The A B C 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, [Pg 72] 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 [Pg 73] 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 [Pg 74] 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 [Pg 75] 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 [Pg 76] 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 [Pg 77] 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? 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 [Pg 78] 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 [Pg 79] 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 [Pg 80] 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