Foundations of Physics

, Volume 32, Issue 10, pp 1525–1556

Space Geometry of Rotating Platforms: An Operational Approach

  • Guido Rizzi
  • Matteo Luca Ruggiero
Article

Abstract

We study the space geometry of a rotating disk both from a theoretical and operational approach; in particular we give a precise definition of the space of the disk, which is not clearly defined in the literature. To this end we define an extended 3-space, which we call “relative space:” it is recognized as the only space having an actual physical meaning from an operational point of view, and it is identified as the “physical space of the rotating platform.” Then, the geometry of the space of the disk turns out to be non Euclidean, according to the early Einstein's intuition; in particular the Born metric is recovered, in a clear and self consistent context. Furthermore, the relativistic kinematics reveals to be self consistent, and able to solve the Ehrenfest's paradox without any need of dynamical considerations or ad hoc assumptions.

special relativity rotating platforms space-geometry Ehrenfest non-time-orthogonal frames 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    P. Ehrenfest, Phys. Z. 10, 918 (1909).Google Scholar
  2. 2.
    M. G. Sagnac, C. R. Acad. Sci. (Paris) 157, 708, 1410 (1913).Google Scholar
  3. 3.
    M. Planck, Phys. Z. 11, 294 (1910).Google Scholar
  4. 4.
    H. A. Lorentz, Nature 106, 793 (1921).Google Scholar
  5. 5.
    A. S. Eddington, Mathematical Theory of Relativity (Cambridge University Press, Cambridge, 1922), Space, Time and Gravitation (Cambridge University Press, Cambridge, 1920).Google Scholar
  6. 6.
    G. L. Clark, Proc. R. Soc. Edinburgh 62A, 434 (1947).Google Scholar
  7. 7.
    C. W. Berenda, Phys. Rev. 62, 280 (1942).Google Scholar
  8. 8.
    G. Cavalleri, Nuovo Cimento LIII B(2), 416 (1968).Google Scholar
  9. 9.
    T. E. Phipps, Jr., Experiment on Relativistic Rigidity of a Rotating Disk, NOLTR 73-9 (Naval Ordnance Laboratory, 1973), p. 47.Google Scholar
  10. 10.
    A. Brotas, C. R. Acad. Sci. (Paris) 267, 57 (1968).Google Scholar
  11. 11.
    W. H. McCrea, Nature 234, 399 (1971).Google Scholar
  12. 12.
    A. Einstein, The Meaning of Relativity (Princeton University Press, Princeton N.J., 1950).Google Scholar
  13. 13.
    J. Stachel, in General Relativity and Gravitation, A. Held, ed. (Plenum, New York, 1980).Google Scholar
  14. 14.
    G. Stead and H. Donaldson, Phil. Mag. 20, 92 (1910).Google Scholar
  15. 15.
    H. E. Ives, J. Opt. Soc. Am. 29, 472 (1939).Google Scholar
  16. 16.
    A. Eagle, Phil. Mag. 28, 592 (1939).Google Scholar
  17. 17.
    M. Galli, Rend. Acc. Lincei 12, 569 (1952).Google Scholar
  18. 18.
    E. L. Hill, Phys. Rev. 69, 488 (1946).Google Scholar
  19. 19.
    N. Rosen, Phys. Rev. 71, 54 (1947).Google Scholar
  20. 20.
    H. Arzeliès, Relativistic Kinematics (Pergamon, New York, 1966).Google Scholar
  21. 21.
    L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, New York, 1971).Google Scholar
  22. 22.
    C. Møller, The Theory of Relativity (Oxford University Press, Oxford, 1972).Google Scholar
  23. 23.
    Ø. Grøn, Found. Phys. 9, 353 (1979).Google Scholar
  24. 24.
    Ø. Grøn, Int. J. Theor. Phys. 16, 603 (1977).Google Scholar
  25. 25.
    Ø. Grøn, Am. J. Phys. 43, 869 (1975).Google Scholar
  26. 26.
    T. A. Weber, Am. J. Phys 65, 946 (1997).Google Scholar
  27. 27.
    D. Dieks, Eur. J. Phys. 12, 253 (1991).Google Scholar
  28. 28.
    J. Anandan, Phys. Rev. D 24, 338 (1981).Google Scholar
  29. 29.
    G. Rizzi and A. Tartaglia, Found. Phys. 28, 1663 (1998).Google Scholar
  30. 30.
    S. Bergia and M. Guidone, Found. Phys. Lett. 11, 549 (1998).Google Scholar
  31. 31.
    F. Selleri, Found. Phys. 26, 641 (1996).Google Scholar
  32. 32.
    F. Selleri, Found. Phys. Lett. 10, 73 (1997).Google Scholar
  33. 33.
    J. Croca and F. Selleri, Nuovo Cimento B 114, 447 (1999).Google Scholar
  34. 34.
    F. Goy and F. Selleri, Found. Phys. Lett. 10, 17 (1997).Google Scholar
  35. 35.
    J. P. Vigier, Phys. Lett. A 234, 75 (1997).Google Scholar
  36. 36.
    P. K. Anastasowksi et al., Found. Phys. Lett. 12, 579 (1999).Google Scholar
  37. 37.
    W. A. Rodrigues, Jr. and M. Sharif, Found. Phys. 31, 1767 (2001).Google Scholar
  38. 38.
    R. D. Klauber, Found. Phys. Lett. 11 (5), 405 (1998).Google Scholar
  39. 39.
    R. D. Klauber, Am. J. Phys. 67 (2), 158 (1999).Google Scholar
  40. 40.
    A. Tartaglia, Found. Phys. Lett 12, 17 (1999).Google Scholar
  41. 41.
    A. Grünbaum and A. J. Janis, Synthèse 34, 281 (1977).Google Scholar
  42. 42.
    M. Strauss, Int. J. Theor. Phys. 11, 107 (1974).Google Scholar
  43. 43.
    C. Cattaneo, Introduzione alla teoria einsteiniana della gravitazione (Veschi, Roma, 1961).Google Scholar
  44. 44.
    C. Cattaneo, Nuovo Cimento 10, 318 (1958).Google Scholar
  45. 45.
    C. Cattaneo, Nuovo Cimento 11, 733 (1959).Google Scholar
  46. 46.
    C. Cattaneo, Nuovo Cimento 13, 237 (1959).Google Scholar
  47. 47.
    C. Cattaneo, Rend. Acc. Lincei 27, 54 (1959).Google Scholar
  48. 48.
    M. Born, Ann. Phys. Leipzig 30, 1 (1909).Google Scholar
  49. 49.
    R. H. Boyer, Proc. Roy. Soc. 283, 343 (1965).Google Scholar
  50. 50.
    W. Pauli, Theory of Relativity (Pergamon, New York, 1958).Google Scholar
  51. 51.
    F. R. Tangherlini, Nuovo Cimento Supp. 20, 1 (1961).Google Scholar
  52. 52.
    E. J. Post, Rev. Mod. Phys. 39 (2), 475 (1967).Google Scholar
  53. 53.
    H. D. Wahlquist and F. B. Estabrook, J. Math. Phys. 7, 894 (1966).Google Scholar
  54. 54.
    H. D. Wahlquist, J. Math. Phys. 33, 304 (1992).Google Scholar
  55. 55.
    G. Rizzi and A. Tartaglia, Found. Phys. Lett. 12, 179 (1999).Google Scholar
  56. 56.
    J. Norton, Found. Phys. 19, 1215 (1989).Google Scholar
  57. 57.
    H. Reichenbach, The Philosophy of Space and Time (Dover, New York, 1957).Google Scholar
  58. 58.
    H. Reichenbach, in Albert Einstein: Philosopher-Scientist, P. A. Schilpp, ed. (Open Court, La Salle, IL, 1949).Google Scholar
  59. 59.
    A. Einstein, in Albert Einstein: Philosopher-Scientist, P. A. Schilpp, ed., Ref. 58.Google Scholar
  60. 60.
    H. Reichenbach, Axiomatization of the Theory of Relativity (University of California Press, Berkeley and Los Angeles, 1969).Google Scholar
  61. 61.
    M. A. Abramowicz, B. Carter, and J. P. Lasota, Gen. Rel. Grav. 29, 1173–1183 (1988).Google Scholar
  62. 62.
    V. Cantoni, Nuovo Cimento B 57, 220 (1968).Google Scholar
  63. 63.
    T. A. Weber, Am. J. Phys. 67 (2), 159 (1999).Google Scholar
  64. 64.
    B. Mashhoon, Phys. Lett. A 145, 147 (1990).Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • Guido Rizzi
    • 1
  • Matteo Luca Ruggiero
    • 1
    • 2
  1. 1.Dipartimento di FisicaPolitecnico di TorinoTorinoItaly
  2. 2.INFNTorinoItaly

Personalised recommendations