Clocks and Rods on a Rotating Disc
Section Twenty-Three
In this section, Einstein gets into the problem of space and time in general relativity. His first example is that of a rotating disc, which is said to have convinced him of the need for non-Euclidean geometry in his generalized theory of relativity.
The observer performs experiments on his circular disc with clocks and measuring-rods. in doing so, it is his intention to arrive at exact definitions for the significance of time- and space-data with reference to the circular disc K', these definitions being based on his observations. What will be his experience in this enterprise? (89)
According to the Euclidean geometry, the ratio of the circumference to the diameter of the disc should be °ë, but the actual ratio on the disc is greater than °ë. Thus, even disregarding time, Euclidean geometry cannot be maintained for the geometry (based on observational data) on this rotating disc. Since general relativity has to treat various sorts of gravitational field, Einstein convinced himself of the need for a more general geometry and mathematics necessary for that. In a word, we have got to be able to treat both time warps and space warps. But, then, how should we set a coordinate system for treating clocks with different rates and rods shrinking and expanding? As an example of this sort of attempt, see Schwarzschild Geometry.
According to John Stachel, the preceding consideration of a rotating disc began almost simultaneously with Paul Ehrenfest's discussion of his "paradox" (Stachel 1989, 49). Ehrenfest raised this question: consider a rapidly rotating disc; then its circumference should show (to an observer in the rest system) a Lorentz contraction, according to the special relativity; but there's no such contraction along the radial direction; then the rotating disc cannot maintain its shape! This argument is fallacious, because the special relativity holds only for inertial (Lorentzian) systems. That is, the object which show a Lorentz contraction must be in a state of free (non-constrained) motion in an inertial system. But the circumference of the disc is certainly constrained, because it is part of the whole disc. Thus, the objects to which Lorentz contraction applies are rods placed along the circumference, not the circumference of the disc itself! As the disc keeps its shape during the rotation, if you count the number of rods (which schrink) along the circumference, this number is larger than the number of rods along the cicumference of the disc when it is at rest (relative to K).
It seems that Einstein came to this conclusion in the late 1912, around the time he came back to Zuerich from Prague. See also Eddington (1920), 75.
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Reference
Eddington, A. S. (1920) Space, Time and Gravitation, Cambridge University Press.
Stachel, John, (1989) "The Rigidly Rotating Disk as the 'Missing Link' in the History of General Relativity", in Einstein and the History of General Relativity (ed. by D. Howard and J. Stachel), Birkhaeser, 1989.
Last modified, September 20, 2002. (c) Soshichi Uchii
suchii@bun.kyoto-u.ac.jp
