Page:The Meaning of Relativity - Albert Einstein (1922).djvu/103

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, $T_{\mu \nu }$, 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 $(T_{\mu \nu })$ denotes the co-variant energy tensor of matter, ${\mathfrak {T}}_{\sigma }^{\nu }$ the corresponding mixed tensor density, then, in accordance with (83), we must require that

$0={\frac {\delta {\mathfrak {T}}_{\sigma }^{\alpha }}{\delta x_{\alpha }}}-\Gamma _{\sigma \beta }^{\alpha }{\mathfrak {T}}_{\alpha }^{\beta }$

(95)

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