The Analysis of Matter

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[Pg 81] 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[Pg 82] 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. FOOTNOTES: [26] This subject is considered again in Chap. XIV. from a somewhat different standpoint. [Pg 84] CHAPTER IX INVARIANTS AND THEIR PHYSICAL INTERPRETATION THERE is a point of view specially associated with Professor Eddington, which it is necessary to consider at this stage, since it arises naturally in the attempt to develop physics as a self-contained deductive system. According to this view, practically all theoretical physics is a vast tautology or convention, the only part excepted, so far, being the part which involves quantum-theory. This is not the whole of Professor Eddington's view on the subject, as he has shown when not writing simply as a technical physicist;[27] but it is what we may call his "professional" view.[28] Let us begin with the conservation of momentum and of energy (or mass). Here we start from a proposition of pure mathematics. To explain this proposition will require certain preliminaries. It will be remembered that we had: We put: And we write for the minor of in this determinant, divided by . Also: which = 0 if and =1 if . The next step is the definition of the "three-index symbols," which are:[Pg 85] We can now define the tensor which Einstein uses for his law of gravitation. It is , where: summed for all values of and from 1 to 4. Einstein takes as the law of gravitation in empty space. For the moment, we are not concerned with the 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