General Relativity and Gravitation

, Volume 2, Issue 1, pp 27–33 | Cite as

Excluded possibilities of geometrodynamical analog to electric charge

  • W. G. Unruh

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References

  1. 1.
    Misner, C. W. and Wheeler, J. A. (1957). Gravitation, Electromagnetism, Unquantized Charge, and Mass as Propeerties of Curved Empty Space,Ann. Phys.,2, 525 603. Reprinted in Wheeler,Geometrodynamics, Academic Press, New York, 1961.MATHMathSciNetCrossRefADSGoogle Scholar
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    InGeometrodynamics, pp. 231, Wheeder cites H. Weyl:Raum Zeit Materie, 4th ed., Sect.34 (Berlin, 192), A. Finstein and N. Rosen (1935).Phys. Rev. 48, 73, and J. L. Synge (1947) as having previously mentioned the idea of multiply connected spaces in Genral Relativity.Google Scholar
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    For the experimental evidence against the existence of magnetic monopoles see R. L. Fleischeret al., Phys. Rev.,177, 2029 (1969) and other references cited therein.CrossRefADSGoogle Scholar
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    For an elementary exposition of homology theory, see for example, note 1or any textbook on homology, theory such as I. M. Singer and J. A. Thorpe,Lecture Notes on Elementary Topology and Geometry, Scott, Foreman and Company, Gleville, Illinois, 1967.MATHGoogle Scholar
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    We assume our space-time to be compact but not boundaryless. In fact for simplicity we assume throughout that our manifold is of the form [0,1]×S whereS is a boundaryless, compact, 3-dimensional manifold (a closed three-universe in physicist terminology). See R. P. Geroch,J. Math. Phys.,8, 782 (1967) for reasons for this assumption.MATHMathSciNetCrossRefADSGoogle Scholar
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    Misner, C. W. and Finklestein, D. (1959).Ann. Phys.,6, 230.MATHCrossRefADSGoogle Scholar
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    Wheeler, J. A., Absence of Geometrical Analog to Electric Charge, inRelativistic Fluid Mechanics and Hydrodynamics. (Ed. Wasserman and Wells).Google Scholar
  8. 8.
    We note that if our tensor is already the curl of an antisymmetric tensor, the charges associated with it are all zero, and we therefore exclude these curl free tensors.Google Scholar
  9. 9.
    IfT (μρ;σ)=0 thent 16 μ=(εμν∞T ρσ);μ=0.Google Scholar
  10. 10.
    For an exposition of the spinor formalism see for example R. Penrose (1960).Ann. Phys.,10, 171–201 or F. Pirani's excellent lectures in the Brandeis Summer Institute Lectures on General Relativity (New Jersey, 1964) pp. 305ff.MATHMathSciNetCrossRefADSGoogle Scholar
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    Mentioned by R. Penrose (1960).Ann. Phys.,10, 183.MathSciNetGoogle Scholar
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    Collinson, C. (1962).Proc. Camb. Phil. Soc.,58, 346.MATHMathSciNetCrossRefGoogle Scholar
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    C. Collinson neglects this consequence of the Bianchi identities, making some of this independent tensors dependent.Google Scholar
  14. 14.
    R. Geroch has suggested a possible method of proving the nonexistence of such curl free tensors for alln for the curl free vector and 2-tensor, and investigation of his suggestion is being carried out at present.Google Scholar

Copyright information

© Plenum Publishing Company Limited 1971

Authors and Affiliations

  • W. G. Unruh
    • 1
  1. 1.Joseph Henry LaboratoriesPrinceton UniversityPrinccton
  2. 2.National Research Council of Canada Postgraduate ScholarCanada

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