Critical phenomenon inside asymptotically flat black holes with spontaneous scalarization

We study the interior dynamics of spontaneously scalarized black holes in Einstein-Maxwell-Scalar theory with zero cosmological constant, revealing novel critical phenomena. We demonstrate that, for a wide range of scalar-electromagnetic couplings, scalarized black holes possess no smooth inner Cauchy horizon and instead evolve into a spacelike Kasner singularity. The scalar hair triggers a rapid collapse of the Einstein-Rosen bridge at the would-be Cauchy horizon. Near the critical point where scalarized black holes bifurcate from the Reissner-Nordstrom solution, we establish a robust scaling relation between the Kasner parameter and the charge-to-mass ratio of the hairy black hole, opening a new window into the remarkable simplicity underlying black hole interiors.

💡 Research Summary

The paper investigates the interior dynamics of black holes that undergo spontaneous scalarization in Einstein‑Maxwell‑Scalar (EMS) theory with zero cosmological constant. In this theory a massless real scalar field (\psi) couples non‑minimally to the Maxwell invariant through a function (Z(\psi)). The authors choose (Z(0)=1) and (\left.\frac{dZ}{d\psi}\right|_{\psi=0}=0) so that the Reissner‑Nordström (RN) solution remains a solution when the scalar vanishes. They consider static, spherically symmetric, purely electric configurations and adopt a compact radial coordinate (z=1/r) that maps the asymptotically flat region to (z=0) and the event horizon to (z=z_H).

A central question is whether the scalarized black holes retain an inner Cauchy horizon like the RN geometry. By integrating the scalar field equation between the event horizon and a hypothetical inner horizon (z_I) the authors derive an identity that contains the term (\frac{dZ}{d\psi}). For any coupling satisfying (\frac{dZ}{d\psi}>0) in the relevant range, the integrand becomes non‑positive, making the identity impossible unless the inner horizon is absent. Consequently, scalar hair inevitably destroys the Cauchy horizon and forces the spacetime to terminate at a spacelike singularity as (z\to\infty). This analytic argument holds for a broad class of coupling functions, including the two representative models studied numerically: (Z(\psi)=e^{\alpha^2\psi^2}) and (Z(\psi)=1+\psi^2/(1+\psi^2)).

The interior evolution is then followed numerically. Near the singularity the fields settle into a Kasner‑type regime: \