The Analysis of Matter

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[Pg 75] 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): "Suppose that an observer has chosen a definite system of space co-ordinates and of time-reckoning (), and that the geometry of these is given by: Let him be under the mistaken impression that the geometry is: —that being the geometry with which he is most familiar in pure mathematics. We use to distinguish his mistaken value of the interval. Since intervals can be compared by experimental methods, he ought soon to discover that his cannot be reconciled with observational results, and so realize his mistake. But the mind does not so readily get rid of an obsession. It is more likely that our observer will continue in his opinion, and attribute the discrepancy of the observations to some influence which is present and affects the behaviour of his test-bodies. He will, so to speak, introduce a supernatural agency which he can blame for the consequences of his mistake. Let us examine what name he would apply to this agency. [Pg 76] "Of the four test-bodies considered the moving particle is in general the most sensitive to small changes of geometry, and it would be by this test that the observer would first discover discrepancies. The path laid down for it by our observer is: —i.e. a straight line in the co-ordinates (). The particle, of course, pays no heed to this, and moves in the different track: Although apparently undisturbed it deviates from 'uniform motion in a straight line.' The name given to any agency which causes deviation from uniform motion in a straight line is force according to the Newtonian definition of force. Hence the agency invoked through our observer's mistake is described as a 'field of force.' "The field of force is not always introduced by inadvertence, as in the foregoing illustration. It is sometimes introduced deliberately by the mathematician—e.g. when he introduces the centrifugal force. There would be little advantage and many disadvantages in banishing the phrase 'field of force' from our vocabulary. We shall therefore regularize the procedure which our observer has adopted. We call (16·2) the abstract geometry of the system of co-ordinates (); it may be chosen arbitrarily by the observer. The natural geometry is (16·1). "A field of force represents the discrepancy between the natural geometry of a co-ordinate system and the abstract geometry arbitrarily ascribed to it. "A field of force thus arises from an attitude of mind. If we do not take our co-ordinate system to be something different from that which it really is, there is no field of force." 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.[Pg 77] 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." [Pg 78] Again, in discussing the structure of space-time, he says (ib., p. 29): "The structure is uniform because of the necessity for knowledge that there be a system of uniform relatedness, in terms of which the contingent relations of natural factors can be expressed. Otherwise we can know nothing until we know every thing." And on p. 64: "Though the character of time and space is not in any sense a priori, the essential relatedness of any perceived field of events to all other events requires that this relatedness of all events should conform to the ascertained disclosure derived from the limited field. For we can only know that distant events are spatio-temporally connected with the events immediately perceived by knowing what these relations are. In other words, these relations must possess a systematic uniformity in order that we may know of nature as extending beyond isolated cases subjected to the direct examination of individual perception.... This doctrine leads to the rejection of Einstein's interpretation of his formulæ, as expressing a casual heterogeneity of spatio-temporal warping, dependent upon contingent adjectives." 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[Pg 79] 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[Pg 80] 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.