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

Understand

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.