Second Gradient Continuum: Role of Electromagnetism Interacting with the Gravitation on the Presence of Torsion and Curvature

  • Lalaonirina R. Rakotomanana
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)


The goal of this paper is to link the geometric variables of strain gradient continuum with electromagnetic fields. For that purpose, we derive the electromagnetic wave equation within a Riemann-Cartan continuum where curvature and torsion are present. We also show that the gravitational and electromagnetic fields are respectively identified as geometric objects of such a continuum, namely the curvature \( \Re_{\alpha \beta \lambda }^{\gamma } \) for gravitation which is a classical result, and the torsion \( \aleph_{\alpha \beta }^{\gamma } \) as source of electromagnetism.


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  1. Antonio Tamarasselvame N, Rakotomanana L (2011) On the form-invariance of Lagrangean function for higher gradient continuum. In: Altenbach H, Maugin GA, Erofeev V (eds) Mechanics of Generalized Continua, Heidelberg, ASM, vol 7, pp 291–322Google Scholar
  2. Charap JM, Duff MJ (1977) Gravitational effects on Yang-Mills topology. Physics Letters 69B(4):445–447Google Scholar
  3. Cho YM (1976) Einstein Lagrangian as the translational Yang-Mills Lagrangian. Physical Review D 14(10):2521–2525Google Scholar
  4. de Andrade VC, Pereira JG (1999) Torsion and the electromagnetic field. International Journal of Modern Physics D 8(2):141–151Google Scholar
  5. Dias L, Moraes F (2005) Effects of torsion on electromagnetic fields. Brazilian Journal of Physics 35(3A):636–640Google Scholar
  6. Fernado J, Giglio T, Rodrigues Jr WA (2012) Gravitation and electromagnetism as geometrical objects of Riemann-Cartan spacetime structure. Advanced Applied Clifford Algebras 22:640–664Google Scholar
  7. Fernandez-Nunez I, Bulashenko O (2016) Anisotropic metamaterial as an anlogue of a black hole. Physics Letters A 380:1–8Google Scholar
  8. Fumeron S, Pereira E, Moraes F (2015) Generation of optical vorticity from topological defects. Physica B 476:19–23Google Scholar
  9. Futhazar G, Le Marrec L, Rakotomanana LR (2014) Covariant gradient continua applied to wave propagation within defected material. Archive for Applied Mechanics 84(9–11):1339–1356Google Scholar
  10. Hammond RT (1987) Gravitation, Torsion, and Electromagnetism. General Relativity and Gravitation 20(8):813–827Google Scholar
  11. Hammond RT (1989) Einstein-Maxwell theory from torsion. Classical Quantum and Gravitation 6:195–198Google Scholar
  12. Hehl FW (2008) Maxwell’s equations in Minkowski’s world: their premetric generalization and the electromagnetic energy-momentum tensor. Annalen der Physik 17(9-10):691–704Google Scholar
  13. Hehl FW, von der Heyde P (1973) Spin and the structure of spacetime. Annales de l’Institut Henri Poincaré, section A 19(2):179–196Google Scholar
  14. Hehl FW, von der Heyde P, Kerlick JM G D ans Nester (1976) General relativity with spin and torsion: Foundations and prospects. Reviews of Modern Physics 48(3):393–416Google Scholar
  15. Kleinert H (2008) Multivalued Fields: in Condensed matter, Electromagnetism, and Gravitation. World Scientific, SingaporeGoogle Scholar
  16. Kovetz A (2000) Electromagnetic Theory. Oxford University Press, OxfordGoogle Scholar
  17. Maugin G (1993) Material Inhomogeneities in Elasticity. Chapman and Hall, LondonGoogle Scholar
  18. Maugin GA (1978) Exact relativistic theory of wave propagation in prestressed nonlinear elastic solids. Annales de l’Institut Henri Poincaré, section A 28(2):155–185Google Scholar
  19. Milonni PW, Boyd RW (2010) Momentum of light in a dielectric medium. Adv Opt Photon 2(4):519–553Google Scholar
  20. Obukhov YN (2008) Electromagnetic energy and momentum in moving media. Annalen der Physik 17(9-10):830–851Google Scholar
  21. Obukhov YN, Hehl FW (2003) Electromagnetic energy-momentum and forces in matter. Physics Letters A 311:277–284Google Scholar
  22. Plebanski J (1960) Electromagnetic waves in gravitational fields. Physical Review 118(5):1396–1408Google Scholar
  23. Poplawski NJ (2010) Torsion as electromagnetism and spin. International Journal of Theoretical Physics 49(7):1481–1488Google Scholar
  24. Prasanna AR (1975) Maxwell’s equations in Riemann-Cartan space U 4. Physics Letters A 54(1):17–18Google Scholar
  25. Puntigam RA, Lämmerzahl C, Hehl FW (1997) Maxwell’s theory on a post-Riemannian spacetime and the equivalence principle. Classical and Quantum Gravitation 14:1347–1356Google Scholar
  26. Rakotomanana RL (2003) A Geometric Approach to Thermomechanics of Dissipating Continua. Progress in Mathematical Physics Series, Birkhaüser, BostonGoogle Scholar
  27. Schutzhold R, Plunien G, Soff G (2002) Dielectric black hole analogs. Physical Review Letters 88(6):061,101/1–061,101/4Google Scholar
  28. Smalley LL (1986) On the extension of geometric optics from Riemaniann to Riemann-Cartan spacetime. Physics Letters A 117(6):267–269Google Scholar
  29. Smalley LL, Krisch JP (1992) Minimal coupling of electromagnetic fields in Riemann-Cartan spacetimes for perfect fluids with spin density. Journal of Mathematical Physics 33(3):1073–1081Google Scholar
  30. Sotiriou TP, Liberati S (2007) Metric-affine f (R) theories of gravity. Annals of Physics 322:935–966Google Scholar
  31. Tiwari RN, Ray S (1997) Static spherical charged dust electromagnetic mass models in Einstein- Cartan theory. General Relativity and Gravitation 29(6):683–690Google Scholar
  32. Vandyck MA (1996) Maxwell’s equations in spaces with non-metricity and torsion. J Physics A : Math Gen 29:2245–2255Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut de Recherche Mathématique de RennesRennesFrance

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