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Conformal orbits of electromagnetic Riemannian curvature tensors electromagnetic implies gravitational radiation

  • Hans Tilgner
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1156)

Abstract

Electromagnetic curvature structures (c.s.) are defined as being bilinear in the two electromagnetic field matrices and electrovac c.s. by having the el.magn. energy-momentum as Einstein tensor. There is a one-parameter familiy of electrovac c.s. having never a component in the space of constant curvatures. Their non-Weyl component is uniquely determined by the el.magn. energy-momentum, and in general they have a Weyl component. It is shown that el.magn. implies gravitational radiation, and coversely that el.magn. gravitational radiation is induced by el.magn. radiation. A structure theory of c.s. is described with morphisms as linear conformal transformations (i.e. Lorentztransformations and dilatations) such that the above properties of c.s., and many others, are orbit properties.

Keywords

Symmetric Space Structure Theory Jordan Algebra Gravitational Radiation Curvature Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [CS]
    C.D. Collinson, R. Shaw, The Rainich Conditions for Neutrino Fields, Int.J.Theor.Phys. 6, 347–357 (1972)CrossRefGoogle Scholar
  2. [F]
    T. Frankel, Gravitational Curvature, Freeman & Co, San Francisco (1979)zbMATHGoogle Scholar
  3. [G]
    R. Geroch, Building things in general relativity, Gen.Rel.Grav. 13, 37–41 (1981)zbMATHGoogle Scholar
  4. [Gy]
    A. Gray, Invariants of Curvature Operators in Four-Dimensional Riemannian Manifolds, in Proceedings of the 13th Biennial Seminar of the Canadian Mathematical Congress, Halifax (1971)Google Scholar
  5. [Gb]
    W.H. Greub, Linear Algebra, Springer-Verlag, Berlin (1967)CrossRefzbMATHGoogle Scholar
  6. [JM]
    P. Jordan, S. Matsushita, Zur Theorie der Lie-Tripel-Algebren, Akad.Wiss. Mainz, Abh. math.naturwissenschaftlicher Klasse 1,123–134 (1967)MathSciNetzbMATHGoogle Scholar
  7. [JEK]
    P. Jordan, J. Ehlers, W. Kundt, Strenge Lösungen der Feldgleichungen der allgemeinen Relativitätstheorie, Akad.Wiss.Mainz, Abh. math. naturwissenschaftlicher Klasse 2, 1–85 (1960)MathSciNetzbMATHGoogle Scholar
  8. [JES]
    P. Jordan, J. Ehlers, K. Sachs, Beiträge zur Theorie der reinen Gravitationsstrahlung. Strenge Lösungen der Feldgleichungen der allgemeinen Relativitätstheorie, Akad.Wiss.Mainz, Abh. math. naturwissenschaftlicher Klasse 1, 1–62 (1961)MathSciNetzbMATHGoogle Scholar
  9. [K]
    O. Kowalski, Partial Curvature Structures and Conformal Transformations, Jour.Diff.Geom. 8, 53–70 (1973)zbMATHGoogle Scholar
  10. [KSMH]
    D. Kramer, H. Stephani, M. MacCallum, E. Herlt, Exact Solutions of Einstein's Field equations, Cambridge Univ. Press, Cambridge (1980)zbMATHGoogle Scholar
  11. [Ku]
    R.S. Kulkarni, Curvature and Metric, Ann.Math. 91, 311–331 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Ku]
    R.S. Kulkarni, Curvature Structures and Conformal Transformations, Jour. Diff.Geom. 4, 53–70 (1970)MathSciNetzbMATHGoogle Scholar
  13. [L]
    A. Lichnerowicz, Ondes et radiations électromagnetiques et gravitationelles en relativité générale, Annali di Mathematica Pura e. Applica 4, 1–95 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Li]
    W.G. Lister, A structure Theory of Lie Triples, Trans.Am.Math.Soc. 72, 217–242 (1972)MathSciNetCrossRefGoogle Scholar
  15. [Lo]
    O. Loos, Symmetric Spaces I,II, Benjamin, N.Y. (1969)zbMATHGoogle Scholar
  16. [Lu]
    G. Ludwig, Geometricdynamics of Electromagnetic Fields in the Newman-Penrose Formalism, Commun.math.Phys. 17, 98–108 (1970)MathSciNetCrossRefGoogle Scholar
  17. [MTW]
    C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, Freeman and Co., San Francisco (1971)Google Scholar
  18. [N]
    K. Nomizu, Invariant Affine Connections on Homogeneous Spaces, Am.Jour. Math. 76, 33–65 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [N]
    K. Nomizu, The Decomposition of Generalized Curvature Tensor Fields, in Differential Geometry, Papers in Honour of K. Yano, Kinukuniya, Tokyo, 335–345 (1972)Google Scholar
  20. [NY]
    K. Nomizu, K. Yano, Some results related to the equivalence problem in Riemannian Geometry, in Proc. of the US-Japan Seminar in Differential Geometry, Kyoto, Japan, 95–100 (1965)Google Scholar
  21. [R]
    G.Y. Rainich, Electrodynamics in the General Relativity Theory, Trans. Am. Math.Soc. 27, 106–136 (1925)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [SW]
    R.K. Sachs, H. Wu, General Relativity for Mathematicians, Springer-Verlag, Berlin (1977)CrossRefzbMATHGoogle Scholar
  23. [Se]
    I.E. Segal, Mathematical Cosmology and Extra-Galactic Astronomy, Academic Press, N.Y (1977)Google Scholar
  24. [ST]
    I.M. Singer, J.A. Thorpe, The Curvature of 4-Dimensional Einstein Spaces, in Papers in honou of K. Kodaira, Princeton Univ. Press, Princeton (1968)Google Scholar
  25. [Sö]
    F. Söler, r-Manifolds and Gravitational Radiation, Gen.Rel.Grav. 13, 37–41 (1981)MathSciNetGoogle Scholar
  26. [S]
    A. Stehney, Principal Null Directions without Spinors, Jour.Math.Phys. 17, 1793–1796Google Scholar
  27. [T]
    J.A. Thorpe, Curvature and the the Petrov Canonical Forms, Jour.Math.Phys. 10, 1–6 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Ti]
    H. Tilgner, The Group Structure of Pseudo-Riemannian Curvature Spaces, Jour.Math.Phys. 19, 1118–1125 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  29. [W]
    S. Weinberg, Gravitation and Cosmology, Wiley, N.Y. (1972)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Hans Tilgner
    • 1
  1. 1.Mathematisches InstitutTechnische Universität BerlinBerlin 37

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