The Analysis of Matter

and and co-punctual with , , , , , , then must be later in the -series than . If this process comes to an end with , then , , , , , , ... together form a point. If it does not come to an end with any finite , it may happen that no outside the series (, , ... ,...) is co-punctual with , , , and all the 's; in that case, , , , and these 's form a point. But if there are 's other than the 's and co-punctual with all of them, let be the first of them. Then is later in the -series than any of the finite 's. We proceed in this way as long as possible, using two principles: (1) given a series of 's ending with , let be the first in the -series after and co-punctual with the group of all the previous 's; (2) given a series of 's having no last term, take as the next the first in the -series which is after all the 's hitherto selected and co-punctual with all of them. If, at any stage, there is no such , the 's already selected form a point. Now this process must end sooner or later; for the 's (other than ) form an ascending series[Pg 301] selected from , and therefore, sooner or later, there will be no 's later than all the 's previously selected. At this stage, if not before, , , , and the 's already selected will form a point. Hence if all events can be well ordered, every event is a member of at least one point, provided every event is a member of a co-punctual quintet. The proof still holds if we only assume that all events co-punctual with a given quintet can be well ordered. Given any class of events , let be the class of those events which are co-punctual with a. Then by definition a is a point if . The necessary and sufficient condition that all the members of a should have a point in common is that a should be contained in . This condition is necessary, for, if is a point and is contained in , it follows that is contained in , and that , so that is contained in . The proof that the condition is sufficient is longer; it is as follows. If , is a point. If not, let denote the part of which is outside . Using again the -series of all events, put and so on, as long as possible. If , precedes in the -order. Hence, as before, there must come a stage when no fresh 's can be constructed. If is the class consisting of a together with all the 's yielded by the method, is a point. For (1) all the quintets in are co-punctual, by the construction; (2) a term co-punctual with all the quartets of cannot be later than all the 's, because if there were such a term we could construct more 's; (3) such a term cannot be[Pg 302] earlier than some member of because, if it were, it would have been chosen as the of that stage in the construction; hence no event outside is co-punctual with every quartet of . Hence is a point. To say that a collection of events have a point in common is to say that the collection is part (or the whole) of the class which is the point. Conversely, a collection of events may contain a sub-class which is a point; the necessary and sufficient condition for this is that should be contained in , where is the collection in question. The proof proceeds exactly as before, if we now make mean the part of which is not contained in . A group of events a is "co-punctual" if is contained in , and a "point" is a co-punctual group which cannot be enlarged without ceasing to be co-punctual. A few purely logical properties of points may be noted. Given any two classes and , if is contained in , then is contained in . Hence if and are points and is contained in , and are identical; for in that case and are respectively identical with and , and therefore if is contained in , is contained in , so that and are identical. Every co-punctual group of events contains at least one point. This has already been proved, since to say that a is a co-punctual group is to say that a is contained in . It may be taken that, in general, there are a number of points of which any given event is a member. Such a set of points will fill a "region," but not every region will be the set of points to which some one event belongs. This topic, however, cannot be dealt with until we have discussed space-time order. FOOTNOTES: [59] In this chapter and the next, I owe much to the criticism and suggestions of Mr M. H. A. Newman of St. John's College, Cambridge, who must not, however, be held responsible for their contents; on me contrary, I am convinced that he could construct a much better theory than that which follows. [60] Stetige Mengen, Monatshafte für Mathematik u. Physik., XXXI., 1921, pp. 173-204. [61] Grundzüge der Mengenlehre, Leipzig, 1914. [62] Ib., p. 211. [63] Ib., p. 213. [64] Zum Metrisationsproblem, Math. Annalen 94 (1925), pp. 309-315. [65] He defines a topological space as "normal" when any two non-overlapping closed manifolds and can be separated by two non-overlapping regions , which respectively contain them and have no boundary-points. Ib., p. 310, and Hausdorff, op. cit., p. 215. A "boundary-point" of a collection is one which has a neighbourhood that is not a sub-class of the collection. [Pg 303] CHAPTER XXIX SPACE-TIME ORDER 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. [Pg 304] 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[Pg 305] 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[Pg 306] 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