Original by Michael Weiss.

# 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 "rigid body".  Pauli's monograph on relativity [1] gives a nice summary of Born's notion, 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 A_1,...,A_n; between any two nearby atoms A_i and A_j there is a "natural distance" d_ij, natural in the sense that if A_i and A_j are pushed together or pulled apart, stresses result, trying to restore the distance d_ij.  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--this leads to Born's definition.  Born used co-ordinates, but I'll try for a co-ordinate-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 worldline.  An "event", as usual, is a point in spacetime.  The events along a particle's worldline are parametrized by tau, the time-like interval.

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

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

Suppose that for any A and B infinitesimally close to one another, d s(tau)/d tau = 0 for all tau.  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 tau 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 problematical 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 co-ordinates inherited from when the ball-bearing was at rest.  Each particle was then labelled with co-ordinates (r,theta), and we can use these to label its whole worldline, and thus also the points in the spatial slice.  The same "transverse versus radial" argument that Ehrenfest used shows that:

ds2 = dr2 + r2 dtheta2/(1 - r2 w2)

where w is angular velocity.  A routine computation shows that the curvature is non-zero.  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-co-ordinate 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,theta be "inherited co-ordinates" 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:

(d tau)2 = dt2 - f(r)dz2 - g(r)dr2 - h(r)(d theta)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 theta 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 w):

(d tau)2 = dt2 - dz2 - dr2 - r2 dtheta2/(1 - r2 w2)

Turn to GR.  Now all sorts of complications appear.

• The mass of the body distorts spacetime, according to the usual formula.  But we can let the mass of the disk tend to zero (it's an ideal rigid disk), and ignore this.
• I presume we can ignore the Lense-Thirring effect for the same reason.
• The stresses in the disk we cannot ignore, since we had to let the stresses tend to infinity to arrive at the Born condition.  Presumably these could warp spacetime.  (Does this conclusion threaten assumption (2)?)

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.

[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)