The Analysis of Matter

such as are too small to be visible or palpable. When we say that the length of an object at the equator is one metre, we do not mean that its length is that which the standard metre would have if moved from Paris to the equator. But the expansion of bodies with temperature would have been difficult to discover if it had not been possible to bring bodies of different temperatures into the same neighbourhood and measure them before their temperatures had become equal; it would also have been difficult if all bodies had expanded equally when their temperatures rose. These elementary considerations, along with many others, make rigidity an ideal, which actual bodies approach without attaining. Mere superposition thus ceases to give a measure of length: it gives still a comparison of the two bodies concerned, but not of either with the standard unit of length. To obtain the latter, we have to adjust the immediate results of the operation of measuring, by means of a mass of physical theory. If the measures which we obtain are mutually consistent, that is all we can ask; but it is possible that a change in physical theory might have given other measures which would also have been mutually consistent. Professor Eddington, in the passage which we quoted partially in introducing this discussion, is careful to say that[Pg 92] he is concerned with measurement by direct comparison. He says: "The statement that the radius of curvature is a constant length requires more consideration before its full significance is appreciated. Length is not absolute, and the result can only mean constant relative to the material standards of length used in all our measurements and in particular in those measurements which verify . In order to make a direct comparison the material unit must be conveyed to the place and pointed in the direction of the length to be measured. It is true that we often use indirect methods, avoiding actual transfer or orientation; but the justification of these indirect methods is that they give the same result as a direct comparison, and their validity depends upon the truth of the fundamental laws of nature. We are here discussing the most fundamental of these laws, and to admit the validity of the indirect methods of comparisons at this stage would land us in a vicious circle." I confess that I am puzzled by this passage. Taken in its plain and obvious sense, it means that the standard metre is to be taken from Paris, and used without any corrections for temperature, etc., because as soon as we introduce such corrections we are assuming a great deal of physics, and thus seem to be making ourselves liable to the vicious circle which, we are told, is to be avoided. It is evident, however, that this is not what Professor Eddington means, since he goes on at once to speak of the electron as making the adjustments concerned. Now the electron may be, theoretically, a perfect spatial unit, but we certainly cannot compare its size with that of larger bodies directly, without assuming any previous physical knowledge. It seems that Professor Eddington is postulating an ideal observer, who can see electrons just as directly as (or, rather, much more directly than) we can see a metre rod. In short, his "direct measurement" is an operation as abstract and theoretical as his mathematical symbolism.[Pg 93] That being admitted, we may take the electron as our spatial unit, and ask ourselves what our ideal observer could do with it. He could not take a lot of electrons and place them end on in a row, with a view to measuring a given length, since an infinite force is required to make two electrons touch. To measure ordinary lengths, he would have to take (say) hydrogen at a given temperature and pressure, enclosed in a balloon whose radius is the length to be measured; he could then count the number of electrons in the balloon and take its cube root as a measure of the said length. But to ascertain the temperature and pressure, he will have to make other measurements; moreover, he will have to assume that his balloon is spherical. Altogether, the method does not seem very practical. I have no complete theory of physical measurements to offer, but it seemed desirable to illustrate how difficult it is to say precisely what measurement means in an advanced science such as physics. We have certain postulates, such as "lengths which are equal to the same length are equal to one another," but actual measurements, when made with sufficient accuracy, are not found to verify these postulates. Therefore we invent physical laws to save the postulates. With each fresh law it becomes more difficult to say exactly what we do mean when, e.g., we give the wave-length of a certain line in the spectrum of hydrogen in terms of the metre. (This is particularly odd in view of the fact that these wave-lengths are given to more significant figures than can be warranted by the operations applicable to the standard metre itself, whose length is only known, in comparison with other lengths, to a very moderate degree of approximation.) In physical theory, measurement should rest upon an integration of the formula for . But in physical practice the of that formula can only be determined by means of measurements. Thus the only thing we seem warranted in saying[Pg 94] is this: It is possible to correct the results of actual measurements according to certain known rules, in such a way that the corrected lengths shall satisfy such postulates as Euclid's first axiom; when this is done, we find, by means of physical theory, that all electrons have the same size. But this is not, considered empirically, at all a simple fact. And considered as a statement in the deductive theory it probably has a good meaning, but one which demands much elucidation. Until this is forthcoming, all use of numbers as measures of physical quantities in theoretical physics raises problems, since we do not know what, in theoretical physics, replaces the operation of measurement as conducted in the laboratory and in daily life. The theory of length-measurement raises problems which bring us naturally to Weyl's relativistic theory of electromagnetism, which we must now briefly consider. FOOTNOTES: [27] See his essay in Science, Religion, and Reality, edited by Needham, 1925. [28] Cf. Mathematical Theory of Relativity, §§ 52, 54, 66. [29] Eddington, op. cit. p. 115. [30] Op. cit., § 66, pp. 152-155. [Pg 95] CHAPTER X WEYL'S THEORY THE theory to be considered in this chapter is, from a geometrical point of view, a natural generalization of Einstein's arbitrariness of co-ordinates; from a physical point of view, it fits electromagnetism into the deductive system, which Einstein's theory does not do. The theory is due to Hermann Weyl, and will be found in his Space, Time, Matter (1922). The puzzles about measurement considered at the end of Chapter IX. naturally suggest the point of view from which Weyl starts. As he says: "The same certainty that characterizes the relativity of motion accompanies the principle of the relativity of magnitude" (op. cit. p. 283). Measurement is a comparison of lengths, and Weyl suggests that, when lengths in different places are to be compared, the result may depend upon the route pursued in passing from the one place to the other. Lengths at the same place (i.e. having one end identical), if small, he regards as directly comparable; also he assumes continuity in the changes accompanying transportation. This is not the sum-total of his assumptions, nor the most general way of stating them; but before we can state them adequately certain explanations are necessary. Reduced to its simplest terms, the conception used by Weyl may be expressed as follows. Given a vector at a point, what are we to mean by the statement that a vector at another point is equal to it? There must be some element of convention in our definition; let us therefore, as a first step, set up a unit of length in each place, and see what limitations it is desirable to impose on our initial arbitrariness. There is, to begin with, an assumption which is made almost[Pg 96] tacitly, and that is, that we can recognize something in one place as the "same" vector as something at another place. We may perhaps take this sameness as being merely analytical: the two are the same function of the co-ordinates at their respective places. I do not think this is all that is meant, since a vector is supposed to have some physical significance; but if more is meant, it is not clear how it is to be defined. We will therefore assume that, given a function of the co-ordinates which is a vector, we shall regard the same function of other values of the co-ordinates as the "same" vector at another place. We next have to define "parallel displacement." This may be defined in various ways. Perhaps the most graphic description is to say that it is displacement along a geodesic (Eddington, op. cit. p. 71). Another definition is that it is a displacement such that the "covariant derivative" vanishes, the covariant derivative of a vector with respect to being defined as , where: For the definition of , see the beginning of Chapter IX. In the tensor calculus, covariant differentiation takes the place of ordinary differentiation for many purposes, since the covariant derivative of a tensor is a tensor, whereas the ordinary derivative is in general not a tensor. We assume that our units of length in different places are so chosen that, when a small displacement is moved to a neighbouring place by parallel displacement, the change in the measure of its length is small, and is proportional to its length. We assume, in short, that the ratio of the increase of length to the initial length for a change of co-ordinates () is: [Pg 97]So that () form a vector, . Now it is possible to express Maxwell's equations in terms of a vector which may be identified with the above vector. Hence it is possible to regard electromagnetic phenomena as explained by the variation of what is taken as the unit as we pass from point to point. I shall not attempt to explain the theory, as it would in any case be necessary to read a full account in order to grasp its significance. Here, perhaps even more than elsewhere in relativity theory, it is difficult to disentangle the conventional elements from those having physical significance. On the face of it, it might seem as though we were attempting to account for actual physical phenomena by means of a mere convention as to choice of units. But this, of course, is not what is meant. The way the unit is assigned in different places is called by Eddington the "gauge-system": this is only partially arbitrary, and is in part the representation of the physical state of the world. This has to do with the fact that vectors are not purely analytical expressions, but also correspond to physical facts. It would seem, however, that the theory has not yet been expressed with the logical purity that is to be desired, chiefly because it is not prefaced by any clear account of what is to be understood by "measurement"—or, what comes to much the same thing from the standpoint of theory, what we are to mean when we talk of "moving" a vector, whether by parallel displacement or in any other way. To "move" something, we must be able to recognize some identity between things in different places. Perhaps all this is quite clear in the minds of competent exponents of the theory, but if so they have not succeeded in conveying their thoughts without loss of clarity to readers who have not their background. When Eddington says: "Take a displacement at and transfer it by parallel displacement to an infinitely near point " (p. 200), I find myself wondering how, exactly, the displacement is to