Ehrenfest Paradox - Common Misconceptions

Common Misconceptions

Surveying the somewhat dismal history sketched in the paper by Grøn (and several recent arXiv preprints which repeat various long-corrected errors originally made by earlier authors), we can identify a number of major conceptual errors which are common to many incorrect claims which have been made over the years concerning "the geometry of a rotating disk":

1. The assumption that there is a unique geometry, even a Riemannian geometry, describing a rotating disk as measured by disk-riding observers. In fact, there are several distinct but operationally significant notions of "distance" which can be employed by accelerating observers (such as the Langevin observers), even in flat spacetime. These are not even symmetric for large distances. However, for small distances they do all agree with a Riemannian metric, the Langevin-Landau-Lifschitz metric. (See Born coordinates for mathematical details.)
2. The assumption that the geometry is defined on some spatial hyperslice. In fact, it is defined on the quotient manifold obtained by collapsing the world lines of the Langevin observers to points.
3. Attempts to compare the geometry of an initially non-rotating disk "before" and "after" a spin-up phase which avoid any consideration of how the material of the disk reacts to being stressed during the spin-up phase are doomed to fail, as Planck already knew in 1910.
4. Ignoring the common-sense expectation that an initially non-rotating disk which is spun-up should exhibit stresses similar to those we would compute in Newtonian physics, but with relativistic corrections which should be small for a slow but steady spin-up.
5. Failure to account for the fact that, because any point of the cylinder not on the axis of rotation experiences centripetal acceleration of v2/r during rotation, some general relativistic corrections are appropriate. (Here r is the perpendicular distance from the point to the axis, and v is the speed at which the point moves.)

Although the theory to resolve the paradox was understood by 1937, many subsequent authors have repeated various conceptual errors which had already been cleared up in earlier work, possibly because some of the explanations were not quite explicit.