Maxwell Equations in Complex Form, Spherical Waves in Spaces of Constant Curvature of Lobachevsky and Riemann

TL;DR

This paper extends the complex formalism of Maxwell equations to arbitrary pseudo-Riemannian spaces, solving them exactly in static cosmological models with constant curvature, and analyzes the spectrum of electromagnetic modes in these geometries.

Contribution

It generalizes Maxwell equations in complex form to curved spaces and provides exact solutions in models of constant curvature, revealing spectral properties of electromagnetic modes.

Findings

Discrete frequency spectrum in Riemann model depends on curvature radius

No discrete spectrum in Lobachevsky (hyperbolic) model

Exact solutions using separation of variables and Wigner D-functions

Abstract

Complex formalism of Riemann - Silberstein - Majorana - Oppenheimer in Maxwell electrodynamics is extended to the case of arbitrary pseudo-Riemannian space - time in accordance with the tetrad recipe of Tetrode - Weyl - Fock - Ivanenko. In this approach, the Maxwell equations are solved exactly on the background of simplest static cosmological models, spaces of constant curvature of Riemann and Lobachevsky parameterized by spherical coordinates. Separation of variables is realized in the basis of Schr\"odinger -- Pauli type, description of angular dependence in electromagnetic complex 3-vectors is given in terms of Wigner D-functions. In the case of compact Riemann model a discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three discrete parameters is found. In the case of hyperbolic Lobachevsky model no discrete spectrum for frequencies of electromagnetic modes arises.