# Black holes in Gauss-Bonnet and Chern-Simons-scalar theory

## TL;DR

This paper analyzes the stability of Schwarzschild black holes in Gauss-Bonnet and Chern-Simons-scalar theories, finding instability and scalarization for one scalar coupling but stability for the other.

## Contribution

It introduces two quadratic scalar couplings to these theories and investigates their effects on black hole stability, revealing conditions for scalarization.

## Key findings

- Schwarzschild black hole unstable against φ₁ perturbation
- Black hole stable against φ₂ and metric perturbations
- Scalarized black holes form due to φ₁ instability

## Abstract

We carry out the stability analysis of the Schwarzschild black hole in Gauss-Bonnet and Chern-Simons-scalar theory. Here, we introduce two quadratic scalar couplings ($\phi_1^2,\phi_2^2$) to Gauss-Bonnet and Chern-Simons terms, where the former term is parity-even, while the latter one is parity-odd. The perturbation equation for the scalar $\phi_1$ is the Klein-Gordon equation with an effective mass, while the perturbation equation for $\phi_2$ is coupled to the parity-odd metric perturbation, providing a system of two coupled equations. It turns out that the Schwarzschild black hole is unstable against $\phi_1$ perturbation, leading to scalarized black holes, while the black hole is stable against $\phi_2$ and metric perturbations, implying no scalarized black holes.