Critical Phenomena in Dynamical Scalarization of Charged Black Holes

We report a new black hole (BH) scalarization mechanism and disclose novel dynamical critical phenomena in the process of the nonlinear accretion of the scalar field into BHs. The accretion process can transform a seed BH into a final scalarized or bald BH, depending on the initial parameter of the scalar field $p$. There is a critical parameter $p_{*}$ and near it all intermediate solutions are attracted to a critical solution (CS) and stay there for a time scaling as $T\propto -\gamma \ln |p-p_{*}|$. At late times, the solutions evolve into scalarized black holes (BHs) if $p>p_{*}$, or bald BHs if $p<p_{*}$. The final masses of the resulting scalarized-bald BHs satisfy power laws $M_p-M_{\pm }\propto |p-p_{*}{|}^{{\gamma }_{\pm }}$, where $M_{\pm }$ are the masses of the scalarized and bald BHs, respectively, when $p\to p_{*}$ from above or below, and ${\gamma }_{\pm }$ the corresponding exponents.

Figure 1

The evolution of ${\varphi }_h$ (upper) and $M_h$ (lower) for various $p$ when $Q/M=0.9$ and $\beta =200$. We show the results obtained from initial data family $\varphi =0.5e^{-(r/pM{)}^2}$. The critical value $p_{*}\approx 2.116895853824$. Other initial data families lead to qualitatively the same behaviors.

Figure 2

The evolution of $1.2+2\ln |d{\varphi }_h/dt|$ (solid) and $\ln (d{M}_h/dt)$ (dashed) for various $p$ when $Q/M=0.9$ and $\beta =200$. Both the evolutions can be divided into six stages, as labeled in the figure. Note that the evolution of $\ln (d{M}_h/dt)$ overlaps with that of $1.2+2\ln |d{\varphi }_h/dt|$ for almost all the stages. The dotted lines are the fitting curves for each stage.

Figure 3

Upper panel: The time $T$ of the intermediate solution that stays near the CS with respect to $\ln |p-p_{*}|$ when $Q/M=0.9$ and $\beta =200$. The lines with circle and square markers are obtained with initial conditions $\varphi =0.5e^{-(r/pM{)}^2}$ and $\varphi =p[1-\tanh (r/2M)]$, respectively. The red lines are for $p<p_{*}$ and the blue lines for $p>p_{*}$. Here all lines have $\gamma \approx 7.4$. We use the time containing the first to the fourth stage as $T$, which can be get easily by counting the time when ${\varphi }_h$ is maximum for $p>p_{*}$ or minimum for $p<p_{*}$, as shown in Fig. 1. Since the first stage takes almost the same time for different $p$, the coefficient $\gamma$ will not be affected. Lower panel: The power-law relations $M_p-M_{\pm }\propto |p-p_{*}{|}^{{\gamma }_{\pm }}$ for $p<p_{*}$ (red) and $p>p_{*}$ (blue) for the two families of initial data.

Figure 4

Radial profiles of the metric functions ${\widehat{g}}_{tt}=(1-{\zeta }^2){\alpha }^2$ (solid), ${\widehat{g}}_{tr}=\zeta \alpha$ (dashed), and $V_{\mathrm{eff}}$, $\varphi$ for bald (orange), scalarized (blue) and critical (purple) solutions. The upper inset shows the tiny difference between the metric functions of the bald and critical solutions. Here $Q/M=0.9$ and $\beta =200$. The horizons of the critical, bald and scalarized solutions locate at $z\approx 0.5880$, 0.5886, 0.5938, respectively.