CHAPTER XXXVITHE MYSTERY OF MATTER
ONE of the greatest merits of the theory of relativity has been to allow us to represent gravitation as a direct consequence of the curvature of space-time. It would certainly be the crowning achievement of this superb theory if it could enable us to interpret all the manifestations of the physical universe in terms of the various types of curvature of one sole fundamental continuum, space-time. But at the present stage two separate reasons render this solution highly improbable.
We remember that in the course of our wanderings we came across two foreign tensors:,the matter-tensor, and,the electromagnetic tensor. It seems impossible to express these tensors in terms of the basic structural tensors of space-time, namely, the’s. However, in the case of the matter-tensor,the case is not yet quite clear; and we shall see that opinion is divided on the subject. In order to understand the point at issue, let us turn once more to the gravitational equations in the interior of matter:
There are ten separate equations corresponding to the ten separate components of the second-order symmetrical tensorsand.If we assume that a certain definite mesh-system has been chosen, or, again, that the observer is specified, each one of the ten components ofwill represent some physical magnitude relevant to matter at each point (seeAppendix II). For instance,will represent the density of mass of the matter at a point;,and,the three components of momentum of the matter at a point, measured in the frame of reference specified;,andwill represent the three components ofvis viva, or energy of motion, coupled with the stresses in the matter at the point considered, etc.
The equations of gravitation thus signify that whenever we recognise the existence of one of these physical magnitudes it is always accompanied by corresponding curvatures of space-time. It is usual to assume that the curvatures are produced by those concrete somethings which we call mass, momentum, energy, pressure. In this way, we must concede a duality to nature; there would exist both matter and space-time, or, better still, matter and the metrical field of space-time. Einstein, when he elaborated his hypothesis of the cylindrical universe, attempted to remove this duality by proving that it was possible to attribute the entire existence of the metrical field, hence of space-time, to the presence of matter. This attitudeled to a matter-moulding conception of the universe, elevating matter over the metrical field of space-time. And, as we recall, only when this attitude was adhered to could Mach’s belief in the relativity of all motion be accepted.
Eddington’s attitude is just the reverse. He prefers to assume that the equations of gravitation are not equations in the ordinary sense of something being equal to something else. In his opinion they are identities. They merely tell us how our senses will recognise the existence of certain curvatures of space-time by interpreting them as matter, motion, and so on. In other words, there is no matter; there is nothing but a variable curvature of space-time. Matter, momentum,vis viva, are the names we give to these curvatures on account of the varying ways they affect our senses.
Of course, Eddington’s attitude, which is reminiscent of that of Clifford, is contrary to Riemann’s ideas, since, according to Riemann, an extension could have no geometry, hence no curvature and no metrical field, in the absence of matter, which would act as an active moulding agent. Furthermore, Eddington’s views lead us into certain difficulties when we attempt to apply the principle of action.
The reason for this is plain. If we regard a certain curvature of the metrical field of space-time as being synonymous with matter, there will be no reason to consider this curvature twice: once as representative of the action of matter, and once again as representative of the action of the gravitational field. By so doing, we should be merely duplicating our results. Hence, according to Eddington’s views, there would be but one action, that of the metrical field; and we could no longer add together thetwo separate actionsin order to obtain atotal action. Calculation then shows that in regions where matter exists the principle of stationary action would break down, so that a rigid application of the principle would connote that the world must be empty of matter. Eddington feels no objection to this limited validity of the principle of action, and defends his attitude with technical arguments which we cannot reproduce here.
Up to this point the motive that has guided us has been the desire to discover unity in nature by reducing all things to some fundamental form of existence. There is, however, another way to approach the problem; and this time our guiding principle will be no longer solely mathematical harmony, but also experimental facts.
We must realise that the tensor,whose components represent those physical magnitudes which our senses perceive and our instruments measure, can at best be considered to portray effects observed only in a macroscopic way. Phenomena such as radioactive explosions and all we have learnt indirectly from the exploration of the atom go to show that matter in the final analysis is constituted by whirling electrons around which intense electromagnetic fields are present. It follows that a microscopic investigation of matter would yield a very different picture from that which we construct from our crude sense perceptions, and that could we but view the atom in an ultra-microscopic way itwould appear to us as a region subjected to intense electromagnetic actions. It would seem, therefore, that our macroscopic tensormust comprise the electromagnetic onein its constitution.
But in order to remove the duality between matter and electricity, we should have to prove thatcould be built up exclusively from some more or less complicated expression containing,together with tensors derived from it; tensors of any other kind being barred. Physically speaking, this would indicate that atomic matter or bound energy, and in particular those forms of matter called electrons and protons, could be built up in terms of electromagnetic or loose energy. Even were we to succeed in this attempt we should not have banished all duality; for though we should have got rid of,there would still remain a duality betweenand the metrical field or’s of space-time.
At any rate, all attempts to reduce matter to electromagnetic fields have thus far presented insuperable difficulties owing to the impossibility of accounting for those ultimate aspects of matter, the electron and proton, in terms ofalone. The difficulty consists in understanding how it is that the electron does not explode under the mutual repulsion of its negatively charged parts. Some kind of counterbalancing pressure, the Poincaré pressure, seems to be necessary, and it appears to be impossible to account for this mysterious pressure in terms of the electromagnetic Maxwellian stressesalone. Following Poincaré, Mie attacked the problem. His procedure was to introduce additional electromagnetic-field magnitudes which would be effective in the interior of the electron, but whose action in outside space would be negligible. By this means, he succeeded in building up matter out of purely electromagnetic quantities. Einstein comments on Mie’s theory in the following words:
“In spite of the beauty of the formal structure of this theory, as erected by Mie, Hilbert and Weyl, its physical results have hitherto been unsatisfactory. On the one hand, the multiplicity of possibilities is discouraging, and, on the other hand, those additional terms have not as yet allowed themselves to be framed in such a simple form that the solution could be satisfactory.”
Following this attempt, Einstein proved that it was possible to obviate the introduction of Mie’s supplementary electromagnetic terms provided the space of the universe were assumed to possess a slight residual curvature, such as would exist in the cylindrical universe. It would then be this residual universal space-curvature which would act as a negative pressure preventing the electron from exploding. If we note that the universal curvature concerns the metrical field, hence the field of gravitation, we see that the energy of the electron must be composed of an electromagnetic and of a gravitational part. In Einstein’s own words: “Of the energy constituting matter, three-quarters is to be ascribed to the electromagnetic field, and one-quarter to the gravitational field.”
Einstein considers that these facts constitute a powerful argument in favour of the finite universe. At all events, it must be concededthat the hypothesis of the finite universe appears to impose itself in a number of different ways. Not only astronomical considerations bearing on the low velocities of the stars, not only mathematical reasons connected with the boundary conditions, not only philosophical urges concerned with the relativity of rotation and the relativity of inertia, but also atomic or electronic phenomena, drive us towards the same solution.
Let us now pass to a study of the electromagnetic tensor itself. We can see that even if the preceding attempts had proved successful, there would still remain a duality between gravitation and matter or electricity, or, we might say, between the metrical field of space-time and the tensor.This latter tensor would still appear as an irreducible foreign entity present in space-time, marring the unity of nature. It was the aim of Weyl’s theory to rid science of this duality.
In order to understand the nature of the problem, we must state that just as the fundamental structural tensors of the metrical field,andat every point could be reduced to highly complicated expressions between the ten fundamental-potentials at every point, so now the electromagnetic tensorcan be reduced to a complicated expression between the four electromagnetic potentials, the four’s (namely,and). The unification demanded by science was therefore to express the’s in terms of the’s, or vice versa. Unfortunately, not the slightest similarity appeared to exist between the tenpotentials and the four’s. The’s appeared as foreign entities embedded in the metrical field of space-time.
All attempts to overcome this duality were unsuccessful so long as we confined our speculations to the space-time of Einstein’s theory. But there always existed the possibility that space-time was of a less simple category than had been assumed by Einstein. If, then, we could conceive of a generalised type of space-time possessing a more complicated metrical field, it might be possible to identify the fundamental’s of electromagnetics with certain particularities of structure of this generalised space-time and of its metrical field. Weyl’s theory is an attempt in this direction; and though it is still in a highly speculative stage, everything seems to suggest that Weyl is on the right track. Einstein himself has accepted a modified form of Weyl’s theory to the extent of contributing to its advancement.