The analysis of matter

IN the present chapter I shall show how to develop spatio-temporal order, in the sense in which it is assumed by the general theory of relativity, without any apparatus beyond that of the preceding chapter, except a few hypotheses of the sort to be expected in founding analysis situs.

The transformations of co-ordinates which are admissible in tensor analysis are not unlimited; they are such, only, as leave relations of neighbourhood unchanged.[66] That is to say, a small displacement in one system of co-ordinates must correspond to a small displacement in any other. This requires that, independently of metrical considerations, the events of the space-time manifold should have certain relations of order. It must be possible, in certain circumstances, to say that is nearer to than to , without presupposing any quantitative measure of distance. It must be possible to construct lines along which there is a definite order, but it must be impossible to distinguish certain lines as "straight." A closed curve will be distinguishable from an open curve, but two open curves will not be distinguishable from each other, provided they have no singularities. Generally, we shall be able to make propositions belonging to analysis situs, at any rate in a sufficiently small region. But propositions about a configuration must, in the geometry we are to construct, be only such as would remain true if the configuration were subjected to any kind of deformation which does not violate continuity. It is this pre-co-ordinate geometry that concerns us in the present chapter.

The order to be introduced is of two sorts, macroscopic and microscopic. We will treat first of the former.

Let us observe, to begin with, that events may be divided into zones with respect to a given event. There are first those that are compresent with a given event, then those not compresent with it, but compresent with an event compresent with it, and so on. The th zone will consist of events that can be reached in steps, but not in , "step" being taken as the passage from an event to another which is compresent with it. We will call two points "connected" when there is an event which is a member of both. The passage from event to event by the relation of compresence may be replaced by the passage from point to point by the relation of connection. Thus points also can be collected into zones. If there is a minimum to the size of events, we may assume that it is always possible to pass from one event to another by a finite number of "steps." If so, there must be a smallest number of steps in which the passage can be made; thus every event will belong to some definite zone with respect to a given event. This is useful in the introduction of order, because we can agree that the th zone is to be nearer the origin than the th if so that it only remains to introduce order among the members of a given zone. And even here we only want such order as is involved in analysis situs, not such more rigid order as is involved, e.g., in projective geometry.

When an event can be reached from another in steps but not in , we may regard the intermediate events as forming a sort of quantized geodesic route between the two events.

In virtue of the above division into zones, which can be effected with respect to any point as origin, we can define a rather small region of space-time by means of four integers, representing the number of steps in which any point in the region can be reached from four given points. It is only within a small region of this sort, therefore, that we need the more delicate methods of microscopic order, to which we shall now proceed.

Given two points and , let us denote by "" their logical product, i.e. the events which are members of both, or, in geometrical language, the events which contain both. It is obvious that, taking the view of events explained at the beginning of the preceding chapter, will be null unless and are fairly near together. As already stated, we say that and are "connected" when is not null. Microscopic order is confined to connected points, at any rate to begin with.

We now define " is between and " as meaning: ", , are points such that is not null and is a proper part of ." An equivalent definition is: ", , are points such that is not null, and is contained in , but is not contained in ." By the help of suitable axioms, "between," so defined, can be made to give rise to the spatio-temporal order presupposed in assigning co-ordinates in the general theory of relativity. What the definition says, in geometrical language, is that every event which contains both and contains , but not every event which contains both and contains .

We must not imagine that all the points between two others lie on one line; each lies on some short route joining the end-points, a "short" route being one composed wholly of points between the end-points; but none lies on all short routes.

Before developing the formal consequences of this definition, it may be as well to consider its geometrical import. In the accompanying figure, will be between and if there are events which contain all three, but there are none which contain and without containing . (I represent events by areas.) Now if events can often be of irregular shapes such as that of the shaded area in the figure, it would seem that one event is not likely ever to be between two others according to the definition. I shall therefore assume that we may picture events as free from re-entrant angles and similar oddities. I imagine them as all oval; but formally it would do just as well if they were all four-dimensional cubes, and it would not matter whether they were large or small, provided they did not differ too much, and were all above a certain minimum. These pictorial requisites are rather for the importance of the theory to be developed than for its truth. In the preceding chapter, we assumed that events are such as to be all spheres according to one possible metric. Formally, we might equally well have assumed that there is a metric in which they are all cubes. Some assumption of this kind, as we saw, is necessary for the success of our definition of points. The other assumptions needed for its truth will be explicitly stated as they are introduced. The assumptions introduced so far in this chapter and its predecessor are:

(1) Compresence is symmetrical.

(2) Defining "events" as the field of compresence, every event is compresent with itself.

(3) Events can be well ordered; or at least those compresent with a given event can be.

(4) Any two events have a relation which is a finite power of compresence. (This is required for mapping space-time into zones.) In other words, the ancestral relation derived from compresence is connected.

We will now define a set of points as "collinear" if every pair of the set are connected, and every triad , , are such that either is contained in , or is contained in . We will define a set of points as a "line" if (1) it is collinear, (2) it is not contained in any larger collinear group with the same extremities. It will be seen that this definition is analogous to that of points. We may define a set of events as "co-punctual" when every quintet of the set are co-punctual; and we can then define a set of events as a "point" when (1) it is co-punctual, (2) it is not contained in any larger co-punctual group. This way of stating our previous definition of "points" brings out the analogy.

The "lines" that we are defining are not to be supposed "straight"; straightness is a notion wholly foreign to the geometry we are developing. Perhaps it might be better to call them "routes"; but there is no harm in calling them "lines" provided we remember that they are not supposed to be straight. For the present, we shall not be concerned with lines, but only with collinear groups of points.

Let us define a set of points as "-collinear" if (1) every pair of the set is connected; (2) given any two, , , either is between or , or is between and . We shall want such axioms as will enable us to show that such a set of points is collinear, not merely -collinear, and that their order is independent of . It is obvious that, if we put before whenever is between and , we obtain a serial order of any set of points which is -collinear. But to insure that the order shall be independent of we require the following three axioms:

(1) If , , , are points, and is contained in , and is contained in , and and are distinct, then is not contained in .

(2) If is contained in , and is contained in , than is contained in the sum of and . (It follows at once that is contained in .)

(3) If is contained in , and is contained in , then is contained in the sum of and . (It follows at once that is contained in .)

The practical effects of these three axioms are:

(1) If and are between and , and is between and , then is not between and .

(2) If is between and , and is between and , then and are between and .

(3) If is between and , and is between and , then is between and .

From these axioms we can deduce that a set of points which is -collinear is collinear. Also that, given a set of -collinear points, if is one of them, the points of the set which are beyond from a are -collinear, and retain the same order when arranged with reference to as they had when arranged with reference to . Also that, if is one of a set of -collinear points, those of the set which are between and are -collinear, and have, when arranged with reference to , the converse order to that which they had when arranged with reference to . These propositions show that we have a satisfactory definition of order among the points of a collinear set.

The above axioms are logically adequate, but regarded as asserting physical truths about events they may perhaps be regarded as more or less doubtful. We have to remember that our lines are not straight, and may therefore return into themselves. Routes with very great curvature are, however, excluded by our definition of collinearity. Consider, e.g., such a route as that in the accompanying figure. We may suppose that , , , are all connected, but and will not be between and according to the definition, because obviously an event may contain and without containing and . Thus if we wish to regard the above route from to as, in some sense, a line, it will have to be in an extended sense, namely, that it can be divided into a number of small finite parts, each of which is a line. And a set of points may be regarded as collinear in an extended sense if it is capable of a serial order such that any sufficiently small consecutive stretch of the series is a collinear set—provided that such stretch must contain not less than four points.

We can now prove, by the help of one further axiom, that any progression of collinear points all lying between two points and must have a limit.

Let our set of points be all lying on a line between and , in an order from towards . Let be the sum of all the points in (i.e. the class of members of members of ), and their product, i.e. the events which belong to every member of . Then is not null, because is contained in it, and , are connected (in virtue of the definition of collinearity).

Let consist of all the 's except , of all the except , etc. Let be the events belonging to all members of and generally let be the events belonging to all members of ; and let be the sum of all the 's. Then consists of all those events which belong to all sufficiently late 's; i.e. to say that an event is a member of is to say that there is an such that the event is a member of for all values of .

It will be observed that is contained in , therefore is contained in . It follows that, if , are two members of , there is an such that , are both members of . Hence they are both members of . Hence any five members of are co-punctual, and therefore there is at least one point which contains the whole of , since is contained in .

If there is a limit, say , to the series of 's, we require:

(1) That should be beyond all the 's, i.e. that for every and we should have contained in i.e. that we should have contained in ;

(2) That there should be no point beyond all the 's but between them and , i.e. that, if is any point such that is contained in , then is contained in .

A sufficient condition is, therefore, . If there is a point fulfilling this condition, it is the required limit.

If there is an event such that every quartet of is co-punctual with and every quartet of which is co-punctual with is a part of , then there is a point which contains and has for a member, and this point will be such that , so that it will be the required limit. But if there is no such event as , we must proceed differently.

In this case we need a new axiom, namely:

If is between and , and is a member of but not of , then there is a quartet which is contained in and but is not co-punctual with .

In the figure, represents a member of such a quartet.

Given this axiom, we proceed as follows.

Since is between and , if is a member of but not of , there is a quartet which is contained in and , but is not co-punctual with . Now is contained in ; therefore there is a quartet which is a part of but is not co-punctual with . It follows by transposition that if is a member of and every quartet of is co-punctual with , then is a member of . It follows that is a member of , , ... so that is a member of . Hence, since may be any member of , it follows that any member of which is co-punctual with the whole of is a member of . Now the terms co-punctual with the whole of constitute the class . Hence the common part of and is contained in , and is therefore equal to , since is contained in and in .

Now if is a point which contains , it follows that is contained in ; hence is contained in , and is therefore equal to , since is contained in and in . Hence is the required limit.

It follows from this that a compact series of points contained within a stretch of collinear points is continuous. It does not follow that there are compact series of points; this would require existence-axioms which there is no object in introducing, since we do not know whether space-time is continuous or not. It is, however, interesting to observe that an initial apparatus of events suffices to generate a continuous space-time of points, by means of the relations of co-punctuality and logical inclusion.

The further development of our geometry, so as to include surfaces, volumes, and four-dimensional regions, obviously presents no difficulty in principle, and I do not propose to enlarge upon it. I will merely observe that it is possible to extend the method by which we have defined points and lines so as to obtain something which we may call surfaces and regions, though not quite in the usual sense. Probably various ways of doing this are possible; the one that I suggest is the following.

A class of lines will be called "co-superficial" when any two intersect, but there is no point common to all the lines of the class.

A "surface" is a co-superficial class of lines which cannot be augmented without ceasing to be co-superficial.

A class of surfaces is "co-regional" when any two have a line in common, but no line is common to all the surfaces of the class.

A "region" is a co-regional class of surfaces which cannot be augmented without ceasing to be co-regional.

It is obvious that this method could be extended to any number of dimensions; also that it requires limitations and extensions. But it seems unnecessary to pursue the matter further, since it is plain that we have what is needed for the pre-co-ordinate geometry of space-time.

Let us now compare our constructed space-time with the spatial manifolds of analysis situs. In the preceding chapter we quoted Hausdorff's definition of a "topological" space, and we saw that, in order to prove the usual propositions about limits, it is necessary that the total number of neighbourhoods should be . Let us now define as a "neighbourhood" of a point any set of points each of which contains as a sub-class a certain finite co-punctual class of events which is a sub-class of . That is to say, if a is a co-punctual class of events each of which is a member of , the set of all the points of which a is a sub-class will be a neighbourhood of . With this definition of a "neighbourhood," it is obvious that our space has the four characteristics by which Hausdorff (loc. cit., p. 213) defines a topological space. In order to insure that our space shall also satisfy his second denumerative axiom (loc. cit., p. 263), it is necessary and sufficient to assume that the total number of events is . With this assumption, the theorems of analysis situs become applicable to our space-time manifold of points.

It remains to say a word on the subject of dimensions. We have not so far said anything explicit on this subject, though our original introduction of co-punctuality as a five-term relation could only prove satisfactory in a four-dimensional manifold. The most suitable definition of dimensions from our point of view is that of Poincaré, which is inductive. He defines a space as one-dimensional if, given any two points , , there is an isolated set of points such that no connected part of -not- contains both and . And he defines a space as -dimensional if, given any two points , , there is an ()-dimensional set of points such that no connected part of -not- contains both and . Using this definition, or any other which is purely topological, we set up the axiom that our topological space-time is to be four-dimensional.[67] This completes the material required for the topological treatment of space-time.