CHAPTERPAGE
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.