Linear stability of black holes with static scalar hair in full Horndeski theories: Generic instabilities and surviving models
Masato Minamitsuji, Kazufumi Takahashi, Shinji Tsujikawa
Abstract
In full Horndeski theories, we show that the static and spherically symmetric black hole (BH) solutions with a static scalar field $\ensuremath{\phi}$ 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 $\ensuremath{\phi}$ 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 ${R}_{\mathrm{GB}}^{2}$, 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 $F({R}_{\mathrm{GB}}^{2})$ gravity are plagued by ghost instabilities. These results imply that the GB coupling of the form $\ensuremath{\xi}(\ensuremath{\phi}){R}_{\mathrm{GB}}^{2}$ plays a prominent role for the existence of asymptotically Minkowski hairy BH solutions free from ghost/Laplacian instabilities.