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diff --git a/old/67104-0.txt b/old/67104-0.txt deleted file mode 100644 index b13edb1..0000000 --- a/old/67104-0.txt +++ /dev/null @@ -1,4798 +0,0 @@ -The Project Gutenberg eBook of The A B C of Relativity, by Bertrand -Russell - -This eBook is for the use of anyone anywhere in the United States and -most other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms -of the Project Gutenberg License included with this eBook or online at -www.gutenberg.org. If you are not located in the United States, you -will have to check the laws of the country where you are located before -using this eBook. - -Title: The A B C of Relativity - -Author: Bertrand Russell - -Release Date: January 4, 2022 [eBook #67104] - -Language: English - -Produced by: Tim Lindell and the Online Distributed Proofreading Team at - https://www.pgdp.net (This book was produced from images - made available by the HathiTrust Digital Library.) - -*** START OF THE PROJECT GUTENBERG EBOOK THE A B C OF RELATIVITY *** - - - - - -Transcriber’s Notes: - -Underscores “_” before and after a word or phrase indicate _italics_ -in the original text. Equal signs “=” before and after a word or -phrase indicate =bold= in the original text. Small capitals have been -converted to SOLID capitals. Illustrations have been moved so they do -not break up paragraphs. Typographical and punctuation errors have been -silently corrected. - - - - - HARPER’S MODERN SCIENCE SERIES - - THE A B C OF RELATIVITY - - BY - BERTRAND RUSSELL - - AUTHOR OF - “THE PRINCIPLES OF MATHEMATICS” - “PROPOSED ROADS TO FREEDOM” - AND “WHY MEN FIGHT” - - PUBLISHERS - HARPER & BROTHERS - NEW YORK AND LONDON - - THE A B C OF RELATIVITY - - Copyright, 1925, by Harper & Brothers - Printed in the United States of America - - - - -_Contents_ - - - CHAPTER PAGE - I. TOUCH AND SIGHT: THE EARTH AND THE HEAVENS 1 - II. WHAT HAPPENS AND WHAT IS OBSERVED 14 - III. THE VELOCITY OF LIGHT 28 - IV. CLOCKS AND FOOT RULES 43 - V. SPACE-TIME 58 - VI. THE SPECIAL THEORY OF RELATIVITY 71 - VII. INTERVALS IN SPACE-TIME 91 - VIII. EINSTEIN’S LAW OF GRAVITATION 111 - IX. PROOFS OF EINSTEIN’S LAW OF GRAVITATION 131 - X. MASS, MOMENTUM, ENERGY AND ACTION 144 - XI. IS THE UNIVERSE FINITE? 163 - XII. CONVENTIONS AND NATURAL LAWS 177 - XIII. THE ABOLITION OF “FORCE” 192 - XIV. WHAT IS MATTER? 206 - XV. PHILOSOPHICAL CONSEQUENCES 219 - - - - -THE A B C OF RELATIVITY - - - - -CHAPTER ONE: TOUCH AND SIGHT: THE EARTH AND THE HEAVENS - - -Everybody knows that Einstein has done something astonishing, but -very few people know exactly what it is that he has done. It is -generally recognized that he has revolutionized our conception of the -physical world, but his new conceptions are wrapped up in mathematical -technicalities. It is true that there are innumerable popular -accounts of the theory of relativity, but they generally cease to -be intelligible just at the point where they begin to say something -important. The authors are hardly to blame for this. Many of the new -ideas can be expressed in non-mathematical language, but they are none -the less difficult on that account. What is demanded is a change in -our imaginative picture of the world—a picture which has been handed -down from remote, perhaps pre-human, ancestors, and has been learned -by each one of us in early childhood. A change in our imagination is -always difficult, especially when we are no longer young. The same sort -of change was demanded by Copernicus, when he taught that the earth -is not stationary and the heavens do not revolve about it once a day. -To us now there is no difficulty in this idea, because we learned it -before our mental habits had become fixed. Einstein’s ideas, similarly, -will seem easy to a generation which has grown up with them; but for -our generation a certain effort of imaginative reconstruction is -unavoidable. - -In exploring the surface of the earth, we make use of all our senses, -more particularly of the senses of touch and sight. In measuring -lengths, parts of the human body are employed in pre-scientific -ages: a “foot,” a “cubit,” a “span” are defined in this way. For -longer distances, we think of the time it takes to walk from one -place to another. We gradually learn to judge distances roughly by -the eye, but we rely upon touch for accuracy. Moreover it is touch -that gives us our sense of “reality.” Some things cannot be touched: -rainbows, reflections in looking-glasses, and so on. These things -puzzle children, whose metaphysical speculations are arrested by the -information that what is in the looking glass is not “real.” Macbeth’s -dagger was unreal because it was not “sensible to feeling as to sight.” -Not only our geometry and physics, but our whole conception of what -exists outside us, is based upon the sense of touch. We carry this even -into our metaphors: a good speech is “solid,“ a bad speech is “gas,” -because we feel that a gas is not quite “real.” - -In studying the heavens, we are debarred from all senses except sight. -We cannot touch the sun, or travel to it; we cannot walk round the -moon, or apply a foot rule to the Pleiades. Nevertheless, astronomers -have unhesitatingly applied the geometry and physics which they found -serviceable on the surface of the earth, and which they had based -upon touch and travel. In doing so, they brought down trouble on -their heads, which it has been left for Einstein to clear up. It has -turned out that much of what we learned from the sense of touch was -unscientific prejudice, which must be rejected if we are to have a true -picture of the world. - -An illustration may help us to understand how much is impossible to -the astronomer as compared to the man who is interested in things on -the surface of the earth. Let us suppose that a drug is administered -to you which makes you temporarily unconscious, and that when you -wake you have lost your memory but not your reasoning powers. Let us -suppose further that while you were unconscious you were carried into -a balloon, which, when you come to, is sailing with the wind in a dark -night—the night of the fifth of November if you are in England, or of -the fourth of July if you are in America. You can see fireworks which -are being sent off from the ground, from trains, and from aeroplanes -traveling in all directions, but you cannot see the ground or the -trains or the aeroplanes be cause of the darkness. What sort of picture -of the world will you form? You will think that nothing is permanent: -there are only brief flashes of light, which, during their short -existence, travel through the void in the most various and bizarre -curves. You cannot touch these flashes of light, you can only see them. -Obviously your geometry and your physics and your metaphysics will be -quite different from those of ordinary mortals. If an ordinary mortal -is with you in the balloon, you will find his speech unintelligible. -But if Einstein is with you, you will understand him more easily than -the ordinary mortal would, because you will be free from a host of -preconceptions which prevent most people from understanding him. - -The theory of relativity depends, to a considerable extent, upon -getting rid of notions which are useful in ordinary life but not to -our drugged balloonist. Circumstances on the surface of the earth, -for various more or less accidental reasons, suggest conceptions -which turn out to be inaccurate, although they have come to seem like -necessities of thought. The most important of these circumstances is -that most objects on the earth’s surface are fairly persistent and -nearly stationary from a terrestrial point of view. If this were not -the case, the idea of going a journey would not seem so definite as it -does. If you want to travel from King’s Cross to Edinburgh, you know -that you will find King’s Cross where it always has been, that the -railway line will take the course that it did when you last made the -journey, and that Waverley Station in Edinburgh will not have walked up -to the Castle. You therefore say and think that you have traveled to -Edinburgh, not that Edinburgh has traveled to you, though the latter -statement would be just as accurate. The success of this common sense -point of view depends upon a number of things which are really of the -nature of luck. Suppose all the houses in London were perpetually -moving about, like a swarm of bees; suppose railways moved and changed -their shapes like avalanches; and finally suppose that material objects -were perpetually being formed and dissolved like clouds. There is -nothing impossible in these suppositions: something like them must have -been verified when the earth was hotter than it is now. But obviously -what we call a journey to Edinburgh would have no meaning in such a -world. You would begin, no doubt, by asking the taxi-driver: “Where -is King’s Cross this morning?“ At the station you would have to ask a -similar question about Edinburgh, but the booking-office clerk would -reply: “What part of Edinburgh do you mean, Sir? Prince’s Street has -gone to Glasgow, the Castle has moved up into the Highlands, and -Waverley Station is under water in the middle of the Firth of Forth.” -And on the journey the stations would not be staying quiet, but some -would be travelling north, some south, some east or west, perhaps much -faster than the train. Under these conditions you could not say where -you were at any moment. Indeed the whole notion that one is always in -some definite “place” is due to the fortunate immovability of most of -the large objects on the earth’s surface. The idea of “place” is only -a rough practical approximation: there is nothing logically necessary -about it, and it cannot be made precise. - -If we were not much larger than an electron, we should not have this -impression of stability, which is only due to the grossness of our -senses. King’s Cross, which to us looks solid, would be too vast to -be conceived except by a few eccentric mathematicians. The bits of it -that we could see would consist of little tiny points of matter, never -coming into contact with each other, but perpetually whizzing round -each other in an inconceivably rapid ballet-dance. The world of our -experience would be quite as mad as the one in which the different -parts of Edinburgh go for walks in different directions. If—to take -the opposite extreme—you were as large as the sun and lived as long, -with a corresponding slowness of perception, you would again find -a higgledy-piggledy universe without permanence—stars and planets -would come and go like morning mists, and nothing would remain in a -fixed position relatively to anything else. The notion of comparative -stability which forms part of our ordinary outlook is thus due to the -fact that we are about the size we are, and live on a planet of which -the surface is no longer very hot. If this were not the case, we should -not find pre-relativity physics intellectually satisfying. Indeed, we -should never have invented such theories. We should have had to arrive -at relativity physics at one bound, or remain ignorant of scientific -laws. It is fortunate for us that we were not faced with this -alternative, since it is almost inconceivable that one man could have -done the work of Euclid, Galileo, Newton, and Einstein. Yet without -such an incredible genius physics could hardly have been discovered -in a world where the universal flux was obvious to non-scientific -observation. - -In astronomy, although the sun, moon, and stars continue to exist year -after year, yet in other respects the world we have to deal with is -very different from that of everyday life. As already observed, we -depend exclusively on sight: the heavenly bodies cannot be touched, -heard, smelt or tasted. Everything in the heavens is moving relatively -to everything else. The earth is going round the sun, the sun is -moving, very much faster than an express train, towards a point in the -constellation “Hercules,” the “fixed” stars are scurrying hither and -thither like a lot of frightened hens. There are no well-marked places -in the sky, like King’s Cross and Edinburgh. When you travel from place -to place on the earth, you say the train moves and not the stations, -because the stations preserve their topographical relations to each -other and the surrounding country. But in astronomy it is arbitrary -which you call the train and which the station: the question is to be -decided purely by convenience and as a matter of convention. - -In this respect, it is interesting to contrast Einstein and Copernicus. -Before Copernicus, people thought that the earth stood still and the -heavens revolved about it once a day. Copernicus taught that “really” -the earth rotates once a day, and the daily revolution of sun and stars -is only “apparent.” Galileo and Newton endorsed this view, and many -things were thought to prove it—for example, the flattening of the -earth at the poles, and the fact that bodies are heavier there than at -the equator. But in the modern theory the question between Copernicus -and his predecessors is merely one of convenience; all motion is -relative, and there is no difference between the two statements: “the -earth rotates once a day” and “the heavens revolve about the earth -once a day.” The two mean exactly the same thing, just as it means the -same thing if I say that a certain length is six feet or two yards. -Astronomy is easier if we take the sun as fixed than if we take the -earth, just as accounts are easier in a decimal coinage. But to say -more for Copernicus is to assume absolute motion, which is a fiction. -All motion is relative, and it is a mere convention to take one body as -at rest. All such conventions are equally legitimate, though not all -are equally convenient. - -There is another matter of great importance, in which astronomy -differs from terrestrial physics because of its exclusive dependence -upon sight. Both popular thought and old-fashioned physics used the -notion of “force,” which seemed intelligible because it was associated -with familiar sensations. When we are walking, we have sensations -connected with our muscles which we do not have when we are sitting -still. In the days before mechanical traction, although people could -travel by sitting in their carriages, they could see the horses -exerting themselves and evidently putting out “force” in the same -way as human beings do. Everybody knew from experience what it is to -push or pull, or to be pushed or pulled. These very familiar facts -made “force” seem a natural basis for dynamics. But Newton’s law of -gravitation introduced a difficulty. The force between two billiard -balls appeared intelligible, because we know what it feels like to bump -into another person; but the force between the earth and the sun, which -are ninety-three million miles apart, was mysterious. Newton himself -regarded this “action at a distance” as impossible, and believed that -there was some hitherto undiscovered mechanism by which the sun’s -influence was transmitted to the planets. However, no such mechanism -was discovered, and gravitation remained a puzzle. The fact is that the -whole conception of “force” is a mistake. The sun does not exert any -force on the planets; in Einstein’s law of gravitation, the planet only -pays attention to what it finds in its own neighborhood. The way in -which this works will be explained in a later chapter; for the present -we are only concerned with the necessity of abandoning the notion of -“force,” which was due to misleading conceptions derived from the sense -of touch. - -As physics has advanced, it has appeared more and more that sight is -less misleading than touch as a source of fundamental notions about -matter. The apparent simplicity in the collision of billiard balls is -quite illusory. As a matter of fact, the two billiard balls never touch -at all; what really happens is inconceivably complicated, but is more -analogous to what happens when a comet penetrates the solar system and -goes away again than to what common sense supposes to happen. - -Most of what we have said hitherto was already recognized by physicists -before Einstein invented the theory of relativity. “Force” was known -to be merely a mathematical fiction, and it was generally held that -motion is a merely relative phenomenon—that is to say, when two -bodies are changing their relative position, we cannot say that one is -moving while the other is at rest, since the occurrence is merely a -change in their relation to each other. But a great labor was required -in order to bring the actual procedure of physics into harmony with -these new convictions. Newton believed in force and in absolute space -and time; he embodied these beliefs in his technical methods, and his -methods remained those of later physicists. Einstein invented a new -technique, free from Newton’s assumptions. But in order to do so he -had to change fundamentally the old ideas of space and time, which had -been unchallenged from time immemorial. This is what makes both the -difficulty and the interest of his theory. But before explaining it -there are some preliminaries which are indispensable. These will occupy -the next two chapters. - - - - -CHAPTER II: WHAT HAPPENS AND WHAT IS OBSERVED - - -A certain type of superior person is fond of asserting that “everything -is relative.” This is, of course, nonsense, because, if _everything_ -were relative, there would be nothing for it to be relative to. -However, without falling into metaphysical absurdities it is possible -to maintain that everything in the physical world is relative to -an observer. This view, true or not, is _not_ that adopted by the -“theory of relativity.” Perhaps the name is unfortunate; certainly -it has led philosophers and uneducated people into confusions. They -imagine that the new theory proves _everything_ in the physical world -to be relative, whereas, on the contrary, it is wholly concerned to -exclude what is relative and arrive at a statement of physical laws -that shall in no way depend upon the circumstances of the observer. It -is true that these circumstances have been found to have more effect -upon what appears to the observer than they were formerly thought to -have, but at the same time Einstein showed how to discount this effect -completely. This was the source of almost everything that is surprising -in his theory. - -When two observers perceive what is regarded as one occurrence, there -are certain similarities, and also certain differences, between their -perceptions. The differences are obscured by the requirements of -daily life, because from a business point of view they are as a rule -unimportant. But both psychology and physics, from their different -angles, are compelled to emphasize the respects in which one man’s -perception of a given occurrence differs from another man’s. Some of -these differences are due to differences in the brains or minds of -the observers, some to differences in their sense organs, some to -differences of physical situation: these three kinds may be called -respectively psychological, physiological, and physical. A remark made -in a language we know will be heard, whereas an equally loud remark -in an unknown language may pass entirely unnoticed. Of two men in the -Alps, one will perceive the beauty of the scenery while the other will -notice the waterfalls with a view to obtaining power from them. Such -differences are psychological. The difference between a long-sighted -and a short-sighted man, or between a deaf man and a man who hears -well, are physiological. Neither of these kinds concerns us, and I have -mentioned them only in order to exclude them. The kind that concerns us -is the purely physical kind. Physical differences between two observers -will be preserved when the observers are replaced by cameras or -phonographs, and can be reproduced on the movies or the gramophone. If -two men both listen to a third man speaking, and one of them is nearer -to the speaker than the other is, the nearer one will hear louder and -slightly earlier sounds than are heard by the other. If two men both -watch a tree falling, they see it from different angles. Both these -differences would be shown equally by recording instruments: they are -in no way due to idiosyncrasies in the observers, but are part of the -ordinary course of physical nature as we experience it. - -The physicist, like the plain man, believes that his perceptions give -him knowledge about what is really occurring in the physical world, -and not only about his private experiences. Professionally, he regards -the physical world as “real,” not merely as something which human -beings dream. An eclipse of the sun, for instance, can be observed -by any person who is suitably situated, and is also observed by the -photographic plates that are exposed for the purpose. The physicist -is persuaded that something has really happened over and above the -experiences of those who have looked at the sun or at photographs of -it. I have emphasized this point, which might seem a trifle obvious, -because some people imagine that Einstein has made a difference in this -respect. In fact he has made none. - -But if the physicist is justified in this belief that a number of -people can observe the “same” physical occurrence, then clearly the -physicist must be concerned with those features which the occurrence -has in common for all observers, for the others cannot be regarded -as belonging to the occurrence itself. At least, the physicist must -confine himself to the features which are common to all “equally -good” observers. The observer who uses a microscope or a telescope is -preferred to one who does not, because he sees all that the latter sees -and more too. A sensitive photographic plate may “see” still more, -and is then preferred to any eye. But such things as differences of -perspective, or differences of apparent size due to difference of -distance, are obviously not attributable to the object; they belong -solely to the point of view of the spectator. Common sense eliminates -these in judging of objects; physics has to carry the same process much -further, but the principle is the same. - -I want to make it clear that I am not concerned with anything that can -be called inaccuracy. I am concerned with genuine physical differences -between occurrences each of which is a correct record of a certain -event, from its own point of view. When a man fires a gun, people who -are not quite close to him see the flash before they hear the report. -This is not due to any defect in their senses, but to the fact that -sound travels more slowly than light. Light travels so fast that, from -the point of view of phenomena on the surface of the earth, it may -be regarded as instantaneous. Anything that we can see on the earth -happens practically at the moment when we see it. In a second, light -travels 300,000 kilometers (about 186,000 miles). It travels from the -sun to the earth in about eight minutes, and from the stars to us in -anything from three to a thousand years. But of course we cannot place -a clock in the sun, and send out a flash of light from it at 12 noon, -Greenwich Mean Time, and have it received at Greenwich at 12.08 P.M. -Our methods of estimating the speed of light have to be more or less -indirect. The only direct method would be that which we apply to sound -when we use an echo. We could send a flash to a mirror, and observe how -long it took for the reflection to reach us; this would give the time -of the double journey to the mirror and back. On the earth, however, -the time would be so short that a great deal of theoretical physics -has to be utilized if this method is to be employed—more even than is -required for the employment of astronomical data. - -The problem of allowing for the spectator’s point of view, we may be -told, is one of which physics has at all times been fully aware; indeed -it has dominated astronomy ever since the time of Copernicus. This is -true. But principles are often acknowledged long before their full -consequences are drawn. Much of traditional physics is incompatible -with the principle, in spite of the fact that it was acknowledged -theoretically by all physicists. - -There existed a set of rules which caused uneasiness to the -philosophically minded, but were accepted by physicists because -they worked in practice. Locke had distinguished “secondary” -qualities—colors, noises, tastes, smells, etc.—as subjective, while -allowing “primary” qualities—shapes and positions and sizes—to be -genuine properties of physical objects. The physicist’s rules were -such as would follow from this doctrine. Colors and noises were -allowed to be subjective, but due to waves proceeding with a definite -velocity—that of light or sound as the case may be—from their source -to the eye or ear of the percipient. Apparent shapes vary according to -the laws of perspective, but these laws are simple and make it easy to -infer the “real” shapes from several visual apparent shapes; moreover, -the “real” shapes can be ascertained by touch in the case of bodies in -our neighborhood. The objective time of a physical occurrence can be -inferred from the time when we perceive it by allowing for the velocity -of transmission—of light or sound or nerve currents according to -circumstances. This was the view adopted by physicists in practice, -whatever qualms they may have had in unprofessional moments. - -This view worked well enough until physicists became concerned with -much greater velocities than those that are common on the surface of -the earth. An express train travels about a mile in a minute; the -planets travel a few miles in a second. Comets, when they are near -the sun, travel much faster, and behave somewhat oddly; but they were -puzzling in various ways. Practically, the planets were the most -swiftly moving bodies to which dynamics could be adequately applied. -With radio-activity a new range of observations became possible. -Individual electrons can be observed, emanating from radium with a -velocity not far short of that of light. The behavior of bodies moving -with these enormous speeds is not what the old theories would lead -us to expect. For one thing, mass seems to increase with speed in a -perfectly definite manner. When an electron is moving very fast, a -bigger force is required to have a given effect upon it than when it -is moving slowly. Then reasons were found for thinking that the size -of a body is affected by its motion—for example, if you take a cube -and move it very fast, it gets shorter in the direction of its motion, -from the point of view of a person who is not moving with it, though -from its own point of view (_i.e._ for an observer traveling with it) -it remains just as it was. What was still more astonishing was the -discovery that lapse of time depends on motion; that is to say, two -perfectly accurate clocks, one of which is moving very fast relatively -to the other, will not continue to show the same time if they come -together again after a journey. It follows that what we discover by -means of clocks and foot rules, which used to be regarded as the acme -of impersonal science, is really in part dependent upon our private -circumstances, _i.e._ upon the way in which we are moving relatively to -the bodies measured. - -This shows that we have to draw a different line from that which is -customary in distinguishing between what belongs to the observer and -what belongs to the occurrence which he is observing. If a man is -wearing blue spectacles he knows that the blue look of everything is -due to his spectacles, and does not belong to what he is observing. -But if he observes two flashes of lightning, and notes the interval -of time between his observations; if he knows where the flashes took -place, and allows, in each case, for the time the light took to reach -him—in that case, if his chronometer is accurate, he naturally thinks -that he has discovered the actual interval of time between the two -flashes, and not something merely personal to himself. He is confirmed -in this view by the fact that all other careful observers to whom he -has access agree with his estimates. This, however, is only due to the -fact that all these observers are on the earth, and share its motion. -Even two observers in aeroplanes moving in opposite directions would -have at the most a relative velocity of 400 miles an hour, which is -very little in comparison with 186,000 miles a second (the velocity -of light). If an electron shot out from a piece of radium with a -velocity of 170,000 miles a second could observe the time between the -two flashes, it would arrive at a quite different estimate, after -making full allowance for the velocity of light. How do you know this? -the reader may ask. You are not an electron, you cannot move at these -terrific speeds, no man of science has ever made the observations which -would prove the truth of your assertion. Nevertheless, as we shall see -in the sequel, there is good ground for the assertion—ground, first -of all, in experiment, and—what is remarkable—ground in reasonings -which could have been made at any time, but were not made until -experiments had shown that the old reasonings must be wrong. - -There is a general principle to which the theory of relativity appeals, -which turns out to be more powerful than anybody would suppose. If -you know that one man is twice as rich as another, this fact must -appear equally whether you estimate the wealth of both in pounds or -dollars or francs or any other currency. The numbers representing their -fortunes will be changed, but one number will always be double the -other. The same sort of thing, in more complicated forms, reappears in -physics. Since all motion is relative, you may take any body you like -as your standard body of reference, and estimate all other motions -with reference to that one. If you are in a train and walking to the -dining-car, you naturally, for the moment, treat the train as fixed -and estimate your motion by relation to it. But when you think of the -journey you are making, you think of the earth as fixed, and say you -are moving at the rate of sixty miles an hour. An astronomer who is -concerned with the solar system takes the sun as fixed, and regards you -as rotating and revolving; in comparison with this motion, that of the -train is so slow that it hardly counts. An astronomer who is interested -in the stellar universe may add the motion of the sun relatively to -the average of the stars. You cannot say that one of these ways of -estimating your motion is more correct than another; each is perfectly -correct as soon as the reference body is assigned. Now just as you can -estimate a man’s fortune in different currencies without altering its -relations to the fortunes of other men, so you can estimate a body’s -motion by means of different reference bodies without altering its -relations to other motions. And as physics is entirely concerned with -relations, it must be possible to express all the laws of physics by -referring all motions to any given body as the standard. - -We may put the matter in another way. Physics is intended to give -information about what really occurs in the physical world, and not -only about the private perceptions of separate observers. Physics must, -therefore, be concerned with those features which a physical process -has in common for all observers, since such features alone can be -regarded as belonging to the physical occurrence itself. This requires -that the _laws_ of phenomena should be the same whether the phenomena -are described as they appear to one observer or as they appear to -another. This single principle is the generating motive of the whole -theory of relativity. - -Now what we have hitherto regarded as the spatial and temporal -properties of physical occurrences are found to be in large part -dependent upon the observer; only a residue can be attributed to the -occurrences in themselves, and only this residue can be involved in -the formulation of any physical law which is to have an _à priori_ -chance of being true. Einstein found ready to his hand an instrument of -pure mathematics, called the theory of tensors, which enabled him to -discover laws expressed in terms of the objective residue and agreeing -approximately with the old laws. Where Einstein’s laws differed from -the old ones, they have hitherto proved more in accord with observation. - -If there were no reality in the physical world, but only a number of -dreams dreamed by different people, we should not expect to find any -laws connecting the dreams of one man with the dreams of another. It -is the close connection between the perceptions of one man and the -(roughly) simultaneous perceptions of another that makes us believe in -a common external origin of the different related perceptions. Physics -accounts both for the likenesses and for the differences between -different people’s perceptions of what we call the “same” occurrence. -But in order to do this it is first necessary for the physicist to -find out just what are the likenesses. They are not quite those -traditionally assumed, because neither space nor time separately can -be taken as strictly objective. What is objective is a kind of mixture -of the two called “space-time.” To explain this is not easy, but the -attempt must be made; it will be begun in the next chapter. - - - - -CHAPTER III: THE VELOCITY OF LIGHT - - -Most of the curious things in the theory of relativity are connected -with the velocity of light. If the reader is to grasp the reasons for -such a serious theoretical reconstruction, he must have some idea of -the facts which made the old system break down. - -The fact that light is transmitted with a definite velocity was -first established by astronomical observations. Jupiter’s moons are -sometimes eclipsed by Jupiter, and it is easy to calculate the times -when this ought to occur. It was found that when Jupiter was unusually -near the earth an eclipse of one of his moons would be observed a few -minutes earlier than was expected; and when Jupiter was unusually -remote, a few minutes later than was expected. It was found that these -deviations could all be accounted for by assuming that light has a -certain velocity, so that what we observe to be happening in Jupiter -really happened a little while ago—longer ago when Jupiter is distant -than when it is near. Just the same velocity of light was found to -account for similar facts in regard to other parts of the solar system. -It was therefore accepted that light _in vacuo_ always travels at a -certain constant rate, almost exactly 300,000 kilometers a second. (A -kilometer is about five-eighths of a mile.) When it became established -that light consists of waves, this velocity was that of propagation -of waves in the ether—at least they used to be in the ether, but now -the ether has grown somewhat shadowy, though the waves remain. This -same velocity is that of the waves used in wireless telegraphy (which -are like light waves, only longer) and in X-rays (which are like light -waves, only shorter). It is generally held nowadays to be the velocity -with which gravitation is propagated, though Eddington considers this -not yet certain. (It used to be thought that gravitation was propagated -instantaneously, but this view is now abandoned.) - -So far, all is plain sailing. But as it became possible to make more -accurate measurements, difficulties began to accumulate. The waves were -supposed to be in the ether, and therefore their velocity ought to -be relative to the ether. Now since the ether (if it exists) clearly -offers no resistance to the motions of the heavenly bodies, it would -seem natural to suppose that it does not share their motion. If the -earth had to push a lot of ether before it, in the sort of way that -a steamer pushes water before it, one would expect a resistance on -the part of the ether analogous to that offered by the water to the -steamer. Therefore the general view was that the ether could pass -through bodies without difficulty, like air through a coarse sieve, -only more so. If this were the case, then the earth in its orbit must -have a velocity relative to the ether. If, at some point of its orbit, -it happened to be moving exactly with the ether, it must at other -points be moving through it all the faster. If you go for a circular -walk on a windy day, you must be walking against the wind part of the -way, whatever wind may be blowing; the principle in this case is the -same. It follows that, if you choose two days six months apart, when -the earth in its orbit is moving in exactly opposite directions, it -must be moving against an ether wind on at least one of these days. - -Now if there is an ether wind, it is clear that, relatively to an -observer on the earth, light signals will seem to travel faster with -the wind than across it, and faster across it than against it. This -is what Michelson and Morley set themselves to test by their famous -experiment. They sent out light signals in two directions at right -angles; each was reflected from a mirror, and came back to the place -from which both had been sent out. Now anybody can verify, either by -trial or by a little arithmetic, that it takes longer to row a given -distance on a river upstream and then back again, than it takes to -row the same distance across the stream and back again. Therefore, if -there were an ether wind, one of the two light signals, which consist -of waves in the ether, ought to have traveled to the mirror and back at -a slower average rate than the other. Michelson and Morley tried the -experiment, they tried it in various positions, they tried it again -later. Their apparatus was quite accurate enough to have detected the -expected difference of speed or even a much smaller difference, if -it had existed, but not the smallest difference could be observed. -The result was a surprise to them as to everybody else; but careful -repetitions made doubt impossible. The experiment was first made as -long ago as 1881, and was repeated with more elaboration in 1887. But -it was many years before it could be rightly interpreted. - -The supposition that the earth carries the neighboring ether with it -in its motion was found to be impossible, for a number of reasons. -Consequently a logical deadlock seemed to have arisen, from which at -first physicists sought to extricate themselves by very arbitrary -hypotheses. The most important of these was that of Fitzgerald, -developed by Lorentz, and known as the Fitzgerald contraction -hypothesis. - -According to this hypothesis, when a body is in motion it becomes -shortened in the direction of motion by a certain proportion depending -upon its velocity. The amount of the contraction was to be just enough -to account for the negative result of the Michelson-Morley experiment. -The journey up stream and down again was to have been really a shorter -journey than the one across the stream, and was to have been just so -much shorter as would enable the slower light wave to traverse it in -the same time. Of course the shortening could never be detected by -measurement, because our measuring rods would share it. A foot rule -placed in the line of the earth’s motion would be shorter than the -same foot rule placed at right angles to the earth’s motion. This -point of view resembles nothing so much as the White Knight’s “plan to -dye my whiskers green, and always use so large a fan that they could -not be seen.” The odd thing was that the plan worked well enough. Later -on, when Einstein propounded his special theory of relativity (1905), -it was found that the theory was in a certain sense correct, but only -in a certain sense. That is to say, the supposed contraction is not -a physical fact, but a result of certain conventions of measurement -which, when once the right point of view has been found, are seen to -be such as we are almost compelled to adopt. But I do not wish yet to -set forth Einstein’s solution of the puzzle. For the present, it is the -nature of the puzzle itself that I want to make clear. - -On the face of it, and apart from hypotheses _ad hoc_, the -Michelson-Morley experiment (in conjunction with others) showed that, -relatively to the earth, the velocity of light is the same in all -directions, and that this is equally true at all times of the year, -although the direction of the earth’s motion is always changing as -it goes round the sun. Moreover, it appeared that this is not a -peculiarity of the earth, but is true of all bodies: if a light signal -is sent out from a body, that body will remain at the center of the -waves as they travel outwards, no matter how it may be moving—at -least, that will be the view of observers moving with the body. This -was the plain and natural meaning of the experiments, and Einstein -succeeded in inventing a theory which accepted it. But at first it was -thought logically impossible to accept this plain and natural meaning. - -A few illustrations will make it clear how very odd the facts are. When -a shell is fired, it moves faster than sound: the people at whom it is -fired first see the flash, then (if they are lucky) see the shell go -by, and last of all hear the report. It is clear that if you could put -a scientific observer on the shell, he would never hear the report, as -the shell would burst and kill him before the sound had overtaken him. -But if sound worked on the same principles as light, our observer would -hear everything just as if he were at rest. In that case, if a screen, -suitable for producing echoes, were attached to the shell and traveling -with it, say a hundred yards in front of it, our observer would hear -the echo of the report from the screen after just the same interval -of time as if he and the shell were at rest. This, of course, is an -experiment which cannot be performed, but others which can be performed -will show the difference. We might find some place on a railway where -there is an echo from a place further along the railway—say a place -where the railway goes into a tunnel—and when a train is traveling -along the railway, let a man on the bank fire a gun. If the train is -traveling towards the echo, the passengers will hear the echo sooner -than the man on the bank; if it is traveling in the opposite direction, -they will hear it later. But these are not quite the circumstances -of the Michelson-Morley experiment. The mirrors in that experiment -correspond to the echo, and the mirrors are moving with the earth, so -that echo ought to move with the train. Let us suppose that the shot -is fired from the guard’s van, and the echo comes from a screen on the -engine. We will suppose the distance from the guard’s van to the engine -to be the distance that sound can travel in a second (about one-fifth -of a mile), and the speed of the train to be one-twelfth of the speed -of sound (about sixty miles an hour). We now have an experiment which -can be performed by the people in the train. If the train were at rest, -the guard would hear the echo in two seconds; as it is, he will hear it -in 2 and ²/₁₄₃ seconds. From this difference, if he knows the velocity -of sound, he can calculate the velocity of the train, even if it is a -foggy night so that he cannot see the banks. But if sound behaved like -light, he would hear the echo in two seconds however fast the train -might be traveling. - -Various other illustrations will help to show how extraordinary—from -the point of view of tradition and common sense—are the facts about -the velocity of light. Every one knows that if you are on an escalator -you reach the top sooner if you walk up than if you stand still. But if -the escalator moved with the velocity of light (which it does not do -even in New York), you would reach the top at exactly the same moment -whether you walked up or stood still. Again: if you are walking along -a road at the rate of four miles an hour, and a motor-car passes you -going in the same direction at the rate of forty miles an hour, if you -and the motor-car both keep going the distance between you after an -hour will be thirty-six miles. But if the motor-car met you, going in -the opposite direction, the distance after an hour would be forty-four -miles. Now if the motor-car were traveling with the velocity of light, -it would make no difference whether it met or passed you: in either -case, it would, after a second, be 186,000 miles from you. It would -also be 186,000 miles from any other motor-car which happened to be -passing or meeting you less rapidly at the previous second. This seems -impossible: how can the car be at the same distance from a number of -different points along the road? - -Let us take another illustration. When a fly touches the surface of -a stagnant pool, it causes ripples which move outwards in widening -circles. The center of the circle at any moment is the point of the -pool touched by the fly. If the fly moves about over the surface of -the pool, it does not remain at the center of the ripples. But if the -ripples were waves of light, and the fly were a skilled physicist, -it would find that it always remained at the center of the ripples, -however it might move. Meanwhile a skilled physicist sitting beside the -pool would judge, as in the case of ordinary ripples, that the center -was not the fly, but the point of the pool touched by the fly. And if -another fly had touched the water at the same spot at the same moment, -it also would find that it remained at the center of the ripples, even -if it separated itself widely from the first fly. This is exactly -analogous to the Michelson-Morley experiment. The pool corresponds to -the ether; the fly corresponds to the earth; the contact of the fly and -the pool corresponds to the light signal which Messrs. Michelson and -Morley send out; and the ripples correspond to the light waves. - -Such a state of affairs seems, at first sight, quite impossible. It -is no wonder that, although the Michelson-Morley experiment was made -in 1881, it was not rightly interpreted until 1905. Let us see what, -exactly, we have been saying. Take the man walking along a road and -passed by a motor-car. Suppose there are a number of people at the same -point of the road, some walking, some in motor-cars; suppose they are -going at varying rates, some in one direction and some in another. I -say that if, at this moment, a light flash is sent out from the place -where they all are, the light waves will be 186,000 miles from each -one of them after a second by his watch, although the travelers will -not any longer be all in the same place. At the end of a second by your -watch it will be 186,000 miles from you, and it will also be 186,000 -miles from a person who met you when it was sent out, but was moving in -the opposite direction, after a second by his watch—assuming both to -be perfect watches. How can this be? - -There is only one way of explaining such facts, and that is, to assume -that watches and clocks are affected by motion. I do not mean that -they are affected in ways that could be remedied by greater accuracy -in construction; I mean something much more fundamental. I mean that, -if you say an hour has elapsed between two events, and if you base -this assertion upon ideally careful measurements with ideally accurate -chronometers, another equally precise person, who has been moving -rapidly relatively to you, may judge that the time was more or less -than an hour. You cannot say that one is right and the other wrong, -any more than you could if one used a clock showing Greenwich time and -another a clock showing New York time. How this comes about, I shall -explain in the next chapter. - -There are other curious things about the velocity of light. One is, -that no material body can ever travel as fast as light, however great -may be the force to which it is exposed, and however long the force -may act. An illustration may help to make this clear. At exhibitions -one sometimes sees a series of moving platforms, going round and round -in a circle. The outside one goes at four miles an hour; the next -goes four miles an hour faster than the first; and so on. You can -step across from each to the next; until you find yourself going at a -tremendous pace. Now you might think that, if the first platform does -four miles an hour, and the second does four miles an hour relatively -to the first, then the second does eight miles an hour relatively to -the ground. This is an error; it does a little less, though so little -less that not even the most careful measurements could detect the -difference. I want to make quite clear what it is that I mean. I will -suppose that, in the morning, when the apparatus is just about to -start, three men with ideally accurate chronometers stand in a row, one -on the ground, one on the first platform, and one on the second. The -first platform moves at the rate of four miles an hour with respect -to the ground. Four miles an hour is 352 feet in a minute. The man on -the ground, after a minute by his watch, notes the place on the ground -opposite the man on the first platform, who has been standing still -while the platform carried him along. The man on the ground measures -the distance on the ground from himself to the point opposite the -man on the first platform, and finds it is 352 feet. The man on the -first platform, after a minute by his watch, notes the point on his -platform opposite to the man on the second platform. The man on the -first platform measures the distance from himself to the point opposite -the man on the second platform; it is again 352 feet. Problem: how far -will the man on the ground judge that the man on the second platform -has traveled in a minute? That is to say, if the man on the ground, -after a minute by his watch, notes the place on the ground opposite -the man on the second platform, how far will this be from the man on -the ground? You would say, twice 352 feet, that is to say, 704 feet. -But in fact it will be a little less, though so little less as to -be inappreciable. The discrepancy is owing to the fact that the two -watches do not keep perfect time, in spite of the fact that each is -accurate from its owner’s point of view. If you had a long series of -such moving platforms, each moving four miles an hour relatively to the -one before it, you would never reach a point where the last was moving -with the velocity of light relatively to the ground, not even if you -had millions of them. The discrepancy, which is very small for small -velocities, becomes greater as the velocity increases, and makes the -velocity of light an unattainable limit. How all this happens, is the -next topic with which we must deal. - - _Note._ The negative result of the - Michelson-Morley experiment has recently been called - in question by Professor Dayton C. Miller, as a - result of observations by what is said to be an - improved method. His claim is set forth by Professor - Silberstein in _Nature_, May 23, 1925, and - discussed unfavorably by Eddington in the issue of - June 6. The matter is _sub judice_, but it seems - highly questionable whether the results bear out the - interpretation which is put upon them. - - - - -CHAPTER IV: CLOCKS AND FOOT RULES - - -Until the advent of the special theory of relativity, no one had -thought that there could be any ambiguity in the statement that -two events in different places happened at the same time. It might -be admitted that, if the places were very far apart, there might -be difficulty in finding out for certain whether the events were -simultaneous, but every one thought the meaning of the question -perfectly definite. It turned out, however, that this was a mistake. -Two events in distant places may appear simultaneous to one observer -who has taken all due precautions to insure accuracy (and, in -particular, has allowed for the velocity of light), while another -equally careful observer may judge that the first event preceded -the second, and still another may judge that the second preceded -the first. This would happen if the three observers were all moving -rapidly relatively to each other. It would not be the case that one -of them would be right and the other two wrong: they would all be -equally right. The time order of events is in part dependent upon the -observer; it is not always and altogether an intrinsic relation between -the events themselves. Einstein has shown, not only that this view -accounts for the phenomena, but also that it is the one which ought -to have resulted from careful reasoning based upon the old data. In -actual fact, however, no one noticed the logical basis of the theory -of relativity until the odd results of experiment had given a jog to -people’s reasoning powers. - -How should we naturally decide whether two events in different places -were simultaneous? One would naturally say: they are simultaneous -if they are seen simultaneously by a person who is exactly half-way -between them. (There is no difficulty about the simultaneity of two -events in the _same_ place, such, for example, as seeing a light -and hearing a noise.) Suppose two flashes of lightning fall in two -different places, say Greenwich Observatory and Kew Observatory. -Suppose that St. Paul’s is half-way between them, and that the flashes -appear simultaneous to an observer on the dome of St. Paul’s. In that -case, a man at Kew will see the Kew flash first, and a man at Greenwich -will see the Greenwich flash first, because of the time taken by -light to travel over the intervening distance. But all three, if they -are ideally accurate observers, will judge that the two flashes were -simultaneous, because they will make the necessary allowance for the -time of transmission of the light. (I am assuming a degree of accuracy -far beyond human powers.) Thus, so far as observers on the earth are -concerned, the definition of simultaneity will work well enough, so -long as we are dealing with events on the surface of the earth. It -gives results which are consistent with each other, and can be used for -terrestrial physics in all problems in which we can ignore the fact -that the earth moves. - -But our definition is no longer so satisfactory when we have two sets -of observers in rapid motion relatively to each other. Suppose we see -what would happen if we substitute sound for light, and defined two -occurrences as simultaneous when they are heard simultaneously by a -man half-way between them. This alters nothing in the principle, but -makes the matter easier owing to the much slower velocity of sound. -Let us suppose that on a foggy night two men belonging to a gang of -brigands shoot the guard and engine driver of a train. The guard is at -the end of the train; the brigands are on the line, and shoot their -victims at close quarters. An old gentleman who is exactly in the -middle of the train hears the two shots simultaneously. You would say, -therefore, that the two shots were simultaneous. But a station master -who is exactly half-way between the two brigands hears the shot which -kills the guard first. An Australian millionaire uncle of the guard -and the engine driver (who are cousins) has left his whole fortune to -the guard, or, should he die first, to the engine driver. Vast sums -are involved in the question of which died first. The case goes to the -House of Lords, and the lawyers on both sides, having been educated at -Oxford, are agreed that either the old gentleman or the station master -must have been mistaken. In fact, both may perfectly well be right. The -train travels away from the shot at the guard, and towards the shot at -the engine driver; therefore the noise of the shot at the guard has -farther to go before reaching the old gentleman than the shot at the -engine driver has. Therefore if the old gentleman is right in saying -that he heard the two reports simultaneously, the station master must -be right in saying that he heard the shot at the guard first. - -We, who live on the earth, would naturally, in such a case, prefer -the view of simultaneity obtained from a person at rest on the earth -to the view of a person traveling in a train. But in theoretical -physics no such parochial prejudices are permissible. A physicist on a -comet, if there were one, would have just as good a right to his view -of simultaneity as an earthly physicist has to his, but the results -would differ, in just the same sort of way as in our illustration of -the train and the shots. The train is not any more “really” in motion -than the earth; there is no “really” about it. You might imagine a -rabbit and a hippopotamus arguing as to whether man is “really” a large -animal; each would think his own point of view the natural one, and -the other a pure flight of fancy. There is just as little substance -in an argument as to whether the earth or the train is “really” in -motion. And, therefore, when we are defining simultaneity between -distant events, we have no right to pick and choose among different -bodies to be used in defining the point half-way between the events. -All bodies have an equal right to be chosen. But if, for one body, the -two events are simultaneous according to the definition, there will -be other bodies for which the first precedes the second, and still -others for which the second precedes the first. We cannot therefore -say unambiguously that two events in distant places are simultaneous. -Such a statement only acquires a definite meaning in relation to a -definite observer. It belongs to the subjective part of our observation -of physical phenomena, not to the objective part which is to enter into -physical laws. - -This question of time in different places is perhaps, for the -imagination, the most difficult aspect of the theory of relativity. We -are accustomed to the idea that everything can be dated. Historians -make use of the fact that there was an eclipse of the sun visible in -China on August 29 in the year 776 B. C.[1] No doubt astronomers could -tell the exact hour and minute when the eclipse began to be total at -any given spot in North China. And it seems obvious that we can speak -of the positions of the planets at a given instant. The Newtonian -theory enables us to calculate the distance between the earth and (say) -Jupiter at a given time by the Greenwich clocks; this enables us to -know how long light takes at that time to travel from Jupiter to the -earth—say half an hour; this enables us to infer that half an hour ago -Jupiter was where we see it now. All this seems obvious. But in fact it -only works in practice because the relative velocities of the planets -are very small compared with the velocity of light. When we judge that -an event on the earth and an event on Jupiter have happened at the -same time—for example, that Jupiter eclipsed one of his moons when -the Greenwich clocks showed twelve midnight—a person moving rapidly -relatively to the earth would judge differently, assuming that both he -and we had made the proper allowance for the velocity of light. And -naturally the disagreement about simultaneity involves a disagreement -about periods of time. If we judged that two events on Jupiter were -separated by twenty-four hours, another person might judge that they -were separated by a longer time, if he were moving rapidly relatively -to Jupiter and the earth. - -[1] A contemporary Chinese ode, after giving the day of the year -correctly, proceeds: - - “For the moon to be eclipsed - Is but an ordinary matter. - Now that the sun has been eclipsed, - How bad it is.” - - -The universal cosmic time which used to be taken for granted is thus no -longer admissible. For each body, there is a definite time order for -the events in its neighborhood; this may be called the “proper” time -for that body. Our own experience is governed by the proper time for -our own body. As we all remain very nearly stationary on the earth, -the proper times of different human beings agree, and can be lumped -together as terrestrial time. But this is only the time appropriate to -_large_ bodies on the earth. For Beta-particles in laboratories, quite -different times would be wanted; it is because we insist upon using -our own time that these particles seem to increase in mass with rapid -motion. From their own point of view, their mass remains constant, -and it is we who suddenly grow thin or corpulent. The history of a -physicist as observed by a Beta-particle would resemble Gulliver’s -travels. - -The question now arises: what really is measured by a clock? When we -speak of a clock in the theory of relativity, we do not mean only -clocks made by human hands: we mean anything which goes through some -regular periodic performance. The earth is a clock, because it rotates -once in every twenty-three hours and fifty-six minutes. An atom is a -clock, because the electrons go round the nucleus a certain number of -times in a second; its properties as a clock are exhibited to us in -its spectrum, which is due to light waves of various frequencies. The -world is full of periodic occurrences, and fundamental mechanisms, -such as atoms, show an extraordinary similarity in different parts of -the universe. Any one of these periodic occurrences may be used for -measuring time; the only advantage of humanly manufactured clocks is -that they are specially easy to observe. One question is: If cosmic -time is abandoned, what is really measured by a clock in the wide sense -that we have just given to the term? - -Each clock gives a correct measure of its own “proper” time, which, -as we shall see presently, is an important physical quantity. But it -does not give an accurate measure of any physical quantity connected -with events on bodies that are moving rapidly in relation to it. It -gives one datum towards the discovery of a physical quantity connected -with such events, but another datum is required, and this has to be -derived from measurement of distances in space. Distances in space, -like periods of time, are in general not objective physical facts, but -partly dependent upon the observer. How this comes about must now be -explained. - -First of all, we have to think of the distance between two events, not -between two bodies. This follows at once from what we have found as -regards time. If two bodies are moving relatively to each other—and -this is really always the case—the distance between them will be -continually changing, so that we can only speak of the distance -between them at a given time. If you are in a train traveling towards -Edinburgh, we can speak of your distance from Edinburgh at a given -time. But, as we said, different observers will judge differently as -to what is the “same” time for an event in the train and an event in -Edinburgh. This makes the measurement of distances relative, in just -the same way as the measurement of times has been found to be relative. -We commonly think that there are two separate kinds of interval between -two events, an interval in space and an interval in time: between your -departure from London and your arrival in Edinburgh, there are 400 -miles and ten hours. We have already seen that another observer will -judge the time differently; it is even more obvious that he will judge -the distance differently. An observer in the sun will think the motion -of the train quite trivial, and will judge that you have traveled the -distance traveled by the earth in its orbit and its diurnal rotation. -On the other hand, a flea in the railway carriage will judge that you -have not moved at all in space, but have afforded him a period of -pleasure which he will measure by his “proper” time, not by Greenwich -Observatory. It cannot be said that you or the sun dweller or the -flea are mistaken: each is equally justified, and is only wrong if he -ascribes an objective validity to his subjective measures. The distance -in space between two events is, therefore, not in itself a physical -fact. But, as we shall see, there is a physical fact which can be -inferred from the distance in time together with the distance in space. -This is what is called the “interval” in space-time. - -Taking any two events in the universe, there are two different -possibilities as to the relation between them. It may be physically -possible for a body to travel so as to be present at both events, or it -may not. This depends upon the fact that no body can travel as fast as -light. Suppose, for example, that it were possible to send out a flash -of light from the earth and have it reflected back from the moon. The -time between the sending of the flash and the return of the reflection -would be about two and a half seconds. No body could travel so fast -as to be present on the earth during any part of those two and a half -seconds and also present on the moon at the moment of the arrival of -the flash, because in order to do so the body would have to travel -faster than light. But theoretically a body could be present on the -earth at any time before or after those two and a half seconds and also -present on the moon at the time when the flash arrived. When it is -physically impossible for a body to travel so as to be present at both -events, we shall say that the interval[2] between the two events is -“space-like”; when it is physically possible for a body to be present -at both events, we shall say that the interval between the two events -is “time-like.” When the interval is “space-like,” it is possible for -a body to move in such a way that an observer on the body will judge -the two events to be simultaneous. In that case, the “interval” between -the two events is what such an observer will judge to be the distance -in space between them. When the interval is “time-like,” a body can -be present at both events; in that case, the “interval” between the -two events is what an observer on the body will judge to be the time -between them, that is to say, it is his “proper” time between the two -events. There is a limiting case between the two, when the two events -are parts of one light flash—or, as we might say, when the one event -is the seeing of the other. In that case, the interval between the two -events is zero. - -[2] I shall define “interval” in a moment. - -There are thus three cases. (1) It may be possible for a ray of light -to be present at both events; this happens whenever one of them is the -seeing of the other. In this case the interval between the two events -is zero. (2) It may happen that no body can travel from one event to -the other, because in order to do so it would have to travel faster -than light. In that case, it is always physically possible for a body -to travel in such a way that an observer on the body would judge the -two events to be simultaneous. The interval is what he would judge to -be the distance in space between the two events. Such an interval is -called “space-like.” (3) It may be physically possible for a body to -travel so as to be present at both events; in that case, the interval -between them is what an observer on such a body will judge to be the -time between them. Such an interval is called “time-like.” - -The interval between two events is a physical fact about them, not -dependent upon the particular circumstances of the observer. - -There are two forms of the theory of relativity, the special and the -general. The former is in general only approximate, but is exact at -great distances from gravitating matter. When the special theory can -be applied, the interval can be calculated when we know the distance -in space and the distance in time between the two events, estimated by -any observer. If the distance in space is greater than the distance -that light would have traveled in the time, the separation is -space-like. Then the following construction gives the interval between -the two events: Draw a line =AB= as long as the distance that light -would travel in the time; round =A= describe a circle whose radius is -the distance in space between the two events; through =B= draw =BC= -perpendicular to =AB=, meeting the circle in =C=. Then =BC= is the -length of the interval between the two events. - -[Illustration] - -When the distance is time-like, use the same figure, but let =AC= be -now the distance that light would travel in the time, while =AB= is the -distance in space between the two events. The interval between them is -now the time that light would take to travel the distance =BC=. - -Although =AB= and =AC= are different for different observers, =BC= is -the same length for all observers, subject to corrections made by the -general theory. It represents the one interval in “space-time” which -replaces the two intervals in space and time of the older physics. So -far, this notion of interval may appear somewhat mysterious, but as we -proceed it will grow less so, and its reason in the nature of things -will gradually emerge. - - - - -CHAPTER V: SPACE-TIME - - -Everybody who has ever heard of relativity knows the phrase -“space-time,” and knows that the correct thing is to use this phrase -when formerly we should have said “space _and_ time.” But very few -people who are not mathematicians have any clear idea of what is meant -by this change of phraseology. Before dealing further with the special -theory of relativity, I want to try to convey to the reader what is -involved in the new phrase “space-time,” because that is, from a -philosophical and imaginative point of view, perhaps the most important -of all the novelties that Einstein has introduced. - -Suppose you wish to say where and when some event has occurred—say -an explosion on an airship—you will have to mention four quantities, -say the latitude and longitude, the height above the ground, and the -time. According to the traditional view, the first three of these -give the position in space, while the fourth gives the position in -time. The three quantities that give the position in space may be -assigned in all sorts of ways. You might, for instance, take the -plane of the equator, the plane of the meridian of Greenwich, and the -plane of the ninetieth meridian, and say how far the airship was from -each of these planes; these three distances would be what are called -“Cartesian co-ordinates,” after Descartes. You might take any other -three planes all at right angles to each other, and you would still -have Cartesian co-ordinates. Or you might take the distance from London -to a point vertically below the airship, the direction of this distance -(northeast, west-southwest, or whatever it might be), and the height of -the airship above the ground. There are an infinite number of such ways -of fixing the position in space, all equally legitimate; the choice -between them is merely one of convenience. - -When people said that space had three dimensions, they meant just this: -that three quantities were necessary in order to specify the position -of a point in space, but that the method of assigning these quantities -was wholly arbitrary. - -With regard to time, the matter was thought to be quite different. The -only arbitrary elements in the reckoning of time were the unit, and -the point of time from which the reckoning started. One could reckon -in Greenwich time, or in Paris time, or in New York time; that made a -difference as to the point of departure. One could reckon in seconds, -minutes, hours, days, or years; that was a difference of unit. Both -these were obvious and trivial matters. There was nothing corresponding -to the liberty of choice as to the method of fixing position in space. -And, in particular, it was thought that the method of fixing position -in space and the method of fixing position in time could be made wholly -independent of each other. For these reasons, people regarded time and -space as quite distinct. - -The theory of relativity has changed this. There are now a number of -different ways of fixing position in time, which do not differ merely -as to the unit and the starting point. Indeed, as we have seen, if one -event is simultaneous with another in one reckoning, it will precede -it in another, and follow it in a third. Moreover, the space and time -reckonings are no longer independent of each other. If you alter the -way of reckoning position in space, you may also alter the time -interval between two events. If you alter the way of reckoning time, -you may also alter the distance in space between two events. Thus space -and time are no longer independent, any more than the three dimensions -of space are. We still need four quantities to determine the position -of an event, but we cannot, as before, divide off one of the four as -quite independent of the other three. - -It is not quite true to say that there is no longer any distinction -between time and space. As we have seen, there are time-like intervals -and space-like intervals. But the distinction is of a different sort -from that which was formerly assumed. There is no longer a universal -time which can be applied without ambiguity to any part of the -universe; there are only the various “proper” times of the various -bodies in the universe, which agree approximately for two bodies which -are not in rapid relative motion, but never agree exactly except for -two bodies which are at rest relatively to each other. - -The picture of the world which is required for this new state of -affairs is as follows: Suppose an event =E= occurs to me, and -simultaneously a flash of light goes out from me in all directions. -Anything that happens to any body after the light from the flash has -reached it is definitely after the event =E= in any system of reckoning -time. Any event anywhere which I could have seen before the event =E= -occurred to me is definitely before the event =E= in any system of -reckoning time. But any event which happened in the intervening time -is not definitely either before or after the event =E=. To make the -matter definite: suppose I could observe a person in Sirius, and he -could observe me. Anything which he does, and which I see before the -event =E= occurs to me, is definitely before =E=; anything he does -after he has seen the event =E= is definitely after =E=. But anything -that he does before he sees the event =E=, but so that I see it after -the event =E= has happened, is not definitely before or after =E=. -Since light takes many years to travel from Sirius to the earth, this -gives a period of twice as many years in Sirius which may be called -“contemporary” with =E=, since these years are not definitely before or -after =E=. - -Dr. A. A. Robb, in his _Theory of Time and Space_, suggests a point -of view which may or may not be philosophically fundamental, but is -at any rate a help in understanding the state of affairs we have -been describing. He maintains that one event can only be said to be -definitely _before_ another if it can influence that other in some -way. Now influences spread from a center at varying rates. Newspapers -exercise an influence emanating from London at an average rate of about -twenty miles an hour—rather more for long distances. Anything a man -does because of what he reads in the newspaper is clearly subsequent -to the printing of the newspaper. Sounds travel much faster: it would -be possible to arrange a series of loud speakers along the main roads, -and have newspapers shouted from each to the next. But telegraphing is -quicker, and wireless telegraphy travels with the velocity of light, -so that nothing quicker can ever be hoped for. Now what a man does in -consequence of receiving a wireless message he does _after_ the message -was sent; the meaning here is quite independent of conventions as to -the measurement of time. But anything that he does while the message -is on its way cannot be influenced by the sending of the message, and -cannot influence the sender until some little time after he sent the -message. That is to say, if two bodies are widely separated, neither -can influence the other except after a certain lapse of time; what -happens before that time has elapsed cannot affect the distant body. -Suppose, for instance, that some notable event happens on the sun: -there is a period of sixteen minutes on the earth during which no event -on the earth can have influenced or been influenced by the said notable -event on the sun. This gives a substantial ground for regarding that -period of sixteen minutes on the earth as neither before nor after the -event on the sun. - -The paradoxes of the special theory of relativity are only paradoxes -because we are unaccustomed to the point of view, and in the habit -of taking things for granted when we have no right to do so. This is -especially true as regards the measurement of lengths. In daily life, -our way of measuring lengths is to apply a foot rule or some other -measure. At the moment when the foot rule is applied, it is at rest -relatively to the body which is being measured. Consequently the length -that we arrive at by measurement is the “proper” length, that is to -say, the length as estimated by an observer who shares the motion of -the body. We never, in ordinary life, have to tackle the problem of -measuring a body which is in continual motion. And even if we did, the -velocities of visible bodies on the earth are so small relatively to -the earth that the anomalies dealt with by the theory of relativity -would not appear. But in astronomy, or in the investigation of atomic -structure, we are faced with problems which cannot be tackled in this -way. Not being Joshua, we cannot make the sun stand still while we -measure it; if we are to estimate its size, we must do so while it is -in motion relatively to us. And similarly if you want to estimate the -size of an electron, you have to do so while it is in rapid motion, -because it never stands still for a moment. This is the sort of problem -with which the theory of relativity is concerned. Measurement with a -foot rule, when it is possible, gives always the same result, because -it gives the “proper” length of a body. But when this method is not -possible, we find that curious things happen, particularly if the -body to be measured is moving very fast relatively to the observer. A -figure like the one at the end of the previous chapter will help us to -understand the state of affairs. - -[Illustration] - -Let us suppose that the body on which we wish to measure lengths -is moving relatively to ourselves, and that in one second it moves -the distance =OM=. Let us draw a circle round =O= whose radius is -the distance that light travels in a second. Through =M= draw =MP= -perpendicular to =OM=, meeting the circle in =P=. Thus =OP= is the -distance that light travels in a second. The ratio of =OP= to =OM= -is the ratio of the velocity of light to the velocity of the body. -The ratio of =OP= to =MP= is the ratio in which apparent lengths are -altered by the motion. That is to say, if the observer judges that -two points in the line of motion on the moving body are at a distance -from each other represented by =MP=, a person moving with the body -would judge that they were at a distance represented (on the same -scale) by =OP=. Distances on the moving body at right angles to the -line of motion are not affected by the motion. The whole thing is -reciprocal; that is to say, if an observer moving with the body were to -measure lengths on the previous observer’s body, they would be altered -in just the same proportion. When two bodies are moving relatively -to each other, lengths on either appear shorter to the other than -to themselves. This is the Fitzgerald contraction, which was first -invented to account for the result of the Michelson-Morley experiment. -But it now emerges naturally from the fact that the two observers do -not make the same judgment of simultaneity. - -The way in which simultaneity comes in is this: We say that two points -on a body are a foot apart when we can _simultaneously_ apply one end -of a foot rule to the one and the other end to the other. If, now, two -people disagree about simultaneity, and the body is in motion, they -will obviously get different results from their measurements. Thus the -trouble about time is at the bottom of the trouble about distance. - -The ratio of =OP= to =MP= is the essential thing in all these matters. -Times and lengths and masses are all altered in this proportion when -the body concerned is in motion relatively to the observer. It will -be seen that, if =OM= is very much smaller than =OP=, that is to say, -if the body is moving very much more slowly than light, =MP= and =OP= -are very nearly equal, so that the alterations produced by the motion -are very small. But if =OM= is nearly as large as =OP=, that is to -say, if the body is moving nearly as fast as light, =MP= becomes very -small compared to =OP=, and the effects become very great. The apparent -increase of mass in swiftly moving particles had been observed, -and the right formula had been found, before Einstein invented his -special theory of relativity. In fact, Lorentz had arrived at the -formulæ called the “Lorentz transformation,” which embody the whole -mathematical essence of the special theory of relativity. But it was -Einstein who showed that the whole thing was what we ought to have -expected, and not a set of makeshift devices to account for surprising -experimental results. Nevertheless, it must not be forgotten that -experimental results were the original motive of the whole theory, -and have remained the ground for undertaking the tremendous logical -reconstruction involved in Einstein’s theories. - -We may now recapitulate the reasons which have made it necessary to -substitute “space-time” for space and time. The old separation of -space and time rested upon the belief that there was no ambiguity in -saying that two events in distant places happened at the same time; -consequently it was thought that we could describe the topography of -the universe at a given instant in purely spatial terms. But now that -simultaneity has become relative to a particular observer, this is -no longer possible. What is, for one observer, a description of the -state of the world at a given instant, is, for another observer, a -series of events at various different times, whose relations are not -merely spatial but also temporal. For the same reason, we are concerned -with _events_, rather than with _bodies_. In the old theory, it was -possible to consider a number of bodies all at the same instant, and -since the time was the same for all of them it could be ignored. But -now we cannot do that if we are to obtain an objective account of -physical occurrences. We must mention the date at which a body is to be -considered, and thus we arrive at an “event,” that is to say, something -which happens at a given time. When we know the time and place of an -event in one observer’s system of reckoning, we can calculate its time -and place according to another observer. But we must know the time as -well as the place, because we can no longer ask what is its place for -the new observer at the “same” time as for the old observer. There is -no such thing as the “same” time for different observers, unless they -are at rest relatively to each other. We need four measurements to -fix a position, and four measurements fix the position of an event in -space-time, not merely of a body in space. Three measurements are not -enough to fix any position. That is the essence of what is meant by the -substitution of space-time for space and time. - - - - -CHAPTER VI: THE SPECIAL THEORY OF RELATIVITY - - -The special theory of relativity arose as a way of accounting for the -facts of electromagnetism. We have here a somewhat curious history. In -the eighteenth and early nineteenth centuries the theory of electricity -was wholly dominated by the Newtonian analogy. Two electric charges -attract each other if they are of different kinds, one positive and -one negative, but repel each other if they are of the same kind; in -each case, the force varies as the inverse square of the distance, -as in the case of gravitation. This force was conceived as an action -at a distance, until Faraday, by a number of remarkable experiments, -demonstrated the effect of the intervening medium. Faraday was no -mathematician; Clerk Maxwell first gave a mathematical form to the -results suggested by Faraday’s experiments. Moreover Clerk Maxwell gave -grounds for thinking that light is an electromagnetic phenomenon, -consisting of electromagnetic waves. The medium for the transmission of -electromagnetic effects could therefore be taken to be the ether, which -had long been assumed for the transmission of light. The correctness -of Maxwell’s theory of light was proved by the experiments of Hertz in -manufacturing electromagnetic waves; these experiments afforded the -basis for wireless telegraphy. So far, we have a record of triumphant -progress, in which theory and experiment alternately assume the leading -role. At the time of Hertz’s experiments, the ether seemed securely -established, and in just as strong a position as any other scientific -hypothesis not capable of direct verification. But a new set of facts -began to be discovered, and gradually the whole picture was changed. - -The movement which culminated with Hertz was a movement for making -everything continuous. The ether was continuous, the waves in it were -continuous, and it was hoped that matter would be found to consist -of some continuous structure in the ether. Then came the discovery -of the electron, a small finite unit of negative electricity, and -the proton, a small finite unit of positive electricity. The most -modern view is that electricity is never found except in the form of -electrons and protons; all electrons have the same amount of negative -electricity, and all protons have an exactly equal and opposite amount -of positive electricity. It appeared that an electric current, which -had been thought of as a continuous phenomenon, consists of electrons -traveling one way and positive ions traveling the other way; it is no -more strictly continuous than the stream of people going up and down -an escalator. Then came the discovery of quanta, which seems to show -a fundamental discontinuity in all such natural processes as can be -measured with sufficient precision. Thus physics has had to digest new -facts and face new problems. - -But the problems raised by the electron and the quantum are not those -that the theory of relativity can solve, at any rate at present; as -yet, it throws no light upon the discontinuities which exist in nature. -The problems solved by the special theory of relativity are typified by -the Michelson-Morley experiment. Assuming the correctness of Maxwell’s -theory of electromagnetism, there should have been certain discoverable -effects of motion through the ether; in fact, there were none. Then -there was the observed fact that a body in very rapid motion appears -to increase its mass; the increase is in the ratio of =OP= to =MP= -in the figure in the preceding chapter. Facts of this sort gradually -accumulated, until it became imperative to find some theory which would -account for them all. - -Maxwell’s theory reduced itself to certain equations, known as -“Maxwell’s equations.” Through all the revolutions which physics has -undergone in the last fifty years, these equations have remained -standing; indeed they have continually grown in importance as well as -in certainty—for Maxwell’s arguments in their favor were so shaky that -the correctness of his results must almost be ascribed to intuition. -Now these equations were, of course, obtained from experiments in -terrestrial laboratories, but there was a tacit assumption that the -motion of the earth through the ether could be ignored. In certain -cases, such as the Michelson-Morley experiment, this ought not to have -been possible without measurable error; but it turned out to be always -possible. Physicists were faced with the odd difficulty that Maxwell’s -equations were more accurate than they should be. A very similar -difficulty was explained by Galileo at the very beginning of modern -physics. Most people think that if you let a weight drop it will fall -vertically. But if you try the experiment in the cabin of a moving -ship, the weight falls, in relation to the cabin, just as if the ship -were at rest; for instance, if it starts from the middle of the ceiling -it will drop onto the middle of the floor. That is to say, from the -point of view of an observer on the shore it does not fall vertically, -since it shares the motion of the ship. So long as the ship’s motion -is steady, everything goes on inside the ship as if the ship were not -moving. Galileo explained how this happens, to the great indignation -of the disciples of Aristotle. In orthodox physics, which is derived -from Galileo, a uniform motion in a straight line has no discoverable -effects. This was, in its day, as astonishing a form of relativity -as that of Einstein is to us. Einstein, in the special theory of -relativity, set to work to show how electromagnetic phenomena could be -unaffected by uniform motion through the ether if there be an ether. -This was a more difficult problem, which could not be solved by merely -adhering to the principles of Galileo. - -The really difficult effort required for solving this problem was in -regard to time. It was necessary to introduce the notion of “proper” -time which we have already considered, and to abandon the old belief in -one universal time. The quantitative laws of electromagnetic phenomena -are expressed in Maxwell’s equations, and these equations are found -to be true for any observer, however he may be moving.[3] It is a -straight-forward mathematical problem to find out what differences -there must be between the measures applied by one observer and the -measures applied by another, if, in spite of their relative motion, -they are to find the same equations verified. The answer is contained -in the “Lorentz transformation,” found as a formula by Lorentz, but -interpreted and made intelligible by Einstein. - -The Lorentz transformation tells us what estimate of distances and -periods of time will be made by an observer whose relative motion is -known, when we are given those of another observer. We may suppose that -you are in a train on a railway which travels due east. You have been -traveling for a time which, by the clocks at the station from which -you started, is _t_. At a distance _x_ from your starting point, as -measured by the people on the line, an event occurs at this moment—say -the line is struck by lightning. You have been traveling all the time -with a uniform velocity _v_. The question is: How far from you will you -judge that this event has taken place, and how long after you started -will it be by your watch, assuming that your watch is correct from the -point of view of an observer on the train? - -[3] So long as he has no considerable acceleration. The treatment of -acceleration belongs to the _general_ theory of relativity. - -Our solution of this problem has to satisfy certain conditions. It has -to bring out the result that the velocity of light is the same for all -observers, however they may be moving. And it has to make physical -phenomena—in particular, those of electromagnetism—obey the same -laws for different observers, however they may find their measures of -distances and times affected by their motion. And it has to make all -such effects on measurement reciprocal. That is to say, if you are in -a train and your motion affects your estimate of distances outside the -train, there must be an exactly similar change in the estimate which -people outside the train make of distances inside it. These conditions -are sufficient to determine the solution of the problem, but the -method of obtaining the solution cannot be explained without more -mathematics than is possible in the present work. - -Before dealing with the matter in general terms, let us take an -example. Let us suppose that you are in a train on a long straight -railway, and that you are traveling at three-fifths of the velocity -of light. Suppose that you measure the length of your train, and find -that it is a hundred yards. Suppose that the people who catch a glimpse -of you as you pass succeed, by skilful scientific methods, in taking -observations which enable them to calculate the length of your train. -If they do their work correctly, they will find that it is eighty -yards long. Everything in the train will seem to them shorter in the -direction of the train than it does to you. Dinner plates, which you -see as ordinary circular plates, will look to the outsider as if they -were oval: they will seem only four-fifths as broad in the direction -in which the train is moving as in the direction of the breadth of the -train. And all this is reciprocal. Suppose you see out of the window a -man carrying a fishing rod which, by his measurement, is fifteen feet -long. If he is holding it upright, you will see it as he does; so you -will if he is holding it horizontally at right angles to the railway. -But if he is pointing it along the railway, it will seem to you to -be only twelve feet long. All lengths in the direction of motion are -diminished by twenty per cent, both for those who look into the train -from outside and for those who look out of the train from inside. - -But the effects in regard to time are even more strange. This matter -has been explained with almost ideal lucidity by Eddington in _Space, -Time and Gravitation_. He supposes an aviator traveling, relatively to -the earth, at a speed of 161,000 miles a second, and he says: - -“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 traveling 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.” - -What a situation for envy! Each man thinks that the other’s cigar lasts -twice as long as his own. It may, however, be some consolation to -reflect that the other man’s visits to the dentist also last twice as -long. - -This question of time is rather intricate, owing to the fact that -events which one man judges to be simultaneous another considers to be -separated by a lapse of time. In order to try to make clear how time -is affected, I shall revert to our railway train traveling due east at -a rate three-fifths of that of light. For the sake of illustration, I -assume that the earth is large and flat, instead of small and round. - -If we take events which happen at a fixed point on the earth, and ask -ourselves how long after the beginning of the journey they will seem to -be to the traveler, the answer is that there will be that retardation -that Eddington speaks of, which means in this case that what seems an -hour in the life of the stationary person is judged to be an hour and a -quarter by the man who observes him from the train. Reciprocally, what -seems an hour in the life of the person in the train is judged by the -man observing him from outside to be an hour and a quarter. Each makes -periods of time observed in the life of the other a quarter as long -again as they are to the person who lives through them. The proportion -is the same in regard to times as in regard to lengths. - -But when, instead of comparing events at the same place on the earth, -we compare events at widely separated places, the results are still -more odd. Let us now take all the events along the railway which, from -the point of view of a person who is stationary on the earth, happen -at a given instant, say the instant when the observer in the train -passes the stationary person. Of these events, those which occur at -points towards which the train is moving will seem to the traveler to -have already happened, while those which occur at points behind the -train will, for him, be still in the future. When I say that events -in the forward direction will seem to have already happened, I am -saying something not strictly accurate, because he will not yet have -seen them; but when he does see them, he will, after allowing for the -velocity of light, come to the conclusion that they must have happened -before the moment in question. An event which happens in the forward -direction along the railway, and which the stationary observer judges -to be now (or rather, will judge to have been now when he comes to know -of it), if it occurs at a distance along the line which light could -travel in a second, will be judged by the traveler to have occurred -three-quarters of a second ago. If it occurs at a distance from the two -observers which the man on the earth judges that light could travel -in a year, the traveler will judge (when he comes to know of it) that -it occurred nine months earlier than the moment when he passed the -earth dweller. And generally, he will ante-date events in the forward -direction along the railway by three-quarters of the time that it would -take light to travel from them to the man on the earth whom he is just -passing, and who holds that these events are happening now—or rather, -will hold that they happened now when the light from them reaches him. -Events happening on the railway behind the train will be post-dated by -an exactly equal amount. - -We have thus a two-fold correction to make in the date of an event when -we pass from the terrestrial observer to the traveler. We must first -take five-fourths of the time as estimated by the earth dweller, and -then subtract three-fourths of the time that it would take light to -travel from the event in question to the earth dweller. - -Take some event in a distant part of the universe, which becomes -visible to the earth dweller and the traveler just as they pass each -other. The earth dweller, if he knows how far off the event occurred, -can judge how long ago it occurred, since he knows the speed of light. -If the event occurred in the direction towards which the traveler is -moving, the traveler will infer that it happened twice as long ago as -the earth dweller thinks. But if it occurred in the direction from -which he has come, he will argue that it happened only half as long -ago as the earth dweller thinks. If the traveler moves at a different -speed, these proportions will be different. - -Suppose now that (as sometimes occurs) two new stars have suddenly -flared up, and have just become visible to the traveler and to the -earth dweller whom he is passing. Let one of them be in the direction -towards which the train is traveling, the other in the direction from -which it has come. Suppose that the earth dweller is able, in some way, -to estimate the distance of the two stars, and to infer that light -takes fifty years to reach him from the one in the direction towards -which the traveler is moving, and one hundred years to reach him from -the other. He will then argue that the explosion which produced the -new star in the forward direction occurred fifty years ago, while the -explosion which produced the other new star occurred a hundred years -ago. The traveler will exactly reverse these figures: he will infer -that the forward explosion occurred a hundred years ago, and the -backward one fifty years ago. I assume that both argue correctly on -correct physical data. In fact, both are right, unless they imagine -that the other must be wrong. It should be noted that both will have -the same estimate of the velocity of light, because their estimates -of the distances of the two new stars will vary in exactly the same -proportion as their estimates of the times since the explosions. -Indeed, one of the main motives of this whole theory is to secure that -the velocity of light shall be the same for all observers, however they -may be moving. This fact, established by experiment, was incompatible -with the old theories, and made it absolutely necessary to admit -something startling. The theory of relativity is just as little -startling as is compatible with the facts. Indeed, after a time, it -ceases to seem startling at all. - -There is another feature of very great importance in the theory we -have been considering, and that is that, although distances and times -vary for different observers, we can derive from them the quantity -called “interval,” which is the same for all observers. The “interval,” -in the special theory of relativity, is obtained as follows: Take -the square of the distance between two events, and the square of the -distance traveled by light in the time between the two events; subtract -the lesser of these from the greater, and the result is defined as -the square of the interval between the events. The interval is the -same for all observers, and represents a genuine physical relation -between the two events, which the time and the distance do not. We -have already given a geometrical construction for the interval at the -end of Chapter IV; this gives the same result as the above rule. The -interval is “time-like” when the time between the events is longer than -light would take to travel from the place of the one to the place -of the other; in the contrary case it is “space-like.” When the time -between the two events is exactly equal to the time taken by light to -travel from one to the other, the interval is zero; the two events are -then situated on parts of one light ray, unless no light happens to be -passing that way. - -When we come to the general theory of relativity, we shall have to -generalize the notion of interval. The more deeply we penetrate into -the structure of the world, the more important this concept becomes; -we are tempted to say that it is the reality of which distances and -periods of time are confused representations. The theory of relativity -has altered our view of the fundamental structure of the world; that is -the source both of its difficulty and of its importance. - -The remainder of this chapter may be omitted by readers who have not -even the most elementary acquaintance with geometry or algebra. But for -the benefit of those whose education has not been _entirely_ neglected, -I will add a few explanations of the general formula of which I have -hitherto given only particular examples. The general formula in -question is the “Lorentz transformation,” which tells, when one body -is moving in a given manner relatively to another, how to infer the -measures of lengths and times appropriate to the one body from those -appropriate to the other. Before giving the algebraical formulæ, I -will give a geometrical construction. As before, we will suppose that -there are two observers, whom we will call =O= and =O=′, one of whom is -stationary on the earth while the other is traveling at a uniform speed -along a straight railway. At the beginning of the time considered, the -two observers were at the same point of the railway, but now they are -separated by a certain distance. A flash of lightning strikes a point -=X= on the railway, and =O= judges that at the moment when the flash -takes place the observer in the train has reached the point =O=′. The -problem is: how far will =O=′ judge that he is from the flash, and -how long after the beginning of the journey (when he was at =O=) will -he judge that the flash took place? We are supposed to know =O=′s -estimates, and we want to calculate those of =O=′. - -[Illustration] - -In the time that, according to =O=, has elapsed since the beginning of -the journey, let =OC= be the distance that light would have traveled -along the railway. Describe a circle about =O=, with =OC= as radius, -and through =O′= draw a perpendicular to the railway, meeting the -circle in =D=. On =OD= take a point =Y= such that =OY= is equal to =OX= -(=X= is the point of the railway where the lightning strikes). Draw -=YM= perpendicular to the railway, and =OS= perpendicular to =OD=. Let -=YM= and =OS= meet in =S=. Also let =DO′= produced and =OS= produced -meet in =R=. Through =X= and =C= draw perpendiculars to the railway -meeting =OS= produced in =Q= and =Z= respectively. Then =RQ= (as -measured by =O=) is the distance at which =O′= will believe himself to -be from the flash, not =O′X= as it would be according to the old view. -And whereas =O= thinks that, in the time from the beginning of the -journey to the flash, light would travel a distance =OC=, =O′= thinks -that the time elapsed is that required for light to travel the distance -=SZ= (as measured by =O=). The interval as measured by =O= is got by -subtracting the square on =OX= from the square on =OC=; the interval -as measured by =O′= is got by subtracting the square on =RQ= from the -square on =SZ=. A little very elementary geometry shows that these are -equal. - -The algebraical formulæ embodied in the above construction are as -follows: From the point of view of =O=, let an event occur at a -distance _x_ along the railway, and at a time _t_ after the beginning -of the journey (when =O′= was at =O=). From the point of view of =O′=, -let the same event occur at a distance _x′_ along the railway, and at a -time _t′_ after the beginning of the journey. Let _c_ be the velocity -of light, and _v_ the velocity of =O′= relative to =O=. Put - - _c_ - β = ———————————— - √(_c_² - _v_²) - -Then - - _x′_ = β(_x_ - _vt_) - - ( _vx_ ) - _t′_ = β(_t_ - —————) - ( _c²_ ) - -This is the Lorentz transformation, from which everything in this -chapter can be deduced. - - - - -CHAPTER VII: INTERVALS IN SPACE-TIME - - -The special theory of relativity, which we have been considering -hitherto, solved completely a certain definite problem: to account for -the experimental fact that, when two bodies are in uniform relative -motion, all the laws of physics, both those of ordinary dynamics and -those connected with electricity and magnetism, are exactly the same -for the two bodies. “Uniform” motion, here, means motion in a straight -line with constant velocity. But although one problem was solved by -the special theory, another was immediately suggested: what if the -motion of the two bodies is not uniform? Suppose, for instance, that -one is the earth while the other is a falling stone. The stone has -an accelerated motion: it is continually falling faster and faster. -Nothing in the special theory enables us to say that the laws of -physical phenomena will be the same for an observer on the stone as for -one on the earth. This is particularly awkward, as the earth itself -is, in an extended sense, a falling body: It has at every moment -an acceleration[4] towards the sun, which makes it go round the sun -instead of moving in a straight line. As our knowledge of physics is -derived from experiments on the earth, we cannot rest satisfied with -a theory in which the observer is supposed to have no acceleration. -The general theory of relativity removes this restriction, and allows -the observer to be moving in any way, straight or crooked, uniformly -or with an acceleration. In the course of removing the restriction, -Einstein was led to his new law of gravitation, which we shall consider -presently. The work was extraordinarily difficult, and occupied him for -ten years. The special theory dates from 1905, the general theory from -1915. - -[4] This does not mean that its velocity is increasing, but that it -is changing its direction. The only sort of motion which is called -“unaccelerated” is motion with uniform velocity _in a straight line_. - -It is obvious from experiences with which we are all familiar that an -accelerated motion is much more difficult to deal with than a uniform -one. When you are in a train which is traveling steadily, the motion -is not noticeable so long as you do not look out of the window; but -when the brakes are applied suddenly you are precipitated forwards, -and you become aware that something is happening without having to -notice anything outside the train. Similarly in a lift everything -seems ordinary while it is moving steadily, but at starting and -stopping, when its motion is accelerated, you have odd sensations -in the pit of the stomach. (We call a motion “accelerated” when it -is getting slower as well as when it is getting quicker; when it is -getting slower the acceleration is negative.) The same thing applies -to dropping a weight in the cabin of a ship. So long as the ship is -moving uniformly, the weight will behave, relatively to the cabin, -just as if the ship were at rest: if it starts from the middle of -the ceiling, it will hit the middle of the floor. But if there is an -acceleration everything is changed. If the boat is increasing its -speed very rapidly, the weight will seem to an observer in the cabin -to fall in a curve directed towards the stern; if the speed is being -rapidly diminished, the curve will be directed towards the bow. All -these facts are familiar, and they led Galileo and Newton to regard an -accelerated motion as something radically different, in its own nature, -from a uniform motion. But this distinction could only be maintained by -regarding motion as absolute, not relative. If all motion is relative, -the earth is accelerated relatively to the lift just as truly as the -lift relatively to the earth. Yet the people on the ground have no -sensations in the pits of their stomachs when the lift starts to go -up. This illustrates the difficulty of our problem. In fact, though -few physicists in modern times have believed in absolute motion, the -technique of mathematical physics still embodied Newton’s belief in it, -and a revolution in method was required to obtain a technique free from -this assumption. This revolution was accomplished in Einstein’s general -theory of relativity. - -It is somewhat optional where we begin in explaining the new ideas -which Einstein introduced, but perhaps we shall do best by taking the -conception of “interval.” This conception, as it appears in the special -theory of relativity, is already a generalization of the traditional -notion of distance in space and time; but it is necessary to generalize -it still further. However, it is necessary first to explain a certain -amount of history, and for this purpose we must go back as far as -Pythagoras. - -Pythagoras, like many of the greatest characters in history, perhaps -never existed: he is a semi-mythical character, who combined -mathematics and priestcraft in uncertain proportions. I shall, however, -assume that he existed, and that he discovered the theorem attributed -to him. He was roughly a contemporary of Confucius and Buddha; he -founded a religious sect, which thought it wicked to eat beans, -and a school of mathematicians, who took a particular interest in -right-angled triangles. The theorem of Pythagoras (the forty-seventh -proposition of Euclid) states that the sum of the squares on the two -shorter sides of a right-angled triangle is equal to the square on -the side opposite the right angle. No proposition in the whole of -mathematics has had such a distinguished history. We all learned to -“prove” it in youth. It is true that the “proof” proved nothing, and -that the only way to prove it is by experiment. It is also the case -that the proposition is not _quite_ true—it is only approximately -true. But everything in geometry, and subsequently in physics, has been -derived from it by successive generalizations. The latest of these -generalizations is the general theory of relativity. - -The theorem of Pythagoras was itself, in all probability, a -generalization of an Egyptian rule of thumb. In Egypt, it had been -known for ages that a triangle whose sides are 3, 4, and 5 units of -length is a right-angled triangle; the Egyptians used this knowledge -practically in measuring their fields. Now if the sides of a triangle -are 3, 4, and 5 inches, the squares on these sides will contain -respectively 9, 16, and 25 square inches; and 9 and 16 added together -make 25. Three times three is written “3²”; four times four, “4²”; five -times five, “5².” So that we have - - 3² + 4² = 5². - -It is supposed that Pythagoras noticed this fact, after he had learned -from the Egyptians that a triangle whose sides are 3, 4 and 5 has a -right angle. He found that this could be generalized, and so arrived -at his famous theorem: In a right-angled triangle, the square on the -side opposite the right angle is equal to the sum of the squares on the -other two sides. - -[Illustration] - -Similarly in three dimensions: if you take a right-angled solid block, -the square on the diagonal (the dotted line in the figure) is equal to -the sum of the squares on the three sides. - -This is as far as the ancients got in this matter. - -[Illustration] - -The next step of importance is due to Descartes, who made the theorem -of Pythagoras the basis of his method of analytical geometry. Suppose -you wish to map out systematically all the places on a plain—we will -suppose it small enough to make it possible to ignore the fact that -the earth is round. We will suppose that you live in the middle of the -plain. One of the simplest ways of describing the position of a place -is to say: starting from my house, go first such and such a distance -east, then such and such a distance north (or it may be west in the -first case, and south in the second). This tells you exactly where -the place is. In the rectangular cities of America, it is the natural -method to adopt: in New York you will be told to go so many blocks east -(or west) and then so many blocks north (or south). The distance you -have to go east is called _x_, and the distance you have to go north -is called _y_. (If you have to go west, _x_ is negative; if you have -to go south, _y_ is negative.) Let =O= be your starting point (the -“origin”); let =OM= be the distance you go east, and =MP= the distance -you go north. How far are you from home in a direct line when you reach -=P=? The theorem of Pythagoras gives the answer. The square on =OP= is -the sum of the squares on =OM= and =MP=. If =OM= is four miles, and -=MP= is three miles, =OP= is 5 miles. If =OM= is 12 miles and =MP= is 5 -miles, =OP= is 13 miles, because 12² + 5² = 13². So that if you adopt -Descartes’ method of mapping, the theorem of Pythagoras is essential in -giving you the distance from place to place. In three dimensions the -thing is exactly analogous. Suppose that, instead of wanting merely -to fix positions on the plain, you want to fix stations for captive -balloons above it, you will then have to add a third quantity, the -height at which the balloon is to be. If you call the height _z_, and -if _r_ is the direct distance from =O= to the balloon, you will have - - _r_² = _x_² + _y_² + _z_², - -and from this you can calculate _r_ when you know _x_, _y_, and _z_. -For example, if you can get to the balloon by going 12 miles east, 4 -miles north, and then 3 miles up, your distance from the balloon in a -straight line is 13 miles, because 12 × 12 = 144, 4 × 4 = 16, 3 × 3 = -9, 144 + 16 + 9 = 169 = 13 × 13. - -But now suppose that, instead of taking a small piece of the earth’s -surface which can be regarded as flat, you consider making a map of -the world. An accurate map of the world on flat paper is impossible. -A globe can be accurate, in the sense that everything is produced -to scale, but a flat map cannot be. I am not talking of practical -difficulties, I am talking of a theoretical impossibility. For example: -the northern halves of the meridian of Greenwich and the ninetieth -meridian of west longitude, together with the piece of the equator -between them, make a triangle whose sides are all equal and whose -angles are all right angles. On a flat surface, a triangle of that sort -would be impossible. On the other hand, it is possible to make a square -on a flat surface, but on a sphere it is impossible. Suppose you try on -the earth: walk 100 miles west, then 100 miles north, then 100 miles -east, then 100 miles south. You might think this would make a square, -but it wouldn’t, because you would not at the end have come back to -your starting point. If you have time, you may convince yourself of -this by experiment. If not, you can easily see that it must be so. When -you are nearer the pole, 100 miles takes you through more longitude -than when you are nearer the equator, so that in doing your 100 miles -east (if you are in the northern hemisphere) you get to a point further -east than that from which you started. As you walk due south after -this, you remain further east than your starting point, and end up at a -different place from that in which you began. Suppose, to take another -illustration, that you start on the equator 4,000 miles east of the -Greenwich meridian; you travel till you reach the meridian, then you -travel northwards along it for 4,000 miles, through Greenwich and up -to the neighborhood of the Shetland Islands; then you travel eastward -for 4,000 miles, and then 4,000 miles south. This will take you to the -equator at a point 4,000 miles further east than the point from which -you started. - -In a sense, what we have just been saying is not quite fair, because, -except on the equator, traveling due east is not the shortest route -from a place to another place due east of it. A ship traveling (say) -from New York to Lisbon, which is nearly due east, will start by going -a certain distance northward. It will sail on a “great circle,” that -is to say, a circle whose centre is the centre of the earth. This -is the nearest approach to a straight line that can be drawn on the -surface of the earth. Meridians of longitude are great circles, and so -is the equator, but the other parallels of latitude are not. We ought, -therefore, to have supposed that, when you reach the Shetland Islands, -you travel 4,000 miles, not due east, but along a great circle which -lands you at a point due east of the Shetland Islands. This, however, -only reinforces our conclusion: you will end at a point even further -east of your starting point than before. - -What are the differences between the geometry on a sphere and the -geometry on a plane? If you make a triangle on the earth, whose sides -are great circles, you will not find that the angles of the triangle -add up to two right angles: they will add up to rather more. The amount -by which they exceed two right angles is proportional to the size of -the triangle. On a small triangle such as you could make with strings -on your lawn, or even on a triangle formed by three ships which can -just see each other, the angles will add up to so little more than two -right angles that you will not be able to detect the difference. But -if you take the triangle made by the equator, the Greenwich meridian, -and the ninetieth meridian, the angles add up to _three_ right angles. -And you can get triangles in which the angles add up to anything up to -six right angles. All this you could discover by measurements on the -surface of the earth, without having to take account of anything in the -rest of space. - -The theorem of Pythagoras also will fail for distances on a sphere. -From the point of view of a traveler bound to the earth, the distance -between two places is their great circle distance, that is to say, the -shortest journey that a man can make without leaving the surface of -the earth. Now suppose you take three bits of great circles which make -a triangle, and suppose one of them is at right angles to another—to -be definite, let one be the equator and one a bit of the meridian of -Greenwich going northward from the equator. Suppose you go 3,000 miles -along the equator, and then 4,000 miles due north; how far will you -be from your starting point, estimating the distance along a great -circle? If you were on a plane, your distance would be 5,000 miles, -as we saw before. In fact, however, your great circle distance will be -considerably less than this. In a right-angled triangle on a sphere, -the square on the side opposite the right angle is less than the sum of -the squares on the other two sides. - -These differences between the geometry on a sphere and the geometry on -a plane are intrinsic differences; that is to say, they enable you to -find out whether the surface on which you live is like a plane or like -a sphere, without requiring that you should take account of anything -outside the surface. Such considerations led to the next step of -importance in our subject, which was taken by Gauss, who flourished a -hundred years ago. He studied the theory of surfaces, and showed how to -develop it by means of measurements on the surfaces themselves, without -going outside them. In order to fix the position of a point in space, -we need three measurements; but in order to fix the position of a point -on a surface we need only two—for example, a point on the earth’s -surface is fixed when we know its latitude and longitude. - -Now Gauss found that, whatever system of measurement you adopt, -and whatever the nature of the surface, there is always a way of -calculating the distance between two not very distant points of the -surface, when you know the quantities which fix their positions. -The formula for the distance is a generalization of the formula of -Pythagoras: it tells you the square of the distance in terms of the -squares of the differences between the measure quantities which fix -the points, and also the product of these two quantities. When you -know this formula, you can discover all the intrinsic properties of -the surface, that is to say, all those which do not depend upon its -relations to points outside the surface. You can discover, for example, -whether the angles of a triangle add up to two right angles, or more, -or less, or more in some cases and less in others. - -But when we speak of a “triangle,” we must explain what we mean, -because on most surfaces there are no straight lines. On a sphere, we -shall replace straight lines by great circles, which are the nearest -possible approach to straight lines. In general, we shall take, -instead of straight lines, the lines that give the shortest route on -the surface from place to place. Such lines are called “geodesics.” -On the earth, the geodesics are great circles. In general, they are -the shortest way of traveling from point to point if you are unable -to leave the surface. They take the place of straight lines in the -intrinsic geometry of a surface. When we inquire whether the angles of -a triangle add up to two right angles or not, we mean to speak of a -triangle whose sides are geodesics. And when we speak of the distance -between two points, we mean the distance along a geodesic. - -The next step in our generalizing process is rather difficult: it is -the transition to non-Euclidean geometry. We live in a world in which -space has three dimensions, and our empirical knowledge of space is -based upon measurement of small distances and of angles. (When I speak -of small distances, I mean distances that are small compared to those -in astronomy; all distances on the earth are small in this sense.) It -was formerly thought that we could be sure _à priori_ that space is -Euclidean—for instance, that the angles of a triangle add up to two -right angles. But it came to be recognized that we could not prove this -by reasoning; if it was to be known, it must be known as the result -of measurements. Before Einstein, it was thought that measurements -confirm Euclidean geometry within the limits of exactitude attainable; -now this is no longer thought. It is still true that we can, by what -may be called a natural artifice, cause Euclidean geometry to _seem_ -true throughout a small region, such as the earth; but in explaining -gravitation Einstein is led to the view that over large regions where -there is matter we cannot regard space as Euclidean. The reasons for -this will concern us later. What concerns us now is the way in which -non-Euclidean geometry results from a generalization of the work of -Gauss. - -There is no reason why we should not have the same circumstances in -three-dimensional space as we have, for example, on the surface of a -sphere. It might happen that the angles of a triangle would always -add up to more than two right angles, and that the excess would be -proportional to the size of the triangle. It might happen that the -distance between two points would be given by a formula analogous -to what we have on the surface of a sphere, but involving three -quantities instead of two. Whether this does happen or not, can only -be discovered by actual measurements. There are an infinite number of -such possibilities. - -This line of argument was developed by Riemann, in his dissertation -“On the hypotheses which underlie geometry” (1854), which applied -Gauss’s work on surfaces to different kinds of three-dimensional -spaces. He showed that all the essential characteristics of a kind -of space could be deduced from the formula for small distances. He -assumed that, from the small distances in three given directions -which would together carry you from one point to another not far from -it, the distances between the two points could be calculated. For -instance, if you know that you can get from one point to another by -first moving a certain distance east, then a certain distance north, -and finally a certain distance straight up in the air, you are to be -able to calculate the distance from the one point to the other. And -the rule for the calculation is to be an extension of the theorem of -Pythagoras, in the sense that you arrive at the square of the required -distance by adding together multiples of the squares of the component -distances, together possibly with multiples of their products. From -certain characteristics in the formula, you can tell what sort of -space you have to deal with. These characteristics do not depend upon -the particular method you have adopted for determining the positions of -points. - -In order to arrive at what we want for the theory of relativity, we -now have one more generalization to make: we have to substitute the -“interval” between events for the distance between points. This takes -us to space-time. We have already seen that, in the special theory -of relativity, the square of the interval is found by subtracting -the square of the distance between the events from the square of the -distance that light would travel in the time between them. In the -general theory, we do not assume this special form of interval, except -at a great distance from matter. Elsewhere, we assume to begin with a -general form, like that which Riemann used for distances. Moreover, -like Riemann, Einstein only assumes his formula for _neighboring_ -events, that is to say, events which have only a small interval -between them. What goes beyond these initial assumptions depends upon -observation of the actual motion of bodies, in ways which we shall -explain in later chapters. - -We may now sum up and re-state the process we have been describing. -In three dimensions, the position of a point relatively to a fixed -point (the “origin”) can be determined by assigning three quantities -(“co-ordinates”). For example, the position of a balloon relatively to -your house is fixed if you know that you will reach it by going first -a given distance due east, then another given distance due north, -then a third given distance straight up. When, as in this case, the -three co-ordinates are three distances all at right angles to each -other, which, taken successively, transport you from the origin to the -point in question, the square of the direct distance to the point in -question is got by adding up the squares of the three co-ordinates. In -all cases, whether in Euclidean or in non-Euclidean spaces, it is got -by adding multiples of the squares and products of the co-ordinates -according to an assignable rule. The co-ordinates may be any quantities -which fix the position of a point, provided that neighboring points -must have neighboring quantities for their co-ordinates. In the general -theory of relativity, we add a fourth co-ordinate to give the time, and -our formula gives “interval” instead of spatial distance; moreover we -assume the accuracy of our formula for small distances only. We assume -further that, at great distances from matter, the formula approximates -more and more closely to the formula for interval which is used in the -special theory. - -We are now at last in a position to tackle Einstein’s theory of -gravitation. - - - - -CHAPTER VIII: EINSTEIN’S LAW OF GRAVITATION - - -Before tackling Einstein’s new law, it is as well to convince -ourselves, on logical grounds, that Newton’s law of gravitation cannot -be quite right. - -Newton said that between any two particles of matter there is a force -which is proportional to the product of their masses and inversely -proportional to the square of their distance. That is to say, ignoring -for the present the question of mass, if there is a certain attraction -when the particles are a mile apart, there will be a quarter as much -attraction when they are two miles apart, a ninth as much when they -are three miles apart, and so on: the attraction diminishes much -faster than the distance increases. Now, of course, Newton, when he -spoke of the distance, meant the distance at a given time: He thought -there could be no ambiguity about time. But we have seen that this -was a mistake. What one observer judges to be the same moment on the -earth and the sun, another will judge to be two different moments. -“Distance at a given moment” is therefore a subjective conception, -which can hardly enter into a cosmic law. Of course, we could make -our law unambiguous by saying that we are going to estimate times as -they are estimated by Greenwich Observatory. But we can hardly believe -that the accidental circumstances of the earth deserve to be taken so -seriously. And the estimate of distance, also, will vary for different -observers. We cannot, therefore, allow that Newton’s form of the law of -gravitation can be quite correct, since it will give different results -according to which of many equally legitimate conventions we adopt. -This is as absurd as it would be if the question whether one man had -murdered another were to depend upon whether they were described by -their Christian names or their surnames. It is obvious that physical -laws must be the same whether distances are measured in miles or in -kilometers, and we are concerned with what is essentially only an -extension of the same principle. - -Our measurements are conventional to an even greater extent than -is admitted by the special theory of relativity. Moreover, every -measurement is a physical process carried out with physical material; -the result is certainly an experimental datum, but may not be -susceptible of the simple interpretation which we ordinarily assign to -it. We are, therefore, not going to assume to begin with that we know -how to measure anything. We assume that there is a certain physical -quantity, called “interval,” which is a relation between two events -that are not widely separated; but we do not assume in advance that we -know how to measure it, beyond taking it for granted that it is given -by some generalization of the theorem of Pythagoras such as we spoke of -in the preceding chapter. - -We do assume, however, that events have an _order_, and that this order -is four-dimensional. We assume, that is to say, that we know what we -mean by saying that a certain event is nearer to another than to a -third, so that before making accurate measurements we can speak of the -“neighborhood” of an event; and we assume that, in order to assign the -position of an event in space-time, four quantities (co-ordinates) are -necessary—_e.g._ in our former case of an explosion on an airship, -latitude, longitude, altitude and time. But we assume nothing about the -way in which these co-ordinates are assigned, except that neighboring -co-ordinates are assigned to neighboring events. - -The way in which these numbers, called co-ordinates, are to be assigned -is neither wholly arbitrary nor a result of careful measurement—it -lies in an intermediate region. While you are making any continuous -journey, your co-ordinates must never alter by sudden jumps. In America -one finds that the houses between (say) Fourteenth Street and Fifteenth -Street are likely to have numbers between 1400 and 1500, while those -between Fifteenth Street and Sixteenth Street have numbers between -1500 and 1600, even if the 1400’s were not used up. This would not do -for our purposes, because there is a sudden jump when we pass from one -block to the next. Or again we might assign the time co-ordinate in the -following way: take the time that elapses between two successive births -of people called Smith; an event occurring between the births of the -3000th and the 3001st Smith known to history shall have a co-ordinate -lying between 3000 and 3001; the fractional part of its co-ordinate -shall be the fraction of a year that has elapsed since the birth of the -3000th Smith. (Obviously there could never be as much as a year between -two successive additions to the Smith family.) This way of assigning -the time co-ordinate is perfectly definite, but it is not admissible -for our purposes, because there will be sudden jumps between events -just before the birth of a Smith and events just after, so that in a -continuous journey your time co-ordinate will not change continuously. -It is assumed that, independently of measurement, we know what a -continuous journey is. And when your position in space-time changes -continuously, each of your four co-ordinates must change continuously. -One, two, or three of them may not change at all; but whatever change -does occur must be smooth, without sudden jumps. This explains what is -_not_ allowable in assigning co-ordinates. - -To explain all the changes that are legitimate in your co-ordinates, -suppose you take a large piece of soft india-rubber. While it is in an -unstretched condition, measure little squares on it, each one-tenth -of an inch each way. Put in little tiny pins at the corners of the -squares. We can take as two of the co-ordinates of one of these pins -the number of pins passed in going to the right from a given pin -until we come just below the pin in question, and then the number of -pins we pass on the way up to this pin. In the figure, let =O= be the -pin we start from and =P= the pin to which we are going to assign -co-ordinates. =P= is in the fifth column and the third row, so its -co-ordinates in the plane of the india-rubber are to be 5 and 3. - -[Illustration: Fig. 1.] - -[Illustration: Fig. 2.] - -Now take the india-rubber and stretch it and twist it as much as -you like. Let the pins now be in the shape they have in Fig. 2. The -divisions now no longer represent distances according to our usual -notions, but they will still do just as well as co-ordinates. We may -still take =P= as having the co-ordinates 5 and 3 in the plane of the -india-rubber; and we may still regard the india-rubber as being in a -plane, even if we have twisted it out of what we should ordinarily -call a plane. Such continuous distortions do not matter. - -To take another illustration: instead of using a steel measuring rod to -fix our co-ordinates, let us use a live eel, which is wriggling all the -time. The distance from the tail to the head of the eel is to count as -one from the point of view of co-ordinates, whatever shape the creature -may be assuming at the moment. The eel is continuous, and its wriggles -are continuous, so it may be taken as our unit of distance in assigning -co-ordinates. Beyond the requirement of continuity, the method of -assigning co-ordinates is purely conventional, and therefore a live eel -is just as good as a steel rod. - -We are apt to think that, for really careful measurements, it is better -to use a steel rod than a live eel. This is a mistake: not because -the eel tells us what the steel rod was thought to tell, but because -the steel rod really tells no more than the eel obviously does. The -point is, not that eels are really rigid, but that steel rods really -wriggle. To an observer in just one possible state of motion, the eel -would appear rigid, while the steel rod would seem to wriggle just -as the eel does to us. For everybody moving differently both from -this observer and ourselves, both the eel and the rod would seem to -wriggle. And there is no saying that one observer is right and another -wrong. In such matters, what is seen does not belong solely to the -physical process observed, but also to the standpoint of the observer. -Measurements of distances and times do not directly reveal properties -of the things measured, but relations of the things to the measurer. -What observation can tell us about the physical world is therefore more -abstract than we have hitherto believed. - -It is important to realize that geometry, as taught in schools since -Greek times, ceases to exist as a separate science, and becomes merged -in physics. The two fundamental notions in elementary geometry were -the straight line and the circle. What appears to you as a straight -road, whose parts all exist now, may appear to another observer to -be like the flight of a rocket, some kind of curve whose parts come -into existence successively. The circle depends upon measurement of -distances, since it consists of all the points at a given distance -from its center. And measurement of distances, as we have seen, is -a subjective affair, depending upon the way in which the observer -is moving. The failure of the circle to have objective validity was -demonstrated by the Michelson-Morley experiment, and is thus, in a -sense, the starting point of the whole theory of relativity. Rigid -bodies, which we need for measurement, are only rigid for certain -observers; for others, they will be constantly changing all their -dimensions. It is only our obstinately earth-bound imagination that -makes us suppose a geometry separate from physics to be possible. - -That is why we do not trouble to give physical significance to our -co-ordinates from the start. Formerly, the co-ordinates used in physics -were supposed to be carefully measured distances; now we realize -that this care at the start is thrown away. It is at a later stage -that care is required. Our co-ordinates now are hardly more than a -systematic way of cataloguing events. But mathematics provides, in -the method of tensors, such an immensely powerful technique that we -can use co-ordinates assigned in this apparently careless way just -as effectively as if we had applied the whole apparatus of minutely -accurate measurement in arriving at them. The advantage of being -haphazard at the start is that we avoid making surreptitious physical -assumptions, which we can hardly help making, if we suppose that our -co-ordinates have initially some particular physical significance. - -We assume that, if two events are close together (but not necessarily -otherwise), there is an interval between them which can be calculated -from the differences between their co-ordinates by some such formula -as we considered in the preceding chapter. That is to say, we take the -squares and products of the differences of co-ordinates, we multiply -them by suitable amounts (which in general will vary from place to -place), and we add the results together. The sum obtained is the -square of the interval. We do not assume in advance that we know the -amounts by which the squares and products must be multiplied; this -is going to be discovered by observing physical phenomena. We know, -however, certain things. We know that the old Newtonian physics is -very nearly accurate when our co-ordinates have been chosen in a -certain way. We know that the special theory of relativity is still -more nearly accurate for suitable co-ordinates. From such facts we can -infer certain things about our new co-ordinates, which, in a logical -deduction, appear as postulates of the new theory. - -As such postulates we take: - - 1. That every body travels in a geodesic in - space-time, except in so far as electromagnetic - forces act upon it. - - 2. That a light ray travels so that the interval - between two parts of it is zero. - - 3. That at a great distance from gravitating matter, - we can transform our co-ordinates by mathematical - manipulation so that the interval shall be what it - is in the special theory of relativity; and that - this is approximately true wherever gravitation is - not very powerful. - -Each of these postulates requires some explanation. - -We saw that a geodesic on a surface is the shortest line that can be -drawn on the surface from one point to another; for example, on the -earth the geodesics are great circles. When we come to space-time, -the mathematics is the same, but the verbal explanations have to be -rather different. In the general theory of relativity, it is only -neighboring events that have a definite interval, independently of -the route by which we travel from one to the other. The interval -between distant events depends upon the route pursued, and has to be -calculated by dividing the route into a number of little bits and -adding up the intervals for the various little bits. If the interval -is space-like, a body cannot travel from one event to the other; -therefore when we are considering the way bodies move, we are confined -to time-like intervals. The interval between neighboring events, when -it is time-like, will appear as the time between them for an observer -who travels from the one event to the other. And so the whole interval -between two events will be judged by a person who travels from one to -the other to be what his clocks show to be the time that he has taken -on the journey. For some routes this time will be longer, for others -shorter; the more slowly the man travels, the longer he will think he -has been on the journey. This must not be taken as a platitude. I am -not saying that if you travel from London to Edinburgh you will take -longer if you travel more slowly. I am saying something much more odd. -I am saying that if you leave London at 10 A.M. and arrive in Edinburgh -at 6.30 P.M. Greenwich time, the more slowly you travel the longer -you will take—if the time is judged by your watch. This is a very -different statement. From the point of view of a person on the earth, -your journey takes eight and a half hours. But if you had been a ray -of light traveling round the solar system, starting from London at 10 -A.M., reflected from Jupiter to Saturn, and so on, until at last you -were reflected back to Edinburgh and arrived there at 6.30 P.M., you -would judge that the journey had taken you exactly no time. And if you -had gone by any circuitous route, which enabled you to arrive in time -by traveling fast, the longer your route the less time you would judge -that you had taken; the diminution of time would be continual as your -speed approached that of light. Now I say that when a body travels, if -it is left to itself, it chooses the route which makes the time between -two stages of the journey as long as possible; if it had traveled from -one event to another by any other route, the time, as measured by its -own clocks, would have been shorter. This is a way of saying that -bodies left to themselves do their journeys as slowly as they can; it -is a sort of law of cosmic laziness. Its mathematical expression is -that they travel in geodesics, in which the total interval between any -two events on the journey is _greater_ than by any alternative route. -(The fact that it is greater, not less, is due to the fact that the -sort of interval we are considering is more analogous to time than to -distance.) For example, if a person could leave the earth and travel -about for a time and then return, the time between his departure and -return would be less by his clocks than by those on the earth: the -earth, in its journey round the sun, chooses the route which makes -the time of any bit of its course by its clocks longer than the time -as judged by clocks which move by a different route. This is what is -meant by saying that bodies left to themselves move in geodesics in -space-time. - -We assume that the body considered is not acted upon by electromagnetic -forces. We are concerned at present with the law of gravitation, not -with the effects of electromagnetism. These effects have been brought -into the framework of the general theory of relativity by Weyl,[5] but -for the present we will ignore his work. The planets, in any case, -are not subject, as wholes, to appreciable electromagnetic forces; it -is only gravitation that has to be considered in accounting for their -motions, with which we are concerned in this chapter. - -[5] See his _Space, Time, Matter_, Methuen, 1922. - -Our second postulate, that a light ray travels so that the interval -between two parts of it is zero, has the advantage that it does not -have to be stated only for _small_ distances. If each little bit of -interval is zero, the sum of them all is zero, and so even distant -parts of the same light ray have a zero interval. The course of a light -ray is also a geodesic according to the definition. Thus we now have -two empirical ways of discovering what are the geodesics in space-time, -namely light rays and bodies moving freely. Among freely-moving -bodies are included all which are not subject to constraints or to -electromagnetic forces, that is to say, the sun, stars, planets and -satellites, and also falling bodies on the earth, at least when they -are falling in a vacuum. When you are standing on the earth, you are -subject to electromagnetic forces: the electrons and protons in the -neighborhood of your feet exert a repulsion on your feet which is just -enough to overcome the earth’s gravitation. This is what prevents you -from falling through the earth, which, solid as it looks, is mostly -empty space. - -The third postulate, which relates the general to the special theory, -is very useful. It is not necessary for the application of the special -theory to a limited region that there should be no gravitation in the -region; it is enough if the intensity of gravitation is practically the -same throughout the region. This enables us to apply the special theory -within any small region. How small it will have to be, depends upon the -neighborhood. On the surface of the earth, it would have to be small -enough for the curvature of the earth to be negligible. In the spaces -between the planets, it need only be small enough for the attraction -of the sun and the planets to be sensibly constant throughout the -region. In the spaces between the stars it might be enormous—say half -the distance from one star to the next—without introducing measurable -inaccuracies. - -At a great distance from gravitating matter, we can so choose our -co-ordinates as to obtain a Euclidean space; this is really only -another way of saying that the special theory of relativity applies. In -the neighborhood of matter, although we can make our space Euclidean -in any small region, we cannot do so throughout any region within -which gravitation varies sensibly—at least, if we do, we shall have -to abandon the view that bodies move in geodesics. In the neighborhood -of a piece of matter, there is, as it were, a hill in space-time; -this hill grows steeper and steeper as it gets nearer the top, like -the neck of a champagne bottle. It ends in a sheer precipice. Now by -the law of cosmic laziness which we mentioned earlier, a body coming -into the neighborhood of the hill will not attempt to go straight -over the top, but will go round. This is the essence of Einstein’s -view of gravitation. What a body does, it does because of the nature -of space-time in its own neighborhood, not because of some mysterious -force emanating from a distant body. - -An analogy will serve to make the point clear. Suppose that on a dark -night a number of men with lanterns were walking in various directions -across a huge plain, and suppose that in one part of the plain there -was a hill with a flaring beacon on the top. Our hill is to be such -as we have described, growing steeper as it goes up, and ending in a -precipice. I shall suppose that there are villages dotted about the -plain, and the men with lanterns are walking to and from these various -villages. Paths have been made showing the easiest way from any one -village to any other. These paths will all be more or less curved, to -avoid going too far up the hill; they will be more sharply curved when -they pass near the top of the hill than when they keep some way off -from it. Now suppose that you are observing all this, as best you can, -from a place high up in a balloon, so that you cannot see the ground, -but only the lanterns and the beacon. You will not know that there is a -hill, or that the beacon is at the top of it. You will see that people -turn out of the straight course when they approach the beacon, and -that the nearer they come the more they turn aside. You will naturally -attribute this to an effect of the beacon; you may think that it is -very hot and people are afraid of getting burnt. But if you wait for -daylight you will see the hill, and you will find that the beacon -merely marks the top of the hill and does not influence the people with -lanterns in any way. - -Now in this analogy the beacon corresponds to the sun, the people with -lanterns correspond to the planets and comets, the paths correspond -to their orbits, and the coming of daylight corresponds to the coming -of Einstein. Einstein says that the sun is at the top of a hill, only -the hill is in space-time, not in space. (I advise the reader not to -try to picture this, because it is impossible.) Each body, at each -moment, adopts the easiest course open to it, but owing to the hill the -easiest course is not a straight line. Each little bit of matter is at -the top of its own little hill, like the cock on his own dung-heap. -What we call a big bit of matter is a bit which is at the top of a big -hill. The hill is what we know about; the bit of matter at the top is -assumed for convenience. Perhaps there is really no need to assume it, -and we could do with the hill alone, for we can never get to the top of -any one else’s hill, any more than the pugnacious cock can fight the -peculiarly irritating bird that he sees in the looking glass. - -I have given only a qualitative description of Einstein’s law of -gravitation; to give its exact quantitative formulation is impossible -without more mathematics than I am permitting myself. The most -interesting point about it is that it makes the law no longer the -result of action at a distance: the sun exerts no force on the planets -whatever. Just as geometry has become physics, so, in a sense, physics -has become geometry. The law of gravitation has become the geometrical -law that every body pursues the easiest course from place to place, but -this course is affected by the hills and valleys that are encountered -on the road. - - - - -CHAPTER IX: PROOFS OF EINSTEIN’S LAW OF GRAVITATION - - -The reasons for accepting Einstein’s law of gravitation rather than -Newton’s are partly empirical, partly logical. We will begin with the -former. - -Einstein’s law of gravitation gives very nearly the same results -as Newton’s, when applied to the calculation of the orbits of the -planets and their satellites. If it did not, it could not be true, -since the consequences deduced from Newton’s law have been found to be -almost exactly verified by observation. When, in 1915, Einstein first -published his new law, there was only one empirical fact to which he -could point to show that his theory was better than Newton’s. This was -what is called the “motion of the perihelion of Mercury.” - -The planet Mercury, like the other planets, moves round the sun in -an ellipse, with the sun in one of the foci. At some points of its -orbit it is nearer to the sun than at other points. The point where -it is nearest to the sun is called its “perihelion.” Now it was found -by observation that, from one occasion when Mercury is nearest to the -sun until the next, Mercury does not go exactly once round the sun, -but a little bit more. The discrepancy is very small; it amounts to -an angle of forty-two seconds in a century. That is to say, in each -year the planet has to move rather less than half a second of angle -after it has finished a complete revolution from the last perihelion -before it reaches the next perihelion. This very minute discrepancy -from Newtonian theory had puzzled astronomers. There was a calculated -effect due to perturbations caused by the other planets, but this small -discrepancy was the residue after allowing for these perturbations. -Einstein’s theory accounted for this residue, as well as for its -absence in the case of the other planets. (In them it exists, but is -too small to be observed.) This was, at first, his only empirical -advantage over Newton. - -His second success was more sensational. According to orthodox -opinion, light in a vacuum ought always to travel in straight lines. -Not being composed of material particles, it ought to be unaffected -by gravitation. However, it was possible, without any serious breach -with old ideas, to admit that, in passing near the sun, light might be -deflected out of the straight path as much as if it were composed of -material particles. Einstein, however, maintained, as a deduction from -his law of gravitation, that light would be deflected twice as much as -this. That is to say, if the light of a star passed very near the sun, -Einstein maintained that the ray from the star would be turned through -an angle of just under one and three-quarters seconds. His opponents -were willing to concede half of this amount. Now it is not every day -that a star almost in line with the sun can be seen. This is only -possible during a total eclipse, and not always then, because there may -be no bright stars in the right position. Eddington points out that, -from this point of view, the best day of the year is May 29, because -then there are a number of bright stars close to the sun. It happened -by incredible good fortune that there was a total eclipse of the sun -on May 29, 1919—the first year after the armistice. Two British -expeditions photographed the stars near the sun during the eclipse, -and the results confirmed Einstein’s prediction. Some astronomers -who remained doubtful whether sufficient precautions had been taken -to insure accuracy were convinced when their own observations in a -subsequent eclipse gave exactly the same result. Einstein’s estimate of -the amount of the deflection of light by gravitation is therefore now -universally accepted. - -The third experimental test is on the whole favorable to Einstein, -though the quantities concerned are so small that it is only just -possible to measure them, and the result is therefore not decisive. But -successive investigations have made it more and more probable that the -small effect predicted by Einstein really occurs. Before explaining the -effect in question, a few preliminary explanations are necessary. The -spectrum of an element consists of certain lines of various shades of -light, separated by a prism, and emitted by the element when it glows. -They are the same (to a very close approximation) whether the element -is in the earth or the sun or a star. Each line is of some definite -shade of color, with some definite wave length. Longer wave lengths are -towards the red end of the spectrum, shorter ones towards the violet -end. When the source of light is moving towards you, the apparent wave -lengths grow shorter, just as waves at sea come quicker when you are -traveling against the wind. When the source of light is moving away -from you, the apparent wave lengths grow longer, for the same reason. -This enables us to know whether the stars are moving towards us or away -from us. If they are moving towards us, all the lines in the spectrum -of an element are moved a little toward violet; if away from us, toward -red. You may notice the analogous effect in sound any day. If you are -in a station and an express comes through whistling, the note of the -whistle seems much more shrill while the train is approaching you than -when it has passed. Probably many people think the note has “really” -changed, but in fact the change in what you hear is only due to the -fact that the train was first approaching and then receding. To people -in the train, there was no change of note. This is _not_ the effect -with which Einstein is concerned. The distance of the sun from the -earth does not change much; for our present purposes, we may regard -it as constant. Einstein deduces from his law of gravitation that -any periodic process which takes place in an atom in the sun (whose -gravitation is very intense) must, as measured by our clocks, take -place at a slightly slower rate than it would in a similar atom on the -earth. The “interval” involved will be the same in the sun and on the -earth, but the same interval in different regions does not correspond -to exactly the same time; this is due to the “hilly” character of -space-time which constitutes gravitation. Consequently any given line -in the spectrum ought, when the light comes from the sun, to seem to -us a little nearer the red end of the spectrum than if the light came -from a source on the earth. The effect to be expected is very small—so -small that there is still some slight uncertainty as to whether it -exists or not. But it now seems highly probable that it exists. - -No other measurable differences between the consequences of Einstein’s -law and those of Newton’s have hitherto been discovered. But the above -experimental tests are quite sufficient to convince astronomers that, -where Newton and Einstein differ as to the motions of the heavenly -bodies, it is Einstein’s law that gives the right results. Even if -the empirical grounds in favor of Einstein stood alone, they would be -conclusive. Whether his law represents the exact truth or not, it is -certainly more nearly exact than Newton’s, though the inaccuracies in -Newton’s were all exceedingly minute. - -But the considerations which originally led Einstein to his law were -not of this detailed kind. Even the consequence about the perihelion of -Mercury, which could be verified at once from previous observations, -could only be deduced after the theory was complete, and could not -form any part of the original grounds for inventing such a theory. -These grounds were of a more abstract logical character. I do not -mean that they were not based upon observed facts, and I do not mean -that they were _à priori_ fantasies such as philosophers indulged in -formerly. What I mean is that they were derived from certain general -characteristics of physical experience, which showed that Newton _must_ -be wrong and that something like Einstein’s law _must_ be substituted. - -The arguments in favor of the relativity of motion are, as we saw in -earlier chapters, quite conclusive. In daily life, when we say that -something moves, we mean that it moves relatively to the earth. In -dealing with the motions of the planets, we consider them as moving -relatively to the sun, or to the center of mass of the solar system. -When we say that the solar system itself is moving, we mean that it is -moving relatively to the stars. There is no physical occurrence which -can be called “absolute motion.” Consequently the laws of physics must -be concerned with relative motions, since these are the only kind that -occur. - -We now take the relativity of motion in conjunction with the -experimental fact that the velocity of light is the same relatively -to one body as relatively to another, however the two may be moving. -This leads us to the relativity of distances and times. This in turn -shows that there is no objective physical fact which can be called “the -distance between two bodies at a given time,” since the time and the -distance will both depend on the observer. Therefore Newton’s law of -gravitation is logically untenable, since it makes use of “distance at -a given time.” - -This shows that we cannot rest content with Newton, but it does not -show what we are to put in his place. Here several considerations -enter in. We have in the first place what is called “the equality -of gravitational and inertial mass.” What this means is as follows: -When you apply a given force[6] to a heavy body, you do not give it -as much acceleration as you would to a light body. What is called the -“inertial” mass of a body is measured by the amount of force required -to produce a given acceleration. At a given point of the earth’s -surface, the “mass” is proportional to the “weight.” What is measured -by scales is rather the mass than the weight: the weight is defined as -the force with which the earth attracts the body. Now this force is -greater at the poles than at the equator, because at the equator the -rotation of the earth produces a “centrifugal force” which partially -counteracts gravitation. The force of the earth’s attraction is also -greater on the surface of the earth than it is at a great height or at -the bottom of a very deep mine. None of these variations are shown by -scales, because they affect the weights used just as much as the body -weighed; but they are shown if we use a spring balance. The mass does -not vary in the course of these changes of weight. - -[6] Although “force” is no longer to be regarded as one of the -fundamental concepts of dynamics, but only as a convenient way of -speaking, it can still be employed, like “sunrise” and “sunset,” -provided we realize what we mean. Often it would require very -roundabout expressions to avoid the term “force.” - -The “gravitational” mass is differently defined. It is capable of two -meanings. We may mean (1), the way a body responds in a situation -where gravitation has a known intensity, for example, on the surface -of the earth, or on the surface of the sun; or (2), the intensity of -the gravitational force produced by the body, as, for example, the sun -produces stronger gravitational forces than the earth does. Newton -says that the force of gravitation between two bodies is proportional -to the product of their masses. Now let us consider the attraction of -different bodies to one and the same body, say the sun. Then different -bodies are attracted by forces which are proportional to their masses, -and which, therefore, produce exactly the same acceleration in all of -them. Thus if we mean “gravitational mass” in sense (1), that is to -say, the way a body responds to gravitation, we find that “the equality -of inertial and gravitational mass,” which sounds formidable, reduces -to this: that in a given gravitational situation, all bodies behave -exactly alike. As regards the surface of the earth, this was one of -the first discoveries of Galileo. Aristotle thought that heavy bodies -fall faster than light ones; Galileo showed that this is not the case, -when the resistance of the air is eliminated. In a vacuum, a feather -falls as fast as a lump of lead. As regards the planets, it was Newton -who established the corresponding facts. At a given distance from the -sun, a comet, which has a very small mass, experiences exactly the -same acceleration towards the sun as a planet experiences at the same -distance. Thus the way in which gravitation affects a body depends only -upon where the body is, and in no degree upon the nature of the body. -This suggests that the gravitational effect is a characteristic of the -locality, which is what Einstein makes it. - -As for the gravitational mass in sense (2), _i.e._, the intensity of -the force produced by a body, this is no longer _exactly_ proportional -to its inertial mass. The question involves some rather complicated -mathematics, and I shall not go into it.[7] - -[7] See Eddington, _The Mathematical Theory of Relativity_, Cambridge -University Press, 2d edition, p. 128. - -We have another indication as to what sort of thing the law of -gravitation _must_ be, if it is to be a characteristic of a -neighborhood, as we have seen reason to suppose that it is. It must -be expressed in some law which is unchanged when we adopt a different -kind of co-ordinates. We saw that we must not, to begin with, regard -our co-ordinates as having any physical significance: they are merely -systematic ways of naming different parts of space-time. Being -conventional, they cannot enter into physical laws. That means to say -that, if we have expressed a law correctly in terms of one set of -co-ordinates, it must be expressed by the same formula in terms of -another set of co-ordinates. Or, more exactly, it must be possible -to find a formula which expresses the law, and which is unchanged -however we change the co-ordinates. It is the business of the theory -of tensors to deal with such formulæ. And the theory of tensors shows -that there is one formula which obviously suggests itself as being -possibly the law of gravitation. When this possibility is examined, -it is found to give the right results; it is here that the empirical -confirmations come in. But if Einstein’s law had not been found to -agree with experience, we could not have gone back to Newton’s law. We -should have been compelled by logic to seek some law expressed in terms -of “tensors,” and therefore independent of our choice of co-ordinates. -It is impossible without mathematics to explain the theory of -tensors; the non-mathematician must be content to know that it is the -technical method by which we eliminate the conventional element from -our measurements and laws, and thus arrive at physical laws which are -independent of the observer’s point of view. Of this method, Einstein’s -law of gravitation is the most splendid example. - - - - -CHAPTER X: MASS, MOMENTUM, ENERGY AND ACTION - - -The pursuit of quantitative precision is as arduous as it is important. -Physical measurements are made with extraordinary exactitude; if -they were made less carefully, such minute discrepancies as form -the experimental data for the theory of relativity could never be -revealed. Mathematical physics, before the coming of relativity, used -a set of conceptions which were supposed to be as precise as physical -measurements, but it has turned out that they were logically defective, -and that this defectiveness showed itself in very small deviations from -expectations based upon calculation. In this chapter I want to show how -the fundamental ideas of pre-relativity physics are affected, and what -modifications they have had to undergo. - -We have already had occasion to speak of mass. For purposes of -daily life, mass is much the same as weight; the usual measures of -weight—ounces, grams, etc.—are really measures of mass. But as -soon as we begin to make accurate measurements, we are compelled to -distinguish between mass and weight. Two different methods of weighing -are in common use, one, that of scales, the other that of the spring -balance. When you go a journey and your luggage is weighed, it is not -put on scales, but on a spring; the weight depresses the spring a -certain amount, and the result is indicated by a needle on a dial. The -same principle is used in automatic machines for finding your weight. -The spring balance shows weight, but scales show _mass_. So long as -you stay in one part of the world, the difference does not matter; -but if you test two weighing machines of different kinds in a number -of different places, you will find, if they are accurate, that their -results do not always agree. Scales will give the same result anywhere, -but a spring balance will not. That is to say, if you have a lump of -lead weighing ten pounds by the scales, it will also weigh ten pounds -by scales in any other part of the world. But if it weighs ten pounds -by a spring balance in London, it will weigh more at the North Pole, -less at the equator, less high up in an aeroplane, and less at the -bottom of a coal mine, if it is weighed in all those places on the same -spring balance. The fact is that the two instruments measure quite -different quantities. The scales measure what may be called (apart from -refinements which will concern us presently) “quantity of matter.” -There is the same “quantity of matter” in a pound of feathers as in a -pound of lead. Standard “weights,” which are really standard “masses,” -will measure the amount of mass in any substance put into the opposite -scales. But “weight” is a properly due to the earth’s gravitation: It -is the amount of the force by which the earth attracts a body. This -force varies from place to place. In the first place, anywhere outside -the earth the attraction varies inversely as the square of the distance -from the center of the earth; it is therefore less at great heights. -In the second place, when you go down a coal mine, part of the earth -is above you, and attracts matter upwards instead of downwards, so -that the net attraction downwards is less than on the surface of the -earth. In the third place, owing to the rotation of the earth, there is -what is called a “centrifugal force,” which acts against gravitation. -This is greatest at the equator, because there the rotation of the -earth involves the fastest motion; at the poles it does not exist, -because they are on the axis of rotation. For all these reasons, the -force with which a given body is attracted to the earth is measureably -different at different places. It is this force that is measured by a -spring balance; that is why a spring balance gives different results -in different places. In the case of scales, the standard “weights” are -altered just as much as the body to be weighed, so that the result is -the same everywhere; but the result is the “mass,” not the “weight.” -A standard “weight” has the same mass everywhere, but not the same -“weight”; it is in fact a unit of mass, not of weight. For theoretical -purposes, mass, which is almost invariable for a given body, is much -more important than weight, which varies according to circumstances. -Mass may be regarded, to begin with, as “quantity of matter”; we shall -see that this view is not strictly correct, but it will serve as a -starting point for subsequent refinements. - -For theoretical purposes, a mass is defined as being determined by the -amount of force required to produce a given acceleration: The more -massive a body is, the greater will be the force required to alter its -velocity by a given amount in a given time. It takes a more powerful -engine to make a long train attain a speed of ten miles an hour at the -end of the first half-minute, than it does to make a short train do so. -Or we may have circumstances where the force is the same for a number -of different bodies; in that case, if we can measure the accelerations -produced in them, we can tell the ratios of their masses: the greater -the mass, the smaller the acceleration. We may take, in illustration -of this method, an example which is important in connection with -relativity. Radio-active bodies emit beta-particles (electrons) with -enormous velocities. We can observe their path by making them travel -through water vapor and form a cloud as they go. We can at the same -time subject them to known electric and magnetic forces, and observe -how much they are bent out of a straight line by these forces. This -makes it possible to compare their masses. It is found that the faster -they travel, the greater is their mass, as measured by the stationary -observer; the increase is greatest as applied to their mass as measured -by the effect of a force in the line of motion. In regard to forces at -right angles to the line of motion, there is a change of mass with -velocity in the same proportion as the changes of length and time. It -is known otherwise that, apart from the effect of motion, all electrons -have the same mass. - -All this was known before the theory of relativity was invented, but -it showed that the traditional conception of mass had not quite the -definiteness that had been ascribed to it. Mass used to be regarded as -“quantity of matter,” and supposed to be quite invariable. Now mass was -found to be relative to the observer, like length and time, and to be -altered by motion in exactly the same proportion. However, this could -be remedied. We could take the “proper mass,” the mass as measured by -an observer who shares the motion of the body. This was easily inferred -from the measured mass, by taking the same proportion as in the case of -lengths and times. - -But there is a more curious fact, and that is, that after we have -made this correction we still have not obtained a quantity which is -at all times exactly the same for the same body. When a body absorbs -energy—for example, by growing hotter—its “proper mass” increases -slightly. The increase is very slight, since it is measured by -dividing the increase of energy by the square of the velocity of -light. On the other hand, when a body parts with energy it loses mass. -The most notable case of this is that four hydrogen atoms can come -together to make one helium atom, but a helium atom has rather less -than four times the mass of one hydrogen atom. - -We have thus two kinds of mass, neither of which quite fulfils the old -ideal. The mass as measured by an observer who is in motion relative -to the body in question is a relative quantity, and has no physical -significance as a property of the body. The “proper mass” is a genuine -property of the body, not dependent upon the observer; but it, also, -is not strictly constant. As we shall see shortly, the notion of mass -becomes absorbed into the notion of energy; it represents, so to speak, -the energy which the body expends internally, as opposed to that which -it displays to the outer world. - -Conservation of mass, conservation of momentum, and conservation of -energy were the great principles of classical mechanics. Let us next -consider conservation of momentum. - -The momentum of a body in a given direction is its velocity in that -direction multiplied by its mass. Thus a heavy body moving slowly may -have the same momentum as a light body moving fast. When a number of -bodies interact in any way, for instance by collisions, or by mutual -gravitation, so long as no outside influences come in, the total -momentum of all the bodies in any direction remains unchanged. This law -remains true in the theory of relativity. For different observers, the -mass will be different, but so will the velocity; these two differences -neutralize each other, and it turns out that the principle still -remains true. - -The momentum of a body is different in different directions. The -ordinary way of measuring it is to take the velocity in a given -direction (as measured by the observer) and multiply it by the mass (as -measured by the observer). Now the velocity in a given direction is -the distance traveled in that direction in unit time. Suppose we take -instead the distance traveled in that direction while the body moves -through unit “interval.” (In ordinary cases, this is only a very slight -change, because, for velocities considerably less than that of light, -interval is very nearly equal to lapse of time.) And suppose that -instead of the mass as measured by the observer we take the proper -mass. These two changes increase the velocity and diminish the mass, -both in, the same proportion. Thus the momentum remains the same, but -the quantities that vary according to the observer have been replaced -by quantities which are fixed independently of the observer—with the -exception of the distance traveled by the body in the given direction. - -When we substitute space-time for time, we find that the measured -mass (as opposed to the proper mass) is a quantity of the same kind -as the momentum in a given direction; it might be called the momentum -in the time direction. The measured mass is obtained by multiplying -the invariant mass by the _time_ traversed in traveling through unit -interval; the momentum is obtained by multiplying the same invariant -mass by the _distance_ traversed (in the given direction) in traveling -through unit interval. From a space-time point of view, these naturally -belong together. - -Although the measured mass of a body depends upon the way the observer -is moving relatively to the body, it is none the less a very important -quantity. For any given observer, the measured mass of the whole -physical universe is constant.[8] The proper mass of all the bodies -in the world is not necessarily the same at one time as at another, -so that in this respect the measured mass has an advantage. The -conservation of measured mass is the same thing as the conservation of -energy. This may seem surprising, since at first sight mass and energy -are very different things. But it has turned out that energy is the -same thing as measured mass. To explain how this comes about is not -easy; nevertheless we will make the attempt. - -[8] This is subject to the explanations given below as regards -conservation of energy. - -In popular talk, “mass” and “energy” do not mean at all the same thing. -We associate “mass” with the idea of a fat man in a chair, very slow to -move, while “energy” suggests a thin person full of hustle and “pep.” -Popular talk associates “mass” and “inertia,” but its view of inertia -is one-sided: it includes slowness in beginning to move, but not -slowness in stopping, which is equally involved. All these terms have -technical meanings in physics, which are only more or less analogous -to the meanings of the terms in popular talk. For the present, we are -concerned with the technical meaning of “energy.” - -Throughout the latter half of the nineteenth century, a great deal was -made of the “conservation of energy,” or the “persistence of force,” -as Herbert Spencer preferred to call it. This principle was not easy -to state in a simple way, because of the different forms of energy; -but the essential point was that energy is never created or destroyed, -though it can be transformed from one kind into another. The principle -acquired its position through Joule’s discovery of “the mechanical -equivalent of heat,” which showed that there was a constant proportion -between the work required to produce a given amount of heat and the -work required to raise a given weight through a given height: in fact, -the same sort of work could be utilized for either purpose according to -the mechanism. When heat was found to consist in motion of molecules, -it was seen to be natural that it should be analogous to other forms of -energy. Broadly speaking, by the help of a certain amount of theory, -all forms of energy were reduced to two, which were called respectively -“kinetic” and “potential.” These were defined as follows: - -The kinetic energy of a particle is half the mass multiplied by the -square of the velocity. The kinetic energy of a number of particles is -the sum of the kinetic energies of the separate particles. - -The potential energy is more difficult to define. It represents any -state of strain, which can only be preserved by the application of -force. To take the easiest case: If a weight is lifted to a height and -kept suspended, it has potential energy, because, if left to itself, it -will fall. Its potential energy is equal to the kinetic energy which it -would acquire in falling through the same distance through which it was -lifted. Similarly when a comet goes round the sun in a very eccentric -orbit, it moves much faster when it is near the sun than when it is far -from it, so that its kinetic energy is much greater when it is near the -sun. On the other hand, its potential energy is greatest when it is -farthest from the sun, because it is then like the stone which has been -lifted to a height. The sum of the kinetic and potential energies of -the comet is constant, unless it suffers collisions or loses matter by -forming a tail. We can determine accurately the _change_ of potential -energy in passing from one position to another, but the total amount of -it is to a certain extent arbitrary, since we can fix the zero level -where we like. For example, the potential energy of our stone may be -taken to be the kinetic energy it would acquire in falling to the -surface of the earth, or what it would acquire in falling down a well -to the center of the earth, or any assigned lesser distance. It does -not matter which we take, so long as we stick to our decision. We are -concerned with a profit-and-loss account, which is unaffected by the -amount of the assets with which we start. - -Both the kinetic and the potential energies of a given set of bodies -will be different for different observers. In classical dynamics, -the kinetic energy differed according to the state of motion of the -observer, but only by a constant amount; the potential energy did not -differ at all. Consequently, for each observer, the total energy was -constant—assuming always that the observers concerned were moving -in straight lines with uniform velocities, or, if not, were able to -refer their motions to bodies which were so moving. But in relativity -dynamics the matter becomes more complicated. We cannot profitably -adapt the idea of potential energy to the theory of relativity, and -therefore the conservation of energy, in a strict sense, cannot -be maintained. But we obtain a property, closely analogous to -conservation, which applies to kinetic energy alone. As Eddington -puts it: the kinetic energy is not always strictly conserved, and the -classical theory therefore introduces a supplementary quantity, the -potential energy, so that the sum of the two is strictly conserved. The -relativity treatment, on the other hand, discovers another formula, -analogous to the one expressing conservation, which holds always for -the kinetic energy. “The relativity treatment adheres to the physical -quantity and modifies the law; the classical treatment adheres to -the law and modifies the physical quantity.” The new formula, he -continues, may be spoken of “as the law of conservation of energy and -momentum, because, though it is not formally a law of conservation, it -expresses exactly the phenomena which classical mechanics attributes to -conservation.”[9] It is only in this modified and less rigorous sense -that the conservation of energy remains true. - -[9] _Mathematical Theory of Relativity_, p. 135. - -What is meant by “conservation” in practice is not exactly what it -means in theory. In theory we say that a quantity is conserved when the -amount of it in the world is the same at any one time as at any other. -But in practice we cannot survey the whole world, so we have to mean -something more manageable. We mean that, taking any given region, if -the amount of the quantity in the region has changed, it is because -some of the quantity has passed across the boundary of the region. If -there were no births and deaths, population would be conserved; in that -case the population of a country could only change by emigration or -immigration, that is to say, by passing across the boundaries. We might -be unable to take an accurate census of China or Central Africa, and, -therefore, we might not be able to ascertain the total population of -the world. But we should be justified in assuming it to be constant if, -wherever statistics were possible, the population never changed except -through people crossing the frontiers. In fact, of course, population -is not conserved. A physiologist of my acquaintance once put four mice -into a thermos. Some hours later, when he went to take them out, there -were eleven of them. But mass is not subject to these fluctuations: -the mass of the eleven mice at the end of the time was no greater than -the mass of the four at the beginning. - -This brings us back to the problem for the sake of which we have been -discussing energy. We stated that, in relativity theory, measured mass -and energy are regarded as the same thing, and we undertook to explain -why. It is now time to embark upon this explanation. But here, as at -the end of Chapter VI, the totally unmathematical reader will do well -to skip, and begin again at the following paragraph. - -Let us take the velocity of light as the unit of velocity; this is -always convenient in relativity theory. Let _m_ be the proper mass of a -particle, _v_ its velocity relative to the observer. Then its measured -mass will be - - _m_ - —————————— - √(1 - _v²_) - -while its kinetic energy, according to the usual formula, will be - - ½ _mv²_ - -As we saw before, energy only occurs in a profit-and-loss account, -so that we can add any constant quantity to it that we like. We may -therefore take the energy to be - - _m_ + ½(_mv²_). - -Now if _v_ is a small fraction of the velocity of light, - - _m_ + ½ _mv²_ - -is almost exactly equal to - - _m_ - ————————— - √(1 - _v²_). - -Consequently, for velocities such as large bodies have, the energy and -the measured mass turn out to be indistinguishable within the limits of -accuracy attainable. In fact, it is better to alter our definition of -energy, and take it to be - - _m_ - —————————— - √(1 - _v²_), - -because this is the quantity for which the law analogous to -conservation holds. And when the velocity is very great, it gives a -better measure of energy than the traditional formula. The traditional -formula must therefore be regarded as an approximation, of which the -new formula gives the exact version. In this way, energy and measured -mass become identified. - -I come now to the notion of “action,” which is less familiar to -the general public than energy, but has become more important in -relativity physics, as well as in the theory of quanta.[10] (The -quantum is a small amount of action.) The word “action” is used to -denote energy multiplied by time. That is to say, if there is one unit -of energy in a system, it will exert one unit of action in a second, -100 units of action in 100 seconds, and so on; a system which has -100 units of energy will exert 100 units of action in a second, and -10,000 in 100 seconds, and so on. “Action” is thus, in a loose sense, -a measure of how much has been accomplished: it is increased both by -displaying more energy and by working for a longer time. Since energy -is the same thing as measured mass, we may also take action to be -measured mass multiplied by time. In classical mechanics, the “density” -of matter in any region is the mass divided by the volume; that is -to say, if you know the density in a small region, you discover the -total amount of matter by multiplying the density by the volume of the -small region. In relativity mechanics, we always want to substitute -space-time for space; therefore a “region” must no longer be taken to -be merely a volume, but a volume lasting for a time; a small region -will be a small volume lasting for a small time. It follows that, given -the density, a small region in the new sense contains, not a small mass -merely, but a small mass multiplied by a small time, that is to say, a -small amount of “action.” This explains why it is to be expected that -“action” will prove of fundamental importance in relativity mechanics. -And so in fact it is. - -[10] On this subject, see the present author’s _A.B.C. of Atoms_, -chaps. VI and XIII. - -All the laws of dynamics have been put together into one principle, -called “The Principle of Least Action.” This states that, in passing -from one state to another, a body chooses a route involving less action -than any slightly different route—again a law of cosmic laziness. The -principle is subject to certain limitations, which have been pointed -out by Eddington,[11] but it remains one of the most comprehensive -ways of stating the purely formal part of mechanics. The fact that -the quantum is a unit of action seems to show that action is also -fundamental in the empirical structure of the world. But at present -there is no bridge connecting the quantum with the theory of relativity. - -[11] _Op. cit._ § 60. - - - - -CHAPTER XI: IS THE UNIVERSE FINITE? - - -We have been dealing hitherto with matters that must be regarded as -acquired scientific results—not that they will never be found to need -improvement, but that further progress must be built upon them, as -Einstein is built upon Newton. Science does not aim at establishing -immutable truths and eternal dogmas: its aim is to approach truth by -successive approximations, without claiming that at any stage final and -complete accuracy has been achieved. There is a difference, however, -between results which are pretty certainly in the line of advance, and -speculations which may or may not prove to be well founded. Some very -interesting speculations are connected with the theory of relativity, -and we shall consider certain of them. But it must not be supposed that -we are dealing with theories having the same solidity as those with -which we have been concerned hitherto. - -One of the most fascinating of the speculations to which I have been -alluding is the suggestion that the universe may be of finite extent. -Two somewhat different finite universes have been constructed, one by -Einstein, the other by De Sitter. Before considering their differences, -we will discuss what they have in common. - -There are, to begin with, certain reasons for thinking that the total -amount of matter in the universe is limited. If this were not the -case, the gravitational effects of enormously distant matter would -make the kind of world in which we live impossible. We must therefore -suppose that there is some definite number of electrons and protons in -the world: theoretically, a complete census would be possible. These -are all contained within a certain finite region; whatever space lies -outside that region is, so to speak, waste, like unfurnished rooms in a -house too large for its inhabitants. This seems futile, but in former -days no one knew of any alternative possibility. It was obviously -impossible to conceive of an edge to space, and therefore, it was -thought, space must be infinite. - -Non-Euclidean geometry, however, showed other possibilities. The -surface of a sphere has no boundary, yet it is not infinite. In -traveling round the earth, we never reach “the edge of the world,” and -yet the earth is not infinite. The surface of the earth is contained -in three-dimensional space, but there is no reason in logic why -three-dimensional space should not be constructed on an analogous plan. -What we imagine to be straight lines going on for ever will then be -like great circles on a sphere: they will ultimately return to their -starting point. There will not be in the universe anything straighter -than these great circles; the Euclidean straight line may remain as -a beautiful dream, but not as a possibility in the actual world. In -particular, light rays in empty space will travel in what are really -great circles. If we could make measurements with sufficient accuracy, -we should be able to infer this state of affairs even from a small part -of space, because the sum of the angles of a triangle would always be -greater than two right angles, and the excess would be proportional to -the size of the triangle. The suggestion we have to consider is the -suggestion that our universe may be spherical in this sense. - -The reader must not confuse this suggestion with the non-Euclidean -character of space upon which the new law of gravitation depends. The -latter is concerned with small regions such as the solar system. The -departures from flatness which it notices are like hills and valleys -on the surface of the earth, local irregularities, not characteristics -of the whole. We are now concerned with the possible curvature of the -universe as a whole, not with the occasional ups and downs due to the -sun and the stars. It is suggested that on the average, and in regions -remote from matter, the universe is not quite flat, but has a slight -curvature, analogous, in three dimensions, to the curvature of a sphere -in two dimensions. - -It is important to realize, in the first place, that there is not the -slightest reason _à priori_ why this should not be the case. People -unaccustomed to non-Euclidean geometry may feel that, even if such a -thing be _logically_ possible, the world simply _cannot_ be so odd -as all that. We all have a tendency to think that the world must -conform to our prejudices. The opposite view involves some effort of -thought, and most people would die sooner than think—in fact, they -do so. But the fact that a spherical universe seems odd to people -who have been brought up on Euclidean prejudices is no evidence that -it is impossible. There is no law of nature to the effect that what -is taught at school must be true. We cannot therefore dismiss the -hypothesis of a spherical universe as in any degree less worthy of -examination than any other. We have to ask ourselves the same two -questions as we should in any other case, namely: (1) Are the facts -consistent with this hypothesis? (2) Is this hypothesis the only one -with which the facts are consistent? - -With regard to the first question, the answer is undoubtedly in the -affirmative. All the known facts are perfectly consistent with the -hypothesis of a spherical universe. A very slight modification of the -law of gravitation—a modification suggested by Einstein himself—leads -to a spherical space, without producing any measurable differences in a -small region such as the solar system. The known stars are all within -a certain distance from us. There is nothing whatever in the stellar -universe as we know it to show that space must be infinite. There can -therefore be no doubt whatever that, so far as our present knowledge -goes, the hypothesis of a finite universe _may_ be true. - -But when we ask whether the hypothesis of a finite universe _must_ -be true, the answer is different. It is obvious, on general grounds, -that we cannot, from what we know, draw conclusive inferences as to -the totality of things. A very slight change in the Newtonian formula -for gravitation would prevent masses beyond the limits of the visible -universe from having appreciable effects if they existed, and would -therefore destroy our reason for supposing that they do not exist. -All arguments as to regions which are too distant to be observed -depend upon extending to them the laws which hold in our part of -the world, and upon assuming that there is not, in these laws, some -inaccuracy which is inappreciable for observable distances, but fatal -to inferences in which very much greater distances are involved. We -cannot, therefore, say that the universe _must_ be finite. We can say -that it may be, and we can even say a little more than this. We can say -that a finite universe fits in better with the laws that hold in the -part we know, and that awkward adjustments of the laws have to be made -in order to allow the universe to be infinite. From the point of view -of choosing the best framework into which to fit what we know—best, I -mean, from a logico-æsthetic point of view—there is no doubt that the -hypothesis of a finite universe is preferable. This, I think, is the -extent of what can be said in its favor. - -Let us now see what the two finite universes are like. The difference -between them is that in Einstein’s world it is only space that -is queer, whereas in De Sitter’s time is queer too. Consequently -Einstein’s world is less puzzling, and we will describe it first. - -In Einstein’s world, light travels round the whole universe in a time -which is supposed to be something like a thousand million years. The -odd thing is that all the rays of light which start (say) from the sun -will meet again, after their enormous journey, in the place where the -sun was when they started. The case is exactly analogous to that of a -number of travelers who set out from London to go round the world in -great circles, all traveling at the same rate in different aeroplanes. -One starts due north, passes the North Pole, then the South Pole, and -finally comes home. Another starts due south, reaches the South Pole -first and then the North Pole. Another starts westward, but he must not -continue to travel due west, because then he would not be traveling on -a great circle. Another starts eastward, and so on. They all meet in -the antipodes of London, and then they all meet again in London. Now -if instead of aeronauts going round the earth you take rays of light -going round the universe, the same sort of thing happens: they all meet -first at the antipodes of their starting point, and then meet again at -their starting point. That means to say that a person who is near the -antipodes of the place where the sun was about five hundred million -years ago will see what is apparently a body as bright as the sun then -was (except for the small amount of light that has been stopped on the -way by opaque bodies), and having the same shape and size. And a person -who is near where the sun was a thousand million years ago will see -what is apparently a body just like what the sun was a thousand million -years ago. And the same applies to the antipodes of the sun fifteen -hundred million years ago, and to the place of the sun two thousand -million years ago, and so on. This series only ends when it carries us -back to a time before the sun existed. - -But all these suns are only ghosts; that is to say, you could pass -through them without experiencing resistance, and they do not exert -gravitation. They are, in fact, like images in a mirror: they exist -only for the sense of sight, not for any other sense. It is rather -disturbing to reflect that, if this theory is true, any number of the -objects we see in the heavens may be merely ghosts. They are like -ghosts in their habit of revisiting the scenes of their past life. -Suppose a star had exploded at a certain place, as stars sometimes -will. Every thousand million years its ghost would return to the scene -of the disaster and explode again in the same place. There is, however, -considerable doubt whether rays of light could perform the journey with -sufficient accuracy to produce a clear image. Some would be stopped by -matter on the way, some would be turned out of the straight course by -passing near heavy bodies, as in the eclipse observations described in -Chapter IX, and for one reason or another their return would not be -punctual and exact. - -There are various reasons for doubting whether Einstein’s universe can -be quite right.[12] Some of these are rather complicated. But there -is one objection which is easily appreciated: in Einstein’s theory, -absolute space and time re-enter by another door. The ghostly sun -is formed in the “place” where it was a thousand million years ago. -Both the “place” and the period of time are in a sense absolute. We -saw as early as Chapter I that “place” is a vague and popular notion, -incapable of scientific precision. It seems hardly worth while to go -through such a vast intellectual labor if the errors we set out to -correct are to reappear at the end. - -[12] See Eddington, _Space, Time and Gravitation_, p. 162ff. - -De Sitter’s world is even odder than Einstein’s, because time goes -mad as well as space. I despair of explaining, in non-mathematical -language, the particular form of lunacy with which time is afflicted, -but some of its manifestations can be described. An observer in this -world, if he observes a number of clocks, each of which is perfectly -accurate from its own point of view, will think that distant clocks -are going slow as compared with those in his neighborhood. They will -seem to go slower and slower, until, at a distance of one quarter of -the circumference of the universe, they will seem to have stopped -altogether. That region will seem to our observer a sort of lotus -land, where nothing is ever done. He will not be able to have any -cognizance of things farther off, because no light waves can get across -the boundary. Not that there is any real boundary: the people who live -in what our observer takes to be lotus land live just as bustling a -life as he does, but get the impression that he is eternally standing -still. As a matter of fact, you would never become aware of the lotus -land, because it would take an infinite time for light to travel from -it to you. You could become aware of places just short of it, but it -would remain itself always just beyond your ken. There will not be the -ghostly suns of Einstein’s world, because light cannot travel so far. - -One of the oddest things about this state of affairs is that empirical -evidence for or against it is possible, and that there is actually -some slight evidence in its favor. If all “clocks” are slowed down at -a great distance from the observer, this will apply to the periodic -motions of atoms, and therefore to the light which they emit. -Consequently all rays of light emitted by distant objects ought, when -they reach us, to look rather more red or less violet than when they -started. This can be tested by the spectroscope. We can compare a -known line, as it appears in the spectrum of a spiral nebula, with -the same line as it appears in a terrestrial laboratory. We find, as a -matter of fact, that in a large majority of spiral nebulæ there is a -considerable displacement of spectral lines towards the red. The spiral -nebulæ are the most distant objects we can see: Eddington states that -their distances “may perhaps be of the order of a million light-years.” -(A light-year is the distance light travels in a year.) The usual -interpretation of a shifting of spectral lines towards the red is that -it is a “Doppler effect,” due to the fact that the source of light is -moving away from us. But one would expect to find the nebulæ just as -often moving towards us as moving away from us, if nothing operated but -the law of chances. If the world is such as De Sitter says it is, the -spectral lines of the spiral nebulæ will be displaced towards the red -owing to the slowing down of distant clocks, even if in fact they are -not moving away from us. This, for what it is worth, is an argument in -favor of De Sitter. - -The same facts afford another argument in favor of De Sitter, for -another reason. If, at a given moment, a body is at rest relatively to -the observer, and at a distance from him, it will (in the absence of -counteracting causes) not remain at rest from his point of view, but -will begin to move away from him, and will continue to move away faster -and faster; the further it is from him, the more its retreat will be -accelerated. For bodies which are not too distant from each other, -gravitation may overcome this tendency; but as this tendency increases -with the distance, while gravitation diminishes, we should expect -to find very distant bodies receding from us if De Sitter’s theory -is right. Thus we have two reasons for the displacement of spectral -lines in spiral nebulæ: one, the slowing down of time; the other, the -movement away from us which we should expect at distances too great -for gravitation to be sensible. However, it cannot be said that the -argument, on either ground, is very strong. Eddington gives a list -of forty-one spiral nebulæ, of which five have their spectral lines -shifted towards the violet, not towards the red. Thus the material is -neither very copious nor quite harmonious. - -Einstein’s and De Sitter’s hypotheses do not exhaust the possibilities -of a finite world: they are merely the two simplest forms of such a -world. There are arguments against each, and it hardly seems probable -that either is quite true. But it does seem probable that something -more or less analogous is true. If the universe is finite, it is -theoretically conceivable that there should be a complete inventory -of it. We may be coming to the end of what physics can do in the way -of stretching the imagination and systematizing the world. The period -since Galileo has been essentially the period of physics, as the age of -the Greeks was the period of geometry. It may be that physics will lose -its attractions through success: if the fundamental laws of physics -come to be fully known, adventurous and inquiring intellects will turn -to other fields. This may alter profoundly the whole texture of human -life, since our present absorption in machinery and industrialism is -the reflection in the practical world of the theorist’s interest in -physical laws. But such speculations are even more rash than those of -De Sitter, and I do not wish to lay any stress upon them. - - - - -CHAPTER XII: CONVENTIONS AND NATURAL LAWS - - -One of the most difficult matters in all controversy is to distinguish -disputes about words from disputes about facts: it ought not to be -difficult, but in practice it is. This is quite as true in physics as -in other subjects. In the seventeenth century there was a terrific -debate as to what “force” is; to us now, it was obviously a debate -as to how the word “force” should be defined, but at the time it was -thought to be much more. One of the purposes of the method of tensors, -which is employed in the mathematics of relativity, is to eliminate -what is purely verbal (in an extended sense) in physical laws. It is -of course obvious that what depends on the choice of co-ordinates is -“verbal” in the sense concerned. A man punting walks along the boat, -but keeps a constant position with reference to the river bed so -long as he does not pick up his pole. The Lilliputians might debate -endlessly whether he is walking or standing still: the debate would -be as to words, not as to facts. If we choose co-ordinates fixed -relatively to the boat, he is walking; if we choose co-ordinates -fixed relatively to the river bed, he is standing still. We want to -express physical laws in such a way that it shall be obvious when we -are expressing the same law by reference to two different systems -of co-ordinates, so that we shall not be misled into supposing we -have different laws when we only have one law in different words. -This is accomplished by the method of tensors. Some laws which seem -plausible in one language cannot be translated into another; these are -impossible as laws of nature. The laws that can be translated into -_any_ co-ordinate language have certain characteristics: this is a -substantial help in looking for such laws of nature as the theory of -relativity can admit to be possible. Combined with what we know of the -actual motions of bodies, it enables us to decide what must be the -correct expression of the law of gravitation: logic and experience -combine in equal proportions in obtaining this expression. - -But the problem of arriving at genuine laws of nature is not to be -solved by the method of tensors alone; a good, deal of careful thought -is wanted in addition. Some of this has been done, especially by -Eddington; much remains to be done. - -To take a simple illustration: Suppose, as in the hypothesis of the -Fitzgerald contraction, that lengths in one direction were shorter than -in another. Let us assume that a foot rule pointing north is only half -as long as the same foot rule pointing east, and that this is equally -true of all other bodies. Does such an hypothesis have any meaning? -If you have a fishing rod fifteen feet long when it is pointing west, -and you then turn it to the north, it will still measure fifteen feet, -because your foot rule will have shrunk too. It won’t “look” any -shorter, because your eye will have been affected in the same way. If -you are to find out the change, it cannot be by ordinary measurement; -it must be by some such method as the Michelson-Morley experiment, in -which the velocity of light is used to measure lengths. Then you still -have to decide whether it is simpler to suppose a change of length -or a change in the velocity of light. The experimental fact would be -that light takes longer to traverse what your foot rule declares to -be a given distance in one direction than in another—or, as in the -Michelson-Morley experiment, that it ought to take longer but doesn’t. -You can adjust your measures to such a fact in various ways; in any -way you choose to adopt, there will be an element of convention. This -element of convention survives in the laws that you arrive at after -you have made your decision as to measures, and often it takes subtle -and elusive forms. To eliminate the element of convention is, in fact, -extraordinarily difficult; the more the subject is studied, the greater -the difficulty is seen to be. - -A more important example is the question of the size and shape of the -electron. We find experimentally that all electrons are the same size, -and that they are symmetrical in all directions. How far is this a -genuine fact ascertained by experiment, and how far is it a result of -our conventions of measurement? We have here a number of different -comparisons to make: (1) between different directions in regard to one -electron at one time; (2) in regard to one electron at different times; -(3) in regard to two electrons at the same time. We can then arrive -at the comparison of two electrons at different times, by combining -(2) and (3). We may dismiss any hypothesis which would affect all -electrons equally; for example, it would be useless to suppose that in -one region of space-time they were all larger than in another. Such a -change would affect our measuring appliances just as much as the things -measured, and would therefore produce no discoverable phenomena. This -is as much as to say that it would be no change at all. But the fact -that two electrons have the same mass, for instance, cannot be regarded -as purely conventional. Given sufficient minuteness and accuracy, we -could compare the effects of two different electrons upon a third; -if they were equal under like circumstances, we should be able to -infer equality in a not purely conventional sense. The question of -the symmetry of the forces exerted by an electron—_i.e._, that these -forces depend only upon the distance from the electron, and not upon -the direction—is more complicated. Eddington finally comes to the -conclusion that this, too, is a matter of convention. The argument -is difficult and I have not fully understood it; but I feel some -hesitation in accepting it as valid. - -Eddington describes the process concerned in the more advanced portions -of the theory of relativity as “world-building.” The structure to be -built is the physical world as we know it; the economical architect -tries to construct it with the smallest possible amount of material. -This is a question for logic and mathematics. The greater our technical -skill in these two subjects, the more real building we shall do, and -the less we shall be content with mere heaps of stones. But before we -can use in our building the stones that nature provides, we have to -hew them into the right shapes: this is all part of the process of -budding. In order that this may be possible, the raw material must -have _some_ structure (which we may conceive as analogous to the -grain in timber), but almost any structure will do. By successive -mathematical refinements, we whittle away our initial requirements -until they amount to very little. Given this necessary minimum of -structure in the raw material, we find that we can construct from it a -mathematical expression which will have the properties that are needed -for describing the world we perceive—in particular, the properties -of conservation which are characteristic of momentum and energy (or -mass). Our raw material consisted merely of events; but when we find -that we can build out of it something which, as measured, will seem -to be never created or destroyed, it seems not surprising that we -should come to believe in “bodies.” These are really mere mathematical -constructions out of events, but owing to their permanence they are -practically important, and our senses (which were presumably developed -by biological needs) are adapted for noticing them, rather than the -crude continuum of events which is theoretically more fundamental. From -this point of view, it is astonishing how little of the real world is -revealed by physical science: our knowledge is limited, not only by the -conventional element, but also by the selectiveness of our perceptual -apparatus. - -We assume that there is an “interval” between two events, in the -sense explained in Chapter VII, but we no longer assume that we can -unambiguously compare the length of an interval in one region with the -length of an interval in another. It is assumed by Weyl, who introduced -this limitation, that we can compare a number of small intervals which -all start from the same point; also that, in a very small journey, -our measuring rod will not alter its length much, so that there will -only be a small error if we compare lengths in neighboring places by -the usual methods. Weyl found that, by diminishing our assumptions as -to interval in this way, it was possible to bring electromagnetism -and gravitation into one system. The mathematics of Weyl’s theory is -complicated, and I shall not attempt to explain it. For the present, -I am concerned with a different consequence of his theory. If lengths -in different regions cannot be compared directly, there is an element -of convention in the indirect comparisons which we actually make. This -element will be at first unrecognized, but will be such as to simplify -to the utmost the expression of the laws of nature. In particular, -conditions of symmetry may be entirely created by conventions as to -measurement, and there is no reason to suppose that they represent any -property of the real world. The law of gravitation itself, according to -Eddington, may be regarded as expressing conventions of measurement. -“The conventions of measurement,” he says, “introduce an isotropy[13] -and homogeneity into measured space which need not originally have any -counterpart in the relation-structure which is being surveyed. This -isotropy and homogeneity is exactly expressed by Einstein’s law of -gravitation.”[14] - -[13] “Isotropy” means being similar in all directions—_e.g._, that a -foot rule is as long when it points north as when it points east. - -[14] _Mathematical Theory of Relativity_, p. 238. - -The limitations of knowledge introduced by the selectiveness of our -perceptual apparatus may be illustrated by the indestructibility -of matter. This has been gradually discovered by experiment, and -seemed a well-founded empirical law of nature. Now it turns out -that, from our original space-time continuum, we can construct a -mathematical expression which will have properties causing it to appear -indestructible. The statement that matter is indestructible then ceases -to be a proposition of physics, and becomes instead a proposition -of linguistics and psychology. As a proposition of linguistics: -“Matter” is the name of the mathematical expression in question. As a -proposition of psychology: Our senses are such that we notice what is -roughly the mathematical expression in question, and we are led nearer -and nearer to it as we refine upon our crude perceptions by scientific -observation. This is much less than physicists used to think they knew -about matter. - -The reader may say: What then is left of physics? What do we really -know about the world of matter? Here we may distinguish three -departments of physics. There is first what is included within the -theory of relativity, generalized as widely as possible. Next, there -are laws which cannot be brought within the scope of relativity. -Thirdly, there is what may be called geography. Let us consider each of -these in turn. - -The theory of relativity, apart from convention, tells us that the -events in the universe have a four-dimensional order, and that, -between any two events which are near together in this order, there -is a relation called “interval,” which is capable of being measured -if suitable precautions are taken. We make also an assumption as to -what happens when a little measuring rod is carried round a closed -circuit in a certain manner; the consequences of this assumption are -such as to make it highly probable that it is true. Beyond this, there -is little in the theory of relativity that can be regarded as physical -laws. There is a great deal of mathematics, showing that certain -mathematically-constructed quantities must behave like the things we -perceive; and there is a suggestion of a bridge between psychology and -physics in the theory that these mathematically-constructed quantities -are what our senses are adapted for perceiving. But neither of these -things is physics in the strict sense. - -The part of physics which cannot, at present, be brought within -the scope of relativity is large and important. There is nothing -in relativity to show why there should be electrons and protons; -relativity cannot give any reason why matter should exist in little -lumps. With this goes the whole theory of the structure of the atom. -The theory of quanta also is quite outside the scope of relativity. -Relativity is, in a sense, the most extreme application of what may -be called next-to-next methods. Gravitation is no longer regarded -as due to the effect of the sun upon a planet, but as expressing -characteristics of the region in which the planet happens to be. -Distance, which used to be thought to have a definite meaning however -far apart two points might be, is now only definite for neighboring -points. The distance between widely separated places depends upon the -route chosen. We may, it is true, define _the_ distance as the geodesic -distance, but that can only be estimated by adding up little bits, -that is to say, by the method we use in estimating the length of a -curve. What applies to distance applies equally to the straight line. -There is nothing in the actual world having exactly the properties -that straight lines were supposed to have; the nearest approach is the -track of a light ray. Straight lines have to be replaced by geodesics, -which are defined by what they do at each point, not all at once, -like Euclidean straight lines. Measurement, in Weyl’s theory, suffers -the same fate. We can only use a measuring rod to give lengths in one -place: when we move it to another region, there is no knowing how it -will alter. We do assume, however, that, if it alters, it alters bit -by bit, gradually, continuously, and not by sudden jumps. Perhaps -this assumption is unjustified. It belongs to the general outlook of -relativity, which is that of continuity. No doubt it is owing to this -outlook that relativity is unable to account for the discontinuities in -physics, such as quanta, electrons and protons. Perhaps relativity will -conquer these domains when it learns to dispense with the assumption of -continuity. - -Finally we come to geography, in which I include history. The -separation of history from geography rests upon the separation of time -from space; when we amalgamate the two in space-time, we need one word -to describe the combination of geography and history. For the sake of -simplicity, I shall use the one word geography in this extended sense. - -Geography, in this sense, includes everything that, as a matter of -crude fact, distinguishes one part of space-time from another. One -part is occupied by the sun, one by the earth; the intermediate -regions contain light waves, but no matter (apart from a very little -here and there). There is a certain degree of theoretical connection -between different geographical facts; to establish this is the purpose -of physical laws. It is thought that a sufficient knowledge of the -geographical facts of the solar system throughout any finite time, -however short, would enable an ideally competent physicist to predict -the future of the solar system so long as it remained remote from other -stars. We are already in a position to calculate the large facts about -the solar system backwards and forwards for vast periods of time. But -in all such calculations we need a basis of crude fact. The facts are -interconnected, but facts can only be inferred from other facts, not -from general laws alone. Thus the facts of geography have a certain -independent status in physics. No amount of physical laws will enable -us to infer a physical fact unless we know other facts as data for our -inference. And here when I speak of “facts” I am thinking of particular -facts of geography, in the extended sense in which I am using the term. - -In the theory of relativity, we are concerned with _structure_, not -with the material of which the structure is composed. In geography, -on the other hand, the material is relevant. If there is to be any -difference between one place and another, there must either be -differences between the material in one place and that in another, or -places where there is material and places where there is none. The -former of these alternatives seems the more satisfactory. We might -try to say: There are electrons and protons, and the rest is empty. -But in the “empty” regions there are light waves, so that we cannot -say nothing happens in them. Some people maintain that the light -waves take place in the ether, others are content to say simply that -they take place; but in any case events are occurring where there are -light waves. That is all that we can really say for the places where -there is matter, since matter has turned out to be a mathematical -construction built out of events. We may say, therefore, that there -are events everywhere in space-time, but they must be of a somewhat -different kind according as we are dealing with a region where there is -an electron or proton or with the sort of region we should ordinarily -call empty. But as to the intrinsic nature of these events we can know -nothing, except when they happen to be events in our own lives. Our own -perceptions and feelings must be part of the crude material of events -which physics arranges into a pattern—or rather, which physics finds -to be arranged in a pattern. As regards events which do not form part -of our own lives, physics tells us the pattern of them, but is quite -unable to tell us what they are like in themselves. Nor does it seem -possible that this should be discovered by any other method. - - - - -CHAPTER XIII: THE ABOLITION OF “FORCE” - - -In the Newtonian system, bodies under the action of no forces move in -straight lines with uniform velocity; when bodies do not move in this -way, their change of motion is ascribed to a “force.” Some forces seem -intelligible to our imagination: those exerted by a rope or string, -by bodies colliding, or by any kind of obvious pushing or pulling. As -explained in an earlier chapter, our apparent imaginative understanding -of these processes is quite fallacious; all that it really means is -that past experience enables us to foresee more or less what is going -to happen without the need of mathematical calculations. But the -“forces” involved in gravitation and in the less familiar forms of -electrical action do not seem in this way “natural” to our imagination. -It seems odd that the earth can float in the void: the natural thing -to suppose is that it must fall. That is why it has to be supported on -an elephant, and the elephant on a tortoise, according to some early -speculators. The Newtonian theory, in addition to action at a distance, -introduced two other imaginative novelties. The first was, that -gravitation is not always and essentially directed what we should call -“downwards,” _i.e._, towards the center of the earth. The second was, -that a body going round and round in a circle with uniform velocity is -not “moving uniformly” in the sense in which that phrase is applied to -the motion of bodies under no forces, but is perpetually being turned -out of the straight course towards the center of the circle, which -requires a force pulling it in that direction. Hence Newton arrived at -the view that the planets are attracted to the sun by a force, which is -called gravitation. - -This whole point of view, as we have seen, is superseded by relativity. -There are no longer such things as “straight lines” in the old -geometrical sense. There are “straightest lines,” or geodesics, but -these involve time as well as space. A light ray passing through -the solar system does not describe the same orbit as a comet, from -a geometrical point of view; nevertheless each moves in a geodesic. -The whole imaginative picture is changed. A poet might say that water -runs down hill because it is attracted to the sea, but a physicist or -an ordinary mortal would say that it moves as it does, at each point, -because of the nature of the ground at that point, without regard to -what lies ahead of it. Just as the sea does not cause the water to run -towards it, so the sun does not cause the planets to move round it. The -planets move round the sun because that is the easiest thing to do—in -the technical sense of “least action.” It is the easiest thing to do -because of the nature of the region in which they are, not because of -an influence emanating from the sun. - -The supposed necessity of attributing gravitation to a “force” -attracting the planets towards the sun has arisen from the -determination to preserve Euclidean geometry at all costs. If we -suppose that our space is Euclidean, when in fact it is not, we shall -have to call in physics to rectify the errors of our geometry. We shall -find bodies not moving in what we insist upon regarding as straight -lines, and we shall demand a cause for this behavior. Eddington has -stated this matter with admirable lucidity. He supposes a physicist -who has assumed the formula for interval which is used in the special -theory of relativity—a formula which still supposes that the -observer’s space is Euclidean. He continues: - - Since intervals can be compared by experimental - methods, he ought soon to discover that his - (formula for the interval) cannot be reconciled - with observational results, and so realize his - mistake. But the mind does not so readily get rid of - an obsession. It is more likely that our observer - will continue in his opinion, and attribute the - discrepancy of the observations to some influence - which is present and affects the behavior of his - test-bodies. He will, so to speak, introduce a - supernatural agency which he can blame for the - consequences of his mistake.... The name given to - any agency which causes deviation from uniform - motion in a straight line is _force_ according - to the Newtonian definition of force. Hence the - agency invoked through our observer’s mistake is - described as a “field of force.”... _A field of - force represents the discrepancy between the natural - geometry of a co-ordinate system and the abstract - geometry arbitrarily ascribed to it._[15] - -[15] _Mathematical Theory of Relativity_, pp. 37-38. Italics in the -original. - -If people were to learn to conceive the world in the new way, without -the old notion of “force,” it would alter not only their physical -imagination, but probably also their morals and politics. The latter -effect would be quite illogical, but is none the less probable on that -account. In Newton’s theory of the solar system, the sun seems like a -monarch whose behests the planets have to obey. In Einstein’s world -there is more individualism and less government than in Newton’s. -There is also far less hustle: we have seen that laziness is the -fundamental law of Einstein’s universe. The word “dynamic” has come to -mean, in newspaper language, “energetic and forceful”; but if it meant -“illustrating the principles of dynamics,” it ought to be applied to -the people in hot climates who sit under banana trees waiting for the -fruit to drop into their mouths. I hope that journalists, in future, -when they speak of a “dynamic personality,” will mean a person who -does what is least trouble at the moment, without thinking of remote -consequences. If I can contribute to this result, I shall not have -written in vain. - -It has been customary for people to draw arguments from the laws of -nature as to what we ought to do. Such arguments seem to me a mistake: -to imitate nature may be merely slavish. But if nature, as portrayed by -Einstein, is to be our model, it would seem that the anarchists will -have the best of the argument. The physical universe is orderly, not -because there is a central government, but because every body minds -its own business. No two particles of matter ever come into contact; -when they get too close, they both move off. If a man were had up -for knocking another man down, he would be scientifically correct in -pleading that he had never touched him. What happened was that there -was a hill in space-time in the region of the other man’s nose, and it -fell down the hill. - -The abolition of “force” seems to be connected with the substitution -of sight for touch as the source of physical ideas, as explained in -Chapter I. When an image in a looking glass moves, we do not think that -something has pushed it. In places where there are two large mirrors -opposite to each other, you may see innumerable reflections of the -same object. Suppose a gentleman in a top-hat is standing between the -mirrors, there may be twenty or thirty top-hats in the reflections. -Suppose now somebody comes and knocks off the gentleman’s hat with a -stick: all the other twenty or thirty top-hats will tumble down at the -same moment. We think that a force is needed to knock off the “real” -top-hat, but we think the remaining twenty or thirty tumble off, so to -speak, of themselves, or out of a mere passion for imitation. Let us -try to think out this matter a little more seriously. - -Obviously something happens when an image in a looking glass moves. -From the point of view of sight, the event seems just as real as if it -were not in a mirror. But nothing has happened from the point of view -of touch or hearing. When the “real” top-hat falls, it makes a noise; -the twenty or thirty reflections fall without a sound. If it falls on -your toe, you feel it; but we believe that the twenty or thirty people -in the mirrors feel nothing, though top-hats fall on their toes too. -But all this is equally true of the astronomical world. It makes no -noise, because sound cannot travel across a vacuum. So far as we know, -it causes no “feelings,” because there is no one on the spot to “feel” -it. The astronomical world, therefore, seems hardly more “real” or -“solid” than the world in the looking glass, and has just as little -need of “force” to make it move. - -The reader may feel that I am indulging in idle sophistry. “After all,” -he may say, “the image in the mirror is the reflection of something -solid, and the top-hat in the mirror only falls off because of the -force applied to the real top-hat. The top-hat in the mirror cannot -indulge in behavior of its own; it has to copy the real one. This shows -how different the image is from the sun and the planets, because _they_ -are not obliged to be perpetually imitating a prototype. So you had -better give up pretending that an image is just as real as one of the -heavenly bodies.” - -There is, of course, some truth in this; the point is to discover -exactly _what_ truth. In the first place, images are not “imaginary.” -When you see an image, certain perfectly real light waves reach your -eye; and if you hang a cloth over the mirror, these light waves cease -to exist. There is, however, a purely optical difference between an -“image” and a “real” thing. The optical difference is bound up with -this question of imitation. When you hang a cloth over the mirror, -it makes no difference to the “real” object; but when you move the -“real” object away, the image vanishes also. This makes us say that the -light rays which make the image are only reflected at the surface of -the mirror, and do not really come from a point behind it, but from -the “real” object. We have here an example of a general principle of -great importance. Most of the events in the world are not isolated -occurrences, but members of groups of more or less similar events, -which are such that each group is connected in an assignable manner -with a certain small region of space-time. This is the case with the -light rays which make us see both the object and its reflection in the -mirror: they all emanate from the object as a center. If you put an -opaque globe round the object at a certain distance, the object and -its reflection are invisible at any point outside the globe. We have -seen that gravitation, although no longer regarded as an action at a -distance, is still connected with a center: there is, so to speak, a -hill symmetrically arranged about its summit, and the summit is the -place where we conceive the body to be which is connected with the -gravitational field we are considering. For simplicity, common sense -lumps together all the events which form one group in the above sense. -When two people see the same object, two different events occur, but -they are events belonging to one group and connected with the same -center. Just the same applies when two people (as we say) hear the -same noise. And so the reflection in a mirror is less “real” than the -object reflected, even from an optical point of view, because light -rays do not spread in _all_ directions from the place where the image -seems to be, but only in directions in front of the mirror, and only so -long as the object reflected remains in position. This illustrates the -usefulness of grouping connected events about a center in the way we -have been considering. - -When we examine the changes in such a group of objects, we find that -they are of two kinds: there are those which affect only some member -of the group, and those which make connected alterations in all the -members of the group. If you put a candle in front of a mirror, and -then hang black cloth over the mirror, you alter only the reflection -of the candle as seen from various places. If you shut your eyes, -you alter its appearance to you, but not its appearance elsewhere. -If you put a red globe round it at a distance of a foot, you alter -its appearance at any distance greater than a foot, but not at any -distance less than a foot. In all these cases, you do not regard the -candle itself as having changed; in fact, in all of them, you find that -there are groups of changes connected with a different center or with -a number of different centers. When you shut your eyes, for instance, -your eyes, not the candle, look different to any other observer: the -center of the changes that occur is in your eyes. But when you blow -out the candle, its appearance _everywhere_ is changed; in this case -you say that the change has happened to the candle. The changes that -happen to an object are those that affect the whole group of events -which center about the object. All this is only an interpretation of -common sense, and an attempt to explain what we mean by saying that the -image of the candle in the mirror is less “real” than the candle. There -is no connected group of events situated all round the place where the -image seems to be, and changes in the image center about the candle, -not about a point behind the mirror. This gives a perfectly verifiable -meaning to the statement that the image is “only” a reflection. And at -the same time it enables us to regard the heavenly bodies, although -we can only see and not touch them, as more “real” than an image in a -looking glass. - -We can now begin to interpret the common sense notion of one body -having an “effect” upon another, which we must do if we are really to -understand what is meant by the abolition of “force.” Suppose you come -into a dark room and switch on the electric light: the appearance of -everything in the room is changed. Since everything in the room is -visible because it reflects the electric light, this case is really -analogous to that of the image in the mirror; the electric light is the -center from which all the changes emanate. In this case, the “effect” -is explained by what we have already said. The more important case is -when the effect is a movement. Suppose you let loose a tiger in the -middle of a Bank Holiday crowd: they would all move, and the tiger -would be the center of their various movements. A person who could -see the people but not the tiger would infer that there was something -repulsive at that point. We say in this case that the tiger has an -effect upon the people, and we might describe the tiger’s action upon -them as of the nature of a repulsive force. We know, however, that they -fly because of something which happens to _them_, not merely because -the tiger is where he is. They fly because they can see and hear him, -that is to say, because certain waves reach their eyes and ears. If -these waves could be made to reach them without there being any tiger, -they would fly just as fast, because the neighborhood would seem to -them just as unpleasant. - -Let us now apply similar considerations to the sun’s gravitation. The -“force” exerted by the sun only differs from that exerted by the tiger -in being attractive instead of repulsive. Instead of acting through -waves of light or sound, the sun acquires its apparent power through -the fact that there are modifications of space-time all round the sun. -Like the noise of the tiger, they are more intense near their source; -as we travel away they grow less and less. To say that the sun “causes” -these modifications of space-time is to add nothing to our knowledge. -What we know is that the modifications proceed according to a certain -rule, and that they are grouped symmetrically about the sun as center. -The language of cause and effect adds only a number of quite irrelevant -imaginings, connected with will, muscular tension, and such matters. -What we can more or less ascertain is merely the formula according to -which space-time is modified by the presence of gravitating matter. -More correctly: we can ascertain what kind of space-time _is_ the -presence of gravitating matter. When space-time is not accurately -Euclidean in a certain region, but has a non-Euclidean character which -grows more and more marked as we approach a certain center, and when, -further, the departure from Euclid obeys a certain law, we describe -this state of affairs briefly by saying that there is gravitating -matter at the center. But this is only a compendious account of what -we know. What we know is about the places where the gravitating matter -is _not_, not about the place where it is. The language of cause -and effect (of which “force” is a particular case) is thus merely -a convenient shorthand for certain purposes; it does not represent -anything that is genuinely to be found in the physical world. - -And how about matter? Is matter also no more than a convenient -shorthand? This question, however, being a large one, demands a -separate chapter. - - - - -CHAPTER XIV: WHAT IS MATTER? - - -The question “What is matter?” is of the kind that is asked by -metaphysicians, and answered in vast books of incredible obscurity. -But I am not asking the question as metaphysician: I am asking it as a -person who wants to find out what is the moral of modern physics, and -more especially of the theory of relativity. It is obvious from what we -have learned of that theory that matter cannot be conceived quite as it -used to be. I think we can now say more or less what the new conception -must be. - -There were two traditional conceptions of matter, both of which have -had advocates ever since scientific speculation began. There were -the atomists, who thought that matter consisted of tiny lumps which -could never be divided; these were supposed to hit each other and then -bounce off in various ways. After Newton, they were no longer supposed -actually to come into contact with each other, but to attract and -repel each other, and move in orbits round each other. Then there -were those who thought that there is matter of some kind everywhere, -and that a true vacuum is impossible. Descartes held this view, and -attributed the motions of the planets to vortices in the ether. The -Newtonian theory of gravitation caused the view that there is matter -everywhere to fall into discredit, the more so as light was thought by -Newton and his disciples to be due to actual particles traveling from -the source of the light. But when this view of light was disproved, and -it was shown that light consisted of waves, the ether was revived so -that there should be something to undulate. The ether became still more -respectable when it was found to play the same part in electromagnetic -phenomena as in the propagation of light. It was even hoped that atoms -might turn out to be a mode of motion of the ether. At this stage, the -atomic view of matter was, on the whole, getting the worst of it. - -Leaving relativity aside for the moment, modern physics has provided -proof of the atomic structure of ordinary matter, while not disproving -the arguments in favor of the ether, to which no such structure is -attributed. The result was a sort of compromise between the two views, -the one applying to what was called “gross” matter, the other to the -ether. There can be no doubt about electrons and protons, though, as we -shall see shortly, they need not be conceived as atoms were conceived -traditionally. As for the ether, its status is very curious: many -physicists still maintain that, without it, the propagation of light -and other electromagnetic waves would be inconceivable, but except in -this way it is difficult to see what purpose it serves. The truth is, -I think, that relativity demands the abandonment of the old conception -of “matter,” which is infected by the metaphysics associated with -“substance,” and represents a point of view not really necessary in -dealing with phenomena. This is what we must now investigate. - -In the old view, a piece of matter was something which survived all -through time, while never being at more than one place at a given time. -This way of looking at things is obviously connected with the complete -separation of space and time in which people formerly believed. When we -substitute space-time for space and time, we shall naturally expect to -derive the physical world from constituents which are as limited in -time as in space. Such constituents are what we call “events.” An event -does not persist and move, like the traditional piece of matter; it -merely exists for its little moment and then ceases. A piece of matter -will thus be resolved into a series of events. Just as, in the old -view, an extended body was composed of a number of particles, so, now, -each particle, being extended in time, must be regarded as composed -of what we may call “event-particles.” The whole series of these -events makes up the whole history of the particle, and the particle is -regarded as _being_ its history, not some metaphysical entity to which -the events happen. This view is rendered necessary by the fact that -relativity compels us to place time and space more on a level than they -were in the older physics. - -This abstract requirement must be brought into relation with the known -facts of the physical world. Now what are the known facts? Let us -take it as conceded that light consists of waves traveling with the -received velocity. We then know a great deal about what goes on in -the parts of space-time where there is no matter; we know, that is to -say, that there are periodic occurrences (light waves) obeying certain -laws. These light waves start from atoms, and the modern theory of -the structure of the atoms enables us to know a great deal about the -circumstances under which they start, and the reasons which determine -their wave lengths. We can find out not only how one light wave -travels, but how its source moves relatively to ourselves. But when I -say this I am assuming that we can recognise a source of light as the -same at two slightly different times. This is, however, the very thing -which had to be investigated. - -We saw, in the preceding chapter, how a group of connected events can -be formed, all related to each other by a law, and all ranged about a -center in space-time. Such a group of events will be the arrival, at -various places, of the light waves emitted by a brief flash of light. -We do not need to suppose that anything particular is happening at the -center; certainly we do not need to suppose that we know _what_ is -happening there. What we know is that, as a matter of geometry, the -group of events in question are ranged about a center, like widening -ripples on a pool when a fly has touched it. We can hypothetically -invent an occurrence which is to have happened at the center, and set -forth the laws by which the consequent disturbance is transmitted. This -hypothetical occurrence will then appear to common sense as the “cause” -of the disturbance. It will also count as one event in the biography of -the particle of matter which is supposed to occupy the center of the -disturbance. - -Now we find not only that one light wave travels outward from a center -according to a certain law, but also that, in general, it is followed -by other closely similar light waves. The sun, for example, does not -change its appearance suddenly; even if a cloud passes across it during -a high wind, the transition is gradual, though swift. In this way a -group of occurrences connected with a center at one point of space-time -is brought into relation with other very similar groups whose centers -are at neighboring points of space-time. For each of these other groups -common sense invents similar hypothetical occurrences to occupy their -centers, and says that all these hypothetical occurrences are part of -one history; that is to say, it invents a hypothetical “particle” to -which the hypothetical occurrences are to have occurred. It is only by -this double use of hypothesis, perfectly unnecessary in each case, that -we arrive at anything that can be called “matter” in the old sense of -the word. - -If we are to avoid unnecessary hypotheses, we shall say that an -electron at a given moment is the various disturbances in the -surrounding medium which, in ordinary language, would be said to be -“caused” by it. But we shall not take these disturbances at what is, -for us, the moment in question, since that would make them depend upon -the observer; we shall instead travel outward from the electron with -the velocity of light, and take the disturbance we find in each place -as we reach it. The closely similar set of disturbances, with very -nearly the same center, which is found existing slightly earlier or -slightly later, will be defined as _being_ the electron at a slightly -earlier or slightly later moment. In this way, we preserve all the -laws of physics, without having recourse to unnecessary hypotheses or -inferred entities, and we remain in harmony with the general principle -of economy which has enabled the theory of relativity to clear away so -much useless lumber. - -Common sense imagines that when it sees a table it sees a table. This -is a gross delusion. When common sense sees a table, certain light -waves reach its eyes, and these are of a sort which, in its previous -experience, has been associated with certain sensations of touch, as -well as with other people’s testimony that they also saw the table. -But none of this ever brought us to the table itself. The light waves -caused occurrences in our eyes, and these caused occurrences in the -optic nerve, and these in turn caused occurrences in the brain. Any one -of these, happening without the usual preliminaries, would have caused -us to have the sensations we call “seeing the table,” even if there had -been no table. (Of course, if matter in general is to be interpreted -as a group of occurrences, this must apply also to the eye, the optic -nerve, and the brain.) As to the sense of touch when we press the table -with our fingers, that is an electric disturbance in the electrons and -protons of our finger tips, produced, according to modern physics, by -the proximity of the electrons and protons in the table. If the same -disturbance in our finger tips arose in any other way, we should have -the same sensations, in spite of there being no table. The testimony -of others is obviously a second-hand affair. A witness in a law court, -if asked whether he had seen some occurrence, would not be allowed to -reply that he believed so because of the testimony of others to that -effect. In any case, testimony consists of sound waves and demands -psychological as well as physical interpretation; its connection with -the object is therefore very indirect. For all these reasons, when -we say that a man “sees a table,” we use a highly abbreviated form -of expression, concealing complicated and difficult inferences, the -validity of which may well be open to question. - -But we are in danger of becoming entangled in psychological questions, -which we must avoid if we can. Let us therefore return to the purely -physical point of view. - -What I wish to suggest may be put as follows. Everything that occurs -elsewhere, owing to the existence of an electron, can be explored -experimentally, at least in theory, unless it occurs in certain -concealed ways. But what occurs within the electron (if anything occurs -there) it is absolutely impossible to know: there is no conceivable -apparatus by which we could obtain even a glimpse of it. An electron is -known by its “effects.” But the word “effects” belongs to a view of -causation which will not fit modern physics, and in particular will -not fit relativity. All that we have a right to say is that certain -groups of occurrences happen together, that is to say, in neighboring -parts of space-time. A given observer will regard one member of the -group as earlier than the other, but another observer may judge the -time order differently. And even when the time order is the same for -all observers, all that we really have is a connection between the two -events, which works equally backwards and forwards. It is not true that -the past determines the future in some sense other than that in which -the future determines the past: the apparent difference is only due to -our ignorance, because we know less about the future than about the -past. This is a mere accident: there might be beings who would remember -the future and have to infer the past. The feelings of such beings -in these matters would be the exact opposite of our own, but no more -fallacious. - -The moral of this is that, if an electron is only known by its -“effects,” there is no reason to suppose that anything exists except -the “effects.” In so far as these “effects” consist of light waves -and other electromagnetic disturbances, we may say that what is -called “empty space” consists of regions where these disturbances are -propagated freely. Every such disturbance, we find, has a center, and -when we get very near the center (though still at a finite distance -from it) we find that the law of propagation of the disturbance ceases -to be valid. This region within which the law does not hold is called -“matter”; it will be an electron or proton according to circumstances. -The region so defined is found to move relatively to other such -regions, and its movements follow the known laws of dynamics. So far, -this theory provides for electromagnetic phenomena and the motions of -matter; and it does so without assuming that “matter” is anything but -systems of electromagnetic phenomena. In order to carry out the theory -fully, it would no doubt be necessary to introduce many complications. -But it seems fairly clear that all the facts and laws of physics -can be interpreted without assuming that “matter” is anything more -than groups of events, each event being of the sort which we should -naturally regard as “caused” by the matter in question. This does not -involve any change in the symbols or formulæ of physics: it is merely -a question of interpretation of the symbols. - -This latitude in interpretation is a characteristic of mathematical -physics. What we know is certain very abstract logical relations, -which we express in mathematical formulæ; we know also that, at -certain points, we arrive at results which are capable of being tested -experimentally. Take, for example, the eclipse observations by which -Einstein’s theory as to the bending of light was established. The -actual observation consisted in the careful measurement of certain -distances on certain photographic plates. The formulæ which were to -be verified were concerned with the course of light in passing near -the sun. Although the part of these formulæ which gives the observed -result must always be interpreted in the same way, the other part of -them may be capable of a great variety of interpretations. The formulæ -giving the motions of the planets are almost exactly the same in -Einstein’s theory as in Newton’s, but the meaning of the formulæ is -quite different. It may be said generally that, in the mathematical -treatment of nature, we can be far more certain that our formulæ are -approximately correct than we can be as to the correctness of this or -that interpretation of them. And so in the case with which this chapter -is concerned: the question as to the nature of an electron or a proton -is by no means answered when we know all that mathematical physics has -to say as to the laws of its motion and the laws of its interaction -with the environment. A definite and conclusive answer to our question -is not possible just because a variety of answers are compatible with -the truth of mathematical physics. Nevertheless some answers are -preferable to others, because some have a greater probability in their -favor. We have been seeking, in this chapter, to define matter so that -there _must_ be such a thing if the formulæ of physics are true. If we -had made our definition such as to secure that a particle of matter -should be what one thinks of as substantial, a hard, definite lump, we -should not have been _sure_ that any such thing exists. That is why -our definition, though it may seem complicated, is preferable from the -point of view of logical economy and scientific caution. - - - - -CHAPTER XV: PHILOSOPHICAL CONSEQUENCES - - -The philosophical consequences of relativity are neither so great nor -so startling as is sometimes thought. It throws very little light on -time-honored controversies, such as that between realism and idealism. -Some people think that it supports Kant’s view that space and time are -“subjective” and are “forms of intuition.” I think such people have -been misled by the way in which writers on relativity speak of “the -observer.” It is natural to suppose that the observer is a human being, -or at least a mind; but he is just as likely to be a photographic -plate or a clock. That is to say, the odd results as to the difference -between one “point of view” and another are concerned with “point of -view” in a sense applicable to physical instruments just as much as to -people with perceptions. The “subjectivity” concerned in the theory of -relativity is a _physical_ subjectivity, which would exist equally if -there were no such things as minds or senses in the world. - -Moreover, it is a strictly limited subjectivity. The theory does -not say that _everything_ is relative; on the contrary, it gives a -technique for distinguishing what is relative from what belongs to a -physical occurrence in its own right. If we are going to say that the -theory supports Kant about space and time, we shall have to say that it -refutes him about space-time. In my view, neither statement is correct. -I see no reason why, on such issues, philosophers should not all stick -to the views they previously held. There were no conclusive arguments -on either side before, and there are none now; to hold either view -shows a dogmatic rather than a scientific temper. - -Nevertheless, when the ideas involved in Einstein’s work have become -familiar, as they will when they are taught in schools, certain changes -in our habits of thought are likely to result, and to have great -importance in the long run. - -One thing which emerges is that physics tells us much less about -the physical world than we thought it did. Almost all the “great -principles” of traditional physics turn out to be like the “great -law” that there are always three feet to a yard; others turn out to -be downright false. The conservation of mass may serve to illustrate -both these misfortunes to which a “law” is liable. Mass used to be -defined as “quantity of matter,” and as far as experiment showed it -was never increased or diminished. But with the greater accuracy of -modern measurements, curious things were found to happen. In the first -place, the mass as measured was found to increase with the velocity; -this kind of mass was found to be really the same thing as energy. This -kind of mass is not constant for a given body, but the total amount of -it in the universe is conserved, or at least obeys a law very closely -analogous to conservation. This law itself, however, is to be regarded -as a truism, of the nature of the “law” that there are three feet to a -yard; it results from our methods of measurement, and does not express -a genuine property of matter. The other kind of mass, which we may call -“proper mass,” is that which is found to be the mass by an observer -moving with the body. This is the ordinary terrestrial case, where -the body we are weighing is not flying through the air. The “proper -mass” of a body is very nearly constant, but not quite, and the total -amount of “proper mass” in the world is not quite constant. One would -suppose that if you have four one-pound weights, and you put them all -together into the scales, they will together weigh four pounds. This is -a fond delusion: they weigh rather less, though not enough less to be -discovered by even the most careful measurements. In the case of four -hydrogen atoms, however, when they are put together to make one helium -atom, the defect is noticeable; the helium atom weighs measurably less -than four separate hydrogen atoms. - -Broadly speaking, traditional physics has collapsed into two portions, -truisms and geography. There are, however, newer portions of physics, -such as the theory of quanta, which do not come under this head, but -appear to give genuine knowledge of laws reached by experiment. - -The world which the theory of relativity presents to our imagination -is not so much a world of “things” in “motion” as a world of _events_. -It is true that there are still electrons and protons which persist, -but these (as we saw in the preceding chapter) are really to be -conceived as strings of connected events, like the successive notes -of a song. It is _events_ that are the stuff of relativity physics. -Between two events which are not too remote from each other there is, -in the general theory as in the special theory, a measurable relation -called “interval,” which appears to be the physical reality of which -lapse of time and distance in space are two more or less confused -representations. Between two distant events, there is not any one -definite interval. But there is one way of moving from one event to -another which makes the sum of all the little intervals along the route -greater than by any other route. This route is called a “geodesic,” and -it is the route which a body will choose if left to itself. - -The whole of relativity physics is a much more step-by-step matter than -the physics and geometry of former days. Euclid’s straight lines have -to be replaced by light rays, which do not quite come up to Euclid’s -standard of straightness when they pass near the sun or any other very -heavy body. The sum of the angles of a triangle is still thought to be -two right angles in very remote regions of empty space, but not where -there is matter in the neighborhood. We, who cannot leave the earth, -are incapable of reaching a place where Euclid is true. Propositions -which used to be proved by reasoning have now become either -conventions, or merely approximate truths verified by observation. - -It is a curious fact—of which relativity is not the only -illustration—that, as reasoning improves, its claims to the power of -proving facts grow less and less. Logic used to be thought to teach -us how to draw inferences; now, it teaches us rather how not to draw -inferences. Animals and children are terribly prone to inference: a -horse is surprised beyond measure if you take an unusual turning. When -men began to reason, they tried to justify the inferences that they -had drawn unthinkingly in earlier days. A great deal of bad philosophy -and bad science resulted from this propensity. “Great principles,” -such as the “uniformity of nature,” the “law of universal causation,” -and so on, are attempts to bolster up our belief that what has often -happened before will happen again, which is no better founded than the -horse’s belief that you will take the turning you usually take. It is -not altogether easy to see what is to replace these pseudo-principles -in the practice of science; but perhaps the theory of relativity gives -us a glimpse of the kind of thing we may expect. Causation, in the -old sense, no longer has a place in theoretical physics. There is, -of course, something else which takes its place, but the substitute -appears to have a better empirical foundation than the old principle -which it has superseded. - -The collapse of the notion of one all-embracing time, in which all -events throughout the universe can be dated, must in the long run -affect our views as to cause and effect, evolution, and many other -matters. For instance, the question whether, on the whole, there is -progress in the universe, may depend upon our choice of a measure of -time. If we choose one out of a number of equally good clocks, we may -find that the universe is progressing as fast as the most optimistic -American thinks it is; if we choose another equally good clock, we may -find that the universe is going from bad to worse as fast as the most -melancholy Slav could imagine. Thus optimism and pessimism are neither -true nor false, but depend upon the choice of clocks. - -The effect of this upon a certain type of emotion is devastating. The -poet speaks of - - One far-off divine event - To which the whole creation moves. - -But if the event is sufficiently far off, and the creation moves -sufficiently quickly, some parts will judge that the event has already -happened, while others will judge that it is still in the future. This -spoils the poetry. The second line ought to be: - - To which some parts of the creation move, - while others move away from it. - -But this won’t do. I suggest that an emotion which can be destroyed by -a little mathematics is neither very genuine nor very valuable. But -this line of argument would lead to a criticism of the Victorian Age, -which lies outside my theme. - -What we know about the physical world, I repeat, is much more -abstract, than was formerly supposed. Between bodies there are -occurrences, such as light waves; of the _laws_ of these occurrences, -we know something—just so much as can be expressed in mathematical -formulæ—but of their _nature_ we know nothing. Of the bodies -themselves, as we saw in the preceding chapter, we know so little -that we cannot even be sure that they are anything: they _may_ be -merely groups of events in other places, those events which we should -naturally regard as their effects. We naturally interpret the world -pictorially; that is to say, we imagine that what goes on is more or -less like what we see. But in fact this likeness can only extend to -certain formal logical properties expressing structure, so that all we -can know is certain general characteristics of its changes. Perhaps an -illustration may make the matter clear. Between a piece of orchestral -music as played, and the same piece of music as printed in the score, -there is a certain resemblance, which may be described as a resemblance -in structure. The resemblance is of such a sort that, when you know the -rules, you can infer the music from the score or the score from the -music. But suppose you had been stone deaf from birth, but had lived -among musical people. You could understand, if you had learned to speak -and to do lip-reading, that the musical scores represented something -quite different from themselves in intrinsic quality, though similar in -structure.[16] The value of music would be completely unimaginable to -you, but you could infer all its mathematical characteristics, since -they are the same as those of the score. Now our knowledge of nature is -something like this. We can read the scores, and infer just so much as -our stone-deaf person could have inferred about music. But we have not -the advantages which he derived from association with musical people. -We cannot know whether the music represented by the scores is beautiful -or hideous; perhaps, in the last analysis, we cannot be quite sure that -the scores represent anything but themselves. But this is a doubt which -the physicist, in his professional capacity, cannot permit himself to -entertain. - -[16] For the definition of “structure,” see the present author’s -_Introduction to Mathematical Philosophy_. - -Assuming the utmost that can be claimed for physics, it does not tell -us what it is that changes, or what are its various states; it only -tells us such things as that changes follow each other periodically, -or spread with a certain speed. Even now we are probably not at the -end of the process of stripping away what is merely imagination, in -order to reach the core of true scientific knowledge. The theory of -relativity has accomplished a very great deal in this respect, and in -doing so has taken us nearer and nearer to bare structure, which is -the mathematician’s goal—not because it is the only thing in which he -is interested as a human being, but because it is the only thing that -he can express in mathematical formulæ. But far as we have traveled in -the direction of abstraction, it may be that we shall have to travel -further still. - -In the preceding chapter, I suggested what may be called a minimum -definition of matter, that is to say, one in which matter has, so -to speak, as little “substance” as is compatible with the truth of -physics. In adopting a definition of this kind, we are playing for -safety: our tenuous matter will exist, even if something more beefy -also exists. We tried to make our definition of matter, like Isabella’s -gruel in Jane Austen, “thin, but not too thin.” We shall, however, fall -into error if we assert positively that matter is nothing more than -this. Leibniz thought that a piece of matter is really a colony of -souls. There is nothing to show that he was wrong, though there is also -nothing to show that he was right: we know no more about it either way -than we do about the flora and fauna of Mars. - -To the non-mathematical mind, the abstract character of our physical -knowledge may seem unsatisfactory. From an artistic or imaginative -point of view, it is perhaps regrettable, but from a practical point -of view it is of no consequence. Abstraction, difficult as it is, is -the source of practical power. A financier, whose dealings with the -world are more abstract than those of any other “practical” man, is -also more powerful than any other practical man. He can deal in wheat -or cotton without needing ever to have seen either: all he needs to -know is whether they will go up or down. This is abstract mathematical -knowledge, at least as compared to the knowledge of the agriculturist. -Similarly the physicist, who knows nothing of matter except certain -laws of its movements, nevertheless knows enough to enable him to -manipulate it. After working through whole strings of equations, in -which the symbols stand for things whose intrinsic nature can never be -known to us, he arrives at last at a result which can be interpreted -in terms of our own perceptions, and utilized to bring about desired -effects in our own lives. What we know about matter, abstract and -schematic as it is, is enough, in principle, to tell us the rules -according to which it produces perceptions and feelings in ourselves; -and it is upon these rules that the _practical_ uses of physics depend. - -The final conclusion is that we know very little, and yet it is -astonishing that we know so much, and still more astonishing that so -little knowledge can give us so much power. - - -THE END - -*** END OF THE PROJECT GUTENBERG EBOOK THE A B C OF RELATIVITY *** - -Updated editions will replace the previous one--the old editions will -be renamed. - -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the -United States without permission and without paying copyright -royalties. 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If you -are not located in the United States, you will have to check the laws of the -country where you are located before using this eBook. -</div> - -<p style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Title: The A B C of Relativity</p> -<p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em'>Author: Bertrand Russell</p> -<p style='display:block; text-indent:0; margin:1em 0'>Release Date: January 4, 2022 [eBook #67104]</p> -<p style='display:block; text-indent:0; margin:1em 0'>Language: English</p> - <p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em; text-align:left'>Produced by: Tim Lindell and the Online Distributed Proofreading Team at https://www.pgdp.net (This book was produced from images made available by the HathiTrust Digital Library.)</p> -<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK THE A B C OF RELATIVITY ***</div> - -<hr class="chap x-ebookmaker-drop" /> -<p class="f150"><b>HARPER’S MODERN SCIENCE SERIES</b></p> -<hr class="chap x-ebookmaker-drop" /> - -<h1>THE A B C OF RELATIVITY</h1> - -<p class="center space-above3">BY</p> -<p class="f150">BERTRAND RUSSELL</p> - -<p class="f90 space-below2">AUTHOR OF<br />“THE PRINCIPLES OF MATHEMATICS”<br /> -“PROPOSED ROADS TO FREEDOM”<br />AND “WHY MEN FIGHT”</p> - -<p class="center">PUBLISHERS</p> -<p class="f120">HARPER & BROTHERS<br />NEW YORK AND LONDON</p> - -<p class="f90 space-above2">THE A B C OF RELATIVITY</p> - -<p class="center">Copyright, 1925, by Harper & Brothers<br /> -Printed in the United States of America</p> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p class="f200"><b><i>Contents</i></b></p> -</div> - -<table border="0" cellspacing="0" summary="TOC" cellpadding="0" > - <tbody><tr> - <td class="tdl" colspan="2"><small>CHAPTER</small></td> - <td class="tdr"><small>PAGE</small></td> - </tr><tr> - <td class="tdr">I.</td> - <td class="tdl_ws1">TOUCH AND SIGHT: THE EARTH AND THE HEAVENS  </td> - <td class="tdr"><a href="#Page_1"> 1</a></td> - </tr><tr> - <td class="tdr">II.</td> - <td class="tdl_ws1">WHAT HAPPENS AND WHAT IS OBSERVED</td> - <td class="tdr"><a href="#Page_14">14</a></td> - </tr><tr> - <td class="tdr">III.</td> - <td class="tdl_ws1">THE VELOCITY OF LIGHT</td> - <td class="tdr"><a href="#Page_28">28</a></td> - </tr><tr> - <td class="tdr">IV.</td> - <td class="tdl_ws1">CLOCKS AND FOOT RULES</td> - <td class="tdr"><a href="#Page_43">43</a></td> - </tr><tr> - <td class="tdr">V.</td> - <td class="tdl_ws1">SPACE-TIME</td> - <td class="tdr"><a href="#Page_58">58</a></td> - </tr><tr> - <td class="tdr">VI.</td> - <td class="tdl_ws1">THE SPECIAL THEORY OF RELATIVITY</td> - <td class="tdr"><a href="#Page_71">71</a></td> - </tr><tr> - <td class="tdr">VII.</td> - <td class="tdl_ws1">INTERVALS IN SPACE-TIME</td> - <td class="tdr"><a href="#Page_91">91</a></td> - </tr><tr> - <td class="tdr">VIII.</td> - <td class="tdl_ws1">EINSTEIN’S LAW OF GRAVITATION</td> - <td class="tdr"><a href="#Page_111">111</a></td> - </tr><tr> - <td class="tdr">IX.</td> - <td class="tdl_ws1">PROOFS OF EINSTEIN’S LAW OF GRAVITATION</td> - <td class="tdr"><a href="#Page_131">131</a></td> - </tr><tr> - <td class="tdr">X.</td> - <td class="tdl_ws1">MASS, MOMENTUM, ENERGY AND ACTION</td> - <td class="tdr"><a href="#Page_144">144</a></td> - </tr><tr> - <td class="tdr">XI.</td> - <td class="tdl_ws1">IS THE UNIVERSE FINITE?</td> - <td class="tdr"><a href="#Page_163">163</a></td> - </tr><tr> - <td class="tdr">XII.</td> - <td class="tdl_ws1">CONVENTIONS AND NATURAL LAWS</td> - <td class="tdr"><a href="#Page_177">177</a></td> - </tr><tr> - <td class="tdr">XIII.</td> - <td class="tdl_ws1">THE ABOLITION OF “FORCE”</td> - <td class="tdr"><a href="#Page_192">192</a></td> - </tr><tr> - <td class="tdr">XIV.</td> - <td class="tdl_ws1">WHAT IS MATTER?</td> - <td class="tdr"><a href="#Page_206">206</a></td> - </tr><tr> - <td class="tdr">XV.</td> - <td class="tdl_ws1">PHILOSOPHICAL CONSEQUENCES</td> - <td class="tdr"><a href="#Page_219">219</a></td> - </tr> - </tbody> -</table> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p class="f200"><b>THE A B C OF RELATIVITY</b></p> -</div> - -<hr class="chap x-ebookmaker-drop" /> -<p><span class="pagenum" id="Page_1">[Pg 1]</span></p> -<div class="chapter"> -<h2 class="nobreak">CHAPTER ONE:<br /> TOUCH AND SIGHT:<br /> THE EARTH AND THE HEAVENS</h2> -</div> - -<p class="drop-cap"><span class="smcap">Everybody</span> knows that -Einstein has done something astonishing, but very few people know -exactly what it is that he has done. It is generally recognized that -he has revolutionized our conception of the physical world, but his -new conceptions are wrapped up in mathematical technicalities. It is -true that there are innumerable popular accounts of the theory of -relativity, but they generally cease to be intelligible just at the -point where they begin to say something important. The authors are -hardly to blame for this. Many of the new ideas can be expressed in -non-mathematical language, but they are none the less difficult on that -account. What is demanded is a change in our imaginative picture of the -<span class="pagenum" id="Page_2">[Pg 2]</span> -world—a picture which has been handed down from remote, perhaps -pre-human, ancestors, and has been learned by each one of us in early -childhood. A change in our imagination is always difficult, especially -when we are no longer young. The same sort of change was demanded -by Copernicus, when he taught that the earth is not stationary and -the heavens do not revolve about it once a day. To us now there is -no difficulty in this idea, because we learned it before our mental -habits had become fixed. Einstein’s ideas, similarly, will seem easy -to a generation which has grown up with them; but for our generation a -certain effort of imaginative reconstruction is unavoidable.</p> - -<p>In exploring the surface of the earth, we make use of all our senses, -more particularly of the senses of touch and sight. In measuring -lengths, parts of the human body are employed in pre-scientific -ages: a “foot,” a “cubit,” a “span” are defined in this way. For -longer distances, we think of the time it takes to walk from one -place to another. We gradually learn to judge distances roughly by -the eye, but we rely upon touch for accuracy. Moreover it is touch -that gives us our sense of “reality.” Some things cannot be touched: -rainbows, reflections in looking-glasses, and so on. These things -<span class="pagenum" id="Page_3">[Pg 3]</span> -puzzle children, whose metaphysical speculations are arrested by the -information that what is in the looking glass is not “real.” Macbeth’s -dagger was unreal because it was not “sensible to feeling as to sight.” -Not only our geometry and physics, but our whole conception of what -exists outside us, is based upon the sense of touch. We carry this even -into our metaphors: a good speech is “solid,“ a bad speech is “gas,” -because we feel that a gas is not quite “real.”</p> - -<p>In studying the heavens, we are debarred from all senses except sight. -We cannot touch the sun, or travel to it; we cannot walk round the -moon, or apply a foot rule to the Pleiades. Nevertheless, astronomers -have unhesitatingly applied the geometry and physics which they found -serviceable on the surface of the earth, and which they had based -upon touch and travel. In doing so, they brought down trouble on -their heads, which it has been left for Einstein to clear up. It has -turned out that much of what we learned from the sense of touch was -unscientific prejudice, which must be rejected if we are to have a true -picture of the world.</p> - -<p>An illustration may help us to understand how much is impossible to -the astronomer as compared to the man who is interested in things on -<span class="pagenum" id="Page_4">[Pg 4]</span> -the surface of the earth. Let us suppose that a drug is administered -to you which makes you temporarily unconscious, and that when you -wake you have lost your memory but not your reasoning powers. Let us -suppose further that while you were unconscious you were carried into -a balloon, which, when you come to, is sailing with the wind in a dark -night—the night of the fifth of November if you are in England, or -of the fourth of July if you are in America. You can see fireworks which -are being sent off from the ground, from trains, and from aeroplanes -traveling in all directions, but you cannot see the ground or the -trains or the aeroplanes be cause of the darkness. What sort of picture -of the world will you form? You will think that nothing is permanent: -there are only brief flashes of light, which, during their short -existence, travel through the void in the most various and bizarre -curves. You cannot touch these flashes of light, you can only see them. -Obviously your geometry and your physics and your metaphysics will be -quite different from those of ordinary mortals. If an ordinary mortal -<span class="pagenum" id="Page_5">[Pg 5]</span> -is with you in the balloon, you will find his speech unintelligible. -But if Einstein is with you, you will understand him more easily than -the ordinary mortal would, because you will be free from a host of -preconceptions which prevent most people from understanding him.</p> - -<p>The theory of relativity depends, to a considerable extent, upon -getting rid of notions which are useful in ordinary life but not to -our drugged balloonist. Circumstances on the surface of the earth, -for various more or less accidental reasons, suggest conceptions -which turn out to be inaccurate, although they have come to seem like -necessities of thought. The most important of these circumstances is -that most objects on the earth’s surface are fairly persistent and -nearly stationary from a terrestrial point of view. If this were not -the case, the idea of going a journey would not seem so definite as it -does. If you want to travel from King’s Cross to Edinburgh, you know -that you will find King’s Cross where it always has been, that the -railway line will take the course that it did when you last made the -journey, and that Waverley Station in Edinburgh will not have walked up -to the Castle. You therefore say and think that you have traveled to -<span class="pagenum" id="Page_6">[Pg 6]</span> -Edinburgh, not that Edinburgh has traveled to you, though the latter -statement would be just as accurate. The success of this common sense -point of view depends upon a number of things which are really of the -nature of luck. Suppose all the houses in London were perpetually -moving about, like a swarm of bees; suppose railways moved and changed -their shapes like avalanches; and finally suppose that material objects -were perpetually being formed and dissolved like clouds. There is -nothing impossible in these suppositions: something like them must have -been verified when the earth was hotter than it is now. But obviously -what we call a journey to Edinburgh would have no meaning in such a -world. You would begin, no doubt, by asking the taxi-driver: “Where -is King’s Cross this morning?“ At the station you would have to ask a -similar question about Edinburgh, but the booking-office clerk would -reply: “What part of Edinburgh do you mean, Sir? Prince’s Street has -gone to Glasgow, the Castle has moved up into the Highlands, and -Waverley Station is under water in the middle of the Firth of Forth.” -And on the journey the stations would not be staying quiet, but some -<span class="pagenum" id="Page_7">[Pg 7]</span> -would be travelling north, some south, some east or west, perhaps much -faster than the train. Under these conditions you could not say where -you were at any moment. Indeed the whole notion that one is always in -some definite “place” is due to the fortunate immovability of most of -the large objects on the earth’s surface. The idea of “place” is only -a rough practical approximation: there is nothing logically necessary -about it, and it cannot be made precise.</p> - -<p>If we were not much larger than an electron, we should not have this -impression of stability, which is only due to the grossness of our -senses. King’s Cross, which to us looks solid, would be too vast to -be conceived except by a few eccentric mathematicians. The bits of it -that we could see would consist of little tiny points of matter, never -coming into contact with each other, but perpetually whizzing round -each other in an inconceivably rapid ballet-dance. The world of our -experience would be quite as mad as the one in which the different -parts of Edinburgh go for walks in different directions. If—to take -the opposite extreme—you were as large as the sun and lived as long, -with a corresponding slowness of perception, you would again find a -<span class="pagenum" id="Page_8">[Pg 8]</span> -higgledy-piggledy universe without permanence—stars and planets -would come and go like morning mists, and nothing would remain in a -fixed position relatively to anything else. The notion of comparative -stability which forms part of our ordinary outlook is thus due to the -fact that we are about the size we are, and live on a planet of which -the surface is no longer very hot. If this were not the case, we should -not find pre-relativity physics intellectually satisfying. Indeed, we -should never have invented such theories. We should have had to arrive -at relativity physics at one bound, or remain ignorant of scientific -laws. It is fortunate for us that we were not faced with this -alternative, since it is almost inconceivable that one man could have -done the work of Euclid, Galileo, Newton, and Einstein. Yet without -such an incredible genius physics could hardly have been discovered -in a world where the universal flux was obvious to non-scientific -observation.</p> - -<p>In astronomy, although the sun, moon, and stars continue to exist year -after year, yet in other respects the world we have to deal with is -very different from that of everyday life. As already observed, we -depend exclusively on sight: the heavenly bodies cannot be touched, -<span class="pagenum" id="Page_9">[Pg 9]</span> -heard, smelt or tasted. Everything in the heavens is moving relatively -to everything else. The earth is going round the sun, the sun is -moving, very much faster than an express train, towards a point in the -constellation “Hercules,” the “fixed” stars are scurrying hither and -thither like a lot of frightened hens. There are no well-marked places -in the sky, like King’s Cross and Edinburgh. When you travel from place -to place on the earth, you say the train moves and not the stations, -because the stations preserve their topographical relations to each -other and the surrounding country. But in astronomy it is arbitrary -which you call the train and which the station: the question is to be -decided purely by convenience and as a matter of convention.</p> - -<p>In this respect, it is interesting to contrast Einstein and Copernicus. -Before Copernicus, people thought that the earth stood still and the -heavens revolved about it once a day. Copernicus taught that “really” -the earth rotates once a day, and the daily revolution of sun and stars -is only “apparent.” Galileo and Newton endorsed this view, and many -things were thought to prove it—for example, the flattening of the -<span class="pagenum" id="Page_10">[Pg 10]</span> -earth at the poles, and the fact that bodies are heavier there than at -the equator. But in the modern theory the question between Copernicus -and his predecessors is merely one of convenience; all motion is -relative, and there is no difference between the two statements: “the -earth rotates once a day” and “the heavens revolve about the earth -once a day.” The two mean exactly the same thing, just as it means the -same thing if I say that a certain length is six feet or two yards. -Astronomy is easier if we take the sun as fixed than if we take the -earth, just as accounts are easier in a decimal coinage. But to say -more for Copernicus is to assume absolute motion, which is a fiction. -All motion is relative, and it is a mere convention to take one body as -at rest. All such conventions are equally legitimate, though not all -are equally convenient.</p> - -<p>There is another matter of great importance, in which astronomy -differs from terrestrial physics because of its exclusive dependence -upon sight. Both popular thought and old-fashioned physics used the -notion of “force,” which seemed intelligible because it was associated -with familiar sensations. When we are walking, we have sensations -connected with our muscles which we do not have when we are sitting -still. In the days before mechanical traction, although people could -<span class="pagenum" id="Page_11">[Pg 11]</span> -travel by sitting in their carriages, they could see the horses -exerting themselves and evidently putting out “force” in the same -way as human beings do. Everybody knew from experience what it is to -push or pull, or to be pushed or pulled. These very familiar facts -made “force” seem a natural basis for dynamics. But Newton’s law of -gravitation introduced a difficulty. The force between two billiard -balls appeared intelligible, because we know what it feels like to bump -into another person; but the force between the earth and the sun, which -are ninety-three million miles apart, was mysterious. Newton himself -regarded this “action at a distance” as impossible, and believed that -there was some hitherto undiscovered mechanism by which the sun’s -influence was transmitted to the planets. However, no such mechanism -was discovered, and gravitation remained a puzzle. The fact is that the -whole conception of “force” is a mistake. The sun does not exert any -force on the planets; in Einstein’s law of gravitation, the planet only -pays attention to what it finds in its own neighborhood. The way in -which this works will be explained in a later chapter; for the present -<span class="pagenum" id="Page_12">[Pg 12]</span> -we are only concerned with the necessity of abandoning the notion of -“force,” which was due to misleading conceptions derived from the sense -of touch.</p> - -<p>As physics has advanced, it has appeared more and more that sight is -less misleading than touch as a source of fundamental notions about -matter. The apparent simplicity in the collision of billiard balls is -quite illusory. As a matter of fact, the two billiard balls never touch -at all; what really happens is inconceivably complicated, but is more -analogous to what happens when a comet penetrates the solar system and -goes away again than to what common sense supposes to happen.</p> - -<p>Most of what we have said hitherto was already recognized by physicists -before Einstein invented the theory of relativity. “Force” was known -to be merely a mathematical fiction, and it was generally held that -motion is a merely relative phenomenon—that is to say, when two -bodies are changing their relative position, we cannot say that one is -moving while the other is at rest, since the occurrence is merely a -change in their relation to each other. But a great labor was required -in order to bring the actual procedure of physics into harmony with -<span class="pagenum" id="Page_13">[Pg 13]</span> -these new convictions. Newton believed in force and in absolute space -and time; he embodied these beliefs in his technical methods, and his -methods remained those of later physicists. Einstein invented a new -technique, free from Newton’s assumptions. But in order to do so he -had to change fundamentally the old ideas of space and time, which had -been unchallenged from time immemorial. This is what makes both the -difficulty and the interest of his theory. But before explaining it -there are some preliminaries which are indispensable. These will occupy -the next two chapters.</p> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_14">[Pg 14]</span></p> - -<h2 class="nobreak">CHAPTER II:<br /> WHAT HAPPENS AND<br /> WHAT IS OBSERVED</h2> -</div> - -<p class="drop-cap"><span class="smcap">A certain</span> type of -superior person is fond of asserting that “everything is relative.” -This is, of course, nonsense, because, if <i>everything</i> were -relative, there would be nothing for it to be relative to. However, -without falling into metaphysical absurdities it is possible to -maintain that everything in the physical world is relative to an -observer. This view, true or not, is <i>not</i> that adopted by the -“theory of relativity.” Perhaps the name is unfortunate; certainly -it has led philosophers and uneducated people into confusions. They -imagine that the new theory proves <i>everything</i> in the physical -world to be relative, whereas, on the contrary, it is wholly concerned -to exclude what is relative and arrive at a statement of physical laws -that shall in no way depend upon the circumstances of the observer. It -is true that these circumstances have been found to have more effect -<span class="pagenum" id="Page_15">[Pg 15]</span> -upon what appears to the observer than they were formerly thought to -have, but at the same time Einstein showed how to discount this effect -completely. This was the source of almost everything that is surprising -in his theory.</p> - -<p>When two observers perceive what is regarded as one occurrence, there -are certain similarities, and also certain differences, between their -perceptions. The differences are obscured by the requirements of -daily life, because from a business point of view they are as a rule -unimportant. But both psychology and physics, from their different -angles, are compelled to emphasize the respects in which one man’s -perception of a given occurrence differs from another man’s. Some of -these differences are due to differences in the brains or minds of -the observers, some to differences in their sense organs, some to -differences of physical situation: these three kinds may be called -respectively psychological, physiological, and physical. A remark made -in a language we know will be heard, whereas an equally loud remark -in an unknown language may pass entirely unnoticed. Of two men in the -Alps, one will perceive the beauty of the scenery while the other will -notice the waterfalls with a view to obtaining power from them. Such -<span class="pagenum" id="Page_16">[Pg 16]</span> -differences are psychological. The difference between a long-sighted -and a short-sighted man, or between a deaf man and a man who hears -well, are physiological. Neither of these kinds concerns us, and I have -mentioned them only in order to exclude them. The kind that concerns us -is the purely physical kind. Physical differences between two observers -will be preserved when the observers are replaced by cameras or -phonographs, and can be reproduced on the movies or the gramophone. If -two men both listen to a third man speaking, and one of them is nearer -to the speaker than the other is, the nearer one will hear louder and -slightly earlier sounds than are heard by the other. If two men both -watch a tree falling, they see it from different angles. Both these -differences would be shown equally by recording instruments: they are -in no way due to idiosyncrasies in the observers, but are part of the -ordinary course of physical nature as we experience it.</p> - -<p>The physicist, like the plain man, believes that his perceptions give -him knowledge about what is really occurring in the physical world, -and not only about his private experiences. Professionally, he regards -<span class="pagenum" id="Page_17">[Pg 17]</span> -the physical world as “real,” not merely as something which human -beings dream. An eclipse of the sun, for instance, can be observed -by any person who is suitably situated, and is also observed by the -photographic plates that are exposed for the purpose. The physicist -is persuaded that something has really happened over and above the -experiences of those who have looked at the sun or at photographs of -it. I have emphasized this point, which might seem a trifle obvious, -because some people imagine that Einstein has made a difference in this -respect. In fact he has made none.</p> - -<p>But if the physicist is justified in this belief that a number of -people can observe the “same” physical occurrence, then clearly the -physicist must be concerned with those features which the occurrence -has in common for all observers, for the others cannot be regarded -as belonging to the occurrence itself. At least, the physicist must -confine himself to the features which are common to all “equally -good” observers. The observer who uses a microscope or a telescope is -preferred to one who does not, because he sees all that the latter sees -and more too. A sensitive photographic plate may “see” still more, -and is then preferred to any eye. But such things as differences of -<span class="pagenum" id="Page_18">[Pg 18]</span> -perspective, or differences of apparent size due to difference of -distance, are obviously not attributable to the object; they belong -solely to the point of view of the spectator. Common sense eliminates -these in judging of objects; physics has to carry the same process much -further, but the principle is the same.</p> - -<p>I want to make it clear that I am not concerned with anything that can -be called inaccuracy. I am concerned with genuine physical differences -between occurrences each of which is a correct record of a certain -event, from its own point of view. When a man fires a gun, people who -are not quite close to him see the flash before they hear the report. -This is not due to any defect in their senses, but to the fact that -sound travels more slowly than light. Light travels so fast that, -from the point of view of phenomena on the surface of the earth, it -may be regarded as instantaneous. Anything that we can see on the -earth happens practically at the moment when we see it. In a second, -light travels 300,000 kilometers (about 186,000 miles). It travels -from the sun to the earth in about eight minutes, and from the stars -to us in anything from three to a thousand years. But of course we -<span class="pagenum" id="Page_19">[Pg 19]</span> -cannot place a clock in the sun, and send out a flash of light from -it at 12 noon, Greenwich Mean Time, and have it received at Greenwich -at 12.08 <span class="smcap">p.m.</span> Our methods of estimating the speed of -light have to be more or less indirect. The only direct method would be that -which we apply to sound when we use an echo. We could send a flash to -a mirror, and observe how long it took for the reflection to reach -us; this would give the time of the double journey to the mirror and -back. On the earth, however, the time would be so short that a great -deal of theoretical physics has to be utilized if this method is to be -employed—more even than is required for the employment of astronomical -data.</p> - -<p>The problem of allowing for the spectator’s point of view, we may be -told, is one of which physics has at all times been fully aware; indeed -it has dominated astronomy ever since the time of Copernicus. This is -true. But principles are often acknowledged long before their full -consequences are drawn. Much of traditional physics is incompatible -<span class="pagenum" id="Page_20">[Pg 20]</span> -with the principle, in spite of the fact that it was acknowledged -theoretically by all physicists.</p> - -<p>There existed a set of rules which caused uneasiness to the -philosophically minded, but were accepted by physicists because -they worked in practice. Locke had distinguished “secondary” -qualities—colors, noises, tastes, smells, etc.—as subjective, while -allowing “primary” qualities—shapes and positions and sizes—to be -genuine properties of physical objects. The physicist’s rules were -such as would follow from this doctrine. Colors and noises were -allowed to be subjective, but due to waves proceeding with a definite -velocity—that of light or sound as the case may be—from their source -to the eye or ear of the percipient. Apparent shapes vary according to -the laws of perspective, but these laws are simple and make it easy to -infer the “real” shapes from several visual apparent shapes; moreover, -the “real” shapes can be ascertained by touch in the case of bodies in -our neighborhood. The objective time of a physical occurrence can be -inferred from the time when we perceive it by allowing for the velocity -of transmission—of light or sound or nerve currents according to -<span class="pagenum" id="Page_21">[Pg 21]</span> -circumstances. This was the view adopted by physicists in practice, -whatever qualms they may have had in unprofessional moments.</p> - -<p>This view worked well enough until physicists became concerned with -much greater velocities than those that are common on the surface of -the earth. An express train travels about a mile in a minute; the -planets travel a few miles in a second. Comets, when they are near -the sun, travel much faster, and behave somewhat oddly; but they were -puzzling in various ways. Practically, the planets were the most -swiftly moving bodies to which dynamics could be adequately applied. -With radio-activity a new range of observations became possible. -Individual electrons can be observed, emanating from radium with a -velocity not far short of that of light. The behavior of bodies moving -with these enormous speeds is not what the old theories would lead -us to expect. For one thing, mass seems to increase with speed in a -perfectly definite manner. When an electron is moving very fast, a -bigger force is required to have a given effect upon it than when it -is moving slowly. Then reasons were found for thinking that the size -<span class="pagenum" id="Page_22">[Pg 22]</span> -of a body is affected by its motion—for example, if you take a cube -and move it very fast, it gets shorter in the direction of its motion, -from the point of view of a person who is not moving with it, though -from its own point of view (<i>i.e.</i> for an observer traveling -with it) it remains just as it was. What was still more astonishing -was the discovery that lapse of time depends on motion; that is to -say, two perfectly accurate clocks, one of which is moving very fast -relatively to the other, will not continue to show the same time if -they come together again after a journey. It follows that what we -discover by means of clocks and foot rules, which used to be regarded -as the acme of impersonal science, is really in part dependent upon our -private circumstances, <i>i.e.</i> upon the way in which we are moving -relatively to the bodies measured.</p> - -<p>This shows that we have to draw a different line from that which is -customary in distinguishing between what belongs to the observer and -what belongs to the occurrence which he is observing. If a man is -wearing blue spectacles he knows that the blue look of everything is -due to his spectacles, and does not belong to what he is observing. -But if he observes two flashes of lightning, and notes the interval -<span class="pagenum" id="Page_23">[Pg 23]</span> -of time between his observations; if he knows where the flashes took -place, and allows, in each case, for the time the light took to reach -him—in that case, if his chronometer is accurate, he naturally thinks -that he has discovered the actual interval of time between the two -flashes, and not something merely personal to himself. He is confirmed -in this view by the fact that all other careful observers to whom he -has access agree with his estimates. This, however, is only due to the -fact that all these observers are on the earth, and share its motion. -Even two observers in aeroplanes moving in opposite directions would -have at the most a relative velocity of 400 miles an hour, which is -very little in comparison with 186,000 miles a second (the velocity -of light). If an electron shot out from a piece of radium with a -velocity of 170,000 miles a second could observe the time between the -two flashes, it would arrive at a quite different estimate, after -making full allowance for the velocity of light. How do you know this? -the reader may ask. You are not an electron, you cannot move at these -terrific speeds, no man of science has ever made the observations which -would prove the truth of your assertion. Nevertheless, as we shall see -<span class="pagenum" id="Page_24">[Pg 24]</span> -in the sequel, there is good ground for the assertion—ground, first -of all, in experiment, and—what is remarkable—ground in reasonings -which could have been made at any time, but were not made until -experiments had shown that the old reasonings must be wrong.</p> - -<p>There is a general principle to which the theory of relativity appeals, -which turns out to be more powerful than anybody would suppose. If -you know that one man is twice as rich as another, this fact must -appear equally whether you estimate the wealth of both in pounds or -dollars or francs or any other currency. The numbers representing their -fortunes will be changed, but one number will always be double the -other. The same sort of thing, in more complicated forms, reappears in -physics. Since all motion is relative, you may take any body you like -as your standard body of reference, and estimate all other motions -with reference to that one. If you are in a train and walking to the -dining-car, you naturally, for the moment, treat the train as fixed -and estimate your motion by relation to it. But when you think of the -journey you are making, you think of the earth as fixed, and say you -<span class="pagenum" id="Page_25">[Pg 25]</span> -are moving at the rate of sixty miles an hour. An astronomer who is -concerned with the solar system takes the sun as fixed, and regards you -as rotating and revolving; in comparison with this motion, that of the -train is so slow that it hardly counts. An astronomer who is interested -in the stellar universe may add the motion of the sun relatively to -the average of the stars. You cannot say that one of these ways of -estimating your motion is more correct than another; each is perfectly -correct as soon as the reference body is assigned. Now just as you can -estimate a man’s fortune in different currencies without altering its -relations to the fortunes of other men, so you can estimate a body’s -motion by means of different reference bodies without altering its -relations to other motions. And as physics is entirely concerned with -relations, it must be possible to express all the laws of physics by -referring all motions to any given body as the standard.</p> - -<p>We may put the matter in another way. Physics is intended to give -information about what really occurs in the physical world, and not -only about the private perceptions of separate observers. Physics -must, therefore, be concerned with those features which a physical -<span class="pagenum" id="Page_26">[Pg 26]</span> -process has in common for all observers, since such features alone -can be regarded as belonging to the physical occurrence itself. This -requires that the <i>laws</i> of phenomena should be the same whether -the phenomena are described as they appear to one observer or as they -appear to another. This single principle is the generating motive of -the whole theory of relativity.</p> - -<p>Now what we have hitherto regarded as the spatial and temporal -properties of physical occurrences are found to be in large part -dependent upon the observer; only a residue can be attributed to the -occurrences in themselves, and only this residue can be involved in the -formulation of any physical law which is to have an <i>à priori</i> -chance of being true. Einstein found ready to his hand an instrument of -pure mathematics, called the theory of tensors, which enabled him to -discover laws expressed in terms of the objective residue and agreeing -approximately with the old laws. Where Einstein’s laws differed from -the old ones, they have hitherto proved more in accord with observation.</p> - -<p>If there were no reality in the physical world, but only a number of -dreams dreamed by different people, we should not expect to find any -<span class="pagenum" id="Page_27">[Pg 27]</span> -laws connecting the dreams of one man with the dreams of another. It -is the close connection between the perceptions of one man and the -(roughly) simultaneous perceptions of another that makes us believe in -a common external origin of the different related perceptions. Physics -accounts both for the likenesses and for the differences between -different people’s perceptions of what we call the “same” occurrence. -But in order to do this it is first necessary for the physicist to -find out just what are the likenesses. They are not quite those -traditionally assumed, because neither space nor time separately can -be taken as strictly objective. What is objective is a kind of mixture -of the two called “space-time.” To explain this is not easy, but the -attempt must be made; it will be begun in the next chapter.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_28">[Pg 28]</span></p> -<h2 class="nobreak">CHAPTER III:<br /> THE VELOCITY OF LIGHT</h2> -</div> - -<p class="drop-cap"><span class="smcap">Most</span> of the curious -things in the theory of relativity are connected with the velocity -of light. If the reader is to grasp the reasons for such a serious -theoretical reconstruction, he must have some idea of the facts which -made the old system break down.</p> - -<p>The fact that light is transmitted with a definite velocity was first -established by astronomical observations. Jupiter’s moons are sometimes -eclipsed by Jupiter, and it is easy to calculate the times when this -ought to occur. It was found that when Jupiter was unusually near the -earth an eclipse of one of his moons would be observed a few minutes -earlier than was expected; and when Jupiter was unusually remote, a few -minutes later than was expected. It was found that these deviations -could all be accounted for by assuming that light has a certain -velocity, so that what we observe to be happening in Jupiter really -happened a little while ago—longer ago when Jupiter is distant than -<span class="pagenum" id="Page_29">[Pg 29]</span> -when it is near. Just the same velocity of light was found to account -for similar facts in regard to other parts of the solar system. It -was therefore accepted that light <i>in vacuo</i> always travels at a -certain constant rate, almost exactly 300,000 kilometers a second. (A -kilometer is about five-eighths of a mile.) When it became established -that light consists of waves, this velocity was that of propagation -of waves in the ether—at least they used to be in the ether, but now -the ether has grown somewhat shadowy, though the waves remain. This -same velocity is that of the waves used in wireless telegraphy (which -are like light waves, only longer) and in X-rays (which are like light -waves, only shorter). It is generally held nowadays to be the velocity -with which gravitation is propagated, though Eddington considers this -not yet certain. (It used to be thought that gravitation was propagated -instantaneously, but this view is now abandoned.)</p> - -<p>So far, all is plain sailing. But as it became possible to make more -accurate measurements, difficulties began to accumulate. The waves were -supposed to be in the ether, and therefore their velocity ought to -be relative to the ether. Now since the ether (if it exists) clearly -offers no resistance to the motions of the heavenly bodies, it would -<span class="pagenum" id="Page_30">[Pg 30]</span> -seem natural to suppose that it does not share their motion. If the -earth had to push a lot of ether before it, in the sort of way that -a steamer pushes water before it, one would expect a resistance on -the part of the ether analogous to that offered by the water to the -steamer. Therefore the general view was that the ether could pass -through bodies without difficulty, like air through a coarse sieve, -only more so. If this were the case, then the earth in its orbit must -have a velocity relative to the ether. If, at some point of its orbit, -it happened to be moving exactly with the ether, it must at other -points be moving through it all the faster. If you go for a circular -walk on a windy day, you must be walking against the wind part of the -way, whatever wind may be blowing; the principle in this case is the -same. It follows that, if you choose two days six months apart, when -the earth in its orbit is moving in exactly opposite directions, it -must be moving against an ether wind on at least one of these days.</p> - -<p>Now if there is an ether wind, it is clear that, relatively to an -observer on the earth, light signals will seem to travel faster with -<span class="pagenum" id="Page_31">[Pg 31]</span> -the wind than across it, and faster across it than against it. This -is what Michelson and Morley set themselves to test by their famous -experiment. They sent out light signals in two directions at right -angles; each was reflected from a mirror, and came back to the place -from which both had been sent out. Now anybody can verify, either by -trial or by a little arithmetic, that it takes longer to row a given -distance on a river upstream and then back again, than it takes to -row the same distance across the stream and back again. Therefore, if -there were an ether wind, one of the two light signals, which consist -of waves in the ether, ought to have traveled to the mirror and back at -a slower average rate than the other. Michelson and Morley tried the -experiment, they tried it in various positions, they tried it again -later. Their apparatus was quite accurate enough to have detected the -expected difference of speed or even a much smaller difference, if -it had existed, but not the smallest difference could be observed. -The result was a surprise to them as to everybody else; but careful -repetitions made doubt impossible. The experiment was first made as -long ago as 1881, and was repeated with more elaboration in 1887. But -<span class="pagenum" id="Page_32">[Pg 32]</span> -it was many years before it could be rightly interpreted.</p> - -<p>The supposition that the earth carries the neighboring ether with it -in its motion was found to be impossible, for a number of reasons. -Consequently a logical deadlock seemed to have arisen, from which at -first physicists sought to extricate themselves by very arbitrary -hypotheses. The most important of these was that of Fitzgerald, -developed by Lorentz, and known as the Fitzgerald contraction hypothesis.</p> - -<p>According to this hypothesis, when a body is in motion it becomes -shortened in the direction of motion by a certain proportion depending -upon its velocity. The amount of the contraction was to be just enough -to account for the negative result of the Michelson-Morley experiment. -The journey up stream and down again was to have been really a shorter -journey than the one across the stream, and was to have been just so -much shorter as would enable the slower light wave to traverse it in -the same time. Of course the shortening could never be detected by -measurement, because our measuring rods would share it. A foot rule -placed in the line of the earth’s motion would be shorter than the -<span class="pagenum" id="Page_33">[Pg 33]</span> -same foot rule placed at right angles to the earth’s motion. This -point of view resembles nothing so much as the White Knight’s “plan to -dye my whiskers green, and always use so large a fan that they could -not be seen.” The odd thing was that the plan worked well enough. Later -on, when Einstein propounded his special theory of relativity (1905), -it was found that the theory was in a certain sense correct, but only -in a certain sense. That is to say, the supposed contraction is not -a physical fact, but a result of certain conventions of measurement -which, when once the right point of view has been found, are seen to -be such as we are almost compelled to adopt. But I do not wish yet to -set forth Einstein’s solution of the puzzle. For the present, it is the -nature of the puzzle itself that I want to make clear.</p> - -<p>On the face of it, and apart from hypotheses <i>ad hoc</i>, the -Michelson-Morley experiment (in conjunction with others) showed that, -relatively to the earth, the velocity of light is the same in all -directions, and that this is equally true at all times of the year, -although the direction of the earth’s motion is always changing as -<span class="pagenum" id="Page_34">[Pg 34]</span> -it goes round the sun. Moreover, it appeared that this is not a -peculiarity of the earth, but is true of all bodies: if a light signal -is sent out from a body, that body will remain at the center of the -waves as they travel outwards, no matter how it may be moving—at -least, that will be the view of observers moving with the body. This -was the plain and natural meaning of the experiments, and Einstein -succeeded in inventing a theory which accepted it. But at first it was -thought logically impossible to accept this plain and natural meaning.</p> - -<p>A few illustrations will make it clear how very odd the facts are. When -a shell is fired, it moves faster than sound: the people at whom it is -fired first see the flash, then (if they are lucky) see the shell go -by, and last of all hear the report. It is clear that if you could put -a scientific observer on the shell, he would never hear the report, as -the shell would burst and kill him before the sound had overtaken him. -But if sound worked on the same principles as light, our observer would -hear everything just as if he were at rest. In that case, if a screen, -suitable for producing echoes, were attached to the shell and traveling -with it, say a hundred yards in front of it, our observer would hear -<span class="pagenum" id="Page_35">[Pg 35]</span> -the echo of the report from the screen after just the same interval -of time as if he and the shell were at rest. This, of course, is an -experiment which cannot be performed, but others which can be performed -will show the difference. We might find some place on a railway where -there is an echo from a place further along the railway—say a place -where the railway goes into a tunnel—and when a train is traveling -along the railway, let a man on the bank fire a gun. If the train is -traveling towards the echo, the passengers will hear the echo sooner -than the man on the bank; if it is traveling in the opposite direction, -they will hear it later. But these are not quite the circumstances -of the Michelson-Morley experiment. The mirrors in that experiment -correspond to the echo, and the mirrors are moving with the earth, so -that echo ought to move with the train. Let us suppose that the shot -is fired from the guard’s van, and the echo comes from a screen on the -engine. We will suppose the distance from the guard’s van to the engine -to be the distance that sound can travel in a second (about one-fifth -of a mile), and the speed of the train to be one-twelfth of the speed -<span class="pagenum" id="Page_36">[Pg 36]</span> -of sound (about sixty miles an hour). We now have an experiment which -can be performed by the people in the train. If the train were at rest, -the guard would hear the echo in two seconds; as it is, he will hear it -in 2 and ²/₁₄₃ seconds. From this difference, if he knows the velocity -of sound, he can calculate the velocity of the train, even if it is a -foggy night so that he cannot see the banks. But if sound behaved like -light, he would hear the echo in two seconds however fast the train -might be traveling.</p> - -<p>Various other illustrations will help to show how extraordinary—from -the point of view of tradition and common sense—are the facts about -the velocity of light. Every one knows that if you are on an escalator -you reach the top sooner if you walk up than if you stand still. But if -the escalator moved with the velocity of light (which it does not do -even in New York), you would reach the top at exactly the same moment -whether you walked up or stood still. Again: if you are walking along -a road at the rate of four miles an hour, and a motor-car passes you -going in the same direction at the rate of forty miles an hour, if you -and the motor-car both keep going the distance between you after an -<span class="pagenum" id="Page_37">[Pg 37]</span> -hour will be thirty-six miles. But if the motor-car met you, going in -the opposite direction, the distance after an hour would be forty-four -miles. Now if the motor-car were traveling with the velocity of light, -it would make no difference whether it met or passed you: in either -case, it would, after a second, be 186,000 miles from you. It would -also be 186,000 miles from any other motor-car which happened to be -passing or meeting you less rapidly at the previous second. This seems -impossible: how can the car be at the same distance from a number of -different points along the road?</p> - -<p>Let us take another illustration. When a fly touches the surface of -a stagnant pool, it causes ripples which move outwards in widening -circles. The center of the circle at any moment is the point of the -pool touched by the fly. If the fly moves about over the surface of -the pool, it does not remain at the center of the ripples. But if the -ripples were waves of light, and the fly were a skilled physicist, -it would find that it always remained at the center of the ripples, -however it might move. Meanwhile a skilled physicist sitting beside the -pool would judge, as in the case of ordinary ripples, that the center -<span class="pagenum" id="Page_38">[Pg 38]</span> -was not the fly, but the point of the pool touched by the fly. And if -another fly had touched the water at the same spot at the same moment, -it also would find that it remained at the center of the ripples, even -if it separated itself widely from the first fly. This is exactly -analogous to the Michelson-Morley experiment. The pool corresponds to -the ether; the fly corresponds to the earth; the contact of the fly and -the pool corresponds to the light signal which Messrs. Michelson and -Morley send out; and the ripples correspond to the light waves.</p> - -<p>Such a state of affairs seems, at first sight, quite impossible. It -is no wonder that, although the Michelson-Morley experiment was made -in 1881, it was not rightly interpreted until 1905. Let us see what, -exactly, we have been saying. Take the man walking along a road and -passed by a motor-car. Suppose there are a number of people at the same -point of the road, some walking, some in motor-cars; suppose they are -going at varying rates, some in one direction and some in another. I -say that if, at this moment, a light flash is sent out from the place -where they all are, the light waves will be 186,000 miles from each -<span class="pagenum" id="Page_39">[Pg 39]</span> -one of them after a second by his watch, although the travelers will -not any longer be all in the same place. At the end of a second by your -watch it will be 186,000 miles from you, and it will also be 186,000 -miles from a person who met you when it was sent out, but was moving in -the opposite direction, after a second by his watch—assuming both to -be perfect watches. How can this be?</p> - -<p>There is only one way of explaining such facts, and that is, to assume -that watches and clocks are affected by motion. I do not mean that -they are affected in ways that could be remedied by greater accuracy -in construction; I mean something much more fundamental. I mean that, -if you say an hour has elapsed between two events, and if you base -this assertion upon ideally careful measurements with ideally accurate -chronometers, another equally precise person, who has been moving -rapidly relatively to you, may judge that the time was more or less -than an hour. You cannot say that one is right and the other wrong, -any more than you could if one used a clock showing Greenwich time and -another a clock showing New York time. How this comes about, I shall -explain in the next chapter. -<span class="pagenum" id="Page_40">[Pg 40]</span></p> - -<p>There are other curious things about the velocity of light. One is, -that no material body can ever travel as fast as light, however great -may be the force to which it is exposed, and however long the force -may act. An illustration may help to make this clear. At exhibitions -one sometimes sees a series of moving platforms, going round and round -in a circle. The outside one goes at four miles an hour; the next -goes four miles an hour faster than the first; and so on. You can -step across from each to the next; until you find yourself going at a -tremendous pace. Now you might think that, if the first platform does -four miles an hour, and the second does four miles an hour relatively -to the first, then the second does eight miles an hour relatively to -the ground. This is an error; it does a little less, though so little -less that not even the most careful measurements could detect the -difference. I want to make quite clear what it is that I mean. I will -suppose that, in the morning, when the apparatus is just about to -start, three men with ideally accurate chronometers stand in a row, one -on the ground, one on the first platform, and one on the second. The -<span class="pagenum" id="Page_41">[Pg 41]</span> -first platform moves at the rate of four miles an hour with respect -to the ground. Four miles an hour is 352 feet in a minute. The man on -the ground, after a minute by his watch, notes the place on the ground -opposite the man on the first platform, who has been standing still -while the platform carried him along. The man on the ground measures -the distance on the ground from himself to the point opposite the -man on the first platform, and finds it is 352 feet. The man on the -first platform, after a minute by his watch, notes the point on his -platform opposite to the man on the second platform. The man on the -first platform measures the distance from himself to the point opposite -the man on the second platform; it is again 352 feet. Problem: how far -will the man on the ground judge that the man on the second platform -has traveled in a minute? That is to say, if the man on the ground, -after a minute by his watch, notes the place on the ground opposite -the man on the second platform, how far will this be from the man on -the ground? You would say, twice 352 feet, that is to say, 704 feet. -But in fact it will be a little less, though so little less as to -be inappreciable. The discrepancy is owing to the fact that the two -watches do not keep perfect time, in spite of the fact that each is -<span class="pagenum" id="Page_42">[Pg 42]</span> -accurate from its owner’s point of view. If you had a long series of -such moving platforms, each moving four miles an hour relatively to the -one before it, you would never reach a point where the last was moving -with the velocity of light relatively to the ground, not even if you -had millions of them. The discrepancy, which is very small for small -velocities, becomes greater as the velocity increases, and makes the -velocity of light an unattainable limit. How all this happens, is the -next topic with which we must deal.</p> - -<p class="blockquot fontsize_90"> <i>Note.</i> The negative result of -the Michelson-Morley experiment has recently been called in question -by Professor Dayton C. Miller, as a result of observations by what is -said to be an improved method. His claim is set forth by Professor -Silberstein in <i>Nature</i>, May 23, 1925, and discussed unfavorably -by Eddington in the issue of June 6. The matter is <i>sub judice</i>, -but it seems highly questionable whether the results bear out the -interpretation which is put upon them.</p> - -<hr class="chap x-ebookmaker-drop" /> -<div class="chapter"> -<p><span class="pagenum" id="Page_43">[Pg 43]</span></p> -<h2 class="nobreak">CHAPTER IV:<br /> CLOCKS AND FOOT RULES</h2> -</div> - -<p class="drop-cap"><span class="smcap">Until</span> the advent of -the special theory of relativity, no one had thought that there could -be any ambiguity in the statement that two events in different places -happened at the same time. It might be admitted that, if the places -were very far apart, there might be difficulty in finding out for -certain whether the events were simultaneous, but every one thought the -meaning of the question perfectly definite. It turned out, however, -that this was a mistake. Two events in distant places may appear -simultaneous to one observer who has taken all due precautions to -insure accuracy (and, in particular, has allowed for the velocity of -light), while another equally careful observer may judge that the first -event preceded the second, and still another may judge that the second -preceded the first. This would happen if the three observers were all -moving rapidly relatively to each other. It would not be the case that -one of them would be right and the other two wrong: they would all be -equally right. The time order of events is in part dependent upon the -observer; it is not always and altogether an intrinsic relation between -the events themselves. Einstein has shown, not only that this view -accounts for the phenomena, but also that it is the one which ought -to have resulted from careful reasoning based upon the old data. In -actual fact, however, no one noticed the logical basis of the theory -of relativity until the odd results of experiment had given a jog to -people’s reasoning powers.</p> - -<p><span class="pagenum" id="Page_44">[Pg 44]</span> -How should we naturally decide whether two events in different places -were simultaneous? One would naturally say: they are simultaneous -if they are seen simultaneously by a person who is exactly half-way -between them. (There is no difficulty about the simultaneity of two -events in the <i>same</i> place, such, for example, as seeing a light -and hearing a noise.) Suppose two flashes of lightning fall in two -different places, say Greenwich Observatory and Kew Observatory. -Suppose that St. Paul’s is half-way between them, and that the flashes -appear simultaneous to an observer on the dome of St. Paul’s. In that -<span class="pagenum" id="Page_45">[Pg 45]</span> -case, a man at Kew will see the Kew flash first, and a man at Greenwich -will see the Greenwich flash first, because of the time taken by -light to travel over the intervening distance. But all three, if they -are ideally accurate observers, will judge that the two flashes were -simultaneous, because they will make the necessary allowance for the -time of transmission of the light. (I am assuming a degree of accuracy -far beyond human powers.) Thus, so far as observers on the earth are -concerned, the definition of simultaneity will work well enough, so -long as we are dealing with events on the surface of the earth. It -gives results which are consistent with each other, and can be used for -terrestrial physics in all problems in which we can ignore the fact -that the earth moves.</p> - -<p>But our definition is no longer so satisfactory when we have two sets -of observers in rapid motion relatively to each other. Suppose we see -what would happen if we substitute sound for light, and defined two -occurrences as simultaneous when they are heard simultaneously by a -man half-way between them. This alters nothing in the principle, but -makes the matter easier owing to the much slower velocity of sound. -<span class="pagenum" id="Page_46">[Pg 46]</span> -Let us suppose that on a foggy night two men belonging to a gang of -brigands shoot the guard and engine driver of a train. The guard is at -the end of the train; the brigands are on the line, and shoot their -victims at close quarters. An old gentleman who is exactly in the -middle of the train hears the two shots simultaneously. You would say, -therefore, that the two shots were simultaneous. But a station master -who is exactly half-way between the two brigands hears the shot which -kills the guard first. An Australian millionaire uncle of the guard -and the engine driver (who are cousins) has left his whole fortune to -the guard, or, should he die first, to the engine driver. Vast sums -are involved in the question of which died first. The case goes to the -House of Lords, and the lawyers on both sides, having been educated at -Oxford, are agreed that either the old gentleman or the station master -must have been mistaken. In fact, both may perfectly well be right. The -train travels away from the shot at the guard, and towards the shot at -the engine driver; therefore the noise of the shot at the guard has -farther to go before reaching the old gentleman than the shot at the -engine driver has. Therefore if the old gentleman is right in saying -<span class="pagenum" id="Page_47">[Pg 47]</span> -that he heard the two reports simultaneously, the station master must -be right in saying that he heard the shot at the guard first.</p> - -<p>We, who live on the earth, would naturally, in such a case, prefer -the view of simultaneity obtained from a person at rest on the earth -to the view of a person traveling in a train. But in theoretical -physics no such parochial prejudices are permissible. A physicist on a -comet, if there were one, would have just as good a right to his view -of simultaneity as an earthly physicist has to his, but the results -would differ, in just the same sort of way as in our illustration of -the train and the shots. The train is not any more “really” in motion -than the earth; there is no “really” about it. You might imagine a -rabbit and a hippopotamus arguing as to whether man is “really” a large -animal; each would think his own point of view the natural one, and -the other a pure flight of fancy. There is just as little substance -in an argument as to whether the earth or the train is “really” in -motion. And, therefore, when we are defining simultaneity between -distant events, we have no right to pick and choose among different -<span class="pagenum" id="Page_48">[Pg 48]</span> -bodies to be used in defining the point half-way between the events. -All bodies have an equal right to be chosen. But if, for one body, the -two events are simultaneous according to the definition, there will -be other bodies for which the first precedes the second, and still -others for which the second precedes the first. We cannot therefore -say unambiguously that two events in distant places are simultaneous. -Such a statement only acquires a definite meaning in relation to a -definite observer. It belongs to the subjective part of our observation -of physical phenomena, not to the objective part which is to enter into -physical laws.</p> - -<p>This question of time in different places is perhaps, for the -imagination, the most difficult aspect of the theory of relativity. We -are accustomed to the idea that everything can be dated. Historians -make use of the fact that there was an eclipse of the sun visible in -China on August 29 in the year 776 B. C.<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">[1]</a> -No doubt astronomers could tell the exact hour and minute when the -<span class="pagenum" id="Page_49">[Pg 49]</span> -eclipse began to be total at any given spot in North China. And it -seems obvious that we can speak of the positions of the planets at -a given instant. The Newtonian theory enables us to calculate the -distance between the earth and (say) Jupiter at a given time by the -Greenwich clocks; this enables us to know how long light takes at that -time to travel from Jupiter to the earth—say half an hour; this -enables us to infer that half an hour ago Jupiter was where we see it -now. All this seems obvious. But in fact it only works in practice -because the relative velocities of the planets are very small compared -with the velocity of light. When we judge that an event on the earth -and an event on Jupiter have happened at the same time—for -example, that Jupiter eclipsed one of his moons when the Greenwich -clocks showed twelve midnight—a person moving rapidly relatively -to the earth would judge differently, assuming that both he and we had -made the proper allowance for the velocity of light. And naturally the -disagreement about simultaneity involves a disagreement about periods -of time. If we judged that two events on Jupiter were separated by -<span class="pagenum" id="Page_50">[Pg 50]</span> -twenty-four hours, another person might judge that they were separated -by a longer time, if he were moving rapidly relatively to Jupiter and -the earth.</p> - -<p>The universal cosmic time which used to be taken for granted is thus no -longer admissible. For each body, there is a definite time order for -the events in its neighborhood; this may be called the “proper” time -for that body. Our own experience is governed by the proper time for -our own body. As we all remain very nearly stationary on the earth, -the proper times of different human beings agree, and can be lumped -together as terrestrial time. But this is only the time appropriate to -<i>large</i> bodies on the earth. For Beta-particles in laboratories, -quite different times would be wanted; it is because we insist upon -using our own time that these particles seem to increase in mass -with rapid motion. From their own point of view, their mass remains -constant, and it is we who suddenly grow thin or corpulent. The history -of a physicist as observed by a Beta-particle would resemble Gulliver’s -travels.</p> - -<p>The question now arises: what really is measured by a clock? When we -speak of a clock in the theory of relativity, we do not mean only -clocks made by human hands: we mean anything which goes through some -<span class="pagenum" id="Page_51">[Pg 51]</span> -regular periodic performance. The earth is a clock, because it rotates -once in every twenty-three hours and fifty-six minutes. An atom is a -clock, because the electrons go round the nucleus a certain number of -times in a second; its properties as a clock are exhibited to us in -its spectrum, which is due to light waves of various frequencies. The -world is full of periodic occurrences, and fundamental mechanisms, -such as atoms, show an extraordinary similarity in different parts of -the universe. Any one of these periodic occurrences may be used for -measuring time; the only advantage of humanly manufactured clocks is -that they are specially easy to observe. One question is: If cosmic -time is abandoned, what is really measured by a clock in the wide sense -that we have just given to the term?</p> - -<p>Each clock gives a correct measure of its own “proper” time, which, -as we shall see presently, is an important physical quantity. But it -does not give an accurate measure of any physical quantity connected -with events on bodies that are moving rapidly in relation to it. It -gives one datum towards the discovery of a physical quantity connected -<span class="pagenum" id="Page_52">[Pg 52]</span> -with such events, but another datum is required, and this has to be -derived from measurement of distances in space. Distances in space, -like periods of time, are in general not objective physical facts, but -partly dependent upon the observer. How this comes about must now be -explained.</p> - -<p>First of all, we have to think of the distance between two events, not -between two bodies. This follows at once from what we have found as -regards time. If two bodies are moving relatively to each other—and -this is really always the case—the distance between them will be -continually changing, so that we can only speak of the distance -between them at a given time. If you are in a train traveling towards -Edinburgh, we can speak of your distance from Edinburgh at a given -time. But, as we said, different observers will judge differently as -to what is the “same” time for an event in the train and an event in -Edinburgh. This makes the measurement of distances relative, in just -the same way as the measurement of times has been found to be relative. -We commonly think that there are two separate kinds of interval between -two events, an interval in space and an interval in time: between your -<span class="pagenum" id="Page_53">[Pg 53]</span> -departure from London and your arrival in Edinburgh, there are 400 -miles and ten hours. We have already seen that another observer will -judge the time differently; it is even more obvious that he will judge -the distance differently. An observer in the sun will think the motion -of the train quite trivial, and will judge that you have traveled the -distance traveled by the earth in its orbit and its diurnal rotation. -On the other hand, a flea in the railway carriage will judge that you -have not moved at all in space, but have afforded him a period of -pleasure which he will measure by his “proper” time, not by Greenwich -Observatory. It cannot be said that you or the sun dweller or the -flea are mistaken: each is equally justified, and is only wrong if he -ascribes an objective validity to his subjective measures. The distance -in space between two events is, therefore, not in itself a physical -fact. But, as we shall see, there is a physical fact which can be -inferred from the distance in time together with the distance in space. -This is what is called the “interval” in space-time.</p> - -<p>Taking any two events in the universe, there are two different -possibilities as to the relation between them. It may be physically -<span class="pagenum" id="Page_54">[Pg 54]</span> -possible for a body to travel so as to be present at both events, or it -may not. This depends upon the fact that no body can travel as fast as -light. Suppose, for example, that it were possible to send out a flash -of light from the earth and have it reflected back from the moon. The -time between the sending of the flash and the return of the reflection -would be about two and a half seconds. No body could travel so fast -as to be present on the earth during any part of those two and a half -seconds and also present on the moon at the moment of the arrival of -the flash, because in order to do so the body would have to travel -faster than light. But theoretically a body could be present on the -earth at any time before or after those two and a half seconds and also -present on the moon at the time when the flash arrived. When it is -physically impossible for a body to travel so as to be present at both -events, we shall say that the interval<a id="FNanchor_2" href="#Footnote_2" class="fnanchor">[2]</a> -between the two events is “space-like”; when it is physically possible -for a body to be present at both events, we shall say that the -interval between the two events is “time-like.” When the interval is -<span class="pagenum" id="Page_55">[Pg 55]</span> -“space-like,” it is possible for a body to move in such a way that an -observer on the body will judge the two events to be simultaneous. -In that case, the “interval” between the two events is what such an -observer will judge to be the distance in space between them. When the -interval is “time-like,” a body can be present at both events; in that -case, the “interval” between the two events is what an observer on the -body will judge to be the time between them, that is to say, it is his -“proper” time between the two events. There is a limiting case between -the two, when the two events are parts of one light flash—or, as -we might say, when the one event is the seeing of the other. In that -case, the interval between the two events is zero.</p> - -<p>There are thus three cases. (1) It may be possible for a ray of light -to be present at both events; this happens whenever one of them is the -seeing of the other. In this case the interval between the two events -is zero. (2) It may happen that no body can travel from one event to -the other, because in order to do so it would have to travel faster -than light. In that case, it is always physically possible for a body -to travel in such a way that an observer on the body would judge the -two events to be simultaneous. The interval is what he would judge to -<span class="pagenum" id="Page_56">[Pg 56]</span> -be the distance in space between the two events. Such an interval is -called “space-like.” (3) It may be physically possible for a body to -travel so as to be present at both events; in that case, the interval -between them is what an observer on such a body will judge to be the -time between them. Such an interval is called “time-like.”</p> - -<p>The interval between two events is a physical fact about them, not -dependent upon the particular circumstances of the observer.</p> - -<p>There are two forms of the theory of relativity, the special and the -general. The former is in general only approximate, but is exact at -great distances from gravitating matter. When the special theory can be -applied, the interval can be calculated when we know the distance in -space and the distance in time between the two events, estimated by any -observer. If the distance in space is greater than the distance that -<span class="pagenum" id="Page_57">[Pg 57]</span> -light would have traveled in the time, the separation is space-like. -Then the <a href="#I056">following construction</a> gives the interval between -the two events: Draw a line <b>AB</b> as long as the distance that light would -travel in the time; round <b>A</b> describe a circle whose radius is -the distance in space between the two events; through <b>B</b> draw -<b>BC</b> perpendicular to <b>AB</b>, meeting the circle in <b>C</b>. -Then <b>BC</b> is the length of the interval between the two events.</p> - -<div id="I056" class="figcenter"> - <img src="images/i_056.jpg" alt="" width="500" height="487" /> -</div> - -<p>When the distance is time-like, use the same figure, but let <b>AC</b> -be now the distance that light would travel in the time, while -<b>AB</b> is the distance in space between the two events. The interval -between them is now the time that light would take to travel the -distance <b>BC</b>.</p> - -<p>Although <b>AB</b> and <b>AC</b> are different for different observers, -<b>BC</b> is the same length for all observers, subject to corrections -made by the general theory. It represents the one interval in -“space-time” which replaces the two intervals in space and time of the -older physics. So far, this notion of interval may appear somewhat -mysterious, but as we proceed it will grow less so, and its reason in -the nature of things will gradually emerge.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_58">[Pg 58]</span></p> -<h2 class="nobreak">CHAPTER V:<br /> SPACE-TIME</h2> -</div> - -<p class="drop-cap"><span class="smcap">Everybody</span> who has ever -heard of relativity knows the phrase “space-time,” and knows that the -correct thing is to use this phrase when formerly we should have said -“space <i>and</i> time.” But very few people who are not mathematicians -have any clear idea of what is meant by this change of phraseology. -Before dealing further with the special theory of relativity, I want -to try to convey to the reader what is involved in the new phrase -“space-time,” because that is, from a philosophical and imaginative -point of view, perhaps the most important of all the novelties that -Einstein has introduced.</p> - -<p>Suppose you wish to say where and when some event has occurred—say -an explosion on an airship—you will have to mention four quantities, -say the latitude and longitude, the height above the ground, and the -time. According to the traditional view, the first three of these -give the position in space, while the fourth gives the position in -<span class="pagenum" id="Page_59">[Pg 59]</span> -time. The three quantities that give the position in space may be -assigned in all sorts of ways. You might, for instance, take the -plane of the equator, the plane of the meridian of Greenwich, and the -plane of the ninetieth meridian, and say how far the airship was from -each of these planes; these three distances would be what are called -“Cartesian co-ordinates,” after Descartes. You might take any other -three planes all at right angles to each other, and you would still -have Cartesian co-ordinates. Or you might take the distance from London -to a point vertically below the airship, the direction of this distance -(northeast, west-southwest, or whatever it might be), and the height of -the airship above the ground. There are an infinite number of such ways -of fixing the position in space, all equally legitimate; the choice -between them is merely one of convenience.</p> - -<p>When people said that space had three dimensions, they meant just this: -that three quantities were necessary in order to specify the position -of a point in space, but that the method of assigning these quantities -was wholly arbitrary.</p> - -<p>With regard to time, the matter was thought to be quite different. The -only arbitrary elements in the reckoning of time were the unit, and -<span class="pagenum" id="Page_60">[Pg 60]</span> -the point of time from which the reckoning started. One could reckon -in Greenwich time, or in Paris time, or in New York time; that made a -difference as to the point of departure. One could reckon in seconds, -minutes, hours, days, or years; that was a difference of unit. Both -these were obvious and trivial matters. There was nothing corresponding -to the liberty of choice as to the method of fixing position in space. -And, in particular, it was thought that the method of fixing position -in space and the method of fixing position in time could be made wholly -independent of each other. For these reasons, people regarded time and -space as quite distinct.</p> - -<p>The theory of relativity has changed this. There are now a number of -different ways of fixing position in time, which do not differ merely -as to the unit and the starting point. Indeed, as we have seen, if one -event is simultaneous with another in one reckoning, it will precede -it in another, and follow it in a third. Moreover, the space and time -reckonings are no longer independent of each other. If you alter the -way of reckoning position in space, you may also alter the time -<span class="pagenum" id="Page_61">[Pg 61]</span> -interval between two events. If you alter the way of reckoning time, -you may also alter the distance in space between two events. Thus space -and time are no longer independent, any more than the three dimensions -of space are. We still need four quantities to determine the position -of an event, but we cannot, as before, divide off one of the four as -quite independent of the other three.</p> - -<p>It is not quite true to say that there is no longer any distinction -between time and space. As we have seen, there are time-like intervals -and space-like intervals. But the distinction is of a different sort -from that which was formerly assumed. There is no longer a universal -time which can be applied without ambiguity to any part of the -universe; there are only the various “proper” times of the various -bodies in the universe, which agree approximately for two bodies which -are not in rapid relative motion, but never agree exactly except for -two bodies which are at rest relatively to each other.</p> - -<p>The picture of the world which is required for this new state of -affairs is as follows: Suppose an event <b>E</b> occurs to me, and -simultaneously a flash of light goes out from me in all directions. -<span class="pagenum" id="Page_62">[Pg 62]</span> -Anything that happens to any body after the light from the flash has -reached it is definitely after the event <b>E</b> in any system of -reckoning time. Any event anywhere which I could have seen before the -event <b>E</b> occurred to me is definitely before the event <b>E</b> -in any system of reckoning time. But any event which happened in the -intervening time is not definitely either before or after the event -<b>E</b>. To make the matter definite: suppose I could observe a person -in Sirius, and he could observe me. Anything which he does, and which -I see before the event <b>E</b> occurs to me, is definitely before -<b>E</b>; anything he does after he has seen the event <b>E</b> is -definitely after <b>E</b>. But anything that he does before he sees -the event <b>E</b>, but so that I see it after the event <b>E</b> has -happened, is not definitely before or after <b>E</b>. Since light takes -many years to travel from Sirius to the earth, this gives a period of -twice as many years in Sirius which may be called “contemporary” with -<b>E</b>, since these years are not definitely before or after <b>E</b>.</p> - -<p>Dr. A. A. Robb, in his <i>Theory of Time and Space</i>, suggests a -point of view which may or may not be philosophically fundamental, -but is at any rate a help in understanding the state of affairs we -<span class="pagenum" id="Page_63">[Pg 63]</span> -have been describing. He maintains that one event can only be said -to be definitely <i>before</i> another if it can influence that other -in some way. Now influences spread from a center at varying rates. -Newspapers exercise an influence emanating from London at an average -rate of about twenty miles an hour—rather more for long distances. -Anything a man does because of what he reads in the newspaper is -clearly subsequent to the printing of the newspaper. Sounds travel -much faster: it would be possible to arrange a series of loud speakers -along the main roads, and have newspapers shouted from each to the -next. But telegraphing is quicker, and wireless telegraphy travels with -the velocity of light, so that nothing quicker can ever be hoped for. -Now what a man does in consequence of receiving a wireless message -he does <i>after</i> the message was sent; the meaning here is quite -independent of conventions as to the measurement of time. But anything -that he does while the message is on its way cannot be influenced by -the sending of the message, and cannot influence the sender until some -little time after he sent the message. That is to say, if two bodies -are widely separated, neither can influence the other except after -a certain lapse of time; what happens before that time has elapsed -<span class="pagenum" id="Page_64">[Pg 64]</span> -cannot affect the distant body. Suppose, for instance, that some -notable event happens on the sun: there is a period of sixteen minutes -on the earth during which no event on the earth can have influenced -or been influenced by the said notable event on the sun. This gives a -substantial ground for regarding that period of sixteen minutes on the -earth as neither before nor after the event on the sun.</p> - -<p>The paradoxes of the special theory of relativity are only paradoxes -because we are unaccustomed to the point of view, and in the habit -of taking things for granted when we have no right to do so. This is -especially true as regards the measurement of lengths. In daily life, -our way of measuring lengths is to apply a foot rule or some other -measure. At the moment when the foot rule is applied, it is at rest -relatively to the body which is being measured. Consequently the length -that we arrive at by measurement is the “proper” length, that is to -say, the length as estimated by an observer who shares the motion of -the body. We never, in ordinary life, have to tackle the problem of -<span class="pagenum" id="Page_65">[Pg 65]</span> -measuring a body which is in continual motion. And even if we did, the -velocities of visible bodies on the earth are so small relatively to -the earth that the anomalies dealt with by the theory of relativity -would not appear. But in astronomy, or in the investigation of atomic -structure, we are faced with problems which cannot be tackled in this -way. Not being Joshua, we cannot make the sun stand still while we -measure it; if we are to estimate its size, we must do so while it is -in motion relatively to us. And similarly if you want to estimate the -size of an electron, you have to do so while it is in rapid motion, -because it never stands still for a moment. This is the sort of problem -with which the theory of relativity is concerned. Measurement with a -foot rule, when it is possible, gives always the same result, because -it gives the “proper” length of a body. But when this method is not -possible, we find that curious things happen, particularly if the -body to be measured is moving very fast relatively to the observer. A -figure like the one at the end of the <a href="#I056">previous chapter</a> -will help us to understand the state of affairs. -<span class="pagenum" id="Page_66">[Pg 66]</span></p> - -<div id="I066" class="figcenter"> - <img src="images/i_066.jpg" alt="" width="500" height="502" /> -</div> - -<p>Let us suppose that the body on which we wish to measure lengths is -moving relatively to ourselves, and that in one second it moves the -distance <b>OM</b>. Let us <a href="#I066">draw a circle</a> round <b>O</b> whose -radius is the distance that light travels in a second. Through <b>M</b> draw -<b>MP</b> perpendicular to <b>OM</b>, meeting the circle in <b>P</b>. -Thus <b>OP</b> is the distance that light travels in a second. The -ratio of <b>OP</b> to <b>OM</b> is the ratio of the velocity of light -to the velocity of the body. The ratio of <b>OP</b> to <b>MP</b> is -the ratio in which apparent lengths are altered by the motion. That is -to say, if the observer judges that two points in the line of motion -on the moving body are at a distance from each other represented by -<b>MP</b>, a person moving with the body would judge that they were at -a distance represented (on the same scale) by <b>OP</b>. Distances on -the moving body at right angles to the line of motion are not affected -by the motion. The whole thing is reciprocal; that is to say, if an -observer moving with the body were to measure lengths on the previous -observer’s body, they would be altered in just the same proportion. -When two bodies are moving relatively to each other, lengths on either -<span class="pagenum" id="Page_67">[Pg 67]</span> -appear shorter to the other than to themselves. This is the Fitzgerald -contraction, which was first invented to account for the result of the -Michelson-Morley experiment. But it now emerges naturally from the fact -that the two observers do not make the same judgment of simultaneity.</p> - -<p>The way in which simultaneity comes in is this: We say that two points -on a body are a foot apart when we can <i>simultaneously</i> apply one -end of a foot rule to the one and the other end to the other. If, now, -two people disagree about simultaneity, and the body is in motion, they -will obviously get different results from their measurements. Thus the -trouble about time is at the bottom of the trouble about distance.</p> - -<p>The ratio of <b>OP</b> to <b>MP</b> is the essential thing in all -these matters. Times and lengths and masses are all altered in this -proportion when the body concerned is in motion relatively to the -observer. It will be seen that, if <b>OM</b> is very much smaller than -<b>OP</b>, that is to say, if the body is moving very much more slowly -than light, <b>MP</b> and <b>OP</b> are very nearly equal, so that the -alterations produced by the motion are very small. But if <b>OM</b> is -<span class="pagenum" id="Page_68">[Pg 68]</span> -nearly as large as <b>OP</b>, that is to say, if the body is moving -nearly as fast as light, <b>MP</b> becomes very small compared to -<b>OP</b>, and the effects become very great. The apparent increase -of mass in swiftly moving particles had been observed, and the right -formula had been found, before Einstein invented his special theory -of relativity. In fact, Lorentz had arrived at the formulæ called the -“Lorentz transformation,” which embody the whole mathematical essence -of the special theory of relativity. But it was Einstein who showed -that the whole thing was what we ought to have expected, and not a set -of makeshift devices to account for surprising experimental results. -Nevertheless, it must not be forgotten that experimental results were -the original motive of the whole theory, and have remained the ground -for undertaking the tremendous logical reconstruction involved in -Einstein’s theories.</p> - -<p>We may now recapitulate the reasons which have made it necessary to -substitute “space-time” for space and time. The old separation of -space and time rested upon the belief that there was no ambiguity in -saying that two events in distant places happened at the same time; -<span class="pagenum" id="Page_69">[Pg 69]</span> -consequently it was thought that we could describe the topography of -the universe at a given instant in purely spatial terms. But now that -simultaneity has become relative to a particular observer, this is no -longer possible. What is, for one observer, a description of the state -of the world at a given instant, is, for another observer, a series -of events at various different times, whose relations are not merely -spatial but also temporal. For the same reason, we are concerned with -<i>events</i>, rather than with <i>bodies</i>. In the old theory, it -was possible to consider a number of bodies all at the same instant, -and since the time was the same for all of them it could be ignored. -But now we cannot do that if we are to obtain an objective account of -physical occurrences. We must mention the date at which a body is to be -considered, and thus we arrive at an “event,” that is to say, something -which happens at a given time. When we know the time and place of an -event in one observer’s system of reckoning, we can calculate its time -and place according to another observer. But we must know the time as -well as the place, because we can no longer ask what is its place for -the new observer at the “same” time as for the old observer. There is -<span class="pagenum" id="Page_70">[Pg 70]</span> -no such thing as the “same” time for different observers, unless they -are at rest relatively to each other. We need four measurements to -fix a position, and four measurements fix the position of an event in -space-time, not merely of a body in space. Three measurements are not -enough to fix any position. That is the essence of what is meant by the -substitution of space-time for space and time.</p> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_71">[Pg 71]</span></p> -<h2 class="nobreak">CHAPTER VI:<br /> THE SPECIAL THEORY<br /> OF RELATIVITY</h2> -</div> - -<p class="drop-cap"><span class="smcap">The</span> special theory -of relativity arose as a way of accounting for the facts of -electromagnetism. We have here a somewhat curious history. In the -eighteenth and early nineteenth centuries the theory of electricity -was wholly dominated by the Newtonian analogy. Two electric charges -attract each other if they are of different kinds, one positive and -one negative, but repel each other if they are of the same kind; in -each case, the force varies as the inverse square of the distance, -as in the case of gravitation. This force was conceived as an action -at a distance, until Faraday, by a number of remarkable experiments, -demonstrated the effect of the intervening medium. Faraday was no -mathematician; Clerk Maxwell first gave a mathematical form to the -results suggested by Faraday’s experiments. Moreover Clerk Maxwell gave -grounds for thinking that light is an electromagnetic phenomenon, -<span class="pagenum" id="Page_72">[Pg 72]</span> -consisting of electromagnetic waves. The medium for the transmission of -electromagnetic effects could therefore be taken to be the ether, which -had long been assumed for the transmission of light. The correctness -of Maxwell’s theory of light was proved by the experiments of Hertz in -manufacturing electromagnetic waves; these experiments afforded the -basis for wireless telegraphy. So far, we have a record of triumphant -progress, in which theory and experiment alternately assume the leading -role. At the time of Hertz’s experiments, the ether seemed securely -established, and in just as strong a position as any other scientific -hypothesis not capable of direct verification. But a new set of facts -began to be discovered, and gradually the whole picture was changed.</p> - -<p>The movement which culminated with Hertz was a movement for making -everything continuous. The ether was continuous, the waves in it were -continuous, and it was hoped that matter would be found to consist -of some continuous structure in the ether. Then came the discovery -of the electron, a small finite unit of negative electricity, and -the proton, a small finite unit of positive electricity. The most -modern view is that electricity is never found except in the form of -<span class="pagenum" id="Page_73">[Pg 73]</span> -electrons and protons; all electrons have the same amount of negative -electricity, and all protons have an exactly equal and opposite amount -of positive electricity. It appeared that an electric current, which -had been thought of as a continuous phenomenon, consists of electrons -traveling one way and positive ions traveling the other way; it is no -more strictly continuous than the stream of people going up and down -an escalator. Then came the discovery of quanta, which seems to show -a fundamental discontinuity in all such natural processes as can be -measured with sufficient precision. Thus physics has had to digest new -facts and face new problems.</p> - -<p>But the problems raised by the electron and the quantum are not those -that the theory of relativity can solve, at any rate at present; as -yet, it throws no light upon the discontinuities which exist in nature. -The problems solved by the special theory of relativity are typified by -the Michelson-Morley experiment. Assuming the correctness of Maxwell’s -theory of electromagnetism, there should have been certain discoverable -effects of motion through the ether; in fact, there were none. Then -<span class="pagenum" id="Page_74">[Pg 74]</span> -there was the observed fact that a body in very rapid motion appears -to increase its mass; the increase is in the ratio of <b>OP</b> to -<b>MP</b> in the <a href="#I066">figure in the preceding chapter</a>. -Facts of this sort gradually accumulated, until it became imperative to -find some theory which would account for them all.</p> - -<p>Maxwell’s theory reduced itself to certain equations, known as -“Maxwell’s equations.” Through all the revolutions which physics has -undergone in the last fifty years, these equations have remained -standing; indeed they have continually grown in importance as well as -in certainty—for Maxwell’s arguments in their favor were so shaky -that the correctness of his results must almost be ascribed to intuition. -Now these equations were, of course, obtained from experiments in -terrestrial laboratories, but there was a tacit assumption that the -motion of the earth through the ether could be ignored. In certain -cases, such as the Michelson-Morley experiment, this ought not to have -been possible without measurable error; but it turned out to be always -possible. Physicists were faced with the odd difficulty that Maxwell’s -equations were more accurate than they should be. A very similar -difficulty was explained by Galileo at the very beginning of modern -<span class="pagenum" id="Page_75">[Pg 75]</span> -physics. Most people think that if you let a weight drop it will fall -vertically. But if you try the experiment in the cabin of a moving -ship, the weight falls, in relation to the cabin, just as if the ship -were at rest; for instance, if it starts from the middle of the ceiling -it will drop onto the middle of the floor. That is to say, from the -point of view of an observer on the shore it does not fall vertically, -since it shares the motion of the ship. So long as the ship’s motion -is steady, everything goes on inside the ship as if the ship were not -moving. Galileo explained how this happens, to the great indignation -of the disciples of Aristotle. In orthodox physics, which is derived -from Galileo, a uniform motion in a straight line has no discoverable -effects. This was, in its day, as astonishing a form of relativity -as that of Einstein is to us. Einstein, in the special theory of -relativity, set to work to show how electromagnetic phenomena could be -unaffected by uniform motion through the ether if there be an ether. -This was a more difficult problem, which could not be solved by merely -adhering to the principles of Galileo.</p> - -<p>The really difficult effort required for solving this problem was in -<span class="pagenum" id="Page_76">[Pg 76]</span> -regard to time. It was necessary to introduce the notion of “proper” -time which we have already considered, and to abandon the old belief in -one universal time. The quantitative laws of electromagnetic phenomena -are expressed in Maxwell’s equations, and these equations are found -to be true for any observer, however he may be moving.<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">[3]</a> -It is a straight-forward mathematical problem to find out what differences -there must be between the measures applied by one observer and the -measures applied by another, if, in spite of their relative motion, -they are to find the same equations verified. The answer is contained -in the “Lorentz transformation,” found as a formula by Lorentz, but -interpreted and made intelligible by Einstein.</p> - -<p>The Lorentz transformation tells us what estimate of distances and -periods of time will be made by an observer whose relative motion is -known, when we are given those of another observer. We may suppose that -you are in a train on a railway which travels due east. You have been -traveling for a time which, by the clocks at the station from which -you started, is <i>t</i>. At a distance <i>x</i> from your starting -point, as measured by the people on the line, an event occurs at this -<span class="pagenum" id="Page_77">[Pg 77]</span> -moment—say the line is struck by lightning. You have been traveling -all the time with a uniform velocity <i>v</i>. The question is: How far -from you will you judge that this event has taken place, and how long -after you started will it be by your watch, assuming that your watch is -correct from the point of view of an observer on the train?</p> - -<p>Our solution of this problem has to satisfy certain conditions. It has -to bring out the result that the velocity of light is the same for all -observers, however they may be moving. And it has to make physical -phenomena—in particular, those of electromagnetism—obey the same -laws for different observers, however they may find their measures of -distances and times affected by their motion. And it has to make all -such effects on measurement reciprocal. That is to say, if you are in -a train and your motion affects your estimate of distances outside the -train, there must be an exactly similar change in the estimate which -people outside the train make of distances inside it. These conditions -are sufficient to determine the solution of the problem, but the -<span class="pagenum" id="Page_78">[Pg 78]</span> -method of obtaining the solution cannot be explained without more -mathematics than is possible in the present work.</p> - -<p>Before dealing with the matter in general terms, let us take an -example. Let us suppose that you are in a train on a long straight -railway, and that you are traveling at three-fifths of the velocity -of light. Suppose that you measure the length of your train, and find -that it is a hundred yards. Suppose that the people who catch a glimpse -of you as you pass succeed, by skilful scientific methods, in taking -observations which enable them to calculate the length of your train. -If they do their work correctly, they will find that it is eighty -yards long. Everything in the train will seem to them shorter in the -direction of the train than it does to you. Dinner plates, which you -see as ordinary circular plates, will look to the outsider as if they -were oval: they will seem only four-fifths as broad in the direction -in which the train is moving as in the direction of the breadth of the -train. And all this is reciprocal. Suppose you see out of the window a -man carrying a fishing rod which, by his measurement, is fifteen feet -long. If he is holding it upright, you will see it as he does; so you -<span class="pagenum" id="Page_79">[Pg 79]</span> -will if he is holding it horizontally at right angles to the railway. -But if he is pointing it along the railway, it will seem to you to -be only twelve feet long. All lengths in the direction of motion are -diminished by twenty per cent, both for those who look into the train -from outside and for those who look out of the train from inside.</p> - -<p>But the effects in regard to time are even more strange. This matter -has been explained with almost ideal lucidity by Eddington in <i>Space, -Time and Gravitation</i>. He supposes an aviator traveling, relatively -to the earth, at a speed of 161,000 miles a second, and he says:</p> - -<p>“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 <i>see</i> 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 -<span class="pagenum" id="Page_80">[Pg 80]</span> -light. But here again reciprocity comes in, because in the aviator’s -opinion it is we who are traveling 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.”</p> - -<p>What a situation for envy! Each man thinks that the other’s cigar -lasts twice as long as his own. It may, however, be some consolation to -reflect that the other man’s visits to the dentist also last twice as long.</p> - -<p>This question of time is rather intricate, owing to the fact that -events which one man judges to be simultaneous another considers to be -separated by a lapse of time. In order to try to make clear how time -is affected, I shall revert to our railway train traveling due east at -a rate three-fifths of that of light. For the sake of illustration, I -assume that the earth is large and flat, instead of small and round.</p> - -<p>If we take events which happen at a fixed point on the earth, and ask -ourselves how long after the beginning of the journey they will seem to -be to the traveler, the answer is that there will be that retardation -that Eddington speaks of, which means in this case that what seems an -<span class="pagenum" id="Page_81">[Pg 81]</span> -hour in the life of the stationary person is judged to be an hour and a -quarter by the man who observes him from the train. Reciprocally, what -seems an hour in the life of the person in the train is judged by the -man observing him from outside to be an hour and a quarter. Each makes -periods of time observed in the life of the other a quarter as long -again as they are to the person who lives through them. The proportion -is the same in regard to times as in regard to lengths.</p> - -<p>But when, instead of comparing events at the same place on the earth, -we compare events at widely separated places, the results are still -more odd. Let us now take all the events along the railway which, from -the point of view of a person who is stationary on the earth, happen -at a given instant, say the instant when the observer in the train -passes the stationary person. Of these events, those which occur at -points towards which the train is moving will seem to the traveler to -have already happened, while those which occur at points behind the -train will, for him, be still in the future. When I say that events -in the forward direction will seem to have already happened, I am -saying something not strictly accurate, because he will not yet have -<span class="pagenum" id="Page_82">[Pg 82]</span> -seen them; but when he does see them, he will, after allowing for the -velocity of light, come to the conclusion that they must have happened -before the moment in question. An event which happens in the forward -direction along the railway, and which the stationary observer judges -to be now (or rather, will judge to have been now when he comes to know -of it), if it occurs at a distance along the line which light could -travel in a second, will be judged by the traveler to have occurred -three-quarters of a second ago. If it occurs at a distance from the two -observers which the man on the earth judges that light could travel -in a year, the traveler will judge (when he comes to know of it) that -it occurred nine months earlier than the moment when he passed the -earth dweller. And generally, he will ante-date events in the forward -direction along the railway by three-quarters of the time that it would -take light to travel from them to the man on the earth whom he is just -passing, and who holds that these events are happening now—or rather, -will hold that they happened now when the light from them reaches him. -Events happening on the railway behind the train will be post-dated by -an exactly equal amount. -<span class="pagenum" id="Page_83">[Pg 83]</span></p> - -<p>We have thus a two-fold correction to make in the date of an event when -we pass from the terrestrial observer to the traveler. We must first -take five-fourths of the time as estimated by the earth dweller, and -then subtract three-fourths of the time that it would take light to -travel from the event in question to the earth dweller.</p> - -<p>Take some event in a distant part of the universe, which becomes -visible to the earth dweller and the traveler just as they pass each -other. The earth dweller, if he knows how far off the event occurred, -can judge how long ago it occurred, since he knows the speed of light. -If the event occurred in the direction towards which the traveler is -moving, the traveler will infer that it happened twice as long ago as -the earth dweller thinks. But if it occurred in the direction from -which he has come, he will argue that it happened only half as long -ago as the earth dweller thinks. If the traveler moves at a different -speed, these proportions will be different.</p> - -<p>Suppose now that (as sometimes occurs) two new stars have suddenly -flared up, and have just become visible to the traveler and to the -earth dweller whom he is passing. Let one of them be in the direction -towards which the train is traveling, the other in the direction from -<span class="pagenum" id="Page_84">[Pg 84]</span> -which it has come. Suppose that the earth dweller is able, in some way, -to estimate the distance of the two stars, and to infer that light -takes fifty years to reach him from the one in the direction towards -which the traveler is moving, and one hundred years to reach him from -the other. He will then argue that the explosion which produced the -new star in the forward direction occurred fifty years ago, while the -explosion which produced the other new star occurred a hundred years -ago. The traveler will exactly reverse these figures: he will infer -that the forward explosion occurred a hundred years ago, and the -backward one fifty years ago. I assume that both argue correctly on -correct physical data. In fact, both are right, unless they imagine -that the other must be wrong. It should be noted that both will have -the same estimate of the velocity of light, because their estimates -of the distances of the two new stars will vary in exactly the same -proportion as their estimates of the times since the explosions. -Indeed, one of the main motives of this whole theory is to secure that -the velocity of light shall be the same for all observers, however they -may be moving. This fact, established by experiment, was incompatible -<span class="pagenum" id="Page_85">[Pg 85]</span> -with the old theories, and made it absolutely necessary to admit -something startling. The theory of relativity is just as little -startling as is compatible with the facts. Indeed, after a time, it -ceases to seem startling at all.</p> - -<p>There is another feature of very great importance in the theory we -have been considering, and that is that, although distances and times -vary for different observers, we can derive from them the quantity -called “interval,” which is the same for all observers. The “interval,” -in the special theory of relativity, is obtained as follows: Take -the square of the distance between two events, and the square of the -distance traveled by light in the time between the two events; subtract -the lesser of these from the greater, and the result is defined as -the square of the interval between the events. The interval is the -same for all observers, and represents a genuine physical relation -between the two events, which the time and the distance do not. We -have already given a <a href="#I088">geometrical construction</a> for the interval -at the end of <a href="#Page_43">Chapter IV</a>; this gives the same result as the -above rule. The interval is “time-like” when the time between the events is longer than -<span class="pagenum" id="Page_86">[Pg 86]</span> -light would take to travel from the place of the one to the place -of the other; in the contrary case it is “space-like.” When the time -between the two events is exactly equal to the time taken by light to -travel from one to the other, the interval is zero; the two events are -then situated on parts of one light ray, unless no light happens to be -passing that way.</p> - -<p>When we come to the general theory of relativity, we shall have to -generalize the notion of interval. The more deeply we penetrate into -the structure of the world, the more important this concept becomes; -we are tempted to say that it is the reality of which distances and -periods of time are confused representations. The theory of relativity -has altered our view of the fundamental structure of the world; that is -the source both of its difficulty and of its importance.</p> - -<p>The remainder of this chapter may be omitted by readers who have not -even the most elementary acquaintance with geometry or algebra. But -for the benefit of those whose education has not been <i>entirely</i> -neglected, I will add a few explanations of the general formula of -which I have hitherto given only particular examples. The general -formula in question is the “Lorentz transformation,” which tells, when -<span class="pagenum" id="Page_87">[Pg 87]</span> -one body is moving in a given manner relatively to another, how to -infer the measures of lengths and times appropriate to the one body -from those appropriate to the other. Before giving the algebraical -formulæ, I will give a geometrical construction. As before, we will -suppose that there are two observers, whom we will call <b>O</b> and -<b>O</b>′, one of whom is stationary on the earth while the other is -traveling at a uniform speed along a straight railway. At the beginning -of the time considered, the two observers were at the same point of the -railway, but now they are separated by a certain distance. A flash of -lightning strikes a point <b>X</b> on the railway, and <b>O</b> judges -that at the moment when the flash takes place the observer in the train -has reached the point <b>O</b>′. The problem is: how far will <b>O</b>′ -judge that he is from the flash, and how long after the beginning of -the journey (when he was at <b>O</b>) will he judge that the flash took -place? We are supposed to know <b>O</b>′s estimates, and we want to -calculate those of <b>O</b>′. -<span class="pagenum" id="Page_88">[Pg 88]</span></p> - -<div id="I088" class="figcenter"> - <img src="images/i_088.jpg" alt="" width="500" height="480" /> -</div> - -<p>In the time that, according to <b>O</b>, has elapsed since the -beginning of the journey, let <b>OC</b> be the distance that light -would have traveled along the railway. Describe a circle about -<b>O</b>, with <b>OC</b> as radius, and through <b>O′</b> draw a -perpendicular to the railway, meeting the circle in <b>D</b>. On -<b>OD</b> take a point <b>Y</b> such that <b>OY</b> is equal to -<b>OX</b> (<b>X</b> is the point of the railway where the lightning -strikes). Draw <b>YM</b> perpendicular to the railway, and <b>OS</b> -perpendicular to <b>OD</b>. Let <b>YM</b> and <b>OS</b> meet in -<b>S</b>. Also let <b>DO′</b> produced and <b>OS</b> produced meet -in <b>R</b>. Through <b>X</b> and <b>C</b> draw perpendiculars to -<span class="pagenum" id="Page_89">[Pg 89]</span> -the railway meeting <b>OS</b> produced in <b>Q</b> and <b>Z</b> -respectively. Then <b>RQ</b> (as measured by <b>O</b>) is the distance -at which <b>O′</b> will believe himself to be from the flash, not -<b>O′X</b> as it would be according to the old view. And whereas -<b>O</b> thinks that, in the time from the beginning of the journey -to the flash, light would travel a distance <b>OC</b>, <b>O′</b> -thinks that the time elapsed is that required for light to travel the -distance <b>SZ</b> (as measured by <b>O</b>). The interval as measured -by <b>O</b> is got by subtracting the square on <b>OX</b> from the -square on <b>OC</b>; the interval as measured by <b>O′</b> is got by -subtracting the square on <b>RQ</b> from the square on <b>SZ</b>. A -little very elementary geometry shows that these are equal.</p> - -<p>The algebraical formulæ embodied in the <a href="#I088">above construction</a> -are as follows: From the point of view of <b>O</b>, let an event occur at a -distance <i>x</i> along the railway, and at a time <i>t</i> after the -beginning of the journey (when <b>O′</b> was at <b>O</b>). From the -point of view of <b>O′</b>, let the same event occur at a distance -<i>x′</i> along the railway, and at a time <i>t′</i> after the -beginning of the journey. Let <i>c</i> be the velocity of light, and -<i>v</i> the velocity of <b>O′</b> relative to <b>O</b>. Put -<span class="pagenum" id="Page_90">[Pg 90]</span></p> - -<table class="fontsize_150" border="0" cellspacing="0" summary=" " cellpadding="0" > - <tbody><tr> - <td class="tdr"> </td> - <td class="tdl_ws1"> </td> - <td class="tdc" colspan="2"><i>c</i></td> - </tr><tr> - <td class="tdr">β</td> - <td class="tdl_ws1">=</td> - <td class="tdl_ws1" colspan="2">————</td> - </tr><tr> - <td class="tdr"> </td> - <td class="tdl_ws1"> </td> - <td class="tdl_ws1">√</td> - <td class="tdl">(<i>c</i>² - <i>v</i>²)</td> - </tr> - </tbody> -</table> - -<p>Then</p> - -<p class="center fontsize_150 space-below2"><i>x′</i> = β(<i>x</i> - <i>vt</i>)</p> - -<table class="fontsize_150" border="0" cellspacing="0" summary=" " cellpadding="0" > - <tbody><tr> - <td class="tdr"> </td> - <td class="tdl_ws1"> </td> - <td class="tdc"> </td> - <td class="tdc" rowspan="3"><img src="images/l_paren.png" alt="" width="29" height="89" /></td> - <td class="tdr"><i>vx</i></td> - <td class="tdc" rowspan="3"><img src="images/r_paren.png" alt="" width="27" height="87" /></td> - </tr><tr> - <td class="tdr"><i>t′</i></td> - <td class="tdl_ws1">=</td> - <td class="tdl_ws1">β </td> - - <td class="tdl"><i>t</i> - —</td> - - </tr><tr> - <td class="tdr"> </td> - <td class="tdl_ws1"> </td> - <td class="tdc"> </td> - - <td class="tdr"><i>c</i>²</td> - </tr> - </tbody> -</table> - -<p>This is the Lorentz transformation, from which everything in this -chapter can be deduced.</p> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_91">[Pg 91]</span></p> -<h2 class="nobreak">CHAPTER VII:<br /> INTERVALS IN SPACE-TIME</h2> -</div> - -<p class="drop-cap"><span class="smcap">The</span> special theory of -relativity, which we have been considering hitherto, solved completely -a certain definite problem: to account for the experimental fact -that, when two bodies are in uniform relative motion, all the laws -of physics, both those of ordinary dynamics and those connected with -electricity and magnetism, are exactly the same for the two bodies. -“Uniform” motion, here, means motion in a straight line with constant -velocity. But although one problem was solved by the special theory, -another was immediately suggested: what if the motion of the two bodies -is not uniform? Suppose, for instance, that one is the earth while -the other is a falling stone. The stone has an accelerated motion: -it is continually falling faster and faster. Nothing in the special -theory enables us to say that the laws of physical phenomena will be -the same for an observer on the stone as for one on the earth. This is -<span class="pagenum" id="Page_92">[Pg 92]</span> -particularly awkward, as the earth itself is, in an extended sense, a -falling body: It has at every moment an acceleration<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">[4]</a> -towards the sun, which makes it go round the sun instead of moving -in a straight line. As our knowledge of physics is derived from -experiments on the earth, we cannot rest satisfied with a theory in -which the observer is supposed to have no acceleration. The general -theory of relativity removes this restriction, and allows the observer -to be moving in any way, straight or crooked, uniformly or with an -acceleration. In the course of removing the restriction, Einstein was -led to his new law of gravitation, which we shall consider presently. -The work was extraordinarily difficult, and occupied him for ten years. -The special theory dates from 1905, the general theory from 1915.</p> - -<p>It is obvious from experiences with which we are all familiar that an -accelerated motion is much more difficult to deal with than a uniform -one. When you are in a train which is traveling steadily, the motion -is not noticeable so long as you do not look out of the window; but -when the brakes are applied suddenly you are precipitated forwards, -<span class="pagenum" id="Page_93">[Pg 93]</span> -and you become aware that something is happening without having to -notice anything outside the train. Similarly in a lift everything -seems ordinary while it is moving steadily, but at starting and -stopping, when its motion is accelerated, you have odd sensations -in the pit of the stomach. (We call a motion “accelerated” when it -is getting slower as well as when it is getting quicker; when it is -getting slower the acceleration is negative.) The same thing applies -to dropping a weight in the cabin of a ship. So long as the ship is -moving uniformly, the weight will behave, relatively to the cabin, -just as if the ship were at rest: if it starts from the middle of -the ceiling, it will hit the middle of the floor. But if there is an -acceleration everything is changed. If the boat is increasing its -speed very rapidly, the weight will seem to an observer in the cabin -to fall in a curve directed towards the stern; if the speed is being -rapidly diminished, the curve will be directed towards the bow. All -these facts are familiar, and they led Galileo and Newton to regard an -accelerated motion as something radically different, in its own nature, -from a uniform motion. But this distinction could only be maintained by -regarding motion as absolute, not relative. If all motion is relative, -<span class="pagenum" id="Page_94">[Pg 94]</span> -the earth is accelerated relatively to the lift just as truly as the -lift relatively to the earth. Yet the people on the ground have no -sensations in the pits of their stomachs when the lift starts to go -up. This illustrates the difficulty of our problem. In fact, though -few physicists in modern times have believed in absolute motion, the -technique of mathematical physics still embodied Newton’s belief in it, -and a revolution in method was required to obtain a technique free from -this assumption. This revolution was accomplished in Einstein’s general -theory of relativity.</p> - -<p>It is somewhat optional where we begin in explaining the new ideas -which Einstein introduced, but perhaps we shall do best by taking the -conception of “interval.” This conception, as it appears in the special -theory of relativity, is already a generalization of the traditional -notion of distance in space and time; but it is necessary to generalize -it still further. However, it is necessary first to explain a certain -amount of history, and for this purpose we must go back as far as Pythagoras.</p> - -<p>Pythagoras, like many of the greatest characters in history, perhaps -<span class="pagenum" id="Page_95">[Pg 95]</span> -never existed: he is a semi-mythical character, who combined -mathematics and priestcraft in uncertain proportions. I shall, however, -assume that he existed, and that he discovered the theorem attributed -to him. He was roughly a contemporary of Confucius and Buddha; he -founded a religious sect, which thought it wicked to eat beans, -and a school of mathematicians, who took a particular interest in -right-angled triangles. The theorem of Pythagoras (the forty-seventh -proposition of Euclid) states that the sum of the squares on the two -shorter sides of a right-angled triangle is equal to the square on -the side opposite the right angle. No proposition in the whole of -mathematics has had such a distinguished history. We all learned to -“prove” it in youth. It is true that the “proof” proved nothing, and -that the only way to prove it is by experiment. It is also the case -that the proposition is not <i>quite</i> true—it is only approximately -true. But everything in geometry, and subsequently in physics, has been -derived from it by successive generalizations. The latest of these -generalizations is the general theory of relativity.</p> - -<p>The theorem of Pythagoras was itself, in all probability, a -<span class="pagenum" id="Page_96">[Pg 96]</span> -generalization of an Egyptian rule of thumb. In Egypt, it had been -known for ages that a triangle whose sides are 3, 4, and 5 units of -length is a right-angled triangle; the Egyptians used this knowledge -practically in measuring their fields. Now if the sides of a triangle -are 3, 4, and 5 inches, the squares on these sides will contain -respectively 9, 16, and 25 square inches; and 9 and 16 added together -make 25. Three times three is written “3²”; four times four, “4²”; five -times five, “5².” So that we have</p> - -<p class="f150">3² + 4² = 5².</p> - -<p>It is supposed that Pythagoras noticed this fact, after he had learned -from the Egyptians that a triangle whose sides are 3, 4 and 5 has a -right angle. He found that this could be generalized, and so arrived -at his famous theorem: In a right-angled triangle, the square on the -side opposite the right angle is equal to the sum of the squares on the -other two sides.</p> - -<div id="I096" class="figcenter"> - <img src="images/i_096.jpg" alt="" width="400" height="370" /> -</div> - -<p>Similarly in three dimensions: if you take a right-angled <a href="#I096">solid block</a>, -<span class="pagenum" id="Page_97">[Pg 97]</span> -the square on the diagonal (the dotted line in the figure) is equal to -the sum of the squares on the three sides.</p> - -<p>This is as far as the ancients got in this matter.</p> - -<div class="figcenter"> - <img src="images/i_097.jpg" alt="" width="500" height="372" /> -</div> - -<p>The next step of importance is due to Descartes, who made the theorem -of Pythagoras the basis of his method of analytical geometry. Suppose -you wish to map out systematically all the places on a plain—we will -suppose it small enough to make it possible to ignore the fact that -the earth is round. We will suppose that you live in the middle of the -plain. One of the simplest ways of describing the position of a place -is to say: starting from my house, go first such and such a distance -east, then such and such a distance north (or it may be west in the -first case, and south in the second). This tells you exactly where -the place is. In the rectangular cities of America, it is the natural -method to adopt: in New York you will be told to go so many blocks -east (or west) and then so many blocks north (or south). The distance -you have to go east is called <i>x</i>, and the distance you have to -<span class="pagenum" id="Page_98">[Pg 98]</span> -go north is called <i>y</i>. (If you have to go west, <i>x</i> is -negative; if you have to go south, <i>y</i> is negative.) Let <b>O</b> -be your starting point (the “origin”); let <b>OM</b> be the distance -you go east, and <b>MP</b> the distance you go north. How far are you -from home in a direct line when you reach <b>P</b>? The theorem of -Pythagoras gives the answer. The square on <b>OP</b> is the sum of the -squares on <b>OM</b> and <b>MP</b>. If <b>OM</b> is four miles, and -<b>MP</b> is three miles, <b>OP</b> is 5 miles. If <b>OM</b> is 12 -miles and <b>MP</b> is 5 miles, <b>OP</b> is 13 miles, because 12² + 5² -= 13². So that if you adopt Descartes’ method of mapping, the theorem -of Pythagoras is essential in giving you the distance from place to -place. In three dimensions the thing is exactly analogous. Suppose -that, instead of wanting merely to fix positions on the plain, you want -to fix stations for captive balloons above it, you will then have to -add a third quantity, the height at which the balloon is to be. If you -call the height <i>z</i>, and if <i>r</i> is the direct distance from -<b>O</b> to the balloon, you will have</p> - -<p class="f150"><i>r</i>² = <i>x</i>² + <i>y</i>² + <i>z</i>²,</p> - -<p class="no-indent">and from this you can calculate <i>r</i> when you know <i>x</i>, -<i>y</i>, and <i>z</i>. For example, if you can get to the balloon by -<span class="pagenum" id="Page_99">[Pg 99]</span> -going 12 miles east, 4 miles north, and then 3 miles up, your distance -from the balloon in a straight line is 13 miles, because</p> - -<ul class="index fontsize_150"> -<li class="isub0">12 × 12 = 144,</li> -<li class="isub0">4 × 4 = 16,</li> -<li class="isub0">3 × 3 = 9,</li> -<li class="isub0">144 + 16 + 9 = 169 = 13 × 13.</li> -</ul> - -<p>But now suppose that, instead of taking a small piece of the earth’s -surface which can be regarded as flat, you consider making a map of -the world. An accurate map of the world on flat paper is impossible. -A globe can be accurate, in the sense that everything is produced -to scale, but a flat map cannot be. I am not talking of practical -difficulties, I am talking of a theoretical impossibility. For example: -the northern halves of the meridian of Greenwich and the ninetieth -meridian of west longitude, together with the piece of the equator -between them, make a triangle whose sides are all equal and whose -angles are all right angles. On a flat surface, a triangle of that sort -would be impossible. On the other hand, it is possible to make a square -on a flat surface, but on a sphere it is impossible. Suppose you try on -the earth: walk 100 miles west, then 100 miles north, then 100 miles -east, then 100 miles south. You might think this would make a square, -but it wouldn’t, because you would not at the end have come back to -<span class="pagenum" id="Page_100">[Pg 100]</span> -your starting point. If you have time, you may convince yourself of -this by experiment. If not, you can easily see that it must be so. When -you are nearer the pole, 100 miles takes you through more longitude -than when you are nearer the equator, so that in doing your 100 miles -east (if you are in the northern hemisphere) you get to a point further -east than that from which you started. As you walk due south after -this, you remain further east than your starting point, and end up at a -different place from that in which you began. Suppose, to take another -illustration, that you start on the equator 4,000 miles east of the -Greenwich meridian; you travel till you reach the meridian, then you -travel northwards along it for 4,000 miles, through Greenwich and up -to the neighborhood of the Shetland Islands; then you travel eastward -for 4,000 miles, and then 4,000 miles south. This will take you to the -equator at a point 4,000 miles further east than the point from which -you started.</p> - -<p>In a sense, what we have just been saying is not quite fair, because, -except on the equator, traveling due east is not the shortest route -from a place to another place due east of it. A ship traveling (say) -<span class="pagenum" id="Page_101">[Pg 101]</span> -from New York to Lisbon, which is nearly due east, will start by going -a certain distance northward. It will sail on a “great circle,” that -is to say, a circle whose centre is the centre of the earth. This -is the nearest approach to a straight line that can be drawn on the -surface of the earth. Meridians of longitude are great circles, and so -is the equator, but the other parallels of latitude are not. We ought, -therefore, to have supposed that, when you reach the Shetland Islands, -you travel 4,000 miles, not due east, but along a great circle which -lands you at a point due east of the Shetland Islands. This, however, -only reinforces our conclusion: you will end at a point even further -east of your starting point than before.</p> - -<p>What are the differences between the geometry on a sphere and the -geometry on a plane? If you make a triangle on the earth, whose sides -are great circles, you will not find that the angles of the triangle -add up to two right angles: they will add up to rather more. The amount -by which they exceed two right angles is proportional to the size of -the triangle. On a small triangle such as you could make with strings -on your lawn, or even on a triangle formed by three ships which can -<span class="pagenum" id="Page_102">[Pg 102]</span> -just see each other, the angles will add up to so little more than two -right angles that you will not be able to detect the difference. But if -you take the triangle made by the equator, the Greenwich meridian, and -the ninetieth meridian, the angles add up to <i>three</i> right angles. -And you can get triangles in which the angles add up to anything up to -six right angles. All this you could discover by measurements on the -surface of the earth, without having to take account of anything in the -rest of space.</p> - -<p>The theorem of Pythagoras also will fail for distances on a sphere. -From the point of view of a traveler bound to the earth, the distance -between two places is their great circle distance, that is to say, the -shortest journey that a man can make without leaving the surface of -the earth. Now suppose you take three bits of great circles which make -a triangle, and suppose one of them is at right angles to another—to -be definite, let one be the equator and one a bit of the meridian of -Greenwich going northward from the equator. Suppose you go 3,000 miles -along the equator, and then 4,000 miles due north; how far will you -be from your starting point, estimating the distance along a great circle? -<span class="pagenum" id="Page_103">[Pg 103]</span> -If you were on a plane, your distance would be 5,000 miles, -as we saw before. In fact, however, your great circle distance will be -considerably less than this. In a right-angled triangle on a sphere, -the square on the side opposite the right angle is less than the sum of -the squares on the other two sides.</p> - -<p>These differences between the geometry on a sphere and the geometry on -a plane are intrinsic differences; that is to say, they enable you to -find out whether the surface on which you live is like a plane or like -a sphere, without requiring that you should take account of anything -outside the surface. Such considerations led to the next step of -importance in our subject, which was taken by Gauss, who flourished a -hundred years ago. He studied the theory of surfaces, and showed how to -develop it by means of measurements on the surfaces themselves, without -going outside them. In order to fix the position of a point in space, -we need three measurements; but in order to fix the position of a point -on a surface we need only two—for example, a point on the earth’s -surface is fixed when we know its latitude and longitude. -<span class="pagenum" id="Page_104">[Pg 104]</span></p> - -<p>Now Gauss found that, whatever system of measurement you adopt, -and whatever the nature of the surface, there is always a way of -calculating the distance between two not very distant points of the -surface, when you know the quantities which fix their positions. -The formula for the distance is a generalization of the formula of -Pythagoras: it tells you the square of the distance in terms of the -squares of the differences between the measure quantities which fix -the points, and also the product of these two quantities. When you -know this formula, you can discover all the intrinsic properties of -the surface, that is to say, all those which do not depend upon its -relations to points outside the surface. You can discover, for example, -whether the angles of a triangle add up to two right angles, or more, -or less, or more in some cases and less in others.</p> - -<p>But when we speak of a “triangle,” we must explain what we mean, -because on most surfaces there are no straight lines. On a sphere, we -shall replace straight lines by great circles, which are the nearest -possible approach to straight lines. In general, we shall take, -instead of straight lines, the lines that give the shortest route on -<span class="pagenum" id="Page_105">[Pg 105]</span> -the surface from place to place. Such lines are called “geodesics.” -On the earth, the geodesics are great circles. In general, they are -the shortest way of traveling from point to point if you are unable -to leave the surface. They take the place of straight lines in the -intrinsic geometry of a surface. When we inquire whether the angles of -a triangle add up to two right angles or not, we mean to speak of a -triangle whose sides are geodesics. And when we speak of the distance -between two points, we mean the distance along a geodesic.</p> - -<p>The next step in our generalizing process is rather difficult: it is -the transition to non-Euclidean geometry. We live in a world in which -space has three dimensions, and our empirical knowledge of space is -based upon measurement of small distances and of angles. (When I speak -of small distances, I mean distances that are small compared to those -in astronomy; all distances on the earth are small in this sense.) It -was formerly thought that we could be sure <i>à priori</i> that space -is Euclidean—for instance, that the angles of a triangle add up to two -right angles. But it came to be recognized that we could not prove this -by reasoning; if it was to be known, it must be known as the result of -<span class="pagenum" id="Page_106">[Pg 106]</span> -measurements. Before Einstein, it was thought that measurements confirm -Euclidean geometry within the limits of exactitude attainable; now -this is no longer thought. It is still true that we can, by what may -be called a natural artifice, cause Euclidean geometry to <i>seem</i> -true throughout a small region, such as the earth; but in explaining -gravitation Einstein is led to the view that over large regions where -there is matter we cannot regard space as Euclidean. The reasons for -this will concern us later. What concerns us now is the way in which -non-Euclidean geometry results from a generalization of the work of Gauss.</p> - -<p>There is no reason why we should not have the same circumstances in -three-dimensional space as we have, for example, on the surface of a -sphere. It might happen that the angles of a triangle would always -add up to more than two right angles, and that the excess would be -proportional to the size of the triangle. It might happen that the -distance between two points would be given by a formula analogous -to what we have on the surface of a sphere, but involving three -quantities instead of two. Whether this does happen or not, can only -<span class="pagenum" id="Page_107">[Pg 107]</span> -be discovered by actual measurements. There are an infinite number of -such possibilities.</p> - -<p>This line of argument was developed by Riemann, in his dissertation -“On the hypotheses which underlie geometry” (1854), which applied -Gauss’s work on surfaces to different kinds of three-dimensional -spaces. He showed that all the essential characteristics of a kind -of space could be deduced from the formula for small distances. He -assumed that, from the small distances in three given directions -which would together carry you from one point to another not far from -it, the distances between the two points could be calculated. For -instance, if you know that you can get from one point to another by -first moving a certain distance east, then a certain distance north, -and finally a certain distance straight up in the air, you are to be -able to calculate the distance from the one point to the other. And -the rule for the calculation is to be an extension of the theorem of -Pythagoras, in the sense that you arrive at the square of the required -distance by adding together multiples of the squares of the component -distances, together possibly with multiples of their products. From -certain characteristics in the formula, you can tell what sort of -<span class="pagenum" id="Page_108">[Pg 108]</span> -space you have to deal with. These characteristics do not depend upon the - particular method you have adopted for determining the positions of points.</p> - -<p>In order to arrive at what we want for the theory of relativity, we -now have one more generalization to make: we have to substitute the -“interval” between events for the distance between points. This takes -us to space-time. We have already seen that, in the special theory -of relativity, the square of the interval is found by subtracting -the square of the distance between the events from the square of the -distance that light would travel in the time between them. In the -general theory, we do not assume this special form of interval, except -at a great distance from matter. Elsewhere, we assume to begin with a -general form, like that which Riemann used for distances. Moreover, -like Riemann, Einstein only assumes his formula for <i>neighboring</i> -events, that is to say, events which have only a small interval -between them. What goes beyond these initial assumptions depends upon -observation of the actual motion of bodies, in ways which we shall -explain in later chapters. -<span class="pagenum" id="Page_109">[Pg 109]</span></p> - -<p>We may now sum up and re-state the process we have been describing. -In three dimensions, the position of a point relatively to a fixed -point (the “origin”) can be determined by assigning three quantities -(“co-ordinates”). For example, the position of a balloon relatively to -your house is fixed if you know that you will reach it by going first -a given distance due east, then another given distance due north, -then a third given distance straight up. When, as in this case, the -three co-ordinates are three distances all at right angles to each -other, which, taken successively, transport you from the origin to the -point in question, the square of the direct distance to the point in -question is got by adding up the squares of the three co-ordinates. In -all cases, whether in Euclidean or in non-Euclidean spaces, it is got -by adding multiples of the squares and products of the co-ordinates -according to an assignable rule. The co-ordinates may be any quantities -which fix the position of a point, provided that neighboring points -must have neighboring quantities for their co-ordinates. In the general -theory of relativity, we add a fourth co-ordinate to give the time, and -our formula gives “interval” instead of spatial distance; moreover we -<span class="pagenum" id="Page_110">[Pg 110]</span> -assume the accuracy of our formula for small distances only. We assume -further that, at great distances from matter, the formula approximates -more and more closely to the formula for interval which is used in the -special theory.</p> - -<p>We are now at last in a position to tackle Einstein’s theory of gravitation.</p> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_111">[Pg 111]</span></p> -<h2 class="nobreak">CHAPTER VIII:<br /> EINSTEIN’S LAW OF GRAVITATION</h2> -</div> - -<p class="drop-cap"><span class="smcap">Before</span> tackling -Einstein’s new law, it is as well to convince ourselves, on logical -grounds, that Newton’s law of gravitation cannot be quite right.</p> - -<p>Newton said that between any two particles of matter there is a force -which is proportional to the product of their masses and inversely -proportional to the square of their distance. That is to say, ignoring -for the present the question of mass, if there is a certain attraction -when the particles are a mile apart, there will be a quarter as much -attraction when they are two miles apart, a ninth as much when they -are three miles apart, and so on: the attraction diminishes much -faster than the distance increases. Now, of course, Newton, when he -spoke of the distance, meant the distance at a given time: He thought -there could be no ambiguity about time. But we have seen that this -was a mistake. What one observer judges to be the same moment on the -<span class="pagenum" id="Page_112">[Pg 112]</span> -earth and the sun, another will judge to be two different moments. -“Distance at a given moment” is therefore a subjective conception, -which can hardly enter into a cosmic law. Of course, we could make -our law unambiguous by saying that we are going to estimate times as -they are estimated by Greenwich Observatory. But we can hardly believe -that the accidental circumstances of the earth deserve to be taken so -seriously. And the estimate of distance, also, will vary for different -observers. We cannot, therefore, allow that Newton’s form of the law of -gravitation can be quite correct, since it will give different results -according to which of many equally legitimate conventions we adopt. -This is as absurd as it would be if the question whether one man had -murdered another were to depend upon whether they were described by -their Christian names or their surnames. It is obvious that physical -laws must be the same whether distances are measured in miles or in -kilometers, and we are concerned with what is essentially only an -extension of the same principle.</p> - -<p>Our measurements are conventional to an even greater extent than -<span class="pagenum" id="Page_113">[Pg 113]</span> -is admitted by the special theory of relativity. Moreover, every -measurement is a physical process carried out with physical material; -the result is certainly an experimental datum, but may not be -susceptible of the simple interpretation which we ordinarily assign to -it. We are, therefore, not going to assume to begin with that we know -how to measure anything. We assume that there is a certain physical -quantity, called “interval,” which is a relation between two events -that are not widely separated; but we do not assume in advance that we -know how to measure it, beyond taking it for granted that it is given -by some generalization of the theorem of Pythagoras such as we spoke of -in the preceding chapter.</p> - -<p>We do assume, however, that events have an <i>order</i>, and that this -order is four-dimensional. We assume, that is to say, that we know what -we mean by saying that a certain event is nearer to another than to -a third, so that before making accurate measurements we can speak of -the “neighborhood” of an event; and we assume that, in order to assign -the position of an event in space-time, four quantities (co-ordinates) -are necessary—<i>e.g.</i> in our former case of an explosion on an -<span class="pagenum" id="Page_114">[Pg 114]</span> -airship, latitude, longitude, altitude and time. But we assume nothing -about the way in which these co-ordinates are assigned, except that -neighboring co-ordinates are assigned to neighboring events.</p> - -<p>The way in which these numbers, called co-ordinates, are to be assigned -is neither wholly arbitrary nor a result of careful measurement—it -lies in an intermediate region. While you are making any continuous -journey, your co-ordinates must never alter by sudden jumps. In America -one finds that the houses between (say) Fourteenth Street and Fifteenth -Street are likely to have numbers between 1400 and 1500, while those -between Fifteenth Street and Sixteenth Street have numbers between -1500 and 1600, even if the 1400’s were not used up. This would not do -for our purposes, because there is a sudden jump when we pass from one -block to the next. Or again we might assign the time co-ordinate in the -following way: take the time that elapses between two successive births -of people called Smith; an event occurring between the births of the -3000th and the 3001st Smith known to history shall have a co-ordinate -lying between 3000 and 3001; the fractional part of its co-ordinate -<span class="pagenum" id="Page_115">[Pg 115]</span> -shall be the fraction of a year that has elapsed since the birth of the -3000th Smith. (Obviously there could never be as much as a year between -two successive additions to the Smith family.) This way of assigning -the time co-ordinate is perfectly definite, but it is not admissible -for our purposes, because there will be sudden jumps between events -just before the birth of a Smith and events just after, so that in a -continuous journey your time co-ordinate will not change continuously. -It is assumed that, independently of measurement, we know what a -continuous journey is. And when your position in space-time changes -continuously, each of your four co-ordinates must change continuously. -One, two, or three of them may not change at all; but whatever change -does occur must be smooth, without sudden jumps. This explains what is -<i>not</i> allowable in assigning co-ordinates.</p> - -<p>To explain all the changes that are legitimate in your co-ordinates, -suppose you take a large piece of soft india-rubber. While it is in an -unstretched condition, measure little squares on it, each one-tenth -of an inch each way. Put in little tiny pins at the corners of the -squares. We can take as two of the co-ordinates of one of these pins -<span class="pagenum" id="Page_116">[Pg 116]</span> -the number of pins passed in going to the right from a given pin until -we come just below the pin in question, and then the number of pins -we pass on the way up to this pin. In the figure, let <b>O</b> be the -pin we start from and <b>P</b> the pin to which we are going to assign -co-ordinates. <b>P</b> is in the fifth column and the third row, so its -co-ordinates in the plane of the india-rubber are to be 5 and 3.</p> - -<div class="figcontainer"> - <div class="figsub"> - <p> </p> - <img src="images/i_116a.jpg" alt="" width="200" height="197" /> - <p class="f120">Fig. 1.</p> - </div> - - <div class="figsub"> - <img id="FIG_02" src="images/i_116b.jpg" alt="" width="300" height="199" /> - <p class="f120">Fig. 2.</p> - </div> -</div> - -<p>Now take the india-rubber and stretch it and twist it as much as -you like. Let the pins now be in the shape they have in <a href="#FIG_02">Fig. 2</a>. -The divisions now no longer represent distances according to our usual -notions, but they will still do just as well as co-ordinates. We may -still take <b>P</b> as having the co-ordinates 5 and 3 in the plane of -the india-rubber; and we may still regard the india-rubber as being in -a plane, even if we have twisted it out of what we should ordinarily -<span class="pagenum" id="Page_117">[Pg 117]</span> -call a plane. Such continuous distortions do not matter.</p> - -<p>To take another illustration: instead of using a steel measuring rod to -fix our co-ordinates, let us use a live eel, which is wriggling all the -time. The distance from the tail to the head of the eel is to count as -one from the point of view of co-ordinates, whatever shape the creature -may be assuming at the moment. The eel is continuous, and its wriggles -are continuous, so it may be taken as our unit of distance in assigning -co-ordinates. Beyond the requirement of continuity, the method of -assigning co-ordinates is purely conventional, and therefore a live eel -is just as good as a steel rod.</p> - -<p>We are apt to think that, for really careful measurements, it is better -to use a steel rod than a live eel. This is a mistake: not because -the eel tells us what the steel rod was thought to tell, but because -the steel rod really tells no more than the eel obviously does. The -point is, not that eels are really rigid, but that steel rods really -wriggle. To an observer in just one possible state of motion, the eel -would appear rigid, while the steel rod would seem to wriggle just -<span class="pagenum" id="Page_118">[Pg 118]</span> -as the eel does to us. For everybody moving differently both from -this observer and ourselves, both the eel and the rod would seem to -wriggle. And there is no saying that one observer is right and another -wrong. In such matters, what is seen does not belong solely to the -physical process observed, but also to the standpoint of the observer. -Measurements of distances and times do not directly reveal properties -of the things measured, but relations of the things to the measurer. -What observation can tell us about the physical world is therefore more -abstract than we have hitherto believed.</p> - -<p>It is important to realize that geometry, as taught in schools since -Greek times, ceases to exist as a separate science, and becomes merged -in physics. The two fundamental notions in elementary geometry were -the straight line and the circle. What appears to you as a straight -road, whose parts all exist now, may appear to another observer to -be like the flight of a rocket, some kind of curve whose parts come -into existence successively. The circle depends upon measurement of -distances, since it consists of all the points at a given distance -from its center. And measurement of distances, as we have seen, is -<span class="pagenum" id="Page_119">[Pg 119]</span> -a subjective affair, depending upon the way in which the observer -is moving. The failure of the circle to have objective validity was -demonstrated by the Michelson-Morley experiment, and is thus, in a -sense, the starting point of the whole theory of relativity. Rigid -bodies, which we need for measurement, are only rigid for certain -observers; for others, they will be constantly changing all their -dimensions. It is only our obstinately earth-bound imagination that -makes us suppose a geometry separate from physics to be possible.</p> - -<p>That is why we do not trouble to give physical significance to our -co-ordinates from the start. Formerly, the co-ordinates used in physics -were supposed to be carefully measured distances; now we realize -that this care at the start is thrown away. It is at a later stage -that care is required. Our co-ordinates now are hardly more than a -systematic way of cataloguing events. But mathematics provides, in -the method of tensors, such an immensely powerful technique that we -can use co-ordinates assigned in this apparently careless way just -as effectively as if we had applied the whole apparatus of minutely -accurate measurement in arriving at them. The advantage of being -<span class="pagenum" id="Page_120">[Pg 120]</span> -haphazard at the start is that we avoid making surreptitious physical -assumptions, which we can hardly help making, if we suppose that our -co-ordinates have initially some particular physical significance.</p> - -<p>We assume that, if two events are close together (but not necessarily -otherwise), there is an interval between them which can be calculated -from the differences between their co-ordinates by some such formula -as we considered in the preceding chapter. That is to say, we take the -squares and products of the differences of co-ordinates, we multiply -them by suitable amounts (which in general will vary from place to -place), and we add the results together. The sum obtained is the -square of the interval. We do not assume in advance that we know the -amounts by which the squares and products must be multiplied; this -is going to be discovered by observing physical phenomena. We know, -however, certain things. We know that the old Newtonian physics is -very nearly accurate when our co-ordinates have been chosen in a -certain way. We know that the special theory of relativity is still -more nearly accurate for suitable co-ordinates. From such facts we can -<span class="pagenum" id="Page_121">[Pg 121]</span> -infer certain things about our new co-ordinates, which, in a logical -deduction, appear as postulates of the new theory.</p> - -<p>As such postulates we take:</p> - -<div class="blockquot"> -<p class="neg-indent">1. That every body travels in a geodesic in -space-time, except in so far as electromagnetic forces act upon it.</p> - -<p class="neg-indent">2. That a light ray travels so that the interval -between two parts of it is zero.</p> - -<p class="neg-indent">3. That at a great distance from gravitating -matter, we can transform our co-ordinates by mathematical manipulation -so that the interval shall be what it is in the special theory of -relativity; and that this is approximately true wherever gravitation is -not very powerful.</p> -</div> - -<p>Each of these postulates requires some explanation.</p> - -<p>We saw that a geodesic on a surface is the shortest line that can be -drawn on the surface from one point to another; for example, on the -earth the geodesics are great circles. When we come to space-time, -the mathematics is the same, but the verbal explanations have to be -rather different. In the general theory of relativity, it is only -neighboring events that have a definite interval, independently of -<span class="pagenum" id="Page_122">[Pg 122]</span> -the route by which we travel from one to the other. The interval -between distant events depends upon the route pursued, and has to be -calculated by dividing the route into a number of little bits and -adding up the intervals for the various little bits. If the interval -is space-like, a body cannot travel from one event to the other; -therefore when we are considering the way bodies move, we are confined -to time-like intervals. The interval between neighboring events, when -it is time-like, will appear as the time between them for an observer -who travels from the one event to the other. And so the whole interval -between two events will be judged by a person who travels from one to -the other to be what his clocks show to be the time that he has taken -on the journey. For some routes this time will be longer, for others -shorter; the more slowly the man travels, the longer he will think he -has been on the journey. This must not be taken as a platitude. I am -not saying that if you travel from London to Edinburgh you will take -longer if you travel more slowly. I am saying something much more odd. -I am saying that if you leave London at 10 <span class="smcap">a.m.</span> and arrive in -Edinburgh at 6.30 <span class="smcap">p.m.</span> Greenwich time, the more slowly you -<span class="pagenum" id="Page_123">[Pg 123]</span> -travel the longer you will take—if the time is judged by your watch. -This is a very different statement. From the point of view of a person -on the earth, your journey takes eight and a half hours. But if you -had been a ray of light traveling round the solar system, starting -from London at 10 <span class="smcap">a.m.</span>, reflected from Jupiter to Saturn, and -so on, until at last you were reflected back to Edinburgh and arrived -there at 6.30 <span class="smcap">p.m.</span>, you would judge that the journey had taken -you exactly no time. And if you had gone by any circuitous route, which -enabled you to arrive in time by traveling fast, the longer your route -the less time you would judge that you had taken; the diminution of -time would be continual as your speed approached that of light. Now -I say that when a body travels, if it is left to itself, it chooses -the route which makes the time between two stages of the journey as -long as possible; if it had traveled from one event to another by any -other route, the time, as measured by its own clocks, would have been -shorter. This is a way of saying that bodies left to themselves do -their journeys as slowly as they can; it is a sort of law of cosmic -laziness. Its mathematical expression is that they travel in geodesics, -in which the total interval between any two events on the journey is -<span class="pagenum" id="Page_124">[Pg 124]</span> -<i>greater</i> than by any alternative route. (The fact that it is -greater, not less, is due to the fact that the sort of interval we are -considering is more analogous to time than to distance.) For example, -if a person could leave the earth and travel about for a time and then -return, the time between his departure and return would be less by his -clocks than by those on the earth: the earth, in its journey round the -sun, chooses the route which makes the time of any bit of its course -by its clocks longer than the time as judged by clocks which move by a -different route. This is what is meant by saying that bodies left to -themselves move in geodesics in space-time.</p> - -<p>We assume that the body considered is not acted upon by electromagnetic -forces. We are concerned at present with the law of gravitation, not -with the effects of electromagnetism. These effects have been brought -into the framework of the general theory of relativity by Weyl,<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">[5]</a> -but for the present we will ignore his work. The planets, in any case, -are not subject, as wholes, to appreciable electromagnetic forces; it -is only gravitation that has to be considered in accounting for their -motions, with which we are concerned in this chapter. -<span class="pagenum" id="Page_125">[Pg 125]</span></p> - -<p>Our second postulate, that a light ray travels so that the interval -between two parts of it is zero, has the advantage that it does not -have to be stated only for <i>small</i> distances. If each little bit -of interval is zero, the sum of them all is zero, and so even distant -parts of the same light ray have a zero interval. The course of a light -ray is also a geodesic according to the definition. Thus we now have -two empirical ways of discovering what are the geodesics in space-time, -namely light rays and bodies moving freely. Among freely-moving -bodies are included all which are not subject to constraints or to -electromagnetic forces, that is to say, the sun, stars, planets and -satellites, and also falling bodies on the earth, at least when they -are falling in a vacuum. When you are standing on the earth, you are -subject to electromagnetic forces: the electrons and protons in the -neighborhood of your feet exert a repulsion on your feet which is just -enough to overcome the earth’s gravitation. This is what prevents you -from falling through the earth, which, solid as it looks, is mostly -empty space. -<span class="pagenum" id="Page_126">[Pg 126]</span></p> - -<p>The third postulate, which relates the general to the special theory, -is very useful. It is not necessary for the application of the special -theory to a limited region that there should be no gravitation in the -region; it is enough if the intensity of gravitation is practically the -same throughout the region. This enables us to apply the special theory -within any small region. How small it will have to be, depends upon the -neighborhood. On the surface of the earth, it would have to be small -enough for the curvature of the earth to be negligible. In the spaces -between the planets, it need only be small enough for the attraction -of the sun and the planets to be sensibly constant throughout the -region. In the spaces between the stars it might be enormous—say half -the distance from one star to the next—without introducing measurable -inaccuracies.</p> - -<p>At a great distance from gravitating matter, we can so choose our -co-ordinates as to obtain a Euclidean space; this is really only -another way of saying that the special theory of relativity applies. In -the neighborhood of matter, although we can make our space Euclidean -in any small region, we cannot do so throughout any region within -<span class="pagenum" id="Page_127">[Pg 127]</span> -which gravitation varies sensibly—at least, if we do, we shall have -to abandon the view that bodies move in geodesics. In the neighborhood -of a piece of matter, there is, as it were, a hill in space-time; -this hill grows steeper and steeper as it gets nearer the top, like -the neck of a champagne bottle. It ends in a sheer precipice. Now by -the law of cosmic laziness which we mentioned earlier, a body coming -into the neighborhood of the hill will not attempt to go straight -over the top, but will go round. This is the essence of Einstein’s -view of gravitation. What a body does, it does because of the nature -of space-time in its own neighborhood, not because of some mysterious -force emanating from a distant body.</p> - -<p>An analogy will serve to make the point clear. Suppose that on a dark -night a number of men with lanterns were walking in various directions -across a huge plain, and suppose that in one part of the plain there -was a hill with a flaring beacon on the top. Our hill is to be such -as we have described, growing steeper as it goes up, and ending in a -precipice. I shall suppose that there are villages dotted about the -plain, and the men with lanterns are walking to and from these various -<span class="pagenum" id="Page_128">[Pg 128]</span> -villages. Paths have been made showing the easiest way from any one -village to any other. These paths will all be more or less curved, to -avoid going too far up the hill; they will be more sharply curved when -they pass near the top of the hill than when they keep some way off -from it. Now suppose that you are observing all this, as best you can, -from a place high up in a balloon, so that you cannot see the ground, -but only the lanterns and the beacon. You will not know that there is a -hill, or that the beacon is at the top of it. You will see that people -turn out of the straight course when they approach the beacon, and -that the nearer they come the more they turn aside. You will naturally -attribute this to an effect of the beacon; you may think that it is -very hot and people are afraid of getting burnt. But if you wait for -daylight you will see the hill, and you will find that the beacon -merely marks the top of the hill and does not influence the people with -lanterns in any way.</p> - -<p>Now in this analogy the beacon corresponds to the sun, the people with -lanterns correspond to the planets and comets, the paths correspond -to their orbits, and the coming of daylight corresponds to the coming -<span class="pagenum" id="Page_129">[Pg 129]</span> -of Einstein. Einstein says that the sun is at the top of a hill, only -the hill is in space-time, not in space. (I advise the reader not to -try to picture this, because it is impossible.) Each body, at each -moment, adopts the easiest course open to it, but owing to the hill the -easiest course is not a straight line. Each little bit of matter is at -the top of its own little hill, like the cock on his own dung-heap. -What we call a big bit of matter is a bit which is at the top of a big -hill. The hill is what we know about; the bit of matter at the top is -assumed for convenience. Perhaps there is really no need to assume it, -and we could do with the hill alone, for we can never get to the top of -any one else’s hill, any more than the pugnacious cock can fight the -peculiarly irritating bird that he sees in the looking glass.</p> - -<p>I have given only a qualitative description of Einstein’s law of -gravitation; to give its exact quantitative formulation is impossible -without more mathematics than I am permitting myself. The most -interesting point about it is that it makes the law no longer the -result of action at a distance: the sun exerts no force on the planets -whatever. Just as geometry has become physics, so, in a sense, physics -<span class="pagenum" id="Page_130">[Pg 130]</span> -has become geometry. The law of gravitation has become the geometrical -law that every body pursues the easiest course from place to place, but -this course is affected by the hills and valleys that are encountered -on the road.</p> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_131">[Pg 131]</span></p> -<h2 class="nobreak">CHAPTER IX:<br /> PROOFS OF EINSTEIN’S<br /> LAW OF GRAVITATION</h2> -</div> - -<p class="drop-cap"><span class="smcap">The</span> reasons for -accepting Einstein’s law of gravitation rather than Newton’s are partly -empirical, partly logical. We will begin with the former.</p> - -<p>Einstein’s law of gravitation gives very nearly the same results -as Newton’s, when applied to the calculation of the orbits of the -planets and their satellites. If it did not, it could not be true, -since the consequences deduced from Newton’s law have been found to be -almost exactly verified by observation. When, in 1915, Einstein first -published his new law, there was only one empirical fact to which he -could point to show that his theory was better than Newton’s. This was -what is called the “motion of the perihelion of Mercury.”</p> - -<p>The planet Mercury, like the other planets, moves round the sun in -an ellipse, with the sun in one of the foci. At some points of its -<span class="pagenum" id="Page_132">[Pg 132]</span> -orbit it is nearer to the sun than at other points. The point where -it is nearest to the sun is called its “perihelion.” Now it was found -by observation that, from one occasion when Mercury is nearest to the -sun until the next, Mercury does not go exactly once round the sun, -but a little bit more. The discrepancy is very small; it amounts to -an angle of forty-two seconds in a century. That is to say, in each -year the planet has to move rather less than half a second of angle -after it has finished a complete revolution from the last perihelion -before it reaches the next perihelion. This very minute discrepancy -from Newtonian theory had puzzled astronomers. There was a calculated -effect due to perturbations caused by the other planets, but this small -discrepancy was the residue after allowing for these perturbations. -Einstein’s theory accounted for this residue, as well as for its -absence in the case of the other planets. (In them it exists, but is -too small to be observed.) This was, at first, his only empirical -advantage over Newton.</p> - -<p>His second success was more sensational. According to orthodox -opinion, light in a vacuum ought always to travel in straight lines. -Not being composed of material particles, it ought to be unaffected -<span class="pagenum" id="Page_133">[Pg 133]</span> -by gravitation. However, it was possible, without any serious breach -with old ideas, to admit that, in passing near the sun, light might be -deflected out of the straight path as much as if it were composed of -material particles. Einstein, however, maintained, as a deduction from -his law of gravitation, that light would be deflected twice as much as -this. That is to say, if the light of a star passed very near the sun, -Einstein maintained that the ray from the star would be turned through -an angle of just under one and three-quarters seconds. His opponents -were willing to concede half of this amount. Now it is not every day -that a star almost in line with the sun can be seen. This is only -possible during a total eclipse, and not always then, because there may -be no bright stars in the right position. Eddington points out that, -from this point of view, the best day of the year is May 29, because -then there are a number of bright stars close to the sun. It happened -by incredible good fortune that there was a total eclipse of the sun -on May 29, 1919—the first year after the armistice. Two British -expeditions photographed the stars near the sun during the eclipse, -and the results confirmed Einstein’s prediction. Some astronomers -<span class="pagenum" id="Page_134">[Pg 134]</span> -who remained doubtful whether sufficient precautions had been taken -to insure accuracy were convinced when their own observations in a -subsequent eclipse gave exactly the same result. Einstein’s estimate of -the amount of the deflection of light by gravitation is therefore now -universally accepted.</p> - -<p>The third experimental test is on the whole favorable to Einstein, -though the quantities concerned are so small that it is only just -possible to measure them, and the result is therefore not decisive. But -successive investigations have made it more and more probable that the -small effect predicted by Einstein really occurs. Before explaining the -effect in question, a few preliminary explanations are necessary. The -spectrum of an element consists of certain lines of various shades of -light, separated by a prism, and emitted by the element when it glows. -They are the same (to a very close approximation) whether the element -is in the earth or the sun or a star. Each line is of some definite -shade of color, with some definite wave length. Longer wave lengths are -towards the red end of the spectrum, shorter ones towards the violet -end. When the source of light is moving towards you, the apparent wave -<span class="pagenum" id="Page_135">[Pg 135]</span> -lengths grow shorter, just as waves at sea come quicker when you are -traveling against the wind. When the source of light is moving away -from you, the apparent wave lengths grow longer, for the same reason. -This enables us to know whether the stars are moving towards us or away -from us. If they are moving towards us, all the lines in the spectrum -of an element are moved a little toward violet; if away from us, toward -red. You may notice the analogous effect in sound any day. If you are -in a station and an express comes through whistling, the note of the -whistle seems much more shrill while the train is approaching you than -when it has passed. Probably many people think the note has “really” -changed, but in fact the change in what you hear is only due to the -fact that the train was first approaching and then receding. To people -in the train, there was no change of note. This is <i>not</i> the -effect with which Einstein is concerned. The distance of the sun from -the earth does not change much; for our present purposes, we may regard -it as constant. Einstein deduces from his law of gravitation that -any periodic process which takes place in an atom in the sun (whose -<span class="pagenum" id="Page_136">[Pg 136]</span> -gravitation is very intense) must, as measured by our clocks, take -place at a slightly slower rate than it would in a similar atom on the -earth. The “interval” involved will be the same in the sun and on the -earth, but the same interval in different regions does not correspond -to exactly the same time; this is due to the “hilly” character of -space-time which constitutes gravitation. Consequently any given line -in the spectrum ought, when the light comes from the sun, to seem to -us a little nearer the red end of the spectrum than if the light came -from a source on the earth. The effect to be expected is very small—so -small that there is still some slight uncertainty as to whether it -exists or not. But it now seems highly probable that it exists.</p> - -<p>No other measurable differences between the consequences of Einstein’s -law and those of Newton’s have hitherto been discovered. But the above -experimental tests are quite sufficient to convince astronomers that, -where Newton and Einstein differ as to the motions of the heavenly -bodies, it is Einstein’s law that gives the right results. Even if -the empirical grounds in favor of Einstein stood alone, they would be -conclusive. Whether his law represents the exact truth or not, it is -<span class="pagenum" id="Page_137">[Pg 137]</span> -certainly more nearly exact than Newton’s, though the inaccuracies in -Newton’s were all exceedingly minute.</p> - -<p>But the considerations which originally led Einstein to his law were -not of this detailed kind. Even the consequence about the perihelion of -Mercury, which could be verified at once from previous observations, -could only be deduced after the theory was complete, and could not -form any part of the original grounds for inventing such a theory. -These grounds were of a more abstract logical character. I do not mean -that they were not based upon observed facts, and I do not mean that -they were <i>à priori</i> fantasies such as philosophers indulged in -formerly. What I mean is that they were derived from certain general -characteristics of physical experience, which showed that Newton -<i>must</i> be wrong and that something like Einstein’s law <i>must</i> -be substituted.</p> - -<p>The arguments in favor of the relativity of motion are, as we saw in -earlier chapters, quite conclusive. In daily life, when we say that -something moves, we mean that it moves relatively to the earth. In -dealing with the motions of the planets, we consider them as moving -<span class="pagenum" id="Page_138">[Pg 138]</span> -relatively to the sun, or to the center of mass of the solar system. -When we say that the solar system itself is moving, we mean that it is -moving relatively to the stars. There is no physical occurrence which -can be called “absolute motion.” Consequently the laws of physics must -be concerned with relative motions, since these are the only kind that occur.</p> - -<p>We now take the relativity of motion in conjunction with the -experimental fact that the velocity of light is the same relatively -to one body as relatively to another, however the two may be moving. -This leads us to the relativity of distances and times. This in turn -shows that there is no objective physical fact which can be called “the -distance between two bodies at a given time,” since the time and the -distance will both depend on the observer. Therefore Newton’s law of -gravitation is logically untenable, since it makes use of “distance at -a given time.”</p> - -<p>This shows that we cannot rest content with Newton, but it does not -show what we are to put in his place. Here several considerations -enter in. We have in the first place what is called “the equality -of gravitational and inertial mass.” What this means is as follows: -<span class="pagenum" id="Page_139">[Pg 139]</span> -When you apply a given force<a id="FNanchor_6" href="#Footnote_6" class="fnanchor">[6]</a> -to a heavy body, you do not give it as much acceleration as you -would to a light body. What is called the “inertial” mass of a body -is measured by the amount of force required to produce a given -acceleration. At a given point of the earth’s surface, the “mass” is -proportional to the “weight.” What is measured by scales is rather the -mass than the weight: the weight is defined as the force with which -the earth attracts the body. Now this force is greater at the poles -than at the equator, because at the equator the rotation of the earth -produces a “centrifugal force” which partially counteracts gravitation. -The force of the earth’s attraction is also greater on the surface of -the earth than it is at a great height or at the bottom of a very deep -mine. None of these variations are shown by scales, because they affect -the weights used just as much as the body weighed; but they are shown -if we use a spring balance. The mass does not vary in the course of -these changes of weight. -<span class="pagenum" id="Page_140">[Pg 140]</span></p> - -<p>The “gravitational” mass is differently defined. It is capable of two -meanings. We may mean (1), the way a body responds in a situation -where gravitation has a known intensity, for example, on the surface -of the earth, or on the surface of the sun; or (2), the intensity of -the gravitational force produced by the body, as, for example, the sun -produces stronger gravitational forces than the earth does. Newton -says that the force of gravitation between two bodies is proportional -to the product of their masses. Now let us consider the attraction of -different bodies to one and the same body, say the sun. Then different -bodies are attracted by forces which are proportional to their masses, -and which, therefore, produce exactly the same acceleration in all of -them. Thus if we mean “gravitational mass” in sense (1), that is to -say, the way a body responds to gravitation, we find that “the equality -of inertial and gravitational mass,” which sounds formidable, reduces -to this: that in a given gravitational situation, all bodies behave -exactly alike. As regards the surface of the earth, this was one of -the first discoveries of Galileo. Aristotle thought that heavy bodies -fall faster than light ones; Galileo showed that this is not the case, -<span class="pagenum" id="Page_141">[Pg 141]</span> -when the resistance of the air is eliminated. In a vacuum, a feather -falls as fast as a lump of lead. As regards the planets, it was Newton -who established the corresponding facts. At a given distance from the -sun, a comet, which has a very small mass, experiences exactly the -same acceleration towards the sun as a planet experiences at the same -distance. Thus the way in which gravitation affects a body depends only -upon where the body is, and in no degree upon the nature of the body. -This suggests that the gravitational effect is a characteristic of the -locality, which is what Einstein makes it.</p> - -<p>As for the gravitational mass in sense (2), <i>i.e.</i>, the intensity -of the force produced by a body, this is no longer <i>exactly</i> -proportional to its inertial mass. The question involves some rather -complicated mathematics, and I shall not go into it.<a id="FNanchor_7" href="#Footnote_7" class="fnanchor">[7]</a></p> - -<p>We have another indication as to what sort of thing the law of -gravitation <i>must</i> be, if it is to be a characteristic of a -neighborhood, as we have seen reason to suppose that it is. It must -<span class="pagenum" id="Page_142">[Pg 142]</span> -be expressed in some law which is unchanged when we adopt a different -kind of co-ordinates. We saw that we must not, to begin with, regard -our co-ordinates as having any physical significance: they are merely -systematic ways of naming different parts of space-time. Being -conventional, they cannot enter into physical laws. That means to say -that, if we have expressed a law correctly in terms of one set of -co-ordinates, it must be expressed by the same formula in terms of -another set of co-ordinates. Or, more exactly, it must be possible -to find a formula which expresses the law, and which is unchanged -however we change the co-ordinates. It is the business of the theory -of tensors to deal with such formulæ. And the theory of tensors shows -that there is one formula which obviously suggests itself as being -possibly the law of gravitation. When this possibility is examined, -it is found to give the right results; it is here that the empirical -confirmations come in. But if Einstein’s law had not been found to -agree with experience, we could not have gone back to Newton’s law. We -should have been compelled by logic to seek some law expressed in terms -of “tensors,” and therefore independent of our choice of co-ordinates. -<span class="pagenum" id="Page_143">[Pg 143]</span> -It is impossible without mathematics to explain the theory of -tensors; the non-mathematician must be content to know that it is the -technical method by which we eliminate the conventional element from -our measurements and laws, and thus arrive at physical laws which are -independent of the observer’s point of view. Of this method, Einstein’s -law of gravitation is the most splendid example.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_144">[Pg 144]</span></p> -<h2 class="nobreak">CHAPTER X:<br /> MASS, MOMENTUM, ENERGY<br /> AND ACTION</h2> -</div> - -<p class="drop-cap"><span class="smcap">The</span> pursuit of -quantitative precision is as arduous as it is important. Physical -measurements are made with extraordinary exactitude; if they were made -less carefully, such minute discrepancies as form the experimental data -for the theory of relativity could never be revealed. Mathematical -physics, before the coming of relativity, used a set of conceptions -which were supposed to be as precise as physical measurements, but -it has turned out that they were logically defective, and that this -defectiveness showed itself in very small deviations from expectations -based upon calculation. In this chapter I want to show how the -fundamental ideas of pre-relativity physics are affected, and what -modifications they have had to undergo.</p> - -<p>We have already had occasion to speak of mass. For purposes of -<span class="pagenum" id="Page_145">[Pg 145]</span> -daily life, mass is much the same as weight; the usual measures of -weight—ounces, grams, etc.—are really measures of mass. But -as soon as we begin to make accurate measurements, we are compelled to -distinguish between mass and weight. Two different methods of weighing -are in common use, one, that of scales, the other that of the spring -balance. When you go a journey and your luggage is weighed, it is not -put on scales, but on a spring; the weight depresses the spring a -certain amount, and the result is indicated by a needle on a dial. The -same principle is used in automatic machines for finding your weight. -The spring balance shows weight, but scales show <i>mass</i>. So long -as you stay in one part of the world, the difference does not matter; -but if you test two weighing machines of different kinds in a number -of different places, you will find, if they are accurate, that their -results do not always agree. Scales will give the same result anywhere, -but a spring balance will not. That is to say, if you have a lump of -lead weighing ten pounds by the scales, it will also weigh ten pounds -by scales in any other part of the world. But if it weighs ten pounds -by a spring balance in London, it will weigh more at the North Pole, -less at the equator, less high up in an aeroplane, and less at the -<span class="pagenum" id="Page_146">[Pg 146]</span> -bottom of a coal mine, if it is weighed in all those places on the same -spring balance. The fact is that the two instruments measure quite -different quantities. The scales measure what may be called (apart from -refinements which will concern us presently) “quantity of matter.” -There is the same “quantity of matter” in a pound of feathers as in a -pound of lead. Standard “weights,” which are really standard “masses,” -will measure the amount of mass in any substance put into the opposite -scales. But “weight” is a properly due to the earth’s gravitation: It -is the amount of the force by which the earth attracts a body. This -force varies from place to place. In the first place, anywhere outside -the earth the attraction varies inversely as the square of the distance -from the center of the earth; it is therefore less at great heights. -In the second place, when you go down a coal mine, part of the earth -is above you, and attracts matter upwards instead of downwards, so -that the net attraction downwards is less than on the surface of the -earth. In the third place, owing to the rotation of the earth, there is -what is called a “centrifugal force,” which acts against gravitation. -This is greatest at the equator, because there the rotation of the -<span class="pagenum" id="Page_147">[Pg 147]</span> -earth involves the fastest motion; at the poles it does not exist, -because they are on the axis of rotation. For all these reasons, the -force with which a given body is attracted to the earth is measureably -different at different places. It is this force that is measured by a -spring balance; that is why a spring balance gives different results -in different places. In the case of scales, the standard “weights” are -altered just as much as the body to be weighed, so that the result is -the same everywhere; but the result is the “mass,” not the “weight.” -A standard “weight” has the same mass everywhere, but not the same -“weight”; it is in fact a unit of mass, not of weight. For theoretical -purposes, mass, which is almost invariable for a given body, is much -more important than weight, which varies according to circumstances. -Mass may be regarded, to begin with, as “quantity of matter”; we shall -see that this view is not strictly correct, but it will serve as a -starting point for subsequent refinements.</p> - -<p>For theoretical purposes, a mass is defined as being determined by the -amount of force required to produce a given acceleration: The more -massive a body is, the greater will be the force required to alter its -<span class="pagenum" id="Page_148">[Pg 148]</span> -velocity by a given amount in a given time. It takes a more powerful -engine to make a long train attain a speed of ten miles an hour at the -end of the first half-minute, than it does to make a short train do so. -Or we may have circumstances where the force is the same for a number -of different bodies; in that case, if we can measure the accelerations -produced in them, we can tell the ratios of their masses: the greater -the mass, the smaller the acceleration. We may take, in illustration -of this method, an example which is important in connection with -relativity. Radio-active bodies emit beta-particles (electrons) with -enormous velocities. We can observe their path by making them travel -through water vapor and form a cloud as they go. We can at the same -time subject them to known electric and magnetic forces, and observe -how much they are bent out of a straight line by these forces. This -makes it possible to compare their masses. It is found that the faster -they travel, the greater is their mass, as measured by the stationary -observer; the increase is greatest as applied to their mass as measured -by the effect of a force in the line of motion. In regard to forces at -right angles to the line of motion, there is a change of mass with -<span class="pagenum" id="Page_149">[Pg 149]</span> -velocity in the same proportion as the changes of length and time. It -is known otherwise that, apart from the effect of motion, all electrons -have the same mass.</p> - -<p>All this was known before the theory of relativity was invented, but -it showed that the traditional conception of mass had not quite the -definiteness that had been ascribed to it. Mass used to be regarded as -“quantity of matter,” and supposed to be quite invariable. Now mass was -found to be relative to the observer, like length and time, and to be -altered by motion in exactly the same proportion. However, this could -be remedied. We could take the “proper mass,” the mass as measured by -an observer who shares the motion of the body. This was easily inferred -from the measured mass, by taking the same proportion as in the case of -lengths and times.</p> - -<p>But there is a more curious fact, and that is, that after we have -made this correction we still have not obtained a quantity which is -at all times exactly the same for the same body. When a body absorbs -energy—for example, by growing hotter—its “proper mass” -increases slightly. The increase is very slight, since it is measured -<span class="pagenum" id="Page_150">[Pg 150]</span> -by dividing the increase of energy by the square of the velocity of -light. On the other hand, when a body parts with energy it loses mass. -The most notable case of this is that four hydrogen atoms can come -together to make one helium atom, but a helium atom has rather less -than four times the mass of one hydrogen atom.</p> - -<p>We have thus two kinds of mass, neither of which quite fulfils the old -ideal. The mass as measured by an observer who is in motion relative -to the body in question is a relative quantity, and has no physical -significance as a property of the body. The “proper mass” is a genuine -property of the body, not dependent upon the observer; but it, also, -is not strictly constant. As we shall see shortly, the notion of mass -becomes absorbed into the notion of energy; it represents, so to speak, -the energy which the body expends internally, as opposed to that which -it displays to the outer world.</p> - -<p>Conservation of mass, conservation of momentum, and conservation of -energy were the great principles of classical mechanics. Let us next -consider conservation of momentum.</p> - -<p>The momentum of a body in a given direction is its velocity in that -direction multiplied by its mass. Thus a heavy body moving slowly may -<span class="pagenum" id="Page_151">[Pg 151]</span> -have the same momentum as a light body moving fast. When a number of -bodies interact in any way, for instance by collisions, or by mutual -gravitation, so long as no outside influences come in, the total -momentum of all the bodies in any direction remains unchanged. This law -remains true in the theory of relativity. For different observers, the -mass will be different, but so will the velocity; these two differences -neutralize each other, and it turns out that the principle still -remains true.</p> - -<p>The momentum of a body is different in different directions. The -ordinary way of measuring it is to take the velocity in a given -direction (as measured by the observer) and multiply it by the mass (as -measured by the observer). Now the velocity in a given direction is -the distance traveled in that direction in unit time. Suppose we take -instead the distance traveled in that direction while the body moves -through unit “interval.” (In ordinary cases, this is only a very slight -change, because, for velocities considerably less than that of light, -interval is very nearly equal to lapse of time.) And suppose that -<span class="pagenum" id="Page_152">[Pg 152]</span> -instead of the mass as measured by the observer we take the proper -mass. These two changes increase the velocity and diminish the mass, -both in, the same proportion. Thus the momentum remains the same, but -the quantities that vary according to the observer have been replaced -by quantities which are fixed independently of the observer—with the -exception of the distance traveled by the body in the given direction.</p> - -<p>When we substitute space-time for time, we find that the measured mass -(as opposed to the proper mass) is a quantity of the same kind as the -momentum in a given direction; it might be called the momentum in -the time direction. The measured mass is obtained by multiplying the -invariant mass by the <i>time</i> traversed in traveling through unit -interval; the momentum is obtained by multiplying the same invariant -mass by the <i>distance</i> traversed (in the given direction) in -traveling through unit interval. From a space-time point of view, these -naturally belong together.</p> - -<p>Although the measured mass of a body depends upon the way the observer -is moving relatively to the body, it is none the less a very important -<span class="pagenum" id="Page_153">[Pg 153]</span> -quantity. For any given observer, the measured mass of the whole -physical universe is constant.<a id="FNanchor_8" href="#Footnote_8" class="fnanchor">[8]</a> -The proper mass of all the bodies in the world is not necessarily the -same at one time as at another, so that in this respect the measured -mass has an advantage. The conservation of measured mass is the same -thing as the conservation of energy. This may seem surprising, since -at first sight mass and energy are very different things. But it has -turned out that energy is the same thing as measured mass. To explain -how this comes about is not easy; nevertheless we will make the attempt.</p> - -<p>In popular talk, “mass” and “energy” do not mean at all the same thing. -We associate “mass” with the idea of a fat man in a chair, very slow to -move, while “energy” suggests a thin person full of hustle and “pep.” -Popular talk associates “mass” and “inertia,” but its view of inertia -is one-sided: it includes slowness in beginning to move, but not -slowness in stopping, which is equally involved. All these terms have -technical meanings in physics, which are only more or less analogous -<span class="pagenum" id="Page_154">[Pg 154]</span> -to the meanings of the terms in popular talk. For the present, we are -concerned with the technical meaning of “energy.”</p> - -<p>Throughout the latter half of the nineteenth century, a great deal was -made of the “conservation of energy,” or the “persistence of force,” -as Herbert Spencer preferred to call it. This principle was not easy -to state in a simple way, because of the different forms of energy; -but the essential point was that energy is never created or destroyed, -though it can be transformed from one kind into another. The principle -acquired its position through Joule’s discovery of “the mechanical -equivalent of heat,” which showed that there was a constant proportion -between the work required to produce a given amount of heat and the -work required to raise a given weight through a given height: in fact, -the same sort of work could be utilized for either purpose according to -the mechanism. When heat was found to consist in motion of molecules, -it was seen to be natural that it should be analogous to other forms of -energy. Broadly speaking, by the help of a certain amount of theory, -all forms of energy were reduced to two, which were called respectively -“kinetic” and “potential.” These were defined as follows: -<span class="pagenum" id="Page_155">[Pg 155]</span></p> - -<p>The kinetic energy of a particle is half the mass multiplied by the -square of the velocity. The kinetic energy of a number of particles is -the sum of the kinetic energies of the separate particles.</p> - -<p>The potential energy is more difficult to define. It represents any -state of strain, which can only be preserved by the application of -force. To take the easiest case: If a weight is lifted to a height and -kept suspended, it has potential energy, because, if left to itself, it -will fall. Its potential energy is equal to the kinetic energy which it -would acquire in falling through the same distance through which it was -lifted. Similarly when a comet goes round the sun in a very eccentric -orbit, it moves much faster when it is near the sun than when it is far -from it, so that its kinetic energy is much greater when it is near -the sun. On the other hand, its potential energy is greatest when it -is farthest from the sun, because it is then like the stone which has -been lifted to a height. The sum of the kinetic and potential energies -of the comet is constant, unless it suffers collisions or loses matter -by forming a tail. We can determine accurately the <i>change</i> of -<span class="pagenum" id="Page_156">[Pg 156]</span> -potential energy in passing from one position to another, but the total -amount of it is to a certain extent arbitrary, since we can fix the -zero level where we like. For example, the potential energy of our -stone may be taken to be the kinetic energy it would acquire in falling -to the surface of the earth, or what it would acquire in falling down -a well to the center of the earth, or any assigned lesser distance. It -does not matter which we take, so long as we stick to our decision. We -are concerned with a profit-and-loss account, which is unaffected by -the amount of the assets with which we start.</p> - -<p>Both the kinetic and the potential energies of a given set of bodies -will be different for different observers. In classical dynamics, -the kinetic energy differed according to the state of motion of the -observer, but only by a constant amount; the potential energy did not -differ at all. Consequently, for each observer, the total energy was -constant—assuming always that the observers concerned were moving -in straight lines with uniform velocities, or, if not, were able to -refer their motions to bodies which were so moving. But in relativity -dynamics the matter becomes more complicated. We cannot profitably -<span class="pagenum" id="Page_157">[Pg 157]</span> -adapt the idea of potential energy to the theory of relativity, and -therefore the conservation of energy, in a strict sense, cannot -be maintained. But we obtain a property, closely analogous to -conservation, which applies to kinetic energy alone. As Eddington -puts it: the kinetic energy is not always strictly conserved, and the -classical theory therefore introduces a supplementary quantity, the -potential energy, so that the sum of the two is strictly conserved. The -relativity treatment, on the other hand, discovers another formula, -analogous to the one expressing conservation, which holds always for -the kinetic energy. “The relativity treatment adheres to the physical -quantity and modifies the law; the classical treatment adheres to -the law and modifies the physical quantity.” The new formula, he -continues, may be spoken of “as the law of conservation of energy and -momentum, because, though it is not formally a law of conservation, it -expresses exactly the phenomena which classical mechanics attributes to -conservation.”<a id="FNanchor_9" href="#Footnote_9" class="fnanchor">[9]</a> -It is only in this modified and less rigorous sense that the conservation -of energy remains true.</p> - -<p>What is meant by “conservation” in practice is not exactly what it -<span class="pagenum" id="Page_158">[Pg 158]</span> -means in theory. In theory we say that a quantity is conserved when the -amount of it in the world is the same at any one time as at any other. -But in practice we cannot survey the whole world, so we have to mean -something more manageable. We mean that, taking any given region, if -the amount of the quantity in the region has changed, it is because -some of the quantity has passed across the boundary of the region. If -there were no births and deaths, population would be conserved; in that -case the population of a country could only change by emigration or -immigration, that is to say, by passing across the boundaries. We might -be unable to take an accurate census of China or Central Africa, and, -therefore, we might not be able to ascertain the total population of -the world. But we should be justified in assuming it to be constant if, -wherever statistics were possible, the population never changed except -through people crossing the frontiers. In fact, of course, population -is not conserved. A physiologist of my acquaintance once put four mice -into a thermos. Some hours later, when he went to take them out, there -were eleven of them. But mass is not subject to these fluctuations: -<span class="pagenum" id="Page_159">[Pg 159]</span> -the mass of the eleven mice at the end of the time was no greater than -the mass of the four at the beginning.</p> - -<p>This brings us back to the problem for the sake of which we have been -discussing energy. We stated that, in relativity theory, measured mass -and energy are regarded as the same thing, and we undertook to explain -why. It is now time to embark upon this explanation. But here, as at -the end of Chapter VI, the totally unmathematical reader will do well -to skip, and begin again at the following paragraph.</p> - -<p>Let us take the velocity of light as the unit of velocity; this is -always convenient in relativity theory. Let <i>m</i> be the proper mass -of a particle, <i>v</i> its velocity relative to the observer. Then its -measured mass will be</p> - -<table class="fontsize_150" border="0" cellspacing="0" summary=" " cellpadding="0" > - <tbody><tr> - <td class="tdc"><i>m</i></td> - </tr><tr> - <td class="tdc">———</td> - </tr><tr> - <td class="tdc">√(1 - <i>v²</i>)</td> - </tr> - </tbody> -</table> - -<p class="no-indent">while its kinetic energy, according to -the usual formula, will be</p> - -<p class="f150">½ <i>mv²</i></p> - -<p>As we saw before, energy only occurs in a profit-and-loss account, -so<span class="pagenum" id="Page_160">[Pg 160]</span> that we can add any constant quantity to it that we like. We may -therefore take the energy to be</p> - -<p class="f150"><i>m</i> + ½ <i>mv²</i></p> - -<p class="no-indent">Now if <span class="fontsize_150"><i>v</i></span> -is a small fraction of the velocity of light,</p> - -<p class="f150"><i>m</i> + ½ <i>mv²</i></p> - -<p class="no-indent">is almost exactly equal to</p> - -<table class="fontsize_150" border="0" cellspacing="0" summary=" " cellpadding="0" > - <tbody><tr> - <td class="tdc"><i>m</i></td> - </tr><tr> - <td class="tdc">———</td> - </tr><tr> - <td class="tdc">√(1 - <i>v²</i>)</td> - </tr> - </tbody> -</table> - -<p>Consequently, for velocities such as large bodies have, the energy and -the measured mass turn out to be indistinguishable within the limits of -accuracy attainable. In fact, it is better to alter our definition of -energy, and take it to be</p> - -<table class="fontsize_150" border="0" cellspacing="0" summary=" " cellpadding="0" > - <tbody><tr> - <td class="tdc"><i>m</i></td> - </tr><tr> - <td class="tdc">———</td> - </tr><tr> - <td class="tdc">√(1 - <i>v²</i>)</td> - </tr> - </tbody> -</table> - -<p class="no-indent">because this is the quantity for which the law analogous -to conservation holds. And when the velocity is very great, it gives a -better measure of energy than the traditional formula. The traditional -formula must therefore be regarded as an approximation, of which the -new formula gives the exact version. In this way, energy and measured -mass become identified.</p> - -<p>I come now to the notion of “action,” which is less familiar to -the general public than energy, but has become more important in -<span class="pagenum" id="Page_161">[Pg 161]</span> -relativity physics, as well as in the theory of quanta.<a id="FNanchor_10" href="#Footnote_10" class="fnanchor">[10]</a> -(The quantum is a small amount of action.) The word “action” is used to -denote energy multiplied by time. That is to say, if there is one unit -of energy in a system, it will exert one unit of action in a second, -100 units of action in 100 seconds, and so on; a system which has -100 units of energy will exert 100 units of action in a second, and -10,000 in 100 seconds, and so on. “Action” is thus, in a loose sense, -a measure of how much has been accomplished: it is increased both by -displaying more energy and by working for a longer time. Since energy -is the same thing as measured mass, we may also take action to be -measured mass multiplied by time. In classical mechanics, the “density” -of matter in any region is the mass divided by the volume; that is -to say, if you know the density in a small region, you discover the -total amount of matter by multiplying the density by the volume of the -small region. In relativity mechanics, we always want to substitute -space-time for space; therefore a “region” must no longer be taken to -<span class="pagenum" id="Page_162">[Pg 162]</span> -be merely a volume, but a volume lasting for a time; a small region -will be a small volume lasting for a small time. It follows that, given -the density, a small region in the new sense contains, not a small mass -merely, but a small mass multiplied by a small time, that is to say, a -small amount of “action.” This explains why it is to be expected that -“action” will prove of fundamental importance in relativity mechanics. -And so in fact it is.</p> - -<p>All the laws of dynamics have been put together into one principle, -called “The Principle of Least Action.” This states that, in passing -from one state to another, a body chooses a route involving less action -than any slightly different route—again a law of cosmic laziness. The -principle is subject to certain limitations, which have been pointed -out by Eddington,<a id="FNanchor_11" href="#Footnote_11" class="fnanchor">[11]</a> -but it remains one of the most comprehensive ways of stating the purely -formal part of mechanics. The fact that the quantum is a unit of -action seems to show that action is also fundamental in the empirical -structure of the world. But at present there is no bridge connecting -the quantum with the theory of relativity.</p> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_163">[Pg 163]</span></p> - -<h2 class="nobreak">CHAPTER XI:<br /> IS THE UNIVERSE FINITE?</h2> -</div> - -<p class="drop-cap"><span class="smcap">We have</span> been dealing -hitherto with matters that must be regarded as acquired scientific -results—not that they will never be found to need improvement, -but that further progress must be built upon them, as Einstein is -built upon Newton. Science does not aim at establishing immutable -truths and eternal dogmas: its aim is to approach truth by successive -approximations, without claiming that at any stage final and complete -accuracy has been achieved. There is a difference, however, between -results which are pretty certainly in the line of advance, and -speculations which may or may not prove to be well founded. Some very -interesting speculations are connected with the theory of relativity, -and we shall consider certain of them. But it must not be supposed that -we are dealing with theories having the same solidity as those with -which we have been concerned hitherto.</p> - -<p>One of the most fascinating of the speculations to which I have been -<span class="pagenum" id="Page_164">[Pg 164]</span> -alluding is the suggestion that the universe may be of finite extent. -Two somewhat different finite universes have been constructed, one by -Einstein, the other by De Sitter. Before considering their differences, -we will discuss what they have in common.</p> - -<p>There are, to begin with, certain reasons for thinking that the total -amount of matter in the universe is limited. If this were not the -case, the gravitational effects of enormously distant matter would -make the kind of world in which we live impossible. We must therefore -suppose that there is some definite number of electrons and protons in -the world: theoretically, a complete census would be possible. These -are all contained within a certain finite region; whatever space lies -outside that region is, so to speak, waste, like unfurnished rooms in a -house too large for its inhabitants. This seems futile, but in former -days no one knew of any alternative possibility. It was obviously -impossible to conceive of an edge to space, and therefore, it was -thought, space must be infinite.</p> - -<p>Non-Euclidean geometry, however, showed other possibilities. The -surface of a sphere has no boundary, yet it is not infinite. In -<span class="pagenum" id="Page_165">[Pg 165]</span> -traveling round the earth, we never reach “the edge of the world,” and -yet the earth is not infinite. The surface of the earth is contained -in three-dimensional space, but there is no reason in logic why -three-dimensional space should not be constructed on an analogous plan. -What we imagine to be straight lines going on for ever will then be -like great circles on a sphere: they will ultimately return to their -starting point. There will not be in the universe anything straighter -than these great circles; the Euclidean straight line may remain as -a beautiful dream, but not as a possibility in the actual world. In -particular, light rays in empty space will travel in what are really -great circles. If we could make measurements with sufficient accuracy, -we should be able to infer this state of affairs even from a small part -of space, because the sum of the angles of a triangle would always be -greater than two right angles, and the excess would be proportional to -the size of the triangle. The suggestion we have to consider is the -suggestion that our universe may be spherical in this sense.</p> - -<p>The reader must not confuse this suggestion with the non-Euclidean -character of space upon which the new law of gravitation depends. The -<span class="pagenum" id="Page_166">[Pg 166]</span> -latter is concerned with small regions such as the solar system. The -departures from flatness which it notices are like hills and valleys -on the surface of the earth, local irregularities, not characteristics -of the whole. We are now concerned with the possible curvature of the -universe as a whole, not with the occasional ups and downs due to the -sun and the stars. It is suggested that on the average, and in regions -remote from matter, the universe is not quite flat, but has a slight -curvature, analogous, in three dimensions, to the curvature of a sphere -in two dimensions.</p> - -<p>It is important to realize, in the first place, that there is not -the slightest reason <i>à priori</i> why this should not be the -case. People unaccustomed to non-Euclidean geometry may feel that, -even if such a thing be <i>logically</i> possible, the world simply -<i>cannot</i> be so odd as all that. We all have a tendency to think -that the world must conform to our prejudices. The opposite view -involves some effort of thought, and most people would die sooner than -think—in fact, they do so. But the fact that a spherical universe -seems odd to people who have been brought up on Euclidean prejudices is -no evidence that it is impossible. There is no law of nature to the -<span class="pagenum" id="Page_167">[Pg 167]</span> -effect that what is taught at school must be true. We cannot therefore -dismiss the hypothesis of a spherical universe as in any degree less -worthy of examination than any other. We have to ask ourselves the same -two questions as we should in any other case, namely: (1) Are the facts -consistent with this hypothesis? (2) Is this hypothesis the only one -with which the facts are consistent?</p> - -<p>With regard to the first question, the answer is undoubtedly in the -affirmative. All the known facts are perfectly consistent with the -hypothesis of a spherical universe. A very slight modification of the -law of gravitation—a modification suggested by Einstein himself—leads -to a spherical space, without producing any measurable differences in a -small region such as the solar system. The known stars are all within -a certain distance from us. There is nothing whatever in the stellar -universe as we know it to show that space must be infinite. There can -therefore be no doubt whatever that, so far as our present knowledge -goes, the hypothesis of a finite universe <i>may</i> be true.</p> - -<p>But when we ask whether the hypothesis of a finite universe -<span class="pagenum" id="Page_168">[Pg 168]</span> -<i>must</i> be true, the answer is different. It is obvious, on -general grounds, that we cannot, from what we know, draw conclusive -inferences as to the totality of things. A very slight change in the -Newtonian formula for gravitation would prevent masses beyond the -limits of the visible universe from having appreciable effects if they -existed, and would therefore destroy our reason for supposing that they -do not exist. All arguments as to regions which are too distant to be -observed depend upon extending to them the laws which hold in our part -of the world, and upon assuming that there is not, in these laws, some -inaccuracy which is inappreciable for observable distances, but fatal -to inferences in which very much greater distances are involved. We -cannot, therefore, say that the universe <i>must</i> be finite. We can -say that it may be, and we can even say a little more than this. We -can say that a finite universe fits in better with the laws that hold -in the part we know, and that awkward adjustments of the laws have -to be made in order to allow the universe to be infinite. From the -point of view of choosing the best framework into which to fit what we -know—best, I mean, from a logico-æsthetic point of view—there is no -<span class="pagenum" id="Page_169">[Pg 169]</span> -doubt that the hypothesis of a finite universe is preferable. This, I -think, is the extent of what can be said in its favor.</p> - -<p>Let us now see what the two finite universes are like. The difference -between them is that in Einstein’s world it is only space that -is queer, whereas in De Sitter’s time is queer too. Consequently -Einstein’s world is less puzzling, and we will describe it first.</p> - -<p>In Einstein’s world, light travels round the whole universe in a time -which is supposed to be something like a thousand million years. The -odd thing is that all the rays of light which start (say) from the sun -will meet again, after their enormous journey, in the place where the -sun was when they started. The case is exactly analogous to that of a -number of travelers who set out from London to go round the world in -great circles, all traveling at the same rate in different aeroplanes. -One starts due north, passes the North Pole, then the South Pole, and -finally comes home. Another starts due south, reaches the South Pole -first and then the North Pole. Another starts westward, but he must not -continue to travel due west, because then he would not be traveling on -<span class="pagenum" id="Page_170">[Pg 170]</span> -a great circle. Another starts eastward, and so on. They all meet in -the antipodes of London, and then they all meet again in London. Now -if instead of aeronauts going round the earth you take rays of light -going round the universe, the same sort of thing happens: they all meet -first at the antipodes of their starting point, and then meet again at -their starting point. That means to say that a person who is near the -antipodes of the place where the sun was about five hundred million -years ago will see what is apparently a body as bright as the sun then -was (except for the small amount of light that has been stopped on the -way by opaque bodies), and having the same shape and size. And a person -who is near where the sun was a thousand million years ago will see -what is apparently a body just like what the sun was a thousand million -years ago. And the same applies to the antipodes of the sun fifteen -hundred million years ago, and to the place of the sun two thousand -million years ago, and so on. This series only ends when it carries us -back to a time before the sun existed.</p> - -<p>But all these suns are only ghosts; that is to say, you could pass -through them without experiencing resistance, and they do not exert -<span class="pagenum" id="Page_171">[Pg 171]</span> -gravitation. They are, in fact, like images in a mirror: they exist -only for the sense of sight, not for any other sense. It is rather -disturbing to reflect that, if this theory is true, any number of the -objects we see in the heavens may be merely ghosts. They are like -ghosts in their habit of revisiting the scenes of their past life. -Suppose a star had exploded at a certain place, as stars sometimes -will. Every thousand million years its ghost would return to the scene -of the disaster and explode again in the same place. There is, however, -considerable doubt whether rays of light could perform the journey with -sufficient accuracy to produce a clear image. Some would be stopped by -matter on the way, some would be turned out of the straight course by -passing near heavy bodies, as in the eclipse observations described in -<a href="#Page_131">Chapter IX</a>, and for one reason or another their -return would not be punctual and exact.</p> - -<p>There are various reasons for doubting whether Einstein’s universe can -be quite right.<a id="FNanchor_12" href="#Footnote_12" class="fnanchor">[12]</a> - Some of these are rather complicated. But there -is one objection which is easily appreciated: in Einstein’s theory, -<span class="pagenum" id="Page_172">[Pg 172]</span> -absolute space and time re-enter by another door. The ghostly sun -is formed in the “place” where it was a thousand million years ago. -Both the “place” and the period of time are in a sense absolute. We -saw as early as Chapter I that “place” is a vague and popular notion, -incapable of scientific precision. It seems hardly worth while to go -through such a vast intellectual labor if the errors we set out to -correct are to reappear at the end.</p> - -<p>De Sitter’s world is even odder than Einstein’s, because time goes -mad as well as space. I despair of explaining, in non-mathematical -language, the particular form of lunacy with which time is afflicted, -but some of its manifestations can be described. An observer in this -world, if he observes a number of clocks, each of which is perfectly -accurate from its own point of view, will think that distant clocks -are going slow as compared with those in his neighborhood. They will -seem to go slower and slower, until, at a distance of one quarter of -the circumference of the universe, they will seem to have stopped -altogether. That region will seem to our observer a sort of lotus -<span class="pagenum" id="Page_173">[Pg 173]</span> -land, where nothing is ever done. He will not be able to have any -cognizance of things farther off, because no light waves can get across -the boundary. Not that there is any real boundary: the people who live -in what our observer takes to be lotus land live just as bustling a -life as he does, but get the impression that he is eternally standing -still. As a matter of fact, you would never become aware of the lotus -land, because it would take an infinite time for light to travel from -it to you. You could become aware of places just short of it, but it -would remain itself always just beyond your ken. There will not be the -ghostly suns of Einstein’s world, because light cannot travel so far.</p> - -<p>One of the oddest things about this state of affairs is that empirical -evidence for or against it is possible, and that there is actually -some slight evidence in its favor. If all “clocks” are slowed down at -a great distance from the observer, this will apply to the periodic -motions of atoms, and therefore to the light which they emit. -Consequently all rays of light emitted by distant objects ought, when -they reach us, to look rather more red or less violet than when they -started. This can be tested by the spectroscope. We can compare a -<span class="pagenum" id="Page_174">[Pg 174]</span> -known line, as it appears in the spectrum of a spiral nebula, with -the same line as it appears in a terrestrial laboratory. We find, as a -matter of fact, that in a large majority of spiral nebulæ there is a -considerable displacement of spectral lines towards the red. The spiral -nebulæ are the most distant objects we can see: Eddington states that -their distances “may perhaps be of the order of a million light-years.” -(A light-year is the distance light travels in a year.) The usual -interpretation of a shifting of spectral lines towards the red is that -it is a “Doppler effect,” due to the fact that the source of light is -moving away from us. But one would expect to find the nebulæ just as -often moving towards us as moving away from us, if nothing operated but -the law of chances. If the world is such as De Sitter says it is, the -spectral lines of the spiral nebulæ will be displaced towards the red -owing to the slowing down of distant clocks, even if in fact they are -not moving away from us. This, for what it is worth, is an argument in -favor of De Sitter.</p> - -<p>The same facts afford another argument in favor of De Sitter, for -another reason. If, at a given moment, a body is at rest relatively to -<span class="pagenum" id="Page_175">[Pg 175]</span> -the observer, and at a distance from him, it will (in the absence of -counteracting causes) not remain at rest from his point of view, but -will begin to move away from him, and will continue to move away faster -and faster; the further it is from him, the more its retreat will be -accelerated. For bodies which are not too distant from each other, -gravitation may overcome this tendency; but as this tendency increases -with the distance, while gravitation diminishes, we should expect -to find very distant bodies receding from us if De Sitter’s theory -is right. Thus we have two reasons for the displacement of spectral -lines in spiral nebulæ: one, the slowing down of time; the other, the -movement away from us which we should expect at distances too great -for gravitation to be sensible. However, it cannot be said that the -argument, on either ground, is very strong. Eddington gives a list -of forty-one spiral nebulæ, of which five have their spectral lines -shifted towards the violet, not towards the red. Thus the material is -neither very copious nor quite harmonious.</p> - -<p>Einstein’s and De Sitter’s hypotheses do not exhaust the possibilities -of a finite world: they are merely the two simplest forms of such a -<span class="pagenum" id="Page_176">[Pg 176]</span> -world. There are arguments against each, and it hardly seems probable -that either is quite true. But it does seem probable that something -more or less analogous is true. If the universe is finite, it is -theoretically conceivable that there should be a complete inventory -of it. We may be coming to the end of what physics can do in the way -of stretching the imagination and systematizing the world. The period -since Galileo has been essentially the period of physics, as the age of -the Greeks was the period of geometry. It may be that physics will lose -its attractions through success: if the fundamental laws of physics -come to be fully known, adventurous and inquiring intellects will turn -to other fields. This may alter profoundly the whole texture of human -life, since our present absorption in machinery and industrialism is -the reflection in the practical world of the theorist’s interest in -physical laws. But such speculations are even more rash than those of -De Sitter, and I do not wish to lay any stress upon them.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_177">[Pg 177]</span></p> -<h2 class="nobreak">CHAPTER XII:<br /> CONVENTIONS AND NATURAL LAWS</h2> -</div> - -<p class="drop-cap"><span class="smcap">One</span> of the most -difficult matters in all controversy is to distinguish disputes about -words from disputes about facts: it ought not to be difficult, but in -practice it is. This is quite as true in physics as in other subjects. -In the seventeenth century there was a terrific debate as to what -“force” is; to us now, it was obviously a debate as to how the word -“force” should be defined, but at the time it was thought to be much -more. One of the purposes of the method of tensors, which is employed -in the mathematics of relativity, is to eliminate what is purely verbal -(in an extended sense) in physical laws. It is of course obvious that -what depends on the choice of co-ordinates is “verbal” in the sense -concerned. A man punting walks along the boat, but keeps a constant -position with reference to the river bed so long as he does not pick up -his pole. The Lilliputians might debate endlessly whether he is walking -<span class="pagenum" id="Page_178">[Pg 178]</span> -or standing still: the debate would be as to words, not as to facts. -If we choose co-ordinates fixed relatively to the boat, he is walking; -if we choose co-ordinates fixed relatively to the river bed, he is -standing still. We want to express physical laws in such a way that -it shall be obvious when we are expressing the same law by reference -to two different systems of co-ordinates, so that we shall not be -misled into supposing we have different laws when we only have one law -in different words. This is accomplished by the method of tensors. -Some laws which seem plausible in one language cannot be translated -into another; these are impossible as laws of nature. The laws that -can be translated into <i>any</i> co-ordinate language have certain -characteristics: this is a substantial help in looking for such laws of -nature as the theory of relativity can admit to be possible. Combined -with what we know of the actual motions of bodies, it enables us to -decide what must be the correct expression of the law of gravitation: -logic and experience combine in equal proportions in obtaining this -expression.</p> - -<p>But the problem of arriving at genuine laws of nature is not to be -solved by the method of tensors alone; a good, deal of careful thought -<span class="pagenum" id="Page_179">[Pg 179]</span> -is wanted in addition. Some of this has been done, especially by -Eddington; much remains to be done.</p> - -<p>To take a simple illustration: Suppose, as in the hypothesis of the -Fitzgerald contraction, that lengths in one direction were shorter than -in another. Let us assume that a foot rule pointing north is only half -as long as the same foot rule pointing east, and that this is equally -true of all other bodies. Does such an hypothesis have any meaning? -If you have a fishing rod fifteen feet long when it is pointing west, -and you then turn it to the north, it will still measure fifteen feet, -because your foot rule will have shrunk too. It won’t “look” any -shorter, because your eye will have been affected in the same way. If -you are to find out the change, it cannot be by ordinary measurement; -it must be by some such method as the Michelson-Morley experiment, in -which the velocity of light is used to measure lengths. Then you still -have to decide whether it is simpler to suppose a change of length -or a change in the velocity of light. The experimental fact would be -that light takes longer to traverse what your foot rule declares to -<span class="pagenum" id="Page_180">[Pg 180]</span> -be a given distance in one direction than in another—or, as in the -Michelson-Morley experiment, that it ought to take longer but doesn’t. -You can adjust your measures to such a fact in various ways; in any -way you choose to adopt, there will be an element of convention. This -element of convention survives in the laws that you arrive at after -you have made your decision as to measures, and often it takes subtle -and elusive forms. To eliminate the element of convention is, in fact, -extraordinarily difficult; the more the subject is studied, the greater -the difficulty is seen to be.</p> - -<p>A more important example is the question of the size and shape of the -electron. We find experimentally that all electrons are the same size, -and that they are symmetrical in all directions. How far is this a -genuine fact ascertained by experiment, and how far is it a result of -our conventions of measurement? We have here a number of different -comparisons to make: (1) between different directions in regard to one -electron at one time; (2) in regard to one electron at different times; -(3) in regard to two electrons at the same time. We can then arrive -at the comparison of two electrons at different times, by combining -<span class="pagenum" id="Page_181">[Pg 181]</span> -(2) and (3). We may dismiss any hypothesis which would affect all -electrons equally; for example, it would be useless to suppose that in -one region of space-time they were all larger than in another. Such a -change would affect our measuring appliances just as much as the things -measured, and would therefore produce no discoverable phenomena. This -is as much as to say that it would be no change at all. But the fact -that two electrons have the same mass, for instance, cannot be regarded -as purely conventional. Given sufficient minuteness and accuracy, we -could compare the effects of two different electrons upon a third; -if they were equal under like circumstances, we should be able to -infer equality in a not purely conventional sense. The question of -the symmetry of the forces exerted by an electron—<i>i.e.</i>, that -these forces depend only upon the distance from the electron, and not -upon the direction—is more complicated. Eddington finally comes to -the conclusion that this, too, is a matter of convention. The argument -is difficult and I have not fully understood it; but I feel some -hesitation in accepting it as valid.</p> - -<p>Eddington describes the process concerned in the more advanced portions -of the theory of relativity as “world-building.” The structure to be -<span class="pagenum" id="Page_182">[Pg 182]</span> -built is the physical world as we know it; the economical architect -tries to construct it with the smallest possible amount of material. -This is a question for logic and mathematics. The greater our technical -skill in these two subjects, the more real building we shall do, and -the less we shall be content with mere heaps of stones. But before we -can use in our building the stones that nature provides, we have to -hew them into the right shapes: this is all part of the process of -budding. In order that this may be possible, the raw material must -have <i>some</i> structure (which we may conceive as analogous to the -grain in timber), but almost any structure will do. By successive -mathematical refinements, we whittle away our initial requirements -until they amount to very little. Given this necessary minimum of -structure in the raw material, we find that we can construct from it a -mathematical expression which will have the properties that are needed -for describing the world we perceive—in particular, the properties -of conservation which are characteristic of momentum and energy (or -mass). Our raw material consisted merely of events; but when we find -<span class="pagenum" id="Page_183">[Pg 183]</span> -that we can build out of it something which, as measured, will seem -to be never created or destroyed, it seems not surprising that we -should come to believe in “bodies.” These are really mere mathematical -constructions out of events, but owing to their permanence they are -practically important, and our senses (which were presumably developed -by biological needs) are adapted for noticing them, rather than the -crude continuum of events which is theoretically more fundamental. From -this point of view, it is astonishing how little of the real world is -revealed by physical science: our knowledge is limited, not only by the -conventional element, but also by the selectiveness of our perceptual -apparatus.</p> - -<p>We assume that there is an “interval” between two events, in the -sense explained in <a href="#Page_91">Chapter VII</a>, but we no longer assume that -we can unambiguously compare the length of an interval in one region with the -length of an interval in another. It is assumed by Weyl, who introduced -this limitation, that we can compare a number of small intervals which -all start from the same point; also that, in a very small journey, -our measuring rod will not alter its length much, so that there will -<span class="pagenum" id="Page_184">[Pg 184]</span> -only be a small error if we compare lengths in neighboring places by -the usual methods. Weyl found that, by diminishing our assumptions as -to interval in this way, it was possible to bring electromagnetism -and gravitation into one system. The mathematics of Weyl’s theory is -complicated, and I shall not attempt to explain it. For the present, -I am concerned with a different consequence of his theory. If lengths -in different regions cannot be compared directly, there is an element -of convention in the indirect comparisons which we actually make. This -element will be at first unrecognized, but will be such as to simplify -to the utmost the expression of the laws of nature. In particular, -conditions of symmetry may be entirely created by conventions as to -measurement, and there is no reason to suppose that they represent any -property of the real world. The law of gravitation itself, according to -Eddington, may be regarded as expressing conventions of measurement. -“The conventions of measurement,” he says, “introduce an isotropy<a id="FNanchor_13" href="#Footnote_13" class="fnanchor">[13]</a> -and homogeneity into measured space which need not originally have any -counterpart in the relation-structure which is being surveyed. This -isotropy and homogeneity is exactly expressed by Einstein’s law of -gravitation.”<a id="FNanchor_14" href="#Footnote_14" class="fnanchor">[14]</a></p> - -<p><span class="pagenum" id="Page_185">[Pg 185]</span> -The limitations of knowledge introduced by the selectiveness of our -perceptual apparatus may be illustrated by the indestructibility -of matter. This has been gradually discovered by experiment, and -seemed a well-founded empirical law of nature. Now it turns out -that, from our original space-time continuum, we can construct a -mathematical expression which will have properties causing it to appear -indestructible. The statement that matter is indestructible then ceases -to be a proposition of physics, and becomes instead a proposition -of linguistics and psychology. As a proposition of linguistics: -“Matter” is the name of the mathematical expression in question. As a -proposition of psychology: Our senses are such that we notice what is -roughly the mathematical expression in question, and we are led nearer -and nearer to it as we refine upon our crude perceptions by scientific -observation. This is much less than physicists used to think they knew -about matter.</p> - -<p>The reader may say: What then is left of physics? What do we really -<span class="pagenum" id="Page_186">[Pg 186]</span> -know about the world of matter? Here we may distinguish three -departments of physics. There is first what is included within the -theory of relativity, generalized as widely as possible. Next, there -are laws which cannot be brought within the scope of relativity. -Thirdly, there is what may be called geography. Let us consider each -of these in turn.</p> - -<p>The theory of relativity, apart from convention, tells us that the -events in the universe have a four-dimensional order, and that, -between any two events which are near together in this order, there -is a relation called “interval,” which is capable of being measured -if suitable precautions are taken. We make also an assumption as to -what happens when a little measuring rod is carried round a closed -circuit in a certain manner; the consequences of this assumption are -such as to make it highly probable that it is true. Beyond this, there -is little in the theory of relativity that can be regarded as physical -laws. There is a great deal of mathematics, showing that certain -mathematically-constructed quantities must behave like the things we -perceive; and there is a suggestion of a bridge between psychology and -<span class="pagenum" id="Page_187">[Pg 187]</span> -physics in the theory that these mathematically-constructed quantities -are what our senses are adapted for perceiving. But neither of these -things is physics in the strict sense.</p> - -<p>The part of physics which cannot, at present, be brought within -the scope of relativity is large and important. There is nothing -in relativity to show why there should be electrons and protons; -relativity cannot give any reason why matter should exist in little -lumps. With this goes the whole theory of the structure of the atom. -The theory of quanta also is quite outside the scope of relativity. -Relativity is, in a sense, the most extreme application of what may -be called next-to-next methods. Gravitation is no longer regarded -as due to the effect of the sun upon a planet, but as expressing -characteristics of the region in which the planet happens to be. -Distance, which used to be thought to have a definite meaning however -far apart two points might be, is now only definite for neighboring -points. The distance between widely separated places depends upon the -route chosen. We may, it is true, define <i>the</i> distance as the -geodesic distance, but that can only be estimated by adding up little -<span class="pagenum" id="Page_188">[Pg 188]</span> -bits, that is to say, by the method we use in estimating the length of -a curve. What applies to distance applies equally to the straight line. -There is nothing in the actual world having exactly the properties -that straight lines were supposed to have; the nearest approach is the -track of a light ray. Straight lines have to be replaced by geodesics, -which are defined by what they do at each point, not all at once, -like Euclidean straight lines. Measurement, in Weyl’s theory, suffers -the same fate. We can only use a measuring rod to give lengths in one -place: when we move it to another region, there is no knowing how it -will alter. We do assume, however, that, if it alters, it alters bit -by bit, gradually, continuously, and not by sudden jumps. Perhaps -this assumption is unjustified. It belongs to the general outlook of -relativity, which is that of continuity. No doubt it is owing to this -outlook that relativity is unable to account for the discontinuities in -physics, such as quanta, electrons and protons. Perhaps relativity will -conquer these domains when it learns to dispense with the assumption of -continuity.</p> - -<p>Finally we come to geography, in which I include history. The -separation of history from geography rests upon the separation of time -<span class="pagenum" id="Page_189">[Pg 189]</span> -from space; when we amalgamate the two in space-time, we need one word -to describe the combination of geography and history. For the sake of -simplicity, I shall use the one word geography in this extended sense.</p> - -<p>Geography, in this sense, includes everything that, as a matter of -crude fact, distinguishes one part of space-time from another. One -part is occupied by the sun, one by the earth; the intermediate -regions contain light waves, but no matter (apart from a very little -here and there). There is a certain degree of theoretical connection -between different geographical facts; to establish this is the purpose -of physical laws. It is thought that a sufficient knowledge of the -geographical facts of the solar system throughout any finite time, -however short, would enable an ideally competent physicist to predict -the future of the solar system so long as it remained remote from other -stars. We are already in a position to calculate the large facts about -the solar system backwards and forwards for vast periods of time. But -in all such calculations we need a basis of crude fact. The facts are -interconnected, but facts can only be inferred from other facts, not -<span class="pagenum" id="Page_190">[Pg 190]</span> -from general laws alone. Thus the facts of geography have a certain -independent status in physics. No amount of physical laws will enable -us to infer a physical fact unless we know other facts as data for our -inference. And here when I speak of “facts” I am thinking of particular -facts of geography, in the extended sense in which I am using the term.</p> - -<p>In the theory of relativity, we are concerned with <i>structure</i>, -not with the material of which the structure is composed. In geography, -on the other hand, the material is relevant. If there is to be any -difference between one place and another, there must either be -differences between the material in one place and that in another, or -places where there is material and places where there is none. The -former of these alternatives seems the more satisfactory. We might -try to say: There are electrons and protons, and the rest is empty. -But in the “empty” regions there are light waves, so that we cannot -say nothing happens in them. Some people maintain that the light -waves take place in the ether, others are content to say simply that -they take place; but in any case events are occurring where there are -light waves. That is all that we can really say for the places where -<span class="pagenum" id="Page_191">[Pg 191]</span> -there is matter, since matter has turned out to be a mathematical -construction built out of events. We may say, therefore, that there -are events everywhere in space-time, but they must be of a somewhat -different kind according as we are dealing with a region where there is -an electron or proton or with the sort of region we should ordinarily -call empty. But as to the intrinsic nature of these events we can know -nothing, except when they happen to be events in our own lives. Our own -perceptions and feelings must be part of the crude material of events -which physics arranges into a pattern—or rather, which physics finds -to be arranged in a pattern. As regards events which do not form part -of our own lives, physics tells us the pattern of them, but is quite -unable to tell us what they are like in themselves. Nor does it seem -possible that this should be discovered by any other method.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_192">[Pg 192]</span></p> -<h2 class="nobreak">CHAPTER XIII:<br /> THE ABOLITION OF “FORCE”</h2> -</div> - -<p class="drop-cap"><span class="smcap">In the</span> Newtonian system, -bodies under the action of no forces move in straight lines with -uniform velocity; when bodies do not move in this way, their change of -motion is ascribed to a “force.” Some forces seem intelligible to our -imagination: those exerted by a rope or string, by bodies colliding, or -by any kind of obvious pushing or pulling. As explained in an earlier -chapter, our apparent imaginative understanding of these processes is -quite fallacious; all that it really means is that past experience -enables us to foresee more or less what is going to happen without -the need of mathematical calculations. But the “forces” involved in -gravitation and in the less familiar forms of electrical action do not -seem in this way “natural” to our imagination. It seems odd that the -earth can float in the void: the natural thing to suppose is that it -must fall. That is why it has to be supported on an elephant, and the -<span class="pagenum" id="Page_193">[Pg 193]</span> -elephant on a tortoise, according to some early speculators. The -Newtonian theory, in addition to action at a distance, introduced -two other imaginative novelties. The first was, that gravitation is -not always and essentially directed what we should call “downwards,” -<i>i.e.</i>, towards the center of the earth. The second was, that a -body going round and round in a circle with uniform velocity is not -“moving uniformly” in the sense in which that phrase is applied to the -motion of bodies under no forces, but is perpetually being turned out -of the straight course towards the center of the circle, which requires -a force pulling it in that direction. Hence Newton arrived at the view -that the planets are attracted to the sun by a force, which is called -gravitation.</p> - -<p>This whole point of view, as we have seen, is superseded by relativity. -There are no longer such things as “straight lines” in the old -geometrical sense. There are “straightest lines,” or geodesics, but -these involve time as well as space. A light ray passing through -the solar system does not describe the same orbit as a comet, from -a geometrical point of view; nevertheless each moves in a geodesic. -The whole imaginative picture is changed. A poet might say that water -<span class="pagenum" id="Page_194">[Pg 194]</span> -runs down hill because it is attracted to the sea, but a physicist or -an ordinary mortal would say that it moves as it does, at each point, -because of the nature of the ground at that point, without regard to -what lies ahead of it. Just as the sea does not cause the water to run -towards it, so the sun does not cause the planets to move round it. The -planets move round the sun because that is the easiest thing to do—in -the technical sense of “least action.” It is the easiest thing to do -because of the nature of the region in which they are, not because of -an influence emanating from the sun.</p> - -<p>The supposed necessity of attributing gravitation to a “force” -attracting the planets towards the sun has arisen from the -determination to preserve Euclidean geometry at all costs. If we -suppose that our space is Euclidean, when in fact it is not, we shall -have to call in physics to rectify the errors of our geometry. We shall -find bodies not moving in what we insist upon regarding as straight -lines, and we shall demand a cause for this behavior. Eddington has -stated this matter with admirable lucidity. He supposes a physicist -who has assumed the formula for interval which is used in the special -<span class="pagenum" id="Page_195">[Pg 195]</span> -theory of relativity—a formula which still supposes that the -observer’s space is Euclidean. He continues:</p> - -<p class="blockquot"> Since intervals can be compared by experimental -methods, he ought soon to discover that his (formula for the interval) -cannot be reconciled with observational results, and so realize his -mistake. But the mind does not so readily get rid of an obsession. It -is more likely that our observer will continue in his opinion, and -attribute the discrepancy of the observations to some influence which -is present and affects the behavior of his test-bodies. He will, so -to speak, introduce a supernatural agency which he can blame for the -consequences of his mistake.... The name given to any agency which -causes deviation from uniform motion in a straight line is <i>force</i> -according to the Newtonian definition of force. Hence the agency -invoked through our observer’s mistake is described as a “field of -force.”... <i>A field of force represents the discrepancy -between the naturalgeometry of a co-ordinate system and the -abstractgeometry arbitrarily ascribed to it.</i><a id="FNanchor_15" href="#Footnote_15" class="fnanchor">[15]</a></p> - -<p>If people were to learn to conceive the world in the new way, without -the old notion of “force,” it would alter not only their physical -imagination, but probably also their morals and politics. The latter -<span class="pagenum" id="Page_196">[Pg 196]</span> -effect would be quite illogical, but is none the less probable on that -account. In Newton’s theory of the solar system, the sun seems like a -monarch whose behests the planets have to obey. In Einstein’s world -there is more individualism and less government than in Newton’s. -There is also far less hustle: we have seen that laziness is the -fundamental law of Einstein’s universe. The word “dynamic” has come to -mean, in newspaper language, “energetic and forceful”; but if it meant -“illustrating the principles of dynamics,” it ought to be applied to -the people in hot climates who sit under banana trees waiting for the -fruit to drop into their mouths. I hope that journalists, in future, -when they speak of a “dynamic personality,” will mean a person who -does what is least trouble at the moment, without thinking of remote -consequences. If I can contribute to this result, I shall not have -written in vain.</p> - -<p>It has been customary for people to draw arguments from the laws of -nature as to what we ought to do. Such arguments seem to me a mistake: -to imitate nature may be merely slavish. But if nature, as portrayed by -Einstein, is to be our model, it would seem that the anarchists will -<span class="pagenum" id="Page_197">[Pg 197]</span> -have the best of the argument. The physical universe is orderly, not -because there is a central government, but because every body minds -its own business. No two particles of matter ever come into contact; -when they get too close, they both move off. If a man were had up -for knocking another man down, he would be scientifically correct in -pleading that he had never touched him. What happened was that there -was a hill in space-time in the region of the other man’s nose, and it -fell down the hill.</p> - -<p>The abolition of “force” seems to be connected with the substitution -of sight for touch as the source of physical ideas, as explained in -<a href="#Page_1">Chapter I</a>. When an image in a looking glass moves, we do not -think that something has pushed it. In places where there are two large mirrors -opposite to each other, you may see innumerable reflections of the -same object. Suppose a gentleman in a top-hat is standing between the -mirrors, there may be twenty or thirty top-hats in the reflections. -Suppose now somebody comes and knocks off the gentleman’s hat with a -stick: all the other twenty or thirty top-hats will tumble down at the -same moment. We think that a force is needed to knock off the “real” -<span class="pagenum" id="Page_198">[Pg 198]</span> -top-hat, but we think the remaining twenty or thirty tumble off, so to -speak, of themselves, or out of a mere passion for imitation. Let us -try to think out this matter a little more seriously.</p> - -<p>Obviously something happens when an image in a looking glass moves. -From the point of view of sight, the event seems just as real as if it -were not in a mirror. But nothing has happened from the point of view -of touch or hearing. When the “real” top-hat falls, it makes a noise; -the twenty or thirty reflections fall without a sound. If it falls on -your toe, you feel it; but we believe that the twenty or thirty people -in the mirrors feel nothing, though top-hats fall on their toes too. -But all this is equally true of the astronomical world. It makes no -noise, because sound cannot travel across a vacuum. So far as we know, -it causes no “feelings,” because there is no one on the spot to “feel” -it. The astronomical world, therefore, seems hardly more “real” or -“solid” than the world in the looking glass, and has just as little -need of “force” to make it move.</p> - -<p>The reader may feel that I am indulging in idle sophistry. “After all,” -he may say, “the image in the mirror is the reflection of something -<span class="pagenum" id="Page_199">[Pg 199]</span> -solid, and the top-hat in the mirror only falls off because of the -force applied to the real top-hat. The top-hat in the mirror cannot -indulge in behavior of its own; it has to copy the real one. This -shows how different the image is from the sun and the planets, because -<i>they</i> are not obliged to be perpetually imitating a prototype. So -you had better give up pretending that an image is just as real as one -of the heavenly bodies.”</p> - -<p>There is, of course, some truth in this; the point is to discover -exactly <i>what</i> truth. In the first place, images are not -“imaginary.” When you see an image, certain perfectly real light waves -reach your eye; and if you hang a cloth over the mirror, these light -waves cease to exist. There is, however, a purely optical difference -between an “image” and a “real” thing. The optical difference is bound -up with this question of imitation. When you hang a cloth over the -mirror, it makes no difference to the “real” object; but when you move -the “real” object away, the image vanishes also. This makes us say that -the light rays which make the image are only reflected at the surface -of the mirror, and do not really come from a point behind it, but from -<span class="pagenum" id="Page_200">[Pg 200]</span> -the “real” object. We have here an example of a general principle of -great importance. Most of the events in the world are not isolated -occurrences, but members of groups of more or less similar events, -which are such that each group is connected in an assignable manner -with a certain small region of space-time. This is the case with the -light rays which make us see both the object and its reflection in the -mirror: they all emanate from the object as a center. If you put an -opaque globe round the object at a certain distance, the object and -its reflection are invisible at any point outside the globe. We have -seen that gravitation, although no longer regarded as an action at a -distance, is still connected with a center: there is, so to speak, a -hill symmetrically arranged about its summit, and the summit is the -place where we conceive the body to be which is connected with the -gravitational field we are considering. For simplicity, common sense -lumps together all the events which form one group in the above sense. -When two people see the same object, two different events occur, but -they are events belonging to one group and connected with the same -center. Just the same applies when two people (as we say) hear the -<span class="pagenum" id="Page_201">[Pg 201]</span> -same noise. And so the reflection in a mirror is less “real” than the -object reflected, even from an optical point of view, because light -rays do not spread in <i>all</i> directions from the place where the -image seems to be, but only in directions in front of the mirror, -and only so long as the object reflected remains in position. This -illustrates the usefulness of grouping connected events about a center -in the way we have been considering.</p> - -<p>When we examine the changes in such a group of objects, we find that -they are of two kinds: there are those which affect only some member -of the group, and those which make connected alterations in all the -members of the group. If you put a candle in front of a mirror, and -then hang black cloth over the mirror, you alter only the reflection -of the candle as seen from various places. If you shut your eyes, -you alter its appearance to you, but not its appearance elsewhere. -If you put a red globe round it at a distance of a foot, you alter -its appearance at any distance greater than a foot, but not at any -distance less than a foot. In all these cases, you do not regard the -candle itself as having changed; in fact, in all of them, you find that -<span class="pagenum" id="Page_202">[Pg 202]</span> -there are groups of changes connected with a different center or with -a number of different centers. When you shut your eyes, for instance, -your eyes, not the candle, look different to any other observer: the -center of the changes that occur is in your eyes. But when you blow out -the candle, its appearance <i>everywhere</i> is changed; in this case -you say that the change has happened to the candle. The changes that -happen to an object are those that affect the whole group of events -which center about the object. All this is only an interpretation of -common sense, and an attempt to explain what we mean by saying that the -image of the candle in the mirror is less “real” than the candle. There -is no connected group of events situated all round the place where the -image seems to be, and changes in the image center about the candle, -not about a point behind the mirror. This gives a perfectly verifiable -meaning to the statement that the image is “only” a reflection. And at -the same time it enables us to regard the heavenly bodies, although -we can only see and not touch them, as more “real” than an image in a -looking glass.</p> - -<p>We can now begin to interpret the common sense notion of one body -<span class="pagenum" id="Page_203">[Pg 203]</span> -having an “effect” upon another, which we must do if we are really to -understand what is meant by the abolition of “force.” Suppose you come -into a dark room and switch on the electric light: the appearance of -everything in the room is changed. Since everything in the room is -visible because it reflects the electric light, this case is really -analogous to that of the image in the mirror; the electric light is the -center from which all the changes emanate. In this case, the “effect” -is explained by what we have already said. The more important case is -when the effect is a movement. Suppose you let loose a tiger in the -middle of a Bank Holiday crowd: they would all move, and the tiger -would be the center of their various movements. A person who could -see the people but not the tiger would infer that there was something -repulsive at that point. We say in this case that the tiger has an -effect upon the people, and we might describe the tiger’s action upon -them as of the nature of a repulsive force. We know, however, that -they fly because of something which happens to <i>them</i>, not merely -because the tiger is where he is. They fly because they can see and -hear him, that is to say, because certain waves reach their eyes and -<span class="pagenum" id="Page_204">[Pg 204]</span> -ears. If these waves could be made to reach them without there being -any tiger, they would fly just as fast, because the neighborhood would -seem to them just as unpleasant.</p> - -<p>Let us now apply similar considerations to the sun’s gravitation. The -“force” exerted by the sun only differs from that exerted by the tiger -in being attractive instead of repulsive. Instead of acting through -waves of light or sound, the sun acquires its apparent power through -the fact that there are modifications of space-time all round the sun. -Like the noise of the tiger, they are more intense near their source; -as we travel away they grow less and less. To say that the sun “causes” -these modifications of space-time is to add nothing to our knowledge. -What we know is that the modifications proceed according to a certain -rule, and that they are grouped symmetrically about the sun as center. -The language of cause and effect adds only a number of quite irrelevant -imaginings, connected with will, muscular tension, and such matters. -What we can more or less ascertain is merely the formula according to -which space-time is modified by the presence of gravitating matter. -<span class="pagenum" id="Page_205">[Pg 205]</span> -More correctly: we can ascertain what kind of space-time <i>is</i> -the presence of gravitating matter. When space-time is not accurately -Euclidean in a certain region, but has a non-Euclidean character which -grows more and more marked as we approach a certain center, and when, -further, the departure from Euclid obeys a certain law, we describe -this state of affairs briefly by saying that there is gravitating -matter at the center. But this is only a compendious account of what -we know. What we know is about the places where the gravitating matter -is <i>not</i>, not about the place where it is. The language of cause -and effect (of which “force” is a particular case) is thus merely -a convenient shorthand for certain purposes; it does not represent -anything that is genuinely to be found in the physical world.</p> - -<p>And how about matter? Is matter also no more than a convenient -shorthand? This question, however, being a large one, demands a -separate chapter.</p> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_206">[Pg 206]</span></p> -<h2 class="nobreak">CHAPTER XIV:<br /> WHAT IS MATTER?</h2> -</div> - -<p class="drop-cap"><span class="smcap">The</span> question “What is -matter?” is of the kind that is asked by metaphysicians, and answered -in vast books of incredible obscurity. But I am not asking the question -as metaphysician: I am asking it as a person who wants to find out what -is the moral of modern physics, and more especially of the theory of -relativity. It is obvious from what we have learned of that theory that -matter cannot be conceived quite as it used to be. I think we can now -say more or less what the new conception must be.</p> - -<p>There were two traditional conceptions of matter, both of which have -had advocates ever since scientific speculation began. There were -the atomists, who thought that matter consisted of tiny lumps which -could never be divided; these were supposed to hit each other and then -bounce off in various ways. After Newton, they were no longer supposed -actually to come into contact with each other, but to attract and -<span class="pagenum" id="Page_207">[Pg 207]</span> -repel each other, and move in orbits round each other. Then there -were those who thought that there is matter of some kind everywhere, -and that a true vacuum is impossible. Descartes held this view, and -attributed the motions of the planets to vortices in the ether. The -Newtonian theory of gravitation caused the view that there is matter -everywhere to fall into discredit, the more so as light was thought by -Newton and his disciples to be due to actual particles traveling from -the source of the light. But when this view of light was disproved, and -it was shown that light consisted of waves, the ether was revived so -that there should be something to undulate. The ether became still more -respectable when it was found to play the same part in electromagnetic -phenomena as in the propagation of light. It was even hoped that atoms -might turn out to be a mode of motion of the ether. At this stage, the -atomic view of matter was, on the whole, getting the worst of it.</p> - -<p>Leaving relativity aside for the moment, modern physics has provided -proof of the atomic structure of ordinary matter, while not disproving -the arguments in favor of the ether, to which no such structure is -attributed. The result was a sort of compromise between the two views, -<span class="pagenum" id="Page_208">[Pg 208]</span> -the one applying to what was called “gross” matter, the other to the -ether. There can be no doubt about electrons and protons, though, as we -shall see shortly, they need not be conceived as atoms were conceived -traditionally. As for the ether, its status is very curious: many -physicists still maintain that, without it, the propagation of light -and other electromagnetic waves would be inconceivable, but except in -this way it is difficult to see what purpose it serves. The truth is, -I think, that relativity demands the abandonment of the old conception -of “matter,” which is infected by the metaphysics associated with -“substance,” and represents a point of view not really necessary in -dealing with phenomena. This is what we must now investigate.</p> - -<p>In the old view, a piece of matter was something which survived all -through time, while never being at more than one place at a given time. -This way of looking at things is obviously connected with the complete -separation of space and time in which people formerly believed. When we -substitute space-time for space and time, we shall naturally expect to -derive the physical world from constituents which are as limited in -<span class="pagenum" id="Page_209">[Pg 209]</span> -time as in space. Such constituents are what we call “events.” An event -does not persist and move, like the traditional piece of matter; it -merely exists for its little moment and then ceases. A piece of matter -will thus be resolved into a series of events. Just as, in the old -view, an extended body was composed of a number of particles, so, now, -each particle, being extended in time, must be regarded as composed -of what we may call “event-particles.” The whole series of these -events makes up the whole history of the particle, and the particle is -regarded as <i>being</i> its history, not some metaphysical entity to -which the events happen. This view is rendered necessary by the fact -that relativity compels us to place time and space more on a level than -they were in the older physics.</p> - -<p>This abstract requirement must be brought into relation with the known -facts of the physical world. Now what are the known facts? Let us -take it as conceded that light consists of waves traveling with the -received velocity. We then know a great deal about what goes on in -the parts of space-time where there is no matter; we know, that is to -say, that there are periodic occurrences (light waves) obeying certain -<span class="pagenum" id="Page_210">[Pg 210]</span> -laws. These light waves start from atoms, and the modern theory of -the structure of the atoms enables us to know a great deal about the -circumstances under which they start, and the reasons which determine -their wave lengths. We can find out not only how one light wave -travels, but how its source moves relatively to ourselves. But when I -say this I am assuming that we can recognise a source of light as the -same at two slightly different times. This is, however, the very thing -which had to be investigated.</p> - -<p>We saw, in the preceding chapter, how a group of connected events can -be formed, all related to each other by a law, and all ranged about a -center in space-time. Such a group of events will be the arrival, at -various places, of the light waves emitted by a brief flash of light. -We do not need to suppose that anything particular is happening at the -center; certainly we do not need to suppose that we know <i>what</i> -is happening there. What we know is that, as a matter of geometry, the -group of events in question are ranged about a center, like widening -ripples on a pool when a fly has touched it. We can hypothetically -invent an occurrence which is to have happened at the center, and set -<span class="pagenum" id="Page_211">[Pg 211]</span> -forth the laws by which the consequent disturbance is transmitted. This -hypothetical occurrence will then appear to common sense as the “cause” -of the disturbance. It will also count as one event in the biography of -the particle of matter which is supposed to occupy the center of the -disturbance.</p> - -<p>Now we find not only that one light wave travels outward from a center -according to a certain law, but also that, in general, it is followed -by other closely similar light waves. The sun, for example, does not -change its appearance suddenly; even if a cloud passes across it during -a high wind, the transition is gradual, though swift. In this way a -group of occurrences connected with a center at one point of space-time -is brought into relation with other very similar groups whose centers -are at neighboring points of space-time. For each of these other groups -common sense invents similar hypothetical occurrences to occupy their -centers, and says that all these hypothetical occurrences are part of -one history; that is to say, it invents a hypothetical “particle” to -which the hypothetical occurrences are to have occurred. It is only by -<span class="pagenum" id="Page_212">[Pg 212]</span> -this double use of hypothesis, perfectly unnecessary in each case, that -we arrive at anything that can be called “matter” in the old sense of -the word.</p> - -<p>If we are to avoid unnecessary hypotheses, we shall say that an -electron at a given moment is the various disturbances in the -surrounding medium which, in ordinary language, would be said to be -“caused” by it. But we shall not take these disturbances at what is, -for us, the moment in question, since that would make them depend -upon the observer; we shall instead travel outward from the electron -with the velocity of light, and take the disturbance we find in each -place as we reach it. The closely similar set of disturbances, with -very nearly the same center, which is found existing slightly earlier -or slightly later, will be defined as <i>being</i> the electron at a -slightly earlier or slightly later moment. In this way, we preserve all -the laws of physics, without having recourse to unnecessary hypotheses -or inferred entities, and we remain in harmony with the general -principle of economy which has enabled the theory of relativity to -clear away so much useless lumber.</p> - -<p>Common sense imagines that when it sees a table it sees a table. This -is a gross delusion. When common sense sees a table, certain light -<span class="pagenum" id="Page_213">[Pg 213]</span> -waves reach its eyes, and these are of a sort which, in its previous -experience, has been associated with certain sensations of touch, as -well as with other people’s testimony that they also saw the table. -But none of this ever brought us to the table itself. The light waves -caused occurrences in our eyes, and these caused occurrences in the -optic nerve, and these in turn caused occurrences in the brain. Any one -of these, happening without the usual preliminaries, would have caused -us to have the sensations we call “seeing the table,” even if there had -been no table. (Of course, if matter in general is to be interpreted -as a group of occurrences, this must apply also to the eye, the optic -nerve, and the brain.) As to the sense of touch when we press the table -with our fingers, that is an electric disturbance in the electrons and -protons of our finger tips, produced, according to modern physics, by -the proximity of the electrons and protons in the table. If the same -disturbance in our finger tips arose in any other way, we should have -the same sensations, in spite of there being no table. The testimony -of others is obviously a second-hand affair. A witness in a law court, -<span class="pagenum" id="Page_214">[Pg 214]</span> -if asked whether he had seen some occurrence, would not be allowed to -reply that he believed so because of the testimony of others to that -effect. In any case, testimony consists of sound waves and demands -psychological as well as physical interpretation; its connection with -the object is therefore very indirect. For all these reasons, when -we say that a man “sees a table,” we use a highly abbreviated form -of expression, concealing complicated and difficult inferences, the -validity of which may well be open to question.</p> - -<p>But we are in danger of becoming entangled in psychological questions, -which we must avoid if we can. Let us therefore return to the purely -physical point of view.</p> - -<p>What I wish to suggest may be put as follows. Everything that occurs -elsewhere, owing to the existence of an electron, can be explored -experimentally, at least in theory, unless it occurs in certain -concealed ways. But what occurs within the electron (if anything occurs -there) it is absolutely impossible to know: there is no conceivable -apparatus by which we could obtain even a glimpse of it. An electron is -known by its “effects.” But the word “effects” belongs to a view of -<span class="pagenum" id="Page_215">[Pg 215]</span> -causation which will not fit modern physics, and in particular will -not fit relativity. All that we have a right to say is that certain -groups of occurrences happen together, that is to say, in neighboring -parts of space-time. A given observer will regard one member of the -group as earlier than the other, but another observer may judge the -time order differently. And even when the time order is the same for -all observers, all that we really have is a connection between the two -events, which works equally backwards and forwards. It is not true that -the past determines the future in some sense other than that in which -the future determines the past: the apparent difference is only due to -our ignorance, because we know less about the future than about the -past. This is a mere accident: there might be beings who would remember -the future and have to infer the past. The feelings of such beings -in these matters would be the exact opposite of our own, but no more -fallacious.</p> - -<p>The moral of this is that, if an electron is only known by its -“effects,” there is no reason to suppose that anything exists except -the “effects.” In so far as these “effects” consist of light waves -<span class="pagenum" id="Page_216">[Pg 216]</span> -and other electromagnetic disturbances, we may say that what is -called “empty space” consists of regions where these disturbances are -propagated freely. Every such disturbance, we find, has a center, and -when we get very near the center (though still at a finite distance -from it) we find that the law of propagation of the disturbance ceases -to be valid. This region within which the law does not hold is called -“matter”; it will be an electron or proton according to circumstances. -The region so defined is found to move relatively to other such -regions, and its movements follow the known laws of dynamics. So far, -this theory provides for electromagnetic phenomena and the motions of -matter; and it does so without assuming that “matter” is anything but -systems of electromagnetic phenomena. In order to carry out the theory -fully, it would no doubt be necessary to introduce many complications. -But it seems fairly clear that all the facts and laws of physics -can be interpreted without assuming that “matter” is anything more -than groups of events, each event being of the sort which we should -naturally regard as “caused” by the matter in question. This does not -<span class="pagenum" id="Page_217">[Pg 217]</span> -involve any change in the symbols or formulæ of physics: it is merely -a question of interpretation of the symbols.</p> - -<p>This latitude in interpretation is a characteristic of mathematical -physics. What we know is certain very abstract logical relations, -which we express in mathematical formulæ; we know also that, at -certain points, we arrive at results which are capable of being tested -experimentally. Take, for example, the eclipse observations by which -Einstein’s theory as to the bending of light was established. The -actual observation consisted in the careful measurement of certain -distances on certain photographic plates. The formulæ which were to -be verified were concerned with the course of light in passing near -the sun. Although the part of these formulæ which gives the observed -result must always be interpreted in the same way, the other part of -them may be capable of a great variety of interpretations. The formulæ -giving the motions of the planets are almost exactly the same in -Einstein’s theory as in Newton’s, but the meaning of the formulæ is -quite different. It may be said generally that, in the mathematical -treatment of nature, we can be far more certain that our formulæ are -<span class="pagenum" id="Page_218">[Pg 218]</span> -approximately correct than we can be as to the correctness of this or -that interpretation of them. And so in the case with which this chapter -is concerned: the question as to the nature of an electron or a proton -is by no means answered when we know all that mathematical physics has -to say as to the laws of its motion and the laws of its interaction -with the environment. A definite and conclusive answer to our question -is not possible just because a variety of answers are compatible -with the truth of mathematical physics. Nevertheless some answers -are preferable to others, because some have a greater probability in -their favor. We have been seeking, in this chapter, to define matter -so that there <i>must</i> be such a thing if the formulæ of physics -are true. If we had made our definition such as to secure that a -particle of matter should be what one thinks of as substantial, a hard, -definite lump, we should not have been <i>sure</i> that any such thing -exists. That is why our definition, though it may seem complicated, is -preferable from the point of view of logical economy and scientific caution.</p> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_219">[Pg 219]</span></p> -<h2 class="nobreak">CHAPTER XV:<br /> PHILOSOPHICAL CONSEQUENCES</h2> -</div> - -<p class="drop-cap"><span class="smcap">The</span> philosophical -consequences of relativity are neither so great nor so startling as -is sometimes thought. It throws very little light on time-honored -controversies, such as that between realism and idealism. Some people -think that it supports Kant’s view that space and time are “subjective” -and are “forms of intuition.” I think such people have been misled by -the way in which writers on relativity speak of “the observer.” It is -natural to suppose that the observer is a human being, or at least a -mind; but he is just as likely to be a photographic plate or a clock. -That is to say, the odd results as to the difference between one “point -of view” and another are concerned with “point of view” in a sense -applicable to physical instruments just as much as to people with -<span class="pagenum" id="Page_220">[Pg 220]</span> -perceptions. The “subjectivity” concerned in the theory of relativity -is a <i>physical</i> subjectivity, which would exist equally if there -were no such things as minds or senses in the world.</p> - -<p>Moreover, it is a strictly limited subjectivity. The theory does not -say that <i>everything</i> is relative; on the contrary, it gives a -technique for distinguishing what is relative from what belongs to a -physical occurrence in its own right. If we are going to say that the -theory supports Kant about space and time, we shall have to say that it -refutes him about space-time. In my view, neither statement is correct. -I see no reason why, on such issues, philosophers should not all stick -to the views they previously held. There were no conclusive arguments -on either side before, and there are none now; to hold either view -shows a dogmatic rather than a scientific temper.</p> - -<p>Nevertheless, when the ideas involved in Einstein’s work have become -familiar, as they will when they are taught in schools, certain changes -in our habits of thought are likely to result, and to have great -importance in the long run.</p> - -<p>One thing which emerges is that physics tells us much less about -the physical world than we thought it did. Almost all the “great -principles” of traditional physics turn out to be like the “great -<span class="pagenum" id="Page_221">[Pg 221]</span> -law” that there are always three feet to a yard; others turn out to -be downright false. The conservation of mass may serve to illustrate -both these misfortunes to which a “law” is liable. Mass used to be -defined as “quantity of matter,” and as far as experiment showed it -was never increased or diminished. But with the greater accuracy of -modern measurements, curious things were found to happen. In the first -place, the mass as measured was found to increase with the velocity; -this kind of mass was found to be really the same thing as energy. This -kind of mass is not constant for a given body, but the total amount of -it in the universe is conserved, or at least obeys a law very closely -analogous to conservation. This law itself, however, is to be regarded -as a truism, of the nature of the “law” that there are three feet to a -yard; it results from our methods of measurement, and does not express -a genuine property of matter. The other kind of mass, which we may call -“proper mass,” is that which is found to be the mass by an observer -moving with the body. This is the ordinary terrestrial case, where -the body we are weighing is not flying through the air. The “proper -<span class="pagenum" id="Page_222">[Pg 222]</span> -mass” of a body is very nearly constant, but not quite, and the total -amount of “proper mass” in the world is not quite constant. One would -suppose that if you have four one-pound weights, and you put them all -together into the scales, they will together weigh four pounds. This is -a fond delusion: they weigh rather less, though not enough less to be -discovered by even the most careful measurements. In the case of four -hydrogen atoms, however, when they are put together to make one helium -atom, the defect is noticeable; the helium atom weighs measurably less -than four separate hydrogen atoms.</p> - -<p>Broadly speaking, traditional physics has collapsed into two portions, -truisms and geography. There are, however, newer portions of physics, -such as the theory of quanta, which do not come under this head, but -appear to give genuine knowledge of laws reached by experiment.</p> - -<p>The world which the theory of relativity presents to our imagination -is not so much a world of “things” in “motion” as a world of -<i>events</i>. It is true that there are still electrons and protons -which persist, but these (as we saw in the preceding chapter) are -really to be conceived as strings of connected events, like the -successive notes of a song. It is <i>events</i> that are the stuff of -<span class="pagenum" id="Page_223">[Pg 223]</span> -relativity physics. Between two events which are not too remote from -each other there is, in the general theory as in the special theory, a -measurable relation called “interval,” which appears to be the physical -reality of which lapse of time and distance in space are two more or -less confused representations. Between two distant events, there is -not any one definite interval. But there is one way of moving from -one event to another which makes the sum of all the little intervals -along the route greater than by any other route. This route is called -a “geodesic,” and it is the route which a body will choose if left to -itself.</p> - -<p>The whole of relativity physics is a much more step-by-step matter than -the physics and geometry of former days. Euclid’s straight lines have -to be replaced by light rays, which do not quite come up to Euclid’s -standard of straightness when they pass near the sun or any other very -heavy body. The sum of the angles of a triangle is still thought to be -two right angles in very remote regions of empty space, but not where -there is matter in the neighborhood. We, who cannot leave the earth, -are incapable of reaching a place where Euclid is true. Propositions -<span class="pagenum" id="Page_224">[Pg 224]</span> -which used to be proved by reasoning have now become either -conventions, or merely approximate truths verified by observation.</p> - -<p>It is a curious fact—of which relativity is not the only -illustration—that, as reasoning improves, its claims to the power -of proving facts grow less and less. Logic used to be thought to teach -us how to draw inferences; now, it teaches us rather how not to draw -inferences. Animals and children are terribly prone to inference: a -horse is surprised beyond measure if you take an unusual turning. When -men began to reason, they tried to justify the inferences that they -had drawn unthinkingly in earlier days. A great deal of bad philosophy -and bad science resulted from this propensity. “Great principles,” -such as the “uniformity of nature,” the “law of universal causation,” -and so on, are attempts to bolster up our belief that what has often -happened before will happen again, which is no better founded than the -horse’s belief that you will take the turning you usually take. It is -not altogether easy to see what is to replace these pseudo-principles -in the practice of science; but perhaps the theory of relativity gives -us a glimpse of the kind of thing we may expect. Causation, in the -<span class="pagenum" id="Page_225">[Pg 225]</span> -old sense, no longer has a place in theoretical physics. There is, -of course, something else which takes its place, but the substitute -appears to have a better empirical foundation than the old principle -which it has superseded.</p> - -<p>The collapse of the notion of one all-embracing time, in which all -events throughout the universe can be dated, must in the long run -affect our views as to cause and effect, evolution, and many other -matters. For instance, the question whether, on the whole, there is -progress in the universe, may depend upon our choice of a measure of -time. If we choose one out of a number of equally good clocks, we may -find that the universe is progressing as fast as the most optimistic -American thinks it is; if we choose another equally good clock, we may -find that the universe is going from bad to worse as fast as the most -melancholy Slav could imagine. Thus optimism and pessimism are neither -true nor false, but depend upon the choice of clocks.</p> - -<p>The effect of this upon a certain type of emotion is devastating. -The poet speaks of</p> - -<div class="poetry-container"> -<div class="poetry"> - <div class="stanza"> - <div class="verse indent0">One far-off divine event</div> - <div class="verse indent0">To which the whole creation moves.</div> - </div> -</div> -</div> - -<p><span class="pagenum" id="Page_226">[Pg 226]</span> -But if the event is sufficiently far off, and the creation moves -sufficiently quickly, some parts will judge that the event has already -happened, while others will judge that it is still in the future. This -spoils the poetry. The second line ought to be:</p> - -<div class="poetry-container"> -<div class="poetry"> - <div class="stanza"> - <div class="verse indent0">To which some parts of the creation move,</div> - <div class="verse indent3">while others move away from it.</div> - </div> -</div> -</div> - -<p>But this won’t do. I suggest that an emotion which can be destroyed by -a little mathematics is neither very genuine nor very valuable. But -this line of argument would lead to a criticism of the Victorian Age, -which lies outside my theme.</p> - -<p>What we know about the physical world, I repeat, is much more abstract, -than was formerly supposed. Between bodies there are occurrences, -such as light waves; of the <i>laws</i> of these occurrences, we -know something—just so much as can be expressed in mathematical -formulæ—but of their <i>nature</i> we know nothing. Of the bodies -themselves, as we saw in the preceding chapter, we know so little that -we cannot even be sure that they are anything: they <i>may</i> be -merely groups of events in other places, those events which we should -<span class="pagenum" id="Page_227">[Pg 227]</span> -naturally regard as their effects. We naturally interpret the world -pictorially; that is to say, we imagine that what goes on is more or -less like what we see. But in fact this likeness can only extend to -certain formal logical properties expressing structure, so that all we -can know is certain general characteristics of its changes. Perhaps an -illustration may make the matter clear. Between a piece of orchestral -music as played, and the same piece of music as printed in the score, -there is a certain resemblance, which may be described as a resemblance -in structure. The resemblance is of such a sort that, when you know the -rules, you can infer the music from the score or the score from the -music. But suppose you had been stone deaf from birth, but had lived -among musical people. You could understand, if you had learned to speak -and to do lip-reading, that the musical scores represented something -quite different from themselves in intrinsic quality, though similar in -structure.<a id="FNanchor_16" href="#Footnote_16" class="fnanchor">[16]</a> -The value of music would be completely unimaginable to -you, but you could infer all its mathematical characteristics, since -<span class="pagenum" id="Page_228">[Pg 228]</span> -they are the same as those of the score. Now our knowledge of nature is -something like this. We can read the scores, and infer just so much as -our stone-deaf person could have inferred about music. But we have not -the advantages which he derived from association with musical people. -We cannot know whether the music represented by the scores is beautiful -or hideous; perhaps, in the last analysis, we cannot be quite sure that -the scores represent anything but themselves. But this is a doubt which -the physicist, in his professional capacity, cannot permit himself to -entertain.</p> - -<p>Assuming the utmost that can be claimed for physics, it does not tell -us what it is that changes, or what are its various states; it only -tells us such things as that changes follow each other periodically, -or spread with a certain speed. Even now we are probably not at the -end of the process of stripping away what is merely imagination, in -order to reach the core of true scientific knowledge. The theory of -relativity has accomplished a very great deal in this respect, and in -doing so has taken us nearer and nearer to bare structure, which is -the mathematician’s goal—not because it is the only thing in which he -<span class="pagenum" id="Page_229">[Pg 229]</span> -is interested as a human being, but because it is the only thing that -he can express in mathematical formulæ. But far as we have traveled in -the direction of abstraction, it may be that we shall have to travel -further still.</p> - -<p>In the preceding chapter, I suggested what may be called a minimum -definition of matter, that is to say, one in which matter has, so -to speak, as little “substance” as is compatible with the truth of -physics. In adopting a definition of this kind, we are playing for -safety: our tenuous matter will exist, even if something more beefy -also exists. We tried to make our definition of matter, like Isabella’s -gruel in Jane Austen, “thin, but not too thin.” We shall, however, fall -into error if we assert positively that matter is nothing more than -this. Leibniz thought that a piece of matter is really a colony of -souls. There is nothing to show that he was wrong, though there is also -nothing to show that he was right: we know no more about it either way -than we do about the flora and fauna of Mars.</p> - -<p>To the non-mathematical mind, the abstract character of our physical -knowledge may seem unsatisfactory. From an artistic or imaginative -<span class="pagenum" id="Page_230">[Pg 230]</span> -point of view, it is perhaps regrettable, but from a practical point -of view it is of no consequence. Abstraction, difficult as it is, is -the source of practical power. A financier, whose dealings with the -world are more abstract than those of any other “practical” man, is -also more powerful than any other practical man. He can deal in wheat -or cotton without needing ever to have seen either: all he needs to -know is whether they will go up or down. This is abstract mathematical -knowledge, at least as compared to the knowledge of the agriculturist. -Similarly the physicist, who knows nothing of matter except certain -laws of its movements, nevertheless knows enough to enable him to -manipulate it. After working through whole strings of equations, in -which the symbols stand for things whose intrinsic nature can never be -known to us, he arrives at last at a result which can be interpreted -in terms of our own perceptions, and utilized to bring about desired -effects in our own lives. What we know about matter, abstract and -schematic as it is, is enough, in principle, to tell us the rules -according to which it produces perceptions and feelings in ourselves; -and it is upon these rules that the <i>practical</i> uses of physics depend. -<span class="pagenum" id="Page_231">[Pg 231]</span></p> - -<p>The final conclusion is that we know very little, and yet it is -astonishing that we know so much, and still more astonishing that so -little knowledge can give us so much power.</p> - -<p class="f150 space-above1 space-below2">THE END</p> - -<div class="footnotes"> -<p class="f150"><b>Footnotes:</b></p> -<div class="footnote"><p class="no-indent"> -<a id="Footnote_1" href="#FNanchor_1" class="label">[1]</a> -A contemporary Chinese ode, after giving the day of the -year correctly, proceeds:</p> -<div class="poetry-container"> -<div class="poetry"> - <div class="stanza"> - <div class="verse indent0">“For the moon to be eclipsed</div> - <div class="verse indent1">Is but an ordinary matter.</div> - <div class="verse indent1">Now that the sun has been eclipsed,</div> - <div class="verse indent1">How bad it is.”</div> -</div></div></div></div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_2" href="#FNanchor_2" class="label">[2]</a> -I shall define “interval” in a moment.</p> -</div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_3" href="#FNanchor_3" class="label">[3]</a> -So long as he has no considerable acceleration. The treatment -of acceleration belongs to the <i>general</i> theory of relativity.</p> -</div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_4" href="#FNanchor_4" class="label">[4]</a> -This does not mean that its velocity is increasing, but that it is -changing its direction. The only sort of motion which is called -“unaccelerated” is motion with uniform velocity <i>in a straight line</i>.</p> -</div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_5" href="#FNanchor_5" class="label">[5]</a> -See his <i>Space, Time, Matter</i>, Methuen, 1922.</p> -</div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_6" href="#FNanchor_6" class="label">[6]</a> -Although “force” is no longer to be regarded as one of the fundamental -concepts of dynamics, but only as a convenient way of speaking, it can -still be employed, like “sunrise” and “sunset,” provided we realize -what we mean. Often it would require very roundabout expressions to -avoid the term “force.”</p> -</div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_7" href="#FNanchor_7" class="label">[7]</a> -See Eddington, <i>The Mathematical Theory of Relativity</i>, -Cambridge University Press, 2d edition, p. 128.</p> -</div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_8" href="#FNanchor_8" class="label">[8]</a> -This is subject to the explanations given below as regards -conservation of energy.</p> -</div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_9" href="#FNanchor_9" class="label">[9]</a> -<i>Mathematical Theory of Relativity</i>, p. 135.</p> -</div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_10" href="#FNanchor_10" class="label">[10]</a> -On this subject, see the present author’s <i>A.B.C. of -Atoms</i>, chaps. <span class="allsmcap">VI</span> and -<span class="allsmcap">XIII</span>.</p></div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_11" href="#FNanchor_11" class="label">[11]</a> -<i>Op. cit.</i> § 60.</p></div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_12" href="#FNanchor_12" class="label">[12]</a> -See Eddington, <i>Space, Time and Gravitation</i>, p. 162ff.</p> -</div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_13" href="#FNanchor_13" class="label">[13]</a> -“Isotropy” means being similar in all directions—<i>e.g.</i>, -that a foot rule is as long when it points north as when it points east.</p> -</div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_14" href="#FNanchor_14" class="label">[14]</a> -<i>Mathematical Theory of Relativity</i>, p. 238.</p> -</div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_15" href="#FNanchor_15" class="label">[15]</a> -<i>Mathematical Theory of Relativity</i>, pp. 37-38. -Italics in the original.</p> -</div> - -<div class="footnote"><p class="no-indent"> -<a id="Footnote_16" href="#FNanchor_16" class="label">[16]</a> -For the definition of “structure,” see the present -author’s <i>Introduction to Mathematical Philosophy</i>.</p> -</div> -</div> - -<div class="transnote bbox space-above2"> -<p class="f120 space-above1">Transcriber’s Notes:</p> -<hr class="r5" /> -<p class="indent">The cover image was created by the transcriber, and is in the public domain.</p> -<p class="indent">The illustrations have been moved so that they do not break up - paragraphs and so that they are next to the text they illustrate.</p> -<p class="indent">Typographical and punctuation errors have been silently corrected.</p> -</div> -<div style='display:block; margin-top:4em'>*** END OF THE PROJECT GUTENBERG EBOOK THE A B C OF RELATIVITY ***</div> -<div style='text-align:left'> - -<div style='display:block; margin:1em 0'> -Updated editions will replace the previous one—the old editions will -be renamed. -</div> - -<div style='display:block; margin:1em 0'> -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the United -States without permission and without paying copyright -royalties. 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