Coupled gravitational and electromagnetic perturbations of Reissner-Nordstr\"om spacetime in a polarized setting

TL;DR

This paper derives a coupled system of equations for gravitational and electromagnetic perturbations of Reissner-Nordström spacetime, extending stability analysis methods from Schwarzschild to charged black holes in a polarized setting.

Contribution

It introduces a gauge-invariant electromagnetic quantity satisfying a Regge-Wheeler equation, generalizing recent stability work to charged black holes with polarization.

Findings

Derived coupled perturbation equations for Reissner-Nordström spacetime.

Established a gauge-invariant electromagnetic quantity obeying a Regge-Wheeler type equation.

Set the foundation for future stability analysis of charged black holes.

Abstract

We derive a system of equations governing the coupled gravitational and electromagnetic perturbations of Reissner-Nordstr\"om spacetime. The equations are derived in the context of global non-linear stability of Reissner-Nordstr\"om under axially symmetric polarized perturbations, as a generalization of the recent work on non-linear stability of Schwarzschild spacetime of Klainerman-Szeftel. The main result consists in deriving, through a Chandrasekhar-type transformation, a gauge invariant quantity associated to the electromagnetic tensor that verifies a Regge-Wheeler equation. In this paper, we present the derivation of the main equations, postponing the estimates of the non-linear system and the construction of the dynamical spacetime to a later time.