Critical behavior in black hole scalar field interaction

We study the critical behavior at the threshold of black hole formation in a model consisting of a scalar field incident to a reflector barrier enclosing a Schwarzschild black hole. Weak incident scalar field waves disturb slightly the black hole spacetime and are completely radiated by the reflector, like water waves striking against the wall of a dam. Strong incident waves produce the formation of an apparent horizon outside the barrier. In this case, a fraction of scalar field crosses the horizon together with the barrier, whereas another fraction escapes to infinity. We have integrated the field equations using a Galerkin collocation code that allowed the necessary accuracy to investigate the behavior of the black hole masses for a broad range of scalar field initial amplitude. We have shown that a scaling law describes the black hole masses for amplitudes very close to the critical value. In the limit of very strong scalar fields, the black hole masses either scale linearly with the initial amplitude or saturate depending on the existence of the initial monopole moment.

Figure 1

Exponential decay of the maximum error in the global energy conservation with the increase of the truncation orders $N$.

Figure 2

Black holes mass as a function of the initial amplitude $\epsilon$ for the initial data (17), where $R/2=1.0$.

Figure 3

Scaling law (20) for black hole formation with masses close to $R/2$. The numerical data can in this region can be fitted with $\gamma \approx 0.113$ and $\gamma \approx 0.115$ for the initial data using Eqs. (17) (upper plot) and (18) (lower plot). The critical parameters are, respectively, ${\epsilon }_{*}\approx 4.4010758346450$ and ${\epsilon }_{*}\approx 1.231345571457$. There is also an oscillatory-like component that superposes the scaling law.

Figure 4

Plots of the black hole masses $\delta M_{\mathrm{BH}}=M_{\mathrm{BH}}-M_{\mathrm{BH}}(\overline{\epsilon })$ with respect to $\delta \epsilon =\epsilon -\overline{\epsilon }$, where $\overline{\epsilon }=4.670$ and 1.2355 for the initial data (17) and (18), respectively. In the first panel $\delta M_{\mathrm{BH}}\propto \delta {\epsilon }^{\nu }$ where $\nu \approx 1.011$, whereas in the second panel the black hole mass saturates for higher amplitudes.