The scalar angular Teukolsky equation and its solution for the Taub-NUT spacetime

TL;DR

This paper analyzes the scalar angular Teukolsky equation in the Taub-NUT spacetime, deriving eigenvalues and solutions, to facilitate analytical studies of wave scattering and imaging of Taub-NUT black holes.

Contribution

It provides a detailed solution to the angular Teukolsky equation in Taub-NUT spacetime using confluent Heun functions, enabling analytical wave scattering analysis.

Findings

Eigenvalues derived for the angular Teukolsky equation.

Solutions expressed in terms of confluent Heun functions.

Re-derivation of the Misner condition for specific parameters.

Abstract

The Taub-NUT spacetime offers many curious insights into the solutions of Einstein's electrovacuum equation. In the Bonnor interpretation, this spacetime possesses so-called Misner strings, which induce phenomena strikingly analogous to Dirac strings in the context of magnetic monopoles. The study of scattering in the latter case leads to a quantization of the product of electric charge and magnetic moment, sometimes called the Dirac condition. To enable a thorough discussion of scattering on the Taub-NUT spacetime, linear perturbations are considered in the Newman-Penrose formalism and separated into angular and radial equations. The angular Teukolsky equation is discussed in detail, and eigenvalues are derived to subsequently solve the differential equation in terms of solutions to the confluent Heun equation. In the Bonnor interpretation of the Taub-NUT spacetime, there is no analog property to the Dirac condition. The choice of spacetime parameters remains unconstrained. However, for a particular parameter choice, one can rederive the well-known \enquote{Misner} condition, in which a product of frequency and NUT charge is of integer value, as well as another product additionally including the Manko-Ruiz parameter. The results of this work will allow us to solve analytically for wave-optical scattering in order to, e.g., examine the wave-optical image of Taub-NUT black holes.