The Analysis of Matter

law of gravitation, but with certain identities. We put: Further, there is a rule for raising or lowering suffixes in any tensor, of which an illustration is: so that— Generalizing the notion of the "divergence" of a vector, we obtain a general definition of the divergence of any tensor. Taking a tensor of the form for purposes of illustration, its "divergence" has four components: where: and similarly for , etc. These definitions have been given in order to enunciate the proposition:[29] [Pg 86]which Eddington calls "the fundamental theorem of mechanics." In order to see the use made of this proposition, we need to introduce the "material energy-tensor," defined as: where is the "proper density" of the matter concerned—i.e. its density relative to axes moving with the matter. From this, by the usual rule for lowering a suffix, we obtain a tensor . The principles of the conservation of mass and momentum are contained in the statement that the divergence of vanishes. This suggests the identification of with , whose divergence vanishes identically—apart from a numerical factor, which, for convenience, is taken as . Thus Eddington puts: which is the law of gravitation for continuous matter. It has been necessary to make the above excursion into mathematical regions in order to be able to understand the observations which succeed to the above in Eddington's exposition (op. cit. p. 119). He says: [Pg 87] "Appeal is now made to a Principle of Identification. Our deductive theory starts with the interval ..., from which the tensor is immediately obtained. By pure mathematics we derive other tensors.... These constitute our world-building material; and the aim of the deductive theory is to construct from this a world which functions in the same way as the known physical world. If we succeed, mass, momentum, stress, etc., must be the vulgar names for certain analytical quantities in the deductive theory; and it is this stage of naming the analytical tensors which is reached in (54·3). If the theory provides a tensor , which behaves in exactly the same way as the tensor summarizing the mass, momentum and stress of matter is observed to behave, it is difficult to see how anything more could be required of it." There are a number of other examples of the same method in Eddington's work, but we may take the above as typical, since it is the simplest mathematically. It is worth while to consider the nature of the method, apart from its technical embodiment. This is the more necessary, as it is not easy to be clear as to the logical and empirical elements in theoretical physics as developed by the above method. Fundamentally, the method is the same as that which has always been pursued when mathematics has been applied to the physical world. The aim has been to obtain mathematical laws which gave correct results wherever they could be tested by observation. The fewer and more general and more comprehensive the laws, the more scientific taste was gratified. Newton's law of gravitation was better than Kepler's laws, both because it was one law instead of three, and because it gave a larger number of correct deductions. But at every stage the subject-matter of physics grows more abstract, and its connection with what we observe grows more remote. Eddington's ideal is to start with only one fundamental law—namely, the formula for is—which, as generalized by Weyl, will give electromagnetic equations as well as gravitation. From this one fundamental law, by pure mathematics, we deduce the existence of quantities behaving in certain ways. Elementary theorizing from observation has led us to believe that there are quantities connected with what we observe which behave in these ways. We therefore identify the observed quantities with the deduced quantities. This is, in essence, the same sort of thing as we do when we associate what we see with light-waves. We may thus regard physics from the two points of view, the inductive and the deductive. In the latter, we start from the formula for interval (together with certain other assumptions), and we deduce by mathematics a world having certain mathematical characteristics. In the inductive view, the same mathematical characteristics[Pg 88] are arrived at, but they are now those which may be supposed to belong to the physical world in its entirety if we supplement observation by means of the postulate that everything happens in accordance with simple general laws. We may thus say that the world of elementary physics is semi-abstract, while that of deductive relativity-theory is wholly abstract. The appearance of deducing actual phenomena from mathematics is delusive; what really happens is that the phenomena afford inductive verification of the general principles from which our mathematics starts. Every observed fact retains its full evidential value; but now it confirms not merely some particular law, but the general law from which the deductive system starts. There is, however, no logical necessity for one fact to follow given another, or a number of others, because there is no logical necessity about our fundamental principles. The question of interpretation, it must be admitted, is somewhat difficult when physics is conceived in this very abstract manner. What, for example, is ? We start from a view which is, to a certain extent, intelligible in terms of observation. In the case of a time-like interval, it is the time which elapses between the two events according to a clock, not subject to constraints, which is present at both events. On the earth's surface, the time measured by a clock can be inferred, with suitable precautions, from the visual perceptions of a careful observer. In the case of a space-like interval, is the distance between two events as estimated by measurements carried out on a body which is present at both, and for which the two events are simultaneous. The elementary operation of measuring lengths is here supposed possible. But when we pass from this initial view to the abstract view which is required by the general theory of relativity, the interval can only be actually estimated by using rather elaborate physics to make deductions from what can be[Pg 89] actually observed by means of clocks and footrules. For logical theory, the interval is primitive, but from the point of view of empirical verification it is a complicated function of empirical data, deduced by means of physics in its semi-abstract form. The unity and simplicity of the deductive edifice, therefore, must not blind us to the complexity of empirical physics, or to the logical independence of its various portions. In particular, when the conservation of mass or of momentum appears as an identity, that is only true in the deductive system; in their empirical meaning, these laws are by no means logical necessities. There might easily be a world in which they were false, and it might be capable of a treatment as unified and mathematical as the general theory of relativity; but, if so, the fundamental laws would be different. What is novel and interesting in the point of view we have been considering is the character of the relation between empirical and deductive physics. But there is no real diminution of the need for empirical observation. I do not for a moment suggest that anything in the above is a criticism of Professor Eddington; indeed, I imagine he would regard it as a string of truisms. I have been concerned only to guard against a possible misunderstanding on the part of those who do not feel for mathematics the contempt which is bred of familiarity. In the foregoing remarks, however, we have neglected one important aspect of Eddington's theory. In addition to the fact that the whole general theory of relativity can be deduced from a few simple assumptions, interest attaches to the manner of the deduction and the considerations by which the substantial import of mathematical formulæ is made less, or at least other, than would naturally be supposed. A good example is afforded by a paragraph headed "Interpretation of Einstein's Law of Gravitation."[30] The law concerned is not[Pg 90] , which is not supposed to be quite accurate where stellar distances are concerned; it is the modified law: where must be very small, so small that within the solar system the new law gives the same results, within the limits of observation, as . The new law is shown to be equivalent to the assumption that, in empty space, the radius of curvature in every direction is everywhere But this is interpreted as a law about our measuring rods—namely, that they adjust themselves to the radius of curvature at any place and in any direction. It is interpreted as meaning: "The length of a specified material structure bears a constant ratio to the radius of curvature of the world at the place and in the direction in which it lies." And the following gloss is added: "The law no longer appears to have any reference to the constitution of an empty continuum. It is a law of material structure showing what dimensions a specified collection of molecules must take up in order to adjust itself to equilibrium with the surrounding conditions of the world." In particular, electrons must make these adjustments, and it is suggested elsewhere that the symmetry of an electron and its equality with other electrons are not substantial facts, but consequences of the method of measurement (pp. 153-4). One cannot complain of an author for not doing everything, but at this point most readers will feel a desire for some discussion of the theory of measurement. The elementary meaning of measurement of lengths is derived from superposition of a supposedly rigid body. A rigid body, as Dr Whitehead has pointed out, is primarily one which seems rigid, such as a steel bar in contradistinction to a piece of putty. When I say that a body "seems" rigid, I mean that it looks and feels as if it were not altering its shape and size.[Pg 91] This, so far as it can be relied upon, implies some constant relation to the human body: if the eye and the hand grew at the same rate as the "rigid" body, it would look and feel as if it were unchanging. But if other objects in our immediate environment did not grow meanwhile, we should infer that we and our measure had grown. There would, however, be no meaning in the supposition that all bodies are bigger in certain places than they are in certain others; at least, if we suppose the alteration to be in a fixed ratio. If we do not add this proviso, there is a good meaning in the supposition; in fact, we do actually believe that all bodies are bigger at the equator than at the North Pole, except 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