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, Volume 44, Issue 1, pp 47–75 | Cite as

E. Cartan’s attempt at bridge-building between Einstein and the Cosserats – or how translational curvature became to be known as torsion

  • Erhard Scholz
Article

Abstract

Élie Cartan’s “généralisation de la notion de courbure” (1922) arose from a creative evaluation of the geometrical structures underlying both, Einstein’s theory of gravity and the Cosserat brothers generalized theory of elasticity. In both theories groups operating in the infinitesimal played a crucial role. To judge from his publications in 1922–24, Cartan developed his concept of generalized spaces with the dual context of general relativity and non-standard elasticity in mind. In this context it seemed natural to express the translational curvature of his new spaces by a rotational quantity (via a kind of Grassmann dualization). So Cartan called his translational curvature “torsion” and coupled it to a hypothetical rotational momentum of matter several years before spin was encountered in quantum mechanics.

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References

  1. 1.
    Badur, J., Stumpf, H. 1989. “On the influence of E. and F. Cosserat on modern continuum mechanics and field theory.” University of Bochum, Institute for Mechanics, Communication number 72. Internet source https://doi.org/www1.am.bi.ruhr-uni-bochum.de/ifm/IFM-072.pdf.
  2. 2.
    Belhoste, B. 1991. Augustin-Louis Cauchy. A Biography. Berlin, etc.: Springer. Google Scholar
  3. 3.
    Bernard, J. 2018. “Riemann’s and Helmholtz-Lie’s problems of space from Weyl’s relativistic perspective.” Studies in History and Philosophy of Modern Physics 61:41–56. ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blagojević, M., Hehl, F. 2013. Gauge Theories of Gravitation. A Reader with Commentaries. London: Imperial College Press. Google Scholar
  5. 5.
    Brocato, M., Chatzis, K. 2009. Les frères Cosserat. Brève introduction à leur vie et à leurs travaux en mécanique. In Théorie des corps déformables, eds. F. Cosserat, E. Cosserat. Paris: Hermann , pp. iii–xlv. Reprint of Cosserat (1909c). Google Scholar
  6. 6.
    Capecchi, D., Ruta, G., Trovaleusci P. 2010. “From classical to Voigt’s molecular models inelasticity.” Archive for History of Exact Sciences 64(5):525–559. MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cartan, É. 1922a. “Sur les équations de structure des espaces généralisés et l’expression analytique du tenseur d’Einstein.” Comptes Rendus Académie des Sciences 174:1104ff. In (Cartan 1952ff., III, 625–628) (submitted 24.04.1922). Google Scholar
  8. 8.
    Cartan, É. 1922b. “Sur les espaces conformes généralisés et l’Univers optique.” Comptes Rendus Académie des Sciences 174:857ff. In (Cartan 1952ff., III, 622–624) (submitted 27.03.1922). zbMATHGoogle Scholar
  9. 9.
    Cartan, É. 1922c. “Sur les espaces généralisés et la théorie de la relativité.” Comptes Rendus Académie des Sciences 174:734–736. In (Cartan 1952ff., III, 619–621) (submitted 13.03.1922). zbMATHGoogle Scholar
  10. 10.
    Cartan, É. 1922d. “Sur les équations de la gravitation d’Einstein.” Journal des Mathématiques pures et appliquées 1:141–203. In (Cartan 1952ff., III, 549–612). zbMATHGoogle Scholar
  11. 11.
    Cartan, É. 1922e. “Sur un théorème fondamental de M.H. Weyl dans la théorie de l’espace métrique.” Comptes Rendus Académie des Sciences 175:82–85. In (Cartan 1952ff., III, 629–632) (submitted 10.07.1922). zbMATHGoogle Scholar
  12. 12.
    Cartan, É. 1922f. “Sur une définition géométrique du tenseur d’énergie d’Einstein.” Comptes Rendus Académie des Sciences 174:437–439. In (Cartan 1952ff., III, 613–615) (submitted 13.02.1922). zbMATHGoogle Scholar
  13. 13.
    Cartan, É. 1922g. “Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion.” Comptes Rendus Académie des Sciences 174:593–595. In (Cartan 1952ff., III, 616–618) (submitted 27.02.1922). zbMATHGoogle Scholar
  14. 14.
    Cartan, É. 1923. “Sur un théorème fondamental de M.H. Weyl.” Journal des Mathématiques pures et appliquées 2:167–192. In (Cartan 1952ff., III, 633–658). zbMATHGoogle Scholar
  15. 15.
    Cartan, É. 1923/1924b. “Sur les variétés à connexion affine et la théorie de la relativité généralisée (prémière partie).” Annales de l’Ecole Normale 40, 41:(vol 40) 325–412, (vol. 41) 1–25. In (Cartan 1952ff., III 1, 659–747, 799–824). Reprint in Cartan (1955), English in Cartan (1986). Google Scholar
  16. 16.
    Cartan, É. 1924a. “Les récentes généralisations de la notion d’espace.” Bulletin Sciences mathématiques 48:294–320. In (Cartan 1952ff., III, 863–890). zbMATHGoogle Scholar
  17. 17.
    Cartan, É. 1924c. “Sur les variétés à connexion projective.” Bulletin Societé Mathématique de France 52:205–241. In (Cartan 1952ff., III, 825–862). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cartan, É. 1925. “Sur les variétés à connexion affine et la théorie de la relativité généralisée (deuxième partie).” Annales de l’Ecole Normale 42:17–88. In (Cartan 1952ff., III 2 921–992). Reprint in Cartan (1955). zbMATHGoogle Scholar
  19. 19.
    Cartan, É. 1929. “Notice historique sur la notion de parallèlisme absolu.” Mathematische Annalen 109:698–706. zbMATHGoogle Scholar
  20. 20.
    Cartan, É. 1952ff. Oeuvres Complètes. Paris: Gauthier-Villars. Google Scholar
  21. 21.
    Cartan, É. 1955. Sur les variétés à connexion affine et la théorie de la relativité généralisée. Paris: Gauthier Villars. Contient les articles (Cartan 1923/1924b, 1925). Google Scholar
  22. 22.
    Cartan, É. 1986. On the Manifolds with an Affine Connection and the Theory of General Relativity. Translated by A. Magnon and A. Ashketar; foreword by A. Trautman. Napoli: Bibliopolis . Translation of Cartan (1955) and (Cartan 1922b). Google Scholar
  23. 23.
    Cartan, É., Einstein, A. 1979. Elie Cartan–Albert Einstein. Letters on Absolute Parallelism 1929–1932. Original text, English translation by Jules Leroy and Jim Ritter, edited by Robert Debever. Princeton: University Press. Google Scholar
  24. 24.
    Chorlay, R. 2010. “Passer aux global: Le cas d’Élie Cartan, 1922–1930.” Revue d’Histoire des Mathématiques 15:231–316. MathSciNetzbMATHGoogle Scholar
  25. 25.
    Cogliati, A., Mastrolia, P. 2018. “Cartan, Schouten and the search for connection.” Historia Mathematica 45(1):39–74. MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Combebiac, G. 1902. “Sur les équations générales de l’élasticité.” Bulletin de la Societé mathématique de France 30:108–110, 242–247. MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Cosserat, E.F. 1896. “Sur la théorie de l’élasticité. Premier Mémoire.” Annales de la Faculté des sciences de Toulouse 1 10 (I.1–I.116). Google Scholar
  28. 28.
    Cosserat, E.F. 1909a. Note sur la théorie de l’action euclidienne. In Paul Appell, Traité de mécanique rationelle, t. III (2e édition). Paris: Gauthier-Villars, pp. 557–629. Reprint 1991. Google Scholar
  29. 29.
    Cosserat, E.F. 1909b. Note sur la théorie des corps déformables. In O.D. Chwolson, Traité de physique. Ouvrage traduit sur les éditions russe et allemand par E. Davaux. Paris: Hermann, pp. 953–1173. Google Scholar
  30. 30.
    Cosserat, E.F. 1909c. Théorie des corps déformables. Paris: Hermann. English by D.H. Delphenich 2007, Internet source https://doi.org/www.uni-due.de/~hm0014/Cosserat_files/Cosserat09_eng.pdf.
  31. 31.
    Cosserat, E.F. 1915. Principes de la mécanique rationelle (d’après l’article allemandde A. Voss). In Encyclopédie des sciences mathématiques pure et appliquées, t.IV, vol. 1. Paris/Leipzig: Gauthier-Villear / Teubner, pp. 1–187. Google Scholar
  32. 32.
    Dahan-Dalmedico, A. 1992. Mathématisations. Augustin-Louis Cauchy et l’École Française. Argenteuil: Edition du Choix. Google Scholar
  33. 33.
    Darrigol, O. 2005. Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl. Oxford: University Press. Google Scholar
  34. 34.
    Darrigol, O. 2012. A History of Optics from Greek Antiquity to the Ninteteenth Century. Oxford: University Press. Google Scholar
  35. 35.
    Dirac, P.A.M. 1928. “The quantum theory of the electron I, II.” Proceedings Royal Society London A 117, 610–624 A 118:351–361. In (Dirac 1995, 303–333). CrossRefzbMATHGoogle Scholar
  36. 36.
    Dirac, P.A.M. 1995. The Collected Works of P.A.M. Dirac. Ed. R.H. Dalitz. Cambridge: University Press. Google Scholar
  37. 37.
    Duhem, P. 1906. Recherches sur l’élasticité. Paris: Gauthier-Villars. Google Scholar
  38. 38.
    Fox, R. 1974. “The rise and fall of Laplacian physics.” Historical Studies in the Physical Sciences 4:89–136. CrossRefGoogle Scholar
  39. 39.
    Gasperini, M. 2017. Theory of Gravitational Interactions, 2nd edn. Berlin: Springer. Google Scholar
  40. 40.
    Goenner, H. 2004. “On the history of unified field theories.” Living Reviews in Relativity 2004-2. https://doi.org/relativity.livingreviews.org/Articles/lrr-2004-2.
  41. 41.
    Graßmann, H.G. 1862. Die Ausdehnungslehre. Vollständig und in strenger Form bearbeitet. Berlin: Ensslin. In (Graßmann 1894–1911, Bd. 1.2, 1–383). Google Scholar
  42. 42.
    Graßmann, H.G. 1894–1911. Gesammelte mathematische und physikalische Werke. Hrsg. Friedrich Engel. 3 Bände in 6 Teilbänden. Leipzig: Teubner. Reprint New York: Johnson 1972. Google Scholar
  43. 43.
    Grattan-Guinness, I. 1990. Convolutions in French Mathematics, 1800–1840. From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics, 3 vols. Vol. 2, 3, 4 of Science Networks Basel: Birkhäuser. Google Scholar
  44. 44.
    Hehl, F. 1966. “Der Spindrehimpuls in der allgemeinen Relativitätstheorie.” Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft 18:98–130. Google Scholar
  45. 45.
    Hehl, F. 2017. Gauge theory of gravity and spacetime. In Towards a Theory of Spacetime Theories, ed. D. Lehmkuhl et al. Springer, pp. 145–170. Google Scholar
  46. 46.
    Hehl, F., McCrea, D. 1986. “Bianchi identities and the automatic conservation of energy-momentum and angular momentum in general-relativistic field theories.” Foundations of Physics 86(3):267–293. ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    Hehl, F., Nitsch, J., von der Heyde P. 1980. Gravitation and the Poincaré gauge field theory with quadratic Lagrangian. In Eneral Relativity and Gravitation – One Hundred Years after the Birth of Albert Einstein, ed. A. Held. Vol. 1, Plenum , pp. 329–355. Reprint in (Blagojević 2013, 183–209). Google Scholar
  48. 48.
    Hehl, F., Obukhov, Y. 2007. “Elie Cartan’s torsion in geometry and in field theory, an essay.” Annales Fondation Louis de Broglie. https://doi.org/arXiv:0711.1535.
  49. 49.
    Kibble, T. 1961. “Lorentz invariance and the gravitational field.” Journal for Mathematical Physics 2:212–221. In (Blagojević 2013, Chap. 4). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Kondo, K. 1952. On the geometrical and physical foundations of the theory of yielding. In Proceedings 2nd Japan National Congress on Applied Mechanics. Tokyo, pp. 41–47. Google Scholar
  51. 51.
    Kröner, E. 1963a. “On the physical reality of torque stresses in continuum mechanics.” International Journal of Engineering Sciences 1:261–278. CrossRefGoogle Scholar
  52. 52.
    Kröner, E. 1963b. “Zum statischen Grundgesetz der Versetzungstheorie.” Annalen der Physik 466(1–6):13–21. ADSCrossRefzbMATHGoogle Scholar
  53. 53.
    Larmor, J. 1891. “On the propagation of a disturbance in a gyrostatically loaded medium.” Proceedings London Mathematical Society 23(1):127–135. MathSciNetCrossRefGoogle Scholar
  54. 54.
    Lazar, M., Hehl F.W. 2010. “Cartan’s spiral staircase in physics and, in particular, in the gauge theory of dislocations.” Foundations of Physics 40:1298–1325. ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Lehmkuhl, D. 2014. “Why Einstein did not believe that general relativity geometrizes gravity.” Studies in History and Philosophy of Modern Physics 46B:316–326. https://doi.org/philsci-archive.pitt.edu/9825/. ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Levy, J.R. 1971. Eugéne and François Cosserat. In Dictionary of Scientific Biography, ed. C.G. Gillispie. Vol. 3. New York, p. 428f. Google Scholar
  57. 57.
    Love, A.E.H. 1892/1906. Treatise on the Mathematical Theory of Elasticity. Cambridge: University Press. 1st edition 1896, 2nd edition 1906. Google Scholar
  58. 58.
    Mauguin, G.A. 2014. Continuum Mechanics Through the Eighteenth and Nineteenth Centuries. Historical Perspectives from John Bernoulli 1727 to Ernst Hellinger (1914). Berlin, etc.: Springer. Google Scholar
  59. 59.
    Mauguin, G.A., Metrikine, A.V. (eds.). 2010. Mechanics of Generalized Continua. One Hundred Years After the Cosserats. Berlin, etc.: Springer. Google Scholar
  60. 60.
    Nabonnand, P. 2009. “La notion d’holonomie chez Élie Cartan.” Revue d’Histoire des Sciences 62:221–245. MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Nabonnand, P. 2016. L’apparition de la notion d’espace généralisé dans les travaux d’Élie Cartan en 1922. In Eléments d’une biographie de l’Espace géométrique, ed. L. Bioesmat-Martagon. Nancy: Editions Universitaires de Lorraine, pp. 313–336. Google Scholar
  62. 62.
    Pauli, W. 1927. “Zur Quantenmechanik des magnetischen Elektrons.” Zeitschrift für Physik 43:601–623. In (Pauli 1964, vol. 1, 306–328). ADSCrossRefzbMATHGoogle Scholar
  63. 63.
    Pauli, W. 1964. Collected Scientific Papers Eds. R. Kronig, V.F. Weisskopf. New York, etc.: Wiley. Google Scholar
  64. 64.
    Pommaret, J.-F. 1997. “François Cosserat et le secret de la théorie mathématique de l’élasticité.” Annales des Ponts et Chaussées 82:59–66. Google Scholar
  65. 65.
    Sauer, T. 2006. “Field equations in teleparallel spacetime: Einstein’s Fernparallelismus approach towards unified field theory.” Historia Mathematica 33:399–439. MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Schaefer, H. 1967. “Das Cosserat-Kontinuum.” Zeitschrift für angewandte Mathematik und Mechanik 47(8):485–498. ADSCrossRefzbMATHGoogle Scholar
  67. 67.
    Scholz, E. 1984. “Hermann Graßmanns Analysis in Vektorräumen.” Mathematische Semesterberichte 31:177–194. MathSciNetzbMATHGoogle Scholar
  68. 68.
    Scholz, E. 2016. The problem of space in the light of relativity: the views of H. Weyl and E. Cartan. In Eléments d’une biographie de l’Espace géométrique, ed. L. Bioesmat-Martagon. Nancy: Edition Universitaire de Lorraine, pp. 255–312. https://doi.org/arXiv:1310.7334.
  69. 69.
    Sciama, D.W. 1962. On the analogy between charge and spin in general relativity. In Recent Developments in General Relativity Festschrift for L. Infeld. Oxford and Warsaw: Pergamon and PWN, pp. 415–439. In (Blagojević 2013, Chap. 4). Google Scholar
  70. 70.
    Sharpe, R.W. 1997. Differential Geometry. Cartan’s Generalization of Klein’s Erlangen program. Berlin, etc.: Springer. Google Scholar
  71. 71.
    Sternberg, S. 2012. Curvature in Mathematics and Physics. New York: Dover. Google Scholar
  72. 72.
    Timoshenko, S.P. 1953. History of Strength of Materials. New York: McGraw-Hill. Google Scholar
  73. 73.
    Trautman, A. 1973. “On the structure of the Einstein-Cartan equations.” Symposia mathematica 12:139–162. Relativitá convegno del Febbraio del 1972. MathSciNetzbMATHGoogle Scholar
  74. 74.
    Trautman, A. 2006. Einstein-Cartan theory. In Encyclopedia of Mathematical Physics, ed. J.-P. Françoise; G.L. Naber; S.T. Tsou. Vol. 2. Oxford: Elsevier, pp. 189–195. In (Blagojević 2013, Chap. 4). Google Scholar
  75. 75.
    Voigt, W. 1887. “Theoretische Studien über die Elasticitätsverhältnisse der Krystalle.” Abhandlungen der Gesellschaft der Wissenschaften zu Göttingen – Math. Classe 34:3–52. Google Scholar
  76. 76.
    Voigt, W. 1900. L’État actuel de nos connaissances sur l’élasticité des cristaux. In Rapports présentés au Congrès International de Physique. Gauthiers-Villars, pp. 277–347. Google Scholar
  77. 77.
    Weitzenböck, R. 1923. Invariantentheorie. Groningen: Noordhoff. Google Scholar
  78. 78.
    Weyl, H. 1921. Raum, - Zeit - Materie. Vorlesungen über allgemeine Relativitätstheorie. Vierte, erweiterte Auflage. Berlin, etc.: Springer. Google Scholar
  79. 79.
    Weyl, H. 1922. Temps, espace, matière. Leçon sur la théorie de la relativité générale. Traduites sur la quatrème édition allemande G. Juret, R. Leroy. Paris: Blanchard. Google Scholar

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© EDP Sciences, Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Erhard Scholz
    • 1
  1. 1.Faculty of Mathematics/Natural Sciences, Interdisciplinary Centre for History and Philosophy of Science, University of WuppertalWuppertalGermany

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