Linear stability of black holes in shift-symmetric Horndeski theories
  with a time-independent scalar field
  We study linear perturbations about static and spherically symmetric black
holes with a time-independent background scalar field in shift-symmetric
Horndeski theories, whose Lagrangian is characterized by coupling functions
depending only on the kinetic term of the scalar field $X$. We clarify
conditions for the absence of ghosts and Laplacian instabilities along the
radial and angular directions in both odd- and even-parity perturbations. For
reflection-symmetric theories described by a k-essence Lagrangian and a
nonminimal derivative coupling with the Ricci scalar, we show that black holes
endowed with nontrivial scalar hair are unstable around the horizon in general.
This includes nonasymptotically flat black holes known to exist when the
nonminimal derivative coupling to the Ricci scalar is a linear function of $X$.
We also investigate several black hole solutions in nonreflection-symmetric
theories. For cubic Galileons with the Einstein-Hilbert term, there exists a
nonasymptotically flat hairy black hole with no ghosts/Laplacian instabilities.
Also, for the scalar field linearly coupled to the Gauss-Bonnet term,
asymptotically flat black hole solutions constructed perturbatively with
respect to a small coupling are free of ghosts/Laplacian instabilities.
