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Classes of Maxwell spaces that admit subgroups of the Poincaré group

Abstract

A Maxwell space is a triple (M, g, F), where M is the four-dimensional Minkowski space or a domain in it, g is a pseudo-Euclidean metric on M, and F is a closed exterior 2-form on M. In this paper, we give an exhaustive description of classes of Maxwell spaces that admit subgroups of the Poincaré group. Representatives of all classes are constructed.

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References

  1. 1.

    H. Bacry, P. Combe, and P. Sorba, “Connected subgroups of the Poincaré group, I,” Rep. Math. Phys., 5, No. 2, 145–186 (1974).

    Article  Google Scholar 

  2. 2.

    H. Bacry, P. Combe, and P. Sorba, “Connected subgroups of the Poincaré group, II, Rep. Math. Phys., 5, No. 3, 361–392 (1974).

    Article  Google Scholar 

  3. 3.

    I. V. Bel’ko, “Subgroups of the Lorentz-Poincaré group,” Izv. Akad. Nauk Belorus. SSR, 1, 5–13 (1971).

    MathSciNet  Google Scholar 

  4. 4.

    O. G. Belova, A. N. Zarembo, M. A. Parinov, O. O. Sergeeva, and Yu. G. Ugarova, “Classification of static electromagnetic fields over subgroups of the Poincaré group,” Nauch. Tr. Ivan. Univ., Ser. Mat., 3, 11–22 (2000).

    Google Scholar 

  5. 5.

    A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin (1987).

    Google Scholar 

  6. 6.

    P. Combe, and P. Sorba, “Electromagnetic fields with symmetry,” Physica, A80,No. 3, 271–286 (1975).

    MathSciNet  Google Scholar 

  7. 7.

    A. S. Ivanova and M. A. Parinov, “First integrals of Lorentz equations for some classes of electromagnetic fields,” Proc. Steklov Inst. Math., 236, 186–192 (2002).

    MathSciNet  Google Scholar 

  8. 8.

    A. Janner and E. Ascher, “Space-time symmetry of linearly polarized electromagnetic plane waves,” Lett. Nuovo Cim., 2, No. 15, 703–705 (1969).

    Google Scholar 

  9. 9.

    A. Janner and E. Ascher, “Space-time symmetry of transverse electromagnetic plane waves,” Helv. Phys. Acta, 43, No. 3, 296–303 (1970).

    MathSciNet  Google Scholar 

  10. 10.

    A. Janner and E. Ascher, “Relativistic symmetry groups of uniform electromagnetic fields, Physica, 48, No. 3, 425–446 (1970).

    MathSciNet  Article  Google Scholar 

  11. 11.

    N. A. Kosheleva, A. K. Kuramshina, and M. A. Parinov, “Group classification of Maxwell spaces admitting elliptic helices,” Nauch. Tr. Ivan. Univ., Ser. Mat., 4, 73–82 (2001).

    Google Scholar 

  12. 12.

    L. D. Landau and E. M. Lifshits, Field Theory [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  13. 13.

    D. A. L’vov and M. A. Parinov, “Group classification of Maxwell spaces admitting parabolic rotations,” Nauch. Tr. Ivan. Univ., Ser. Mat., 5, 51–62 (2002).

    Google Scholar 

  14. 14.

    E. G. Morokhova, “Group classification of Maxwell spaces admitting proportional bi-rotations,” Nauch. Tr. Ivan. Univ., Ser. Mat., 5, 63–70 (2002).

    Google Scholar 

  15. 15.

    E. V. Morozova and M. A. Parinov, “Group classification of Maxwell spaces admitting translations along isotropic straight lines,” Nauch. Tr. Ivan. Univ., Ser. Mat., 4, 87–94 (2001).

    Google Scholar 

  16. 16.

    M. A. Parinov, “Problem of group classification of electromagnetic fields,” in: Contemporary Methods of Theory of Functions and Related Problems. Voronezh Winter Math. School [in Russian], Voronezh (1999), p. 156.

  17. 17.

    M. A. Parinov, “Group classification of Maxwell spaces,” in: Contemporary Analysis and Its Applications. Voronezh Winter Math. School [in Russian], Voronezh (2000), pp. 129–130.

  18. 18.

    M. A. Parinov, “Classes of Maxwell spaces admitting translations,” Sovr. Mat. Prilozh., 10, 157–166 (2003).

    Google Scholar 

  19. 19.

    M. A. Parinov, Einstein-Maxwell spaces and Lorentz equations [in Russian], Ivanovo (2003).

  20. 20.

    N. S. Polezhaeva and M. A. Parinov, Group classification of 4-potentials admitting parabolic rotations [in Russian], Preprint, Ivanovo (2003); deposited at the All-Russian Institute for Scientific and Technical Information, Moscow (2003), No. 1489-V2003.

  21. 21.

    A. I. Vorob’ev, “Group classification of Maxwell spaces admitting hyperbolic helices,” Nauch. Tr. Ivan. Univ., Ser. Mat., 4, 35–42 (2001).

    Google Scholar 

  22. 22.

    A. I. Vorob’ev, “On the classification of Maxwell spaces admitting hyperbolic helices and first integrals of Lorentz equations,” Sovr. Mat. Prilozh., 10, 39–47 (2003).

    Google Scholar 

  23. 23.

    A. I. Vorob’ev, “Classification of potential stuctures invariant relative to hyperbolic helices,” Mat. Prilozh. (Ivanovo), 1, 41–50 (2004).

    Google Scholar 

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Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 1, Geometry of Integrable Models, 2004.

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Parinov, M.A. Classes of Maxwell spaces that admit subgroups of the Poincaré group. J Math Sci 136, 4419–4458 (2006). https://doi.org/10.1007/s10958-006-0235-2

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Keywords

  • Symmetry Group
  • Dimensional Group
  • Maxwell Space
  • Parabolic Rotation
  • Isotropic Hyperplane