. of the second rank, 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