Maxwell Equations in Complex Form of Majorana - Oppenheimer, Solutions with Cylindric Symmetry in Riemann S_{3} and Lobachevsky H_{3} Spaces

TL;DR

This paper extends the complex formalism of Maxwell equations to arbitrary pseudo-Riemannian spaces, solving them exactly in cosmological models with cylindrical symmetry and analyzing the resulting electromagnetic mode spectra.

Contribution

It provides exact solutions for Maxwell equations in curved spaces with cylindrical symmetry, including quantized spectra in Einstein models and complete solutions in Lobachevsky space.

Findings

Discrete frequency spectrum in Einstein model depends on curvature and parameters

Complete basis of solutions in Lobachevsky space without frequency quantization

Part of solutions in elliptical model are rejected by continuity

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 static cosmological Einstein model, parameterized by special cylindrical coordinates and realized as a Riemann space of constant positive curvature. A discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three parameters is found, and corresponding basis electromagnetic solutions have been constructed explicitly. In the case of elliptical model a part of the constructed solutions should be rejected by continuity considerations. Similar treatment is given for Maxwell equations in hyperbolic Lobachevsky model, the complete basis of electromagnetic solutions in corresponding cylindrical coordinates has been constructed as well, no quantization of frequencies of electromagnetic modes arises.