The analysis of matter

THE importance of geodesics arises through the law that, in the general theory of relativity, a particle not subject to constraints moves in a geodesic. But let us first consider what a geodesic is.

An adventurous pedestrian in the Alps may wish to go from a place in one valley to a place in another by the shortest route—i.e. the shortest compatible with remaining all the time on the earth's surface. He cannot determine the shortest route by looking at a large-scale map and drawing a straight line between the two places, for if this line involves a greater average gradient than another it may be longer, in distance as well as in time, than another route which slopes gradually to the head of a pass and then down again. What the traveller is seeking is a "geodesic"—i.e. the shortest line that can be drawn on the earth's surface between the two points. In the absence of hills—e.g. on the sea—the shortest route is by a great circle. On complicated surfaces, geodesics may become very complicated curves. The definition is not exactly "the shortest route between two points." The definition is that the distance along a geodesic from any one of its points to any other must be "stationary"—i.e. such that either all very slightly different paths are longer, or all very slightly different paths are shorter. This means that, for small variations of path, the first-order change of length is zero. In effect, in the ordinary geometry of surfaces the geodesic distance is a minimum, and in relativity theory it is a maximum. This is not so great a difference as it may seem to the non-mathematical reader, since the geodesic distance concerned in relativity theory is more analogous to what would ordinarily count as lapse of time than to what would ordinarily count as distance in space.

Let us try to make the matter a little more concrete. The earth, in its annual revolution, travels from place to place in space-time; between the positions of Greenwich Observatory on two occasions six months apart, there is a certain interval. From the point of view of an observer in the sun, the interval would formerly have been divided into two parts—namely, six months and about 186,000,000 miles. But from the point of view of the observer at Greenwich there is only one interval—namely, time—since the place concerned is the same on both occasions. Given a clock which travels without constraint from one point of space-time to another, the interval between these two points is what that clock registers as the time between them. I say that if a clock were constrained to travel by some other slightly different route, so as to be present at Greenwich Observatory on two occasions six months apart, but absent from the earth in the meantime, the time which that clock would register as having been taken by its journey would be less than six months. The interval between distant points is not, like distance in geometry, something which can be defined independently of the route chosen. The interval must be obtained by integration along a specified route, and a geodesic route is one which makes the interval greater than it is by any slightly different route. The time between two given events at which a man is present seems less if he has spent the intervening time in rapid travel than if he has let himself drift passively; this is a sort of law of cosmic boredom. All bodies, left to themselves, choose the course which is at each moment the most boring, in the sense that it makes the time between two given events seem longest. However, it is time to have done with these irrelevancies, and return to seriousness.

Since the small interval is independent of the co-ordinates, a geodesic also is independent of them. We can easily obtain the differential equations which a geodesic must satisfy, and these equations must be satisfied by the same lines whatever system of co-ordinates we are employing. From a given point, geodesics start in all directions. Some of these are the paths of freely moving particles; others are not. The law that the path of a particle is a geodesic does not tell us quite as much as it seems to do, since it is only by observation of the motions of bodies that we discover what paths are geodesics. Assuming that the orbit of the earth is a geodesic, we can draw inferences as to the nature of the formula for in the sun's gravitational field. For we have no a priori knowledge about the coefficients which appear in the formula for ; their values are to be deduced from observation. What we can say is that it is possible, compatibly with observed facts, so to determine the that the path of a body in a gravitational field shall be a geodesic. In fact, we get in this way a more accurate representation of the facts than we got from the Newtonian law, but the observable differences between the two are few and minute.

Although the new law of gravitation and the old do not lead to very different results—as, indeed, they could not, since the old law accorded closely with observed facts—yet the difference in the ideas involved is very great. A planet, in the new theory, is moving freely, whereas in the old theory it was subject to a central force directed towards the sun. In the old theory, the planet moved in an ellipse; in the new theory, it moves in the nearest possible approach to a straight line—to wit, a geodesic. In the old theory, the sun was like a despotic government, emitting decrees from the metropolis; in the new, the solar system is like the society of Kropotkin's dreams, in which everybody does what he prefers at each moment, and the result is perfect order. The odd thing is that, as far as observation goes, the difference between these two theories is exceedingly minute. To the plain man, it would seem impossible to reconcile the statement that the earth moves in an ellipse with the statement that it moves in a sort of straight line, however queer the sort may be. And yet almost the whole of the difference between these two statements is a matter of convention. It is possible to adhere to Euclidean space even now; this requires a different way of stating Einstein's law of gravitation, but does not demand the rejection of anything that has been proved true. Dr Whitehead considers this plan preferable to Einstein's. What may be called the new orthodoxy, per contra, is set forth by Professor Eddington. It will be worth while to consider the point at issue between them.

Professor Eddington says (op. cit., p. 37):

It is not quite clear why the man who uses forces with a conventional geometry should be regarded as making a "mistake," while the man who says that free particles travel in geodesics, and to justify himself has a queer geometry, is thought to be saying something substantially more accurate. It is true that we must not conceive "force" as an actual agency, as the older mechanics did; it is merely part of the method of describing how bodies move. But as soon as this is recognized, it is a mere question of convenience whether we speak of forces or not. Let it be conceded that the method of the general theory of relativity is better from a logico-æsthetic point of view; I do not see, however, why we should regard it as any more "true." I am not considering, at the moment, the fact that Einstein's law of gravitation gives a slightly more accurate picture of the phenomena than Newton's, since this is not really relevant to the particular point at issue.

Let us now consider Dr Whitehead's view, which is, on this point, the opposite of Professor Eddington's. In the Preface to The Principle of Relativity,[1] he says:

"As the result of a consideration of the character of our knowledge in general, and of our knowledge of nature in particular, ... I deduce that our experience requires and exhibits a basis of uniformity, and that in the case of nature this basis exhibits itself as the uniformity of spatio-temporal relations. This conclusion entirely cuts away the casual heterogeneity of these relations which is the essential of Einstein's later theory. It is this uniformity which is essential to my outlook, and not the Euclidean geometry which I adopt as lending itself to the simplest exposition of the facts of nature. I should be very willing to believe that each permanent space is either uniformly elliptic or uniformly hyperbolic, if any observations are more simply explained by such a hypothesis. It is inherent in my theory to maintain the old division between physics and geometry. Physics is the science of the contingent relations of nature, and geometry expresses its uniform relatedness."

Again, in discussing the structure of space-time, he says (ib., p. 29):

And on p. 64:

Thus whereas Eddington seems to regard it as necessary to adopt Einstein's variable space, Whitehead regards it as necessary to reject it. For my part, I do not see why we should agree with either view: the matter seems to be one of convenience in the interpretation of formulæ. Nevertheless, Dr Whitehead's arguments deserve careful examination.

The main force of the above passages is epistemological: the question involved is the Kantian one, How is knowledge possible? I do not wish to deal with this question in its general form. But without going into theory of knowledge, there is what may be called a common-sense answer. Einstein enables us to predict what in fact can be predicted about astronomical occurrences, and that seems all that ought to be demanded of him. Dr Whitehead objects to the "casual" heterogeneity of space-time in Einstein's system. In a sense, this adjective is justified, since the character of space-time in any region depends upon circumstances which can only be ascertained empirically—namely, the distribution of matter in the neighbourhood. But in another sense the adjective is not justified, since Einstein's law of gravitation gives the rule according to which space-time is affected by the neighbourhood of matter. To say that we cannot, by the help of this rule, know in advance the geometry of a region we have not explored, seems an insufficient objection, since we also cannot know what astronomical occurrences will take place unless we know the distribution of matter. Einstein, like other people, assumes the permanence of matter; this is a point to be considered in another connection, but it has no particular relevance to the present issue. The way the heavenly bodies move depends upon the distribution of matter in their neighbourhood, which is, in Dr Whitehead's phrase, "casual." Even by assuming Euclidean geometry we cannot make astronomical predictions unless we assume that we know the important facts about the distribution of matter in the region concerned. Whether we put the consequences of these facts into our geometry or not does not seem to make any real difference to the possibility of physical knowledge. In all theoretical physics, there is a certain admixture of facts and calculations; so long as the combination is such as to give results which observation confirms, I cannot see that we can have any a priori objection. Dr Whitehead's view seems to rest upon the assumption that the principles of scientific inference ought to be in some sense "reasonable." Perhaps we all make this assumption in one form or another. But for my part I should prefer to infer "reasonableness" from success, rather than set up in advance a standard of what can be regarded as credible.

I do not therefore see any ground for rejecting a variable geometry such as Einstein's. But equally I see no ground for supposing that the facts necessitate it. The question is, to my mind, merely one of logical simplicity and comprehensiveness. From this point of view, I prefer the variable space in which bodies move in geodesics to a Euclidean space with a field of force. But I cannot regard the question as one concerning the facts.

The conclusion would seem to be, therefore, that, when physics is considered, as we are now considering it, as a deductive system, we do well to adopt the Einsteinian interpretation: free particles move in geodesics, and the law of gravitation is a law as to how geodesics are shaped in the l neighbourhood of matter. This view is essentially simple, though it leads to complicated mathematics. It accords with the facts, and it puts the law of gravitation in a recognizable place among physical principles, instead of leaving it, as heretofore, an isolated and unrelated law. I propose, therefore, to continue to adopt Einstein's view as to the best way of interpreting the principles of physics, without suggesting that no other way is logically possible.

There is one matter of great theoretical importance, which is not very clear in the usual accounts of relativity. How do we know whether two events are to be regarded as happening to the same piece of matter? An electron or a proton is supposed to preserve its identity throughout time; but our fundamental continuum is a continuum of events. One must therefore suppose that one unit of matter is a series of events, or a series of sets of events. It is not clear what is the theoretical criterion for determining whether two events both belong to one such series. We may assume, I suppose, that two events which overlap—i.e. which are both present at some point of space-time—must belong to one unit of matter. (It is not to be assumed that an event which belongs to one unit of matter belongs to no other.) We may also assume that two events which have a space-like interval, or have a zero interval without overlapping, do not belong to one unit of matter. But when two events have a time-like interval, there is no obvious criterion. Any two such events can be connected by a geodesic in which any two points have a time-like separation; therefore, so far as the laws of dynamics are concerned, they might both belong to the same material unit. Yet sometimes we think they do, and sometimes we think they do not. It is evidently part of the business of physics to tell us how we are to decide this question in a given case. What can we say about it?[26]

The decision must depend upon intermediate history—i.e. upon the existence of some series of intermediate events (or sets of events) following each other according to some law. If there exists any law which is in fact obeyed by strings of events, such a law can be used to define what we mean by one material unit. We know that there are such laws, but their importance in this connection is not emphasized, because it has hardly been realized that there is a problem owing to the substitution of events for bits of matter as the fundamental stuff of physics. For common sense, there is a more or less vague law of what may be called qualitative continuity. If you look persistently in a given direction, what you see, as a rule, alters gradually; there are exceptions, such as explosions, but they are rare. (I am not talking of a theoretical gradualness, but of one that is obvious to untrained perception.) If you see, say, a well-defined red patch, whose shape and tint do not alter greatly while you are looking, you conclude that there is a material object there, especially if you can touch it whenever you choose. Common sense achieves in this way a considerable measure of constancy in its objects. More is achieved by reducing matter to molecules, more still by reducing it to atoms, and yet more by reducing it to protons and electrons. But physicists would not feel pleased with electrons and protons but for the fact that their tables and chairs, their laboratories and their books, consist, on the whole, of the same electrons and protons on different occasions. Qualitative continuity remains the basis of the whole proceeding. Suppose, one evening, you were to say to an astronomer: How do you know that that white patch in the sky is the moon? He would stare at you, and think you mad. He would not reply: because the course and phases of the moon have been worked out by astronomical theory, and that is where the moon ought to be, and the shape it ought to have, at the present moment in this latitude and longitude. What he would say is: Why, can't you see it's the moon? To which the right answer would be: Yes, I can, but I didn't suppose you could, because you ought to have got beyond such a crude criterion.

Moreover, there are identities in physics which are not material. A wave has a certain identity; if this were not the case, our visual perceptions would not have the intimate connection they in fact do have with physical objects. Suppose we see several lamps simultaneously: we are able to distinguish them because each sends out its own light-waves, which preserve their individuality until they reach the eye. Our chief reason for not regarding a wave as a physical object seems to be that it is not indestructible. But this is not our only reason, since, if it were, we might regard the energy of a wave as a physical object. We do not regard energy as a "thing," because it is not connected with the qualitative continuity of common-sense objects: it may appear as light or heat or sound or what not. But now that energy and mass have turned out to be identical, our refusal to regard energy as a "thing" should incline us to the view that what possesses mass need not be a "thing." We seem driven, therefore, to the view advocated by Eddington, that there are certain invariants, and that (with some degree of inaccuracy) our senses and our common sense have singled them out as deserving names. The correct theoretical definition of a single piece of matter will thus depend upon the mathematical invariants resulting from our formula for interval. This topic, however, demands a new chapter.