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

Abstract

In full Horndeski theories, we show that the static and spherically symmetric black hole (BH) solutions with a static scalar field~$\varphi$ 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 $\varphi$ 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 $\xi (\varphi )R_{\mathrm{GB}}^2$ plays a prominent role for the existence of asymptotically Minkowski hairy BH solutions free from ghost/Laplacian instabilities.

29 pages, no figures

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