Page:Eddington A. Space Time and Gravitation. 1920.djvu/98

which specifically refer to a particular mesh-system, although the mesh-system can have nothing to do with the matter. It is worrying not to be able to express the differences of space in a purer form without mixing them up with irrelevant differences of potential. But we have neither the vocabulary nor the imagination for a description of absolute properties as such. All physical knowledge is relative to space and time partitions; and to gain an understanding of the absolute it is necessary to approach it through the relative. The absolute may be defined as a relative which is always the same no matter what it is relative to^[1]. Although we think of it as self-existing, we cannot give it a place in our knowledge without setting up some dummy to relate it to. And similarly the absolute differences of space always appear as related to some mesh-system, although the mesh-system is only a dummy and has nothing to do with the problem.

The results for two dimensions can be generalised, and applied to four-dimensional space-time. Distance must be replaced by interval, which it will be remembered, is an absolute quantity, and therefore independent of the mesh-system used. Partitioning space-time by any system of meshes, a mesh being given by the crossing of four channels, we must specify a point in space-time by four coordinate numbers, $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$. By analogy the general formula will be ${\begin{aligned}ds^{2}&=g_{11}d{x_{1}}^{2}+g_{22}d{x_{2}}^{2}+g_{33}d{x_{3}}^{2}+g_{44}d{x_{4}}^{2}+2g_{12}dx_{1}dx_{2}\\&\qquad {}+2g_{13}dx_{1}dx_{3}+2g_{14}dx_{1}dx_{4}+2g_{23}dx_{2}dx_{3}\\&\qquad {}+2g_{24}dx_{2}dx_{4}+2g_{34}dx_{3}dx_{4}\quad {\text{........................... (3).}}\end{aligned}}$ The only difference is that there are now ten $g$'s, or potentials, instead of three, to summarise the metrical properties of the mesh-system. It is convenient in specifying special values of the potentials to arrange them in the standard form ${\begin{matrix}g_{11}&g_{12}&g_{13}&g_{14}\\&g_{22}&g_{23}&g_{24}\\&&g_{33}&g_{34}\\&&&g_{44}\end{matrix}}$

1. Cf. p. 31, where a distinction was drawn between knowledge which does not particularise the observer and knowledge which does not postulate an observer at all.