Codimension-Two Critical Behavior in Vacuum Gravitational Collapse

We consider the critical behavior at the threshold of black-hole formation for the five-dimensional vacuum Einstein equations satisfying the cohomogeneity-two triaxial Bianchi type-IX ansatz. Exploiting a discrete symmetry present in this model we predict the existence of a codimension-two attractor. This prediction is confirmed numerically and the codimension-two attractor is identified as a discretely self-similar solution with two unstable modes.

Figure 1

Snapshots of the evolution of near-critical initial data of the form (5) for $a=1/2$. In the last frame the squashing modes become synchronized and the solution coincides with the ${\mathrm{DSS}}_1$.

Figure 2

Snapshots of the evolution of near double critical initial data ($a=0.1411\dots$, $p=0.0952\dots$). The new (codimension-two) intermediate attractor (${\mathrm{DSS}}_2$) is seen in the second frame. The departure from this attractor (proceeding from the origin) and the approach to the old (codimension-one) intermediate attractor ($X_1^{(1)}$ copy of the ${\mathrm{DSS}}_1$) in seen in the third frame. In the last frame the ${\mathrm{DSS}}_1$ regime is well developed near the origin while the ${\mathrm{DSS}}_2$ regime is still visible for large $r$. To produce this figure we had to fine-tune $p$ to the critical value with a relative accuracy of ${10}^{-26}$ (using quadruple precision arithmetics).

Figure 3

(a) Evidence for discrete self-similarity of the codimension-two attractor. We plot the mode $B$ at some central proper time $T_0$ during the intermediate regime shown in the second frame in Fig. 2 and superimpose the next four echoes which subsequently develop. (b) The close-up of the ${\mathrm{DSS}}_2$ solution. It is evident that this solution is not biaxial. Note that the mode $C$ oscillates with twice the frequency of the mode $B$.

Figure 4

The black-hole mass scaling law for near double critical initial data. The linear fit yields different slopes for small and large distances from the threshold.