# Spontaneous scalarization of charged black holes at the approach to extremality

## TL;DR

This paper investigates how charged black holes can spontaneously develop scalar fields near extremality due to tachyonic instabilities, revealing new parameter domains influenced by electric charge and near-horizon geometry.

## Contribution

It demonstrates the existence of a new domain for scalarization at negative coupling values enabled by the near-horizon $AdS_2$ geometry of nearly extremal charged black holes.

## Key findings

- Scalarization occurs for large scalar-tensor coupling $oldsymbol{ extgamma}$.
- Two domains of scalarization in the $oldsymbol{ extgamma}$-$oldsymbol{Q}$ plane for charged black holes.
- A new negative-$oldsymbol{ extgamma}$ domain appears for charges above 74 ext{\%} of extremality.

## Abstract

We study static, spherically symmetric and electrically charged black hole solutions in a quadratic Einstein-scalar-Gauss-Bonnet gravity model. Very similar to the uncharged case, black holes undergo spontaneous scalarization for sufficiently large scalar-tensor coupling $\gamma$ - a phenomenon attributed to a tachyonic instability of the scalar field system. While in the uncharged case, this effect is only possible for positive values of $\gamma$, we show that for sufficiently large values of the electric charge $Q$ two independent domains of existence in the $\gamma$-$Q$-plane appear: one for positive $\gamma$ and one for negative $\gamma$. We demonstrate that this new domain for negative $\gamma$ exists because of the fact that the near-horizon geometry of a nearly extremally charged black hole is $AdS_2\times S^2$.This new domain appears for electric charges larger than approximately 74$\%$ of the extremal charge. For positive $\gamma$ we observe that a singularity with diverging curvature invariants forms outside the horizon when approaching extremality.