Critical phenomenon inside asymptotically flat black holes with spontaneous scalarization

TL;DR

We address the interior dynamics of spontaneously scalarized black holes in four-dimensional Einstein-Maxwell-scalar theory with $\Lambda=0$. Using static, spherically symmetric backgrounds and a family of scalar–electromagnetic couplings $Z(\psi)$, we show there is no smooth inner horizon and the interior collapses to a Kasner singularity, with an ER‑bridge collapse triggered by the scalar hair. Near the critical point $q_c$ where hairy solutions bifurcate from RN, we uncover a universal scaling law $\beta = c_0\left(\frac{q}{q_c}-1\right)^{\gamma}$ with $\gamma=-\tfrac{1}{2}$ and derive the Kasner exponents $p_t=\frac{\beta^2-1}{\beta^2+3}$, $p_s=\frac{2}{\beta^2+3}$, $p_\psi=\frac{2\sqrt{2}\,\beta}{\beta^2+3}$, which satisfy $p_t+2p_s=1$ and $p_t^2+2p_s^2+p_\psi^2=1$. We also relate the interior parameter to exterior observables, e.g., the photon-sphere radius obeys $r_{\mathrm{ph}}(q)-r_{\mathrm{ph}}(q_c)\sim (q-q_c)$, and discuss implications for probing black hole interiors via imaging. Collectively, these results reveal universal interior dynamics of strong-field GR in EMS theories and highlight subtle connections—and limits—between interior structure and external observations.

Abstract

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.

Figures (7)

• Figure 1: An illustration of spontaneously scalarized black holes under two representative coupling functions, showing the reduced event horizon area $A_H/ M^2$ as a function of the charge-to-mass ratio $q$. The scalarized black holes bifurcate smoothly from the RN solution (green curve) at the critical charge-to-mass ratio $q_c$ (vertical dashed line).
• Figure 2: The behaviors of $g_{tt}$ near the would-be inner horizon (vertical dashed line) for $Z=e^{\alpha^2\psi^2}$. For clearer visualization, the horizontal axis has been linearly scaled as $\tilde{z}=(z/z_H-1)/(z_I/z_H-1)$ such that the position of the would-be inner horizon horizon $z_I$ always corresponds to $\tilde{z}_I=1$. Here $z_I$ is determined from the inner horizon of the RN solution at $q=q_c$. Top panel: Results for $q/q_c-1=10^{-6}$ with varying $\alpha^2$. Curves correspond (from bottom to top) to $\alpha^2=1, 5, 10, 20, 100$. Bottom panel: Results for fixed $\alpha^2=10$ with varying $q$. Curves correspond (from top to bottom) to $q/q_c-1=\{10^{-6},10^{-7},10^{-8},10^{-9}\}$. The inset in each panel displays the coupling function $Z(\psi)$ as a function of $z$.
• Figure 3: The dynamical behavior of the scalar field inside the hairy black hole for $Z=e^{\psi^2}$ (top panel) and $Z=1+\psi^2/(1+\psi^2)$ (bottom panel). The plateaus correspond to Kasner geometries, for which $(q/q_c-1)$ for the successive plateaus (from top to bottom) are $\{0.33, 0.60, 1.28, 5.38\} \times 10^{-4}$ in the top panel, and $\{0.56, 0.94, 1.89, 6.64\} \times 10^{-4}$ in the bottom panel. The dependence of the Kasner parameter $\beta$ on $q$ is denoted by the dotted green curves.
• Figure 4: The critical behavior between the Kasner parameter $\beta$ and the charge-to-mass ratio $q$ of hairy black holes. Top panel: Results for different coupling functions with $\alpha=1$. Bottom panel: Dependence on $\alpha$ for the coupling $Z=e^{\alpha^2\psi^2}$. Symbols of varying shapes represent numerical data, while the solid line corresponds to the scaling relation given in \ref{['BetaToQ']}.
• Figure 5: The variation of the photon ring (bright orange annulus) and the shadow (central dark region) with the Kasner parameter $\beta$ for $Z=e^{0.9\psi^2}$. Green dashed circles denote critical curves associated with unstable photon spheres, and white dashed circles indicate the boundaries of the inner shadow. The top three panels illustrate the behavior near $q_c$, while the bottom three show the variation far from $q_c$. Two photon spheres are observed for $\beta=0.45$ (bottom left) and $\beta=0.40$ (bottom middle), whereas only one photon sphere appears in the remaining cases.