Existence and instability of hairy black holes in shift-symmetric Horndeski theories

Shift-symmetric Horndeski theories admit an interesting class of Schwarzschild black hole solutions exhibiting time-dependent scalar hair. By making use of Lemaitre co-ordinates, we analyze perturbations around these types of black holes, and demonstrate that scalar perturbations around black hole backgrounds inevitably suffer from instabilities. Specifically, we demonstrate the existence of a tachyonic instability in dipole perturbations of the even-sector. We also offer evidence (for example, in the decoupling limit) that the instability may persist at higher multipoles as well, indicating that the instability is a gradient instability. Taken together with previously established results, this newly-discovered instability casts doubt on the theoretical viability black holes with time-dependent scalar hair in Horndeski theories.