100 GTR: Uniformly Rotating Disk and the Hole Argument

The “Ehrenfest paradox“: Ehrenfest imagined a rigid cylinder set in motion from rest and rotating around its axis of symmetry. Consider an observer at rest measuring the circumference and radius of the rotating cylinder. The observer arrives at two contradictory requirements relating to the cylinder’s radius:

  1. Every point in the circumference of the cylinder moves with radial velocity ωR, and thus, the circumference of the cylinder  should appear Lorentz contracted to a smaller value than at rest, by the usual “relativistic” factor γ: 2πR‘ < 2πR.
  2. The radius R’ is always perpendicular to its motion and suffers no contraction at all; it should therefore be equal to its value R: R’ = R.

Einstein wrote to Vladimir Varićak either in 1909 or in 1910 (Febuary 28): “The rotation of the rigid body is the most interesting problem currently provided by the theory of relativity, because the only thing that causes the contradiction is the Lorentz contraction”.  CPAE 5, Doc. 197b.

Ehrenfest imagined a rigid cylinder gradually set into rotation (from rest) around its axis until it reaches a state of constant rotation.

In 1919 Einstein explained why this was impossible: (CPAE 9, Doc. 93)

“One must take into account that a rigid circular disk at rest would have to snap when set into rotation, because of the Lorentz shortening of the tangential fibers and the non-shortening of the radial ones. Similarly, a rigid disk in rotation (made by casting) would have to shatter as a result of the inverse changes in length if one attempts to bring it to the state of rest. If you take these facts fully into consideration, your paradox disappears”.

Assuming that the cylinder does not expand or contract, its radius stays the same. But measuring rods laid out along the circumference 2πR should be Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the paradox that the rigid measuring rods would have to separate from one another due to Lorentz contraction; the discrepancy noted by Ehrenfest seems to suggest that a rotated Born rigid disk should shatter. According to special relativity an object cannot be spun up from a non-rotating state while maintaining Born rigidity, but once it has achieved a constant nonzero angular velocity it does maintain Born rigidity without violating special relativity, and then (as Einstein showed in 1912) a disk riding observer will measure a circumference.

Hence, in 1912, Einstein discussed what came to be known as the uniformly rotating disk thought experiment in general relativity. Thinking about Ehrenfest’s paradox and taking into consideration the principle of equivalence, Einstein considered a disk (already) in a state of uniform rotation observed from an inertial system.

We take a great number of small measuring rods (all equal to each other) and place them end-to-end across the diameter 2R and circumference 2πR of the uniformly rotating disk. From the point of view of a system at rest all the measuring rods on the circumference are subject to the Lorentz contraction. Since measuring rods aligned along the periphery and moving with it should appear contracted, more would fit around the circumference, which would thus measure greater than 2πR. An observer in the system at rest concludes that in the uniformly rotating disk the ratio of the circumference to the diameter is different from π:

circumference/diameter = 2π(Lorentz contracted by a factor…)/2R = π (Lorentz contracted by a factor….).

According to the equivalence principle the disk system is equivalent to a system at rest in which there exists a certain kind of static gravitational field. Einstein thus arrived at the conclusion that a system in a static gravitational field has non-Euclidean geometry.

Soon afterwards, from 1912 onwards, Einstein adopted the metric tensor as the mathematical respresentation of gravitation.

Indeed Einstein’s first mention of the rotating disk in print was in his paper dealing with the static gravitational fields of 1912; and after the 1912 paper, the rotating-disk thought experiment occurred in Einstein’s writings only in a 1916 review article on general relativity: “The Foundation of the General Theory of Relativity”.

He now understood that in the general theory of relativity the method of laying coordinates in the space-time continuum (in a definite manner) breaks down, and one cannot adapt coordinate systems to the four-dimensional space.

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My new paper deals with Einstein’s 1912 and 1916 rotating disk problem, Einstein’s hole argument, the 1916 point coincidence argument and Mach’s principle; a combined-into-one deal (academic paper) for the readers of this blog.

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Sitting: Sir Arthur Stanley Eddington and Hendrik Antoon Lorentz. Standing: Albert Einstein, Paul Ehrenfest and Willem de Sitter. September 26, 1923.

Further reading: Ehrenfest paradox

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4 thoughts on “100 GTR: Uniformly Rotating Disk and the Hole Argument

  1. Just to recall it: Why did we have to assume that the circumference must shrink while the radius may not? To my guess, rotation itself does not cause to shorten something.

    1. According to Einstein, the circumference of the uniformly rotating disk would seem to be longer because the measuring rods would be contracted (Lorentz contraction), whereas its radius would remain uncontracted (would apparently be measured with uncontracted measuring rods).

      1. From Weinstein, Galina, “Einstein’s Uniformly Rotating Disk and the Hole Argument” , 2015, p.4: Einstein wrote:
        “Physical meaning of spatial determinations. The propositions of Euclidean geometry acquire a physical content through our assumption that there exist objects that possess the properties of the basic structures of Euclidean geometry. We assume that appropriately constructed edges of solid bodies not subjected to external influences have the >>definitional properties<< that straight lines have (material straight line), and that the part of a material straight line between two marked material points has the properties of a segment. Then the propositions of geometry turn into propositions concerning the arrangements of material straight lines and segments that are possible when these structures are at relative rest".

        p.4: Einstein wrote (concerning the snapping Ehrenfest-disc):
        “If you take these facts fully into consideration, your paradox disappears.”
        “The measuring rod as well as the coordinate axes are to be conceived as rigid bodies. This is permitted despite the fact that, according to the [special] theory of relativity, the rigid body cannot really exist.”

        Within our surrounding reality it seems impossible to compare moving rods with a measuring rod that stays at rest. Hence it is impossible to detect what Paul Ehrenfest stated to contradict the Euclidian geometry.

        In general, if somebody states that we have to distinguish between bodies at rest and others in motion because of hidden effects then we do not have to follow him (because it is an assumption only) but the proponent of this message has to obey his own principles.

        All the paradoxes in relativity are made like this:
        Compare a length at rest with the same length at high speed.
        Clearly, all these attempts to compare are impossible to undertake.
        Then we should not waste our life time just by doing exactly this kind of reasoning.
        Galina: “If we adhere to Euclidean geometry, then we arrive at a contradiction because one and the same measuring rod produces what looks like two different lengths due to the Lorentz contraction.”

        The conclusion of Einstein in 1919 was that RRBs do not exist:
        They “would have to shatter” and therefore the “paradox disappears” as such rigid rotating bodies (RRBs) cannot exist [at all].

        The very same mystery [as that of rotating disks being unable to rigidly rotate] occurs when applying Euklid’s geometry to the well known light clock which is very often used to introduce the time dilatation in course books (Langevin,Feynman):
        (1) In the transversal direction (of +x) we have to use the uncontracted measuring rods while observing diagonally lines of the light [observer is at rest].
        (2) If we were not allowed (because of the very first law of proper measurements) to do this comparison, we could not have seen any difference between the vertical (short) length and the diagonal (longer) length and we could never give an idea of or an argument in favor of the time dilatation.
        (*) According to the rigidity of Einstein’s argumentation we must agree that we are unable to compare the vertical and the diagonal distances because there is no common measuring rod available. Hence the difference between the vertical and the diagonal distances is a “disappearing paradox” which is improper to found a theory [except it is a theory of contradiction].

        “The diameter of the uniformly rotating disk would apparently be measured with uncontracted measuring bodies, but the circumference would seem to be longer because the measuring bodies would be contracted. The possibility of Euclidean geometry on a uniformly rotating disk is thus unacceptable.”

        Proposition: Einstein did not obey his own principles (maybe he did not want to obey any principles).
        To break your own principles [or to leave some the self created language] is according to Kurt Gödel the topmost sin. Einstein introduced [as Lorentz did with local time] independent time lines [for each moving particle] that are coupled by the actual [not the theoretical] distance that the light takes to travel. In order to release some conclusions he stated “der Kürze halber” and “sei es widerspruchsfrei lösbar” that light starts at t=t’=0. But, don’t hesitate to read this twice, this is the fixpoint of the identity of time. From that onward there is no argument supporting the idea of independent time measures.

        Finally:
        There is given an object in motion A crossing two other objects B and C, all having different velocities. If all objects use the same light signals [traversing some waves], they will get different measures of length. As A has two partner objects to scan while passing by, he will get two different measures. A will be able to compare all the measures as they are local. Hence A would detect that there is a difference in length and time that makes no sense. And, finally again, if Einstein was right, A would never come to the conclusion, that the ideas of Einstein are working.

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