[Physics FAQ] - [Copyright]

Original by Michael Weiss.
End note added by Don Koks, 2013.

The Rigid Rotating Disk in Relativity

This old chestnut predates GR.  The trail starts with Max Born's notion of an "SR rigid body", a relativistic replacement for the classical notion.  It leads past Einstein's discussion in his great 1916 paper on GR, where he uses the rotating disk to introduce non-euclidean geometry.  It then twists and turns as Eddington, Lorentz, and lesser lights attempt to compute the fate of the rigid disk.

In this entry, I summarize what I know of the literature, and invite others to fill in the gaps.  Surely such a celebrated problem should have found a definitive resolution by now.  But the tale I have to tell ends on an incomplete note.  We look at SR first, then GR.

Special Relativity

Born rigidity: In 1909, Born proposed a Lorentz-invariant definition of a "rigid body".  Pauli's monograph on relativity [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 (*) 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.  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.  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.  More generally, rigid motion without any twisting corresponds to so-called Fermi–Walker transport (see MTW [3]).  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.)

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.

The issue of spatial slices deserves a few words.  The particles in a rotating disk (not assumed rigid) cannot agree on a global notion of simultaneity.  For if you make a circuit around the edge, joining up the infinitesimal planes of simultaneity, when you return to your starting point, the planes no longer match up.  This makes it problematic to talk about geometry "as seen by the particles" (or by observers standing on the disk).

I've talked about making complete circuits about the center, but both Ehrenfest's argument and the simultaneity problem have local versions.  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.  Nor can the particles of the ball bearing agree on a notion of simultaneity.

The proofs are not hard.  I'll sketch the local version of Ehrenfest's argument.  Take a spatial slice in the rest frame and look at the metric.  The simplest way to express it is to use polar coordinates inherited from when the ball bearing was at rest.  Each particle was then labelled with coordinates (r,θ), and we can use these to label its whole world line, and thus also the points in the spatial slice.  The same "transverse versus radial" argument that Ehrenfest used shows that:

ds2 = dr2 + r22/(1 − r2ω2)

where ω is angular speed.  A routine computation shows that the curvature is nonzero.  This contradicts the fact that the spatial slice is 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.

General Relativity

Einstein's 1916 paper on GR [5] makes no mention of elevators; instead, the Equivalence Principle is introduced via the rotating disk.  Einstein reproduces Ehrenfest's argument, but with a different conclusion: since we are no longer assuming flat Minkowski space, Einstein asserts that geometry for the rigid rotating disk is non-euclidean.  The Equivalence Principle now implies that geometry in a gravitational field will also be non-euclidean.  (By "geometry", I mean spatial geometry, i.e., we're not concerned with the temporal components of the spacetime metric.)

Can we make any sense of Einstein's argument?  The simplest interpretation makes a couple of assumptions:

  1. The stresses in the rigid disk warp spacetime.  This is plausible, even assuming the mass of the disk is negligible.  Recall that we had to allow the stress/strain ratio to approach infinity to obtain Born rigidity.
  2. The time coordinate of the original rest frame survives undistorted.  In other words, let t be the time as measured by observers in the original rest frame, and let z, r, θ be "inherited coordinates" from when the disk was at rest (see the notion of marking points introduced in the previous section).  Then we assume that the metric looks like this once the disk has been spun up to a steady speed:

    2 = dt2f(r) dz2g(r) dr2h(r)2

Assumption (2) seems a lot more dubious than (1), but it does allow us to talk about spatial slices (t = constant), and hence the geometry of the spinning disk.  We appeal to axial symmetry and steady-state conditions in making f, g, and h independent of θ and t.  We have no such justification for leaving out z.

Einstein doesn't give an explicit formula for the spacetime metric of the rigid spinning disk, but here's one obvious candidate (assuming the angular speed is ω):

2 = dt2 − dz2 − dr2r22/(1 − r2ω2)

Turn to GR.  Now all sorts of complications appear.

To settle the question definitively, it seems one has to perform a full-blown, hairy GR calculation.  Perhaps someone has done this; perhaps someone has turned the vague notion of "infinitely rigid" into a formula for a stress–energy tensor, plugged that into the Einstein field equations, and solved.  If the Gentle Reader knows of a reference, please let me know.

Note added by the Editor in 2013–2015

I have retained the above essay for its historical interest, but I have always felt that the real question that the physicists above meant to address is the description of a series of events in flat spacetime from the viewpoint of a rotating frame.  (Sure, after sorting that out, we could then turn our attention to curved spacetime, but flat spacetime is tough enough.)  I think the discussion described above that was held by physicists of some decades ago got derailed from the start when it began to focus on the structure of the disk.  Did they really mean to focus on the disk's structure?  Maybe, but I doubt it.  The rotating disk is merely a device to help us picture and think about the rotating frame, but discussion of how the disk rotates, how it gets accelerated and so on, are completely irrelevant to the analysis of a rotating frame within the context of special relativity, which has to do with what is simultaneous with what; for that, we must appeal to the Clock Postulate to construct a plane of simultaneity at each event of interest.  This has nothing to do with general relativity, which is a theory of gravity.  If one draws planes of simultaneity at various events in a rotating system such as a disk, then one can form a halfway coherent picture of those events.  It turns out be impossible to construct a proper set of coordinates, but we can at least analyse the events in some way.  Nevertheless, that analysis simply has nothing to do with questions of what the disk is made of and how it got spun up.

The same idea is universally taken for granted in the standard Lorentz transform studied throughout special relativity.  There too, we don't concern ourselves—no one does—with how the "primed frame" was ever made to move.  It is simply taken as having always been moving, forever.  Any other treatment of its motion would only mask and complicate the underlying ideas, which concern the Lorentz transform and not questions of how to accelerate physical objects.  We are content to treat a constant-velocity primed frame as having had its state of motion forever, so we should do likewise for the rotating frame and not allow discussion of how the disk was set into motion derail the core issue, which is the analysis of events in a rotating frame.

Of course, a study of how objects are accelerated is certainly important and fruitful in special relativity.  For example, when discussing Bell's Spaceship Paradox we would do well to ask ourselves how the string connecting the two spacecraft behaves; it has mass and it needs to be accelerated, and we can't just treat it as a series of massless events on a spacetime diagram that merrily go along for the ride as the spaceships accelerate.  The string requires the leading spaceship to accelerate it, so we shouldn't be surprised to find that eventually it breaks as it becomes more difficult to accelerate, when the ships' speeds get close to the speed of light.  (Notice: this is distinct from the commonly found saying that "the string breaks because it wants to Lorentz-contract".  Rather, it breaks because its atoms are becoming harder to accelerate as the leading spaceship keeps incessantly pulling ahead; these atoms' relativistic mass increases, so that they are simply tougher to accelerate, and eventually the string's physical makeup can't support the force required to be applied by the leading ship to accelerate the string's atoms to have them keep up with the leading ship.  So they get the left behind—meaning, the string falls apart.)  So a study of accelerating physical objects has its place in special relativity, and there's nothing wrong with asking how a disk is accelerated.  But just as we do for the standard Lorentz transform, let's treat such an advanced dynamical question as secondary to simply asking how a rotating frame views the world.  The above discussion of the rotating disk seems never to have managed that.

It might at first be thought that the rotating frame gives problems with the speed of light.  For instance if we spin around on the spot, we see the Moon whizz around us in a huge circle, so isn't it travelling faster than light in our frame?  It is travelling faster than the tabulated value of "c", but it is not travelling faster than light; after all, we agree that light is still escaping from its surface.  In the language of special relativity, the Moon's world line always remains within its local light cone; it remains "timelike".  In fact, this behaviour is no different to the well-accepted behaviour of light in an accelerated frame, where the measured speed of light depends upon where in that frame it currently is.  This speed can in fact have any value, from zero to infinity.  (See the FAQ entry Do moving clocks always run slowly? for further discussion on this.)

Historically, a central question about the spinning disk was known as Ehrenfest's Paradox: what is the length of the disk's circumference?  Some think the circumference should be Lorentz contracted; but that implies its length is no longer 2πr, which plays havoc with the idea of spacetime geometry because changes in that geometry are tied to the presence of real gravity (not acceleration, but real gravity), and there's no gravity in the usual scenario of a spinning disk; in the absence of gravity, spacetime is as flat for a disk observer as it is for everyone else.  But if we want to set up a spinning disk in the same way that we set up the "primed frame" that moves with constant velocity in any textbook on relativity, then we should do so in a way that involves no consideration of the structure of matter—after all, how often, if ever, do you see discussion of the structure of the matter comprising, say, the railroad train that constitutes the primed frame moving at constant velocity in any relativity textbook?  We could have such a discussion, but it's universally held as not relevant to those primed frames.  So let's not have that discussion for a disk either.  In that case, there are two simpler ways to get the disk spinning.  Each involves grabbing each point on the disk and moving it in such a way that the whole disk spins up.  Physically impossible, sure, but that's of no relevance.

  1. You could move each point such that all points at equal distance from the centre trace out congruent helical world lines.  The circumference's length cannot change: it's 2πr by construction.  What do the observers at rest on the disk measure for its circumference?  If you draw various planes of simultaneity at a selection of events on the circumference, you'll find that because these planes are inclined and themselves spinning around, these observers measure their neighbours on the disk's perimeter to move apart in a non-trivial way.  They also notice that some of those neighbours suddenly age rapidly, while other neighbours get younger.  It all means that the observers at rest on the disk cannot construct a decent set of coordinates that they can use to agree on the simultaneity of events.  And that means that they cannot construct a frame (by definition of the word "frame").
  2. You could move each disk point such that not all points at equal distance from the centre trace out congruent helical world lines: you could choose to create world lines so as to give the observers on the disk a different experience of simultaneity to what was described in item 1 above.  But owing to the see-sawing nature of those observers' planes of simultaneity, what results will still not comprise a frame.  And while you'll move different points on the circumference in different ways, they'll all remain on a circle, and its length will still be 2πr.  If you need proof of that, imagine having the disk sit tightly inside a tube.  As you spin the disk up in whatever non-trivial way you choose, it still fits tightly within the tube.
Either way, Ehrenfest's Paradox is resolved, because there's a lot more going on here than the simple (and simplistic) idea of Lorentz contraction around the disk's circumference that is so often thought to be occurring.  If you analyse the measurements of disk observers by drawing planes of simultaneity, you won't fall into the trap of thinking that the perimeter of the disk can be treated as if it were moving at constant velocity.  It isn't: the rotation sets up planes of simultaneity that see-saw helically, and the whole thing is just not the same as the non-see-sawing planes of simultaneity constructed by inertial observers.

The Born metric that you'll find in the "General Relativity" discussion above sets t' = t on the disk.  This is nothing more than the Galilei transform dressed up in relativity language, and it has no relativistic significance.  (A related metric is the Langevin metric, which does the same thing.)  Of course, one can always define a coordinate t' in that way, but that coordinate has no relativistic significance, just as the Galilei transform t' = t doesn't give you a real time in the case of inertial frames (obviously; you have to use the Lorentz transform t' = γ (t − vx/c2) + constant, which is the whole point of special relativity).  Actually, in the Global Positioning System, what is called GPS time on the clocks of the "Earth-Centred Earth-Fixed frame" (meaning the number that our GPS receiver shows us on its clock) is set by the Global Positioning System standard in such a way that GPS time t' is defined as t' = t, where t is the time on the momentarily coincident clock in the "Earth-centred Inertial Frame".  And of course this t' is very useful for GPS.  But as far as relativity is concerned, t' is not a real time coordinate for a GPS satellite, even though it is true time in the Earth-centred Inertial Frame: t' approximates a true time coordinate very well because Earth doesn't spin quickly.  You can appreciate that the time shown by a GPS receiver is not a true time when you ask what the famous twins of the "Twin Paradox" would see if they each carried such a receiver with them on their separate trips.  Before parting, their receivers would show identical times, as they should.  The stay-at-home twin then sits for a year, while the space-bound twin goes travelling at high speed.  When they re-unite and again consult their receivers' clocks, those receivers will again calculate identical times, as they should, because that is the time of that event in the Earth-centred Inertial Frame.  But if the twins didn't know any better, they would conclude that they had aged identically, even though one is clearly older than the other. However, they are not confused because they know that the GPS time on their receivers is not "proper time"; yes, it's the time in the Earth-centred Inertial Frame, but that's not appropriate for them to measure their own ageing.  The moral of the story is that you shouldn't be misled into thinking that the Born and Langevin metrics are deep or even useful.

So setting t' = t on a spinning disk in a discussion of real relativity (unhampered by technical details of running satellites) to arrive at the Born metric is—on the face of it—a na´ve thing to do, akin to using a Galilei transform in place of a Lorentz transform in a relativity task.  The Born metric uses this artificial coordinate t', but that coordinate is not time.  If the Born metric has some use outside of GPS in a real relativity discussion, I don't know what that use is.  It's certainly not the metric on a spinning disk if its "time" coordinate is to be interpreted as time on the disk.  Time as measured by disk observers is far more complicated than that.

There are some who would represent spacetime on the disk's edge by a cylinder (which is certainly valid as far as simple pictures go), then cut the cylinder on a line parallel to the time axis, flatten the cylinder out, draw lines through events, and call those "lines of simultaneity".  But this procedure is something make-believe that has nothing to do with proper relativity; it's not consistent with ideas of simultaneity already established by the Lorentz transform.  The cylinder should be cut by a plane of simultaneity, and that plane certainly doesn't produce a line on the cylinder's surface.  We don't have the freedom to re-arrange spacetime into pieces that suit some arbitrary definition of simultaneity.  Simultaneity's definition is already established, and while in principle it's straightforward to analyse in a rotating frame, in practice the details are difficult.


[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, pg 214.

[3] Misner, Thorne, and Wheeler, Gravitation, pg 170.

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

[5] The Principle of Relativity, Dover.

[6] G. Cavalleri, Nuovo Cimento 53B pg 415.

[7] O. Gron, AJP Vol. 43 No. 10 pg 869 (1975)

[8] C. Berenda Phys. Rev. 62 pg 280 (1942)