# Constitutive tensor in the geometrized Maxwell theory

**Authors:**Korolkova A.V.^{1}-
**Affiliations:**- Peoples’ Friendship University of Russia (RUDN University)

**Issue:**Vol 30, No 4 (2022)**Pages:**305-317**Section:**Articles**URL:**https://journals.rudn.ru/miph/article/view/33012**DOI:**https://doi.org/10.22363/2658-4670-2022-30-4-305-317

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## Abstract

It is generally accepted that the main obstacle to the application of Riemannian geometrization of Maxwell’s equations is an insufficient number of parameters defining a geometrized medium. In the classical description of the equations of electrodynamics in the medium, a constitutive tensor with 20 components is used. With Riemannian geometrization, the constitutive tensor is constructed from a Riemannian metric tensor having 10 components. It is assumed that this discrepancy prevents the application of Riemannian geometrization of Maxwell’s equations. It is necessary to study the scope of applicability of the Riemannian geometrization of Maxwell’s equations. To determine whether the lack of components is really critical for the application of Riemannian geometrization. To determine the applicability of Riemannian geometrization, the most common variants of electromagnetic media are considered. The structure of the dielectric and magnetic permittivity is written out for them, the number of significant components for these tensors is determined. Practically all the considered types of electromagnetic media require less than ten parameters to describe the constitutive tensor. In the Riemannian geometrization of Maxwell’s equations, the requirement of a single impedance of the medium is critical. It is possible to circumvent this limitation by moving from the complete Maxwell’s equations to the approximation of geometric optics. The Riemannian geometrization of Maxwell’s equations is applicable to a wide variety of media types, but only for approximating geometric optics.

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1. Introduction With the advent of the model Cayley-Klein [1, 2] the formalism of nonEuclidean spaces became used to describe physical models. This approach received popularity after the creation Einstein’s general theory of relativity [3]. At the same time, there were attempts to geometrize Maxwell’s electrodynamics [4-6]. However, this approach remained quite marginal until the golden age of theory of relativity [7, 8]. This direction became popular again in the new century and gave rise to the development of transformational optics [9-12]. However, it became visible that Riemannian geometry is insufficient for geometrization of Maxwell’s equations [13, 14]. In this paper, the author expects to figure out what could hinder the application of Riemannian geometrization of Maxwell’s equations and what is the scope of its applicability. To do this, we consider different electromagnetic media options and the limitations imposed by them are studied for possible geometrizations. 1.1. Article structure In paragraph 1.2 we provide basic notation and conventions used in the article. In the section 1.3 we consider the limitation only for the case of a local linear medium. In the section 2 the constitutive tensor is formulated in a six-dimensional space. This is being done for clarity, to represent it as a matrix 6 × 6. In the section 3 the reader is reminded of Riemannian geometrization of Maxwell’s equations. 1.2. Notations and conventions 1. Greek indexes (## About the authors

### Anna V. Korolkova

Peoples’ Friendship University of Russia (RUDN University)
**Author for correspondence.**

Email: korolkova-av@rudn.ru

ORCID iD: 0000-0001-7141-7610

Docent, Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation## References

- F. C. Klein, “Ueber die sogenannte Nicht-Euklidische Geometrie,” German, in Gau und die Anfnge der nicht-euklidischen Geometrie, ser. Teubner-Archiv zur Mathematik, vol. 4, Wien: Springer-Verlag Wien, 1985, pp. 224-238. doi: 10.1007/978-3-7091-9511-6_5.
- F. C. Klein, “A comparative review of recent researches in geometry,” Bulletin of the American Mathematical Society, vol. 2, no. 10, pp. 215- 249, 1893. doi: 10.1090/S0002-9904-1893-00147-X.
- C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation. San Francisco: W. H. Freeman, 1973.
- I. Y. Tamm, “Crystal optics theory of relativity in connection with geometry biquadratic forms,” vol. 57, no. 3-4, pp. 209-240, 1925, in Russian.
- I. Y. Tamm, “Electrodynamics of an anisotropic medium in a special theory of relativity,” Russian Journal of Physical and Chemical Society. Part physical, vol. 56, no. 2-3, pp. 248-262, 1924, in Russian.
- L. I. Mandelstam and I. Y. Tamm, “Elektrodynamik der anisotropen Medien in der speziellen Relativittstheorie,” German, Mathematische Annalen, vol. 95, no. 1, pp. 154-160, 1925.
- J. Plebanski, “Electromagnetic waves in gravitational fields,” Physical Review, vol. 118, no. 5, pp. 1396-1408, 1960. doi: 10.1103/PhysRev. 118.1396.
- F. Felice, “On the gravitational field acting as an optical medium,” General Relativity and Gravitation, vol. 2, no. 4, pp. 347-357, 1971. doi: 10.1007/BF00758153.
- A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” Journal of Modern Optics, vol. 43, no. 4, pp. 773-793, Apr. 1996. doi: 10.1080/09500349608232782.
- U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” in Progress in Optics. 2009, vol. 53, ch. 2. doi: 10.1016/S0079-6638(08)00202-3.
- U. Leonhardt and T. G. Philbin, “General Relativity in Electrical Engineering,” vol. 8, 247, pp. 247.1-18, 2006. doi: 10.1088/13672630/8/10/247.
- D. S. Kulyabov, A. V. Korolkova, L. A. Sevastianov, M. N. Gevorkyan, and A. V. Demidova, “Geometrization of Maxwell’s Equations in the Construction of Optical Devices,” in Proceedings of SPIE. Saratov Fall Meeting 2016: Laser Physics and Photonics XVII and Computational Biophysics and Analysis of Biomedical Data III, vol. 10337, SPIE, 2017. doi: 10.1117/12.2267959.
- D. S. Kulyabov, A. V. Korolkova, and T. R. Velieva, “The Riemannian Geometry is not Sufficient for the Geometrization of the Maxwell’s Equations,” in Proceedings of SPIE. Saratov Fall Meeting 2017: Laser Physics and Photonics XVIII; and Computational Biophysics and Analysis of Biomedical Data IV, vol. 10717, Saratov: SPIE, Apr. 2018. doi: 10.1117/12.2315204.
- D. S. Kulyabov, A. V. Korolkova, T. R. Velieva, and A. V. Demidova, “Finslerian representation of the Maxwell equations,” in Proceedings of SPIE. Saratov Fall Meeting 2018: Laser Physics, Photonic Technologies, and Molecular Modeling, vol. 11066, Saratov: SPIE, Jun. 2019. doi: 10.1117/12.2525534.
- D. V. Sivukhin, “The international system of physical units,” Soviet Physics Uspekhi, vol. 22, no. 10, pp. 834-836, Oct. 1979. doi: 10.1070/pu1979v022n10abeh005711.
- D. S. Kulyabov, A. V. Korolkova, and V. I. Korolkov, “Maxwell’s equations in arbitrary coordinate system,” Bulletin of Peoples’ Friendship University of Russia. Series “Mathematics. Information Sciences. Physics”, no. 1, pp. 96-106, 2012.
- L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Course of Theoretical Physics, 2nd. Butterworth-Heinemann, 1984, vol. 8, 460 pp.
- S. Bolioli, “Bi-Isotropic and Bi-Anisotropic Media,” in Advances in Complex Electromagnetic Materials. NATO ASI Series. Springer Netherlands, 1997, vol. 28, ch. 3, pp. 33-51. doi: 10.1007/978-94-011-5734-6_3.