Bertrand Russell

“ Now just as you can estimate a man’s fortune in different currencies without altering its relations to the fortunes of other men, so you can estimate a body’s motion by means of different reference bodies without altering its relations to other motions. And as physics is entirely concerned with relations, it must be possible to express all the laws of physics by referring all motions to any given body as the standard. ”

“ It is a curious fact—of which relativity is not the only illustration—that, as reasoning improves, its claims to the power of proving facts grow less and less. Logic used to be thought to teach us how to draw inferences; now, it teaches us rather how not to draw inferences. Animals and children are terribly prone to inference: a horse is surprised beyond measure if you take an unusual turning. When men began to reason, they tried to justify the inferences that they had drawn unthinkingly in earlier days. ”

“ 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. ”

“ We have seen that gravitation, although no longer regarded as an action at a distance, is still connected with a center: there is, so to speak, a hill symmetrically arranged about its summit, and the summit is the place where we conceive the body to be which is connected with the gravitational field we are considering. For simplicity, common sense lumps together all the events which form one group in the above sense. When two people see the same object, two different events occur, but they are events belonging to one group and connected with the same center. ”

“ All the known facts are perfectly consistent with the hypothesis of a spherical universe. A very slight modification of the law of gravitation—a modification suggested by Einstein himself—leads to a spherical space, without producing any measurable differences in a small region such as the solar system. The known stars are all within a certain distance from us. There is nothing whatever in the stellar universe as we know it to show that space must be infinite. There can therefore be no doubt whatever that, so far as our present knowledge goes, the hypothesis of a finite universe may be true. ”

“ We saw, in the preceding chapter, how a group of connected events can be formed, all related to each other by a law, and all ranged about a center in space-time. Such a group of events will be the arrival, at various places, of the light waves emitted by a brief flash of light. We do not need to suppose that anything particular is happening at the center; certainly we do not need to suppose that we know what is happening there. What we know is that, as a matter of geometry, the group of events in question are ranged about a center, like widening ripples on a pool when a fly has touched it. ”

“ A financier, whose dealings with the world are more abstract than those of any other “practical” man, is also more powerful than any other practical man. He can deal in wheat or cotton without needing ever to have seen either: all he needs to know is whether they will go up or down. This is abstract mathematical knowledge, at least as compared to the knowledge of the agriculturist. Similarly the physicist, who knows nothing of matter except certain laws of its movements, nevertheless knows enough to enable him to manipulate it. ”

“ THE EARTH AND THE HEAVENS Everybody knows that Einstein has done something astonishing, but very few people know exactly what it is that he has done. It is generally recognized that he has revolutionized our conception of the physical world, but his new conceptions are wrapped up in mathematical technicalities. It is true that there are innumerable popular accounts of the theory of relativity, but they generally cease to be intelligible just at the point where they begin to say something important. ”

“ If the universe is finite, it is theoretically conceivable that there should be a complete inventory of it. We may be coming to the end of what physics can do in the way of stretching the imagination and systematizing the world. The period since Galileo has been essentially the period of physics, as the age of the Greeks was the period of geometry. It may be that physics will lose its attractions through success: if the fundamental laws of physics come to be fully known, adventurous and inquiring intellects will turn to other fields. ”

“ What is, for one observer, a description of the state of the world at a given instant, is, for another observer, a series of events at various different times, whose relations are not merely spatial but also temporal. For the same reason, we are concerned with events, rather than with bodies. In the old theory, it was possible to consider a number of bodies all at the same instant, and since the time was the same for all of them it could be ignored. But now we cannot do that if we are to obtain an objective account of physical occurrences. ”

“ 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 ”

“ It is important to realize that geometry, as taught in schools since Greek times, ceases to exist as a separate science, and becomes merged in physics. The two fundamental notions in elementary geometry were the straight line and the circle. What appears to you as a straight road, whose parts all exist now, may appear to another observer to be like the flight of a rocket, some kind of curve whose parts come into existence successively. The circle depends upon measurement of distances, since it consists of all the points at a given distance from its center. ”

“ 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. ”

“ There is a general principle to which the theory of relativity appeals, which turns out to be more powerful than anybody would suppose. If you know that one man is twice as rich as another, this fact must appear equally whether you estimate the wealth of both in pounds or dollars or francs or any other currency. The numbers representing their fortunes will be changed, but one number will always be double the other. The same sort of thing, in more complicated forms, reappears in physics. ”

“ But although one problem was solved by the special theory, another was immediately suggested: what if the motion of the two bodies is not uniform? Suppose, for instance, that one is the earth while the other is a falling stone. The stone has an accelerated motion: it is continually falling faster and faster. Nothing in the special theory enables us to say that the laws of physical phenomena will be the same for an observer on the stone as for one on the earth. ”

“ 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 [Pg 48] bodies to be used in defining the point half-way between the events. All bodies have an equal right to be chosen. ”

“ We may say, therefore, that there are events everywhere in space-time, but they must be of a somewhat different kind according as we are dealing with a region where there is an electron or proton or with the sort of region we should ordinarily call empty. But as to the intrinsic nature of these events we can know nothing, except when they happen to be events in our own lives. Our own perceptions and feelings must be part of the crude material of events which physics arranges into a pattern—or rather, which physics finds to be arranged in a pattern. ”

“ We need four measurements to fix a position, and four measurements fix the position of an event in space-time, not merely of a body in space. Three measurements are not enough to fix any position. That is the essence of what is meant by the substitution of space-time for space and time. ”

“ We should have been compelled by logic to seek some law expressed in terms of “tensors,” and therefore independent of our choice of co-ordinates. [Pg 143] It is impossible without mathematics to explain the theory of tensors; the non-mathematician must be content to know that it is the technical method by which we eliminate the conventional element from our measurements and laws, and thus arrive at physical laws which are independent of the observer’s point of view. Of this method, Einstein’s law of gravitation is the most splendid example. ”

“ There will not be in the universe anything straighter than these great circles; the Euclidean straight line may remain as a beautiful dream, but not as a possibility in the actual world. In particular, light rays in empty space will travel in what are really great circles. If we could make measurements with sufficient accuracy, we should be able to infer this state of affairs even from a small part of space, because the sum of the angles of a triangle would always be greater than two right angles, and the excess would be proportional to the size of the triangle. ”

“ The name given to any agency which causes deviation from uniform motion in a straight line is force according to the Newtonian definition of force. Hence the agency invoked through our observer’s mistake is described as a “field of force.”... A field of force represents the discrepancy between the naturalgeometry of a co-ordinate system and the abstractgeometry arbitrarily ascribed to it. [15]
If people were to learn to conceive the world in the new way, without the old notion of “force,” it would alter not only their physical imagination, but probably also their morals and politics. ”

“ We have already seen that, in the special theory of relativity, the square of the interval is found by subtracting the square of the distance between the events from the square of the distance that light would travel in the time between them. In the general theory, we do not assume this special form of interval, except at a great distance from matter. Elsewhere, we assume to begin with a general form, like that which Riemann used for distances. Moreover, like Riemann, Einstein only assumes his formula for neighboring events, that is to say, events which have only a small interval between them. ”

“ Let us now apply similar considerations to the sun’s gravitation. The “force” exerted by the sun only differs from that exerted by the tiger in being attractive instead of repulsive. Instead of acting through waves of light or sound, the sun acquires its apparent power through the fact that there are modifications of space-time all round the sun. Like the noise of the tiger, they are more intense near their source; as we travel away they grow less and less. ”

“ Now if instead of aeronauts going round the earth you take rays of light going round the universe, the same sort of thing happens: they all meet first at the antipodes of their starting point, and then meet again at their starting point. ”

“ Measurement, in Weyl’s theory, suffers the same fate. We can only use a measuring rod to give lengths in one place: when we move it to another region, there is no knowing how it will alter. We do assume, however, that, if it alters, it alters bit by bit, gradually, continuously, and not by sudden jumps. Perhaps this assumption is unjustified. It belongs to the general outlook of relativity, which is that of continuity. No doubt it is owing to this outlook that relativity is unable to account for the discontinuities in physics, such as quanta, electrons and protons. ”