Page:Eddington A. Space Time and Gravitation. 1920.djvu/120

104 its non-Euclidean character could be ascertained by sufficiently precise measures with rigid scales.

If we lay our measuring scale transversely and proceed to measure the circumference of a circle of nominal radius $$r$$, we see from the formula that the measured length $$ds$$ is equal to $$r\,d\theta$$, so that, when we have gone right round the circle, $$\theta$$ has increased by $$2\pi$$ and the measured circumference is $$2\pi r$$. But when we lay the scale radially the measured length $$ds$$ is equal to $$dr/\sqrt{\gamma}$$, which is always greater than $$dr$$. Thus, in measuring a diameter, we obtain a result greater than $$2r$$, each portion being greater than the corresponding change of $$r$$.

Thus if we draw a circle, placing a massive particle near the centre so as to produce a gravitational field, and measure with a rigid scale the circumference and the diameter, the ratio of the measured circumference to the measured diameter will not be the famous number $$\pi = 3.141592653589793238462643383279\ldots$$ but a little smaller. Or if we inscribe a regular hexagon in this circle its sides will not be exactly equal to the radius of the circle. Placing the particle near, instead of at, the centre, avoids measuring the diameter through the particle, and so makes the experiment a practical one. But though practical, it is not practicable to determine the non-Euclidean character of space in this way. Sufficient refinement of measures is not attainable. If the mass of a ton were placed inside a circle of 5 yards radius, the defect in the value of $$\pi$$ would only appear in the twenty-fourth or twenty-fifth place of decimals.

It is of value to put the result in this way, because it shows that the relativist is not talking metaphysics when he says that space in the gravitational field is non-Euclidean. His statement has a plain physical meaning, which we may some day learn how to test experimentally. Meanwhile we can test it by indirect methods.

Suppose that a plane field is uniformly studded with hurdles. The distance between any two points will be proportional to the number of hurdles that must be passed over in getting from one point to the other by the straight route—in fact the minimum number of hurdles. We can use counts of hurdles as the equivalent of distance, and map the field by these counts. The map can be drawn on a plane sheet of paper without any