We therefore conclude from these considerations that the energy per unit volume has the character of a tensor. This has been proved directly only for an electromagnetic field, although we may claim universal validity for it. Maxwell's equations determine the electromagnetic field when the distribution of electric charges and currents is known. But we do not know the laws which govern the currents and charges. We do know, indeed, that electricity consists of elementary particles (electrons, positive nuclei), but from a theoretical point of view we cannot comprehend this. We do not know the energy factors which determine the distribution of electricity in particles of definite size and charge, and all attempts to complete the theory in this direction have failed. If then we can build upon Maxwell's equations in general, the energy tensor of the electromagnetic field is known only outside the charged particles.[13] In these regions, outside of charged particles, the only regions in which we can believe that we have the complete expression for the energy tensor, we have, by (47),
[13]It has been attempted to remedy this lack of knowledge by considering the charged particles as proper singularities. But in my opinion this means giving up a real understanding of the structure of matter. It seems to me much better to give in to our present inability rather than to be satisfied by a solution that is only apparent.
General Expressions for the Conservation Principles. We can hardly avoid making the assumption that in all other cases, also, the space distribution of energy is given by a symmetrical tensor,
, and that this complete energy tensor everywhere satisfies the relation (47c). At any rate we shall see that by means of this assumption we obtain the correct expression for the integral energy principle.
Let us consider a spatially bounded, closed system, which, four-dimensionally, we may represent as a strip, outside of which the
vanish. Integrate equation (47c) over a space section. Since the integrals of
