The Riemannian geometry is not sufficient for the geometrization of the Maxwell's equations

TL;DR

This paper argues that quadratic Riemannian geometry is inadequate for geometrizing Maxwell's equations and suggests Finslerian geometry, specifically Berwald-Moor metric, as a more suitable alternative.

Contribution

The paper identifies the limitations of Riemannian geometry in representing Maxwell's constitutive tensor and proposes Finslerian geometry as a potential solution.

Findings

Riemannian metric components are fewer than constitutive tensor components

Riemannian geometry cannot fully describe Maxwell's constitutive relations

Finslerian geometry may provide a better structural match for electromagnetic fields

Abstract

The transformation optics uses geometrized Maxwell's constitutive equations to solve the inverse problem of optics, namely to solve the problem of finding the parameters of the medium along the paths of the electromagnetic field propagation. The quadratic Riemannian geometry is usually used for the geometrization of Maxwell's constitutive equations, because of the usage of the general relativity approaches. However, the problem of the insufficiency of the Riemannian structure for describing the constitutive tensor of the Maxwell's equations arises. The authors analyze the structure of the constitutive tensor and correlate it with the structure of the metric tensor of Riemannian geometry. It was concluded that the use of the quadratic metric for the geometrization of Maxwell's equations is insufficient, since the number of components of the metric tensor is less than the number of components of the constitutive tensor. The possible solution to this problem may be a transition to Finslerian geometry, in particular, the use of the Berwald-Moor metric to establish the structural correspondence between the field tensors of the electromagnetic field.