How Does a Spinning Disc Appear at Near-Light Speeds?

[Oldfart]

If I had a spoked disc made of unobtainium that could be spun up to .9999c outer rim speed, what would it look like as compared to the disc at rest? Assume I can photo the spinning wheel with an ultra fast camera.

 

[pervect]

There are a lot of relativistic visualizations at http://www.anu.edu.au/physics/Searle/
I'm not sure if your spoked disk is one of them.

 

[WannabeNewton]

I'm going to assume that your exact phrase "could be spun up to..." translates over to the assumption of a material that can withstand the internal stresses that result from applying a torque to speed up a disk into rotation. It's easier to just consider a disk that has been spinning indefinitely into the past, and will be spinning indefinitely into the future, with some constant angular velocity.

Anyways, let me make two points:

(1) You have to specify with respect to what observer you want to describe "what the disk looks like".

(2) I put "what the disk looks like" in quotes because the statement has two meanings. Each point on the rotating disk has its own worldline in space-time. The entire family of worldlines so obtained from each point on the disk is called the worldtube of the rotating disk. The spatial shape of the disk as determined by an observer is simply the set of all points obtained by intersecting this worldtube with the simultaneity plane of the observer at any given instant of the observer's local clock time.

This is not the same thing as the image of the disk obtained from (for example) a super-snapshot taken by this observer because this is the intersection of the rotating disk's worldtube with the past light-cone at a given instant of the observer's local clock time.

You must specify which of these you are talking about.

The spatial shape of the rotating disk is trivial to derive relative to inertial observers; on the other hand the spatial shape of the rotating disk is noticeably harder to derive relative to observers who are sitting on the disk.

See here for more examples: https://www.physicsforums.com/showthread.php?t=730506

 

[pervect]

I"m assuming that "looks like" means "looks like to a camera".

http://phys.org/news/2012-08-effect-rotating-relevance-everyday-applications.html

appears credible at the pop-sci level if this is the question. There's also a link to a Physical Review letter paper in the pop-sci article.

 

[WannabeNewton]

In that case, check these out as well: http://www.spacetimetravel.org/ueberblick/ueberblick1.html

 

[bcrowell]

The locus of points that forms the circumference of the disk is a circle, regardless of whether the disk is rotating. One could ask whether an optical observation such as a camera snapshot would show the various *parts* of the disk as distorted to more or less than their normal size. But this implicitly assumes that it's possible to observe the disk first without rotation and then later after it has been spun up while maintaining its rigidity. This is impossible by the Noether-Herglotz theorem, which is essentially the resolution of the Ehrenfest paradox. (There is also a theorem in relativistic optics that says that a sphere still appears to be a sphere, regardless of the motion of the observer relative to the sphere's center. For an animation showing this, see the end of this video: http://youtube.com/watch?v=JQnHTKZBTI4 .)

 

[Oldfart]

Thanks to all for your replies! Unfortunately, I've been trying to study the Noether-Herglotz theorem and your other comments without much luck, the math exceeds my paygrade, so to speak, I'm an ME, class of 1958. But I appreciate your efforts, really do!

For whatever it's worth, I envisioned a souped-up ultracentrifuge thingy in a lab setting, the observer is a superspeed camera.

Would it help if I change my question: Does a real-life ultra centrifuge exhibit any relativistic distortions when up to full speed, no matter how tiny those distortions might be?

Thanks again, OF

 

[PAllen]

The locus of points that forms the circumference of the disk is a circle, regardless of whether the disk is rotating. One could ask whether an optical observation such as a camera snapshot would show the various *parts* of the disk as distorted to more or less than their normal size. But this implicitly assumes that it's possible to observe the disk first without rotation and then later after it has been spun up while maintaining its rigidity. This is impossible by the Noether-Herglotz theorem, which is essentially the resolution of the Ehrenfest paradox. (There is also a theorem in relativistic optics that says that a sphere still appears to be a sphere, regardless of the motion of the observer relative to the sphere's center. For an animation showing this, see the end of this video: http://youtube.com/watch?v=JQnHTKZBTI4 .)

Well, Herglotz-Noether allows no rotation, spin up without rigidity, then settling into rigid uniform rotation. It presents a more severe problem to ever-increasing spin up because the stresses increase without bound, as in the Bell spaceship where the ships are uniformly accelerating in the inertial frame. So if you allow the spun up disk to be deformed (materially, not optically) version of the original, your are ok. If you don't want this, you are screwed.

 

[pervect]

The locus of points that forms the circumference of the disk is a circle, regardless of whether the disk is rotating. One could ask whether an optical observation such as a camera snapshot would show the various *parts* of the disk as distorted to more or less than their normal size. But this implicitly assumes that it's possible to observe the disk first without rotation and then later after it has been spun up while maintaining its rigidity. This is impossible by the Noether-Herglotz theorem, which is essentially the resolution of the Ehrenfest paradox. (There is also a theorem in relativistic optics that says that a sphere still appears to be a sphere, regardless of the motion of the observer relative to the sphere's center. For an animation showing this, see the end of this video: http://youtube.com/watch?v=JQnHTKZBTI4 .)

Most of the visualizations show a disk that is not just rotating, but moving and rotating as if the disk were rolling, a combination of linear motion and rotation.

Reading back, it's not clear if this is what the OP wants. In the case where the disk is rolling (and thus rotating and also moving in a translational manner), the locus of points won't be a circle anymore (either for the camera or for an observer who corrects for light propagation delays). But it appears the OP originally asked about a disk that was rotating, rather than rolling, so the visualizations given may not be the ones they wanted, and Ben's remark about the motion being circular would then be what they asked for.

I would agree that the disk must deform, so a detailed analysis of how the disk looks would depend on the exact nature of the unobtanium used to construct it. My initial notion was that if the disk were originally round and constructed in a manner that had symmetry under rotation around the $\phi$ axis, it would be reasonable to assume that it retained said symmetry - so the distortion would be one of the radius of the disk, not it's circular shape.

However, thinking about this further, the presence of the spokes (mentioned in the original post) does spoils the perfect symmetry under rotation, so it's quite possible and perhaps even probable that the disk would deform to a non-round "flower" shape, that is symmetrical under rotation by the angle $\phi$ between the spokes but not round in the sense that it was symmetrical under $\phi$ for all values, unless the disk were designed and pre-stressed in some manner so that it would become round when it did rotate.

A few of the simulations had some fine print on this, mentioning that their disks as simulated were assumed to be designed to be round when rolling.

Talk about what a perfectly rigid disk would do is rightly pointed out as non-productive, as one can't ask what happens within a theory when one violates the theory without generating nonsense, and SR doesn't allow perfectly rigid materials.

 

[bcrowell]

Talk about what a perfectly rigid disk would do is rightly pointed out as non-productive, as one can't ask what happens within a theory when one violates the theory without generating nonsense, and SR doesn't allow perfectly rigid materials.

It isn't really relevant that SR doesn't allow perfect rigidity. Born rigidity isn't a property of a material, it's the result of external forces acting in a manner that is planned so as to produce the Born-rigid motion. The issue is kinematic, not dynamic: Born-rigid angular acceleration is impossible essentially because clock synchronization isn't transitive in a rotating frame.

 

[PAllen]

It isn't really relevant that SR doesn't allow perfect rigidity. Born rigidity isn't a property of a material, it's the result of external forces acting in a manner that is planned so as to produce the Born-rigid motion. The issue is kinematic, not dynamic: Born-rigid angular acceleration is impossible essentially because clock synchronization isn't transitive in a rotating frame.

Born rigid rotation at constant angular speed is fine. It is only during change of angular speed that you have to give up on Born rigidity.

 

[pervect]

Thanks to all for your replies! Unfortunately, I've been trying to study the Noether-Herglotz theorem and your other comments without much luck, the math exceeds my paygrade, so to speak, I'm an ME, class of 1958. But I appreciate your efforts, really do!

For whatever it's worth, I envisioned a souped-up ultracentrifuge thingy in a lab setting, the observer is a superspeed camera.

Would it help if I change my question: Does a real-life ultra centrifuge exhibit any relativistic distortions when up to full speed, no matter how tiny those distortions might be?

Thanks again, OF

If I am envisioning the setup correctly, you'll mainly get light speed delay effects. For instance, if you have a camera mounted above the central rotor point, light will take longer to get to the camera from the rim than from the center.

 

[Oldfart]

Thanks. Pervect, I think I'm now starting to get it! Seems now that there are two types of distortion: (1) Everyday distortion due to centrifugal forces acting on the spinning object, something that even an old ME can understand, and (2) Distortion due to light delays like you mentioned.

So to sort of answer my original question, the spocked wheel will become flower-shaped and its radius will increase due to the physical forces, and if the wheel is dish-shaped so the distance between the camera and the wheel is constant, then there will be no relativistic distortion of the wheel. That seems so simple now, assuming I'm right!

Many thanks! OF

 

[WannabeNewton]

Thanks. Pervect, I think I'm now starting to get it! Seems now that there are two types of distortion: (1) Everyday distortion due to centrifugal forces acting on the spinning object, something that even an old ME can understand, and (2) Distortion due to light delays like you mentioned.

Don't forget there are also "distortions" due to the relativity of simultaneity when you calculate the spatial geometry of the object in two different inertial frames. I put "distortions" in quotes because I think it's too strong a word when referring to (2) or the effects of relative simultaneity wherein the effects are purely optical and frame-dependent respectively, unlike the actual distortions one gets due to centrifugal forces, or internal stresses arising from the inability to remain Born-Rigid when given a tangential acceleration.

 

[pervect]

Thanks. Pervect, I think I'm now starting to get it! Seems now that there are two types of distortion: (1) Everyday distortion due to centrifugal forces acting on the spinning object, something that even an old ME can understand, and (2) Distortion due to light delays like you mentioned.

So to sort of answer my original question, the spocked wheel will become flower-shaped and its radius will increase due to the physical forces, and if the wheel is dish-shaped so the distance between the camera and the wheel is constant, then there will be no relativistic distortion of the wheel. That seems so simple now, assuming I'm right!

Many thanks! OF

Lorentz contractions, etc do "exist" - but given the circular symmetry of the problem, I don't see a way that they'll show up in photographs. You'd need other means to see them, like strain gauges on the wheel (and you'd have to separate out the relativistic effects from the Newtonian ones).

The most obvious effect of rotation is an effect on clocks rather than rulers - an effect present and meaasurable on the Earth - the Sagnac effect, one of the main contributors to the Haefel-Keating experiment results.