The A B C of Relativity

by Bertrand Russell

As such postulates we take:

1. That every body travels in a geodesic in space-time, except in so far as electromagnetic forces act upon it.

2. That a light ray travels so that the interval between two parts of it is zero.

3. That at a great distance from gravitating matter, we can transform our co-ordinates by mathematical manipulation so that the interval shall be what it is in the special theory of relativity; and that this is approximately true wherever gravitation is not very powerful.

Each of these postulates requires some explanation.

We saw that a geodesic on a surface is the shortest line that can be drawn on the surface from one point to another; for example, on the earth the geodesics are great circles. When we come to space-time, the mathematics is the same, but the verbal explanations have to be rather different. In the general theory of relativity, it is only neighboring events that have a definite interval, independently of the route by which we travel from one to the other. The interval between distant events depends upon the route pursued, and has to be calculated by dividing the route into a number of little bits and adding up the intervals for the various little bits. If the interval is space-like, a body cannot travel from one event to the other; therefore when we are considering the way bodies move, we are confined to time-like intervals. The interval between neighboring events, when it is time-like, will appear as the time between them for an observer who travels from the one event to the other. And so the whole interval between two events will be judged by a person who travels from one to the other to be what his clocks show to be the time that he has taken on the journey. For some routes this time will be longer, for others shorter; the more slowly the man travels, the longer he will think he has been on the journey. This must not be taken as a platitude. I am not saying that if you travel from London to Edinburgh you will take longer if you travel more slowly. I am saying something much more odd. I am saying that if you leave London at 10 a.m. and arrive in Edinburgh at 6.30 p.m. Greenwich time, the more slowly you travel the longer you will take—if the time is judged by your watch. This is a very different statement. From the point of view of a person on the earth, your journey takes eight and a half hours. But if you had been a ray of light traveling round the solar system, starting from London at 10 a.m., reflected from Jupiter to Saturn, and so on, until at last you were reflected back to Edinburgh and arrived there at 6.30 p.m., you would judge that the journey had taken you exactly no time. And if you had gone by any circuitous route, which enabled you to arrive in time by traveling fast, the longer your route the less time you would judge that you had taken; the diminution of time would be continual as your speed approached that of light. Now I say that when a body travels, if it is left to itself, it chooses the route which makes the time between two stages of the journey as long as possible; if it had traveled from one event to another by any other route, the time, as measured by its own clocks, would have been shorter. This is a way of saying that bodies left to themselves do their journeys as slowly as they can; it is a sort of law of cosmic laziness. Its mathematical expression is that they travel in geodesics, in which the total interval between any two events on the journey is greater than by any alternative route. (The fact that it is greater, not less, is due to the fact that the sort of interval we are considering is more analogous to time than to distance.) For example, if a person could leave the earth and travel about for a time and then return, the time between his departure and return would be less by his clocks than by those on the earth: the earth, in its journey round the sun, chooses the route which makes the time of any bit of its course by its clocks longer than the time as judged by clocks which move by a different route. This is what is meant by saying that bodies left to themselves move in geodesics in space-time.

We assume that the body considered is not acted upon by electromagnetic forces. We are concerned at present with the law of gravitation, not with the effects of electromagnetism. These effects have been brought into the framework of the general theory of relativity by Weyl,[5] but for the present we will ignore his work. The planets, in any case, are not subject, as wholes, to appreciable electromagnetic forces; it is only gravitation that has to be considered in accounting for their motions, with which we are concerned in this chapter.

Our second postulate, that a light ray travels so that the interval between two parts of it is zero, has the advantage that it does not have to be stated only for small distances. If each little bit of interval is zero, the sum of them all is zero, and so even distant parts of the same light ray have a zero interval. The course of a light ray is also a geodesic according to the definition. Thus we now have two empirical ways of discovering what are the geodesics in space-time, namely light rays and bodies moving freely. Among freely-moving bodies are included all which are not subject to constraints or to electromagnetic forces, that is to say, the sun, stars, planets and satellites, and also falling bodies on the earth, at least when they are falling in a vacuum. When you are standing on the earth, you are subject to electromagnetic forces: the electrons and protons in the neighborhood of your feet exert a repulsion on your feet which is just enough to overcome the earth’s gravitation. This is what prevents you from falling through the earth, which, solid as it looks, is mostly empty space.

The third postulate, which relates the general to the special theory, is very useful. It is not necessary for the application of the special theory to a limited region that there should be no gravitation in the region; it is enough if the intensity of gravitation is practically the same throughout the region. This enables us to apply the special theory within any small region. How small it will have to be, depends upon the neighborhood. On the surface of the earth, it would have to be small enough for the curvature of the earth to be negligible. In the spaces between the planets, it need only be small enough for the attraction of the sun and the planets to be sensibly constant throughout the region. In the spaces between the stars it might be enormous—say half the distance from one star to the next—without introducing measurable inaccuracies.

At a great distance from gravitating matter, we can so choose our co-ordinates as to obtain a Euclidean space; this is really only another way of saying that the special theory of relativity applies. In the neighborhood of matter, although we can make our space Euclidean in any small region, we cannot do so throughout any region within which gravitation varies sensibly—at least, if we do, we shall have to abandon the view that bodies move in geodesics. In the neighborhood of a piece of matter, there is, as it were, a hill in space-time; this hill grows steeper and steeper as it gets nearer the top, like the neck of a champagne bottle. It ends in a sheer precipice. Now by the law of cosmic laziness which we mentioned earlier, a body coming into the neighborhood of the hill will not attempt to go straight over the top, but will go round. This is the essence of Einstein’s view of gravitation. What a body does, it does because of the nature of space-time in its own neighborhood, not because of some mysterious force emanating from a distant body.