Harmonic Analysis of Linear Fields on the Nilgeometric Cosmological Model

TL;DR

This paper develops harmonic functions for a specific cosmological model based on the Bianchi II group, enabling the reduction of complex field equations to simpler ordinary differential equations.

Contribution

It provides a complete set of harmonics for the nilgeometric Bianchi II model, including vector harmonics, and demonstrates their application to field equations.

Findings

Harmonics explicitly derived for the Bianchi II group

Reduction of Klein-Gordon and Maxwell equations to ODEs

Asymptotic solutions obtained for a special case

Abstract

To analyze linear field equations on a locally homogeneous spacetime by means of separation of variables, it is necessary to set up appropriate harmonics according to its symmetry group. In this paper, the harmonics are presented for a spatially compactified Bianchi II cosmological model -- the nilgeometric model. Based on the group structure of the Bianchi II group (also known as the Heisenberg group) and the compactified spatial topology, the irreducible differential regular representations and the multiplicity of each irreducible representation, as well as the explicit form of the harmonics are all completely determined. They are also extended to vector harmonics. It is demonstrated that the Klein-Gordon and Maxwell equations actually reduce to systems of ODEs, with an asymptotic solution for a special case.