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 Lemaître coordinates, 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.