[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. [Pg 53] 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 , and vanish because the vanish at the limits of integration, we obtain Inside the parentheses are the expressions for the momentum of the whole system, multiplied by , together with the negative energy of the system, so that (49) expresses the conservation principles in their integral form. That this gives the right conception of energy and the conservation principles will be seen from the following considerations. PHENOMENOLOGICAL REPRESENTATION OF THE ENERGY TENSOR OF MATTER. Hydrodynamical Equations. We know that matter is built up of electrically charged particles, but we do not know the laws which govern the constitution of these particles. In treating mechanical problems, we are therefore obliged to make use of an [Pg 54] inexact description of matter, which corresponds to that of classical mechanics. The density , of a material substance and the hydrodynamical pressures are the fundamental concepts upon which such a description is based. FIG. 3. Let be the density of matter at a place, estimated with reference to a system of co-ordinates moving with the matter. Then , the density at rest, is an invariant. If we think of the matter in arbitrary motion and neglect the pressures (particles of dust in vacuo, neglecting the size of the particles and the temperature), then the energy tensor will depend only upon the [Pg 55] velocity components, and . We secure the tensor character of by putting in which the , in the three-dimensional representation, are given by (41). In fact, it follows from (50) that for , (equal to the negative energy per unit volume), as it should, according to the theorem of the equivalence of mass and energy, and according to the physical interpretation of the energy tensor given above. If an external force (four-dimensional vector, ) acts upon the matter, by the principles of momentum and energy the equation