Radial Perturbations of Black Holes in DHOST Theories

We study radial perturbations of static black holes with primary hair in a subfamily of degenerate higher-order scalar-tensor (DHOST) theories. We recast the equation of motion for the monopole degree of freedom into a flat radial wave equation and show that the associated operator can be extended, through appropriate boundary conditions, to a positive self-adjoint operator which ensures the stability of the radial mode. Remarkably, the coordinate choice that leads to the flat wave equation corresponds to the unitary gauge, in which the scalar field is uniform. As a result, the radial coordinate extends beyond the event horizon, into the black hole interior, in contrast with the tortoise coordinate in General Relativity. The same wave equation with the same coordinate choice applies to all solutions that are connected by disformal transformations. We also examine stealth black hole solutions, with either a constant or non constant kinetic term. In the former case, we find, to linear order, the absence of a propagating degree of freedom. In the latter case, we identify a stable radial degree of freedom, except for special values of the theory coupling constants.