The initial data were either time-symmetric or approximately ingoing, with the scalar field either symmetric or antisymmetric in z. In the symmetric case, even strongly non-spherical data were attracted to the known spherical critical solution for the massless scalar field. Scaling with the known was observed in the Ricci scalar. However, with sufficiently good fine-tuning to the black hole threshold, the approximately spherical region that approaches the critical solution suffers an l = 2 (and by ansatz m = 0) instability and splits into two new spherical regions which again approach the critical solution. The spatial separation of the two new centres is related to the smallest length scale that developed prior to the branching. There is evidence that with increasing fine-tuning each of these centres splits again. The antisymmetric initial data cannot approach a single spherical critical solution, but the solution splits initially into two approximately spherical regions where the critical solution is approached (up to an overall sign in the scalar field). The separation of these initial two centres is determined by the initial data, but there is evidence that they in turn split.
All this is consistent with the assumption that the spherical critical solution has, besides the known one spherical unstable mode, precisely one further l = 2 unstable mode. (Without the restriction to axisymmetry, if such a mode exists, it would be 5-fold degenerate with m = =–2, ..., 2.) This contradicts the calculation of the perturbation spectrum in . Choptuik and co-workers do not state with certainty that the mode they see in numerical evolutions is a continuum mode, although they have no indication that it is a numerical artifact. The growth rate of the putative mode is measured to be , which should be compared with the growth rate of the spherical mode and the relatively small decay rate of claimed in  for the least damped mode, which is also an l = 2 mode. We observe that the range of in Figures 6 and 7 of  is about 10, and over this range the plot of the amplitude of the l = 2 perturbation against the log-scale coordinate seems equally consistent with linear growth in as with exponential growth. (In the notation of , denotes proper time and the accumulation point of echos, so that the log-scale coordinate used in this review corresponds to in the notation of .)
An interesting extension was made in  by considering a complex scalar field giving rise to an axisymmetric spacetime with angular momentum. The stress-energy tensor of a complex scalar field is
For any tangent to , in particular a hypersurface t = constant, the angular momentum density measured by a normal observer becomes
For all initial data in numerical evolutions, a critical solution is approached that is discretely self-similar with log-scale period . (By ansatz this solution is axisymmetric but spherical symmetry is ruled out and so the critical solution cannot be the Choptuik solution.) The same critical solution is approached in particular for initial data with and hence no angular momentum, and initial data where and hence with large angular momentum. A scaling exponent of is observed in the Ricci scalar in subcritical evolutions. The critical solution is purely real (up to an initial data-dependent constant phase) and hence has no angular momentum. Only m = 1 was investigated, but it is plausible that a different critical solution exists for each integer m.
Far from the black hole threshold in the final black hole, but nearer the black hole threshold, , where and are measured on the apparent horizon when it first forms. is compatible with a non-rotating critical solution.
In the absence of angular momentum, the wave equations for the real and imaginary part of decouple. Assuming that the background critical solution is purely real with and , and angular momentum is provided by a perturbation with , one would expect and . Integrating over a region of size when the black hole forms, we find and . Then would imply .
Olabarrieta et al.  study a similar system in spherical symmetry, by arranging 2l+1 scalar fields given by for with the same for all values of m so that the total stress-energy tensor and the spacetime are spherically symmetric. Note that not the are added but their stress-energies, and each value of l describes a different matter content. then sees a centrifugal barrier in its evolution equation, but there is no angular momentum in the spacetime. DSS critical behaviour is found, and the logarithmic echoing period and mass scaling exponent both decrease approximately exponentially with l. It is observed empirically that the radius of maximum compactness during evolution (a measure of the scale of the initial data) and the accumulation time of echos (measured from the initial data) obey for all l and families of initial data. Update
Lai  has studied type I critical phenomena for boson (massive complex scalar field) stars in axisymmetry, the first study of type I in axisymmetry. He finds that the subcritical end state is a boson star with a large amplitude fundamental mode oscillation.
This work is licensed under a Creative Commons License.