General Relativity and Gravitation

, Volume 38, Issue 8, pp 1241–1252

The hole argument for covariant theories

Research Article

Abstract

The hole argument was developed by Einstein in 1913 while he was searching for a relativistic theory of gravitation. Einstein used the language of coordinate systems and coordinate invariance, rather than the language of manifolds and diffeomorphism invariance. He formulated the hole argument against covariant field equations and later found a way to avoid it using coordinate language. In this paper we shall use the invariant language of categories, manifolds and natural objects to give a coordinate-free description of the hole argument and a way of avoiding it. Finally we shall point out some important implications of further extensions of the hole argument to sets and relations for the problem of quantum gravity.

Keywords

General relativity Differential geometry 

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References

  1. 1.
    Kerner R., Barbosa A.L., Gal’tsov D.V. Topics in Born–Infeld Electrodynamics, hep-th/0108026 (2005)Google Scholar
  2. 2.
    Earman J., Norton J. (1997) What prince spacetime substantivalism? the hole story. Br. J. Philos. Sci. 38, 515–525CrossRefMathSciNetGoogle Scholar
  3. 3.
    Einstein, A.: Die formale Grundlage der allgemeinen Relativitätstheorie, pp. 1030–1085 Königlich Preussische Akademie der Wissenschaften (Berlin), Sitzungsberichte (1914)Google Scholar
  4. 4.
    Einstein, A.: Reltivity, the Special and General Theory, 15th edn. Translation by Robert W. Lawson, University of Sheffield. Crown Publishers Inc., New York (1952)Google Scholar
  5. 5.
    Hawking S.W., Ellis G. (1973) The Large Scale Structure of the Universe. Cambridge University Press, CambridgeGoogle Scholar
  6. 6.
    Hermann R. Gauge Fields and Cartan-Ehresmann Connections, Part A. Math Sci Press, Brookline (1975)Google Scholar
  7. 7.
    Hodges W. (1993) Model Theory. Cambridge University Press, CambridgeMATHGoogle Scholar
  8. 8.
    Geroch R. (2004) Gauge, Diffeomorphisms, Initial-Value Formulation, etc. The Einstein equations and the large scale behavior of gravitational fields, pp. 441–477. Birkhuser, BaselGoogle Scholar
  9. 9.
    Fisher, A.: The Theory of Superspace. In: Relativity: Proceedings of the Relativity Conference in the Midwest. Plenum Press, New York/London (1970)Google Scholar
  10. 10.
    Stachel, J., Iftime, M.: Fibered Manifolds, Natural Bundles, Structured Sets, G-Sets and all that: The Hole Story from Space Time to Elementary Particles, gr-qc/0505138 (2005)Google Scholar
  11. 11.
    Isenberg J., Marsden J.E. (1982) A slice theorem for the space of solutions of Einstein’s equations, Physics Report, vol 89, no 2. North-Holland Publishing Company, Amsterdam, pp. 179–222Google Scholar
  12. 12.
    Klainerman S., Christodoulou D. (1993) The Global Nonlinear Stability of the Minkowski Space. Princeton University Press, New JerseyMATHGoogle Scholar
  13. 13.
    Kolár I., Michor P., Slovák J. (1993) Natural Operations in Differential Geometry. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  14. 14.
    Kamiński, W., Lewandowski, J., Bobieński, M.: Background independent quantizations: the scalar field I, gr-qc/0508091 (2005)Google Scholar
  15. 15.
    Stephani H. et al. (2003) Exact Solution of Einstein’s Field Equations 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
  16. 16.
    MacLane S. (1992) Sheaves in Geometry and Logic: a first introduction to topos theory. Springer, Berlin Heidelberg New YorkGoogle Scholar
  17. 17.
    Palais, R.: Notes on the slice theorem for the space of Riemannian metrics. Letter circulated in 1969 and notes written at Santa Cruz in 1975 (1970)Google Scholar
  18. 18.
    Sachs R.K., Wu H. (1977) General Relativity for Mathematicians. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  19. 19.
    Sardanashvily G., Zakharov O. (1991) Gauge Gravitation Theory. World Scientific, SingaporeGoogle Scholar
  20. 20.
    Sorkin R.D. (2000) Indications of causal set cosmology Int. J. Theor. Phys. 39, 1731–1736CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Stachel J. (1969) Specifying sources in general relativity. Phys. Rev. 180, 1256–1261CrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Stachel J. (1986) What a Physicist Can Learn From the Discovery of General Relativity. In: Remo Ruffini (ed). Proceedings of the Fourth Marcel Grossmann Meeting on General Relativity. Elsevier, Amsterdam, pp. 1857–1862Google Scholar
  23. 23.
    Stachel, J.: Einstein’s Search for General Covariance, 1912–1915. In: Don Howard, John Stachel (eds.). Einstein and the History of General Relativity, Einstein Studies, vol. 1. Birkhäuser, Boston/Basel/Berlin (1989), pp. 63–100. Reprinted in Stachel, John, Einstein from B to Z, Boston/Basel/Berlin: Birkhäuser 2002, pp. 301–337Google Scholar
  24. 24.
    Stachel J. (1993) The Meaning of General Covariance: the Hole Story. In: John Earman, et al. (eds). Philosophical Problems of the Internal and External Worlds. University of Pittsburgh Press Pittsburgh/Universitätsverlag, Konstanz, pp. 129–160Google Scholar
  25. 25.
    Stachel J. (1994) Changes in the Concepts of Space and Time Brought about by Relativity. In: Carol C. Gould, Robert S. Cohen (eds). Artifacts, Representations, and Social Practice/Essays for Marx Wartofsky. Kluwer, Dordrecht/Boston/London, pp. 141–162Google Scholar
  26. 26.
    Stachel J. New light on the Einstein–Hilbert priority question. J. Astrophys. Astron. 20, 91–101 (1999). Reprinted in Stachel, John, Einstein from B to Z. (see ref 23), pp. 353–364Google Scholar
  27. 27.
    Stachel, J.: The story of Newstein: or is gravity just another pretty force? In Renn, J., Schimmel, M. (eds.). The Genesis of General Relativity: Sources and Interpretations, vol. 4, Gravitation in the Twilight of Classical Physics: The Promise of Mathematics (Berlin, Springer 2006), pp. 1041–1078Google Scholar
  28. 28.
    Stachel J. (2004) Structural Realism and Contextual Individuality. In: Yemima Ben-Menahem, (ed). Hilary Putnam. Cambridge University Press, CambridgeGoogle Scholar
  29. 29.
    Stachel J. (2006) Structure, individuality and quantum gravity. In: Rickles D., French S., Saatsi J. (eds) The Structural Foundations of Quantum Gravity. Oxford University Press, OxfordGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.School of Arts and SciencesMCPHSBostonUSA
  2. 2.Department of Physics and Center for Philosophy and History of ScienceBostonUSA

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