# Stability of scalarized charged black holes in the Einstein-Maxwell-Scalar theory

## TL;DR

This paper investigates the stability of scalarized charged black holes in Einstein-Maxwell-Scalar theory, finding the fundamental black hole stable while excited states are unstable, and discusses implications for black hole evolution.

## Contribution

It provides a stability analysis of scalarized charged black holes in EMS theory with quadratic coupling, highlighting differences from other theories and identifying the endpoint of unstable black holes.

## Key findings

- The fundamental black hole ($n=0$) is stable against perturbations.
- Excited black holes ($n=1,2$) are unstable against scalar perturbations.
- Unstable Reissner-Nordström black holes evolve into the stable $n=0$ black hole.

## Abstract

We analyze the stability of scalarized charged black holes in the Einstein-Maxwell-Scalar (EMS) theory with quadratic coupling. These black holes are labelled by the number of $n=0,1,2,\cdots$, where $n=0$ is called the fundamental black hole and $n=1,2,\cdots$ denote the $n$-excited black holes. We show that the $n=0$ black hole is stable against full perturbations, whereas the $n=1,2$ excited black holes are unstable against the $s(l=0)$-mode scalar perturbation. This is consistent with the EMS theory with exponential coupling, but it contrasts to the $n=0$ scalarized black hole in the Einstein-Gauss-Bonnet-Scalar theory with quadratic coupling. This implies that the endpoint of unstable Reissner-Nordstr\"{o}m black holes with $\alpha>8.019$ is the $n=0$ black hole with the same $q$. Furthermore, we study the scalarized charged black holes in the EMS theory with scalar mass $m^2_\phi=\alpha/\beta$.