Updated by Michael Weiss 2017.
Original by Michael Weiss.

The Rotating Disk in Relativity

Two questions arose in the first few decades after the birth of relativity: (a) What would a rotating disk look like, particularly a rigid disk?  (b) What is a good set of spacetime coordinates for life on a rotating platform?  In other words, we can enquire into the physics of a rotating physical disk, and we can do likewise for systems of rotating coordinates.

This page is about an actual disk.  (To spoil the surprise: there is no such beast as a rigid disk in relativity, but some related concepts do make sense and can illuminate the theory.)  Rotating coordinates have increased in importance in the era of GPS; the page Rotating Coordinates in Relativity addresses that topic.  The two topics have less to do with other than you might guess, although of course they are related.

Born rigidity

As special relativity started to filter through the physics community in the early 1900s, people realized the need to re-examine all the traditional concepts of Newtonian mechanics.  Idealized rigid bodies pervade elementary mechanics.  In 1909, Max Born took a fresh look at the idea of rigidity.

Born proposed a Lorentz-invariant definition of a "rigid body".  Pauli's monograph [1] gives a nice summary of Born's idea, and the responses it drew from Ehrenfest, Herglotz, Noether, and von Laue.  (Pais's Einstein bio suggests that Born's 1909 paper may have helped set Einstein on the road to Riemannian geometry [2].)

We know already that rigidity and SR don't mix—just think of the barn and the pole!  How could a physicist like Born, mathematically sophisticated, have made such an elementary error?  A simple remark by Pauli clarifies things considerably:

If thus the concept of a RIGID BODY has no place in relativistic mechanics, it is nevertheless useful and natural to introduce the concept of a RIGID MOTION of a body.  We shall denote those motions as rigid for which Born's condition (*) [see below] is satisfied.

Born thought he was defining a rigid body, but Pauli's rephrasing saves the mathematics while improving the physics.  We have no rigid rods in SR, but if you accelerate every atom of an ordinary rod in just the right way, you can move the rod rigidly.  (In fact, this was realized shortly after Ehrenfest's paper (see below) by a number of physicists.)  And Born's definition is Lorentz invariant.

I won't plunge right into Born's definition (as Pauli does).  Instead I'll approach it by thinking atomistically.  Imagine our solid as made up of a large number of atoms A1, ..., An.  Between any two nearby atoms Ai and Aj there is a "natural distance" dij, natural in the sense that if Ai and Aj are pushed together or pulled apart, stresses result, trying to restore the distance dij.  Of course there are propagation delays, but if we start with the solid at rest in some inertial frame, and accelerate it gently, the resulting elastic waves in the solid should die out pretty quickly.  (To use the right jargon, we wait until the transients have subsided.)  Or we can pretend that exactly the right force is applied to each atom at all times, so that natural distances are preserved.  Let the number of atoms tend to infinity (continuum approximation), let the stress/strain ratio tend to infinity, and apply forces gently enough so that the elastic waves can be ignored.  In such a case we arrive at Born's definition.  Born used coordinates, but I'll try for a coordinate-free rephrasing.

First, what is a solid?  Is it just a swath in spacetime?  This is not enough: the atomistic viewpoint suggests we should be able to "mark" a point inside the solid (call it a particle) and follow its world line.  An "event", as usual, is a point in spacetime.  The events along a particle's world line are parametrized by τ, the time on that particle's clock.

(*) Pick a particle (call it A) in the solid.  Pick an event p at time τ on the world line of A.  Draw the spatial plane orthogonal to the world line at p (i.e., the plane of simultaneity at that event).

Now pick another particle B.  The plane of simultaneity intersects the world lines of these two particles A and B at two events; let s(τ,A,B) be the interval between these two events.

Suppose that for any A and B infinitesimally close to one another, ds(τ)/dτ = 0 for all τ.  Then we say the body MOVES RIGIDLY.

If you don't like the notion of "infinitesimally close", there are ways to get around it, but I won't go into that.  (The basic idea is that we are using particle world lines to transport the metric from one tangent plane to another.)

OK, now what?  First, it should be pretty clear that Born's definition captures the idea that the body moves without internal stresses.  Or if you prefer, you can say that we have nearly rigid motion when Young's modulus is large enough, and the accelerations are gentle enough, so that infinitesimal pieces of the body are barely deformed, when viewed in the comoving frame of reference.  Born rigidity is then the limiting case.

If we accelerate a rod rigidly in the longitudinal direction, then the rod suffers the usual Lorentz contraction (in the original rest frame).  More generally, rigid motion without any twisting corresponds to so-called Fermi–Walker transport (see [3, page 170]).  The acceleration of the front of the rod is less than the acceleration of the rear; this is a variation on Bell's Spaceship Paradox (see [4] and the Relativity FAQ entry.)

In a two-page note in 1909, Paul Ehrenfest noted that a disk cannot be brought from rest into a state of rotation without violating Born's condition.  Integrating τ out of Born's condition, we see that infinitesimally close particles must keep the same proper distance.  So in the original rest frame, they suffer Lorentz contraction in the transverse direction but none in the radial direction.  The circumference contracts but the radius doesn't.  But in the original rest frame, the circumference is a circle, sitting in a spatial slice (t = constant) of ordinary flat Minkowski spacetime.  In other words, we would have a "non-euclidean circle" sitting in ordinary euclidean space.  This is a contradiction.

I emphasize that Ehrenfest's argument is a proof by contradiction.  We assume it is possible to "spin up" the disk, maintaining Born rigidity all the time.  We conclude that we have a non-euclidean circle sitting in ordinary euclidean space.  Contradiction, QED!  The argument does not say that a spinning disk actually has a circumference less than 2πr.  Nor does it say that it is impossible to spin up a stationary disk, just that this can't be done without violating Born rigidity at some point.  Once the disk has been spun up, it can keep spinning merrily along, Born-rigidly, forever.

Ehrenfest's argument involves making a complete circuit about the center.  The argument also has a local version.  Say we have a ball bearing a light-year from the Sun.  We cannot put the ball bearing in orbit around the Sun, keeping one face to the Sun, without violating Born's condition.  The proof is not hard, but I'll omit the details.  The key point: the Lorentz contraction factor is not uniform throughout the ball, because some particles have a larger radial distance from the Sun than others.  The assumption of Born rigidity implies that, once in motion, all tangential distances will be Lorentz-contracted as measured in the original rest frame, while the radial distances will be unaffected.  This is enough to give us a little "pocket of non-euclideaness" sitting in ordinary euclidean space.

Pauli states: "It was further proved, independently, by Herglotz and Noether that a rigid body in the Born sense has only three degrees of freedom...  Apart from exceptional cases, the motion of the body is completely determined when the motion of a single of its points is prescribed."  I haven't looked up the papers by Herglotz and Noether, but I dare say it's similar to the argument above.

Max von Laue pointed out that a body made up of n point particles must have at least 3n degrees of freedom.  Say we give an impulse to each particle at t = 0.  Because of the finite speed of light, there can be no constraints relating the velocities of different particles.  Rigid motion can occur in SR only through a conspiracy of forces.

So much for the rotating rigid disk in SR.

Einstein's argument

OK, let's say we start with a disk at rest.  We spin it up (not Born-rigidly).  Once it's spinning nicely and all temporary distortions have had time to settle down (transients have died out), what can we say about it?

Einstein started looking at this in 1909; a letter from 1910 shows he was familiar with Herglotz's results.  Some context: Einstein had a research program based on the Principle of General Relativity: "All reference frames are equally valid."  His famous elevator argument had already paid dividends in his Principle of Equivalence, which resulted in his discoveries of gravitational time dilation and his first (incorrect) computation of the bending of light.  The elevator argument dealt with a linearly accelerated frame; naturally, he turned next to uniform circular motion.  By the Equivalence Principle, this should be the same as a stationary frame plus some kind of "gravitational field".  Einstein asked, what would life be like to inhabitants of the rotating disk?

As a side note, most experts today regard the Principle of General Relativity as formally vacuous: any law of physics can be expressed in any reference frame if you're willing to add in "fudge forces".  The history of science has shown, time and time again, that guiding ideas don't have to stand up to strict logical scrutiny in order to prove fruitful.  Einstein's famous 1916 paper on GR [5] makes no mention of elevators; instead, he introduces the Equivalence Principle via the rotating disk.  The issue continued to absorb him and probably played a decisive role on his road to general relativity [6,7,8].

Einstein's argument for the rotating disk covers similar ground to Ehrenfest's, but with different conclusions.  First, Einstein asserts that the circumference of the rotating disk will be greater than 2πr.  Second, the non-euclidean geometry does not faze him; instead, he invokes the Equivalence Principle to conclude that the geometry of a gravitational field will also be non-euclidean.

Can we make any sense of Einstein's argument?  It takes some unpacking.  His description in the 1916 paper could have been clearer; he did better in a letter in 1919 [10, page 289].  Imagine an army of observers spread out across the disk ("disk observers").  Each is equipped with a meter stick.  Crucial assumption: the dimensions of the sticks are unaffected by the acceleration.  To make this precise, we need the notion of a comoving observer.  That's an observer in an inertial (i.e., unaccelerated) frame, who matches velocity with a disk observer for an instant.  Speaking in spacetime diagrams, her worldline is a straight line tangent to the disk observer's worldline.  Our crucial assumption is that the comoving observer agrees that the disk guy's "meter stick" is really a meter long.

Now, any real meter stick would be at least a little bit affected by acceleration.  But we can idealize.  If you make your stick out of rubber, well, that's bad.  The stiffer the material, the better the stick.  Or we can use more abstract sticks, say, 1,650,763.73 wavelengths of the orange-red emission line of the krypton-86 atom in a vacuum, as was the SI standard between 1960 and 1982.  Anyway, say the comoving observer deems the stick to be a true meter long.  If the stick is radial, the folks in the original rest frame agree: Lorentz contraction doesn't affect transverse motion.  If the stick is tangential to the circumference, the rest frame observers say it's Lorentz contracted.  So more of them would fit around the rim!  Just to be concrete, let's say the contraction factor is 1/3; then the disk observers will measure a radius r and a circumference 6πr.  That's Einstein's argument for a non-euclidean geometry.

It's worth mentioning two things right off the bat.  First, in the original rest frame, the geometry of the disk remains euclidean.  According to rest frame observers, the spinning disk occupies a perfectly normal cylindrical volume.  If you like, picture the disk as enclosed in a snug-fitting, motionless, but infinitely slippery case (zero friction).  The geometry of the case is, of course, euclidean.

Second, Grøn [10, page 288] has given a pithy summation of the difference between Ehrenfest's and Einstein's arguments:

Ehrenfest considered a hypothetical, but impossible situation where a disc had been put into rotational motion in a Born-rigid way, while Einstein considered a situation in which the disk had been put into rotation in an arbitrary way, but the measuring rods were required to be Born rigid.

People raised (and occasionally still raise) various objections to Einstein's argument.  One such objection cannot be denied: Einstein has proposed a definition of a spinning disk geometry, but not the only possible definition.  This objection applies to every possible definition.  Einstein's definition has this in its favor: it's an operational answer to the question, what is the geometry as observed by the spinning disk's inhabitants?  Let's look at more specific complaints.

First, we have the matter of simultaneity.  Let's say the disk observers try to use the famous Einstein synchronization procedure to settle on a common definition of simultaneity.  They will fail!  In mathematical terms, if we try to set up a rotating coordinate system adapted to the disk observers, we inevitably find we can't have all the properties we'd like.  For something like the GPS system, this issue has to be faced, but for Einstein's argument, it's a red herring!  Once the disk is in steady-state rotation, things are time independent.  To quote one recent paper [13, page 144] :

It may be objected that measurements around the periphery of the disk are not done simultaneously in the standard way.  But an observer on the disk need not measure any lengths simultaneously; he merely lays a standard rod of length much smaller than [the radius of the disk] on the surface and marks off the ends, say, with a piece of chalk.  The chalk marks on either end of the rod can be made at any time since the rod is at rest with respect to the surface.  Repeating the procedure, the circumference of the disk is measured.

Second, we have the assumption that the comoving observer and the disk observer agree about length measurements at the instant they match velocities.  This is sometimes called the locality hypothesis or the generalized clock postulate, and is fundamental to relativity.  You could also say it's the definition of an ideal measuring stick; [3, section 16.4] has a nice discussion.  Without this hypothesis, it's hard see what real-world significance SR could have, since some acceleration is ubiquitous.  A full discussion would involve talking about "agreement to first order".

Finally, what does Einstein's argument prove, mathematically speaking?  It would be nice to have a rigorous mathematical definition of the "space" of Einstein's disk, whose geometry is supposed to be non-euclidean.  It's natural to think about a spatial slice.  That runs into difficulties, because of the synchronization issue.  A better solution builds on time independence.  Replace each disk observer with a point.  Formally speaking, we are defining a manifold whose elements (called points) correspond to the worldlines of the disk observers.  Still more formally, we are constructing a quotient manifold of a part of Minkowski spacetime.  We define the metric on the manifold using the Born recipe: if Aragorn and Boromir are neighbors on the disk, then (*) defines s(τ, Aragorn, Boromir), the interval from Aragorn to Boromir.  But once the disk is in steady-state Born-rigid rotation, s is independent of τ.  So we have a definition of the distance s(Aragorn, Boromir).  Now to cross all the t's we have to take limits and stuff, but when the dust settles, we have a perfectly nicely defined Riemannian manifold.  Its geometry turns out to be non-euclidean, just like Einstein said.

I should emphasize that this is not rocket science for a differential geometer.  The idea of a manifold whose "points" are really geometric figures of some sort goes back to the 19th century.  The quotient construction is also not super-sophisticated.  I am not sure who first did this construction explicitly for Einstein's rotating disk, but [10, page 306] cites [11].

Let me return very briefly to the genesis of GR.  A potted summary of Stachel [8]: Einstein had "the happiest thought of [his] life" (the elevator argument) in 1907.  The full theory didn't arrive until 1915.  What took him so long?  Two quotes from Stachel:

It is worth emphasizing that Einstein attributed his success in formulating the special theory in 1905 in no small measure to his insistence on physically defining coordinate systems that allow one to attach direct physical significance to coordinate differences:
The theory to be developed—like every other electrodynamics—rests upon on the kinematics of rigid bodies, since the assertions of each such theory concern relations between rigid bodies (coordinate systems), clocks and electromagnetic processes.  Taking this into account insufficiently is the root of the difficulties, with which the electrodynamics of moving bodies currently has to contend.
Little wonder that Einstein was "tormented" by the problem of "just what coordinates are actually supposed to mean in physics" once they lose their direct physical significance!
A litte further on:

[T]he aim of interpreting rotation as rest-plus-a-gravitational-field appears to have loomed large in Einstein's motivation.  This motive led him to consider uniformly rotating systems of reference soon after his 1907 treatment of uniformly linearly accelerated systems.  But only after the clarification of the question of rigid motions [by Born and Herglotz] do we find any signs of progress on the rotation problem.

The study of uniformly rotating reference systems then led him to the conclusion that, in this case, the spatial coordinates cannot be given a direct physical meaning.

Uniformly linear acceleration was just a bit too easy: it got Einstein partway down the path, but didn't force him to reconsider some hampering assumptions.  The rotating disk problem offered just enough additional resistance.

Stachel mentions a more technical aspect.  Einstein noted the analogy between the Coriolis force and magnetism, which are both forces that depend on velocity.  Now, the Newtonian potential is (to use the proper jargon) a scalar; the electromagnetic potential is a vector; the potential in GR is a symmetric rank-2 tensor (see [3, sec. 7.1]).  Once again, studying the rotating disk pushed Einstein towards the right kind of complexity.

Elastic disks

Physicists continued to write papers about relativity and rotating disks.  Grøn's paper [10] provides an extensive survey; the bibliography includes items up through 2002.  From this large literature I select just two more items.

Lorentz and later Eddington [11] asked about a disk made of "incompressible" material.  Eddington put it this way:

Now the meaning of the term incompressible is that no stress-system can make any difference in the closeness of packing of the molecules; hence the particle-density σ (referred to proper-measure) is the same as for an element of the non-rotating disk.  But the particle-density σ' referred to axes fixed in space may be different.
Eddington's idea is that the Lorentz contraction acts as a kind of tension acting on each concentric ring of disk material; just as tightening a belt makes the diameter smaller, so too the Lorentz contraction.  By crowding in closer to the center, the disk particles relieve some of this "Lorentz tension".  Not all of it—that's impossible, as shown by Ehrenfest's argument.  The upshot, in Eddington's words: "We see that the contraction is one quarter of that predicted by a crude application of the FitzGerald formula to the circumference."  The geometry of the disk remains euclidean.

It's a bit strange that Eddington took into account the "Lorentz stress", but ignored centrifugal force.  After all, if we rotate a disk at ordinary non-relativistic speeds, we expect it to stretch out, not contract!  How much depends on the stiffness of the material, which (as it happens) also helps determine the speed of sound csound for that material.  G.L. Clark obtained a formula taking both effects into account.  It turns out that you get Eddington's result if you take csound to be infinite; if you let csound be the speed of light, then the two effects cancel out and the radius of the disk doesn't change at all.

Finally, in GR you would expect the tensions in the disk to contribute to the stress–energy tensor and so curve spacetime.  I am not aware of any work that has been done on this.

References:

[1] Wolfgang Pauli, Theory of Relativity, pages 130–134, Pergamon Press, 1958.

[2] Abraham Pais, Subtle is the Lord: the Science and Life of Albert Einstein, page 214.

[3] Misner, Thorne, and Wheeler, Gravitation.

[4] J.S. Bell, Speakable and Unspeakable in Quantum Mechanics, page 67, Cambridge University Press, 1987.

[5] A. Einstein, The Principle of Relativity, Dover.

[6] John Stachel, "Einstein and the Rigidly Rotating Disk", in General Relativity and Gravitation One Hundred Years After the Birth of Albert Einstein, pages 1–15, editor A. Held, 1980.
Note: I haven't yet gotten a hold of this or [7]; my remarks are based on [8].

[7] John Stachel, "The Rigidly Rotating Disk as the 'Missing Link' in the History of General Relativity", in Einstein and the History of General Relativity, editors D. Howard and J. Stachel, 1989.

[8] John Stachel, "The First Two Acts", in The Genesis of General Relativity, vol 1, pages 81–111, editor Jürgen Renn, 2007

[9] Ø. Grøn, AJP Vol. 43 No. 10 page 869 (1975)

[10] Ø. Grøn, "Space Geometry in rotating reference frames: A historical appraisal", in Relativity in Rotating Frames, edited by Guido Rizzi and Matteo Luca Ruggiero, pages 285–325 (2004).

[11] Guido Rizzi and Matteo Luca Ruggiero, "Space geometry of rotating platforms: an operational approach", in the arXiv.

[12] Eddington, The Mathematical Theory of Relativity, Cambridge University Press, 1924. Sec.50, pages 112–113.

[13] Thomas A. Weber, "Elementary Considerations of the Time and Geometry of Rotating Frames", in Relativity in Rotating Frames, edited by Guido Rizzi and Matteo Luca Ruggiero, pages 139–153 (2004).