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**Space geometry in rotating reference frames: a historical appraisal.**
*(English)*
Zbl 1075.83001

Rizzi, Guido (ed.) et al., Relativity in rotating frames. Relativistic physics in rotating reference frames. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1805-3/hbk). Fundamental Theories of Physics 135, 285-333 (2004).

The problem of providing a mathematical description of the phoronomy of a rotating solid disk admits relativistic as well as quantum physical aspects. The relativistic description of a uniformly spinning disk has a history nearly as old as that of the theory of relativity itself. Already in 1909 Ehrenfest formulated his paradox of relativistic rigidity within the framework of special relativity: On the one hand the periphery of a relativistically rotating cylinder is Lorentz contracted, and on the other hand a radial line on the spinning cylinder is not. The result is that a solid disk of radius \(r > 0\) might be brought to spin with an arbitrarily great uniform angular velocity \(\omega\), and that, according to the prediction of the Terrell effect, its shape should not undergo any distortion with spinning, but should appear to be globally contracted by a scale factor which depends on \(\omega\).

A few years later, Einstein who actually did not participate in the discussion of the Ehrenfest paradox, made heuristically use of the relativistic rigidity problem in order to motivate the introduction of non-Euclidean geometry in the context of a relativistic theory of gravity [P. M. Schwarz, J. H. Schwarz, Special Relativity: From Einstein to Strings. Cambridge University Press, (Cambridge, New York, Melbourne) (2004; Zbl 1104.83008)]. Einstein considered the spinning disk as a rotating reference frame and imagined this frame filled by radial and tangential standard rods. The definition of a standard rod is that it is Born rigid, so that it undergoes a Lorentz contraction under motion. In this way he made it clear that the length of the periphery of a spinning disk of radius \(r > 0\) is longer than \(2 \pi r\), and not shorter as initially stated in Ehrenfest’s formulation of his paradox. Einstein wrote in a letter dated August 19, 1919: “A rigid circular disk must break up if it is set into rotation, on account of the Lorentz contraction of the tangential fibers and the non-contraction of the radial ones. Similarly, a rigid disk in rotation must explode as a consequence of the inverse changes in lengths, if one attempts to bring it to the rest state” [D. Howard, J. Stachel (eds.), Einstein and the History of General Relativity. Birkhäuser Verlag, (Basel, Boston) (1989; Zbl 0846.01007)].

Born noted an interesting consequence of the non-Euclidean geometry on a spinning disk combined with the principle of equivalence: In a gravitational field a standard measuring rod is longer or shorter according to the position at which it is located. The paper under review follows the conceptual evolution of this topic from Ehrenfest via Born, Planck, Einstein, Eddington to the present time discussion of the spinning disk, the Thomas precession and Sagnac effect. Specifically, the paper emphasizes the importance of taking the relativity of simultaneity properly into account which implies that in the neighborhood of an arbitrary point on a spinning disk a local coordinate system may be introduced by means of differential transformations from coordinates in the inertial rest frame of the axis. These transformations may be chosen so that the spatial line element at constant time in the rotating reference frame defines a Euclidean metric. A description of the quantum physical aspects of the planar rotations of a crystalline disk has recently be given by [A. Leggett, Superfluidity in a crystal? Science 305, 1921–1922 (2004)].

Due to the fact that the symplectic group SL(2,\(\mathbb R\)) acting as a group of automorphisms of the three-dimensional real Heisenberg nilpotent Lie group forms a double cover of the positive light cone preserving the Lorentz subgroup SO\(_0(2,1)\), the global metaplectic symmetries inherent to the harmonic analysis of the Heisenberg group model solve the problem of non-classical rotational inertia and help to obtain an understanding of the issues associated with Ehrenfest’s paradox of a uniformly spinning disk, and the cosmological physics of rotating black holes.

For the entire collection see [Zbl 1050.83005].

A few years later, Einstein who actually did not participate in the discussion of the Ehrenfest paradox, made heuristically use of the relativistic rigidity problem in order to motivate the introduction of non-Euclidean geometry in the context of a relativistic theory of gravity [P. M. Schwarz, J. H. Schwarz, Special Relativity: From Einstein to Strings. Cambridge University Press, (Cambridge, New York, Melbourne) (2004; Zbl 1104.83008)]. Einstein considered the spinning disk as a rotating reference frame and imagined this frame filled by radial and tangential standard rods. The definition of a standard rod is that it is Born rigid, so that it undergoes a Lorentz contraction under motion. In this way he made it clear that the length of the periphery of a spinning disk of radius \(r > 0\) is longer than \(2 \pi r\), and not shorter as initially stated in Ehrenfest’s formulation of his paradox. Einstein wrote in a letter dated August 19, 1919: “A rigid circular disk must break up if it is set into rotation, on account of the Lorentz contraction of the tangential fibers and the non-contraction of the radial ones. Similarly, a rigid disk in rotation must explode as a consequence of the inverse changes in lengths, if one attempts to bring it to the rest state” [D. Howard, J. Stachel (eds.), Einstein and the History of General Relativity. Birkhäuser Verlag, (Basel, Boston) (1989; Zbl 0846.01007)].

Born noted an interesting consequence of the non-Euclidean geometry on a spinning disk combined with the principle of equivalence: In a gravitational field a standard measuring rod is longer or shorter according to the position at which it is located. The paper under review follows the conceptual evolution of this topic from Ehrenfest via Born, Planck, Einstein, Eddington to the present time discussion of the spinning disk, the Thomas precession and Sagnac effect. Specifically, the paper emphasizes the importance of taking the relativity of simultaneity properly into account which implies that in the neighborhood of an arbitrary point on a spinning disk a local coordinate system may be introduced by means of differential transformations from coordinates in the inertial rest frame of the axis. These transformations may be chosen so that the spatial line element at constant time in the rotating reference frame defines a Euclidean metric. A description of the quantum physical aspects of the planar rotations of a crystalline disk has recently be given by [A. Leggett, Superfluidity in a crystal? Science 305, 1921–1922 (2004)].

Due to the fact that the symplectic group SL(2,\(\mathbb R\)) acting as a group of automorphisms of the three-dimensional real Heisenberg nilpotent Lie group forms a double cover of the positive light cone preserving the Lorentz subgroup SO\(_0(2,1)\), the global metaplectic symmetries inherent to the harmonic analysis of the Heisenberg group model solve the problem of non-classical rotational inertia and help to obtain an understanding of the issues associated with Ehrenfest’s paradox of a uniformly spinning disk, and the cosmological physics of rotating black holes.

For the entire collection see [Zbl 1050.83005].

Reviewer: Walter Schempp (Siegen)