Page:Eddington A. Space Time and Gravitation. 1920.djvu/209

terms ordinarily used in mechanics. If we write it in the slightly modified, but equivalent, form $G_{\mu \nu }-{\tfrac {1}{2}}g_{\mu \nu }G=-8\pi \,T_{\mu \nu }$ we have the following scheme of interpretation ${\begin{matrix}T_{11}{\text{,}}&T_{12}{\text{,}}&T_{13}{\text{,}}&T_{14}&=p_{11}+\rho u^{2}{\text{,}}&p_{12}+\rho uv{\text{,}}&p_{13}+\rho uw{\text{,}}&-\rho u{\text{,}}\\&T_{22}{\text{,}}&T_{23}{\text{,}}&T_{24}&&p_{22}+\rho v^{2}{\text{,}}&p_{23}+\rho vw{\text{,}}&-\rho v{\text{,}}\\&&T_{33}{\text{,}}&T_{34}&&&p_{33}+\rho w^{2}{\text{,}}&-\rho w{\text{,}}\\&&&T_{44}&&&&\rho {\text{.}}\end{matrix}}$ Here we are using the partitions of space and time adopted in ordinary mechanics; $\rho$ is the density of the matter, $u$, $v$, $w$ its component velocities, and $p_{11}$, $p_{12}$, $\ldots p_{33}$, the components of the internal stresses which are believed to be analysable into molecular movements.

Now the question arises, is it legitimate to make identifications on such a wholesale scale? Having identified $T_{44}$ as density, can we go on to identify another quantity $T_{34}$ as density multiplied by velocity? It is as though we identified one "thing" as air, and a quite different "thing" as wind. Yes, it is legitimate, because we have not hitherto explained what is to be the counterpart of velocity in our scheme of the world; and this is the way we choose to introduce it. All identifications are at this stage provisional, being subject to subsequent test by observation.

A definition of the velocity of matter in some such terms as "wind divided by air" does not correspond to the way in which motion primarily manifests itself in our experience. Motion is generally recognised by the disappearance of a particle at one point of space and the appearance of an apparently identical particle at a neighbouring point. This manifestation of motion can be deduced mathematically from the identifying definition here adopted. Remembering that in physical theory it is necessary to proceed from the simple to the complex, which is often opposed to the instinctive desire to proceed from the familiar to the unfamiliar, this inversion of the order in which the manifestations of motion appear need occasion no surprise. Permanent identity of particles of matter (without which the ordinary notion of velocity fails) is a very familiar idea, but it appears to be a very complex feature of the world.