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

TL;DR

This work extends the master-function formalism for black-hole perturbations to Reissner–Nordström spacetimes with a cosmological constant by allowing master functions to be linear combinations of metric and electromagnetic perturbations and their derivatives, leading to a wave equation with an undetermined potential V(r). The authors derive a set of algebraic-differential constraints that yield a four-branch structure per parity: two standard branches that reduce to the Zerilli–Moncrief formalism and two Darboux branches with potentials fixed by nonlinear differential equations; within each parity, a Darboux transformation relates the branches, preserving the quasinormal-mode spectrum and establishing isospectrality between descriptions. The analysis provides explicit forms for the standard master functions and their Darboux counterparts, and demonstrates how the Darboux transformation maps between different but physically equivalent potential systems, extending the Lenzi–Sopuerta program to charged black holes with Λ. These results offer a unified, gauge-robust framework for black-hole perturbations in realistic environments and may impact waveform modeling in strong-field gravity.

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.