Page:The Meaning of Relativity - Albert Einstein (1922).djvu/67

energy tensor of the electromagnetic field is known only outside the charged particles.^[1] 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),

${\frac {\delta T_{\mu \nu }}{\delta x_{\nu }}}=0$

(47c)

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, $T_{\mu }$, 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 $T_{\mu \nu }$ vanish. Integrate equation (47c) over a space section. Since the integrals of ${\frac {\delta T_{\mu 1}}{\delta x_{1}}},{\frac {\delta T_{\mu 2}}{\delta x_{2}}},{\frac {\delta T_{\mu 3}}{\delta x_{3}}}$ vanish because the $T_{\mu \nu }$ vanish at the limits of integration, we obtain

${\frac {\delta }{\delta l}}\left\{\int T_{\mu 4}dx_{1}dx_{2}d_{3}\right\}=0$

(49)

Inside the parentheses are the expressions for the

1. 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.