The events in the physical world have relations to each other which are of the sort that have led to the notions of space and time. They have relations of order, so that we can say that one event is nearer to a second than to a third. In this way we can arrive at the notion of the “neighbourhood” of an event: it will consist roughly speaking of all the events that are very near the given event. When we say that neighbouring events have a certain relation, we shall mean that the nearer two events are to each other, the more nearly they have this relation, and that they approximate to having it without limit as they are taken nearer and nearer together.
Two neighbouring events have a measurable quantitative relation called “interval”, which is sometimes analogous to distance in space, sometimes to lapse of time. In the former case it is called space-like, in the latter time-like. The interval between two events is time-like when one body might be present at both—for example, when both are parts of the history of your body. The interval is space-like in the contrary case. In the marginal case between the two, the interval is zero; this happens when both are parts of one light-ray.
The interval between two neighbouring events is something objective, in the sense that any two careful observers will arrive at the same estimate of it. They will not arrive at the same estimate for the distance in space or the lapse of time between the two events, but the interval is a genuine physical fact, the same for all. If a body can travel freely from one event to the other, the interval between the two events will be the same as the time between them as measured by a clock travelling with the body. If such a journey is physically impossible, the interval will be the same as the distance as estimated by an observer to whom the two events are simultaneous. But the interval is only definite when the two events are very near together; otherwise the interval depends upon the route chosen for travelling from the one event to the other.
Four numbers are needed to fix the position of an event in the world; these correspond to the time and the three dimensions of space in the old reckoning. These four numbers are called the co-ordinates of the event. They may be assigned on any principle which gives neighbouring co-ordinates to neighbouring events; subject to this condition, they are merely conventional. For example, suppose an aeroplane has had an accident. You can fix the position of the accident by four numbers: latitude, longitude, altitude above sea-level, and Greenwich Mean Time. But you cannot fix the position of the explosion in space-time by means of less than four numbers.
Everything in relativity-theory goes (in a sense) from next to next; there are no direct relations between distant events, such as distance in time or space. And of course there are no forces acting at a distance; in fact, except as a convenient fiction, there are no “forces” at all. Bodies take the course which is easiest at each moment, according to the character of space-time in the particular region where they are; this course is called a geodesic.
Now it will be observed that I have been speaking freely of bodies and motion, although I said that bodies were merely certain strings of events. That being so, it is of course necessary to say what strings of events constitute bodies, since not all continuous strings of events do so, not even all geodesics. Until we have defined the sort of thing that makes a body, we cannot legitimately speak of motion, since this involves the presence of one body on different occasions. We must therefore set to work to define what we mean by the persistence of a body, and how a string of events constituting a body differs from one which does not. This topic will occupy the next chapter.
But it may be useful, as a preliminary, to teach our imagination to work in accordance with the new ideas. We must give up what Whitehead admirably calls the “pushiness” of matter. We naturally think of an atom as being like a billiard-ball; we should do better to think of it as like a ghost, which has no “pushiness” and yet can make you fly. We have to change our notions both of substance and of cause. To say that an atom persists is like saying that a tune persists. If a tune takes five minutes to play, we do not conceive of it as a single thing which exists throughout that time, but as a series of notes, so related as to form a unity. In the case of the tune, the unity is æsthetic; in the case of the atom, it is causal. But when I say “causal” I do not mean exactly what the word naturally conveys. There must be no idea of compulsion or “force”, neither the force of contact which we imagine we see between billiard-balls nor the action at a distance which was formerly supposed to constitute gravitation. There is merely an observed law of succession from next to next. An event at one moment is succeeded by an event at a neighbouring moment, which, to the first order of small quantities, can be calculated from the earlier event. This enables us to construct a string of events, each, approximately, growing out of a slightly earlier event according to an intrinsic law. Outside influences only affect the second order of small quantities. A string of events connected, in this way, by an approximate intrinsic law of development is called one piece of matter. This is what I mean by saying that the unity of a piece of matter is causal. I shall explain this notion more fully in later chapters.
[CHAPTER XI]
CAUSAL LAWS IN PHYSICS
In the last chapter we spoke about the substitution of space-time for space and time, and the effect which this has had in substituting strings of events for “things” conceived as substances. In this chapter we will deal with cause and effect as they appear in the light of modern science. It is at least as difficult to purge our imagination of irrelevances in this matter as in regard to substance. The old-fashioned notion of cause appeared in dynamics as “force”. We still speak of forces just as we still speak of the sunrise, but we recognise that this is nothing but a convenient way of speaking, in the one case as in the other.
Causation is deeply embedded in language and common sense. We say that people build houses or make roads: to “build” and to “make” are both notions involving causality. We say that a man is “powerful”, meaning that his volitions are causes over a wide range. Some examples of causation seem to us quite natural, others less so. It seems natural that our muscles should obey our will, and only reflection makes us perceive the necessity of finding an explanation of this phenomenon. It seems natural that when you hit a billiard-ball with a cue it moves. When we see a horse pulling a cart, or a heavy object being dragged by a rope, we feel as if we understood all about it. It is events of this sort that have given rise to the common-sense belief in causes and forces.