Page:Eddington A. Space Time and Gravitation. 1920.djvu/143

VIII] We come now to another kind of test. In the statement of the law of gravitation just given, a quantity $$s$$ is mentioned; and, so far as that statement goes, $$s$$ is merely an intermediary quantity defined mathematically. But in our theory we have been identifying $$s$$ with interval-length, measured with an apparatus of scales and clocks; and it is very desirable to test whether this identification can be confirmed—whether the geometry of scales and clocks is the same as the geometry of moving particles and light-pulses.

The question has been mooted whether we may not divide the present theory into two parts. Can we not accept the law of gravitation in the form suggested above as a self-contained result proved by observation, leaving the further possibility that $$s$$ is to be identified with interval-length open to debate? The motive is partly a desire to consolidate our gains, freeing them from the least taint of speculation; but perhaps also it is inspired by the wish to leave an opening by which clock-scale geometry, i.e. the space and time of ordinary perception, may remain Euclidean. Disregarding the connection of $$s$$ with interval-length, there is no object in attributing any significance of length to it; it can be regarded as a dynamical quantity like Action, and the new law of gravitation can be expressed after the traditional manner without dragging in strange theories of space and time. Thus interpreted, the law perhaps loses its theoretical inevitability; but it remains strongly grounded on observation. Unfortunately for this proposal, it is impossible to make a clean division of the theory at the point suggested. Without some geometrical interpretation of $$s$$ our conclusions as to the courses of planets and light- waves cannot be connected with the astronomical measurements which verify them. The track of a light-wave in terms of the coordinates $$r$$, $$\theta$$, $$t$$ cannot be tested directly; the coordinates afford only a temporary resting-place; and the measurement of the displacement of the star-image on the photographic plate involves a reconversion from the coordinates to $$s$$, which here appears in its significance as the interval in clock-scale geometry.

Thus even from the experimental standpoint, a rough correspondence of the quantity $$s$$ occurring in the law of gravitation with the clock-scale interval is an essential feature. We have