Nonlinear electrodynamics and stability of spherically symmetric space-times in scalar-tensor gravity

We study linear perturbations of static, spherically symmetric solutions of scalar-tensor theories (STT) of gravity from the Bergmann-Wagoner-Nordtvedt class, sourced by nonlinear electrodynamics (NED). We obtain a general expression for the effective potential $V_{\rm eff}$ governing the perturbation dynamics for theories with arbitrary scalar-electromagnetic interaction of the form $L(\psi, F)$, where $\psi$ is a scalar field and $F = F_{\mu\nu} F^{\mu\nu}$ the electromagnetic invariant. This consideration includes, in particular, arbitrary scalar self-interaction potentials and scalar fields that can be phantom in some regions of space-time (the so-called trapped ghosts). Only radial (monopole) perturbations are considered here as the most likely ones to cause an instability. It is shown, in particular, that if NED has a correct Maxwell weak field limit, the zero charge limit of $V_{\rm eff}$ does not contain any trace of NED, and the perturbation dynamics is the same as for vacuum STT solutions. The previously obtained stability results for STT-Maxwell solutions are shown to be extended without change to STT-NED solutions with equal electric and magnetic charges, implying $F =0$.