The

Ehrenfest paradox was to apply special relativity to a rotating disc, and finding some geometrical oddities. It requires non-Euclidean geometry to resolve it. It is a good test of how well early physicists understood relativity. Einstein wrote about it many times.

Einstein historian Tilman Sauer wrote in 2007 about Einstein correspondence with Vladimir Varićak:

These were rebutted by
Einstein in a response of 28 February 1910 in which he also, with reference to
Ehrenfest’s paradox, referred to the rigidly rotating disk as the “most interesting
problem” that the theory of relativity would presently have to offer. In his next
two letters, dated 5 and 11 April 1910 respectively, Einstein argued against the
existence of rigid bodies invoking the impossibility of superluminal signalling,
and also discussed the rigidly rotating disk. A resolution of Ehrenfest’s paradox,
suggested by Vari´cak, in terms of a distortion of the radial lines so as to preserve
the ratio of π with the Lorentz contracted circumference, was **called interesting
but not viable**. The radial and tangential lines would not be orthogonal in spite
of the fact that an inertial observer comoving with a circumferential point would
only see a pure rotation of the disk’s neighborhood.
About a year later, Einstein and Vari´cak corresponded once more. Vari´cak
had contributed to the polemic between Ehrenfest and von Ignatowsky by suggesting a **distinction between ‘real’ and ‘apparent’ length contraction**. The reality of relativistic length contraction was discussed in terms of Ehrenfest’s tracing
paper experiment, but for linear relative motion. According to Vari´cak, the experiment would show that **the contraction is only a psychological effect** whereas
Einstein argued that the effect will be observable in the distance of the recorded
marker positions. When Vari´cak published his note, **Einstein responded with a
brief rebuttal**.17

Sauer is trying to be favorable to Einstein. Varicak wrote several papers on applying non-Euclidean geometry
to special relativity, and

Einstein rejected this approach. Varicak's explanation was viable, and better than Einstein's.

The idea that the Lorentz contraction was "psychological" appears to have originated in this 1909 American article:

Let us emphasize once more, that these changes in the units of time and length, as well as the changes in the units of mass, force, and energy which we are about to discuss, possess in a certain sense a purely factitious significance; although, as we shall show, this is equally true of other universally accepted physical conceptions. We are only justified in speaking of a body in motion when we have in mind some definite though arbitrarily chosen point as a point of rest. The **distortion of a moving body is not a physical change in the body itself, but is a scientific fiction**.
**When Lorentz first advanced the idea that an electron, or in fact any moving body, is shortened in the line of its motion, he pictured a real distortion of the body** in consequence of a real motion through a stationary ether, and his theory has aroused considerable discussion as to the nature of the forces which would be necessary to produce such a deformation. The point of view first advanced by Einstein, which we have here adopted, is radically different. Absolute motion has no significance. Imagine an electron and a number of observers moving in different directions with respect to it. To each observer, naïvely considering himself to be at rest, the electron will appear shortened in a different direction and by a different amount; but the physical condition of the electron obviously does not depend upon the state of mind of the observers.

Although these **changes in the units of space and time appear in a certain sense psychological**, we adopt them rather than abandon completely the fundamental conceptions of space, time, and velocity, upon which the science of physics now rests. At present there appears **no other alternative**.

This is all completely correct, but rejected by Einstein. Based on this paper, Varicak attributes the new view of space and time to Einstein, but Einstein published a rebuttal denying that his viewpoint was any different from Lorentz's.

The issue here is: Do rigid bodies really contract, or is the apparent contraction just an artifact of the non-euclidean geometry of spacetime?

Lorentz would say the former, while Minkowski proposed the latter in 1907 and that has been the preferred interpretation in textbooks ever since.

Poincare also proposed the latter in 1905, but said that the views were mathematically equivalent, so he would say that they were both correct. In his view, the contraction is "only apparent, something which would be due to our methods of measurement".

What would Einstein say? The latter view seemed to be attributed to him in the above 1909 paper, and repeated by Varicak in his 1911 paper on the Ehrenfest paradox, which says, "contraction is only an apparent, subjective phenomenon, caused by the manner of our clock-regulation and length-measurement." We can be pretty sure the attribution is incorrect, because Einstein published a rebuttal to that 1911 paper. Einstein corresponded with Varicak and was fascinated by the subject, so I think he was clearly favoring Lorentz's view during 1905-1911, at least. He could have accepted credit for the geometrical view, but he vigorously denied it.

This is the clearest evidence that Einstein did not understand and accept special relativity, as it has been explained by Minkowski in 1907 and every textbook since.

Marco Giovanelli has written a new paper on Appearance and Reality: Einstein and the Early Debate on the Reality of Length Contraction. It has a lot of historical info on this issue.

In Einstein’s theory, length contraction is a kinematic effect that depends on the
definition of simultaneity; however, it is just as real as length contraction in Lorentz’s
theory, where it is conceived as a dynamic effect due to the motion of a rod through the
ether. The two theories derive the same quantitative measure for the contraction through
different routes. To explain this point, Einstein resorts to his beloved comparison between
relativity theory and thermodynamics:
One cannot ask whether the contraction should be understood as a consequence of the
modification of molecular forces caused by motion or as a kinematic consequence arising
from the foundations of the theory of relativity. Both points of view are justified. [letter to Varicak, 1911]

He also relates it to

Bell's spaceship paradox.

Einstein is correct that different points of view about the contraction are justified. The first view, "a consequence of the
modification of molecular forces caused by motion", is usually attributed to Lorentz. The second view refers to Einstein's 1905 two-postulate approach. Einstein appears to say that it is meaningless to say which is better.

The approaches were logically equivalent. Lorentz started from the Michelson-Morley experiment and Maxwell's equations, and deduced the contraction. Einstein postulated the constant speed of light and the Poincare relativity principle, and made the same deductions. Neither really analyzed the molecular forces. Lorentz did correctly believe that the forces were electromagnetic, and hence subject to his transformations.

What is missing from Einstein's 1911 comments is any recognition of the non-euclidean geometry view put forward by Poincare in 1905 and Minkowski in 1907.

In the writings of those years, Einstein appears to have still been reluctant to embrace
Minkowski’s (1909) reduction of kinematics to geometry. Indeed, he presented the key
result of relativity as the distinction between the geometric and the kinematic configuration
of a body (Einstein, 1908, 1910, 1911a).29 (In modern terms, the distinction between the proper and the coordinate shape of a body.)

Einstein would often argue that his approach was not ad hoc. and hence superior to Lorentz's.

Lorentz complained that, in a popular article, Einstein
had referred to the Lorentz-Fitzgerald contraction as a “hypotheses invented ad hoc”
(Einstein, 1915, 707) to neutralize Michelson’s result (Lorentz to Einstein, Jan. 1, 1915;
CPAE, Vol. 8, Doc. 43). Lorentz argued that such an objection might have applied to
his first formulation of the contraction hypothesis. At a later stage, however, reacting
to Poincaré’s criticism, Lorentz provided a coherent theory of matter from which length
contraction can be derived as a consequence. Lorentz regretted not having stressed this
more, as it would have left less of an impression of being an ad hoc hypothesis (Lorentz to
Einstein, Jan. 1, 1915; CPAE, Vol. 8, Doc. 43).
Lorentz argued that Einstein’s approach was somewhat misleading from a “didactical”
point of view (Lorentz to Einstein, Jan. 1, 1915; CPAE, Vol. 8, Doc. 43). If the contrac-
tion is derived as a consequence of the new kinematics “and nothing more is added in
commentary”, it could give rise to the suspicion that “only ‘apparent’ [scheinbare] things
were involved here and not a real [wirkliche] physical phenomenon” (Lorentz to Einstein,
Jan. 1, 1915; CPAE, Vol. 8, Doc. 43). ...

Once again, Einstein replied by alluding to a more subtle dialectic between the real
and the apparent:

... Regarding the erroneous view that the Lorentz contraction was ‘merely apparent,’
[scheinbar] I am not free from guilt, without ever having myself lapsed into that error. It
is real [wirklich], i.e., measurable with rods and clocks, and at the same time apparent
[scheinbar] to the extent that it is not present for the co-moving observers.39 (Einstein to
Lorentz, Jan. 23, 1915; CPAE, Vol. 8, Doc. 47)

So Einstein's differences with Lorentz were slight, and mostly have to do with Einstein trying to take credit for what Lorentz had already done. Einstein never says Lorentz was wrong, but he does say that the geometrical view is wrong:

Perhaps Mr. Varičak might admit—and thus in a way retract his assertion—that the Lorentz
contraction is a ‘subjective phenomenon.’ But perhaps he might cling to the view that the
Lorentz contraction has its roots solely in the arbitrary stipulations about the ‘manner of
our clock regulation and length measurement.’ The following thought experiment shows the
extent to which this view cannot be maintained. (Einstein, 1911d, 509)

Einstein is wrong here. The modern view is that our manner of clock regulation and length measurement
corresponds to a non-euclidean geometry on spacetime. The contraction is subjective in the sense that it only
shows up in the comparison between the true non-euclidean geometry and the more intuitive Euclidean geometry.
That is what Poincare said in 1905, Minkowski in 1907, and Varicak in 1911. Einstein did not understand it.

I have posted many criticisms of Einstein's lack of originality. Many of these are not new, as Whittaker argued
in a 1953 book that Lorentz and Poincare had all of special relativity. Lorentz said back in 1909 that Einstein just postulated what was previously proved. But I have not seen anyone else
make the point I make here. That the modern geometrical view of relativity was explicitly rejected by Einstein
as late as 1911.

Even when experts were starting to credit Einstein with the new geometrical view of relativity, he was adamantly denying it.

There are Einstein fans who claim that Lorentz and Poincare never really understood special relativity,
based on post-1905 lectures or writings that supposedly showed confusion about fundamentals. Usually
the argument is that Poincare occasionally chose an preferred reference frame. But of course choosing
a preferred frame is not incorrect or contrary to modern thinking. Everyone chooses preferred frames all the time.

Einstein is not wrong either when he clings to a Lorentzian anti-geometry view. But he is contrary to modern thinking,
and he was wrong to say that Varicak's "view cannot be maintained."

Einstein did eventually accept non-euclidean geometry, as Grossmann, Levi-Civita, and Hilbert convinced him that it
was necessary for general relativity in 1913-1916. But he never really accepted the geometric view, and never accepted
Varicak's argument.

If you are a physicist reading this, you might complain that I am a mathematician siding with other mathematicians -- Poincare, Varicak, Hilbert, Whittaker -- against the great physicist Einstein. Einstein's genius was in Physics, not Mathematics, and maybe it is unfair to judge him by mathematicians. Maybe so, but I am discussing the mathematical understanding of relativity, and Einstein's was deficient.

Einstein's special relativity did not have anything physically new. The physical predictions were the same as Lorentz's, and physicists called it the Lorentz-Einstein theory. The only appeal was his mathematical derivation. So yes, I think it is fair to judge his mathematics by mathematical standards.

It is hard to understand just what Einstein's view was. Giovanelli writes:

What
is clear is that in the following months, Einstein made the first published reference to
Ehrenfest’s thought experiment in a paper on gravitation published in February, where he
pointed out that the geometry of the rotating disk is non-Euclidean (Einstein, 1912a, 356).
Since a rotating system is equivalent to a system at rest in a suitable gravitational field,
Einstein (1912b, 1064) soon began to realize that the traditional physical interpretation
of coordinates as readings on rods and clocks could not be maintained in the presence of
gravitation (see Stachel, 1989, for more detail).
After returning to Zurich, Einstein famously found a solution to the conundrum with
the help of his friend Marcel Grossman. However, his struggles with the meaning of
coordinates in physics continued during the Berlin period (Giovanelli, 2021).

In modern terminology, spacetime is a 4-dimensional manifold, with many coordinate systems possible,
not necessarily having physical significance. Grossmann and others tried to convince him to use covariant tensors,
but during 1913-15 he was persuaded by his

Hole Argument
that such things were impossible. It appears that Hilbert enlightened him to use covariant equations.

In the Lorentzian view, bodies really contract. In the Poincare-Minkowski-Varicak mathematician view, the contraction
is an artifact of using coordinates that do not match the geometry. Einstein did not seem to be fully in either camp,
and saying only that the contraction is required by the kinematics.

Here is an argument from the above 1909 article:

If our ideas possess a certain degree of artificiality, this is also true of others which have long since been adopted into mechanics. The apparent change in rate of a moving clock, and the apparent change in length and mass of a moving body, are completely analogous to that apparent change in energy of a body in motion, which we have long been accustomed to call its kinetic energy.

An object at rest has no kinetic energy. If you watch it from a moving frame, all of a sudden it has kinetic energy. Where did that energy come from? The energy is not real. It is just an artifact of the coordinates being used. It is just psychological. Not imaginary. If a brick hits you in the head, your pain will be real. The energy is measurable.

The best way to make sense of this is to say spacetime is a manifold with a non-euclidean geometry.