This paper studies small radial disturbances of static, spherically symmetric black holes that carry “primary hair” in a class of modified gravity models. The models are a subfamily of degenerate higher-order scalar-tensor (DHOST) theories, which add a single extra scalar field to gravity. The authors show that the combined scalar-and-metric radial disturbance can be written as a simple flat radial wave equation. Under reasonable boundary conditions the associated operator can be made positive and self-adjoint, which implies the radial mode is stable.

To get this result the authors linearized the field equations around static black hole backgrounds where the scalar field has a mild time dependence of the form ϕ = q t + ψ(r). They focus on the monopole, or ℓ = 0, perturbation—this is the purely radial fluctuation that mixes the scalar and the metric. By fixing a gauge (setting the scalar perturbation to zero when q is nonzero) and introducing new time and radial coordinates, they transform the perturbation equation into a Schrödinger-like (flat wave) equation. Notably, the new time coordinate matches the scalar field itself (unitary gauge), and the new radial coordinate passes through the event horizon into the black hole interior. This behavior contrasts with the familiar “tortoise” coordinate in General Relativity, which maps the exterior to an infinite real line.

The work pays attention to solutions related by conformal or disformal transformations. These transformations change the metric in a way that depends on the scalar field and generate new solutions in related theories. The authors find that the same Schrödinger-like equation and the same new coordinates describe the radial mode for all solutions connected by such transformations. This invariance simplifies the stability discussion across a range of related models.