3 The basic scenario2 The phenomena2.1 Case study: The spherically

2.2 Other matter models

Results similar to Choptuik's were subsequently found for a variety of other matter models. In some of these, qualitatively new phenomena were discovered, and we have reviewed this body of work by phenomena rather than by matter models. The number of matter models is now so large that a presentation by matter models is given only in the form of Table  1 . The second column specifies the type of critical phenomena that is seen (compare Sections  4.1 and  5.1). The next column gives references to numerical evolutions of initial data, while the last two columns give references to the semi-analytic approach.

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 [110Jump To The Next Citation Point In The Article]. 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 [60]. Results are similar to four spacetime dimensions.

Matter model                               Type of phenomena   Collapse simulations         Critical solution         Perturbations        
Perfect fluid
  tex2html_wrap_inline2413   k =1/3 II [53Jump To The Next Citation Point In The Article] CSS [53Jump To The Next Citation Point In The Article] [91Jump To The Next Citation Point In The Article]
  tex2html_wrap_inline2413  general k II [100Jump To The Next Citation Point In The Article] CSS [98Jump To The Next Citation Point In The Article, 100Jump To The Next Citation Point In The Article] [98Jump To The Next Citation Point In The Article, 92], [70Jump To The Next Citation Point In The Article] Popup Footnote
Real scalar field
  tex2html_wrap_inline2413  massless, min. coupled II [36Jump To The Next Citation Point In The Article, 37Jump To The Next Citation Point In The Article, 38Jump To The Next Citation Point In The Article] DSS [71Jump To The Next Citation Point In The Article] [74Jump To The Next Citation Point In The Article], [99Jump To The Next Citation Point In The Article] Popup Footnote
  tex2html_wrap_inline2413  massive I [22Jump To The Next Citation Point In The Article] oscillating [113Jump To The Next Citation Point In The Article]
II [38Jump To The Next Citation Point In The Article] DSS [82Jump To The Next Citation Point In The Article, 78Jump To The Next Citation Point In The Article] Popup Footnote [82Jump To The Next Citation Point In The Article, 78Jump To The Next Citation Point In The Article] Popup Footnote
  tex2html_wrap_inline2413  conformally coupled II [37Jump To The Next Citation Point In The Article] DSS
2-d sigma model
  tex2html_wrap_inline2413  complex scalar (tex2html_wrap_inline2433) II [35Jump To The Next Citation Point In The Article] DSS [74Jump To The Next Citation Point In The Article] Popup Footnote, [84Jump To The Next Citation Point In The Article] Popup Footnote [74Jump To The Next Citation Point In The Article] Popup Footnote, [83Jump To The Next Citation Point In The Article] Popup Footnote
  tex2html_wrap_inline2413  axion-dilaton (tex2html_wrap_inline2441) II [80Jump To The Next Citation Point In The Article] CSS [51Jump To The Next Citation Point In The Article, 80Jump To The Next Citation Point In The Article] [80Jump To The Next Citation Point In The Article]
  tex2html_wrap_inline2413  scalar-Brans-Dicke (tex2html_wrap_inline2445) II [97Jump To The Next Citation Point In The Article, 95Jump To The Next Citation Point In The Article] CSS, DSS
  tex2html_wrap_inline2413  general tex2html_wrap_inline2449 including tex2html_wrap_inline2451 II CSS, DSS [85Jump To The Next Citation Point In The Article] [85Jump To The Next Citation Point In The Article]
Massless scalar electrodynamics II [86Jump To The Next Citation Point In The Article] DSS [78Jump To The Next Citation Point In The Article] Popup Footnote [78Jump To The Next Citation Point In The Article] Popup Footnote
SU (2) Yang-Mills I [40Jump To The Next Citation Point In The Article] static [8Jump To The Next Citation Point In The Article] [94]
II [40Jump To The Next Citation Point In The Article] DSS [73Jump To The Next Citation Point In The Article] [73Jump To The Next Citation Point In The Article]
``III'' [42Jump To The Next Citation Point In The Article] colored BH [10, 118] [114, 117]
SU (2) Skyrme model I [13Jump To The Next Citation Point In The Article] static [13Jump To The Next Citation Point In The Article] [13Jump To The Next Citation Point In The Article]
II [15Jump To The Next Citation Point In The Article] static [15Jump To The Next Citation Point In The Article] Popup Footnote
SO (3) mexican hat II [96] DSS
Axisymmetric vacuum II [1Jump To The Next Citation Point In The Article, 2Jump To The Next Citation Point In The Article] DSS
Vlasov none? [110]
  
Table 1: An overview of numerical and semi-analytic work in critical collapse.

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.



3 The basic scenario2 The phenomena2.1 Case study: The spherically

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
http://www.livingreviews.org/lrr-1999-4
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