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Space Geometry in Rotating Reference Frames: A Historical Appraisal

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Relativity in Rotating Frames

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 135))

Abstract

The problem of giving a relativistic description of the geometry of a rotating disk has a history nearly as old as that of the theory of relativity itself. Already in 1909 Ehrenfest formulated his famous paradox in the context of the special theory of relativity. A few years later Einstein made heuristic use of this problem in order to motivate the introduction of non-Euclidean geometry in a relativistic theory of gravity. We shall here follow the conceptual evolution of this topic from Ehrenfest and Einstein to the present time. In particular we emphasize the importance of taking the relativity of simultaneity properly into account in order to obtain a full understanding of the issues connected with Ehrenfest’s paradox.

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References

  1. P. Ehrenfest, Gleichförmige Rotation starrer Körper und Relativitätstheorie, Physik Zeitschrift 10, 918 (1909).

    Google Scholar 

  2. M. Planck, Gleichformige Rotation und Lorentz-Kontraktion, Physik. Zeitschrift 11294 (1910).

    Google Scholar 

  3. A. Einstein, Zum Ehrenfestschen Paradoxon, Physik. Zeitschrift, 12, 509 (1911).

    MATH  Google Scholar 

  4. G Rizzi and M. L. Ruggiero, Space Geometry of Rotating Platforms: An Operational Approach, Found. Phys. 32, 1525 (2002).

    MathSciNet  Google Scholar 

  5. M. Born, Die Theorie des starren Elektrons in der Kinematik des RelativitätPrinzipes, Ann.d.Phys. 30, 1 (1909).

    Google Scholar 

  6. V. Variéak, Zum Ehrenfestschen Paradoxon, Physik. Zeitschrift, 12, 169 (1911).

    Google Scholar 

  7. M. J. Klein, Paul Ehrenfest: volume 1: The making of a Theoretical Physicist, North-Holland Publishing Company (1970), p. 152–154.

    Google Scholar 

  8. A. I. Miller, Albert Einstein’s Special Theory of Relativity, Addison-Wesley Publishing Co. (1981) section 2.4. 8, p. 245.

    Google Scholar 

  9. Th. Kaluza, Zur Relativitätstheorie, Physik.Zeitschrift, 11, 977 (1910).

    Google Scholar 

  10. W. von Ignatowsky, Der starre Körper und das Relativitätsprinzip, Ann.Phys. 33,607 (1910).

    Google Scholar 

  11. P. Ehrenfest, Zu Herrn v. Ignatowskys Behandlung der Bornschen Starrheits-definition, Physik. Zeitschrift 11, 1127 (1910).

    MATH  Google Scholar 

  12. W. von Ignatowsky, Zur Elastizitätstheorie vom Standpunkte des Relativitätsprinzips, Physik. Zeitschrift 12, 164 (1911).

    Google Scholar 

  13. P. Ehrenfest, Zu Herrn v. Ignatowskys Behandlung der Bornschen Starrheitsdefinition. II, Physik. Zeitschrift 12, 412 (1911)

    Google Scholar 

  14. J. Stachel, 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, Birkhäuser (1989) p. 48.

    MATH  Google Scholar 

  15. J. Stachel, Einstein and the Rigidly Rotating Disk, in General Relativity and Gravitation, ed. A. Held, Plenum (1980) p. 1.

    Google Scholar 

  16. A. Einstein, Die Grunlage der allgemeinen Relativitätstheorie, Annalen der Physik 49, 769 (1916).

    ADS  MATH  Google Scholar 

  17. A. Einstein, Über die spezielle und die allgemeine Relativitätstheorie. (Gemeinverständlich). Braunschweig: Friedrich Vieweg & Sohn. (1917).

    Google Scholar 

  18. A. Einstein, The Meaning of Relativity: Four Lectures Delivered at Princeton University, Princeton University Press (1922).

    Google Scholar 

  19. M. Born, Einstein’s Theory of Relativity, first edition in German 1929, reference is to revised edition, Dover (1962), p.319.

    Google Scholar 

  20. A. Einstein and L. Infeld, The Evolution of Physics, Cambridge Univ. Press, p. 239–242 (1938).

    Google Scholar 

  21. A. Metz, Les Problèmes Relatifs a la Rotation Dans la Théorie de la Relativité, Le Journal de Physique et le Radium, 13, 224 (1952).

    Article  Google Scholar 

  22. K. Kraus, Note on Lorentz Contractions and the Space Geometry of the Rotating Disk, Zeitschrift für Naturforschung, 25a, 792 (1970).

    ADS  Google Scholar 

  23. J. Becquerel, Le Principe de Relativité et la Théorie de la Gravitation, Gauthier-Villars, Paris, 1922, page V III.

    MATH  Google Scholar 

  24. S. Shu, Critical Studies on The Theory of Relativity, Princeton, 1945.

    Google Scholar 

  25. W. Glaser, Gielt af der rotierende Scheibe die nichteuklidische Geometrie? Physik. Zeitschrift 35, 867 (1934).

    Google Scholar 

  26. P. Langevin, Remarques au sujet de la Note de Prunier, Comptes rendus 200, 48 (1935).

    Google Scholar 

  27. N. Rosen, Notes on Rotation and Rigid Bodies in Relativity Theory, Phys.Rev. 71, 54 (1947).

    Google Scholar 

  28. F. Goy and F. Selleri, Time on a rotating platform, Found. Phys. Lett. 10, 17 (1997).

    MathSciNet  Google Scholar 

  29. H. Reichenbach, Axiomatization of the Theory of Relativity, University of California Press 1969 p.172–178. Translated from Axiomatik der relativistischen Raum-Zeit-Lehre (1924).

    Google Scholar 

  30. J. Weyssenhoff, Metrisches field und Gravitatationsfield, Bull. Acad. Polonaise Sc. Lett. Serie A, 252 (1937).

    Google Scholar 

  31. G. Sagnac, L’ether lumineux démontré par l’effect du vent relatif d’ether dans un intérferomètre an rotation uniforme, Comptes rendus 157, 708 (1913).

    Google Scholar 

  32. E. J. Post, Sagnac Effect, Rev. Mod. Phys. 39, 475 (1967).

    Google Scholar 

  33. H. Thirring, Begriffsystem und Grundgezetze der Feldphysik, Handbuch der Physik, 4, Kapitel 2 (1929).

    Google Scholar 

  34. H. A. Atwater, Transforming to Rotating coordinates, Nature Phys. Sci. 228, 272 (1970).

    Google Scholar 

  35. M. Suzuki, Transformations to a Rotating Frame, Nature Phys. Sci. 230, 13 (1971).

    Google Scholar 

  36. Letters by G. E. Marsh and T. W. Noonan together with reply from H. A. Atwater under the common heading Transformation to Rotating Coordinates, reply to Atwater, Nature Phys. Sci. 230, 197 (1971).

    Google Scholar 

  37. C. Møller, The Theory of Relativity, Clarendon Press, Oxford 1952, §90.

    Google Scholar 

  38. H. Arzeliès, Relativistic kinematics, Pergamon Press 1966, p. 204–244.

    Google Scholar 

  39. D. G. Ashworth and P. A. Davies, Geodesics in Rotating Systems, Int. J. Th. Phys. 16, 845 (1977).

    Article  Google Scholar 

  40. P. Franklin The Meaning of Rotation in the Special Theory of RelativityProc.Nat.Acad.Sc. 8 265 (1922).

    Google Scholar 

  41. M.G.Trocheris Electrodynamics in a Rotating Frame of ReferencePhil.Mag. ser.7, vol.40 1143 (1949).

    Google Scholar 

  42. H. Takeno, On Relativistic Theory of Rotating Disk, Progr.Th.Phys. 7, 367 (1952).

    Article  ADS  Google Scholar 

  43. L. Herrera, The relativistic transformation to rotating frames, Il Nuovo Cimento 115B, 307 (2000).

    Google Scholar 

  44. B. Chakraborty and S. Sarkar, Physics in Rotating Frames. I. On Uniform Rotation about a Fixd axis in Some Holonomic and Anholonomic Frames, Ann. Phys. 163, 167 (1985).

    Article  ADS  Google Scholar 

  45. T. A. Weber Measurements on a rotating frame in relativity and the Wilson and Wilson experimentAm. J. Phys. 65 946 (1997).

    Google Scholar 

  46. J. F. Corum, Relativistic rotation and the anholonomic object, J. Math. Phys. 18, 770 (1977).

    Article  ADS  Google Scholar 

  47. H. Nikoli´c, Relativistic contraction and related effects in noninertial frames, Phys. Rev. A61, 032109 (2000).

    Google Scholar 

  48. V. Bashkov and M. Malakhaltsev Geometry of rotating disk and the Sagnac effectgr-qc/00 1106 1.

    Google Scholar 

  49. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W. H. Freeman and Company, New York (1973) p. 223, 224.

    Google Scholar 

  50. G. Stead and H. Donaldson The Problem of Uniform Rotation treated on the Principle of RelativityPhil.Mag. 20 92 (1910).

    Google Scholar 

  51. H. E. Ives, Theory of the Double Fizeau Toothed Wheel. J. Opt. Soc. Am. 29, 472 (1939).

    Article  ADS  Google Scholar 

  52. M. Galli, Le deformazioni relativistiche di un cilindro rotante, Lincei–Rend. Sc. Fis. Mat. E nat. 12, Nota I, p.86 and Nota II, p. 569 (1952).

    Google Scholar 

  53. A. S. Eddington, Space, Time and Gravitation, Cambridge University Press (1920) p. 75.

    Google Scholar 

  54. H. A. Lorentz The Michelson-Morley Experiment and the Dimensions of Moving BodiesNature 17 793 (1921).

    Google Scholar 

  55. A. S. Eddington, The Mathematical Theory of Relativity, Cambridge Univ. Press (1923) section 50, p. 112.

    Google Scholar 

  56. L. I. Schiff, A question in general relativity, Proceeding of National Academy of Sciences 25, 391 (1939).

    Article  ADS  Google Scholar 

  57. Ø. Grøn and K. Vøyenli, On the Foundation of the Principle of Relativity, Found. Phys. 29, 1695 (1999).

    Google Scholar 

  58. F. Winterberg, Resolution of the Ehrenfest Paradox in the Dynamic Interpretation of Lorentz InvarianceZ. Naturforsch. 53a 751 (1998).

    Google Scholar 

  59. C. W. Berenda, The Problem of the Rotating Disk, Physical Review 62, 280 (1942).

    Article  ADS  Google Scholar 

  60. E. L. Hill, A Note on the Relativistic Problem of Uniform Rotation, Phys. Rev. 69, 488 (1946).

    Article  MATH  Google Scholar 

  61. A. Prechtl, Remarks on Steady Rotation About a Fixed Axis in Relativity, Acta Mechanica 29, 283 (1978).

    Article  MATH  Google Scholar 

  62. N. Rosen, Note on the Problem of Uniform Rotation, Phys. Rev. 70, 93 (1946).

    Google Scholar 

  63. G. L. Clark, The Mechanics of Continuous Matter in the Relativity Theory, Proc. Roy. Soc. Edinburgh 62A, 434 (1947–8).

    Google Scholar 

  64. G.L.Clark, The Problem of a Rotating Incompressible Disk, Proc. Cambr. Phil. Soc. 45, 405 (1949).

    Google Scholar 

  65. G. Cavalleri, Solution of Ehrenfest’s Paradox for a Relativistic Rotating Disk, Il Nuovo Cimento, 53B, 415 (1968).

    Article  Google Scholar 

  66. A. Brotas, Sur le problème du disque tournant, C.R. Acad. Sc. Paris, 267, 57A (1968).

    Google Scholar 

  67. W. H. McCrea, Rotating Relativistic Ring, Nature, 234, 399 (1971).

    Article  ADS  Google Scholar 

  68. R. G. Newburgh, A Practical Proposal for the Experimental Study of Born Rigid Motions, Il Nuovo Cimento 23B, 365 (1974).

    Article  Google Scholar 

  69. P. A. Davies and R. C. Jennison, Experiments involving mirror transponders in rotating frames, J. Phys. A. Math. Gen. 8, 1390 (1975).

    Article  ADS  Google Scholar 

  70. D. G. Ashworth, Surveying in rotating systems, J. Phys. A. Math. Gen. 9, 35 (1976).

    Google Scholar 

  71. ] P. A. Davies, Measurements in rotating systems, J. Phys. A. Math. Gen. 9, 951 (1976).

    Article  ADS  Google Scholar 

  72. R. C. Jennison and D. G. Ashworth, Surveying in rotating systems II. Triangulation Radius. J. Phys. A. Math. Gen. 9, 1257 (1976).

    Article  MathSciNet  ADS  Google Scholar 

  73. R. C. Jennison, Relativity on a turntable, New Scientist 73, 283 (1977).

    Google Scholar 

  74. D. G. Ashworth and P. A. Davies, Transformations between inertial and rotating frames of reference, J. Phys. A. Math. Gen. 12, 1425 (1979).

    Article  MathSciNet  ADS  Google Scholar 

  75. P. F. Browne, Relativity of Rotation, J. Phys. A. Math. Gen. 10, 727 (1977).

    Article  ADS  Google Scholar 

  76. K. McFarlane and N. C. McGill, Light ray and particle paths on a rotating disk, J. Phys. A. Math. Gen. 11, 2191 (1978).

    Article  ADS  Google Scholar 

  77. D. G. Ashworth, P. A. Davies and R. C. Jennison, “Light ray and particle paths on a rotating disk”: a reply to criticism therein, J. Phys. A. Math. Gen. 11, L259 (1978).

    Article  ADS  Google Scholar 

  78. K. McFarlane and N. C. McGill, “Light ray and particle paths on a rotating disk”: a reply to comments by Ashworth, Davies and Jennison, J. Phys. A. Math. Gen. 12, L71 (1979).

    Article  ADS  Google Scholar 

  79. K. McFarlane and N. C. McGill, Comment on “Geodesics in Rotating Systems”, Int. J. Th. Phys. 18, 151 (1979).

    Article  Google Scholar 

  80. V. Cantoni, What is Wrong with Relativistic Kinematics, Il Nuovo Cimento, 57B, 220 (1968).

    Article  Google Scholar 

  81. G. Rizzi and A. Tartaglia, Speed of Light on Rotating Platforms, Found. Phys. 28, 1663 (1998).

    MathSciNet  Google Scholar 

  82. W. A. Rodrigues, The Standard of Length in the Theory of Relativity and Ehrenfest Paradox, Il Nuovo Cimento, 74B, 199 (1983).

    Article  Google Scholar 

  83. T. E. Phipps, Jr. Do Metric Standards Contract? Found. Phys. 10, 289 (1980).

    MathSciNet  Google Scholar 

  84. V. Cantoni, Comments on Do Metric Standards Contract?, Found. Phys. 10, 809 (1980).

    Google Scholar 

  85. T. E. Phipps, Jr. Do Metric Standards Contract? –A Reply to Cantoni, Found. Phys. 10, 811 (1980).

    MathSciNet  Google Scholar 

  86. Ø. Grøn, Special-Relativistic Resolution of Ehrenfest’s Paradox: Comments on Some Recent Statements by T. E. Phipps, Jr. Found. Phys. 11, 623 (1981).

    Article  ADS  Google Scholar 

  87. Ø. Grøn, Covariant formulation of Hooke’s law, Am. J. Phys. 49, 28 (1981).

    Article  ADS  Google Scholar 

  88. T. A. Weber, A note on rotating coordinates in relativity, Am. J. Phys. 65, 486 (1997).

    Article  ADS  MATH  Google Scholar 

  89. D. Dieks, Time in special relativity and its philosophical significance, Eur. J. Phys. 12, 253 (1991).

    Article  Google Scholar 

  90. E. M. Kelly, Resolution of the Ehrenfest Paradox by New Contraction (Expansion) Criteria, Z. Naturforsch. 43a, 865 (1988).

    Google Scholar 

  91. R. D. Klauber, New Perspectives on the Relativistically Rotating Disk and Non-Time-Orthogonal Reference Frames, Found. Phys. Lett. 11, 405 (1998).

    Google Scholar 

  92. R. D. Klauber, Comment regarding recent articles on relativistically rotating frames, Am. J. Phys. 67, 158 (1999).

    Article  ADS  Google Scholar 

  93. T. A. Weber, Response to “Comment regarding recent articles on relativistic rotating frames”, Am. J. Phys. 67, 159 (1999).

    Article  ADS  Google Scholar 

  94. A. Tartaglia, Lengths on Rotating Platforms, Found. Phys. Lett. 12, 17 (1999).

    Google Scholar 

  95. B. Kur¸sunoglu, Spacetime on The Rotating Disk, Proc. Cambr. Phil. Soc. 47, 177 (1951).

    Article  ADS  Google Scholar 

  96. Ø. Grøn, The Metric in a Static Cylindrical Elastic Medium and in an Empty Rotating Frame as Solutions of Einstein’s Field Equations, Found. Phys. 12, 509 (1982).

    Google Scholar 

  97. W. Wilson, Space-Time Manifolds and corresponding Gravitational Fields, Phil. Mag. 40, 703 (1920).

    MATH  Google Scholar 

  98. R. J. Adler, M. J. Bazin, and M. Schiffer, General Relativity, 2nd. Ed. (McGraw Hill, 1975) §4.2.

    Google Scholar 

  99. H. Arzeliès, Relativitè gènèralisèe, Fasc. I (Gauthier-Villars 1961), §198.

    Google Scholar 

  100. N. Sama, On the Ehrenfest Paradox, Am. J. Phys. 40, 415 (1972).

    Article  ADS  Google Scholar 

  101. Ø. Grøn, Relativistic description of a rotating disk, Am. J. Phys. 43, 869 (1975).

    Article  ADS  Google Scholar 

  102. P. D. Noerdlinger, Cylindrical cardboard modell for a rotating system in special relativity, Am. J. Phys. 47, 218 (1979).

    Article  ADS  Google Scholar 

  103. W. Pauli, Theory of Relativity, Pergamon Press (1958), p. 131. Translated from the article Relativitätstheorie in Encyklopädie der mathematischen Wissenschaften, 19 (1921).

    Google Scholar 

  104. M. H. Mac Gregor, Do Dewan-Beran Relativistic Stresses Actually Exist?, Lett. Nuovo Cimento 30, 417 (1981).

    Google Scholar 

  105. Ø. Grøn, Energy considerations in connection with a relativistic rotating ring, Am. J. Phys. 50, 1144 (1982).

    Article  ADS  Google Scholar 

  106. M. Strauss, Rotating Frames in Special Relativity, Int. J. Th. Phys. 11, 107 (1974).

    Article  Google Scholar 

  107. Ø. Grøn, Rotating Frames in Special Relativity Analyzed in Light of a Recent Article by M. Strauss, Int. J. Th. Phys. 16, 603 (1977).

    Article  Google Scholar 

  108. Ø. Grøn, Relativistic Description of a Rotating Disk with Angular Acceleration, Found. Phys. 9, 353 (1979).

    Google Scholar 

  109. K. Brown, A Rotating Disk in Translation, Math. Pages. Internet adress

    Google Scholar 

  110. ] K. MacFarlane, Appearance of a Relativistically Rotating Disk, Int. J. Th. Phys. 20, 397 (1981).

    Article  Google Scholar 

  111. Ø. Grøn, “Optical Appearance” of a Rolling Ring, Int. J. Th. Phys. 22, 821 (1983).

    Article  ADS  Google Scholar 

  112. K. Vøyenli, Alternative derivation of the circumference of a relativistic, rotating disk, Am. J. Phys. 9, 875 (1977).

    Google Scholar 

  113. D. H. Weinstein, Ehrenfest’s Paradox, Nature Phys. Sci. 232, 548 (1971).

    Google Scholar 

  114. G. Cavalleri, On Some Recent Papers Regarding the Ehrenfest’s Paradox, Lett. Nuovo Cimento 3, 608 (1972).

    Google Scholar 

  115. R. G. Newburgh, Thomas Precession and Extended Structures, Lett. Nuovo Cimento 5, 387 (1972).

    Google Scholar 

  116. D. P. Whitmire, Thomas Precession and the Relativistic Disk, Nature Phys. Sci. 235, 175 (1972).

    Google Scholar 

  117. A. Grünbaum and A. I. Janis, The Geometry of the Rotating Disk in the Special Theory of Relativity, Synthese 34, 281 (1977).

    Article  MathSciNet  Google Scholar 

  118. A. Grünbaum and A. I. Janis, The Rotating Disk: Reply to Grøn, Found. Phys. 10, 495 (1980).

    Google Scholar 

  119. Ø. Grøn, The Rotating Disk: Reply to Grünbaum and Janis, Found. Phys. 10, 499 (1980).

    Google Scholar 

  120. G. Ziino, An Improved Approach to Relativistic Rotational Kinematics, Helv. Phys. Acta 69, 1 (1996).

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of engineering, Oslo College, Cort Adelers gt. 30, 0254, Oslo, Norway

    Øvind Grøn

  2. Institute of Physics, University of Oslo, P. O. Box 1048, 0316, Blindern, Oslo, Norway

    Øvind Grøn

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  1. Politecnico di Torino and Istituto Nazionale di Fisica Nucleare, Torino, Italy

    Guido Rizzi  & Matteo Luca Ruggiero  & 

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Grøn, Ø. (2004). Space Geometry in Rotating Reference Frames: A Historical Appraisal. In: Rizzi, G., Ruggiero, M.L. (eds) Relativity in Rotating Frames. Fundamental Theories of Physics, vol 135. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0528-8_17

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  • DOI: https://doi.org/10.1007/978-94-017-0528-8_17

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