General Framework for the Spontaneous Scalarization of Regular Black Holes

TL;DR

The paper develops a general framework to study spontaneous scalarization of static, regular black holes sourced by nonlinear electrodynamics using the P-dual formalism to reconstruct the EM sector and a nonminimal scalar coupling with $f(\phi)=e^{-\alpha \phi^2}$. It shows that scalar-free and scalarized BHs can co-exist within a domain bounded by an existence line and a horizon-area constraint, with scalarized solutions being entropically preferred in that region, and provides a concrete worked example on Balart–Vagenas BHs. The method yields explicit scalarized configurations and reveals percent-level imprints on shadows and scalar QNMs, compatible with current EHT and GW observations, thereby offering a general route to test beyond-GR physics in strong-field regimes. The framework is readily extensible to other seed geometries and coupling choices, enabling broad exploration of scalarization in nonlinear electrodynamics-supported spacetimes and potential observational signatures in future data.

Abstract

We investigate the spontaneous scalarization of generic, static, and spherically symmetric regular black holes supported by nonlinear electrodynamics. Starting from an arbitrary seed metric, we employ the P-dual formalism to reconstruct the electromagnetic sector and subsequently couple a real scalar field nonminimally. As a worked example, we apply the framework to the regular Balart-Vagenas black hole, showing that scalarized and scalar-free branches can coexist in a region where the scalarized configurations are entropically preferred. We further assess possible observational imprints, finding percent-level deviations in both the shadow size and the fundamental scalar quasi-normal modes ($< 10\%$ for small charge-to-mass ratios), indicating that current electromagnetic and gravitational-wave observations do not rule out these solutions. Our construction thus provides a general route to explore scalarization on top of nonlinear-electrodynamics-supported spacetimes, extending beyond specific Reissner-Nordström-like cases.

Figures (6)

• Figure 1: Normalized scalar field, $\phi$, mass function, $m$, and lapse, $\delta$ (see figure legends) as a function of the radius for $Q=0.1$, $M=0.9$, $\alpha=-30$ and $r_H=0.3$.
• Figure 2: Domain of existence for spherical scalarized regular black holes as a function of $-\alpha$ and $q=\frac{Q}{m_{\infty}}$, where $m_\infty=m(r\to\infty)$.
• Figure 3: Virial identity as a function of the numerical step size. The results show that the virial identity converges to zero in the limit of vanishing step size.
• Figure 4: Normalized area, $a_H=\frac{A_H}{4\pi m_\infty}$, as a function of $q$, where $A_H=4\pi r_H^2$. The entropy is proportional to the area in the theories under consideration.
• Figure 5: Relative differences between the critical impact parameter for scalarized solutions relative to unscalarized ones, $\delta b_{\rm cr}$, as a function of charge, $Q$, and coupling constant, $\alpha$. Redder shades indicate greater departures.