# Nonlinear perturbations of Reissner-Nordstr\"om black holes

## TL;DR

This paper develops a nonlinear perturbation theory for Reissner-Nordström black holes, reducing Einstein-Maxwell equations to wave equations with sources, and constructs solutions for various angular momentum modes.

## Contribution

It introduces a systematic nonlinear perturbation framework for charged black holes, extending classical linear theories to include source-driven wave equations and mode-specific analyses.

## Key findings

- Reduction of Einstein-Maxwell equations to four wave equations at each perturbation level.
- Explicit construction of solutions for inhomogeneous wave equations with sources.
- Separate analysis of $ ext{l}=0$ and $ ext{l}=1$ perturbation modes.

## Abstract

We develop a nonlinear perturbation theory of Reissner-Nordstr\"om black holes. We show that, at each perturbation level, Einstein-Maxwell equations can be reduced to four inhomogeneous wave equations, two for polar and two for axial sector. Gravitational part of these equations is similar to Regge-Wheeler and Zerilli equations with source and additional coupling to the electromagnetic sector. We construct solutions to the inhomogeneous part of wave equations in terms of sources for Einstein-Maxwell equations. We discuss $\ell=0$ and $ \ell=1$ cases separately.