Page:Eddington A. Space Time and Gravitation. 1920.djvu/186

170 mesh a gauge whose precise length must be arbitrary. Having done this the next step is to make measurements of intervals (using our gauges). This connects the absolute properties of the world with our arbitrarily drawn mesh-system and gauge-system. And so by measurement we determine the $$g$$'s and the new additional quantities, which determine the geometry of our chosen system of reference, and at the same time contain within themselves the absolute geometry of the world—the kind of space-time which exists in the field of our experiments.

Having laid down a unit-gauge at every point, we can speak quite definitely of the change in interval-length of a measuring-rod moved from point to point, meaning, of course, the change compared with the unit-gauges. Let us take a rod of interval-length $$l$$ at $$P$$, and move it successively through the displacements $$dx_1$$, $$dx_2$$, $$dx_3$$, $$dx_4$$; and let the result be to increase its length in terms of the gauges by the amount $$\lambda l$$. The change depends as much on the difference of the gauges at the two points as on the behaviour of the rod; but there is no possibility of separating the two factors. It is clear that $$\lambda$$ will not depend on $$l$$, because the change of length must be proportional to the original length—unless indeed our whole idea of measurement by comparison with a gauge is wrong. Further it will not depend on the direction of the rod either in its initial or final positions because the interval-length is independent of direction. (Of course, the space-length would change, but that is already taken care of by the $$g$$'s.) $$\lambda$$ can thus only depend on the displacements $$dx_1$$, $$dx_2$$, $$dx_3$$, $$dx_4$$, and we may write it so long as the displacements are small. The coefficients $$\kappa_1$$, $$\kappa_2$$, $$\kappa_3$$, $$\kappa_4$$ apply to the neighbourhood of $$P$$, and will in general be different in different parts of space.

This indeed assumes that the result is independent of the order of the displacements $$dx_1$$, $$dx_2$$, $$dx_3$$, $$dx_4$$—that is to say that the ambiguity of the comparison by different routes disappears in the limit when the whole route is sufficiently small. It is parallel with our previous implicit assumption that although the length of the track from a point $$P$$ to a distant point $$Q$$