Most models in the table are restricted to spherical symmetry, and their matter content is described by a few functions of space (radius) and time. Two models in the table are quite different, and therefore particularly interesting. The axisymmetric vacuum model (see Section 4.6.1) is unique in going beyond spherical symmetry nonperturbatively and in being vacuum rather than containing matter. The fact that similar phenomena to Choptuik's were found in that model strongly suggests that critical phenomena are not artifacts of spherical symmetry or a specific matter model.
The second exceptional model, a collisionless matter (Vlasov equation) model, is distinguished by having a much larger number of matter degrees of freedom. Here, the matter content is described by a function not only of space and time but also momentum. Remarkably, no scaling phenomena of the kind seen in the scalar field were discovered in numerical collapse simulations. Collisionless matter appears to show a mass gap in critical collapse that depends on the initial matter - black hole formation turns on with a mass that is a large part of the ADM mass of the initial data . Therefore universality is not observed either. It is important to both confirm and further investigate this phenomenology, in order to understand it better. The explanation may be that the numerical precision was not high enough to find critical phenomena, or they may be genuinely absent, perhaps because the space of possible matter configurations is so much bigger than the space of metrics in this case.
Critical collapse of a massless scalar field in spherical symmetry in six spacetime dimensions was investigated in . Results are similar to four spacetime dimensions.
|Matter model||Type of phenomena||Collapse simulations||Critical solution||Perturbations|
|k =1/3||II||||CSS |||
|general k||II||||CSS [98, 100]||[98, 92], |
|Real scalar field|
|massless, min. coupled||II||[36, 37, 38]||DSS ||, |
|II||||DSS [82, 78]||[82, 78]|
|2-d sigma model|
|complex scalar ()||II||||DSS  , || , |
|axion-dilaton ()||II||||CSS [51, 80]|||
|scalar-Brans-Dicke ()||II||[97, 95]||CSS, DSS|
|general including||II||CSS, DSS |||
|Massless scalar electrodynamics||II||||DSS |||
|SU (2) Yang-Mills||I||||static |||
|``III''||||colored BH [10, 118]||[114, 117]|
|SU (2) Skyrme model||I||||static |||
|SO (3) mexican hat||II||||DSS|
|Axisymmetric vacuum||II||[1, 2]||DSS|
Related results not listed in the table concern spherically symmetric dust collapse. Here, the entire spacetime, the Tolman-Bondi solution, is given in closed form from the initial velocity and density profiles. Excluding shell crossing singularities, there is a ``phase transition'' between initial data forming naked singularities at the center and data forming black holes. Which of the two happens depends only the leading terms in an expansion of the initial data around r =0 [43, 90]. One could argue that this fact also makes the matter model rather unphysical.
|Critical Phenomena in Gravitational Collapse
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