4.2 CSS and DSS critical 4 Extensions of the basic 4 Extensions of the basic

4.1 Black hole thresholds with a mass gap 

The spherical SU (2) Einstein-Yang-Mills system [40Jump To The Next Citation Point In The Article, 13, 14, 12] shows two different kinds of critical phenomena, dominated by two different critical solutions. Which kind of behavior arises appears to depend on the qualitative shape of the initial data. In one kind of behavior, black hole formation turns on at an infinitesimal mass with the familiar power-law scaling, dominated by a DSS critical solution. In the other kind, black hole formation turns on at a finite mass, and the critical solution is now a static, asymptotically flat solution which had been found before by Bartnik and McKinnon [8]. Choptuik, Chmaj and Bizon labelled the two kinds of critical behavior type II and type I respectively, corresponding to a second- and a first-order phase transition. The newly found, type I critical phenomena show a scaling law that is mathematically similar to the black hole mass scaling observed in type II critical phenomena. Let tex2html_wrap_inline2673 be the static Killing vector of the critical solution. Then the perturbed critical solution is of the form

  equation860

This is similar to Eqn. (33Popup Equation), but the growth of the unstable mode is now exponential in t, not in tex2html_wrap_inline2677 . In a close parallel to tex2html_wrap_inline2611, we define a time tex2html_wrap_inline2681 by

equation867

so that the initial data at tex2html_wrap_inline2681 are

equation871

and so the final black hole mass is independent of tex2html_wrap_inline2291 Popup Footnote . The scaling is only apparent in the lifetime of the critical solution, which we can take to be tex2html_wrap_inline2681 . It is

equation874

The type I critical solution can also have a discrete symmetry, that is, can be periodic in time instead of being static. This behavior was found in collapse situations of the massive scalar field by Brady, Chambers and Gonçalves [22Jump To The Next Citation Point In The Article]. Previously, Seidel and Suen [113] had constructed periodic, asymptotically flat, spherically symmetric self-gravitating massive scalar field solutions they called oscillating soliton stars. By dimensional analysis, the scalar field mass m sets an overall scale of 1/ m (in units G = c =1). For given m, Seidel and Suen found a one-parameter family of such solutions with two branches. The more compact solution for a given ADM mass is unstable, while the more extended one is stable to spherical perturbations. Brady, Chambers and Gonçalves (BCG) report that the type I critical solutions they find are from the unstable branch of the Seidel and Suen solutions. They see a one-parameter family of (type I) critical solutions, rather than an isolated critical solution. BCG in fact report that the black hole mass gap does depend on the initial data. As expected from the discrete symmetry, they find a small wiggle in the mass of the critical solution which is periodic in tex2html_wrap_inline2657 . Whether type I or type II behavior is seen appears to depend mainly on the ratio of the length scale of the initial data to the length scale 1/ m .

In the critical phenomena that were first observed, with an isolated critical solution, only one number's worth of information, namely the separation tex2html_wrap_inline2291 of the initial data from the black hole threshold, survives to the late stages of the time evolution. Recall that our definition of a critical solution is one that has exactly one unstable perturbation mode, with a black hole formed for one sign of the unstable mode, but not for the other. This definition does not exclude an n -dimensional family of critical solutions. Each solution in the family then has n marginal modes leading to neighboring critical solutions, as well as the one unstable mode. n +1 numbers' worth of information survive from the initial data, and the mass gap in type I, or the critical exponent for the black hole mass in type II, for example, depend on the initial data through n parameters. In other words, universality exists in diminished form. The results of BCG are an example of a one-parameter family of type I critical solutions. Recently, Brodbeck et al. [25] have shown, under the assumption of linearization stability, that there is a one-parameter family of stationary, rotating solutions beginning at the (spherically symmetric) Bartnik-McKinnon solution. This could turn out to be a second one-parameter family of type I critical solutions, provided that the Bartnik-McKinnon solution does not have any unstable modes outside spherical symmetry (which has not yet been investigated) [111].

Bizon and Chmaj have studied type I critical collapse of an SU (2) Skyrme model coupled to gravity, which in spherical symmetry with a hedgehog ansatz is characterized by one field F (r, t) and one dimensionless coupling constant tex2html_wrap_inline2323 . Initial data tex2html_wrap_inline2717, tex2html_wrap_inline2719 surprisingly form black holes for both large and small values of the parameter p, while for an intermediate range of p the endpoint is a stable static solution called a skyrmion. (If F was a scalar field, one would expect only one critical point on this family.) The ultimate reason for this behavior is the presence of a conserved integer ``baryon number'' in the matter model. Both phase transitions along this one-parameter family are dominated by a type I critical solution, that is a different skyrmion which has one unstable mode. In particular, an intermediate time regime of critical collapse evolutions agrees well with an ansatz of the form (40Popup Equation), where tex2html_wrap_inline2565, tex2html_wrap_inline2729 and tex2html_wrap_inline2573 were obtained independently. It is interesting to note that the type I critical solution is singular in the limit tex2html_wrap_inline2733, which is equivalent to tex2html_wrap_inline2735, because the known type II critical solutions for any matter model also do not have a weak gravity limit.

Apparently, type I critical phenomena can arise even without the presence of a scale in the field equations. A family of exact spherically symmetric, static, asymptotically flat solutions of vacuum Brans-Dicke gravity given by van Putten was found by Choptuik, Hirschmann and Liebling [41] to sit at the black hole threshold and to have exactly one growing mode. This family has two parameters, one of which is an arbitrary overall scale.



4.2 CSS and DSS critical 4 Extensions of the basic 4 Extensions of the basic

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
http://www.livingreviews.org/lrr-1999-4
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de