Clocks and Foot Rules

The A B C of Relativity by Bertrand Russells, is part of the HackerNoon Books Series. You can jump to any chapter in this book here. IV. CLOCKS AND FOOT RULES

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. 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.

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 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.

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.

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.

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