General Relativity and Gravitation

, Volume 45, Issue 7, pp 1411–1431 | Cite as

A Generalization of the Goldberg–Sachs theorem and its consequences

Research Article

Abstract

The Goldberg–Sachs theorem is generalized for all four-dimensional manifolds endowed with torsion-free connection compatible with the metric, the treatment includes all signatures as well as complex manifolds. It is shown that when the Weyl tensor is algebraically special severe geometric restrictions are imposed. In particular it is demonstrated that the simple self-dual eigenbivectors of the Weyl tensor generate integrable isotropic planes. Another result obtained here is that if the self-dual part of the Weyl tensor vanishes in a Ricci-flat manifold of (2,2) signature the manifold must be Calabi–Yau or symplectic and admits a solution for the source-free Einstein–Maxwell equations.

Keywords

Goldberg–Sachs theorem Weyl tensor Integrable distributions  Petrov classification General relativity 

References

  1. 1.
    Petrov, A.Z.: The classification of spaces definig gravitational fields. Gen. Relativ. Gravit. 32, 1665 (2000). This is a translated republication of original 1954 paperGoogle Scholar
  2. 2.
    Stephani, H., et al.: Exact Solutions of Einstein’s field Equations. Cambridge University Press, Cambridge (2009)Google Scholar
  3. 3.
    Kerr, R.P.: Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11, 237 (1963)MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Goldberg, J., Sachs, R.: A theorem on Petrov types. Gen. Relativ. Gravit. 41, 433 (2009). This is a republication of original 1962 paperGoogle Scholar
  5. 5.
    Batista, C.: Weyl tensor classifcation in four-dimensional manifolds of all signatures. Gen. Relativ. Gravit. (2013). doi:10.1007/s10714-013-1499-8. Available at arXiv:1204.5133
  6. 6.
    Plebański, J.F., Hacyan, S.: Null geodesic surfaces and Goldberg-Sachs theorem in complex Riemannian spaces. J. Math. Phys. 16, 2403 (1975)ADSMATHCrossRefGoogle Scholar
  7. 7.
    Przanowski, M., Broda, B.: Locally Kähler gravitational instantons. Acta Physica Polonica B14, 637 (1983)MathSciNetGoogle Scholar
  8. 8.
    Nurowski, P., Trautman, A.: Robinson manifolds as the Lorentzian analogs of Hermite Manifolds. Differ Geom Appl 17, 175 (2002)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Gover, A., Hill, C., Nurowski, P.: Sharp version of the Goldberg–Sachs theorem. Annali di Matematica Pura ed Applicata 190 Number 2, 295 (2011). Available at arXiv:0911.3364Google Scholar
  10. 10.
    Durkee, M., Reall, H.S.: A higher-dimensional generalization of the geodesic part of the Goldberg–Sachs theorem. Class. Quantum Grav. 26, 245005 (2009). Available at arXiv:0908.2771Google Scholar
  11. 11.
    Ortaggio, M., et al.: On a five-dimensional version of the Goldberg-Sachs theorem. Available at arXiv:1205.1119Google Scholar
  12. 12.
    Taghavi-Chabert, A.: Optical structures, algebraically special spacetimes and the Goldberg-Sachs theorem in five dimensions. Class. Quantum Grav. 28, 145010 (2011). Available at arXiv:1011.6168Google Scholar
  13. 13.
    Taghavi-Chabert, A.: The complex Goldberg-Sachs theorem in higher dimensions. J. Geom. Phys. 62, 981 (2012). Available at arXiv:1107.2283Google Scholar
  14. 14.
    Law, P.R.: Neutral Einstein metrics in four dimensions. J. Math. Phys. 32, 3039 (1991)MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Coley, A., Hervik, S.: Higher dimensional bivectors and classification of the Weyl operator. Class. Quantum Grav. 27, 015002 (2010). Available at arXiv:0909.1160Google Scholar
  16. 16.
    Hervik, S., Coley, A.: Curvature operators and scalar curvature invariants. Class. Quantum Grav. 27, 095014 (2010). Available at arXiv:1002.0505Google Scholar
  17. 17.
    Plebański, J.: Some solutions of complex Einstein equations. J. Math. Phys. 16, 2395 (1975)ADSCrossRefGoogle Scholar
  18. 18.
    Hacyan, S.: Gravitational instantons in H-spaces. Phys. Lett. 75A, 23 (1979)MathSciNetADSGoogle Scholar
  19. 19.
    Law, P.R.: Classification of the Weyl curvature spinors of neutral metrics in four dimensions. J. Geom. Phys. 56, 2093 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    Hervik, S., Coley, A.: On the algebraic classification of pseudo-Riemannian spaces. Int. J. Geom. Methods Mod. Phys. 8, 1679 (2011). Available at arXiv:1008.3021Google Scholar
  21. 21.
    Penrose, R., Rindler, W.: Spinors and space-time vol. 1 and 2, Cambridge University Press, Cambridge (1984 and 1986)Google Scholar
  22. 22.
    Bel, L.: Radiation states and the problem of energy in general relativity. Gen. Relativ. Gravit. 32, 2047 (2000) Reprint of a 1962 paperGoogle Scholar
  23. 23.
    Robinson, I., Schild, A.: Generalization of a theorem by Goldberg and Sachs. J. Math. Phys. 4, 484 (1963)MathSciNetADSMATHCrossRefGoogle Scholar
  24. 24.
    Kopczynski, W., Trautman, A.: Simple spinors and real structures. J. Math. Phys. 33, 550 (1992)MathSciNetADSMATHCrossRefGoogle Scholar
  25. 25.
    Nakahara, M.: Geometry, Topology and Physics. Taylor & Francis, London (2003)MATHGoogle Scholar
  26. 26.
    Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. Math. 65, 391 (1957)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Wald, R.M.: General Relativity. The University of Chicago Press, Chicago (1984)MATHCrossRefGoogle Scholar
  28. 28.
    Kinnersley, W.: Type D vacuum metrics. J. Math. Phys. 10, 1195 (1969)MathSciNetADSMATHCrossRefGoogle Scholar
  29. 29.
    Ivanov, S., Zamkovoy, S.: Parahermitian and paraquaternionic manifolds. Differ. Geom. Appl. 23, 205 (2005). Available at arXiv:math/0310415Google Scholar
  30. 30.
    McIntosh, C., Hickman, M.: Complex relativity and real solutions. I: Introduction. Gen. Relativ. Gravit. 17, 111 (1985)MathSciNetADSMATHCrossRefGoogle Scholar
  31. 31.
    Mason, L., Woodhouse, N.: Integrability, self-duality and twistor theory. Oxford University Press, Oxford (1996)MATHGoogle Scholar
  32. 32.
    Coley, A., Milson, R., Pravda, V., Pravdová, A.: Classification of the Weyl Tensor in higher dimensions. Class. Quantum Grav. 21, L-35 (2004). Available at arXiv:gr-qc/0401008Google Scholar
  33. 33.
    Frolov, V., Stojković, D.: Particle and light motion in a space-time of a five-dimensional black hole. Phys. Rev. D 68, 064011 (2003). Available at arXiv:gr-qc/0301016Google Scholar
  34. 34.
    Berkovits, N., Marchioro, D.: Relating the Green–Schwarz and pure spinor formalisms for the superstring. J. High Energy Phys. 01(2005). Available at arXiv:hep-th/0412198.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidade Federal de PernambucoRecifeBrazil

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