Maxwell equations in Riemannian space-time, geometry effect on material equations in media

TL;DR

This paper explores how the geometry of curved space-time influences electromagnetic material equations by deriving effective constitutive tensors from Riemannian metrics, with applications to anisotropic media and moving frames.

Contribution

It explicitly derives the form of constitutive tensors in Riemannian space-time and demonstrates their application to modeling anisotropic media and media in motion.

Findings

Derived explicit forms of four constitutive tensors in Riemannian space-time.

Showed geometrical modeling of anisotropic media like magnetic crystals.

Analyzed effective material equations in constant curvature spaces.

Abstract

The known possibility to consider the (vacuum) Maxwell equations in a curved space-time as Maxwell equations in flat space-time(Mandel'stam L.I., Tamm I.E.) taken in an effective media the properties of which are determined by metrical structure of the curved model is studied. Metrical structure of the curved space-time generates effective constitutive equations for electromagnetic fields, the form of four corresponding symmetrical tensors is found explicitly for general case of an arbitrary Riemannian space - time. Four constitutive tensors are not independent and obey some additional constraints between them. Several simple examples are specified in detail:itis given geometrical modeling of the anisotropic media (magnetic crystals) and the geometrical modeling of a uniform media in moving reference frame in the Minkowsky 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.