The Analysis of Matter

no extraneous source from which the luminous event can discover that it is lasting a long time, and there is, in fact, no meaning in saying that it is lasting a long time. But when we say that it is reflected back to its starting-point, we mean that it has undergone a transaction which has turned it into a new luminous event, and that this new event is compresent with causal descendants of events compresent with the earlier one, these compresent events being not luminous, but of the kinds associated with matter. Now, given any two events and , neither of which is an ancestor of the other, it is possible to find a luminous event compresent with and with a descendant of . We then say that the events and have a space-like separation, whose measure is that of the time-like separation between and . In the above theory, it is assumed that, in all cases where one process or piece of matter has an effect upon another, there is at least one event which is compresent with both. This is the form taken by the denial of action at a distance. If we assume, as we have been doing, that change is discontinuous, a single period of a rhythm will contain some finite number of points. Suppose, now, that there are two rhythms such that the initial event of a period in the one is always identical with the initial event of a period in the other, but the other events are diverse; and suppose that the first rhythm contains event in a period while the second contains . Then a period of the first rhythm will contain points, and one of the second will contain . We said that the "interval" between two events was to be the number of points in the longest causal route from one to the other; hence the interval between the beginning and end of a period in either rhythm is measured by the greater of the two numbers and . Suppose this is . Then we may regard the -rhythm as having a smaller "velocity" than the -rhythm, while the frequencies[Pg 375] of the two rhythms would be the same. This suggests, in a certain class of cases, a possibility of defining "velocity" otherwise than by relative motion. How far the resulting properties of "velocity" would resemble those resulting from the usual definition, I do not know. There is no difficulty in defining what is to be meant by saying that a steady event "moves." An event occupies a number of points of space-time, which can be regarded as a four-dimensional tube divisible into sections such that all the points in one section are simultaneous, and are all later or all earlier than all the points in another section. We shall then regard our event as moving along the tube, and occupying the various instantaneous sections successively. But this does not imply any process or change within ; it merely implies transitions among events compresent with but not all compresent with each other. It seems, therefore, that everything essential to theoretical physics can be stated in terms of our theory. According to the above theory, motion is discontinuous. But this hypothesis is required for one purpose only, namely for the definition of interval. It is easy to introduce such axioms as shall make our space-time continuous, and secure, as in current physics, that discontinuity shall be confined to quantum phenomena, i.e. to what we have called "transactions." But if this is done, our definition of interval must be abandoned, and interval resumes its place as something mysterious and unaccountable. There is no logical reason why it should not have such a place; the laws of transactions have such a place in our account. But it is always intellectually satisfying when we can reduce the number of inexplicabilities. So far as I can discover, there is no good ground for supposing that motion is continuous; it is therefore worth while to develop a discontinuous hypothesis if we can thereby increase the unity and diminish the arbitrariness in our account of the physical world. [Pg 376] CHAPTER XXXVI THE GENESIS OF SPACE-TIME SPACE-TIME, as it appears in mathematical physics, is obviously an artefact, i.e. a structure in which materials found in the world are compounded in such a manner as to be convenient for the mathematician. In the present chapter, I wish to collect what has already been said on this subject in various parts of the book, and to consider the resulting metaphysical status of space-time. In the general theory of relativity, space-time appears in two ways: first, as providing a four-dimensional order; secondly, as giving rise to the metrical concept of "interval." Both are relations between "points," but both are treated mathematically as differential relations. This requires us to solve a purely mathematical problem: what is the function or process which tends towards these relations as a limit? This is on the assumption that space-time is continuous, which we do not know it to be. Let us begin with this hypothesis, and proceed afterwards to the hypothesis of discreteness. In the absence of evidence, it is necessary to develop both. For the present, therefore, I assume space-time to be continuous. This involves, or at least renders natural, the assumption that there is an infinite number of events compresent with any given event; I shall make this assumption also so long as I assume continuity. "Compresence" is assumed to be a symmetrical relation, which every term in its field has to itself, and whose field is capable of being well ordered. A group of five events is capable of a relation called "co-punctuality," which means, in effect, that there is a region common to all five. A group[Pg 377] of more than five events is called "co-punctual" when every quintet chosen out of it is co-punctual. A "point" is defined as a co-punctual group of events which cannot be added to without ceasing to be co-punctual. "Events" are defined as the field of the relation of compresence. Hence, by means of not implausible axioms, we arrive at the space-time order presupposed in the assignment of co-ordinates. This part of the theory is straightforward. When we come to "interval" there is more difficulty. In the discussion of measurement we decided, following Eddington, that equality of two intervals is what has to be defined, and that this has to be defined as a limit when both intervals tend towards zero. For this purpose, we supposed a relation of five points , , , , ' which we may express in the words: " is more nearly a parallelogram than ." From this, by means of a certain apparatus of axioms, we can arrive at what seems to be metrically necessary for mathematical physics. But this procedure is somewhat artificial. It seems natural to suppose that our relation of five points arises as follows: between any two points there is a relation, which for the moment we will call "separation," and the separation of and is more like that of and than like that of and . Thus we shall have to do with degrees of resemblance between separations of point-pairs; these separations, however, cannot exist only for infinitesimal distances, but must exist for finite distances, at any rate if they are sufficiently small. We have therefore to ask ourselves whether any physical meaning can be found for "separation," remembering that in the limit it is to have the properties of a small interval . This means to say that a separation may be of two sorts, space-like and time-like; also that the separation between two parts of a light-ray is zero. Now the separation will be time-like if there is any event at the one point which is a causal ancestor of an event at the other point; and the separation will[Pg 378] be space-like if some event at the one point but not at the other and some event at the other but not at the one have a common ancestor or descendant, but no event at either is an ancestor or descendant of, or identical with, any event at the other. We shall assume that every pair of points has some causal relation, direct or indirect; that is to say, given any two events, and there will be somewhere in space-time two compresent events of which one is an ancestor or descendant of and the other of . This is hardly more than a definition of the "world of physics"; for if an event had no causal relation, however indirect, to the part of the world which we know, it could never be inferred by us, and would in effect belong to a different universe. It follows that if two diverse points have neither a time-like nor a space-like separation, there is an event which is a member of both, but nothing at either is an effect of anything at the other. This happens with parts of a light-ray, if we suppose, as we have done, that it consists of steady events which persist until the light-ray is transformed into some other form of energy. Thus we are led to the view that the relation of separation is somehow connected with the amount of causal action intervening between the two points concerned. It is easy to give a precise meaning to this idea when we assume a discrete space-time, but it is much more difficult in a continuous space-time. Nevertheless, it is perhaps not impossible. Causality, for these purposes, may be confined to rhythms and transactions; mere relative motion, whether accelerated or uniform, will be regarded as not involving causality in the sense in which we mean it. Indirectly causality will be involved, since there will be a change of space-like separation; but the causality will be primarily concerned with other events, not with those constituting the biographies of the bodies in relative motion. In saying this, we are, I think, only interpreting the Einsteinian theory of gravitation. [Pg 379] In the preceding chapter, when we were considering a discrete space-time, we defined a time-like interval as the number of intervening points on the longest causal route connecting the two given points. The natural way to generalize this so as to become applicable to a continuous space-time would be to regard the number of points as the measure of geodesic distance; this would enable us to say that the geodesic distance traversed by a unit of matter measures the amount of causal action which it has undergone. If we further assume that, in comparing different units of matter, we must multiply by the mass to obtain a measure of the amount of causal action, then the amount in a finite motion is the integral of . But this is the amount of "action" in the technical sense.[71] It seems therefore—though this is only a tentative suggestion—that we can regard a time-like separation as the measure of the maximum amount of causal action on the various causal routes which lead from one point to another. It is to be observed that, since points are classes of events, motion from one point to another consists in the cessation of certain events and the coming into existence of others; every such change is causal when it happens along the route of a piece of matter, since the unity of a piece of matter at different times is defined by means of the concept of a causal route. There is, therefore, so far as I can see, no fundamental objection to regarding time-like separations as measuring amounts of intervening causal action, and small time-like intervals as limits of separations. Space-like intervals, as we have seen, are derivative from time-like intervals; hence they, also, depend upon amount of causal action. Passing now to the hypothesis of a discrete space-time, in which each point consists of a finite number of events, we find that a similar analysis to the above is still possible, and is in fact considerably easier than when we assume continuity.[Pg 380]