The A B C of Relativity

One of the most difficult matters in all controversy is to distinguish disputes about words from disputes about facts: it ought not to be difficult, but in practice it is. This is quite as true in physics as in other subjects. In the seventeenth century there was a terrific debate as to what “force” is; to us now, it was obviously a debate as to how the word “force” should be defined, but at the time it was thought to be much more. One of the purposes of the method of tensors, which is employed in the mathematics of relativity, is to eliminate what is purely verbal (in an extended sense) in physical laws. It is of course obvious that what depends on the choice of co-ordinates is “verbal” in the sense concerned. A man punting walks along the boat, but keeps a constant position with reference to the river bed so long as he does not pick up his pole. The Lilliputians might debate endlessly whether he is walking [Pg 178] or standing still: the debate would be as to words, not as to facts. If we choose co-ordinates fixed relatively to the boat, he is walking; if we choose co-ordinates fixed relatively to the river bed, he is standing still. We want to express physical laws in such a way that it shall be obvious when we are expressing the same law by reference to two different systems of co-ordinates, so that we shall not be misled into supposing we have different laws when we only have one law in different words. This is accomplished by the method of tensors. Some laws which seem plausible in one language cannot be translated into another; these are impossible as laws of nature. The laws that can be translated into any co-ordinate language have certain characteristics: this is a substantial help in looking for such laws of nature as the theory of relativity can admit to be possible. Combined with what we know of the actual motions of bodies, it enables us to decide what must be the correct expression of the law of gravitation: logic and experience combine in equal proportions in obtaining this expression. But the problem of arriving at genuine laws of nature is not to be solved by the method of tensors alone; a good, deal of careful thought [Pg 179] is wanted in addition. Some of this has been done, especially by Eddington; much remains to be done. To take a simple illustration: Suppose, as in the hypothesis of the Fitzgerald contraction, that lengths in one direction were shorter than in another. Let us assume that a foot rule pointing north is only half as long as the same foot rule pointing east, and that this is equally true of all other bodies. Does such an hypothesis have any meaning? If you have a fishing rod fifteen feet long when it is pointing west, and you then turn it to the north, it will still measure fifteen feet, because your foot rule will have shrunk too. It won’t “look” any shorter, because your eye will have been affected in the same way. If you are to find out the change, it cannot be by ordinary measurement; it must be by some such method as the Michelson-Morley experiment, in which the velocity of light is used to measure lengths. Then you still have to decide whether it is simpler to suppose a change of length or a change in the velocity of light. The experimental fact would be that light takes longer to traverse what your foot rule declares to [Pg 180] be a given distance in one direction than in another—or, as in the Michelson-Morley experiment, that it ought to take longer but doesn’t. You can adjust your measures to such a fact in various ways; in any way you choose to adopt, there will be an element of convention. This element of convention survives in the laws that you arrive at after you have made your decision as to measures, and often it takes subtle and elusive forms. To eliminate the element of convention is, in fact, extraordinarily difficult; the more the subject is studied, the greater the difficulty is seen to be. A more important example is the question of the size and shape of the electron. We find experimentally that all electrons are the same size, and that they are symmetrical in all directions. How far is this a genuine fact ascertained by experiment, and how far is it a result of our conventions of measurement? We have here a number of different comparisons to make: (1) between different directions in regard to one electron at one time; (2) in regard to one electron at different times; (3) in regard to two electrons at the same time. We can then arrive at the comparison of two electrons at different times, by combining [Pg 181] (2) and (3). We may dismiss any hypothesis which would affect all electrons equally; for example, it would be useless to suppose that in one region of space-time they were all larger than in another. Such a change would affect our measuring appliances just as much as the things measured, and would therefore produce no discoverable phenomena. This is as much as to say that it would be no change at all. But the fact that two electrons have the same mass, for instance, cannot be regarded as purely conventional. Given sufficient minuteness and accuracy, we could compare the effects of two different electrons upon a third; if they were equal under like circumstances, we should be able to infer equality in a not purely conventional sense. The question of the symmetry of the forces exerted by an electron—i.e., that these forces depend only upon the distance from the electron, and not upon the direction—is more complicated. Eddington finally comes to the conclusion that this, too, is a matter of convention. The argument is difficult and I have not fully understood it; but I feel some hesitation in accepting it as valid. Eddington describes the process concerned in the more advanced portions of the theory of relativity as “world-building.” The structure to be [Pg 182] built is the physical world as we know it; the economical architect tries to construct it with the smallest possible amount of material. This is a question for logic and mathematics. The greater our technical skill in these two subjects, the more real building we shall do, and the less we shall be content with mere heaps of stones. But before we can use in our building the stones that nature provides, we have to hew them into the right shapes: this is all part of the process of budding. In order that this may be possible, the raw material must have some structure (which we may conceive as analogous to the grain in timber), but almost any structure will do. By successive mathematical refinements, we whittle away our initial requirements until they amount to very little. Given this necessary minimum of structure in the raw material, we find that we can construct from it a mathematical expression which will have the properties that are needed for describing the world we perceive—in particular, the properties of conservation which are characteristic of momentum and energy (or mass). Our raw material consisted merely of events; but when we find [Pg 183] that we can build out of it something which, as measured, will seem to be never created or destroyed, it seems not surprising that we should come to believe in “bodies.” These are really mere mathematical constructions out of events, but owing to their permanence they are practically important, and our senses (which were presumably developed by biological needs) are adapted for noticing them, rather than the crude continuum of events which is theoretically more fundamental. From this point of view, it is astonishing how little of the real world is revealed by physical science: our knowledge is limited, not only by the conventional element, but also by the selectiveness of our perceptual apparatus. We assume that there is an “interval” between two events, in the sense explained in Chapter VII, but we no longer assume that we can unambiguously compare the length of an interval in one region with the length of an interval in another. It is assumed by Weyl, who introduced this limitation, that we can compare a number of small intervals which all start from the same point; also that, in a very small journey, our measuring rod will not alter its length much, so that there will [Pg 184] only be a small error if we compare lengths in neighboring places by the usual methods. Weyl found that, by diminishing our assumptions as to interval in this way, it was possible to bring electromagnetism and gravitation into one system. The mathematics of Weyl’s theory is complicated, and I shall not attempt to explain it. For the present, I am concerned with a different consequence of his theory. If lengths in different regions cannot be compared directly, there is an element of convention in the indirect comparisons which we actually make. This element will be at first unrecognized, but will be such as to simplify to the utmost the expression of the laws of nature. In particular, conditions of symmetry may be entirely created by conventions as to measurement, and there is no reason to suppose that they represent any property of the real world. The law of gravitation itself, according to Eddington, may be regarded as expressing conventions of measurement. “The conventions of measurement,” he says, “introduce an isotropy[13] and homogeneity into measured space which need not originally have any counterpart in the relation-structure which is being surveyed. This isotropy and homogeneity is exactly expressed by Einstein’s law of gravitation.”[14] [Pg 185] The limitations of knowledge introduced by the selectiveness of our perceptual apparatus may be illustrated by the indestructibility of matter. This has been gradually discovered by experiment, and seemed a well-founded empirical law of nature. Now it turns out that, from our original space-time continuum, we can construct a mathematical expression which will have properties causing it to appear indestructible. The statement that matter is indestructible then ceases to be a proposition of physics, and becomes instead a proposition of linguistics and psychology. As a proposition of linguistics: “Matter” is the name of the mathematical expression in question. As a proposition of psychology: Our senses are such that we notice what is roughly the mathematical expression in question, and we are led nearer and nearer to it as we refine upon our crude perceptions by scientific observation. This is much less than physicists used to think they knew about matter. The reader may say: What then is left of physics? What do we really [Pg 186] know about the world of matter? Here we may distinguish three departments of physics. There is first what is included within the theory of relativity, generalized as widely as possible. Next, there are laws which cannot be brought within the scope of relativity. Thirdly, there is what may be called geography. Let us consider each of these in turn. The theory of relativity, apart from convention, tells us that the events in the universe have a four-dimensional order, and that, between any two events which are near together in this order, there is a relation called “interval,” which is capable of being measured if suitable precautions are taken. We make also an assumption as to what happens when a little measuring rod is carried round a closed circuit in a certain manner; the consequences of this assumption are such as to make it highly probable that it is true. Beyond this, there is little in the theory of relativity that can be regarded as physical laws. There is a great deal of mathematics, showing that certain mathematically-constructed quantities must behave like the things we perceive; and there is a suggestion of a bridge between psychology and [Pg 187] physics in the theory