Constitutive tensor in the geometrized Maxwell theory

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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 (
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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

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