The final fate of instability of Reissner-Nordstr\"om-anti-de Sitter black holes by charged complex scalar fields

TL;DR

This paper studies the instability of Reissner-Nordström-anti-de Sitter black holes under charged scalar perturbations, demonstrating that below a critical temperature they become unstable and evolve into black holes with scalar hair, which have higher entropy.

Contribution

It provides a numerical and analytical analysis of the instability and constructs static hairy black hole solutions near the critical temperature, proposing the hairy black hole as the final state.

Findings

Black holes become unstable below a critical temperature.

Charge extraction occurs during the unstable evolution.

Hairy black holes have higher entropy than unstable RN-AdS black holes.

Abstract

We investigate instability of 4-dimensional Reissner-Nordstr\"om-anti-de Sitter (RN-AdS$_4$) black holes with various topologies by charged scalar field perturbations. We numerically find that the RN-AdS$_4$ black holes become unstable against the linear perturbations below a critical temperature. It is analytically shown that charge extraction from the black holes occurs during the unstable evolution. To explore the end state of the instability, we perturbatively construct static black hole solutions with the scalar hair near the critical temperature. It is numerically found that the entropy of the hairly black hole is always larger than the one of the unstable RN-AdS$_4$ black hole in the microcanonical ensemble. Our results support the speculation that the black hole with charged scalar hair always appears as the final fate of the instability of the RN-AdS$_4$ black hole.