# Instability of Reissner-Nordstr\"om-AdS black hole under perturbations of a scalar field coupled to Einstein tensor

## TL;DR

This paper investigates how a scalar field coupled to Einstein tensor can induce instability in Reissner-Nordström-AdS black holes, revealing a critical coupling strength that triggers instability and depends on the black hole's charge.

## Contribution

It demonstrates the instability mechanism caused by scalar-Einstein tensor coupling and identifies the critical coupling value related to black hole charge.

## Key findings

- Potential develops a negative well indicating instability
- Existence of a critical coupling triggering instability
- Critical coupling decreases as black hole charge approaches extremality

## Abstract

We study the instability of a Reissner-Nordstr\"om-AdS (RNAdS) black hole under perturbations of a massive scalar field coupled to Einstein tensor. Calculating the potential of the scalar perturbations we find that as the strength of the coupling of the scalar to Einstein tensor is increasing, the potential develops a negative well outside the black hole horizon, indicating an instability of the background RNAdS. We then investigate the effect of this coupling on the quasinormal modes. We find that there exists a critical value of the coupling which triggers the instability of the RNAdS. We also find that as the charge of the RNAdS is increased towards its extremal value, the critical value of the derivative coupling is decreased.