Clocks ON a rotating disk: What happens?

[JulianMau]

I was reading about the Ehrenfest Paradox and it got me thinking about something (that I think is) similar:

Suppose we take a large, flat, and rigid disk, and we attach to various parts of it a number of clocks (some very close to the center of the disk, some along the edge, others in between). We synchronize all these clocks using some master clock that is at reference with respect to them, and then set the disk spinning very, very fast. Suppose we then stop the disk after the master clock reads, say, 12 hours, and compare the times of the clocks on the disk with the master clock. According to Special Relativity, what should we expect to find, and what are the implications about the structure of spacetime? How do we even know it wasn't the reference frame of the master clock that was spinning the whole time?

Am I right in suspecting that less time will have ticked off the clocks on the disk because they are moving relative to the master clock? And that the clocks nearer the center of the disk will show less time having past than those at the outer edge, due to their higher rotational velocity? What would you say any of this implies about how spacetime is structured and how I should know which frame is at rest?

 

[Ibix]

Welcome to PF.

I was reading about the Ehrenfest Paradox and it got me thinking about something (that I think is) similar:

It's actually closer to the twin paradox, with the master clock being the stay-at-home twin and the rotating clocks being hordes of traveling "twins".

Am I right in suspecting that less time will have ticked off the clocks on the disk because they are moving relative to the master clock?

Yes.

And that the clocks nearer the center of the disk will show less time having past than those at the outer edge, due to their higher rotational velocity?

No. The clocks near the centre are moving slower relative to the master clock than those at the rim, so will be tick at nearer the same rate as the master clock. The way you have it you could see extreme relativistic effects just from dropping your wristwatch onto an old school 45rpm record player.

What would you say any of this implies about how spacetime is structured and how I should know which frame is at rest?

I'm not sure that the experiment on its own tells you anything about spacetime. Although highly precise measurements of the clock rates in a real test might lead you towards the Lorentz transforms and hence Minkowski geometry.

The easiest way to spot which frame is at rest is with an accelerometer. The rotating clocks will notice a centrifugal force which the master clock will not. Note that this is different from the frames you usually encounter in special relativity, which are inertial and can always be considered at rest. So in this case you do genuinely have a frame that might be described as "moving" in an absolute sense.

 

[JulianMau]

Welcome to PF.
It's actually closer to the twin paradox, with the master clock being the stay-at-home twin and the rotating clocks being hordes of traveling "twins".
Yes.
No. The clocks near the centre are moving slower relative to the master clock than those at the rim, so will be tick at nearer the same rate as the master clock. The way you have it you could see extreme relativistic effects just from dropping your wristwatch onto an old school 45rpm record player.
I'm not sure that the experiment on its own tells you anything about spacetime. Although highly precise measurements of the clock rates in a real test might lead you towards the Lorentz transforms and hence Minkowski geometry.

The easiest way to spot which frame is at rest is with an accelerometer. The rotating clocks will notice a centrifugal force which the master clock will not. Note that this is different from the frames you usually encounter in special relativity, which are inertial and can always be considered at rest. So in this case you do genuinely have a frame that might be described as "moving" in an absolute sense.

Thank you, Ibix, for your response! I think I understand where I went wrong with my second supposition. As regards your last bit: can you elaborate a little? I'm supposing this to be taking place in a special relativistic setting. Why is it, then, that the rotating disk might be considered a frame absolutely in motion? Without an accelerometer, wouldn't we not be able to determine which frame -- the master clock's or the disk's -- is "at rest"? When does this become a meaningless question, if ever it does?

 

[Ibix]

Maybe "absolutely in motion" isn't a helpful way to phrase it. I tend to think, though, that if your accelerometer is reading non-zero then you are changing your state of motion, so you can't possibly be at rest all the time.

What is definitely true is that the elapsed time on a clock is a measure of the "length" of the path it took through spacetime (the interval) - similar to the way your odometer measures the length of your car's path through space. Put your master clock near the rim of the disc and, just as a clock on the rim passes by it, make a note of the readings on both clocks. The next time the clock on the disc comes past, you will notice that it has not advanced as far as your master clock. So the clock on the disc took a shorter route through spacetime than the master clock. An odd property of spacetime is that the straightline distance between two time-like separated events is the longest one, not the shortest. That tells you that the disc clock wasn't inertial - it didn't follow a straight line path.

The question of "which clock took a longer path through spacetime" for two clocks in inertial motion is meaningless, however. They can't meet more than once - so since they both start at point A but don't both end at point B, there's no significance to which took the longer path.

 

[JulianMau]

Maybe "absolutely in motion" isn't a helpful way to phrase it. I tend to think, though, that if your accelerometer is reading non-zero then you are changing your state of motion, so you can't possibly be at rest all the time.

What is definitely true is that the elapsed time on a clock is a measure of the "length" of the path it took through spacetime (the interval) - similar to the way your odometer measures the length of your car's path through space. Put your master clock near the rim of the disc and, just as a clock on the rim passes by it, make a note of the readings on both clocks. The next time the clock on the disc comes past, you will notice that it has not advanced as far as your master clock. So the clock on the disc took a shorter route through spacetime than the master clock. An odd property of spacetime is that the straightline distance between two time-like separated events is the longest one, not the shortest. That tells you that the disc clock wasn't inertial - it didn't follow a straight line path.

The question of "which clock took a longer path through spacetime" for two clocks in inertial motion is meaningless, however. They can't meet more than once - so since they both start at point A but don't both end at point B, there's no significance to which took the longer path.

Thanks, this cleared things up for me quite a bit. Going to consider it a bit more.

 

[pervect]

I was reading about the Ehrenfest Paradox and it got me thinking about something (that I think is) similar:

Suppose we take a large, flat, and rigid disk, and we attach to various parts of it a number of clocks (some very close to the center of the disk, some along the edge, others in between). We synchronize all these clocks using some master clock that is at reference with respect to them, and then set the disk spinning very, very fast.

Basically, you'll find that when you spin up the disk, the disk isn't rigid - at least it's not Born rigid. And you'll find that clocks that were synchronized in one frame of reference aren't synchronized anymore when you change frames of references.

To be precise, you need a precise way to describe how you spin up the disk. Saying it's rigid won't work - it won't be, at least not by the usual definitions of rigidit. But there is a reasonably simple, though somewhat messy, way to specify how you spin up the disk without making a lot of intuitive assumptions that turn out to be false . What you do is draw a space-time diagram of all the particles- or a selected subset of particles- of the disk. This particular case will be a bit hard to draw the diagram of, because the diagram will have three dimensions and not two, but it's something that can be visualized. It's hard and messy enough that it's not something I'm going to do for a PF post.

You'll also find that the clocks near the center show more elapsed time than the clocks near the edge, not less.

From the sounds of things, though, if you want to learn special relativity, you'd do better off not studying the rotating disk at this point in your studies. You seem to be mainly confusing yourself by making intuitive assumptions that are incorrect in the context of special relativity. This won't do much for learning, though perhaps you potentially could learn that when studying new things, sometimes you might have to forget a few of the old things in the process if they get in the way of learning the new things.

Instead of studying the rotating disk, the course I'd recommend is to study cases where you can draw the needed space-time diagrams on a flat sheet of two-dimensional paper. The first step is to learn how to draw space-time diagrams, it seems surprisingly difficult to get people who are learning SR to actualy do this. (I've never quite understood the resistance, perhaps it's an effortr to cling to the old and familiar?). Along with the general theoretical framework that allows you to actually calculate things, it would be helpful to study some specific well-known problems, which would be the twin paradox (covering the issue of differential aging), and Einstein's Train, which would cover the issue that different frames of reference have different notions of simultaneity. The relativity of simultaneity may be the hardest thing to grasp, but it's vital.

On the topic of learning theoretical framework, I'm rather fond of Bondi's aproach. Robphy has a PF insight article on the topic, https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/, there are various other sources available for this approach including Bondi's rather old book, "Relativity and Common Sense".

There is another approach that is particularly simple, also covered by robphy, though it's (afaik) fairly new in comparison to Bondi's approach. That's https://www.physicsforums.com/insights/relativity-rotated-graph-paper/, it's based on drawing space-time diagrams, and understanding how "light clocks" are represented on such diagrams.

 

[Dale]

Without an accelerometer, wouldn't we not be able to determine which frame -- the master clock's or the disk's -- is "at rest"?

You can always take any perfectly solvable physics problem and delete information until you wind up with an unsolvable problem. That doesn't imply that the original problem didn't have have a solution.

In this case, an inertial frame is one where accelerometers at rest read 0. If you don't have an accelerometer that doesn't mean that all frames suddenly become inertial, just that you have too little information.