1 upvotes, 0 direct replies (showing 0)
View submission: Ask Anything Wednesday - Physics, Astronomy, Earth and Planetary Science
This is a doozy of a problem that has been debated for a long time. /u/twoTheta gives a good response but I wanted to add some things.
Naive special relativity says that objects moving at relativistic speeds will be contracted along their direction of travel as measured by a stationary observer. Thus, we'd (again, naively) expect the circumference of the disk to shrink, as each "piece" of the disk would shrink along its direction of motion. However, we'd also expect the radius to *stay the same*, since none of those pieces are moving in the radial direction. That leads us to a paradox - we can't change the circumference without changing the radius. This is called the Ehrenfest Paradox!
As with all paradoxes in relativity, the issue lies with making assumptions and approximations which turn out to make your conclusions invalid. Key things to consider here: what circular motion implies, what "length contraction" really means, and the effects of time lags on what you *perceive*.
Last piece first: Terrell rotation[1] is what you get when you try to work out what an observer would actually *see* when a macroscopic object passes by at relativistic speed. Long story short: you don't actually perceive the length contraction, due to light travel time differences between the different parts of the object. Instead, it appears to *rotate*, because the light that you see from the far side of the object are emitted later than the light you see from the edge closer to you. See also here[2] for some fancier illustrations. However, all those examples look at cubes moving laterally - for a spinning disk viewed face-on, any distortion would be on the surface of the disk. Although I'm not sure if the disk spinning would produce any Terrell rotation.
1: https://personal.math.ubc.ca/~cass/courses/m309-01a/cook/terrell1.html
2: https://physicsworld.com/a/the-invisibility-of-length%E2%80%AFcontraction/
On length contraction: length contraction is purely an observational effect where the apparent distance between two *events* changes depending on the observer's relative velocity. "Events" in relativity are points in space-time - think of, maybe, a little spark going off at some specific time and location. Let's say you have two events that are stationary relative to each other and simultaneous in some reference frame. An observer who is moving relative to that frame will see them closer together and will see that they don't happen at the same time.
So when we talk about length contraction, we're talking about straight-line distances and we're talking about inertial reference frames. Spinning disks are not straight lines and their edges are not in inertial reference frames, so we cannot take the the simple approach. In fact, if you place a measuring rod on the edge of a rotating disk, its two ends will be in *different reference frames*! In fact, only an infinitesimally-small wedge of the disk is in a single reference frame in the spinning disk - as soon as you rotate around the disk at all, you're in a different frame.
Putting it another way: if you travel tangentially to the disk at the same speed as its tangential velocity, at the moment you pass by the edge of the disk, you will be at rest relative to the infinitesimally small wedge of the disk that is perpendicular to you, and only for an infinitesimally small amount of time.
This post[3] goes through a very detailed look at the spinning disk problem (note, though, that I cannot vouch for anything else this user has posted and I don't know their credentials - this specific analysis seems sound to me, though). They point out that, via the logic in my last two paragraphs, a non-rotating observer in fact shares simultaneity with *all* points along the edge of the disk, and since those wedges are all infinitesimally thin, length contraction does not come into play, and the circumference measured by the observer is not impacted by the rotation of the disk. I am not 100% sure about how good their analysis is at the end, but it's a much more robust analysis compared to a lot of the material I've seen on this subject, and they do a good job of showing why various assumptions that are often made are incorrect.
There's nothing here!