Dipole perturbations of the Reissner-Nordstrom solution. II. The axial case

TL;DR

This paper investigates axial dipole perturbations of the Reissner-Nordstrom black hole, identifies limitations of existing methods for dipole modes, and introduces new functions to extend the analysis to all multipoles.

Contribution

It develops a generalized perturbation formalism that includes dipole modes by defining new radial functions, completing the analysis of axial perturbations.

Findings

Standard analysis fails for dipole perturbations.

New radial functions enable a complete perturbation solution.

Perturbed metric and Maxwell tensors are explicitly constructed.

Abstract

We study the linear metric perturbations of the Reissner-Nordstr\"{o}m solution for the case of axial perturbation modes. We find that the well-known perturbative analysis fails for the case of dipole $(l=1)$ perturbations, although valid for higher multipoles. We define new radial functions, with which the perturbation formalism is generalized to all multipole orders, including the dipole. We then complete the solution by constructing the perturbed metric and Maxwell tensors.