The Analysis of Matter

there is an important constant . In the first place, only exists, or at any rate is only important, in the case of periodic processes, and it is a characteristic of one complete period. In the second place, only integral multiples of occur. In the third place, when a transaction involves the loss by one system of a certain multiple of , another system may acquire another multiple of : what is transferred always unaltered in amount is energy. These seem to be the most significant facts about . It seems impossible to resist the view that represents something of fundamental importance in the physical world, which, in turn, involves the conclusion that periodicity is an element in physical laws, and that one period of a periodic[Pg 366] process must be treated as, in some sense, a unit. This follows from the fact that processes arrange themselves so as to secure that a period shall have an important property. This property is simplest in the case of a light-wave: the energy of one light-wave multiplied by the time it takes to pass a given material point is . If we take the velocity of light as unity, the time a light-wave takes to pass a given point is equal to the spatial distance between the beginning and end of the wave; therefore this distance multiplied by the energy is . This form might seem preferable for our purposes, since it does not involve reference to an extraneous material point. At least, it does not obviously involve such reference; but perhaps the reference is concealed in the process of estimating spatial distance. We have seen that this process must be indirect; one part of a light-wave cannot catch up another, so that the space-like interval between them can only be estimated by means of some process taking place in matter. If it should be found that quantum phenomena are not physically fundamental, much of what has been said in this chapter will become unnecessary. It should be said, however, that relativity should prepare our minds for the oddest feature of the quantum theory, namely the existence of causal laws involving whole periods. The causal unit, on relativity principles, should be expected to occupy a small region of space-time, not only of space; it should not therefore be instantaneous, as in pre-relativity dynamics. If we combine this with the hypothesis of a discrete space-time, we can imagine a theoretical physics which would make the existence of the quantum no longer seem surprising. I have to confess, reluctantly, that the theory developed in the present chapter, inadequate as it is, is the best that I know how to suggest on the topic of quanta. Perhaps the progress of physics will make a better philosophy of the subject possible before long. Meanwhile I commend the matter to the consideration of the reader. [Pg 367] CHAPTER XXXV CAUSALITY AND INTERVAL THE conception of "interval," upon which the mathematical theory of relativity depends, is very hard to translate, even approximately, into non-technical terms. Yet it is difficult to resist the conviction that it has some connection with causality. Perhaps a discontinuous theory of interval might diminish the obstacles to such an interpretation. Let us try to discover whether this is the case. The view which naturally suggests itself as a point of departure is something like this: Given two groups of co-punctual events, it may happen that at least one member of one group has a causal relation to at least one member of the other group; in that case, the interval between the two groups is time-like. If causality is a matter of discontinuous transitions, one might expect that the magnitude of the interval would be measured by the number of intermediate transitions. Again, it may happen that no member of one group has a causal relation to any member of the other, but that both contain members having causal relations to a member of a third group. In that case, the interval will be space-like, and again one might suppose that the number of intermediate links would determine the magnitude of the interval. This represents what might be hoped, but as it stands it is unduly simple, and open to obvious objections. Let us see, therefore, whether it is possible to answer the objections, or to introduce such modifications as will obviate them. First, let us be clear as to what we mean by a causal relation. There is a causal relation whenever two events, or two groups of events of which one at least is co-punctual, are related[Pg 368] by a law which allows something to be inferred about the one from the other. Formerly, one would have supposed that everything about the later event could be inferred from a sufficient number of antecedents; but in view of the explosive and apparently spontaneous character of radio-activity and quantum changes, we must be content with a more modest definition so far as this point is concerned. In another respect, however, our definition is less modest than it would formerly have been. In classical dynamics, causal laws connect accelerations with configurations, so that from the present state of a small region we cannot accurately infer anything as to what will be happening there after a finite time. Quanta have altered this: we can associate the light radiated from an atom with its causal origin, until it hits other matter; we can associate the state of the atom after the emission of the light with its state before, until it undergoes another quantum change. In fact, as we saw in the preceding chapter, we can analyze the course of nature into a set of steady events and rhythms with causal relations governing the "transactions" in which rhythms undergo changes. The above definition was framed with these considerations in mind. We shall say, then, that all causal relations consist of a series of rhythms or steady events separated by "transactions." If such a series connects a rhythm or steady event with a rhythm or steady event , we shall say that is a "causal ancestor" of , and is a "causal descendant" of . We may assume that, in such a case, the number of transactions between and is always finite, since one supposes that the time between two transactions cannot fall below a certain minimum, or at any rate that the number of causally connected transactions in a finite time is never infinite. Perhaps we may assume that a rhythm must last long enough to achieve an amount of action ; perhaps, even, we could construct a discrete theory of time from which[Pg 369] this result would follow. All this, however, is very speculative. Now let us consider the stock case of a light-signal sent from to , and reflected back from to . Only two transactions are involved, namely the emission and reflection of the light; perhaps we ought to add the final transaction, namely the re-absorption of the light by . In any case, there need be only two steady events, one in the outward beam and one in the returning beam. But the interval between the departure and return of the light may have any magnitude. This is all the more curious, as the interval between the departure of the light from and its arrival at is zero, and so is the interval between its departure from and its return to . This suggests that too much effort has been made to regard interval as analogous to distance in conventional geometry and time in conventional kinematics. Suppose we say that, if an event is a causal ancestor of an event , we take all the possible causal routes from to , and choose that which contains the greatest number of events: then the "interval" from to is defined as the number of events in this longest route. It is obvious that, if a measurable time elapses between the departure of the light from and its return to , there must have been a variety of events at meanwhile. When I say "at" , I have a meaning to be considered shortly; but for the moment it is enough to say that this meaning includes causal inheritance. Thus we have a meaning for the view that the interval at is quite long, and also for the view that the interval between the departure of the light from and its arrival at is zero. This latter statement means that it is the very same event that starts from and arrives at , and moreover that there is no longer causal route connecting the two transactions of starting from and arriving at . This event which starts from and arrives at I call a "luminous event." [Pg 370] But we must deal with space-like intervals before we can decide whether the above theory of time-like intervals will do. It is to be observed that space-like intervals are obtained by calculation from time-like intervals. Let us imagine the following ideal experiment: An astronomer on the sun sends a message to an earthly mirror, and an astronomer on the earth sends one to a solar mirror. Each observes the time of departure and return of his own message, and the time of arrival of the other's message. Each finds that the other's message is received at a time half-way between the arrival and departure of his own message. They compare notes, and discover this fact about each other's observations. They will conclude that, according to the reckonings of both, the two messages were despatched simultaneously, and that the measure of the space-like interval between the despatch of the two messages is half the time between the despatch and return of either, i.e. about eight minutes. We may re-state the general method involved as follows: Let us have two transactions and connected by a number of causal routes, all going straight from to ; and let the longest of these consist of events. Suppose that there is another transaction such that its later event extends to , and that there is no longer causal route from to , nor any causal route at all from to . Here corresponds to the sending of the signal from the earth, to the sending of the signal from the sun, and to the arrival of the solar signal at the terrestrial observatory. The question is: What is to be the interval between and ? There cannot be a causal route from to , because if there were it could be prolonged to , and would be longer than the single event which extends from to , contra hyp. Thus no causal series connects and ; there is a causal series connecting and ; and is a transaction that begins an event which ends in the transaction . In these circumstances, we say that the interval between and [Pg 371] is of a different kind from that between and , but has the same numerical measure. The fact that this definition works is what appears as the constant velocity of light. Difficulties, however, still suggest themselves. What are we to do with the bending of light in a gravitational field? And what are we to say about the connected theory, according to which the velocity of light in vacuo is not strictly constant? We have been attempting to regard the passage of light from one body to another as a single static occurrence, involving no change within itself, and therefore having zero for its proper time, since time must be measured by changes. If we have to suppose that the light from a star alters its direction as it passes near the sun, we shall have to think of the journey of the light as a process, not as a mere continuing event. I do not believe, however, that this would be regarded as the correct account of the influence of gravitation on light. Gravitation consists in the fact that a geodesic is geometrically different from what it would be in the absence of a gravitational field; the course of the light is not "really" bent, but is "really" the straightest course geometrically possible. In any case,