The analysis of matter

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 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 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.

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. In a discrete space-time, if and are two points containing events which belong to the biography of one material unit, the number of points on the route of this unit between and is always finite. If several geodesic routes lead from to , there will be a maximum to the number of points on such routes; this maximum will be the measure of the interval between and , which will therefore always be an integer. A longer route means a greater number of intermediate events, and therefore a greater amount of causal action. Thus again the interval measures the greatest amount of causal action on any causal route from to . And causal routes consist of a succession of rhythms or steady events separated by transactions.

It will be observed that, in our theory, spatial distance does not directly represent any physical fact, but is a rather complicated way of speaking about the possibility of a common causal ancestry or posterity. For example, while a light-wave is supposed to be travelling away from an atom, it has no physical relation to anything in the atom subsequent to its emission. It may be reflected back to the atom after reaching some other atom, and then half the time of the double journey (as measured at the first atom) is called the spatial distance between the two atoms (taking the velocity of light as unity). But there is no adequate ground for asserting that at every moment of the intervening time the light-ray is at a certain spatial distance from the atom; indeed, the theory of relativity vetoes such a suggestion. There is therefore, so far as I can see, no reason in physics for believing in continuous motion, except as a convenient symbolic device for dealing with the time-relations of various discontinuous changes. And whether we regard space-time as continuous or discontinuous, motion loses its fundamental character, being replaced by successions of events belonging to the biographies of bits of matter. This is inevitable if we are to hold that motion is relative and action at a distance is a fiction.

There remains a question which is of some interest. Can time be derived from causality, or must we retain temporal order as fundamental, and distinguish cause and effect as the earlier and later terms in a causal relation?[72] This question is bound up with that as to the reversibility of physical processes. If causal relations are symmetrical, so that whenever and are related as cause and effect it is physically possible that, on another occasion, and may be so related, then we must regard the time-order as something additional to the causal relation, not derivative from it. If, on the other hand, causal laws are irreversible, then we can define the time-order in terms of them, and need not introduce it as a logically separate factor. The question of reversibility is still sub judice, and I will not venture an opinion. The second law of thermodynamics asserts an irreversible process, but is purely statistical. All radiation of energy in spherical waves is prima facie irreversible, but we do not know that it really takes place. Dr Jeans suggests that there may also be converging spherical waves, and that these can be used to explain quantum phenomena.[73] For him, reversibility is a fundamental postulate.[74] I do not know whether he would maintain that the ejection of an electron or helium nucleus from a radio-active atom is a reversible process; but it must be confessed that, if it is not, the existence of radio-active elements becomes a mystery. Quantum theory has, on the whole, increased the arguments in favour of reversibility; but it cannot be said that there is as yet conclusive evidence on either side. We must, therefore, leave open the question whether the time-order of events in one causal route can be defined in terms of causal laws.