Foundations of Physics

, Volume 47, Issue 4, pp 505–531 | Cite as

Simultaneity on the Rotating Disk

  • Don Koks


The disk that rotates in an inertial frame in special relativity has long been analysed by assuming a Lorentz contraction of its peripheral elements in that frame, which has produced widely varying views in the literature. We show that this assumption is unnecessary for a disk that corresponds to the simplest form of rotation in special relativity. After constructing such a disk and showing that observers at rest on it do not constitute a true rotating frame, we choose a “master” observer and calculate a set of disk coordinates and spacetime metric pertinent to that observer. We use this formalism to resolve the “circular twin paradox”, then calculate the speed of light sent around the periphery as measured by the master observer, to show that this speed is a function of sent-direction and disk angle traversed. This result is consistent with the Sagnac Effect, but constitutes a finer analysis of that effect, which is normally expressed using an average speed for a full trip of the periphery. We also use the formalism to give a resolution of “Selleri’s paradox”.


Rotating disk Clock synchronisation Precise timing Sagnac Effect Accelerated frame Circular twin paradox Selleri’s paradox 

Mathematics Subject Classification




I wish to thank Scott Foster, Belinda Pickett, Andy Rawlinson, Alice von Trojan, David Wiltshire, and an anonymous referee for discussions and comments on the manuscript.


  1. 1.
    Grøn, Ø.: Space geometry in rotating reference frames: a historical appraisal. In: Relativity in Rotating Frames, pp. 285–333. Springer, Dordrecht (2004)Google Scholar
  2. 2.
    Ehrenfest, P.: Gleichförmige Rotation starrer Körper und Relativitätstheorie. Phys. Z. 10, 918 (1909)MATHGoogle Scholar
  3. 3.
    Einstein, A.: The Meaning of Relativity, pp. 58 and 59. Methuen and Co., London (1950)Google Scholar
  4. 4.
    Becquerel, J.: Le Principe de Relativité et la Théorie de la Gravitation, p. VIII. Gauthier-Villars, Paris (1922)MATHGoogle Scholar
  5. 5.
    Langevin, P.: Remarques au sujet de la Note de Prunier. Comptes Rendus 200, 48 (1935)MATHGoogle Scholar
  6. 6.
    Franklin, P.: The meaning of rotation in the special theory of relativity. Proc. Natl Acad. Sci. USA 8, 265 (1922)CrossRefMATHADSGoogle Scholar
  7. 7.
    Kurşunoğlu, B.: Spacetime on the rotating disk. Proc. Camb. Philos. Soc. 47, 177 (1951)MathSciNetCrossRefMATHADSGoogle Scholar
  8. 8.
    Grøn, Ø., Vøyenli, K.: On the foundation of the principle of relativity. Found. Phys. 29, 1695 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Schutz, B.: A First Course in General Relativity, p. 44. Cambridge University Press, Cambridge (1988)Google Scholar
  10. 10.
    Cranor, M., Heider, E., Price, R.: A circular twin paradox. Am. J. Phys. 68, 11 (2000)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cantoni, V.: What is wrong with relativistic kinematics? Il Nuovo Cim. 57B, 220 (1968)CrossRefADSGoogle Scholar
  12. 12.
    Cook, R.: Physical time and physical space in general relativity. Am. J. Phys. 72, 214–219 (2004)MathSciNetCrossRefMATHADSGoogle Scholar
  13. 13.
    Misner, C., Thorne, K., Wheeler, J.: Gravitation, Section 6.6. W.H. Freeman, New York (1973)Google Scholar
  14. 14.
    Koks, D.: Explorations in Mathematical Physics, Chapter 7. Springer, New York (2006)MATHGoogle Scholar
  15. 15.
    Desloge, E., Philpott, R.: Uniformly accelerated reference frames in special relativity. Am. J. Phys. 55, 252–261 (1987)MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Grøn, Ø.: Relativistic description of a rotating disk. Am. J. Phys. 43, 869–876 (1975)CrossRefADSGoogle Scholar
  17. 17.
    Rosen, N.: Notes on rotation and rigid bodies in relativity theory. Phys. Rev. 71, 54 (1947)MathSciNetCrossRefMATHADSGoogle Scholar
  18. 18.
    Brown, K.: A Rotating Disk in Translation. Cited 5th September 2016. This essay and its picture are cited in [1]. Brown himself maintains a negligible Internet footprint, so discussions of the details of his essays must presumably remain just discussions.
  19. 19.
    Carroll, S.: Spacetime and Geometry: An Introduction to General Relativity, p. 11. Addison Wesley, San Francisco (2004)MATHGoogle Scholar
  20. 20.
    Brown, K.: Vis Inertiae. Cited 5th September 2016
  21. 21.
    Selleri, F.: Noninvariant one-way speed of light and locally equivalent reference frames. Found. Phys. Lett. 10, 73–83 (1997)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Selleri, F.: Sagnac effect: end of the mystery. In: Relativity in Rotating Frames, p. 57. Kluwer, Dordrecht (2004)Google Scholar

Copyright information

© Her Majesty the Queen in Right of Australia 2017

Authors and Affiliations

  1. 1.Defence Science and Technology GroupEdinburghAustralia

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