURE

V. WHAT is A NATURAL LAW?. . . 39VI. THE WORK OF NEWTON . . .45

VII. RELATIVITY AND THE MOVEMENTS OFBODIES . . . . .53

VIII. SOME PROBLEMS OF RELATIVITY . . 63INDEX . . . . .71


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RELATIVITY FOR ALLPART I

THE FOUNDATIONS OF SCIENCECHAPTER I

HOW THE THEORY AROSE"Space is thought's, and the wonders thereof, and the

secret of space;Is thought not more than the thunders and lightnings ?

shall thought give place ?Tune, father of life, and more great than the life it begatand began,Earth's keeper and heaven's and their fate, lives, thinksand hath substance in man."

WHEN Swinburne wrote these words, hewas thinking what a wonderful beinghe was. That they would ever come tobe a poetical expression of cold, scientific ideasabout matter, time, and space, was probably thethought farthest from his inaccessible mind. Yetso it is. The new doctrine of Relativity entailsa complete uprooting of the conceptions that

I


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y FOR ALLhave formerly been held to lie inviolable at thefoundations of thought and experience. Thetheory is not merely a metaphysical speculation.It has arisen in order to explain certain facts ofobservation, which seem to point to it as the mostprobable statement of the nature of the Universewhich we perceive.Let us think for a moment of the way in whichwe are accustomed tacitly, almost subcon-

sciously to regard the physical world. We thinkof it as a number of pieces of matter. Thesepieces of matter exist in space. We do not gener-ally take the trouble to define to ourselves exactlywhat we mean by " space," but we understandone another quite well when we refer to it inconversation. It is a sort of receptacle, withoutlimit in any direction, in which the material of theworld exists and moves about. When we say acertain object is

"there," we have a clear idea ofwhat we mean, and we feel confident that, pro-

vided the object does not move, it will always be" there," no matter what we do ourselves. If wecould take the wings of the morning, and dwellin the uttermost parts of the sea, the object wouldstill be " there." Then we have also an idea oftime. We do not define this either, but we knowwhat it means. When something has happened,it belongs to the past. Nothing can ever bringthe same happening into the present or the futureagain. It happened at some definite time, and


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HOW THE THEORY AROSE 3every person in the Universe who observed itwould agree with every other person as to whatthat time was, supposing every one had an accurateclock.These three " things " matter, space, and timeare the three independent, immovable founda-

tion-stones of the World, as we are accustomed toregard it, and Science has hitherto adopted themas the only possible data in terms of which toexpress its discoveries. For instance, the law ofgravitation expresses the way in which matterwill move near other matter, i.e. it describes howthe position of matter in space changes as timeadvances. All other physical laws have beenessentially of the same kind.But recently, scientists have had reason toquestion whether space, time, and matter are reallythe absolute and fundamental things we havesupposed. The doubt arises in the followingway. As the result of a considerable accumula-tion of experience, it has been impressed uponphysicists that space is not empty, but is filled,in every nook and cranny of its infinite extent,with a kind of invisible, intangible super-matter,which has been called the " ether." The firstand still one of the most convincing of theindications of this substance came from the studyof the propagation of light through space. Cer-tain remarkable laboratory experiments seemedto assert that light could travel from a luminous


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4 RELATIVITY FOR ALLbody to the eye in no other form than that ofa train of waves, like the ripples spreading outin a pool of water when a stone is thrown into it.The facts pointed unanimously in this direction,and their combined force was almost irresistible.But waves are unthinkable without some mediumin which they exist. There must obviously besomething through which light waves travel :what is that something ? It is not the air, becauselight reaches us from stars millions of miles away,and there is, to the best of our knowledge, no airor matter of any kind reaching all the way fromthe stars to us. It must be something of whoseexistence we have not previously been aware ;something that fills all space, for light comes tous from all directions and from unimaginabledistances. It must, moreover, penetrate thepores and secret places of matter itself, for doesnot light pass through some bodies, which aresaid to be " transparent " ? Scientists, then,were led to the idea of a space filled with thisinfinite, all-permeating ether.Now there is nothing in all this to challengeour common sense. The conception of an omni-present ether offers no difficulties to the imagina-tion. Space might as well be full as empty, sofar as mere possibility is concerned. Neverthe-less, we should feel more satisfied on the matterif we had some direct sign of the ether's existence.An experiment giving immediate evidence of it


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HOW THE THEORY AROSE 5would be more convincing than its appearanceas the last link in a chain of reasoning. Thiswas recognized by physicists, and many attemptswere made to betray the ether into a declarationof its reality. One of the most promising of thesewas the search for the velocity of the earth as ittravels through the ether. Since tte ether filledall space, it had to be regarded as being at restas a whole. It could not move bodily becauseit was infinite and there was nothing for it to moveinto. Consequently, the velocity of the earththrough the ether could be looked upon as its" absolute" velocity something more funda-mental than its velocity of revolution round the \Sun, which ignores any possible motion of theSun itself.To understand the most famous of all experi-

ments made to measure this absolute velocity,we must picture the earth swimming through theether at some speed which we are to find out.Suppose that, on the earth's surface, and travellingwith it, there are two objects a lamp and amirror (A), represented in Fig. i. Suppose alsothat these objects he in the line of absolute motionof the earth whatever that may be the mirrorin advance of the lamp. Let the lamp be uncoveredfor an instant, so that it sends a beam towardsthe mirror (A). Now light travels with a definitevelocity (186,000 miles a second). It moves atthis speed through the ether towards the mirror (A).


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6 RELATIVITY FOR ALLBut the mirror (A) is running away from the beam,since it is fixed to the earth, and the earth is movingthrough the ether. Consequently, the light shouldtake longer to reach the mirror (A) than it wouldif the earth were not moving. On reaching themirror (A), the light is reflected back to the lamp.But now the lamp is moving to meet it, so that

Mirror (B)

< 7J> Lamp

Mirror (A)__Direction of Motion ofEarbh through Ether

FIG. i.

the light will make the return journey more quicklythan it would have done if the earth had beenat rest. It is a very simple matter to calculatewhat the time should be for the total journey toand fro, in terms of the unknown absolute velocityof the earth. But now suppose that, at the sametime as the beam of light left the lamp, another


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HOW THE THEORY AROSE 7beam left the same point at right angles to tlvefirst, towards the mirror (B). This beam wouldmove across the line of motion of the earth, andthe time it would take to perform its completejourney to the mirror (B) and back, can be cal-culated also. On making the calculations, wefind that the second beam should return to thelamp before the first, and we can tell exactly howmuch sooner it should arrive in terms, of course,of the unknown velocity. The interval shouldvary throughout the year, owing to the changein the direction and rate of motion of the earth.Now an experiment on these lines the famousMichelson-Morley experiment was actually per-formed in 1887. The apparatus used was sodelicate that it was capable of detecting a farsmaller quantity than that which it was expectedto measure. Every one awaited the result withconfidence. But when the experiment was made,it was found that the two beams arrived backat the same time. The apparatus was turnedround, so that the mirrors were in different positionsrelative to the lamp ; the experiment was repeatedat different times of the year ; but always theresult was the same the two beams took preciselythe same time for their respective journeys.Now it must be recognized at once that thiswas a most extraordinary thing. Here was anexperiment, performed with every care andapparently with full understanding of what was


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8 RELATIVITY FOR ALLbeing done, which completely failed to give theresult that common sense would have thoughtinevitable. For what the experiment seems toimply is this. We know that, if a bird flies fromone end of a train to the other, he will completethe journey sooner if the train is moving towardshim than he would do if it were at rest. Theexperiment suggests that, if he only moves asquickly as light, he will appear to the engine-driver to reach the end in the same time, no matterwhether the train is at rest, or moving towardshim, or moving away from him. It seems im-possible, but experience shows it to be true.If any explanation is to be given, therefore, itmust necessarily involve something revolutionary.Various suggestions were offered, but, in the lightof future investigations at any rate, none of themwas so satisfactory or far-reaching as the mostrevolutionary of all the principle of relativity.Let us ask ourselves why common sense saysthat the bird cannot reach the end of the trainin the same time, when the train is moving, as hedoes when it is at rest. We reply that he hasto travel different distances in space in the twocases. If, during his flight, the train has moved,its far end, when he reaches it, will be at a point inspace different from that which it occupied at thebeginning. The times taken by the two journeyswill therefore be different. But, in saying this,we are assuming that " space " and " time "


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HOW THE THEORY AROSE 9mean the same things for the engine-driver atrest as they do for the engine-driver in motion.What if they are different ? In that case, of course,we shall not know what to expect. If what oneman calls an hour, another calls a minute, andwhat the first pronounces a yard, the second assertsto be a mile and if there is no possible criterionfor testing their statements, so that both are equallyright or equally wrong then it will not be surprisingif results are obtained which would otherwisebe considered impossible. This is, in essence,just what the principle of relativity says. It\ideclares that the conceptions of space and time, I -

and, as will subsequently appear, of matter Ialso are not absolute and independent, but are Irelative to the observer. What do we mean by *this ? We will try to explain it in the next chapter.


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CHAPTER IISPACE, TIME, AND MATTER

SUPPOSEa being, endowed with full human

intelligence, but without any experience orknowledge of the world, were suddenly

created and placed, say, on Hampstead Heath :what would he perceive ? The answer we shouldnaturally give to this question is contained inthe first chapter he would perceive materialthings in space and time. The answer of therelativist, however, is different. According tohim, the man would perceive a number of happen-ings, occurrences, events. Their interpretation asmaterial objects in space and time would comelater, and would be the result of his intelligentordering of the events among themselves.

Let us take an example. Suppose our visitorsees a wasp alight on a flower. That is an event.Next, suppose the wasp alights on his hand.That is another event. We have here, then, twoevents, and to the man they would at first bemerely two events and nothing more. But now,suppose he begins to use his intelligence, and triesto impose some order or arrangement on the


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SPACE, TIME, AND MATTER 11circumstances in which he finds himself. Henotices that there is something common to thetwo events, and also to a number of intermediateevents, with which we need not concern ourselves.He has an impression of an "object" with blackand yellow bands, which characterizes the wholeseries of events from the wasp on the flower tothe wasp on the hand. This " character " of theevents he calls " matter/' and the particularexample with which we are dealing, a "wasp."He has now the first of the three entities whichwe supposed were his original perceptions matter.But that is not enough. If he confines himselfto what is common to the two events, he willnot be able to distinguish them, one from theother. He must construct some other relationbetween them. He does this by saying that theyare " in different places " : the flower is in one"

place," and the hand in another. In this wayhe forms an idea of place, and by extending thesame relation to other events which he perceives,he becomes conscious of " infinite space." Matter,space two types of relation between eventshave arisen as conceptions derived from a commonsource the events themselves.Are these conceptions sufficient to enable ourobserver to think clearly and to comprehend theworld around him ? Not quite : matter andspace will not relate all the events which he per-ceives. Consider a third event : suppose he feels


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12 RELATIVITY FOR ALLa stinging sensation in his hand. How can herelate this to the second of the events we havealready considered the arrival of the wasp onhis hand ? The space relations are the sameif we make the legitimate assumption that thereis no movement between the two events. Thematerial relations are the same the wasp andthe hand. He must find a third type of relation.He therefore says that one of the events occurs

." before " the other. By generalizing this rela-^ tion, he forms the conception of " time."Matter, space, and time, then, according to the

relativist, are types of relation between events.Together they appear to be capable of relatingthe whole of inanimate Nature in a consistentand orderly way. Our visitor employs them forpurposes of thought ; he hands them down tohis successors, generation after generation, until,ultimately, they come to be regarded as the funda-mental perceptions of the human mind, and thepoor event, the legitimate father of them all, sinksto the rank of a dependent.

This idea of the derivative character of matter,\ space, and time lies at the heart of the modernj principle of relativity. It deserves particularemphasis, for, if it is once firmly grasped, thegreater part of the difficulty of the subject dis-appears. It is the event that is the immediate^ entity of perception ; Nature is the sum-totalV of events, and every instrument of thought that


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SPACE, TIME, AND MATTER 13our minds employ can be traced back to its ultimateorigin in events. Two observers of Nature see,not necessarily the same matter, but the same r"events, because events finally constitute the ex-ternal physical world. What about the spatial,temporal, and material relations the observersimpose on the events : will they be the same ?Evidently it is not necessary that they should be.We have no right to say, without experimentaltest, that a man on the Earth and a man on Mars,say, who are moving relatively to one another,will both declare Regent Street to be half a milelong. What they may both be immediatelyaware of are the two events which are the existenceof the two ends of the street during their perceptionof them. If one man relates them spatially bysaying that they are half a mile apart, there isno fundamental necessity, so far as we know, forthe other man to do the same. It is essentiallya matter for experiment.

Let us illustrate this point, which is of basicimportance, by an example which, however,must not be pressed too close. Consider twoevents : first, a young man sees a young maiden ;second, he shows signs of agitation. Consider,further, two observers of these events the youngman himself and another young maiden. Each ofthem relates the events in a certain way. Theyoung man calls his relation, "love" ; the secondyoung maiden supposing she is honest with her-


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14 RELATIVITY FOR ALL "self calls her relation " jealousy." Here, then,we have a type of relation between events whichwe definitely recognize as relative* Why is thisso ? We do not know : it is a complete, mystery.But we certainly do know that the events arerelated differently by different persons. May not,then, the space and time relations also be relative ?" Ah ! " yo^i exclaim, " but the tyo' observers inyour example are in different circumstances.They have different predispositions, differenthistories, different emotional states. Conse-quently, their emotional relations between eventswill inevitably be different. But 'space' and timeare independent of our predispositions, ourhistories, our emotional states. Tittiey dre on anentirely different footing." That i^ quite true :space and time do not change with dyr emotions.But it does not follow that they do not changewith anything. Emotions are relative becausethey depend on our emotional state. Might notspace and time depend on our spatio-temporalstate ? Might they not be modified by motion,for example, i.e., by a change of our position inspace as our position in time advances ? It seemsto be possible, at any rate. If I am movingrelatively to you, it does not seem to be imperativethat my spatial and my temporal relations,between events that are observed by both of us,shall be the same as yours. "But," you reply,"it is idle to talk of what seems to be possible.


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SPACE, TIME, AND MATTER 15Is it not a fact that there is no such differencebetween us ? If my watch does not keep meansolar time when I am on an express train, have 'I not legitimate ground of complaint against mywatchmaker ? If the road becomes longer orshorter when I am travelling along it on a bus,shall not my habits justly be open to suspicion ?You may be right with your possibilities, butexperience shows that they are not actual."But then, after all, experience has its limitations.Perhaps, in a journey from London to Manchester,your watch keeps " perfect " time, so far as youcan judge : it may yet have varied by an amounttoo small for you to detect. If you could travelat the same rate for 100,000 years, or if youcould move at a speed of 100,000 miles a second,might not your extended experience show thatthe former conclusion was too hasty ? Experienceis certainly the final judge in the case before us,but it gives a verdict strictly according to thefacts in its possession. If we wish to get at thetruth, we must elicit all the facts.The question, then, comes down to this : Canwe make an experiment that will decide definitely ,

' whether space and time are different for observersin relative motion ? Such an experiment is notinconceivable, but it would be one of colossaldifficulty. At present, all we can hope to do isto see if there are any facts which receive a simpleexplanation if we assume the relativity of time


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16 RELATIVITY FOR ALLand space, but which can be interpreted onlywith difficulty, or not at all, if those relations areabsolute. Now facts of this kind are presentedto us in the Michelson-Morley experiment, and inother attempts to observe the drift of matterthrough ether. They are to be found also incertain astronomical observations, to which weshall refer later. The theory of relativity, in fact,

. has passed with honours every test it has so farbeen found possible to apply. It is only theabsence of direct experimental confirmation that

^prevents it from being recognized as a proven lawof the Universe.We shall consider the precise nature of therelativity in the next chapter, but it will be usefulto state at once the kind of effect the relativistrequires. The table opposite shows the loss ofa watch during one day, and the shortening of ai-foot rule in the direction of motion, as they wouldappear to an observer moving relatively to themat different speeds, who tests them by standardinstruments moving with him. 1

1 As we have said, the principle from which theseresults are obtained will be stated in the next chapter.The reader, however, may, at this point, be somewhatcurious as to its nature. We will, therefore, say inadvance that the theory of relativity assumes that it is

1 impossible for any observer ever to obtain experimentalevidence of relative motion between matter and ether.This means (cf. the Michelson-Morley experiment) thatall observers, whatever their state of relative motion, will


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SPACE, TIME, AND MATTER 1.7


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38 RELATIVITY FOR ALLIt is clear nowwhy the relativity of space and time,

if it is true, did not declare itself long ago. The effectis so small for speeds which we ordinarily use thatit is quite impossible to detect it. High velocities,or observations extending over long periods, arenecessary before its importance begins to appear.We have still to consider the nature of matter.Is that relative also ? If two of the primary rela-tions between events vary with the observer, itseems probable that the third will do so as well.As a matter of fact, the principle of relativityrequires that there shall be a change in matterarising from motion. The quantity of matter ina body (not its " size," which is an attribute ofspace, but the actual amount of matter in it ;what we call its " mass," and usually measure byweighing the body) should be different for differentobservers moving relatively to one another. Or,obtain the same measure of the velocity of light. Sucha result can only occur if the spaces and times used bythe observers are related in one particular way, which,granted the principle, is readily determined by mathe-matical calculation. In this manner the foregoing table hasbeen constructed. Thus, to a hypothetical observer onArcturus, a beam of light would travel i^oop- inch perfoot less than the same beam to an observer on the Earth.But the former observer would find that it did so in aperiod of time less than that measured by the Earth manby -g^th second per day. These numbers are such thatboth observers would obtain the same value for the velocityof the beam, and are the only ones that are consistent withthe application of the same principle to all possible relativevelocities.


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20 RELATIVITY FOR ALLprecision, or if we could compare the mass of abody at rest with its mass when moving at anextremely high speed. As with time and space,the demands of relativity are so humble atordinary velocities that they might be granted infull without giving us the slightest suspicion thatthey are made. As the body moves faster andfaster, the theory gathers confidence, and asks formore and more. Table 2 shows, for the samevelocities as before, the increase in mass of a bodycontaining i Ib. of matter when it is at rest withrespect to the observer.

TABLE 2.Speed. Increase of Mass.

60 miles per hour

67,000 miles per hour

700,000 miles per hour

93,000 miles per second .161,000 miles per second .186,000 miles per second .

250.000,000,000,000 Ib.

1,730,000

lhU*

Ib.

i Ib.

Infinitely great.

Can we test this by experiment ? It is ex-ceedingly difficult. We must rely again onindirect evidence. Nature has provided us withextremely small electrified particles which move


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SPACE, TIME, AND MATTER 21with enormous speed. It has been found possibleto measure their mass at different rates of motion.The result is that they actually do show a change'of mass with speed, of just the amount requiredby the theory. Moreover, theoretical resultsdeduced from the assumption of such a changewhen these bodies move inside material atoms,have been verified with almost incredible accuracy.The theory again has been successful in everytest to which it has been subjected. But wemust remark that these are highly special cases.The particles are electrically charged they arebelieved, in fact, to be particles of electricityitself and the effects observed can be explainedequally well on the ordinary electro-magnetic theory,without reference to relativity. We cannot, there-fore, put them forward as unequivocal evidence forrelativity. But we can say that, so far as experi-ment has yet gone, there is nothing that has putthe theory in the slightest difficulty, or necessitatedany modification of its fundamental principles.

Let us see, then, just where we stand. We havesaid that, since the only element in Nature is theevent, space, time, and matter may be relative tothe observer. We have considered experimentswhich seem to make it very probable that theyare relative. It remains for us now to investigatethe principles governing the magnitude of theirvariation with change of speed. It is to thisquestion that we turn in the next chapter.


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CHAPTER IIITHE "FOUR-DIMENSIONAL CON-

TINUUM"IT has already been pointed out and it isso important that it will bear repetitionthat the principle of relativity is a de-duction from facts of observation. It is emphati-cally not an arm-chair doctrine, proceeding fromthe inner recesses of the brain without referenceto the results of experience. When the modernrelativist says that space, time, and matter aredifferent ideas for different observers, he does sobecause he believes that the interpretation ofexperimental facts which thereby becomes pos-sible is the simplest and, on the whole, the mostplausible that he can devise. Consequently, heis not content with the mere statement that theserelations change with one's state of motion. Hemust say exactly by how much they change. Aswe implied in the first chapter, if a yard and anhour to one observer become a quite indiscrimi-nate length and time to another, anything mighthappen. But, actually, " anything " does nothappen : particular things happen for example,


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24 RELATIVITY FOR ALLthe velocity of light, the state of affairs wouldappear to be inconceivable.We shall return to this point in the nextchapter. For the moment we will direct ourattention to a very simple relation betweenthe space and time used by any one observera relation which summarizes the quantitativerequirements of the principle. It is not broughtout in the tables, and, as it is of very great im-portance, we will deal with it at length.

All who have tried to get at the meaning ofrelativity have, at one time or another, comeacross the blessed phrase, " the four-dimensionalcontinuum." What does it mean ? Let us .bequite clear, first of all, as to the meaning of theword " dimension." Suppose we have a room,enclosed by four square, vertical walls, and asquare floor and ceiling. Suppose an electriclamp globe is suspended somewhere in its interior.How can we describe to an architect, say theexact position of the globe in the room ? (Weare not concerning ourselves now with the questionof what we mean by " position," which occupiedus in the last chapter. We are using the word inits ordinary, everyday sense, just as we mighthave done if we had had no reason to doubt theabsolute nature of space.) We should tell himsomething about it if we said the globe was 7 feetabove the ground. But that would not be enough.If that were all, the globe might be anywhere in a


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" FOUR-DIMENSIONAL CONTINUUM " 25movable floor placed at that height. Suppose,then, we say, further, that it is 6 feet from thewall containing the door. That is better, butstill he is not satisfied. There may be a wholeline of objects on our imaginary movable floorwhich satisfy this condition, and the globe mightbe at any point in it. But let us now say thatit is 5 feet from the adjacent wall containing thewindow. We have then determined its positioncompletely. There is only one point in the linethat is 5 feet from the wall containing the window :or, in other words, there is only one point in theroom that is 7 feet above the ground, 6 feet fromone specified wall, and 5 feet from another. Wehave definitely fixed the position of the globe bythese measurements.Now there are other ways in which we couldhave done this, but, in all of them, three inde-pendent measurements are necessary and sufficient.This is expressed, in technical language, by sayingthat space has three dimensions. Another aspectof the same property of space is embodied in thestatement that three independent measurementsare necessary to calculate the spatial volumeoccupied by a body e.g. its length, breadth, andheight.Three independent measurements, we say, will

define the position of a body in relation to a *"'given structure (e.g. a room), and these threemeasurements may be made in different ways.


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" FOUR-DIMENSIONAL CONTINUUM " 27Now suppose as the result of an earthquake,say the room is twisted into the position shownby the dotted lines in the figure, in such a waythat the points A and B remain in exactly the samepositions as before. The distance AB will thenbe unaltered about loj feet. But the distancesof the lamp from the walls and floor will now bequite different. Yet and this is the point weare trying to illustrate provided the walls andfloor still remain at right angles to one another,those distances must be such that the sum of theirsquares is equal to no. Thus, if the lamp is 9 feetand 5 feet from the two walls, it must be 2 feetfrom the floor, for 92 + 52 + 22= no. It cannotpossibly be at any other distance, however theroom is twisted within the restrictions we havementioned.Now we need not have had the earthquake toalter the position of the room. We could haveimagined the walls and floor to be anywhere weliked, and defined the position of the globe relativeto the point B by reference to the imaginary room.Or, without supposing the room twisted at all,we could have fixed the position by three othermeasurements of a different kind. The essentialpoint is that, however we do it, there is a definiteway of combining the measurements so as to givethe number no, and the measurements are boundto be related among themselves so that the com-bination will give this number.


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" FOUR-DIMENSIONAL CONTINUUM " 29time, we must introduce something containing thetemporal quality. Actually, the perceived ex-istence of the corner and the lamp are events, forperception itself takes time. It will simplifymatters, however, if our data are more readilyrecognized as events. Let us consider the totalinterval (which we usually analyse into space in-terval and time interval) between the lightingof the lamp, A, and the arrival of a spider atthe corner, B. Suppose, first, that we are atrest in the room. Then the square of the spatialdistance between these two events is, as we haveseen, f + 62 + 52 , i.e. no. Suppose the time inter-val is 10 units ; l so that its square is 100. Thedifference between the squares of the space andtime intervals (which, we shall soon see, is animportant quantity) will then be 72 + 62 + 52 - io2=no - 100 = 10. Now consider how the eventswould appear to us if we were moving relativelyto the room. As we know, both the spaceinterval and the time interval would be different.Suppose our speed to be such as to give usa space interval of 9 feet between the events.What would the time interval be ? It is foundthat it must be just large enough for its square todiffer from the square of the space interval byexactly the same amount (namely, 10) as in theformer case. It must therefore be nearly 8J units,

1 The magnitude of a time unit for this purpose is onethousand-millionth of a second.


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30 RELATIVITY FOR ALLfor 92 -(8J)2 =io, very nearly. And, however wemove, this condition must always be satisfied.1And that is all that the four-dimensional con-tinuum means. There is nothing essentially

mysterious about it. The mental panic that itsometimes creates seems to arise from the completeillusion that there is actually a fourth dimensionin space a sort of additional direction of spatialextension, of which we could obtain experience ifwe possessed an additional sense. In reality, thefourth dimension does not exist, except as anacademic expression of our familiar experience oftime. Nothing exists fundamentally but events.It is not true to say that they take place in a four-dimensional continuum. Strictly speaking, theydo not take place at all : they simply exist in them-selves. We do not say that Nature " takes place,"and Nature is simply the aggregate of events. Thefour-dimensional continuum is merely the mathe-matician's shorthand way of saying, first, thatfour measurements are necessary to define thecomplete interval between two events, and second,

1 Strictly speaking, this particular combination (spaceinterval) 2 - (time interval) 2 is invariant only in freespace. In the neighbourhood of heavy bodies or, touse scientific language, in strong gravitational fields aslightly different combination must be taken. This willbe dealt with in a later chapter. In all circumstances,the general statement that there is a particular combina-tion of the four measurements that is constant for allobservers is strictly true.


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34 RELATIVITY FOR ALLrelativity are not to be believed. They may keepthe word of promise to our ear ; they will certainlybreak it to our hope. The " mass " that growswith velocity is not " size " ; the size of thematter, in fact, actually diminishes, owing to theshortening in the direction of motion. Whatmass really means (see Chapter VI) is inertia,or resistance of matter to change of motion. Sothat, all that Table 2 implies is that the faster abody moves past an observer, the more difficultdoes it become to increase the relative speed :the body itself gets smaller and smaller all thetime.From the practical point of view, then, any

anticipations of increased wealth or novel ordersof experience that the theory might have aroused,are likely to meet with disappointment. Itmight, however, be some consolation to the in-tellect to know that it is not called upon toconceive the possibility of such anticipations.But we have yet to indicate another property ofthis remarkable velocity of light : it is the highestvelocity that one body can have relative to an-other a natural maximum of speed, to exceedwhich is for ever impossible. Let us look oncemore at Table 2. At the velocity of light wesee that the mass of a body is " infinitely great."Remembering that " mass " means resistance tochange of motion, it follows that there is an in-finitely great resistance to the change of speed of


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36 RELATIVITY FOR ALLpoints out that when we say that two thingshappen

"at the same time," we may have ageneral idea of what we mean, but we should be

in difficulties if we were asked to give a rigidexplanation of the phrase. In the light of thetheory of relativity, he suggests a definition on thefollowing lines. Suppose we have two events,and an observer situated midway in his spatialdistance between them. Suppose light signalsreach the observer from the two events at thesame time. Then, according to Einstein, the twoevents themselves were simultaneous. To an-other observer, in motion relative to the first,the events would not necessarily be simultaneous,because the same light signals might reach him,at the mid-point of his spatial distance, at differenttimes. This, it should be noted, is not a thing tobe proved. It is a definition, not a proposition.If we were to try to test it, we should requiresome means of determining whether the twoevents actually were simultaneous, and we couldnot do this without knowing what the simul-taneity of events means, i.e., without falling backon the statement itself. Also, it is not a completephilosophical definition of simultaneity. It canonly be applied to events separated in space fromthe observer, and it assumes, moreover, that theobserver knows what he means when he says thelight signals reach him " at the same time."

This book is concerned solely with experimental


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42 RELATIVITY FOR ALLwe need, but there are tongues whose soundsthey certainly will not fit, and these tongues, aswell as ours, belong to Nature. We must goback to the primitive sounds for the expression ofnatural laws. We can afterwards spell them outin our own characters for our own special use,but the laws themselves must be universal.

In other words, the only possible terms for thestatement of a law of Nature are events. Anymateria-spatio-temporal statement that is peculiarto a particular observer, and has not its exactequivalent in the relations of other observers, isnot a natural law.

Relativity demands, therefore, a review ofexisting laws. Only those which can be generalizedin the way we have suggested can survive ; theothers must be restated. Since it is the inevit-able custom of physicists to express their con-clusions in mathematical form, the test must bea mathematical one. We therefore pass over itsdetails, which are of interest only to the specialist.The general idea, however, we hope has beenmade clear.The new point of view is of especial interest

because it suggests the possibility of a morecomplete unification of Nature than any previouslyimagined. With one hand relativity destroysthe throne of matter and motion : with the otherit erects an altar to the event. Matter, space,and time, even if they could have explained


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WHAT IS A NATURAL LAW ? 43everything in Nature, were, after all, three in-dependent things : the event is one thing. But,as we have seen, the three things failed : can theone thing succeed ? It has a better chance, forthe following reason. Matter, space, and time,when they were thought to be fundamental,were, on that very account, incapable of modifica-tion to meet new discoveries. If they did theunexpected, it was not they, but something else,that was responsible. They were absolute, wrappedin immutability as in an impenetrable garment.Force had to be invented ; electricity, magnetismwere postulated all because matter, space, andtime were held to be above caprice. But if westart with the event, there is only one deity onour Olympus. Matter, space, and time are hisbroken lights, and there is no sacrilege in suppos-ing them liable to change. Consequently, whatwe formerly attributed to force might perhaps bederived from a modification of space or time ormatter in which case, force can be dispensedwith. Possibly, also, the other extraneous entitiesthat we have called into being might be treatedin the same way.

It should particularly be noted that it is notsufficient merely to conceive an idea of this kind.It must be something more than a philosophicalpossibility before it can apply for recognition asa scientific hypothesis. A particular modifica-tion, that will tally exactly with experiment, must


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CHAPTER VITHE WORK OF NEWTON

WHEN Newton began his work, the timewas ripe for a great generalization.Galileo had studied the manner inwhich bodies on the earth's surface fell to theground, and had shown how the distances fallenthrough increased with the time of flight. Keplerhad succeeded, after many failures, in expressing,in three famous laws, how the planets moved roundthe Sun. The manner in which different bodiesmoved was known with great exactitude. Whatwas lacking was a co-ordination between the fallof a body to the Earth and the journey of a planetround the Sun.

It was not obvious to all that such a co-ordina-tion was necessary. One could regard the ellipticmotion of the Earth and the linear motion of afalling stone as distinct phenomena. It wasnatural, said some, for the Earth to move in anellipse, and it was natural for the stone to fall ina straight line. To attempt to get behind thesefacts was absurd. They were ultimate facts ofNature.

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46 RELATIVITY FOR ALLNewton, however, regarded them in a different

light. Matter was matter, and if it moved in acertain way in one set of circumstances, and in adifferent way in another set, the reason must besought in the circumstances, and not placed tothe account of the moving bodies. We mustremember that Newton thought in terms ofabsolute space, time, and matter.From this point of view he was faced with adouble problem. He had, first of all, to determinewhat was the natural tendency of matter in nocircumstances at all, i.e., when it was entirelyuninfluenced by anything outside itself ; secondly,he had to consider the effect on this tendency ofthe various circumstances occurring in Nature.For the second task he was guided by observedfacts : the effect must be such as to produce thephenomena we actually find. But for the firsthe had to fall back on his own inspired imagina-tion. There was no experience to guide him,because he could never be quite sure that anymovements he observed were free from externalinfluence.

In making his fundamental assumption, Newtontook as his starting-point the essential deadness,or " inertia," of matter. He put forward thehypothesis that matter by itself could do nothingto change its state of motion. If it was at rest,it would remain at rest until something moved it.If it was moving, it would continue to move, in


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48 RELATIVITY FOR ALLwhich it changes motion. But he recognized thathe would probably not achieve much success witha definition of this kind, unless he understood bymotion something more than mere velocity. Theidea that the agency producing a given change ofvelocity was of the same magnitude, no matterwhat was the bulk of the moving body, did notappear very promising as the basis of a universallaw. He therefore defined the quantity of motiona body as the product of its "mass" and itsvelocity. It followed that the change of velocityproduced in a body by a given force was greater,the smaller the mass of the body, the change ofmotion being the same in both cases.

It was in this way that the idea of mass firstbecame definite in physics. From its derivation,its meaning is simply inertia, or resistance tochange of velocity. Since inertia istaken tp_j2-the fundamental property of matterTtEe mass ofa body can also be interpreted as the quantityof matter in it. The motion of the body, then,according to Newton, arises from the quantity ofmatter it contains and the velocity with which itis moving. A change in either of these things willalter the motion and reveal the existence of aforce.

Later experiments showed that, to the degreeof accuracy attainable, the mass of a body nevervaried. If disintegration took place, the surr^of the masses of the various parts was always


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THE WORK OF NEWTON 49exactly equal to the mass of the original body,whatever treatment the body or its parts received.In this way the idea of matter became absolute.Change of motion was due entirely to change of'velocity, the mass remaining constant all thetime. Newton's law of force, therefore, amountedto a statement that the force acting on a body wasequal to the product of the mass and the changeof velocity (or, the " acceleration ") which itproduced.Newton had now provided himself with generallaws expressing the possible motions of matter.He had next to apply them to the facts of Nature,and see if it were possible to devise a single forcethat would give rise to the varied motions actuallyobserved. The problem was no easy one ; itrequired a Newton to solve it. A stone fell in aconstant direction, but with varying speed ; theEarth revolved with almost uniform speed, butwith continually changing direction. Moreover,a heavy body fell from a given height to the Earthin the same time as a light one. All these experi-mental facts had to be the inevitable result ofone simple hypothesis.As every one knows, Newton was almost com-pletely successful. He assumed that, between

every two pieces of matter in the Universeor, at any rate, in our own Solar System thereexisted a force of attraction (gravitation) whichwas proportional to the product of the masses of

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THE WORK OF NEWTON 51movement can be said to come within the scopeof gravitation. Electricity, magnetism, radia-tion all have had to be recognized as origins offorce. Nevertheless, observations have been con-sistent with the idea that the forces they produceare in accordance with Newton's definition.Almost every observed change of motion inNature can be explained as the result of a New-tonian force, arising from particular physicalconditions. But there are one or two that havedefied such explanation. They are exceedinglysmall so small, in fact, that one might be in-clined at first to neglect them. But astronomicalobservations are very exact, and they leave nodoubt at all that there are motions in the Solar ^System that so far it has not been possibleto bring under the sway of Newton's laws. Oneof the most important of these is exhibited bythe planet Mercury the nearest to the Sun of all /the planets yet discovered. Mercury, like all theSun's satellites, revolves round its primary in anellipse. There is one point in its orbit (its" perihelion ") which is nearer to the Sun thanany other. Now Mercury is, of course, attractedby the other planets as well as by the Sun, andcalculation shows that, as a result, its perihelionshould gradually change its position in space(the absolute space of the Newtonian system).Mercury has been under observation for manyyears, and it is found that the position of the


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52 RELATIVITY FOR ALLperihelion does change, but npt by quite the sameamount as the calculations require. The differ-ence in one century amounts only to the apparentlength of a i-foot rule placed one mile away, butthis is much greater than the possible errors ofobservation. It must have some cause not yetrevealed, or else the Newtonian laws are not quiteexact. Until the advent of the theory of rela-tivity, it can be said that there was no explana-tion of this phenomenon.


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THE MOVEMENTS OF BODIES 55Now, in dealing with events, we must make

use of the only relation among them known tous so far ; namely, that their complete separationwhat we have called, for convenience, the in-terval between them in the four-dimensionalcontinuum is constant for all observers. Thisseparation, as we have seen, is obtained by sub-tracting the square of the time interval from thesquare of the space interval as measured by thesame observer, and finding the square root ofthe difference.The complete separation, we say, is constant

for all observers. But that does not tell us any-thing about the course of Nature. It does nottell us why (speaking in our own terms, for brevity)the Moon should travel from its position on the8th February to its position on the 22nd Februaryin an ellipse, as seen from the Earth, and not ina straight line or a circle. If it moved in eitherof these paths, the complete separation betweenthe two events which we describe as its positionson the dates given, would still have the samevalue for all observers, though a different valuefrom that which it actually has. Clearly, to obtaina law of Nature we must make some hypothesisas to the actual value of the interval betweenevents.

Einstein assumed Nature to be such that thetotal four-dimensional interval between any twoevents, when computed from event to event


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56 RELATIVITY FOR ALLalong the actual succession, has a maximum value.That is to say to refer to our example againif the Moon moved in any path slightly differentfrom that which it chooses, then the total intervalbetween the two events which we call its positionson the 8th February and the 22nd Februarywould be smaller than it is. This will probablyappear very surprising at first, because, accustomedas we are to intervals in space, it seems incon-ceivable that there can be a path which has nota slightly greater neighbouring one. But wemust note that the interval with which we aredealing is to be calculated in the hypotheticalfour-dimensional continuum, and not in space.Referring to Chapter III, we see that it dependson the difference between the squares of the spaceand time intervals. This difference can be in-creased by diminishing the time interval as wellas by augmenting the space interval, so that weshall be getting nearer to the idea of Einsteinif we think of the actual path of the Moon asbeing that in which it can cover the greatestspatial distance in the shortest time.

It must be recognized that Einstein's hypothesiswas a guess, though an inspired one. It dependedfor its justification on its ability to explain thefacts of observation. It is very like Newton'sguess about the movement of matter in the freestate. Newton assumed that free matter wouldmove in a straight line, i.e. that it would take


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THE MOVEMENTS OF BODIES 57the minimum spatial distance between any twopoints in its path. Einstein assumed that anactual event would be separated from its neigh-bour by the maximum four-dimensional distance.There is this important difference, however,between the two assumptions. Newton wasthinking of an ideal case, which hardly occursin Nature, for matter is never quite free. Toaccount for actual motions he had to introduceforce. But Einstein dealt with actual events.If his assumption was successful, he would there-fore have no need of force or any agency at alloutside the events themselves.Now this assumption of Einstein's can be

tested, for, from our knowledge of the Moon'spath, for instance, we can calculate the four-dimensional interval between any two events init, and compare the result with the intervalbetween the same two events, supposing theMoon had travelled in a slightly different path ;i.e. supposing the series of events we call theMoon's successive positions in space had beenslightly different from what they are. This hasbeen done, using the geometry of Euclid in thecalculation. The result shows that the actualpath does not give the maximum four-dimensionalinterval.We are faced, then, with two possibilities.Either Einstein's assumption is contrary to Nature,or else the definitions and axioms of Euclid are


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THE MOVEMENTS OF BODIES 59lead us to expect some, at least, of the propositionsof Euclid to be falsified by exact measurements /,in space. For example, the sum of the threeangles of a triangle might not be exactly equalto two right angles.The test seems an easy one, but, as before, weare baffled by the extreme minuteness of thecrucial effect. The differences between practi-cable measurements in the space of Euclid and inthe space which must be assumed in order tojustify the hypothesis of Einstein, are beyondthe power of existing instruments to detect. Ouronly hope at present, at least is to assumeEinstein's hypothesis, and see if the consequencesagree with fact better than those of any otherassumption.The result of investigations of this kind hasbeen all in favour of Einstein. There are twopoints on which the Einstein and the Newtoniantheories give definitely conflicting results. Thefirst is connected with the orbit of the planet ^Mercury. The Newtonian laws, as we have seen, J\leave a small movement of the perihelion of thisplanet unexplained. Einstein's modified assump-tion gives a motion equal to that observed, withinthe limits of experimental error. It does not needany additional modification for this explanation,nor was the hypothesis constructed for the pur-pose of explaining the motion. The result followsdirectly from the assumption that Mercury moves


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60 RELATIVITY FOR ALLin the maximum four-dimensional path, and theconsequent supposition of non-Euclidean space.This is the only explanation of the phenomenonthat has not been devised ad hoc and foundinapplicable to, or conflicting with, other ob-servations.The second point at issue between the Einstein^Vand Newtonian theories involves observations

/ that had not been made previous to the formula-tion of the principle of relativity. A ray of light,passing close to a heavy body, should, on Einstein'sassumption, suffer a slight change of direction, asif it were pulled towards the body. Accordingto Newton's

principles,there seems to be no

reason why the light should be bent at all. It ispossible, however, that light possesses the equi-valent of weight in a material body, and, if so,the gravitational force should cause a bendingsimilar to that predicted by the theory of rela-tivity, but of only about half the amount. Thetwo theories are therefore definitely at variancein their predictions, and an experimental testbecomes possible. This test was made at thetotal solar eclipse of 2Qth May 1919. The heavybody chosen was the Sun, and the light examinedwas that emitted by stars which were almostdirectly behind the Sun as seen from the Earth.During the eclipse the sunlight was extinguished,and the stars became visible, apparently veryclose to the obscured Sun. Now these stars


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62 RELATIVITY FOR ALL

>formto the definitions and axioms of Euclid,

explains all the movements of bodies that areaccounted for by the Newtonian law of gravitation.In addition, it explains a movement of the planetMercury that stands outside the Newtonian law,and it has predicted the true path of a ray oflight, which no other theory seems able to do.It requires the supposition of no imaginary exist-ences, such as force, but proceeds entirely andcompletely from a single hypothesis as to theassociation of events. There is no known pheno-menon with which it is at variance.


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CHAPTER VIIISOME PROBLEMS OF RELATIVITYA GREAT idea invariably creates as manyproblems as it solves : that is a sign ofits greatness. The thoughtful student

will not be baffled by its novelty or lose himselfin its details. He will patiently probe it to thecore, and lay bare to his mind its inner meaningand its relation to the world of experience. Andin so doing he will meet with difficulties not,perhaps, the difficulties that are dealt with inbooks, for they were the authors', and may notbe his own. The attainment of a new point ofview is hindered, not so much by the roughnessof the road as by the tendency to return to theold standpoint. It is our prepossessions that holdus back, drawing us, like a magnet, to themselves.Each of us has his own standpoint and prejudices ;each will have his own difficulties.

It may be said of the principle of relativity that,for the most part, it offers the same problems toall plain men. Its point of view is so remotefrom that to which most of us are accustomedthat we are relatively together, and approach it63


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ALLalong parallel roads. This, our concluding chapter,is devoted to a few general comments on some of themore prominent difficulties that are our commonlot. It makes no claim to be exhaustive orfinal ; its sole purpose is to help.

It is important that we should recognize thatthe principle of relativity is not a complex, fantastictheory a sort of last hope, called in to save thehuman mind from defeat by the manoeuvres ofphenomena. It is, on the contrary, a straight-forward attack on the problems of Nature, anattempt to see them as they are. It is a questafter the simple. It is inevitable that it shouldpursue its ends at the expense of plausibility.We are accustomed to the complex. We think interms of three things matter, space, and timeand we are so much at home with them that wedo not, perhaps, give our race full credit for theconsistency and success with which it has appliedthem to the interpretation of the Universe. Jug-gling with three balls is not an easy matter, andwe have done it almost to perfection. It is notsurprising that, when we are left with one, we areat a loss to know how to perform our tricks. Butlet u? once clearly realize the conditions; let ussuitably arrange our mirrors to make up for thelost balls, and we shall find our repertoire augmentedand our own effort simplified. That is, in essence,the central meaning of relativity. It takes us


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66 .. i i. itELOTjtVITY. FOR ALL

,j

or

possesses some physical property, other than thepower to transmit waves, for which no compensa-tion is made by the relativity transformations.The essence of relativity is the universal character

the event, and the subordination of space, time,and matter. A subordinate ether in additionwould appear not to be inconsistent with this.The ether may, at any rate, have an existence asreal as that of matter, and that is all that isdemanded of it by the physical facts which calledit into existence.

We have already dealt more than once with theidea that relativity is not a physical theory, buta metaphysical one. This book has failed in itspurpose if it has not made it clear that the entiretheory is built up with the one object of accountingfor actual physical facts, and stands or falls at thedictate of experiment. The theory of relativityis no more metaphysical than the wave theory oflight, or Newton's laws of motion. It partakesof their nature, and is vulnerable to the sameweapons. It appears, perhaps, at first blush to bemetaphysical, because it deals with the nature ofmatter, space, and time, which are part of theplayground of metaphysics. But it is with thesethings as objective entities that relativity isconcerned. Its assertions about them are sus-ceptible to actual physical tests with clocks,scales, and balances. They may be considered


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SOME PROBLEMS OF RELATIVITY 67metaphysically, it is true, but relativity doesnot so consider them. Love is fair game forthe metaphysician, but we do not fall in lovemetaphysically. It is essentially a matter ofexperience.

But perhaps the greatest difficulty of relativityis presented to the imagination. The conse-quences of the theory are so extraordinary thatwe cannot picture them. It seems impossible,for instance, that the order of events in time canbe different for different persons. We must re-member, however, that relativity does not entailanything that is contrary to experience. Itwould have no right to exist if that were so.Its predictions, that appear to us so strange, arerelated to matters as yet outside our experience,about which we can only form conjectures.Everything that we come across in our everydaylife is left untouched. Space, time, and matterhave an absolute meaning for observers relativelyat rest. For them there is a real and definitemeaning in the statements that a man is 6 feethigh, that is it nine o'clock, and that sugar is 8d.a pound. With all terrestrial movements, even,the statements are, for practical purposes, exact.They are quite as true as the statement that, insunlight, grass is green. It is only when weget into conditions that are far beyond our commonexperience that the reasoned effects belie our


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68 RELATIVITY FOR ALLanticipations. With a Sun composed of sodium,grass would no longer be green, and space, time,and matter, in appropriate circumstances, suffera corresponding change. It is not true to saythat relativity is revolting to our experience. Allwe can say is that we did not expect it. But,after all, Nature has nothing to do with ourexpectations.It is well with Science when reason and imagina-tion go hand in hand. One assists the other,and the mind is satisfied. But the history ofScience shows that it is not in this way that thegreatest advances have been made. The imagina-tion of Faraday saw the electric field threadedby "lines of force," to which bodies were harnessedand in obedience to which they moved in theircourses. But the mind was baffled : reasoncould find no lines of force until Clerk-Maxwellbrought them within its light. In the words ofHelmholtz : " Now that the mathematical inter-pretation of Faraday's conceptions regarding thenature of electric and magnetic forces has beengiven by Clerk-Maxwell, we see how great adegree of exactness and precision was reallyhidden behind the words, which to Faraday'scontemporaries appeared either vague or ob-scure. ... I confess that many times I havemyself sat hopelessly looking upon some para-graph of Faraday's descriptions of lines of force,or of the galvanic current being an axis of power."


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SOME PROBLEMS OF RELATIVITY 69Would it not have been unwise to restrain theimagination of Faraday because reason could notfollow it ?Or to take an example more closely allied to

the present discussion think of the dawn of theidea that the Earth was round and rotated on anaxis. Reason demanded it; there was no Botherexplanation of facts. But where was the imagina-tion ? Was it conceivable that we were whirlingthrough space with breathless speed, and yet wereinsensible of the motion ? Could one really be-lieve that there were people on the Earth whowere upside down and yet did not fall off ? Itwas impossible ; common sense scorned the sugges-tion. Yet, would progress have been possible ifwe had listened to common sense and silenced thevoice of reason ?We are in much the same position to-day.Can we not take heart from the experience of thepast ? To-day the dullest schoolboy knows ourplace in the Solar System, and finds it not hardto accept. Surely it is not impossible that theparadox of relativity will one day become a partof our common knowledge and fashion our viewof Nature. Reason calls for it : if it is true, theimagination will not be left far behind.

Meanwhile, reason will march along and explorefresh country, and we can at least meditate on itsconquests. Whatever our attitude towards themmay be, we must recognize that the world is a far


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70 RELATIVITY FOR ALLmore wonderful thing than we have ever imagined.Conceptions which we thought were universal ap-pear as merely one set of an infinite number ofpossible conceptions. Our idea of Nature con-sistent though it has been, and, therefore, tosome extent, true is yet not the whole truth.We have " swayed about upon a rocking-horse,and thought it Pegasus." The theory of relativitydoes not countenance the bombast of Swinburne,with which we opened : it should make us veryhumble.

Finally, we must not imagine that the theory ofrelativity is complete and self-sufficient. Ratheris it the beginning of a new chapter in Science.Some pages of that chapter have already beenwritten, and the work is even now in progress.Electro-magnetism is being examined ; the natureof space, and its extent, are being subjected tosearching inquiry. We have tried, in this book,to indicate the view-point ; the view we must/eave till our eyes are adjusted to the new dis-tances. But the view is there, and the reward ofour labours will be limited only by the qualityof our vision.


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INDEXAcceleration, 49Astronomy, 16, 51Atoms, 21Chemistry, 19Clerk-Maxwell, 68Conservation of mass, 19, 48Dimension, 24Earth, the, 5, 45, 49, 50, 54,

55, 60, 6 1, 69Eclipse, solar, 60Economics, 33Einstein, v, 35, 36, 55 et seq.Electricity, 21, 40, 41, 43, 51,68Electro-magnetism, 21, 44,

70Emotions, 14Ether, the, 3, 40, 65Euclid, 57 et seq., 62Events, 10, 28, 30, 42, 54, 57,62,66Faraday, 68, 69Force, 40, 43, 47 et seq., 62" Four - Dimensional Con-

tinuum," 22, 24, 28, 30, 55,56Galileo, 45Geometry, 57Gravitation, 3, 30, 40, 44, 49,60

Helmholtz. 68History, 33Imagination, 67Inertia, 34, 46, 48Interval, total, 29, 30, 55,

58, 60Kepler, 45Lavoisier, 19Laws, natural, 3, 19, 39, 49,Light, 3, 5, 1 8, 32, 36, 37, 40,

60, 61, 62, 66Lines of force, 68Magnetism, 43, 51, 68Mass, 18, 19, 34, 48Materialism, 40, 41Mathematics, 18, 19, 23, 42,61Matter, i, 10, n, 18, 22, 39,

4i 46, 53, 57, 64Mercury, 51, 59, 62Metaphysics, 2, 37, 66Michelson - Morley experi-

ment, 5, 7, 16, 19, 23, 35Moon, the, 54, 55, 57Motion, 8, 1 6, 18, 22, 29, 34,40, 41, 45, 46, 66

Newton, 40, 44, 45 et seq., 53,54. 56, 59, 62, 66Non-Euclidean space, 58, 61


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Relativity For All

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