On simulating a medium with special reflecting properties by Lobachevsky geometry (One exactly solvable electromagnetic problem)

TL;DR

This paper demonstrates how Lobachevsky geometry can simulate a medium with unique reflective properties, providing exact solutions to Maxwell's equations in curved space and revealing quantum-like behavior of electromagnetic waves.

Contribution

It presents the first exact solutions of Maxwell's equations in Lobachevsky space, linking curved geometry to electromagnetic reflection properties and quantum analogies.

Findings

Lobachevsky geometry acts as an ideal mirror for electromagnetic waves.

The penetration depth depends on wave parameters and curvature radius.

The problem reduces to a Schrödinger-like differential equation with an exponential potential.

Abstract

Lobachewsky geometry simulates a medium with special constitutive relations. The situation is specified in quasi-cartesian coordinates (x,y,z). Exact solutions of the Maxwell equations in complex 3-vector form, extended to curved space models within the tetrad formalism, have been found in Lobachevsky space. The problem reduces to a second order differential equation which can be associated with an 1-dimensional Schrodinger problem for a particle in external potential field U(z) = U_{0} e^{2z}. In quantum mechanics, curved geometry acts as an effective potential barrier with reflection coefficient R=1; in electrodynamic context results similar to quantum-mechanical ones arise: the Lobachevsky geometry simulates a medium that effectively acts as an ideal mirror. Penetration of the electromagnetic field into the effective medium, depends on the parameters of an electromagnetic wave, \omega, k_{1}^{2} + k_{2}^{2}, and the curvature radius \rho.