The A B C of Relativity

by Bertrand Russell

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[2] 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.