Skip to main content
SpringerLink
Log in

On Metric and Matter in Unconnected, Connected, and Metrically Connected Manifolds

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

From Einstein's point of view, his General Relativity Theory had strengths as well as failings. For him, its shortcoming mainly was that it did not unify gravitation and electromagnetism and did not provide solutions to field equations which can be interpreted as particle models with discrete mass and charge spectra, As a consequence, General Relativity did (and does) not solve the quantum problem, either. Einstein tried to get rid of the shortcomings without losing the achievements of General Relativity Theory. Stimulated by papers of Weyl (Sitzungsber. Preuss. Akad. Wiss (1918) 465) and Eddington (Proc. R. Soc. Hond. 99 (1921) 194), from 1923 onward, he believed that, to reach this goal, one has to transit to space–times which possess more comprehensive geometrical structures than the Riemann space–time. This was the beginning of a decade's lasting search for a unitary field theory. We describe this exciting part of the history of physics, discuss achievements and failures of this development, and ask how these early attempts of a unified theory strike us today. Taking into account the fact that the Equivalence Principle only speaks for a geometrization of gravitation, we consider an alternative way to give those non-Riemannian structures which were introduced by the unitary field approach a physical meaning, namely the meaning of a generalized gravitational field. This is interesting since there are arguments in favor of such a generalization of General Relativity Theory, e.g., the problems the latter theory meets with if one tries to quantize it and to unify gravitation with other interactions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. A. Einstein, unpublished paper; cited by A. Pais (1921), in "Subtle is the Lord...". The Science and the Life of Albert Einstein, (Oxford University Press, Oxford, 1982).

    Google Scholar 

  2. A. Einstein, The Meaning of Relativity, Appendix II:Relativistic theory of the non-symmetric field (University Press Princeton, Princeton, 1955).

    Google Scholar 

  3. A. Einstein, "Die Feldgleichungen der Gravitation," Sitzungsber. Preuss. Akad. Wiss., pp. 844-847 (1915).

  4. D. Hilbert, "Die Grundlagen der Physik," Nachr. K¨onigl. Ges. Wiss. G¨ottingen, Math.-Phys. Kl., pp. 395-407 (1915).

  5. A. Einstein, "Hamiltonsches Prinzip und allgemeine Relativit¨atstheorie," Sitzungsber. Preuss. Akad. Wiss., pp. 1111-1116 (1916).

  6. P. G. Bergmann, "Unitary eld theory: yesterday, today, tomorrow," in Einstein-Centenarium, H.-J. Treder, ed. (Akademie-Verlag, Berlin, 1979).

    Google Scholar 

  7. A. Einstein and P. G. Bergmann, Ann. of Math. 39, 683 (1938).

    Google Scholar 

  8. A. Einstein, V. Bargmann, and P. G. Bergmann, "Five-Dimensional Representation of Gravitation and Electricity," in Theodore von K´arm´an Anniversary Volume (Pasadena University Press, 1941), p. 212.

  9. A. Einstein and J. Grommer, "Allgemeine Relativit¨atstheorie und Bewegungsgesetz," Sitzungsber. Preuss. Akad. Wiss., pp. 2-13 (1927).

  10. P. Havas, "The early history of the 'problem of motion' in General Relativity," in Einstein and the History of General Relativity, D. Howard and J. Stachel, eds. (Birkh¨auser, Boston, 1989).

    Google Scholar 

  11. A. Einstein and M. Groß mann, Entwurf einer verallgemeinerten Relativit¨atstheorie und einer Theorie der Gravitation (Teubner, Leipzig, 1913);cf. Collected papers of Albert Einstein, Vol. 4, No. 13.

    Google Scholar 

  12. H.-H. v. Borzeszkowski and H.-J. Treder, Found. Phys. 26, 829 (1996).

    Google Scholar 

  13. G. Mie, Ann. Phys. (Leipzig) 37, 511-134 (1912).

    Google Scholar 

  14. G. Mie, Ann. Phys. (Leipzig) 39, 1. 1-40 (1912).

    Google Scholar 

  15. G. Mie, Ann. Phys. (Leipzig) 40, 1-65 (1913).

    Google Scholar 

  16. A. Einstein, "Kosmologische Betrachtungenzur allgemeinen Relativit¨atstheorie," Sitzungsber. Preuss. Akad. Wiss., pp. 142-152 (1917).

  17. A. Einstein, "Spielen Gravitationsfelder im Aufbau der materiellen Elementarteilchen eine wesentliche Rolle?," Sitzungsber. Preuss. Akad. Wiss., pp. 349-356 (1919).

  18. A. Einstein, "Bietet die Feldtheorie M¨oglichkeiten f¨ur die L¨osung des Quantenproblems?," Sitzungsber. Preuss. Akad. Wiss., pp. 359-364 (1923).

  19. H. Weyl, "Gravitation und Elektrizit¨at," Sitzungsber. Preuss. Akad. Wiss., pp. 465-480 (1918).

  20. A. S. Eddington, "A generalization of Weyl's theory of the electromagnetic and gravitation fields," Proc. R. Soc. (London) 99, 194 (1921).

    Google Scholar 

  21. J. A. Schouten, Ricci-Calculus, 2nd edn. (Springer, Berlin, 1954).

    Google Scholar 

  22. A. S. Eddington, Space, Time and Gravitation (University Press, Cambridge, 1920).

    Google Scholar 

  23. A. S. Eddington, The Mathematical Theory of Relativity (Cambridge University Press, Cambridge, 1922).

    Google Scholar 

  24. J. A. Schouten, ¨Uber die Entwicklung der Begriffe des Raumes und der Zeit und ihre Beziehungen zum Relativit¨atsprinzip (Teubner, Leipzig, 1924).

    Google Scholar 

  25. H. Weyl, "¨Uber die physikalischen Grundlagen der erweiterten Gravitationstheorie," Phys. Z. 22, 2 (1921).

    Google Scholar 

  26. H. Weyl, Raum, Zeit, Materie (Springer, Berlin; (1st edn. 1918, 5th edn. 1923).

    Google Scholar 

  27. A. Einstein, "Zur af nen Feldtheorie, Sitzungsber. Preuss. Akad. Wiss., pp. 137-140 (1923).

  28. A. Einstein, "Einheitliche Feldtheorie von Gravitation und Elektrizit¨at," Sitzungsber. Preuss. Akad. Wiss., pp. 414-419 (1925).

  29. A. Einstein, "Eddingtons Theorie und Hamiltonsches Prinzip," Appendix in the German edition of Ref. 23 (Springer, Berlin, 1925).

    Google Scholar 

  30. A. Einstein, A. (1925). "Elektron und allgemeine Relativit¨atstheorie," Physica 5, 330 (1925).

    Google Scholar 

  31. H.-J. Treder, Astr. Nachr. 296, 149 (1975).

    Google Scholar 

  32. H.-J. Treder, "Die Geometrisierung der Physik und die Physikalisierung der Geometrie," Sitzungsber. Akad. Wiss. der DDR, 14 N (1975).

  33. A. Einstein, A. "Riemann-Geometrie mit Aufrechterhaltung des Begriffes des Fernpar-allelismus," Sitzungsber. Preuss. Akad. Wiss., pp. 217-221 (1928).

  34. A. Einstein and E. G. Straus, Ann. Math. Stud. 47, 731 (1946).

    Google Scholar 

  35. M. A. Tonnelat, La Th ´eorie du Champ uni ´e d'Einstein, (Gauthier-Villars, Paris, 1955).

    Google Scholar 

  36. V. Hlav´aty, Geometry of Einstein's Uni ed Field theory (Noordhoff, Groningen, 1957).

    Google Scholar 

  37. E. Schr¨odinger, "The nal af ne eld laws III," Proc. R. Irish Acad. 52A, 1 (1948).

    Google Scholar 

  38. E. Schr¨odinger, Space-Time Structure (Cambridge University Press, Cambridge, 1950).

    Google Scholar 

  39. A. Lichnerowicz, Th´eories relativistes de la gravitation et de l'electromagnitisme (Masson, Paris, 1955).

    Google Scholar 

  40. A. Einstein and B. Kaufman, Ann. Math. 59, 230 (1954).

    Google Scholar 

  41. B. Kaufman, "Mathematical structure of the non-symmetric field theory," in F¨unfzig Jahre Relativit¨atstheorie, A. Mercier, ed. (Birkh¨auser, Basel, 1956).

    Google Scholar 

  42. E. Schr¨odinger, "The general unitary theory of the physial elds," Proc. R. Irish Acad. 49A, 43 (1943).

    Google Scholar 

  43. M. Born and L. Infeld, Proc. R. Soc. A144, 425-451 (1934).

    Google Scholar 

  44. E. Schr¨odinger, "Contributions to Born's new theory of the electromagnetic field," Proc. R. Soc. (London) A150, 465-477 (1935).

    Google Scholar 

  45. O. Hittmair, "Schr¨odinger's unified field theory seen 40 years later," in Schr¨odinger. Centenary celebration of a polymath, I. Kilmister, ed. (Cambridge University Press, Cambridge, 1987).

    Google Scholar 

  46. E. Schr¨odinger, "The earth's and the sun's permanent magnetic fields in the unitary field theory," Proc. R. Irish Acad. 49A, 135 (1943).

    Google Scholar 

  47. E. Schr¨odinger, "The union of the three fundamental fields (gravitation, meson, electromagnetism)," Proc. R. Irish Acad. 49A, 275 (1944).

    Google Scholar 

  48. E. Schr¨odinger, "The general affine field laws," Proc. R. Irish Acad. 51A, 41 (1946).

    Google Scholar 

  49. E. Schr¨odinger, "The final affine field laws I," Proc. R. Irish Acad. 49A, 163 (1947).

    Google Scholar 

  50. E. Schr¨odinger, "The final affine field laws II," Proc. R. Irish Acad. 51A, 205 (1948).

    Google Scholar 

  51. M. A. Tonnelat, Histoire du Principe de Relativit´e (Flammarion, Paris, 1971).

    Google Scholar 

  52. V. P. Vizgin, Unified Field Theories in the First Third of the Twentieth Century (Birkh¨auser, Basel, 1994).

    Google Scholar 

  53. H. Goenner, in Sir Artur Eddington Centenary Symposium, Vol. I, V. de Sabbata and T. M. Karade, eds. (World Scientific, Singapore, 1984). After nishing thid paper, the authors became aware of: H. Goenner, "On the History of Unified Field Theories (1914-1933)," http://relativity. livingreviews. org/lrr-2004-2.

  54. C. Goldstein and J. J. Ritter, in Revisiting the Foundations of Relativistic Physics, A. Ashtekar et al., eds. (Kluwer Academic, Dordrecht, 2003), pp. 93-149.

    Google Scholar 

  55. A. S. Eddington, Fundamental Theory (Cambridge University Press, Cambridge, 1946).

    Google Scholar 

  56. H. Weyl, Space-Time-Matter (Dover, New York, 1952).

    Google Scholar 

  57. M. Ikeda, Progr. Theor. Phys. 12, 17-30 (1954).

    Google Scholar 

  58. M. Ikeda, Progr. Theor. Phys. 13, 265-275 (1955).

    Google Scholar 

  59. A. Papapetrou, Phys. Rev. 73, 1105 (1948).

    Google Scholar 

  60. H. Weyl, Math. Z. 12, 114 (1922).

    Google Scholar 

  61. H. Weyl, Mathematische Analyse des Raumproblems (Springer, Berlin, 1923).

    Google Scholar 

  62. E. Cartan, Jour. d. math. p. et a. 2, 167 (1923); see also Œuvres compl´etes, (Gauthier-Villars, Paris, p. 633).

    Google Scholar 

  63. E. Cartan, Ann. Ec. norm. 40, 325 (1923).

    Google Scholar 

  64. H.-H. v. Borzeszkowski and H.-J. Treder, Gen. Rel. Grav. 33, 1351 (2001).

    Google Scholar 

  65. H.-H. v. Borzeszkowski and H.-J. Treder, Gen. Rel. Grav. 34, 1909 (2002).

    Google Scholar 

  66. W. Szczyrba, Comm. Math. Phys. 51, 163 (1976).

    Google Scholar 

  67. J. Kijowski, Gen. Rel. Grav. 9, 857 (1978).

    Google Scholar 

  68. H.-J. Treder, H.-J. (1955). "Der Materietensor in der unsymmetrischen Feldtheorie Einsteins," Wiss. Z. Humboldt-Universit¨at Berlin, Math.-Naturwiss. R. 4, No. 1, pp. 9-10 (1955).

    Google Scholar 

  69. H.-J. Treder, Ann. Phys. (Leipzig) 45, 47 (1988).

    Google Scholar 

  70. M. Ferraris and J. Kijowski, Gen. Rel. Grav. 1, 37 (1982).

    Google Scholar 

  71. F. W. Hehl, Y. N. Obukhov and G. F. Rubilar, Ann. Phys. (Leipzig) 11, 71 (2000).

    Google Scholar 

  72. Y. N. Obukhov and F. W. Hehl, Phys. Lett. B458, 466 (1999).

    Google Scholar 

  73. R. A. Puntigam, C. L¨ammerzahl and F. W. Hehl, Class. Quantum Grav. 14, 1347 (1997).

    Google Scholar 

  74. T. W. B. Kibble, J. Math. Phys. 2, 212 (1961).

    Google Scholar 

  75. D. W. Sciama, in Recent Developments of General Relativity (Pergamon, London, 1962).

    Google Scholar 

  76. F. W. Hehl, P. van der Heide and D. G. Kerlick, Rev. Mod. Phys. 48, 393 (1976).

    Google Scholar 

  77. M. Carmeli, F. Leibowitz, and N. Nissani, Gravitation:SL(2, C)Gauge Theory and Conservation Laws (World Scientific, Singapore, 1990).

  78. J. W. Moffat, preprint gr-qc/9401001vl (1994).

  79. J. L. Cervantes-Cota and D.-E. Liebscher, Purely Af ne Theories with Matter, (submitted for publication); http//www.aip.de/`lie/PUBLICATIONS/CCL04.pdf

  80. A. Jakubiec and J. Kijowski, J. Math. Phys. 30, 1073 (1989); J. Math. Phys. 30, 1077 (1989).

    Google Scholar 

  81. P. G. Bergmann, "General relativity theory," in Encyclopedia of Physics, Vol. IV, S. Fl¨ugge, ed. (Springer, Berlin, 1962).

    Google Scholar 

  82. H.-J. Treder, Astr. Nachr. 315, 1 (1994).

    Google Scholar 

  83. S. B. Edgar, J. Math. Phys. 33, 3716 (1992).

    Google Scholar 

  84. M. Planck, "¨Uber irreversible Strahlungsvorg¨ange. 5. Mitteilung," Sitzungsber. Preuss. Akad. Wiss., pp. 440-480 (1899).

  85. N. Bohr and L. Rosenfeld, Det. Kgl. Danske Vidensk. Selskab., Math.-fys. Medd. XII, No. 8 (1933).

  86. H. Weyl, "Universum und Atom," Naturwiss. 22, 145 (1934).

    Google Scholar 

  87. H.-H. v. Borzeszkowski, Quantum Gravity, M. A. Markov, V. A. Berezin, and V. P. Frolov, eds. (World Scientific, Singapore, 1985).

  88. H.-H. v. Borzeszkowski and H.-J. Treder, On the Meaning of Quantum Gravity (Reidel, Dordrecht, 1988).

    Google Scholar 

  89. L. Rosenfeld, in Entstehung, Entwicklung und Persektiven der Einsteinschen Gravitations-theorie, H.-J. Treder, ed. (Akademie-Verlag, Berlin, 1966).

    Google Scholar 

  90. R. C. Tolman, Relativity, Thermodynamics, and Cosmology (Claredon Press, Oxford, 1934).

    Google Scholar 

  91. V. de Sabbata, Nuovo Cimento 107A, 363 (1994).

    Google Scholar 

  92. V. de Sabbata and L. Ronchetti, in Advances in the Interplay between Quantum and Gravity Physics, P. G. Bergmann and V. de Sabbata, eds. (Kluwer Academic, Dordrecht, 2002), pp. 85-1021.

    Google Scholar 

  93. V. de Sabbata, L. Ronchetti, and Alfred Yu, Found. Phys. 29, 1099 (1999).

    Google Scholar 

  94. H. Freudenthal, Math. Z. 64, 374 (1956).

    Google Scholar 

  95. H. Helmholtz, "¨Uber die Thatsachen, die der Geometrie zum Grunde liegen," Nachr. Ges. Wiss. G¨ottingen, p. 193 (1868).

  96. S. Lie, "¨Uber die Grundlagen der Geometrie", Nachr. Ges. Wiss. Leipzig 42, p. 284, and 355 (1890).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Theoretical Physics, Technical University Berlin, Hardenbergstrasse 36, D-10623, Berlin, Germany

    Horst-Heino von Borzeszkowski

  2. Rosa-Luxemburg-Str. 17a, D-14482, Potsdam, Germany

    Hans-Jürgen Treder

Authors
  1. Horst-Heino von Borzeszkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hans-Jürgen Treder
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

von Borzeszkowski, HH., Treder, HJ. On Metric and Matter in Unconnected, Connected, and Metrically Connected Manifolds. Foundations of Physics 34, 1541–1572 (2004). https://doi.org/10.1023/B:FOOP.0000044104.36647.ef

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:FOOP.0000044104.36647.ef

Search

Navigation