On the wave equation in spacetimes of Goedel type

TL;DR

This paper investigates the solutions of the wave equation in G"odel-type spacetimes with spherical and Lobachevsky geometries, revealing discrete and continuous spectra influenced by the spacetime's rotation and symmetry properties.

Contribution

It provides a group-theoretical framework for solving the wave equation in G"odel-type spacetimes with explicit formulas for solutions in both spherical and Lobachevsky cases.

Findings

Solutions have discrete frequencies in spherical case.

Formulas for solutions in Lobachevsky case are derived.

Restrictions on spectrum due to wave equation are identified.

Abstract

We analyze the d'Alembert equation in the Goedel-type spacetimes with spherical and Lobachevsky sections (with sufficiently rapid rotation). By separating the $t$ and $x_3$ dependence we reduce the problem to a group-theoretical one. In the spherical case solutions have discrete frequencies, and involve spin-weighted spherical harmonics. In the Lobachevsky case we give simple formulas for obtaining all the solutions belonging to the $D^\pm_\la$ sectors of the irreducible unitary representations of the reduced Lorentz group. The wave equation enforces restrictions on $\la$ and the allowed (here: continuous) spectrum of frequencies.