The Problem of the Rotating Disk

Abstract

In Part I of this paper, the spatial geometry of the surface of a rotating disk is examined from the standpoint of general relativity theory. Eddington's argument for a homaloidal surface is shown to be in error, and Einstein's "relative" geometry is correlated with the "intrinsic" geometry of the disk (i.e., the geometry as determined by an observer at rest on the rotating disk). The Gaussian measure of hypercurvature of the surface, at any point on the disk at radius $r$ is found to be $-3\mathrm{}\frac{{\omega }^2}{c^2}{(1-\frac{{\omega }^2{r}^2}{c^2})}^{-2\mathrm{}}$. In Part II, the temporal aspects of the rotating disk are examined and a new test of general relativity, by use of the cyclotron, is proposed: an artificially radioactive element of low atomic weight is revolved, as ions, within the cyclotron. Upon being brought to rest, the element should be found more radioactive than an equivalent sample of that element remaining at rest.