Page:Eddington A. Space Time and Gravitation. 1920.djvu/185

XI] unit carried by different routes all agree on arrival at $$Q$$, this method is at any rate explicit. The question whether the unit at $$Q$$ defined in this way is really the same as that at $$P$$ is mere metaphysics. But if the units carried by different routes disagree, there is no unambiguous means of identifying a unit at $$Q$$ with the unit at $$P$$. Suppose $$P$$ is an event at Cambridge on March 1, and $$Q$$ at London on May 1; we are contemplating the possibility that there will be a difference in the results of measures made with our standard in London on May 1, according as the standard is taken up to London on March 1 and remains there, or is left at Cambridge and taken up on May 1. This seems at first very improbable; but our reasons for allowing for this possibility will appear presently. If there is this ambiguity the only possible course is to lay down (1) a mesh-system filling all the space and time considered, (2) a definite unit of interval, or gauge, at every point of space and time. The geometry of the world referred to such a system will be more complicated than that of Riemann hitherto used; and we shall see that it is necessary to specify not only the 10 $$g$$'s, but four other functions of position, which will be found to have an important physical meaning.

The observer will naturally simplify things by making the units of gauge at different points as nearly as possible equal, judged by ordinary comparisons. But the fact remains that, when the comparison depends on the route taken, exact equality is not definable; and we have therefore to admit that the exact standards are laid down at every point independently.

It is the same problem over again as occurs in regard to mesh-systems. We lay down particular rectangular axes near a point $$P$$; presently we make some observations near a distant point $$Q$$. To what coordinates shall the latter be referred? The natural answer is that we must use the same coordinates as we were using at $$P$$. But, except in the particular case of flat space, there is no means of defining exactly what coordinates at $$Q$$ are the same as those at $$P$$. In many cases the ambiguity may be too trifling to trouble us; but in exact work the only course is to lay down a definite mesh-system extending throughout space, the precise route of the partitions being necessarily arbitrary. We now find that we have to add to this by placing in each