Semilinear wave model for critical collapse

In spherical symmetry compelling numerical evidence suggests that in general relativity solutions near the threshold of black hole formation exhibit critical behavior. One aspect of this is that threshold solutions themselves are self-similar and are, in a certain sense, unique. To an extent yet to be fully understood, the same phenomena persist beyond spherical symmetry. It is therefore desirable to construct models that exhibit such symmetry at the threshold of blowup. Starting with deformations of the wave equation, we discuss models which have discretely self-similar threshold solutions. We study threshold solutions in the past light cone of the blowup point. In spherical symmetry there is a sense in which a unique critical solution exists. Spherical numerical evolutions are also presented for more general models, and exhibit similar behavior. Away from spherical symmetry threshold solutions attain more freedom. Different topologies of blowup are possible, and even locally the critical solution needs reinterpretation as a parametrized family.

Figure 1

A contour plot of a threshold solution of model 3 with $A_3=1/15$, see Eq. (24). The parameter here was chosen simply for clarity of plotting. This threshold solution blows up in $H^1$ (and not in $L^2$) in a discretely self-similar fashion at $(t_{\star },r_{\star })=(1,0)$. In the inset the solution is plotted along the red curve $t+2r(t)=1$ indicated in the main plot. The figure is naturally compared with Fig. 1 of [15].

Figure 2

Plots of the $E^1$ norm for spherical solutions of various models up to the time at which some field quantity explodes in $L^{\infty }$. On the top left we have a threshold solution for model 1. On the top right a supercritical solution for the same model is shown, demonstrating that a variation of behavior is possible at blowup. On the bottom left we show the result for ${\varphi }_1$ from model 3 at the threshold. Finally in the lower right panel we show the same for the composite deformation function $\sin \circ \mathcal{C}$, with the compactification (49) and $n=1/4$, which can be used in practice within model 5. These examples are compatible with our consideration of self-similar functions and our naive norm estimates.

Figure 3

A contour plot of an axisymmetric threshold solution for model 3 shown on the symmetry axis. Despite shared attributes with the spherical solution of Fig. 1, there are obvious differences too, as the data here leading to blowup is mostly outgoing. For this model therefore the conjecture that there is in general a unique threshold solution regardless of initial data is false.

Figure 4

Here we plot a pure $l=2$, $m=0$ threshold solution for model 3 shortly before blowup. Special in this case is that the blowup occurs on a ring in the $z=0$ plane. This was achieved with the family (71), which we may think of as the same data as (67), but evolved backwards in time. This shows that away from spherical symmetry, even when building threshold solutions purely from a single harmonic, there exist fundamentally different threshold solutions, although the number of such branches for each harmonic is always presumably finite. This story becomes even more involved with higher harmonics.

Figure 5

Plots of the blowup threshold amplitude starting from either a pure spherical solution (blue curve) or an $l=1$, $m=0$ solution (red dashed curve), and adding in each case by $l=2$, $m=0$ spherical harmonic parametrized by $\epsilon$. See the main text following Eq. (66) for details. There is a neighborhood around the spherical solution in which the nonspherical deformation makes absolutely no difference to critical amplitude, although the asymptotic solution in the past light cone of the blowup point is modified. Once the perturbation is sufficiently large however the blue curve does bend away. At this point the threshold solution takes a structure similar to that illustrated in Fig. 3. We expect that when the blue curve is extended to the left, eventually the threshold solution will take the form illustrated in 4. In contrast, the pure $l=1$ threshold amplitude is immediately affected by the perturbation.

Figure 6

$L^2$ and $E^1$ for model 1 for subcritical, critical and supercritical data computed from our numerical simulations and the exact solution for model 1 with $A_1=-1$. The numerical data agree extremely well with the values computed from the exact solution. This indicates that, with suitable care, numerical evolutions can be of real value in determining regularity even at blowup.

Figure 7

In the left panel we plot the scaling law obtained close to the threshold by taking the maximum of the time derivatives of the evolved fields ${\varphi }_1,{\varphi }_2$ for model 2. We have chosen $A_2=B_2=-1$, and used initial data as stated in (90). The threshold amplitude $a_{\star }=1.5103468$ was obtained by numerical bisection. In the legend $r^2$ refers to the square of the Pearson correlation coefficient, which we computed using the scipy python library [22]. A best fit on the data at this resolution returns the gradient 0.49594 with standard error 0.00018. On the right we plot snapshots of the same fields close to blowup for the threshold solution itself. Observe that the fields lie on top of each other at late times, indicating that the threshold solution is in fact described by the same critical solution of model 1. Identical results are obtained with other families of initial data.

Figure 8

Scaling plot for ${\mathrm{\Pi }}_1(t,0{)}_{\max }$ for model 3 with $A_3=1$ for the family of initial data (94). The threshold amplitude for this family is $a_{\star }=-\sqrt{2}$. For comparison the analytic result is also given. The drift between the numerical and analytic curves is caused by numerical error, but does converge away with resolution, as can be understood from the higher resolution data. In the legend $r^2$ again refers to the square of Pearson correlation coefficient, which was computed from the lower resolution data and is close to unity. Linear regression on the numerical data gives the gradient 0.4945, with standard error 0.0049, close to the expected value $1/2$ seen in Sec. 4.

Figure 9

Representative plots obtained with model 4 with $A_4=B_4=1$ and the initial data family (96). This is obtained with $a_{\star }\simeq -2.4122$. On the left we give the now familiar scaling plot for ${\mathrm{\Pi }}_1$. As in model 3 the curve looks like a straight line plus a periodic wiggle, indicating that we are in a DSS regime. Linear regression on the numerical data gives a slope 0.499, with standard error 0.019. On the right we plot the maximum of the absolute value at the origin of the quantity that serves as a constraint in model 3. In fact this quantity is small in a neighborhood around the origin, so that near the threshold, solutions of model 4 are close to solutions of model 3.