Figure 1

(a) For a black-hole solution we plot a series of late-time snapshots of the mass function $m(t,r)$. The initial mass function (solid line) has the form of a kink asymptoting the total mass $m(\infty )=0.575$. During the evolution the horizon develops at $r_h=0.606$ which corresponds to the black-hole mass $m_{BH}=r_h^6=0.05$. Thus, only a small fraction of the total mass gets trapped inside the horizon while the remaining mass is being radiated away to spatial infinity, as is clearly seen from the plot. (b) We plot the time series $\ln |B(t,r_0)|$ at $r_0=5$ for the same solution as in (a), and superimpose the fundamental quasinormal mode with frequency $k_0=(3.4488-0.8601i)/r_h$. On the time interval $10<t<16$ we get very good agreement between these two curves which confirms a well-known fact that quasinormal modes encode an intermediate time behavior of solutions at a fixed point in space [14].Reuse & Permissions

Figure 2

(a) For a near-critical evolution we plot the function $B$ at some late central proper time $T_0$ and superimpose the next three echoes (at times $T_n$) shifted by $\ln (r)\to \ln (r)+n\Delta$. Minimization of the discrepancy between the profiles yields $\Delta \approx 0.78$. (b) For supercritical solutions the logarithm of black-hole mass $m_{BH}$ is plotted versus the logarithmic distance to criticality. A fit of the power law $\ln (m_{BH})=\gamma \ln (p-p^{*})+\mathrm{\text{const.}}$ yields $\gamma \approx 1.64$. The small wiggles around the linear fit are the imprints of discrete self-similarity; their period is equal to 2.87 in agreement with the theoretical prediction $6\Delta /\gamma$.Reuse & Permissions