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

Abstract

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

governing the perturbation dynamics for theories with arbitrary scalar-electromagnetic interaction of the form

$$L(\psi , F)$$ L ( ψ , F )

, where

$$\psi $$ ψ

is a scalar field and

$$F = F_{\mu \nu }F^{\mu \nu }$$ F = F μ ν F μ ν

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

$$V_\textrm{eff}$$ V 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

$$F =0$$ F = 0

. As an example of a solution with

$$F\not \equiv 0$$ F ≢ 0

, we consider a system with a massless scalar and truncated Born-Infeld NED, both minimally coupled to gravity, in GR, and show that almost all results previously obtained for

$$F=0$$ F = 0

are qualitatively preserved.