# How does SR apply to rotating bodies?

• mrspeedybob

#### mrspeedybob

Suppose I have a disk that is 100,000 km in diameter. I attempt to rotate it at 1 revolution per second.

Am I unsuccessful because the material on the outside edge would have to travel faster then light or am I successful because length contraction at the outside edge reduces the circumference?

At any given time is a point on the edge traveling twice as fast or less then twice as fast as a point 50,000 km from the center?

Is the geometry of a rotating object inherently non Euclidean?

I think that's 3 different ways of asking the same question. If not I'm particularly interested in what the difference is?

What happens to an actual disk depends on the material model it's made out of. It's tough to find a well-behaved model where the disk won't break apart first.

For more info try http://www.gregegan.net/SCIENCE/Rings/Rings.html (for attempts to solve what happens with a particular model for a hoop - note that the model has some definite limits which are discussed in the artice.)

For a general discussion of the relativistic rotating disk try http://www.desy.de/user/projects/Physics/Relativity/SR/rigid_disk.html

The basic issue in talking about "the geometry of the disk" is how you do the time-slice. This tends to cause no ends of confusion, alas, but there isn't any simpler way that I know of than to say that the 3-geometry you get when you slice a 4 dimensional object depends on exactly how you slice it.

I suspect that one of the difficulties in this is realizing that space-time really is inherently four dimensional, that space and time intermix via the Lorentz transform. This is explained in many introductory textbooks, and there's a particularly clear explanation in "Space-time Physics" by Taylor and Wheeler (the first chapter of the older edition is online at http://www.eftaylor.com/special.html). People have an unfortunate tendency to jump right to the rotating disk, rather than to learn the basics of special relativity first as well, this does not help avoid confusion :(.

http://en.wikipedia.org/wiki/Ehrenfest_paradox
"1911: Max von Laue shows, that an accelerated body has an infinite amount of degrees of freedom, thus no rigid bodies can exist in special relativity"

So if the disk isn't a rigid body then the results depend on what elastic properties it has.

Those references just show that rotation cannot be treated in special relativity, the concepts are mutually exclusive, and has not yet been treated successfully in general relativity.

Rotation certainly can be treated in special relativity. What can't be done in special relativity is to have rigid bodies. And it's also noteworthy that Born Rigid bodies (which aren't perfectly rigid, but rigid in a matter that's compatible with special relativity) can't rotate either. So one can't talk about rotating rigid bodies in SR, and one can't talk about rotating Born Rigid bodies either.

But certainly SR can be used to analyze non-rigid rotating bodies. The result of any such analysis for a realistic body will be that it will fly apart due to internal stresses before the speed of light is reached, but this observation may not be particularly helpful.

Greg Egan's paper shows the results you get when you assume Hooke's law applies to the force, though it's worth noting that this seeingly innocuous assumption also can start to run into problems (such as singular equations of motions) if you push it too far.

The behavior of clocks and rulers on a rotating body such as the Earth can also be studied in detail, but it's difficult to know in advance what sort of questions one wants to ask.

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Those references just show that rotation cannot be treated in special relativity, the concepts are mutually exclusive, and has not yet been treated successfully in general relativity.
Nothing in that sentence is true. Rotation can be treated in SR as long as the stress-energy is small enough that the curvature of spacetime can be neglected, the concepts are not mutually exclusive, and it has been treated in GR also (particularly since SR is a subset of GR).

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