I MEAN by an "extrinsic" causal law any formula in which one piece of matter is mentioned as concerned in the behaviour of another. Newtonian gravitation afforded a perfect example of an extrinsic causal law, but Einsteinian gravitation, prima facie, does not. The question I want to consider is: Can we, in the last analysis, dispense with such laws altogether, and regard each piece of matter as completely self-determined? Or must we admit them, and, if so, in what form? And what are we to say of such matters as the emission and absorption of light?

Let us first consider Einsteinian gravitation. The theory consists in ascribing to every region of space-time a metrical structure which is obtained (roughly speaking) by superposing a number of structures which are symmetrical about centres, the centres being portions of matter; and, given the structure, each piece of matter moves in a geodesic, or rather is a geodesic. It is not very easy to see what this means when it is translated from the technical language of theoretical physics into the language of groups of events. Nevertheless, we must make the attempt.

To begin with: Can we make "matter" into a mere law according to which events occur in the places where there is no matter? This question is analogous to that of phenomenalism as discussed in Chapter XX. We there considered the possibility of explaining unperceived "things" as laws concerning the behaviour of perceived "things." Similarly we might take events which occur in empty space, and find that they were subject to laws symmetrical about centres, and define each such law as a piece of matter situated at the centre. Conversely, we might regard the supposed events in empty space as mere laws connecting events in different pieces of matter; this becomes phenomenalism if we confine the pieces of matter to human brains. There are many possible ways of turning some things hitherto regarded as "real" into mere laws concerning the other things. Obviously there must be a limit to this process, or else all the things in the world will merely be each other's washing. But the only obvious final limit is that set by phenomenalism—perhaps one ought to say, rather, that set by solipsism. If we have once admitted unperceived events, there is no very obvious reason for picking and choosing among the events which physics leads us to infer.

This argument, however, hardly warrants us in assuming events inside an electron. If we assume an electron of the Rutherford type, we shall have to say that, if anything does take place inside the electron, we can know nothing about it. No physical process passes through the electron, so that the inside, if it exists, is a prison from which nothing can escape. No event inside an electron can be compresent with an event outside it; consequently, according to the theory of Chapter XXIX., no line can cross the boundary of an electron. What goes on inside, if anything does, is irrelevant to the rest of the universe, and is not really in the same space-time as what-goes on outside. Now the world of physics is intended to be a causally interconnected world, and must be such if it is not to be a groundless fairy tale, since our inferences depend upon causal laws. Therefore if anything occurs which is causally isolated, we cannot include it in physics. We have no ground whatever for saying that nothing is causally isolated, but we can never have ground for saying: Such-and-such a causally isolated event exists. The physical world is the world which is causally continuous with percepts, and what is not so continuous lies outside physics. Thus if anything occurs inside an electron, such an occurrence does not belong to the world of physics. It would seem to follow that, if the electron is to have a definite position in space-time, it must be either a point or a hole. The former, however, is physically unsatisfactory, and the latter seems scarcely capable of an intelligible interpretation. Thus the Rutherford type of electron raises problems, however we may interpret it.

The Heisenberg electron offers a way out of these difficulties. This electron is not in a definite place, and nothing happens inside it. It is essentially a collection of radiations observable in other places than that in which the electron would formerly have been said to be. Thus the electron is reduced to a law as to occurrences in a certain region. We cannot say, on this view, that the electron is a point, or that it is a certain finite region, or that it is a hole; it is, so to speak, something of a different logical type, connected with a region through the fact that the radiations concerned have diminishing intensity as we pass away from this region, but not capable of accurate correlation with either a region or a point. Thus on this view matter consists merely of laws as to occurrences in "empty" space.

Owing to the fact that an electron at one time cannot be identified with an electron at another time where quantum changes have intervened, the conception of motion loses its definiteness where electrons are concerned. This, however, only raises difficulties when we are concerned with very minute phenomena, such as those which occur within an atom. For large-scale phenomena, such as those with which astronomy is concerned, we may still regard the electron as persisting and as moving in space-time.

We can now return to the Einsteinian theory of gravitation, which necessitated this long digression. According to this theory, each electron is associated with a crinkle, which grows less marked as we get away from the electron, but extends theoretically throughout space. The actual metrical structure of space-time in any region is obtained (roughly speaking) by superposing these crinkles. Now the metrical properties of space-time are nothing but a method of stating causal laws. In the case of gravitation, these laws have to do with the way in which the movement of one electron is connected with the positions of the others. We must suppose that the formula for interval represents something in the state of affairs at each place, and that bodies left to themselves move in geodesics, and that, so long as electromagnetic phenomena are left out of account, the formula for interval at any place is found approximately by superposing a number of spherically symmetrical formulæ, each of which corresponds to an electron in its central region. It is natural to ask, at this point, whether interval has any more physical reality than force. But I do not wish to raise this question yet, as I propose to consider it in later chapters. For the present we may say (a) that we can recognize peculiar regions in space-time, which are those that would naturally be regarded as in the immediate neighbourhood of matter; (b) that the formula for interval at any place is a function of the geodesic distances from that place to neighbouring pieces of matter; (c) that pieces of matter travel along geodesics.

The question whether, in such a theory, there is "action at a distance" is really one of words. The formula by which we determine what will happen in a given region will contain references to distant regions, and it may be said that this is all we can mean by "action at a distance." To mean more, it may be said, is to regard causality as something more than correlation, which there can be no reason for doing. If what happens in one place is correlated with what happens in another, we may be told, nothing more could be imagined in the way of action at a distance. But this is not quite what in fact occurs. What happens in one place is not correlated with what happens in another place, but with another place, which is a different thing. Different neighbourhoods have different characters, and the differences can be represented by a combination of formulæ which are spherically symmetrical. This is not action at a distance, but action according to a distance; there is nothing that cam properly be called an effect of one thing upon another at a distance from it. Thus so far, pending the discussion of interval, we have found nothing that cam properly be described as am extrinsic causal law.

Electromagnetic phenomena, if we accept Weyl's theory, will not differ importantly, so far as our present question is concerned, from gravitation. An electromagnetic field will be represented by gauge-relations between points in a neighbourhood, and there will be no ground for supposing that one piece of matter influences another; all that we can say is that a piece of matter corresponds to a metrical state of affairs which makes the geodesics different from what they would otherwise be. The motion of an electron or proton is then due to the peculiarities of the metrical state of affairs where it is, not to something even so near as the hydrogen nucleus is to its planetary electron.

But what are we to say of the emission and absorption of light? It is clear that whenever we perceive light we absorb it, that is to say, the energy in the waves of light (or light-quanta?) that hit the eye is transformed into a different kind of energy, though I should not venture to say what kind. Therefore all visual percepts involve this process of absorbing light. And if perception can ever be a source of knowledge as to things outside the percipient's body, there must be causal laws connecting what happens to the percipient with what goes on outside. It is, of course, obvious that there are such laws; we cannot revive Leibniz's windowless monads. The process of absorption and emission of light will serve as a special case, about which we have considerable knowledge, in which we can hope to analyze exactly what occurs.

Let us take, for simplicity, two hydrogen atoms, of which one emits energy which the other absorbs. But for the theory of quanta, and such phenomena as the photo-electric effect, a supposition of this sort would be impossible. If the energy radiated from a hydrogen atom in the form of light really has the shape of a spherical wave, it is impossible that the whole of it should be absorbed by one other atom, any more than the whole of the light radiated from the sun can fall on the earth. But if the light emitted by a single atom travels in a straight line (approximately), like a material particle, then it may happen to hit one atom and be absorbed whole, just as Jonah might have been swallowed by another whale. We shall have to suppose, in this case, that the spherical distribution of light round a radiating body is a statistical phenomenon, like bullets fired from a fort in all directions. This suggests the hypothesis which we have already considered in Chapter XIII., according to which nothing at all happens between the emission of light by one body and its absorption by another. In that case, empty space collapses just as the electron did, and only the surface of the electron remains. This, however, seems hardly a tenable view. The intervening space might be described as non-existent from a metrical point of view, since the interval between the emission and the absorption of a light-ray is zero; but from an ordinal point of view this is not the case, since, if and are two points on a light-ray, we can distinguish the case in which the ray goes from to from that in which it goes from to . This difference can be stated in metrical terms. For example: Let us take as our time co-ordinate the proper time of no matter what body; whatever body we choose, will be earlier than , or else, whatever body we choose, will be earlier than . Again: Suppose that at and there are mirrors, which reflect part of the ray in such a way that an observer sees both reflected rays. Then either every such observer will see the reflection from before that from , or else every such observer will see the reflection from before that from . We can free this from dependence on an observer by the following method of statement: Let be a point on the ray reflected from , and a point on the ray reflected from , so chosen that the interval between and is time-like. Then, however and may be chosen, either is always before , or is always before . This is stated in the language of the special theory, but it is still valid, mutatis mutandis, in the general theory. Thus when we say that the interval between two points on a light-ray is zero we are not denying that there is an important sense in which one is earlier than the other, and in which one can be regarded as cause and the other as effect. This suggests that the zero interval is not quite so significant as it might seem to be, and I cannot therefore accept the view that there are no events along the path of a light-ray in empty space.

Let us now return to the emission of light, ignoring absorption for the present; and let us still consider a single hydrogen atom. We are told to suppose that the electron revolves about the proton for a certain time, say in a circular orbit four times as large as the minimum orbit; then, suddenly, it decides to revolve in the minimum orbit. When this change occurs, the atom loses a certain amount of energy, which is transformed into light whose frequency is obtained by dividing the loss of energy by (Planck's constant). Whether the light travels only in one direction, or in a spherical wave, we are compelled, in the present state of physical knowledge, to leave an open question. But we do assume that something travels away from the electron, and that, if light is absorbed by another atom, that light has traversed a route from its place or places of origin. We assume also that the light has a frequency, i.e. that what travels is a periodic process. When the light is absorbed, it ceases to exist as light, although it may reappear (in fluorescence). But often its energy exists in discoverable forms—chemical forms in chlorophyl, for example. When, however, the energy exists in the form of a steady motion of the electron in its orbit, it is not discoverable until there is a change of orbit. If we had sufficiently powerful microscopes, we could see a glowing gas dissolving into a comparatively small number of spots of light, while the atoms in steady motion would be invisible. Thus we seem to reach the conclusion that the causal laws which genuinely connect one piece of matter with another are quantum laws, in which there are various stages: first, a periodic process having no outside effect; secondly, a sudden disruption of the energy of this process into two parts, one being a new periodic process in the original body, the other a periodic process travelling in empty space; thirdly, the arrival of the travelling process at another body; fourthly, a quantum change in this other body, involving absorption of the radiant energy in the production of a new steady state in the absorbing body. All genuine causal relations between different bodies, we may suppose, involve this process of sudden loss of energy by one body and its sudden acquisition, later, by another body. The older physical laws, as reinterpreted by relativity, can apparently be so stated as to leave bodies independent of each other; but I cannot see how the quantum laws can be so stated.

If one could adopt what may be called the "parcels-post" theory of radiation, according to which, when energy leaves an atom, it does so with a definite destination in view, we could simplify our account of the matter. In that case, atoms would, at most times, live a self-contained life, "the world forgetting, by the world forgot." But sometimes they would give a parcel of energy to the postman, and sometimes they would receive one from him. The postman (who is perhaps not a teetotaller) sways from side to side as he travels, and the bigger the parcel the faster he sways. But he travels at the same rate whether his parcel is big or small; and he is the only link between the atom and the rest of the world.

For the present, we dare not assume that the question is as simple as in the parcels-post illustration. Energy may (as the orthodox theory supposes) be lost by radiation into the void—lost, I mean, not mathematically, but practically. The difficulty is that we cannot put an instrument into the void to see what happens there; the attempt is just like trying to go and see what things look like from a place where there is no eye. All our actual knowledge is concerned with the boundary surfaces between matter and empty space: what is inside and outside these surfaces is conjectural. I cannot help believing that some far simpler logical scheme of physics is possible than any yet evolved, and that the simplification is most likely to come through giving up the attempt to make physical space resemble the space of percepts, of which a beginning has been made by the Heisenberg quantum mechanics. The theory of space-time developed in Chapters XXVIII. and XXIX. was, perhaps, unduly orthodox and unimaginative. Perhaps a great deal of apparatus could be cut away if we could free ourselves from the belief that we must preserve, in physics, characteristics which we find in psychological space and time. To this topic I shall devote the next chapter.