Page:The Meaning of Relativity - Albert Einstein (1922).djvu/69

know the laws which govern the constitution of these particles. In treating mechanical problems, we are therefore obliged to make use of an inexact description of matter, which corresponds to that of classical mechanics. The density $\sigma$, of a material substance and the hydrodynamical pressures are the fundamental concepts upon which such a description is based.

Let $\sigma _{0}$ be the density of matter at a place, estimated with reference to a system of co-ordinates moving with the matter. Then $\sigma _{0}$, the density at rest, is an invariant. If we think of the matter in arbitrary motion and neglect the pressures (particles of dust in vacuo, neglecting the size of the particles and the temperature), then the energy tensor will depend only upon the velocity components, $u_{\nu }$ and $\sigma _{0}$. We secure the tensor character of $T_{\mu \nu }$ by putting

$T_{\mu \nu }=\sigma _{0}u_{\mu }u_{\nu }$

(50)

in which the $u_{\mu }$, in the three-dimensional representation, are given by (41). In fact, it follows from (50) that for $q=0,T_{44}=-\sigma _{0}$ (equal to the negative energy per unit volume), as it should, according to the theorem of the equivalence of mass and energy, and according to the physical interpretation of the energy tensor given above. If an external force (four-dimensional vector, $K_{\mu }$) acts upon the matter, by the principles of momentum and energy the equation

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