Master functions of Reissner-Nordstrom black hole perturbations and their Darboux transformation

TL;DR

This paper extends a method for black hole perturbations to charged black holes with cosmological constant, identifying four master functions and a Darboux transformation linking different potential formulations.

Contribution

It introduces a generalized approach to construct master functions for Einstein-Maxwell perturbations, revealing new Darboux branches and transformations.

Findings

Identified four master function branches per parity sector.

Established Darboux transformations linking standard and Darboux branches.

Confirmed the physical equivalence of different potential formulations.

Abstract

Lenzi and Sopuerta developed a new method to construct master functions for the perturbation of vacuum black holes. We extend this method to black holes coupled with electromagnetic field and cosmological constant by allowing the master functions to be linear combinations of the metric and electromagnetic-potential perturbations, as well as their first-order derivatives. Requiring these master functions satisfy wave equations with yet-to-be-determined effective potentials, we reduce the linearized Einstein--Maxwell system to a set of algebraic-differential constraints. Solving these constraints reveals four master function branches in each parity sector: two standard branches, which coincide with the Zerilli-Moncrief formalism, and two Darboux branches, characterized by their effective potentials. Within each parity sector, a Darboux transformation exists which connects the standard and Darboux branches, preserving the quasinormal mode spectrum and confirming their physical equivalence.