Published in The European Physical Journal C as a regular article in Theoretical Physics, the work examines the Bergmann-Wagoner-Nordtvedt class of scalar-tensor theories coupled to nonlinear electromagnetic fields through a Lagrangian of the form L(ψ,F), where ψ is a scalar field and F is the electromagnetic invariant. The framework permits arbitrary scalar self-interaction potentials and scalar fields that are canonical, phantom, or partly phantom in different regions of spacetime. By working in the Einstein conformal frame, the researchers derived a single Schrödinger-like equation for radial perturbation modes, together with a general effective potential intended to determine their behavior.

It incorporates the scalar field’s kinetic function, the electromagnetic Lagrangian and its derivatives, background metric functions, electric and magnetic charges, and the spatial variation of the scalar field. After separating a perturbation into a time-dependent factor and a radial mode, the master equation takes the form Yzz + [Ω² − Veff(z)]Y = 0, where z is a tortoise-like radial coordinate and Ω is the mode frequency. A nontrivial solution with Im Ω less than or equal to zero that satisfies physically reasonable boundary conditions, including the absence of ingoing waves at the ends of the coordinate range, indicates instability.

If no such solution exists, the system is stable in the linear approximation. Stability therefore has to be assessed as a boundary-value problem with conditions appropriate to the background space-time. A striking result emerges when the nonlinear electromagnetic Lagrangian has the correct Maxwell limit at weak fields.

In the zero-charge limit, every trace of the nonlinear theory disappears from the effective potential, so the perturbation dynamics coincide with those of the corresponding vacuum scalar-tensor solution. This conclusion also holds for the exceptional Brans-Dicke wormhole studied by the authors: it remains stable under the radial perturbations considered when it carries nonzero electric charge, magnetic charge, or both, but is unstable in the uncharged, symmetric case. The finding means that nonlinear electrodynamics, under the assumptions examined, does not remove the instability of that neutral configuration.

The same result applies to the family of scalar-tensor electrovacuum configurations in which the electromagnetic invariant vanishes everywhere and the electric and magnetic charge magnitudes are equal. Under a Maxwell-like weak-field condition, L(F) and its first derivative at F = 0 reduce to the corresponding Maxwell values. The electromagnetic stress-energy tensor and the effective potential governing monopole perturbations are then identical to those of Maxwell-coupled scalar-tensor gravity.

Consequently, established stability results for these Maxwell configurations extend directly to their nonlinear-electrodynamics counterparts without modification. This equivalence is an analytical result under the stated conditions rather than a numerical approximation. The authors also considered cases in which the nonlinear electromagnetic field is not trivial.

Their worked example lies within general relativity rather than a nonminimally coupled scalar-tensor theory: a massless scalar field and truncated Born-Infeld electrodynamics are minimally coupled to gravity. The Born-Infeld Lagrangian includes a parameter b that controls the departure from Maxwell behavior, while the scalar field’s radial flux supplies an integration constant. The researchers numerically solved the coupled background equations and then evaluated the radial effective potential.

The study was conducted in units with the Schwarzschild mass fixed at m = 1, and the numerical integration began at a large radius using asymptotic expansions whose neglected terms were estimated to be no greater than 10^−12. For these example configurations, the nonlinear theory changes the central matter distribution without erasing the characteristic instability inherited from the corresponding scalar-Maxwell system. With only electric charge or only magnetic charge, the electromagnetic stress-energy diverges near the center more slowly than in Maxwell electrodynamics, and the background develops a scalar-type naked singularity.

When both charge types are present but unequal, the balance between scalar and electromagnetic contributions produces regions with either scalar-type or Reissner-Nordström-type singularities. The division between the two depends on the Born-Infeld parameter and the total charge, while the results remain symmetric under exchanging electric and magnetic charge. The equal-charge case has F = 0 and coincides with the previously described Penney solution.

Numerically, the effective potential approaches zero at large radius but develops an infinitely deep well close to the center. For scalar-type singularities it approaches −1/(4z²), while for Reissner-Nordström-type singularities it approaches −2/(9z²). These are the same leading forms found for the scalar-Maxwell configurations.

Changing the Born-Infeld parameter altered the potential only slightly: at one representative radius, the reported value shifted from −2.049 in the Maxwell limit to −2.072 at b = 10, a difference of about one percent. The authors therefore interpret the Born-Infeld result as a quantitative refinement rather than a qualitative change in the radial stability properties. The implication is that radial perturbations in the studied nonlinear-electrodynamics configurations can exhibit the same catastrophic linear instability as their Maxwell counterparts, even when the initial perturbations are small.

Such growth makes a linear calculation alone inadequate for describing the subsequent evolution and requires a nonperturbative treatment. These results do not amount to a universal verdict on nonlinear-electrodynamics black holes or other compact objects: the analysis is restricted to static spherical backgrounds and radial perturbations, and its Born-Infeld example is a singular general-relativistic system rather than a broad survey of scalar-tensor black holes. The authors identify quasinormal modes, shadows and other observational properties as subjects for subsequent work.

Subject of Research: Linear radial stability of scalar-tensor gravity coupled to nonlinear electrodynamics

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

Article References: Bronnikov, K. A., Ibadov, R., Shaymanova, F. Y., & Shukurullokhon, N. M. (2026). Nonlinear electrodynamics and stability of spherically symmetric space-times in scalar-tensor gravity. The European Physical Journal C, 86(10), Article 1190. https://doi.org/10.1140/epjc/s10052-026-16469-3

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DOI: 10.1140/epjc/s10052-026-16469-3

Keywords: scalar-tensor gravity, nonlinear electrodynamics, radial perturbations, linear stability, effective potential, Born-Infeld electrodynamics, Bergmann-Wagoner-Nordtvedt theory, Brans-Dicke gravity, spherical symmetry, gravitational instability, Nonlinear, electrodynamics