Critical Behavior in Vacuum Gravitational Collapse in $4+1$ Dimensions

We show that the ($4+1$)-dimensional vacuum Einstein equations admit gravitational waves with radial symmetry. The dynamical degrees of freedom correspond to deformations of the three-sphere orthogonal to the $(t,r)$ plane. Gravitational collapse of such waves is studied numerically and shown to exhibit discretely self-similar type II critical behavior at the threshold of black hole formation.

Figure 1

Formation of a black hole for highly supercritical initial data. We plot the mass function $m(t,r)$ at the initial time and at four late times. The total mass (measured in units $3\pi /8G$) is 0.67 (see the plateau of the initial profile). During the evolution the function $m(t,r)$ develops a second inner plateau that indicates formation of the Schwarzschild black hole with mass $M_{\mathrm{BH}}=0.33$.

Figure 2

The plot of $B(t,r)$ for the same data as in Fig. 1. Outside the horizon developing at $r_h=0.57$, $B$ tends to zero.

Figure 3

Quasinormal ringing of the Schwarzschild black hole. We plot the time series $\ln |B(t,r_0)|$ at $r_0=5$ for the same data as in Fig. 2 but using the proper time at infinity. The horizon forms at $r_h=0.575$; hence according to 6, the least damped quasinormal mode has eigenvalue $k=2.62-0.62i$. The linear perturbation regime sets in at $t\sim 11$. The fit of an exponentially damped sinusoid to the data on the time interval $13<t<18$ gives $k=2.56-0.61i$, which is in good agreement with the theoretical prediction.

Figure 4

The late time profile of $B$ for a near-critical solution. We easily get many echoes even for a moderate fine-tuning because the product of the eigenvalue $\lambda$ of the growing mode about the critical solution and the echoing period $\Delta$ is relatively small, $\lambda \Delta \approx 2.86$ (for comparison, this product is about 3 times greater for the massless scalar field critical collapse [3]).

Figure 5

Evidence for discrete self-similarity of the critical solution. For a near-critical evolution we plot the profile of $B$ at some arbitrary central proper time $T_0$ and superimpose the next three echoes that subsequently develop. The times $T_1,T_2,T_3$ and the echoing period $\Delta$ were chosen to minimize $B\mathbf{(}\ln (r)+n\Delta ,T_n\mathbf{)}-B\mathbf{(}\ln (r),T_0\mathbf{)}$.

Figure 6

Black hole mass scaling. For supercritical solutions the logarithm of black hole mass $M_{\mathrm{BH}}$ (measured in units $3\pi /8G$) is plotted versus the logarithmic distance to criticality. The slope of the linear fit yields $\gamma \approx 0.3289$. The wiggles, which are imprints of discrete self-similarity, are shown in more detail in Fig. 7.

Figure 7

Fine structure of black hole mass scaling. The linear fit from Fig. 6 is subtracted from the data. The period of the wiggles agrees to two decimal places with the theoretical prediction $\Delta /\gamma =1.43$.