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

Rh energy tensor of the electromagnetic field is known only outside the charged particles. 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),

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

Inside the parentheses are the expressions for the