A spinning loop isn't related to a non-spinning loop by a global Lorentz transformation that's the mistake, that's the source of the confusion.

Furthermore, the statement about it shrinking-or not-must be specified: Is it meant to imply that the loop stays circular, but its radius changes size, or that it becomes elliptical? In both cases (and in any case, in fact) what is interesting is that the statements involve dealing with points that are separated by a space-like distance, which implies, immediately, that they can't be related by a time-like or light-like signal, therefore no global statement is possible.

This is an exercise that's more appropriate to general, rather than special, relativity. There, too, space-like separations can't become time-like or light-like, however (in the absence of horizons, but that's another story).

The more fundamental error is the statement that Lorentz contraction isn't a coordinate artifact. So the real answer is that an object can't change its physical properties, in particular its shape, in any intrinsic sense, just be changing the coordinate system. The fact that a slanted bar can fit through a door whose height is smaller than the length of the bar is something similar: The bar didn't change its length to do this. It changed one coordinate, by a rotation. But coordinates aren't invariant quantities, under rotations, the length of the bar is.

The confusion, once more, is in focusing on the quantities that are not invariant under the symmetries of the physical system and not focusing on the quantities that are.

Thank you Stam Nicolis but I'm not sure if the answer is "yes" or "no". I asked myself about ellipses but convinced myself that this is not the answer because of symmetry seen by an observer on the stationary center line. I asked myself about weird geometries that allow a circumference to change without the diameter changing, but convinced myself otherwise for two reasons. One is that the Lorentz transformation applies to flat space and is not suppose to change the ratio of a circles circumference to its radius (the spinning loop is still a circle not an ellipse). The other reason is what would a weird geometry look like to the observer?

I don't know if the answer is yes or no but I am more convinced by the shrinking distance between closely spaced tick marks than by the non-shrinking diameter (requires a global transformation without a second coordinate system being clearly defined). So it looks to me that the loop shrinks.

But a shrinking loop leads to another problem. Suppose the loop is a steal pipe filled with water. The pipe is stationary but by initial conditions (so we don't need pumps or motors upsetting the pipe being a continuous loop) the water inside is flowing in a loop. If the circumference of the water stream shrinks, how does the water fit in the pipe?

The same way a ladder fits through a door whose height is less than the length of the ladder. Rotations can't change the length of the ladder, Lorentz boosts can't do it, either, because they can't change the spacetime interval between two events, which, in the present case, is space-like, that's all.

I think I am getting the picture. I see your mathematical reasoning, spacetime distances are invariant and when they are only spatial distances then spatial distances become invariant. The conclusion then is that the loop does not shrink. Good, or water pipes forming a loop could not hold water. I need to think some more to see more intuitively why the Lorentz length contraction does not apply to the spacing between adjacent tick marks. I think the reason is that both tick marks are at the same distance from the observer on the center line. I have to go through the math to see if that is correct, but either way I believe your conclusion. Thank you Stam Nicolis for answering the question.

I believe the answers given above but I still need help understanding why the Lorentz length contraction does not apply to the distance between closely-spaced tick marks. Consider an inertial coordinate system, call it the prime system, that is instantaneously at rest with a selected pair of adjacent tick marks. I assume that the tick spacing seen in this system is the same as seen in the laboratory system (the observer is on the center line) when the loop is not spinning. We will call this the nominal distance. The prime system sees the nominal distance for this selected pair of tick marks (other tick marks move in a more complicated way in this system but that is not relevant when focusing on the selected pair). Using the Lorentz transformation between coordinate systems predicts that the laboratory system sees a smaller spacing between tick marks. Where is my mistake? Is it that the length contraction is invalid due to the fact that the selected pair of tick marks is accelerating in the prime system? Or, is my mistake the assumption that the prime system sees the same spacing that the laboratory system would see if the loop was not spinning?

I'd say that it's the fact that the primed system, precisely, isn't an inertial frame, because it's accelerating. So it can't be related to the unprimed system by a Lorentz boost.

Frames that can be related by Lorentz boosts are equivalent, those that can't, aren't. Furthrrmore, I think it's more confusing than enlightening not to try and extract the invariant quantities in each frame. The presentation as made, doesn't make the distinction between quantities that depend on the frame, primed or unprimed and quantities that don't. In an accelerating frame the framework is that of general relativity, so there it's a new game.

But the primed system is not accelerating. It is only momentarily at rest with the selected tick mark pair. The only complication is that the selected tick mark pair is accelerating in the prime system (also accelerating in the laboratory system). There is no acceleration between coordinate systems so the coordinate systems should be related by the Lorentz transformation. But my interpretation of this transformation seems to have a mistake that I need help to understand it.

Maybe this is my mistake. Suppose an object is accelerating as seen by intertial frames. We compare two inertial frames. The first is called the laboratory frame. The other frame is at rest with the accelerating object at some selected point in time. We will call this the "instantaneously attached frame". The object accelerates in all initial frames including this one. The two reference frames are both inertial and connected by the Lorentz transformation. But to make this connection we must agree on a space-time point in order to compare the space-time coordinates of that point in the different frames. And, we must agree on which frame is the instantaneously attached frame because it is a different frame at different time points (the object has different velocities at different times). Different frames can disagree on time coordinates and this becomes relevant because we must ask which time coordinate is used to define the instantaneously attached frame. Ambiguity in the answer to this question might invalidate the use of the Lorentz length contraction when applied to accelerating objects. So my mistake is applying the Lorentz length contraction to accelerating objects. Is that correct?

But if the loop is not spinning in one frame and spinning in the other, the two frames can't both be inertial, so they can't be related by a Lorentz boost. That's the point. Another way to see this is by realizing that a point on a spinning loop-even a uniformly spinning loop-is accelerating in the frame, where the loop is spinning about the axis that's perpendicular to its plane, that goes through its center-else the circle would be a straight line... The points on the loop, that's not spinning in the laboratory frame, aren't subject to any acceleration, the points on the spinning loop are subject to centripetal acceleration. Therefore there can't exist a global Lorentz boost that relates the points of the two loops.

But the loop is spinning in both frames and both frames are inertial. A simple circular motion in the laboratory frame and something more complicated (producing an elliptical loop) in the instantaneously attached frame. The latter frame is attached to a single point on the loop at a single point in time, not the entire loop. My focus is on that single point on the loop the frame is attached to, which is the point between adjacent tick marks at the time assigned to the frame. I am concerned with the distance between these tick marks as seen in the different inertial frames.

An embedding diagram of the shape of your spinning wheel shows how the circumference of the wheel varies as the radius increases. "For a given radius, the circumference is the distance around the upside-down "bowl" at a particular height. This is called an "embedding diagram."”

https://phys.libretexts.org/Bookshelves/Relativity/Supplemental_Modules_%28Relativity%29/Miscellaneous_Relativity_Topics/General_Theory_of_Relativity

Wow! So there is a weird geometry (circumference of a circle divided by diameter does not equal pi). Thank you George Soli. I never expected this. I expected that special relativity could handle this topic, even though objects of interest are accelerating, because we can find inertial coordinate systems to make comparisons between. It also seems that my initial thoughts, as stupid as I am, were on target. The circumference of the spinning loop shrinks but the diameter does not shrink (I thought that was a contradiction but now see that it is not). This also explains how water flowing through a loop-shaped water pipe can fit inside the pipe even though the circumference of the flowing water stream is smaller than the circumference of the pipe. Thanks again.

But even with the loop spinning, how can the circumference divided by diameter not be pi in this flat space that we live in. Suppose someone takes a photograph of the spinning loop. What kind of geometry would be seen in the photo?

I thought I was having a mental block for not being able to figure out an obvious answer but I see that this topic is very involved. And, for me, quite mysterious. I have a lot to learn that I didn't know that I still need to learn.

I checked the internet and found that my question has a name. It is called "the Ehrenfest Paradox". Explanations and resolutions can be found in the literature. I'm still studying it because it is a little subtle and not as simple or obvious as I hoped. But at least I'm not stupid for not immediately knowing the answer to my question.

Of course there’s no ``paradox", just the usual ambiguities that result, when formulas are applied inappropriately. It’s simply wrong to claim that the transformation from a non-spinning to a spinning loop can be a global Lorentz boost. It can’t. The statement that the circumference contracts, since it's, apparently, parallel to the boost, but the radius doesn't since it's, apparently, perpendicular, is, unfortunately, an example of the typical errors that are made when things aren't thought through. A quick way to spot the error is by recalling that Lorentz transformations are global, they apply to all points on the loop, not on some.

A standard calculation shows that a particle whose trajectory is given, in the lab frame, by (x,y)=(cosωt,sinωt) has non-zero acceleration-and it ought to be obvious that it's not possible to eliminate acceleration by a global Lorentz boost.

An observer in the lab frame can deduce that a loop that's a circle when not spinning, remains a circle upon spinning: The points of the non-spinning loop can be parametrized as R(cosθ,sinθ), where θ runs from 0 to 2π. The points of the spinning loop, in the same frame, can be parametrized as R(cos(θ+ωt),sin(θ+ωt)). The circle's spinning, with angular velocity ω and it's definitely a circle, whose radius doesn't depend on the angular velocity. In particular it doesn't shrink.

What does shrink-and doesn't remain a circle-is the trajectory of an electric charge that starts to move in a circle, since it loses energy through electromagnetic radiation; any particle, indeed, that starts to move on a circle will not stay on it, since it will lose energy through gravitational radiation (at a very slower rate, of course). That's where the confusion comes from-from the assumption, first of all, that the trajectory of the radiating charge remains a circle, of shrinking radius, while it's losing energy (which is a good approximation, when the rate is slow enough) but, also, that the frames of the charge are inertial; they're not, since a uniformly moving charge doesn't radiate.

So to ``ray trace" a ``relativistic" loop, it suffices to calculate the trajectory of a relativistically moving charge. It won't stay a loop and the trajectory does ``shrink''; but there's no problem with π...

Thank you again Stam Nicolis. You gave the first direct answer: It doesn't shrink.

While studying the literature on the Ehrenfest Paradox I see some of the subtleties involved. We can refer to inertial coordinate systems when focusing on a single little piece of the loop and arrive at a Lorentz length contraction, but the difficulty comes from assembling all of the little pieces together to make a complete loop. There is no inertial coordinate system momentarily attached to each little piece (only a rotating coordinate system can do that) so the whole analysis gets complicated. I learned this much but the literature did not tell me the bottom line, that it does not shrink. Thanks for that information.