The Analysis of Matter

not "really" bent, but is "really" the straightest course geometrically possible. In any case, this point arises at an advanced stage in the theory of relativity, and the considerations involved are so numerous that it would almost certainly be possible to find an interpretation consistent with our suggestion if no other obstacle existed. When an interval is space-like, it is always theoretically possible to send a light-signal from one of the events concerned to a causal descendant of the other; consequently our definition of the measure of a space-like interval is always possible. To say that the greatest velocity in nature is that of light is to say that, when two transitions are the beginning and end, respectively, of one luminous event, there is no transition which is a causal descendant of the one and a causal ancestor of the other. To say that a causal chain of transitions belongs[Pg 372] to the history of one piece of matter is to say that no two members of the chain can be connected by a chain longer than the portion of the given chain which lies between the two transitions. This is our translation of the law that the history of a piece of matter is a geodesic. The fact that the interval between two points of one light-ray is zero appears, on the above theory, to be just what might be expected. For when an event has temporal extension, that means that two events which are compresent with it have a causal relation to each other; while when an event has spatial extension, that means that two events compresent with it have a common causal ancestry or posterity. Neither happens in the case of a luminous event, which therefore has neither temporal nor spatial extension, in spite of the fact that it covers a whole region of space-time points. It will be seen that, according to the above, intervals are discrete, and are always measured by integers. There is, so far as I know, no empirical evidence for or against this view. If the integers concerned were very large, the phenomena would be sensibly the same as if intervals could vary continuously. I do not put forward the theory with any confidence in it as it stands, but rather to suggest to men with more physical competence the possibility of great changes in our picture of the world without rejecting anything probably true. In order to bring out this point, I shall now re-state the theory without interposing argumentative justifications. The world, it is suggested, consists of a number of events, each involving no change within itself, but each connected with earlier and later events by quantum or other laws which enable us to regard the earlier as the cause and the later as the effect. The quantum transition I call a "transaction." A transaction is subject to laws as to the conservation of energy and as to action. Events may be compresent, and one event may be compresent with a number of others which are[Pg 373] separated by transitions; in that case, the one event is said to last for a long time. We can even obtain a continuous time in our theory, if the number of events compresent with a given event is infinite, and their beginnings and ends do not synchronize, i.e. one of them may be compresent with two others which are not compresent with each other. But I see no reason to suppose that the number of events compresent with a given event is infinite, or to desire a theory which makes time continuous; I therefore lay no stress upon this possibility. In a transaction, or during a rhythm, the causal antecedent may consist of more than one event, and so may the causal consequent; but the events which constitute the causal antecedent must all be co-punctual, and so must those which constitute the causal consequent. Any event of the antecedent group will be called a "parent" of any event of the consequent group. When two events are connected by a chain of events, each of which is a parent of the next, the one is said to be an "ancestor" of the other, and the other a "descendant" of the one. Two events may be connected by many causal chains, but all will consist of a finite number of events, and we assume that, in the case of any two given events, there is a maximum to the number of generations in the various lines of descent connecting them. This maximum number is the measure of "interval" when the interval is time-like. When the interval is space-like, the definition of interval is slightly more complicated. To define space-like intervals, we must first say a few words about light. When a luminous event travels from one body to another, I regard the whole as one static event, involving no internal change or process. Consequently, from the standpoint of the event itself, if one could imagine a being of whose biography it formed a part, there is no time between the beginning and the end. Since nothing travels faster than light, it is impossible that two parts of one luminous event should[Pg 374] be compresent with two events of which one is a causal descendant of another; therefore there is 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