Is there a Lorentz contraction of the radius of the rotating disk, after all?

Question

In every relativistic discussion of the uniformly rotating disk and/or of a uniformly rotating system of coordinates, we are told that the radius of the disk does not suffer Lorentz contraction by virtue of being orthogonal to the direction of the motion. This is true of every textbook on relativity, or of any popular exposition thereof, and in particular that's what happens in Einstein's celebrated 1916 review paper on general relativity, and also in his three books, Relativity, the Special and General Theory: a Popular Exposition (1920), The Meaning of Relativity (1921), and The Evolution of Physics (1938). Not only is this that we are told, as we agree that it could not be otherwise. How then to interpret the following paragraph of a letter from Einstein to the positivist Philosopher J. Petzoldt, written in 1919? Here it goes:

«Now you believe that a rigidly rotating circular line must have a circumference which is less than $2r\pi$ because of the Lorentz contraction. The basic error here is that you instinctively set the radius $r$ of the rotating circular line equal to the radius $r_0$ that the circular line has in the case when it is at rest. This however is not correct; because of the Lorentz contraction rather $2 \pi r=2 \pi r_0 \sqrt{1-v^2/c^2}$."

This letter appears in John Stachel's article "Einstein and the rigidly rotating disk", included in the book "General Relativity and Gravitation - One Hundred Years After the Birth of Albert Einstein", Ed. A. Held, Plenum Press, New York, 1980.

I'd really appreciate someone's help because I don't know how to understand this paragraph. Not only Einstein denies one of the most reliable assertions of special relativity, the one that claims that there is Lorentz contraction only along the diretion of motion, as he ends agreeing with the Petzoldt claim that he intended to refute, that the circunference of the rotating line undergoes Lorentz contraction.

Let me just add that astonishingly Stachel doesn't comment on this strange paragraph, as if it was perfectly ok.

I think that I have found a first clue to the solution of the problem that I have described above. In the letter to Petzoldt, the $r_0$ in the paragraph that I have quoted is not the same $r_0$ of just two lines below. The first is "the radius $r_0$ that the circular line has in the case when it is at rest". The second is "$r_0$ the radius of the rotating disk, considered from the standpoint of $K_0$ (that is, the rest frame)".

Answer 1

I suspect the problem is that we are missing context, and Einstein is referring to some of the specific variables used in Petzoldt's derivation. Looking at the whole letter (this is Einstein to Joseph Petzoldt, August 19, 1919, CPAE 9, Doc. 93) we see that Einstein's aim is apparently to point out that the length of the circumference of the rotating disc when observed from the stationary reference frame is actually measured to be longer, not shorter, because the co-moving rulers used to measure it are Lorentz-contracted.

There is a fairly subtle issue of reference frames here. Part of the problem is that we cannot consistently define a rotating coordinate system of space and time like we can with inertial frames. Each part of the circumference is moving relative to every other part. Rather than make the attempt and suffer the ambiguity, Einstein instead chooses to consider matters from the stationary frame of reference. Then each part of the circumference is moving instantaneously in a particular direction, and a ruler passing the disc tangentially, moving uniformly at its instantaneous velocity, is Lorentz-contracted in accordance with special relativity. In this way, Einstein avoids all the worrying uncertainty of how to handle non-inertial coordinate systems, and reduces everything to simple, inertial reference frames moving uniformly with respect to one another; a case we know well how to handle.

We take a snapshot in time (as defined in the stationary observer frame) and set a separate ruler flying tangentially past each point on the circumference at this moment. The unit-length (when at rest) standard rulers are all contracted to length $\sqrt{1-\frac{v^2}{c^2}}$, so the ratio of moving circumference $U$ to radius $r$ is related to the at-rest circumference $U_0$ and radius $r_0$ by:

$$\frac{U}{r}=\frac{U_0}{r_0}\cdot\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

The rest of Einstein's letter makes it very clear that he is Lorentz-contracting tangentially, not radially. I think the phrasing of the paragraph you quote is intended to refer to the length of the rulers arranged around the circumference - that Petzoldt is somehow assuming the moving disc to be shrunk but the moving rulers to be their normal size, that he has derived the moving circumferential ruler's length from the rest-frame radius, and not Lorentz-shrunk it. I agree though that it's not clear, and my German isn't good enough to pick apart the nuances of his choice of words.