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?
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.