Nonlinear Phenomena in Complex Systems, 2009, Vol.12, No.3, pp.232-250

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2009, Vol.12, No.3, pp.232-250

Maxwell Equations in Riemannian Space-Time, Geometry Effect on Constitutive Relations in Media.
V. M. Red'kov, N. G. Tokarevskaya, E. M. Bychkouskaya, and George J. Spix

In the paper, the known possibility to consider the (vacuum) Maxwell equations in a curved space-time as Maxwell equations in flat space-time (Gordon W., Mandel'stam L.I., Tamm I.E.) as taken in an effective media the properties of which are determined by metrical structure of the initial curved model is studied

is 4-rank tensor; metrical structure of the curved space-time generates effective constitutive equations for electromagnetic fields:

the form of four symmetrical tensors is found explicitly for general case of an arbitrary Riemannian space-time geometry . The main peculiarity of the geometrical generating for effective electromagnetic medias characteristics consists in the following: four tensors are not independent and obey some additional constraints between them. Several, the most simple examples are specified in detail: it is given geometrical modeling of the anisotropic media (magnetic crystals) and the geometrical modeling of a uniform media in moving reference frame in the background of Minkowski electrodynamics - the latter is realized trough the use of a non-diagonal metrical tensor determined by 4-vector velocity of the moving uniform media . Also the effective material equations generated by geometry of space of constant curvature (Lobachevsky and Riemann models) are determined. General problem of geometrical transforming arbitrary (linear) material equations, given by , has been studied - corresponding formulas have been produced.
Key words: Maxwell equations, media, constitutive relations, Riemannian geometry

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