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, [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 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 [Pg 81]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 [Pg 82]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.[Pg 83]

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 [Pg 84]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 [Pg 85]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 [Pg 86]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 [Pg 87]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′.

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 [Pg 89]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

This is the Lorentz transformation, from which everything in this chapter can be deduced.