Page:Eddington A. Space Time and Gravitation. 1920.djvu/91

(2) that the scales and clocks used for measuring it must be falling freely. The second proviso is natural, because, if we do not leave our apparatus to fall freely, we must allow for the strain that it undergoes. The first is not a serious disadvantage, because a larger interval can be split up into a number of small intervals and the parts measured separately. In mathematical problems the same device is met with under the name of integration. To emphasise that the formula is strictly true only for infinitesimal intervals, it is written with a new notation $ds^{2}=-dx^{2}-dy^{2}-dz^{2}+dt^{2}\qquad \ldots \ldots \ldots \ldots$(2) where $dx$ stands for the small difference $x_{2}-x_{1}$, etc.

The condition that the measuring appliances must not be subjected to a field of force is illustrated by Ehrenfest's paradox. Consider a wheel revolving rapidly. Each portion of the circumference is moving in the direction of its length, and might be expected to undergo the FitzGerald contraction due to its velocity; each portion of a radius is moving transversely and would therefore have no longitudinal contraction. It looks as though the rim of the wheel should contract and the spokes remain the same length, when the wheel is set revolving. The conclusion is absurd, for a revolving wheel has no tendency to buckle—which would be the only way of reconciling these conditions. The point which the argument has overlooked is that the results here appealed to apply to unconstrained bodies, which have no acceleration relative to the natural tracks in space. Each portion of the rim of the wheel has a radial acceleration, and this affects its extensional properties. When accelerations as well as velocities occur a more far-reaching theory is needed to determine the changes of length.

To sum up—the interval between two (near) events is something quantitative which has an absolute significance in nature. The track between two (distant) events which has the longest interval-length must therefore have an absolute significance. Such tracks are called geodesics. Geodesics can be traced practically, because they are the tracks of particles undisturbed by material impacts. By the practical tracing of these geodesics we have the best means of studying the character of the natural geometry of the world. An auxiliary method is by scales and