Skip to main content
Log in

Development of the concept of natural geometry for physical interactions

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We analyze the development of the concept of natural geometry for the gravitational field in Logunov’s works. We discuss the application of this concept to vacuum nonlinear electrodynamics and show that defining the natural geometry for a nonlinear theory and finding its metric tensor permit obtaining sufficiently complete information about the propagation laws for electromagnetic field pulses in background electromagnetic fields.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. N. I. Lobachevsky, “On the foundations of geometry [in Russian],” Kazan Vestnik, 25, 178–187 (1829).

    Google Scholar 

  2. A. Einstein, “Die Feldgleichungen der Gravitation,” Sitzungsber. Königl. Preuß. Akad. Wiss. (Berlin), 48, 844–847 (1915).

    MATH  Google Scholar 

  3. A. A. Logunov and V. N. Folomeshkin, “Energy–momentum of gravitational waves in the general theory of relativity,” Theor. Math. Phys., 32, 667–672 (1977).

    Article  MATH  Google Scholar 

  4. A. A. Logunov and V. N. Folomeshkin, “The energy–momentum problem and the theory of gravitation,” Theor. Math. Phys., 32, 749–771 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. A. Logunov and V. N. Folomeshkin, “Geometrized theories of gravitation,” Theor. Math. Phys., 32, 653–666 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. A. Logunov, V. I. Denisov, A. A. Vlasov, M. A. Mestvirishvili, and V. N. Folomeshkin, “New concepts of space–time and gravitation,” Theor. Math. Phys., 40, 753–777 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. A. Logunov and M. A. Mestvirishvili, “Relativistic theory of gravitation,” Prog. Theor. Phys., 74, 31–50 (1985).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. A. A. Logunov, “Fundamental principles of the relativistic theory of gravitation,” Theor. Math. Phys., 80, 785–789 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. A. Logunov, “Basic equations for the massive gravitational field,” Theor. Math. Phys., 92, 826–836 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  10. S. S. Gershtein, A. A. Logunov, and M. A. Mestvirishvili, “Gravitational field self-limitation and its role in the Universe,” Phys. Usp., 49, 1179–1195 (2006).

    Article  ADS  Google Scholar 

  11. S. S. Gershtein, A. A. Logunov, and M. A. Mestvirishvili, “Gravitational waves in the relativistic theory of gravity,” Theor. Math. Phys., 160, 1096–1100 (2009).

    Article  MATH  Google Scholar 

  12. A. A. Logunov, Relativistic Theory of Gravitation [in Russian], Nauka, Moscow (2012).

    MATH  Google Scholar 

  13. M. Born and L. Infeld, “Foundations of the new field theory,” Proc. Roy. Soc. Ser. A, 144, 425–430 (1934).

    Article  ADS  MATH  Google Scholar 

  14. W. Heisenberg and H. Euler, “Folgerungen aus der Diracschen Theorie des Positrons,” Z. Phys., 98, 714–720 (1936).

    Article  ADS  MATH  Google Scholar 

  15. G. Boillat, “Nonlinear electrodynamics: Lagrangians and equations of motion,” J. Math. Phys., 11, 941–950 (1970).

    Article  ADS  Google Scholar 

  16. V. I. Denisov and I. P. Denisova, “Verifiable post-Maxwellian effect of the nonlinear electrodynamics in vacuum,” Opt. Spectrosc., 90, 282–287 (2001).

    Article  ADS  Google Scholar 

  17. V. I. Denisov and I. P. Denisova, “Interaction of intense laser radiation with weak electromagnetic waves in an evacuated section of a ring laser,” Opt. Spectrosc., 90, 928–930 (2001).

    Article  ADS  Google Scholar 

  18. I. P. Denisova, I. V. Krivchenkov, P. A. Vshivtseva, and A. A. Zubrilo, “Letter: Nonlinear gravitationalelectromagnetic bending of the rays of weak electromagnetic waves in the fields of pulsars and magnetars,” Gen. Relativ. Gravit., 36, 889–897 (2004).

    Article  ADS  MATH  Google Scholar 

  19. K. V. Zhukovsky, “A method of inverse differential operators using orthogonal polynomials and special functions for solving some types of differential equations and physical problems,” Moscow Univ. Phys. Bulletin, 70, 93 (2015).

    Article  ADS  Google Scholar 

  20. B. M. Barbashov and N. A. Chernikov, “Solution of the two plane wave scattering problem in a nonlinear scalar field theory of the Born–Infeld type,” JETP, 24, 437–442 (1967).

    ADS  Google Scholar 

  21. V. I. Denisov and I. P. Denisova, “Interaction effect of plane electromagnetic waves in the Born–Infeld nonlinear electrodynamics,” Theor. Math. Phys., 129, 1421–1427 (2001).

    Article  MATH  Google Scholar 

  22. V. I. Denisov and V. A. Sokolov, “Analysis of regularizing properties of nonlinear electrodynamics in the Einstein–Born–Infeld theory,” JETP, 113, 926–933 (2011).

    Article  ADS  Google Scholar 

  23. I. P. Denisova, “Development of the method of spin coefficients for the integration of equations of bimetric gravitational theories,” Differ. Equ., 35, 942–948 (1999).

    MathSciNet  MATH  Google Scholar 

  24. I. V. Krivtchenkiv, V. A. Sokolov, P. A. Vshivtseva, and A. A. Zubrilo, Moscow Univ. Phys. Bull., 5, 6–8 (2006).

    Google Scholar 

  25. G. A. Alekseev and V. I. Khlebnikov, “Newman–Penrose formalism and its application in general relativity [in Russian],” PEPAN, 9, 790–870 (1978).

    Google Scholar 

  26. A. A. Logunov, Lectures on the Theory of Relativity and Gravitation [in Russian], Nauka, Moscow (1987); English transl.: Lectures in Relativity and Gravitation: A Modern Look, Pergamon Press, Oxford (1990).

    MATH  Google Scholar 

  27. R. Courant, Methods of Mathematical Physics, Vol. 2, Partial Differential Equations, Interscience, New York (1962).

    Google Scholar 

  28. W.-T. Ni, “Space–time structure and asymmetric metric from the premetric formulation of electromagnetism,” Phys. Lett. A, 379, 1297–1303 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. V. I. Denisov and I. P. Denisova, “The eikonal equation in parametrized nonlinear electrodynamics of vacuum,” Dokl. Phys., 46, 377–379 (2001).

    Article  ADS  Google Scholar 

  30. V. I. Denisov, “Investigation of the effective space–time of the vacuum nonlinear electrodynamics in a magnetic dipole field,” Theor. Math. Phys., 132, 1071–1079 (2002).

    Article  MATH  Google Scholar 

  31. M. G. Gapochka, M. M. Denisov, I. P. Denisova, N. V. Kalenova, and A. F. Korolev, “Mathematical modeling of the nonlinear electrodynamics effect of signal delay in the magnetic field of pulsars,” Comput. Math. Math. Phys., 55, 1857–1866 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  32. V. I. Denisov and M. I. Denisov, Phys. Rev. D, 60, 047301 (1999).

    Article  ADS  Google Scholar 

  33. J. Y. Kim, “Bending of electromagnetic wave in an ultra-strong magnetic field,” J. Cosmol. Astropart. Phys., 10, 056 (2012); arXiv:1208.1319v1 [hep-ph] (2012).

    Article  ADS  Google Scholar 

  34. V. I. Denisov, I. V. Krivchenkov, and I. P. Denisova, “Nonlinear electrodynamic lag of electromagnetic signals in a magnetic dipole field,” JETP, 95, 194–198 (2002).

    Article  ADS  Google Scholar 

  35. V. I. Denisov and S. I. Svertilov, “Nonlinear electromagnetic and gravitational actions of neutron star fields on electromagnetic wave propagation,” Phys. Rev. D, 71, 063002 (2005).

    Article  ADS  Google Scholar 

  36. M. I. Vasil’ev and V. A. Sokolov, “Nonlinear-electrodynamic effects in the electromagnetic field of a rotating pulsar,” Moscow Univ. Phys. Bulletin, 67, 418–422 (2012).

    Article  ADS  Google Scholar 

  37. V. I. Denisov, V. A. Sokolov, and M. I. Vasili’ev, “Nonlinear vacuum electrodynamics birefringence effect in a pulsar’s strong magnetic field,” Phys. Rev. D., 90, 023011 (2014).

    Article  ADS  Google Scholar 

  38. V. I. Denisov, “Nonlinear effect of quantum electrodynamics for experiments with a ring laser,” J. Opt. A, 2, 372–379 (2000).

    Article  ADS  Google Scholar 

  39. V. A. Sokolov, “Electromagnetic wave interactions in a strong permanent magnetic field,” Moscow Univ. Phys. Bulletin, 64, 349–351 (2009).

    Article  ADS  Google Scholar 

  40. L. Elsgolts, Differential Equations and the Calculus of Variations [in Russian], Nauka, Moscow (1965); English transl., Mir, Moscow (1970).

    Google Scholar 

  41. P. A. Vshivtseva and M. M. Denisov, “Mathematical modeling of the propagation of electromagnetic waves in nonlinear electrodynamics,” Comput. Math. Math. Phys., 49, 2092–2102 (2009).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. I. Denisov.

Additional information

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 191, No. 2, pp. 205–211, May, 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Denisov, V.I. Development of the concept of natural geometry for physical interactions. Theor Math Phys 191, 649–654 (2017). https://doi.org/10.1134/S004057791705004X

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S004057791705004X

Keywords

Navigation