Geometrization of Maxwell's Equations in Extended Riemannian Geometry

Abstract

Different variants of Riemannian geometry can be classified according to three Schoutens: nonmetricity, torsion, difference between covariant and contravariant connections. Thus, 27 variants of the Riemannian space can be considered.
The Riemannian space of general relativity is degenerate, since all three schoutens are equal to zero.
The standard geometrization of Maxwell's electrodynamics uses the standard Riemannian space:
- the metric is consistent with covariant differentiation;
- the connection and metric of the given Riemannian space are symmetric;
- the Riemannian metric tensor has only 10 components.
This is clearly not enough to set the permittivity tensor. To set the permittivity tensor, 36 components are required. However, for the classical (non-quantum) field theory, one can limit oneself to only 20 components, discarding the remaining 16 as non-physical ones.
When geometrizing Maxwell's equations on the basis of such a Riemannian space, one has to implicitly impose some physical restrictions. Thus, in the general case of connections in the tangent and cotangent bundles of the electromagnetic field manifold, it is impossible to translate into each other by means of a symmetric metric tensor.
It seems to us that the problem of the geometrization of Maxwellian electrodynamics can be solved radically only with the help of Finsler geometry (for example, the biquadratic metric). However, it is of interest to consider all the possibilities of quadratic geometry.
The paper investigates the introduction of nontrivial schoutens in the geometrization of Maxwell's equations.

Speaker

Dmitry S. Kulyabov
RUDN University
Russia

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