of the Newtonian theory must serve as a model. This equation has its foundation in the idea that the gravitational field arises from the density of ponderable matter. It must also be so in the general theory of relativity. But our investigations of the special theory of relativity have shown that in place of the scalar density of matter we have the tensor of energy per unit volume. In the latter is included not only the tensor of the energy of ponderable matter, but also that of the electromagnetic energy. We have seen, indeed, that in a more complete analysis the energy tensor can be regarded only as a provisional means of representing matter. In reality, matter consists of electrically charged particles, and is to be regarded itself as a part, in fact, the principal part, of the electromagnetic field. It is only the circumstance that we have not sufficient knowledge of the electromagnetic field of concentrated charges that compels us, provisionally, to leave undetermined in presenting the theory, the true form of this tensor. From this point of view our problem now is to introduce a tensor, . of the second rank, [Pg 87] whose structure we do not know provisionally, and which includes in itself the energy density of the electromagnetic field and of ponderable matter; we shall denote this in the following as the "energy tensor of matter." According to our previous results, the principles of momentum and energy are expressed by the statement that the divergence of this tensor vanishes (47c). In the general theory of relativity, we shall have to assume as valid the corresponding general co-variant equation. If () denotes the co-variant energy tensor of matter, the corresponding mixed tensor density, then, in accordance with (83), we must require that be satisfied. It must be remembered that besides the energy density of the matter there must also be given an energy density of the gravitational field, so that there can be no talk of principles of conservation of energy and momentum for matter alone. This is expressed mathematically by the presence of the second term in (95), which makes it impossible to conclude the existence of an integral equation of the form of (49). The gravitational field transfers energy and momentum to the "matter," in that it exerts forces upon it and gives it energy; this is expressed by the second term in (95). If there is an analogue of Poisson's equation in the general theory of relativity, then this equation must be a tensor equation for the tensor of the gravitational potential; the energy tensor of matter must appear on the right-hand side of this equation. On the left-hand side of the equation there must be a differential tensor in the . We have to find this differential [Pg 88] tensor. It is completely determined by the following three conditions:— 1. It may contain no differential coefficients of the higher than the second.