Linear stability of black holes with static scalar hair in full Horndeski theories: generic instabilities and surviving models

TL;DR

This paper investigates the stability of black holes with scalar hair in Horndeski theories, finding that certain couplings, especially with the Gauss-Bonnet term, enable stable hairy black hole solutions while others lead to instabilities.

Contribution

It identifies conditions under which asymptotically Minkowski hairy black holes are stable in Horndeski theories, highlighting the critical role of Gauss-Bonnet couplings.

Findings

Generic static scalar hair black holes are unstable due to ghost/Laplacian instabilities.

Gauss-Bonnet coupling enables stable asymptotically Minkowski hairy black holes.

Power-law F(R_GB^2) models are unstable due to ghost instabilities.

Abstract

In full Horndeski theories, we show that the static and spherically symmetric black hole (BH) solutions with a static scalar field~$\phi$ whose kinetic term~$X$ is nonvanishing on the BH horizon are generically prone to ghost/Laplacian instabilities. We then search for asymptotically Minkowski hairy BH solutions with a vanishing $X$ on the horizon free from ghost/Laplacian instabilities. We show that models with regular coupling functions of $\phi$ and $X$ result in no-hair Schwarzschild BHs in general. On the other hand, the presence of a coupling between the scalar field and the Gauss-Bonnet (GB) term $R_{\rm GB}^2$, even with the coexistence of other regular coupling functions, leads to the realization of asymptotically Minkowski hairy BH solutions without ghost/Laplacian instabilities. Finally, we find that hairy BH solutions in power-law $F(R_{\rm GB}^2)$ gravity are plagued by ghost instabilities. These results imply that the GB coupling of the form $\xi(\phi)R_{\rm GB}^2$ plays a prominent role for the existence of asymptotically Minkowski hairy BH solutions free from ghost/Laplacian instabilities.