The A B C of Relativity

Before tackling Einstein’s new law, it is as well to convince ourselves, on logical grounds, that Newton’s law of gravitation cannot be quite right. Newton said that between any two particles of matter there is a force which is proportional to the product of their masses and inversely proportional to the square of their distance. That is to say, ignoring for the present the question of mass, if there is a certain attraction when the particles are a mile apart, there will be a quarter as much attraction when they are two miles apart, a ninth as much when they are three miles apart, and so on: the attraction diminishes much faster than the distance increases. Now, of course, Newton, when he spoke of the distance, meant the distance at a given time: He thought there could be no ambiguity about time. But we have seen that this was a mistake. What one observer judges to be the same moment on the [Pg 112] earth and the sun, another will judge to be two different moments. “Distance at a given moment” is therefore a subjective conception, which can hardly enter into a cosmic law. Of course, we could make our law unambiguous by saying that we are going to estimate times as they are estimated by Greenwich Observatory. But we can hardly believe that the accidental circumstances of the earth deserve to be taken so seriously. And the estimate of distance, also, will vary for different observers. We cannot, therefore, allow that Newton’s form of the law of gravitation can be quite correct, since it will give different results according to which of many equally legitimate conventions we adopt. This is as absurd as it would be if the question whether one man had murdered another were to depend upon whether they were described by their Christian names or their surnames. It is obvious that physical laws must be the same whether distances are measured in miles or in kilometers, and we are concerned with what is essentially only an extension of the same principle. Our measurements are conventional to an even greater extent than [Pg 113] is admitted by the special theory of relativity. Moreover, every measurement is a physical process carried out with physical material; the result is certainly an experimental datum, but may not be susceptible of the simple interpretation which we ordinarily assign to it. We are, therefore, not going to assume to begin with that we know how to measure anything. We assume that there is a certain physical quantity, called “interval,” which is a relation between two events that are not widely separated; but we do not assume in advance that we know how to measure it, beyond taking it for granted that it is given by some generalization of the theorem of Pythagoras such as we spoke of in the preceding chapter. We do assume, however, that events have an order, and that this order is four-dimensional. We assume, that is to say, that we know what we mean by saying that a certain event is nearer to another than to a third, so that before making accurate measurements we can speak of the “neighborhood” of an event; and we assume that, in order to assign the position of an event in space-time, four quantities (co-ordinates) are necessary—e.g. in our former case of an explosion on an [Pg 114] airship, latitude, longitude, altitude and time. But we assume nothing about the way in which these co-ordinates are assigned, except that neighboring co-ordinates are assigned to neighboring events. The way in which these numbers, called co-ordinates, are to be assigned is neither wholly arbitrary nor a result of careful measurement—it lies in an intermediate region. While you are making any continuous journey, your co-ordinates must never alter by sudden jumps. In America one finds that the houses between (say) Fourteenth Street and Fifteenth Street are likely to have numbers between 1400 and 1500, while those between Fifteenth Street and Sixteenth Street have numbers between 1500 and 1600, even if the 1400’s were not used up. This would not do for our purposes, because there is a sudden jump when we pass from one block to the next. Or again we might assign the time co-ordinate in the following way: take the time that elapses between two successive births of people called Smith; an event occurring between the births of the 3000th and the 3001st Smith known to history shall have a co-ordinate lying between 3000 and 3001; the fractional part of its co-ordinate [Pg 115] shall be the fraction of a year that has elapsed since the birth of the 3000th Smith. (Obviously there could never be as much as a year between two successive additions to the Smith family.) This way of assigning the time co-ordinate is perfectly definite, but it is not admissible for our purposes, because there will be sudden jumps between events just before the birth of a Smith and events just after, so that in a continuous journey your time co-ordinate will not change continuously. It is assumed that, independently of measurement, we know what a continuous journey is. And when your position in space-time changes continuously, each of your four co-ordinates must change continuously. One, two, or three of them may not change at all; but whatever change does occur must be smooth, without sudden jumps. This explains what is not allowable in assigning co-ordinates. To explain all the changes that are legitimate in your co-ordinates, suppose you take a large piece of soft india-rubber. While it is in an unstretched condition, measure little squares on it, each one-tenth of an inch each way. Put in little tiny pins at the corners of the squares. We can take as two of the co-ordinates of one of these pins [Pg 116] the number of pins passed in going to the right from a given pin until we come just below the pin in question, and then the number of pins we pass on the way up to this pin. In the figure, let O be the pin we start from and P the pin to which we are going to assign co-ordinates. P is in the fifth column and the third row, so its co-ordinates in the plane of the india-rubber are to be 5 and 3. Fig. 1. Fig. 2. Now take the india-rubber and stretch it and twist it as much as you like. Let the pins now be in the shape they have in Fig. 2. The divisions now no longer represent distances according to our usual notions, but they will still do just as well as co-ordinates. We may still take P as having the co-ordinates 5 and 3 in the plane of the india-rubber; and we may still regard the india-rubber as being in a plane, even if we have twisted it out of what we should ordinarily [Pg 117] call a plane. Such continuous distortions do not matter. To take another illustration: instead of using a steel measuring rod to fix our co-ordinates, let us use a live eel, which is wriggling all the time. The distance from the tail to the head of the eel is to count as one from the point of view of co-ordinates, whatever shape the creature may be assuming at the moment. The eel is continuous, and its wriggles are continuous, so it may be taken as our unit of distance in assigning co-ordinates. Beyond the requirement of continuity, the method of assigning co-ordinates is purely conventional, and therefore a live eel is just as good as a steel rod. We are apt to think that, for really careful measurements, it is better to use a steel rod than a live eel. This is a mistake: not because the eel tells us what the steel rod was thought to tell, but because the steel rod really tells no more than the eel obviously does. The point is, not that eels are really rigid, but that steel rods really wriggle. To an observer in just one possible state of motion, the eel would appear rigid, while the steel rod would seem to wriggle just [Pg 118] as the eel does to us. For everybody moving differently both from this observer and ourselves, both the eel and the rod would seem to wriggle. And there is no saying that one observer is right and another wrong. In such matters, what is seen does not belong solely to the physical process observed, but also to the standpoint of the observer. Measurements of distances and times do not directly reveal properties of the things measured, but relations of the things to the measurer. What observation can tell us about the physical world is therefore more abstract than we have hitherto believed. 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. And measurement of distances, as we have seen, is [Pg 119] a subjective affair, depending upon the way in which the observer is moving. The failure of the circle to have objective validity was demonstrated by the Michelson-Morley experiment, and is thus, in a sense, the starting point of the whole theory of relativity. Rigid bodies, which we need for measurement, are only rigid for certain observers; for others, they will be constantly changing all their dimensions. It is only our obstinately earth-bound imagination that makes us suppose a geometry separate from physics to be possible. That is why we do not trouble to give physical significance to our co-ordinates from the start. Formerly, the co-ordinates used in physics were supposed to be carefully measured distances; now we realize that this care at the start is thrown away. It is at a later stage that care is required. Our co-ordinates now are hardly more than a systematic way of cataloguing events. But mathematics provides, in the method of tensors, such an immensely powerful technique that we can use co-ordinates assigned in this apparently careless way just as effectively as if we had applied the whole apparatus of minutely accurate measurement in arriving at them. The advantage of being [Pg 120] haphazard at the start is that we avoid making surreptitious physical assumptions, which we can hardly help making, if we suppose that our co-ordinates have initially some particular physical significance. We assume that, if two events are close together (but not necessarily otherwise), there is an interval between them which can be calculated from the differences between their co-ordinates by some such formula as we considered in the preceding chapter. That is to say, we take the squares and products of the differences of co-ordinates, we multiply them by suitable amounts (which in general will vary from place to place), and we add the results together. The sum obtained is the square of the interval. We do not assume in advance that we know the amounts by which the squares and products must be multiplied; this is going to be discovered by observing physical phenomena. We know, however, certain things. We know that the old Newtonian physics is very nearly accurate when our co-ordinates have been chosen in a certain way. We know that the special theory of relativity is still more nearly accurate for suitable co-ordinates. From such facts we can [Pg 121] infer certain things about our new co-ordinates, which, in a logical deduction, appear as postulates of the new theory. 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